Mellin Moments of Heavy Flavor
Contributions to F2(x, Q2) at NNLO
Dissertation
zur Erlangung des wissenschaftlichen Grades
Dr. rer. nat.
der Fakulta¨t Physik der Technischen Universita¨t Dortmund
vorgelegt von
Sebastian Werner Gerhard Kleina,b
geboren am 23.01.1980 in Karlsruhe
Betreuer: PD Dr. habil. Johannes Blu¨mleina,b
aTechnische Universita¨t Dortmund, Fakulta¨t Physik
Otto-Hahn-Str. 4, D-44227 Dortmund
bDeutsches Elektronen–Synchrotron, DESY
Platanenallee 6, D–15738 Zeuthen
eingereicht am 5. Juni 2009
Contents
1 Introduction 5
2 Deeply Inelastic Scattering 12
2.1 Kinematics and Cross Section . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The Parton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Validity of the Parton Model . . . . . . . . . . . . . . . . . . . . . . 20
2.3 The Light–Cone Expansion . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Light–Cone Dominance . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.3 The Light–Cone Expansion applied to DIS . . . . . . . . . . . . . . 26
2.4 RGE–improved Parton Model and Anomalous Dimensions . . . . 29
3 Heavy Quark Production in DIS 33
3.1 Electroproduction of Heavy Quarks . . . . . . . . . . . . . . . . . . . 34
3.2 Asymptotic Heavy Quark Coefficient Functions . . . . . . . . . . . 37
3.3 Heavy Quark Parton Densities . . . . . . . . . . . . . . . . . . . . . . 41
4 Renormalization of Composite Operator Matrix Elements 44
4.1 Regularization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Renormalization of the Mass . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Renormalization of the Coupling . . . . . . . . . . . . . . . . . . . . . 49
4.5 Operator Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6 Mass Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.7 General Structure of the Massive Operator Matrix Elements . . . 57
4.7.1 Self–energy contributions . . . . . . . . . . . . . . . . . . . . . . . . 58
4.7.2 ANSqq,Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.7.3 APSQq and APSqq,Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.7.4 AQg and Aqg,Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.7.5 Agq,Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.7.6 Agg,Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Representation in Different Renormalization Schemes 69
5.1 Scheme Dependence at NLO . . . . . . . . . . . . . . . . . . . . . . . . 71
6 Calculation of the Massive Operator Matrix Elements up to O(a2sε) 74
6.1 Representation in Terms of Hypergeometric Functions . . . . . . . 74
6.2 Difference Equations and Infinite Summation . . . . . . . . . . . . . 78
6.2.1 The Sigma-Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2.2 Alternative Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.4 Checks on the Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 93
1
7 Calculation of Moments at O(a3s) 95
7.1 Generation of Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2 Calculation of Fixed 3–Loop Moments Using MATAD . . . . . . . . 98
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8 Heavy Flavor Corrections to Polarized Deep-Inelastic Scattering 106
8.1 Polarized Scattering Cross Sections . . . . . . . . . . . . . . . . . . . 107
8.2 Polarized Massive Operator Matrix Elements . . . . . . . . . . . . . 109
8.2.1 ∆A(2),NSqq,Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.2.2 ∆A(2)Qg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.2.3 ∆A(2),PSQq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9 Heavy Flavor Contributions to Transversity 118
10 First Steps Towards a Calculation of A(3)ij for all Moments. 123
10.1 Results for all–N Using Generalized Hypergeometric Functions . 123
10.2 Reconstructing General–N Relations
from a Finite Number of Mellin–Moments . . . . . . . . . . . . . . . 127
10.2.1 Single Scale Feynman Integrals as Recurrent Quantities . . . . . . . 128
10.2.2 Establishing and Solving Recurrences . . . . . . . . . . . . . . . . . 128
10.2.3 Determination of the 3-Loop Anomalous Dimensions
and Wilson Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 130
11 Conclusions 132
A Conventions 137
B Feynman Rules 139
C Special Functions 142
C.1 The Γ–function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C.2 The Generalized Hypergeometric Function . . . . . . . . . . . . . . . . . . 143
C.3 Mellin–Barnes Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
C.4 Harmonic Sums and Nielsen–Integrals . . . . . . . . . . . . . . . . . . . . . 144
D Finite and Infinite Sums 147
E Moments of the Fermionic Contributions to the
3–Loop Anomalous Dimensions 149
F The O(ε0) Contributions to ˆˆA
(3)
ij 155
G 3–loop Moments for Transversity 171
List of Figures
1 Schematic graph of deeply inelastic scattering for single boson exchange. . . . 13
2 Deeply inelastic electron-proton scattering in the parton model. . . . . . . . . 18
2
3 Schematic picture of the optical theorem. . . . . . . . . . . . . . . . . . . . . 22
4 Integration contour in the complex x′-plane. . . . . . . . . . . . . . . . . . . 25
5 LO intrinsic heavy quark production. . . . . . . . . . . . . . . . . . . . . . . 34
6 LO extrinsic heavy quark production. . . . . . . . . . . . . . . . . . . . . . . 36
7 O(a2s) virtual heavy quark corrections. . . . . . . . . . . . . . . . . . . . . . . 71
8 Examples for 2–loop diagrams contributing to the massive OMEs. . . . . . . . 75
9 Basic 2–loop massive tadpole . . . . . . . . . . . . . . . . . . . . . . . . . . 75
10 Examples for 3–loop diagrams contributing to the massive OMEs. . . . . . . . 95
11 Diagrams contributing to H(1)g,(2,L) via the optical theorem. . . . . . . . . . . . 95
12 Diagrams contributing to A(1)Qg. . . . . . . . . . . . . . . . . . . . . . . . . . 96
13 Generation of the operator insertion. . . . . . . . . . . . . . . . . . . . . . . 96
14 2–Loop topologies for MATAD . . . . . . . . . . . . . . . . . . . . . . . . . 97
15 Master 3–loop topology for MATAD. . . . . . . . . . . . . . . . . . . . . . . 97
16 Additional 3–loop topologies for MATAD. . . . . . . . . . . . . . . . . . . . . 98
17 Basic 3–loop topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
18 3–loop ladder graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
19 Example 3–loop graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
20 Feynman rules of QCD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
21 Feynman rules for quarkonic composite operators. . . . . . . . . . . . . . . . 140
22 Feynman rules for gluonic composite operators. . . . . . . . . . . . . . . . . . 141
List of Tables
1 Complexity of the results for the individual diagrams contributing to A(2)Qg . . . 86
2 Number of diagrams contributing to the 3–loop massive OMEs. . . . . . . . 96
3 Numerical values for moments of individual diagrams of ∆ ˆˆA
(2)
Qg. . . . . . . . . 115
3
1 Introduction
Quantum Chromodynamics (QCD) has been established as the theory of the strong in-
teraction and explains the properties of hadrons, such as the proton or the neutron, in
particular at short distances. Hadrons are composite objects and made up of quarks and
antiquarks, which are bound together by the exchange of gluons, the gauge field of the
strong force. The corresponding charge is called color, leading to a SU(3)c gauge theory.
This is analogous to the electric charge, which induces the U(1) gauge group of electro-
magnetism.
The path to the discovery of QCD started in the 1960ies. By that time, a large
amount of hadrons had been observed in cosmic ray and accelerator experiments. Hadrons
are strongly interacting particles which occur as mesons (spin = 0, 1) or baryons
(spin = 1/2, 3/2). In the early 1960ies investigations were undertaken to classify all
hadrons, based on their properties such as flavor– and spin quantum numbers and masses.
In 1964, M. Gell-Mann, [1], and G. Zweig, [2], proposed the quark model as a mathemat-
ical description for these hadrons. Three fractionally charged quark flavors, up (u), down
(d) and strange (s), known as valence quarks, were sufficient to describe the quantum
numbers of the hadron spectrum which had been discovered by then. Baryons are thus
considered as bound states of three quarks and mesons of a quark-antiquark pair. Assum-
ing an approximate SU(3) flavor symmetry, “the eightfold way”, [3–5], mass formulas for
hadrons built on the basis of quark states could be derived. A great success for the quark
model was marked by the prediction of the mass of the Ω−-baryon before it was finally
observed, [6]. In the same year, Gu¨rsey and Radicati, [7], introduced spin into the model
and proposed a larger SU(6)spin−flavor = SU(2)spin⊗SU(3)flavor symmetry. This allowed
the unification of the mass formulas for the spin–1/2 and spin–3/2 baryons and provided
the tool to calculate the ratio of the magnetic moments of the proton and the neutron to
be ≈ −3/2, which is in agreement with experiment within 3%, [8,9]. However, this theory
required the quarks that gave the correct low-lying baryons to be in a symmetric state
under permutations, which contradicts the spin–statistics theorem, [10], since quarks have
to be fermions. Greenberg, [11], resolved this contradiction by introducing a “symmetric
quark model”. It allows quarks to have a new hidden three–valued charge, called color,
which is expressed in terms of parafermi statistics. Finally, in 1965, Nambu, [12], and
Han and Nambu, [13], proposed a new symmetry, SU(3)color, which makes the hidden
three–valued charge degree of freedom explicit and is equivalent to Greenberg’s descrip-
tion. Since there was no explicit experimental evidence of this new degree of freedom, the
assumption was made that all physical bound states must be color-neutral, [12–14].
The possibility to study the substructure of nucleons arose at the end of the 1960ies
with the advent of the Stanford Linear Accelerator SLAC, [15]. This facility allowed to
perform deeply inelastic lepton-nucleon scattering (DIS) experiments at much higher res-
olutions than previously possible. The cross section can be parametrized quite generally
in terms of several structure functions Fi of the nucleon, [16]. These were measured for
the proton by the SLAC-MIT experiments and depend both on the energy transfer ν and
the 4-momentum transfer q2 = −Q2 from the lepton to the nucleon in the nucleon’s rest
frame. In the Bjorken limit, {Q2, ν → ∞, Q2/ν = fixed}, [17], it was found that the
structure functions depend on the ratio of Q2 and ν only, Fi(ν,Q2) = Fi(Q2/ν). This
5
phenomenon was called scaling, [18] cf. also [19], and had been predicted by Bjorken
in his field theoretic analysis based on current algebra, [17]. As the relevant parame-
ter in the deep-inelastic limit he introduced the Bjorken-scaling variable x = Q2/2Mν,
where M is the mass of the nucleon. After scaling was discovered, R. Feynman gave
a phenomenological explanation for this behavior of the structure functions within the
parton model, [20–22]. According to this model, the proton consists of several point-like
constituents, the partons. His assumption was that during the interaction time - which
is very short since high energies are involved - these partons behave as free particles
off which the electrons scatter elastically. Therefore, the total cross section is just the
incoherent sum of the individual electron-parton cross-sections, weighted by the prob-
ability to find the particular parton inside the proton. The latter is described by the
parton density fi(z). It denotes the probability to find parton i in the proton, carrying
the fraction z of the total proton momentum P . In the limit considered by Feynman,
z becomes equal to x, giving an explanation for scaling. This is a direct consequence
of the rigid correlation Mν = q.P , as observed in experiment. Even more important
for the acceptance of the quark parton model was the observation that the Callan-Gross
relation, [23], holds, namely that the longitudinal structure function FL vanishes in the
situation of strict scaling. This experimental result favored the idea of the proton con-
taining spin–1/2, point-like constituents and ruled out different approaches, such as the
algebra of fields, [24], or explanations assuming vector–meson dominance, [25]. Finally,
Bjorken and Paschos, [26], linked the parton model to the group theoretic approach by
identifying quarks and partons.
Today QCD forms one part of the Standard Model of elementary particle physics, sup-
plementing the electroweak SUL(2)×UY (1) sector, which had been proposed by S. Wein-
berg in 1967, [27], extending earlier work by S. Glashow, [28], cf. also [29], for the leptonic
sector. This theory was proved to be renormalizable by G. t’Hooft and M. Veltman in
1972, [30], see also [31], if anomalies are canceled, [32,33], requiring an appropriate repre-
sentation for all fermions. G. t’ Hooft also proved renormalization for massless Yang-Mills
theories, [34]. These gauge theories had first been studied by C.N. Yang and R.L. Mills
in 1954, [35], and have the distinctive property that their gauge group is non-abelian,
leading to interactions between the gauge–bosons, [14], contrary to the case of Quan-
tum Electrodynamics. In 1972/73, M. Gell-Mann, H. Fritzsch and H. Leutwyler, [36], cf.
also [12], proposed to gauge color which led to an extension of the Standard Model to
SUL(2)×UY (1)×SUc(3), including the strongly interacting sector. The dynamical theory
of quarks and gluons, Quantum Chromodynamics, is thus a massless Yang-Mills theory
which describes the interaction of different quark flavors via massless gluons. Among the
semi-simple compact Lie-groups, SU(3)c turns out to be the only possible gauge group
for this theory, cf. [37, 38]. In 1973, D. Gross and F. Wilczek, [39], and H. Politzer, [40],
proved by a 1-loop calculation that Quantum Chromodynamics is an asymptotically free
gauge theory, cf. also [41], which allows to perform perturbative calculations for processes
at large enough scales. There, the strong coupling constant becomes a sufficiently small
perturbative parameter.
In the beginning, QCD was not an experimentally well–established theory, which was
mainly due to its non–perturbative nature. The large value of the strong coupling con-
stant over a wide energy range prevents one from using perturbation theory. In the
course of performing precision tests of QCD, the operator product expansion near the
6
light–cone, the light–cone expansion (LCE), [42], proved to be important. By applying
it to deep–inelastic processes, one facilitates a separation of hadronic bound state effects
and the short distance effects. This is possible, since the cross sections of deeply inelastic
processes receive contributions from two different resolution scales µ2. One is the short
distance region, where perturbative techniques can be applied. The other describes the
long distance region. Here bound state effects are essential and a perturbative treatment
is not possible due to the large coupling involved. By means of the LCE, the two energy
scales of the process are associated with two different quantities: the Wilson coefficients
and the hadronic operator matrix elements or parton densities. The former contain the
large scale contributions and can therefore be calculated perturbatively, whereas the lat-
ter describe the low scale behavior and are quantities which have to be extracted from
experimental data or can be calculated by applying rigorous non–perturbative methods.
Using the LCE, one may derive Feynman’s parton model and show the equivalence of the
approaches by Feynman and Bjorken in the twist–2 approximation, [43]. The LCE also
allows to go beyond the naive partonic description, which is formulated in the renormal-
ization group improved parton model. Shortly after the formulation of QCD, logarithmic
scaling violations of the deep inelastic cross section where observed, [44], which had to be
expected since QCD is not an essentially free field theory, neither is it conformally invari-
ant, [45]. The theoretical explanation involves the calculation of higher order corrections
to the Wilson coefficients as well as to the anomalous dimensions of the composite oper-
ators emerging in the LCE, [46], and predicts the correct logarithmic Q2 dependence of
the structure functions. In fact, the prediction of scaling violations is one of the strongest
experimental evidences for QCD.
Thus deeply inelastic scattering played a crucial role in formulating and testing QCD
as the theory governing the dynamics of quark systems. Its two most important prop-
erties are the confinement postulate - all physical states have to be color singlets - and
asymptotic freedom - the strength of the interaction becomes weaker at higher scales, i.e.
at shorter distances, cf. e.g. [37,47–56].
An important step toward completing the Standard Model were the observations of the
three heavy quarks charm (c), bottom (b) and top (t). In 1974, two narrow resonances,
called Ψ and Ψ′, were observed at SLAC in e+e− collisions at 3.1 GeV and 3.7 GeV, respec-
tively, [57]. At the same time another resonance called J was discovered in proton-proton
collisions at BNL, [58], which turned out to be the same particle. Its existence could not be
explained in terms of the three known quark flavors and was interpreted as a meson con-
sisting of a new quark, the charm quark. This was an important success of the Standard
Model since the existence of the charm had been postulated before, [59]. It is necessary
to cancel anomalies for the 2nd family as well as for the GIM–mechanism, [60], in order to
explain the absence of flavor changing neutral currents. With its mass of mc ≈ 1.3 GeV it
is much heavier than the light quarks, mu ≈ 2 MeV ,md ≈ 5 MeV ,ms ≈ 104 MeV, [61],
and heavier than the nucleons. In later experiments, two other heavy quarks were de-
tected. In 1977, the Υ (= bb) resonance was observed at FERMILAB, [62], and interpreted
as a bound state of the even heavier bottom quark, with mb ≈ 4.2 GeV, [61]. Ultimately,
the quark picture was completed in case of three fermionic families by the discovery of
the heaviest quark, the top-quark, in pp collisions at the TEVATRON in 1995, [63]. Its
mass is given by roughly mt ≈ 171 GeV, [61]. Due to their large masses, heavy quarks
cannot be considered as constituents of hadrons at rest or bound in atomic nuclei. They
are rather excited in high energy experiments and may form short-lived hadrons, with the
7
exception of the top-quark, which decays before it can form a bound state.
The theoretical calculation in this thesis relates to the production of heavy quarks
in unpolarized deeply inelastic scattering via single photon exchange. In this case, the
double differential scattering cross-section can be expressed in terms of the structure
functions F2(x,Q2) and FL(x,Q2). Throughout the last forty years, many DIS experi-
ments have been performed, [44, 64–74]. The proton was probed to shortest distances at
the Hadron-Elektron-Ring-Anlage HERA at DESY in Hamburg, [75–79]. In these experi-
ments, a large amount of data has been acquired, and in the case of HERA it is still being
processed, especially for those of the last running period, which was also devoted to the
measurement of FL(x,Q2), [80]. Up to now, the structure function F2(x,Q2) is measured
in a wide kinematic region, [61], whereas FL(x,Q2) was mainly measured in fixed target
experiments, [81], and determined in the region of large ν, [82]. In the analysis of DIS
data, the contributions of heavy quarks play an important role, cf. e.g. [83–87]. One
finds that the scaling violations of the heavy quark contributions differ significantly from
those of the light partons in a rather wide range starting from lower values of Q2. This
demands a detailed description. Additionally, it turns out that the heavy quark contri-
butions to the structure functions may amount up to 25-35%, especially in the small–x
region, [85,86,88,89], which requires a more precise theoretical evaluation of these terms.
Due to the kinematic range of HERA and the previous DIS experiments, charm is pro-
duced much more abundantly and gives a higher contribution to the cross section than
bottom, [86]. Therefore we subsequently limit our discussion to one species of a heavy
quark. Intrinsic heavy quark production is not considered, since data from HERA show
that this production mechanism hardly gives any contribution, cf. [90, 91]. The need for
considering heavy quark production has several aspects. One of them is to obtain a better
description of heavy flavor production and its contribution to the structure functions of
the nucleon. On the other hand, increasing our knowledge on the perturbative part of
deep–inelastic processes allows for a more precise determination of the QCD–scale ΛQCD
and the strong coupling constant αs, as well as of the parton–densities from experimental
data. For the former, sufficient knowledge of the NNLO massive corrections in DIS is
required to control the theory–errors on the level of the experimental accuracy and be-
low, [92–100]. The parton distribution functions are process independent quantities and
can be used to describe not only deeply inelastic scattering, but also a large variety of scat-
tering events at (anti–)proton–proton colliders such as the TEVATRON at FERMILAB, and
the Large–Hadron–Collider (LHC) at CERN, [87]. Heavy quark production is well suited
to extract the gluon density since at leading order (LO) only the photon–gluon fusion pro-
cess contributes to the cross section, [101,102]. Next-to-leading order (NLO) calculations,
as performed in Refs. [103], showed that this process is still dominant, although now other
processes contribute, too. The gluon density plays a special role, since it carries roughly
50 % of the proton momentum, as data from FERMILAB and CERN showed already in the
1970ies, [104]. Improved knowledge on the gluon distribution G(x,Q2) is also necessary
to describe gluon-initiated processes at the TEVATRON and at the LHC. The study of
heavy quark production will also help to further understand the small-x behavior of the
structure functions, showing a steep rise, which is mainly attributed to properties of the
gluon density.
8
The perturbatively calculable contributions to the DIS cross section are the Wilson
coefficients. In case of light flavors only, these are denoted by C(q,g),(2,L)(x,Q2/µ2) 1 and at
present they are known up to the third order in the strong coupling constant, [105–115].
Including massive quarks into the analysis, the corresponding terms are known exactly
at NLO. The LO terms have been derived in the late seventies, [101, 102], and the NLO
corrections semi–analytically in z–space in the mid–90ies, [103]. A fast numerical im-
plementation was given in [116]. In order to describe DIS at the level of twist τ = 2,
also the anomalous dimensions of the local composite operators emerging in the LCE are
needed. These have to be combined with the Wilson coefficients and describe, e.g., the
scaling violations of the structure functions and parton densities, [46]. This description
is equivalent to the picture in z–space in terms of the splitting functions, [117]. The
unpolarized anomalous dimensions are known up to NNLO 2. At leading, [46], and at
next–to–leading–order level, [119–123], they have been known for a long time and were
confirmed several times. The NNLO anomalous dimension were calculated by Vermaseren
et. al. First, the fixed moments were calculated in Refs. [111, 112, 114] and the complete
result was obtained in Refs. [124,125].
The main parts of this thesis are the extension of the description of the contributions
of heavy quark mass–effects to the deep–inelastic Wilson coefficients to NNLO. In course
of that, we also obtain a first independent calculation of fixed moments of the fermionic
parts of the NNLO anomalous dimensions given in Refs. [111,112] before.
The calculation of the 3-loop heavy flavor Wilson coefficients in the whole Q2 region
is currently not within reach. However, as noticed in Ref. [126], a very precise description
of the heavy flavor Wilson coefficients contributing to the structure function F2(x,Q2) at
NLO is obtained for Q2 >∼ 10 m2Q, disregarding the power corrections ∝ (m2Q/Q2)k, k ≥ 1. If
one considers the charm quark, this covers an important region for deep–inelastic physics
at HERA. In this limit, the massive Wilson coefficients factorize into universal massive
operator matrix elements (OMEs) Aij(x, µ2/m2Q) and the light flavor Wilson coefficients
C(q,g),(2,L)(x,Q2/µ2). The former are process independent quantities and describe all quark
mass effects. They are given by matrix elements of the leading twist local composite op-
erators Oi between partonic states j (i, j = q, g), including quark masses. The process
dependence is described by the massless Wilson coefficients. This factorization has been
applied in Ref. [127] to obtain the asymptotic limit for F ccL (x,Q2) at NNLO. However, un-
like the case for F cc2 , the asymptotic result in this case is only valid for much higher values
Q2 >∼ 800 m2Q, outside the kinematic domain at HERA for this quantity. An analytic result
for the NLO quarkonic massive operator matrix elements Aqj needed for the description of
the structure functions at this order was derived in Ref. [126] and confirmed in Ref. [128].
A related application of the massive OMEs concerns the formulation of a variable flavor
number scheme (VFNS) to describe parton densities of massive quarks at sufficiently high
scales. This procedure has been described in detail in Ref. [129], where the remaining
gluonic massive OMEs Agj were calculated up to 2–loop order, thereby giving a full NLO
description. This calculation was confirmed and extended in [130].
In this work, fixed moments of all contributing massive OMEs at the 3–loop level are
calculated and presented, which is a new result, [131–134]. The OMEs are then matched
1q=quark, g=gluon
2In Ref. [118], the 2nd moment of the 4–loop NS+ anomalous dimension was calculated.
9
with the corresponding known O(α3s) light flavor Wilson coefficients to obtain the heavy
flavor Wilson coefficients in the limit Q2 ≫ m2, which leads to a precise description for
Q2/m2 >∼ 10 in case of F2(x,Q2). It is now possible to calculate all logarithmic contribu-
tions ∝ ln(Q2/m2)k to the massive Wilson coefficients in the asymptotic region for general
values of the Mellin variable N . This applies as well for a large part of the constant term,
where also the O(ε) contributions at the 2–loop level occur. The first calculation of the
latter for all–N forms a part of this thesis, too, [130, 131, 133, 135–137]. Thus only the
constant terms of the unrenormalized 3–loop results are at present only known for fixed
moments. Since the OMEs are given by the twist τ = 2 composite operators between
on–shell partonic states, also fixed moments of the fermionic contributions to the NNLO
unpolarized anomalous are obtained, which are thereby confirmed for the first time in an
independent calculation, [131–134].
A more technical aspect of this thesis is the study of the mathematical structure of
single scale quantities in renormalizable quantum field theories, [138–141]. One finds that
the known results for a large number of different hard scattering processes are most simply
expressed in terms of nested harmonic sums, cf. [142, 143]. This holds at least up to 3–
loop order for massless Yang–Mills theories, cf. [95, 115, 124, 125, 138, 144, 145], including
the 3–loop Wilson coefficients and anomalous dimensions. By studying properties of
harmonic sums, one may thus obtain significant simplifications, [121], since they obey
algebraic, [146], and structural relations, [147,148]. Performing the calculation in Mellin–
space one is naturally led to harmonic sums, which is an approach we thoroughly adopt
in our calculation. In course of this, new types of infinite sums occur if compared to
massless calculations. In the latter case, summation algorithms such as presented in
Refs. [143, 149, 150] may be used to calculate the respective sums. The new sums which
emerge were calculated using the recent summation package Sigma, [151–154], written in
MATHEMATICA, which opens up completely new possibilities in symbolic summation and
has been tested extensively through this work, [139].
For fixed values of N , single scale quantities reduce to zero–scale quantities, which
can be expressed by rational numbers and certain special numbers as multiple zeta values
(MZVs), [155,156], and related quantities. Zero scale problems are much easier to calcu-
late than single scale problems. By working in Mellin–space, single scale quantities are
discrete and one can seek a description in terms of difference equations. One may think
of an automated reconstruction of the all–N relation out of a finite number of Mellin
moments given in analytic form. This is possible for recurrent quantities. At least up
to 3-loop order, presumably to even higher orders, single scale quantities belong to this
class. In this work, [140,141], we report on a general algorithm for this purpose, which we
applied to a problem being currently one of the most sophisticated ones: the determina-
tion of the anomalous dimensions and Wilson coefficients to 3–loop order for unpolarized
deeply-inelastic scattering, [115,124,125].
The thesis is based on the publications Refs. [130, 134, 137, 141], the conference con-
tributions [131–133, 135, 136, 138–140, 157, 158] and the papers in preparation [159, 160].
It is organized as follows. Deeply inelastic scattering within the parton model, the LCE
and how one obtains improved results using the renormalization group are described in
Section 2. Section 3 is devoted to the production mechanisms of heavy quarks and their
contributions to the cross section. We also discuss the framework of obtaining the heavy
flavor Wilson coefficients using massive OMEs in the asymptotic limit Q2 ≫ m2Q and com-
10
ment on the different schemes one may apply to treat heavy quark production, [130,134].
The massive operator matrix elements are considered in Section 4 and we describe in detail
the renormalization of these objects to 3–loop order, cf. [130–134,137]. Section 5 contains
transformation formulas between the different renormalization schemes. We clarify an
apparent inconsistency which we find in the renormalization of the massive contributions
to the NLO Wilson coefficients given in Refs. [103] and the massive OMEs as presented
in Refs. [126, 129]. This is due to the renormalization scheme chosen, cf. Ref. [130, 134].
In Section 6 the calculation and the results for the 2–loop massive operator matrix ele-
ments up to O(ε) in dimensional regularization are presented. This confirms the results
of Ref. [129], cf. [130]. The O(ε) terms are new results and are needed for renormalization
at O(α3s), cf. [130, 131, 133, 135–137]. We describe the calculation using hypergeometric
functions to set up infinite sums containing the parameter N as well. These sums are
solved using the summation package Sigma, cf. [137,139]. All sums can then be expressed
in terms of nested harmonic sums. The same structure is expected for the 3–loop terms, of
which we calculate fixed moments (N = 2, ..., 10(14)) using the programs QGRAF, [161],
FORM, [162,163], and MATAD, [164] in Section 7, cf. [131–134]. Thus we confirm the cor-
responding moments of the fermionic contributions to all unpolarized 3–loop anomalous
dimensions which have been calculated before in Refs. [111,112,114,124,125]. In Section 8
we calculate the asymptotic heavy flavor Wilson coefficients for the polarized structure
function g1(x,Q2) to O(α2s) following Ref. [165] and compare them with the results given
there. We newly present the terms of O(α2sε) which contribute to the polarized massive
OMEs at O(α3s) through renormalization, [157–159]. One may also consider the local
flavor non–singlet tensor operator for transversity, [166]. This is done in Section 9. We
derive the corresponding massive OMEs for general values of N up to O(α2sε) and for
the fixed moments N = 1 . . . 13 at O(α3s), [160]. A calculation keeping the full N depen-
dence has not been performed yet. In Section 10 we describe several steps which have
been undertaken in this direction so far. This involves the calculation of several non–
trivial 3–loop scalar integrals for all N and the description of a technique to reconstruct
the complete result starting from a fixed number of moments, cf. [140, 141]. Section 11
contains the conclusions. Our conventions are summarized in Appendix A. The set of
Feynman–rules used, in particular for the composite operators, is given in Appendix B.
In Appendix C we summarize properties of special functions which frequently occurred
in this work. Appendix D contains examples of different types of infinite sums which had
to be computed in the present calculation. The main results are shown in Appendices E–
G: various anomalous dimensions and the constant contributions of the different massive
OMEs for fixed values of N at O(α3s). All Figures in this work have been drawn using
Axodraw, [167].
11
2 Deeply Inelastic Scattering
Deep–inelastic scattering experiments provide one of the cleanest possibilities to probe
the space–like short distance structure of hadrons through the reactions
l±N → l± +X (2.1)
νl(νl)N → l∓ +X (2.2)
l∓N → νl(νl) +X , (2.3)
with l = e, µ, νl = νe,µ,τ , N = p, d or a nucleus, and X the inclusive hadronic final state.
The 4-momentum transfers q2 = −Q2 involved are at least of the order of Q2 ≥ 4 GeV2
and one may resolve spatial scales of approximately 1/
√
Q2. The different deep inelastic
charged– and neutral current reactions offer complementary sensitivity to unfold the quark
flavor and gluonic structure of the nucleons. Furthermore, polarized lepton scattering off
polarized targets is studied in order to investigate the spin structure of the nucleons.
The electron–proton experiments performed at SLAC in 1968, [15, 18], cf. also [19],
and at DESY, [168], found the famous scaling behavior of the structure functions which
had been predicted by Bjorken before, [17]. These measurements led to the creation
of the parton model, [20, 21, 26]. Several years later, after a series of experiments had
confirmed its main predictions, the partons were identified with the quarks, anti-quarks
and gluons as real quantum fields, which are confined inside hadrons. Being formerly
merely mathematical objects, [1,2], they became essential building blocks of the Standard
Model of elementary particle physics, besides the leptons and the electroweak gauge fields,
thereby solving the anomaly-problem, [32,33].
In the following years, more studies were undertaken at higher energies, such as the
electron–proton/neutron scattering experiments at SLAC, [64]. Muons were used as probes
of the nucleons by EMC, [65], BCDMS, [66], and NMC, [67], at the SPS, [169], at CERN, as
well as by the E26–, [44,170], CHIO–, [68], and E665–, [69], collaborations at FERMILAB.
For a general review of µ± N–scattering, see [171]. The latter experiments were augmented
by several high energy neutrino scattering experiments by the CHARM– and CDHSW–
collaborations, [70, 71, 172], and the WA21/25–experiments, [72, 173], at the SPS, and
by the CCFR–collaboration, [73, 174], at FERMILAB. Further results on neutrinos were
reported in Refs. [74], cf. also [175]. The data of these experiments confirmed QCD as the
theory describing the strong interactions within hadrons, most notably by the observation
of logarithmic scaling violations of the structure functions at higher energies and lower
values of x, which had been precisely predicted by theoretical calculations, [46].
All these experiments had in common that they were fixed target experi-
ments and therefore could only probe a limited region of phase space, up to
x ≥ 10−3, Q2 ≤ 500GeV2. The first electron–proton collider experiments became
possible with the advent of the HERA facility, which began operating in the beginning
of the 1990ies at DESY, [75]. This allowed measurements at much larger values of Q2
and at far smaller values of x than before, x ≥ 10−4, Q2 ≤ 20000GeV2. The physics
potential for the deep–inelastic experiments at HERA was studied during a series of work-
shops, see [176–180]. HERA collected a vast amount of data until its shutdown in 2007,
a part of which is still being analyzed, reaching unprecedented experimental precisions
below the level of 1 %. Two general purpose experiments to study inclusive and various
semi-inclusive unpolarized deep–inelastic reactions, H1, [76], and ZEUS, [77], were per-
formed. Both experiments measured the structure functions F2,L(x,Q2) as well as the
12
heavy quark contributions to these structure functions to high precision. The theoretical
calculations in this thesis are important for the analysis and understanding of the latter,
as will be outlined in Section 3. The HERMES–experiment, [78], studied scattering of
polarized electrons and positrons off polarized gas–targets. HERA-B, [79], was dedicated
to the study of CP–violations in the B–sector.
In the following, we give a brief introduction into the theory of DIS and the theoretical
tools which are used to predict the properties of structure functions, such as asymptotic
scaling and scaling violations. In Section 2.1, we discuss the kinematics of the DIS process
and derive the cross section for unpolarized electromagnetic electron-proton scattering.
In Section 2.2, we give a description of the naive parton model, which was employed to
explain the results obtained at SLAC and gave a first correct qualitative prediction of the
observed experimental data. A rigorous treatment of DIS can be obtained by applying
the light–cone expansion to the forward Compton amplitude, [42], which is described in
Section 2.3. This is equivalent to the QCD–improved parton model at the level of twist τ =
2, cf. e.g. [37,38,54,181]. One obtains evolution equations for the structure functions and
the parton densities with respect to the mass scales considered. The evolution is governed
by the splitting functions, [117], or the anomalous dimensions, [46], cf. Section 2.4.
2.1 Kinematics and Cross Section
The schematic diagram for the Born cross section of DIS is shown in Figure 1 for single
gauge boson exchange. A lepton with momentum l scatters off a nucleon of mass M
P
}
PF
q
l
l′
Figure 1: Schematic graph of deeply inelastic scattering for single boson exchange.
and momentum P via the exchange of a virtual vector boson with momentum q. The
momenta of the outgoing lepton and the set of hadrons are given by l′ and PF , respec-
tively. Here F can consist of any combination of hadronic final states allowed by quantum
number conservation. We consider inclusive final states and thus all the hadronic states
contributing to F are summed over. The kinematics of the process can be measured from
the scattered lepton or the hadronic final states, cf. e.g. [182–184], depending on the re-
spective experiment. The virtual vector boson has space-like momentum with a virtuality
Q2
Q2 ≡ −q2 , q = l − l′ . (2.4)
13
There are two additional independent kinematic variables for which we choose
s ≡ (P + l)2 , (2.5)
W 2 ≡ (P + q)2 = P 2F . (2.6)
Here, s is the total cms energy squared and W denotes the invariant mass of the hadronic
final state. In order to describe the process, one usually refers to Bjorken’s scaling variable
x, the inelasticity y, and the total energy transfer ν of the lepton to the nucleon in the
nucleon’s rest frame, [185]. They are defined by
ν ≡ P.qM =
W 2 +Q2 −M2
2M , (2.7)
x ≡ −q
2
2P.q =
Q2
2Mν =
Q2
W 2 +Q2 −M2 , (2.8)
y ≡ P.qP.l =
2Mν
s−M2 =
W 2 +Q2 −M2
s−M2 , (2.9)
where lepton masses are disregarded. In general, the virtual vector boson exchanged can
be a γ, Z or W±–boson with the in– and outgoing lepton, respectively, being an electron,
muon or neutrino. In the following, we consider only unpolarized neutral current charged
lepton–nucleon scattering. In addition, we will disregard weak gauge boson effects caused
by the exchange of a Z–boson. This is justified as long as the virtuality is not too large,
i.e. Q2 < 500 GeV2, cf. [186]. We assume the QED- and electroweak radiative corrections
to have been carried out, [183,184,187].
The kinematic region of DIS is limited by a series of conditions. The hadronic mass
obeys
W 2 ≥M2 . (2.10)
Furthermore,
ν ≥ 0 , 0 ≤ y ≤ 1 , s ≥M2 . (2.11)
From (2.10) follows the kinematic region for Bjorken-x via
W 2 = (P + q)2 = M2 −Q2
(
1− 1x
)
≥M2 =⇒ 0 ≤ x ≤ 1 . (2.12)
Note that x = 1 describes the elastic process, while the inelastic region is defined by x < 1.
Additional kinematic constraints follow from the design parameters of the accelerator,
[188]. In the case of HERA, these were 820(920) GeV for the proton beam and 27.5 GeV
for the electron beam, resulting in a cms–energy
√s of 300.3(319) GeV 3. This additionally
imposes kinematic constraints which follow from
Q2 = xy(s−M2) , (2.13)
correlating s and Q2. For the kinematics at HERA, this implies
Q2 ≤ sx ≈ 105x . (2.14)
3During the final running period of HERA, low–energy measurements were carried out with Ep =
460 (575) GeV in order to extract the longitudinal structure function FL(x,Q2), [80].
14
In order to calculate the cross section of deeply inelastic ep–scattering, one considers
the tree–level transition matrix element for the electromagnetic current. It is given by,
cf. e.g. [37,38,54],
Mfi = e2u(l′, η′)γµu(l, η)
1
q2 〈PF | J
em
µ (0) | P, σ〉 . (2.15)
Here, the spin of the charged lepton or nucleon is denoted by η(η′) and σ, respectively.
The state vectors of the initial–state nucleons and the hadronic final state are | P, σ〉
and | PF 〉. The Dirac–matrices are denoted by γµ and bi–spinors by u, see Appendix A.
Further e is the electric unit charge and Jemµ (ξ) the quarkonic part of the electromagnetic
current operator, which is self-adjoint :
J†µ(ξ) = Jµ(ξ) . (2.16)
In QCD, it is given by
Jemµ (ξ) =
∑
f,f ′
Ψf (ξ)γµλemff ′Ψf ′(ξ) , (2.17)
where Ψf (ξ) denotes the quark field of flavor f . For three light flavors, λem is given by the
following combination of Gell–Mann matrices of the flavor group SU(3)flavor, cf. [189,190],
λem = 1
2
(
λ3flavor +
1√
3
λ8flavor
)
. (2.18)
According to standard definitions, [37, 38, 54, 191], the differential inclusive cross section
is then given by
l′0
dσ
d3l′ =
1
32(2π)3(l.P )
∑
η′,η,σ,F
(2π)4δ4(PF + l′ − P − l)|Mfi|2 . (2.19)
Inserting the transition matrix element (2.15) into the relation for the scattering cross
section (2.19), one notices that the trace over the leptonic states forms a separate tensor,
Lµν . Similarly, the hadronic tensor Wµν is obtained,
Lµν(l, l′) =
∑
η′,η
[
u(l′, η′)γµu(l, η)
]∗[
u(l′, η′)γνu(l, η)
]
, (2.20)
Wµν(q, P ) =
1
4π
∑
σ,F
(2π)4δ4(PF − q − P )〈P, σ | Jemµ (0) | PF 〉〈PF | Jemν (0) | P, σ〉 .
(2.21)
Thus one arrives at the following relation for the cross section
l′0
dσ
d3l′ =
1
4P.l
α2
Q4L
µνWµν =
1
2(s−M2)
α2
Q4L
µνWµν , (2.22)
where α denotes the fine-structure constant, see Appendix A. The leptonic tensor in (2.22)
can be easily computed in the context of the Standard Model,
Lµν(l, l′) = Tr[l/γµl′/γν ] = 4
(
lµl′ν + l′µlν −
Q2
2
gµν
)
. (2.23)
15
This is not the case for the hadronic tensor, which contains non–perturbative hadronic con-
tributions due to long-distance effects. To calculate these effects a priori, non-perturbative
QCD calculations have to be performed, as in QCD lattice simulations. During the last
years these calculations were performed with increasing systematic and numerical accu-
racy, cf. e.g. [192,193].
The general structure of the hadronic tensor can be fixed using S–matrix theory and
the global symmetries of the process. In order to obtain a form suitable for the subsequent
calculations, one rewrites Eq. (2.21) as, cf. [38, 194],
Wµν(q, P ) =
1
4π
∑
σ
∫
d4ξ exp(iqξ)〈P | [Jemµ (ξ), Jemν (0)] | P 〉
=
1
2π
∫
d4ξ exp(iqξ)〈P | [Jemµ (ξ), Jemν (0)] | P 〉 . (2.24)
Here, the following notation for the spin-average is introduced in Eq. (2.24)
1
2
∑
σ
〈P, σ | X | P, σ〉 ≡ 〈P | X | P 〉 . (2.25)
Further, [a, b] denotes the commutator of a and b. Using symmetry and conservation laws,
the hadronic tensor can be decomposed into different scalar structure functions and thus
be stripped of its Lorentz–structure. In the most general case, including polarization, there
are 14 independent structure functions, [195, 196], which contain all information on the
structure of the proton. However, in the case considered here, only two structure functions
contribute. One uses Lorentz– and time–reversal invariance, [42], and additionally the
fact that the electromagnetic current is conserved. This enforces electromagnetic gauge
invariance for the hadronic tensor,
qµW µν = 0 . (2.26)
The leptonic tensor (2.23) is symmetric and thus Wµν can be taken to be symmetric as
well, since all antisymmetric parts are canceled in the contraction. By making a general
ansatz for the hadronic tensor using these properties, one obtains
Wµν(q, P ) =
1
2x
(
gµν +
qµqν
Q2
)
FL(x,Q2)
+
2x
Q2
(
PµPν +
qµPν + qνPµ
2x −
Q2
4x2 gµν
)
F2(x,Q2) . (2.27)
The dimensionless structure functions F2(x,Q2) and FL(x,Q2) depend on two variables,
Bjorken-x and Q2, contrary to the case of elastic scattering, in which only one variable,
e.g. Q2, determines the cross section. Due to hermiticity of the hadronic tensor, the
structure functions are real. The decomposition (2.27) of the hadronic tensor leads to the
differential cross section of unpolarized DIS in case of single photon exchange
dσ
dxdy =
2πα2
xyQ2
{[
1 + (1− y)2
]
F2(x,Q2)− y2FL(x,Q2)
}
. (2.28)
A third structure function, F1(x,Q2),
F1(x,Q2) =
1
2x
[
F2(x,Q2)− FL(x,Q2)
]
, (2.29)
16
which is often found in the literature, is not independent of the previous ones.
For completeness, we finally give the full Born cross section for the neutral current,
including the exchange of Z–bosons, cf. [184]. Not neglecting the lepton mass m, it is
given by
d2σNC
dxdy =
2πα2
xyQ2
{[
2 (1− y)− 2xyM
2
s +
(
1− 2m
2
Q2
)(
1 + 4x2M
2
Q2
)
× y
2
1 +R(x,Q2)
]
F2(x,Q2) + xy(2− y)F3(x,Q2)
}
. (2.30)
Here, R(x,Q2) denotes the ratio
R(x,Q2) = σLσT
=
(
1 + 4x2M
2
Q2
) F2(x,Q2)
2xF1(x,Q2)
− 1 , (2.31)
and the effective structure functions Fl(x,Q2), l = 1...3 are represented by the structure
functions Fl, Gl and Hl via
F1,2(x,Q2) = F1,2(x,Q2) + 2|Qe| (ve + λae)χ(Q2)G1,2(x,Q2)
+ 4
(
v2e + a2e + 2λveae
)
χ2(Q2)H1,2(x,Q2) , (2.32)
xF3(x,Q2) = −2 sign(Qe)
{
|Qe| (ae + λve)χ(Q2)xG3(x,Q2)
+
[
2veae + λ
(
v2e + a2e
)]
χ2(Q2)xH3(x,Q2)
}
. (2.33)
Here, Qe = −1, ae = 1 in case of electrons and
λ = ξ sign(Qe) , (2.34)
ve = 1− 4 sin2 θeffW , (2.35)
χ(Q2) = Gµ√
2
M2Z
8πα(Q2)
Q2
Q2 +M2Z
, (2.36)
with ξ the electron polarization, θeffW the effective weak mixing angle, Gµ the Fermi constant
and MZ the Z–boson mass.
2.2 The Parton Model
The structure functions (2.27) depend on two kinematic variables, x and Q2. Based on
an analysis using current algebra, Bjorken predicted scaling of the structure functions,
cf. [17],
lim
{Q2, ν} → ∞, x=const.
F(2,L)(x,Q2) = F(2,L)(x) . (2.37)
This means that in the Bjorken limit {Q2, ν } → ∞, with x fixed, the structure functions
depend on the ratio Q2/ν only. Soon after this prediction, approximate scaling was
17
observed experimentally in electron-proton collisions at SLAC (1968), [18], cf. also [19] 4.
Similar to the α−particle scattering experiments by Rutherford in 1911, [197], the cross
section remained large at high momentum transfer Q2, a behavior which is known from
point–like targets. This was found in contradiction to the expectation that the cross
section should decrease rapidly with increasing Q2, since the size of the proton had been
determined to be about 10−13 cm with a smooth charge distribution, [198]. However,
only in rare cases a single proton was detected in the final state, instead it consisted of
a large number of hadrons. A proposal by Feynman contained the correct ansatz. To
account for the observations, he introduced the parton model, [20,21], cf. also [22,26,37,
54, 181]. He assumed the proton as an extended object, consisting of several point-like
particles, the partons. They are bound together by their mutual interaction and behave
like free particles during the interaction with the highly virtual photon in the Bjorken-
limit 5. One arrives at the picture of the proton being “frozen” while the scattering takes
place. The electron scatters elastically off the partons and this process does not interfere
with the other partonic states, the “spectators”. The DIS cross section is then given
by the incoherent sum over the individual virtual electron–parton cross sections. Since
no information on the particular proton structure is known, Feynman described parton i
by the parton distribution function (PDF) fi(z). It gives the probability to find parton
i in the “frozen” proton, carrying the fraction z of its momentum. Figure 2 shows a
schematic picture of the parton model in Born approximation. The in– and outgoing
parton momenta are denoted by p and p′, respectively.
P
p = zP
l′
l
p′
︸ ︷︷ ︸
spectators
Figure 2: Deeply inelastic electron-proton scattering in the parton model.
Similar to the scaling variable x, one defines the partonic scaling variable τ ,
τ ≡ Q
2
2p.q . (2.38)
It plays the same role as the Bjorken-variable, but for the partonic sub-process. In the
4The results obtained at DESY, [168], pointed in the same direction, but were less decisive, because
not as large values of Q2 as at SLAC could be reached.
5Asymptotic freedom, which was discovered later, is instrumental for this property.
18
collinear parton model 6, which is applied throughout this thesis, p = zP holds, i.e., the
momentum of the partons is taken to be collinear to the proton momentum. From (2.38)
one obtains
τz = x . (2.39)
Feynman’s original parton model, referred to as the naive parton model, neglects the mass
of the partons and enforces the strict correlation
δ
(q.p
M −
Q2
2M
)
, (2.40)
due to the experimentally observed scaling behavior, which leads to z = x. The naive parton
model then assumes, in accordance with the quark hypothesis, [1,2,26], that the proton is
made up of three valence quarks, two up and one down type, cf. e.g. [5]. This conclusion
was generally accepted only several years after the introduction of the parton model, when
various experiments had verified its predictions.
Let us consider a simple example, which reproduces the naive parton model at LO and
incorporates already some aspects of the improved parton model. The latter allows virtual
quark states (sea-quarks) and gluons as partons as well. In the QCD–improved parton
model, cf. [37,54,181], besides the δ-distribution, (2.40), a functionW iµν(τ,Q2) contributes
to the hadronic tensor. It is called partonic tensor and given by the hadronic tensor, Eq.
(2.24), replacing the hadronic states by partonic states i. The basic assumption is that the
hadronic tensor can be factorized into the PDFs and the partonic tensor, cf. e.g. [51,203].
The PDFs are non-perturbative quantities and have to be extracted from experiment,
whereas the partonic tensors are calculable perturbatively. A more detailed discussion of
this using the LCE is given in Section 2.3. The hadronic tensor reads, cf. [56],
Wµν(x,Q2) =
1
4π
∑
i
∫ 1
0
dz
∫ 1
0
dτ (fi(z) + fi(z))W iµν(τ,Q2)δ(x− zτ) . (2.41)
Here, the number of partons and their respective type are not yet specified and we have
included the corresponding PDF of the respective anti-parton, denoted by fi(z). Let us
assume that the electromagnetic parton current takes the simple form
〈i | jiµ(τ) | i〉 = −ieiuiγµui , (2.42)
similar to the leptonic current, (2.15). Here ei is the electric charge of parton i. At LO
one finds
W iµν(τ,Q2) =
2πe2i
q.pi δ(1− τ)
[
2piµpiν + piµqν + piνqµ − gµνq.pi
]
. (2.43)
The δ-distribution in (2.43), together with the δ-distribution in (2.41), just reproduces
Feynman’s assumption of the naive parton model, z = x. Substitution of (2.43) into
the general expression for the hadronic tensor (2.27) and projecting onto the structure
functions yields
FL(x,Q2) = 0 ,
F2(x,Q2) = x
∑
i
e2i (fi(x) + fi(x)) . (2.44)
This result, at LO, is the same as in the naive parton model. It predicts
6For other parton models, as the covariant parton model, cf. [199–202].
19
• the Callan-Gross relation, cf. [23],
FL(x,Q2) = F2(x,Q2)− 2xF1(x,Q2) = 0 . (2.45)
• the structure functions are scale-independent.
These findings were a success of the parton model, since they reproduced the general
behavior of the data as observed by the MIT/SLAC experiments.
Finally, we present for completeness the remaining structure functions G2,3 and H2,3
at the Born level for the complete neutral current, cf. Eq. (2.30),
G2(x,Q2) = x
∑
i
|ei|vi (fi(x) + fi(x)) , (2.46)
H2(x,Q2) = x
∑
i
1
4
(
v2i + a2i
)
(fi(x) + fi(x)) , (2.47)
xG3(x,Q2) = x
∑
i
|ei|ai (fi(x)− fi(x)) , (2.48)
xH3(x,Q2) = x
∑
i
1
2
viai (fi(x)− fi(x)) , (2.49)
with ai = 1 and
vi = 1− 4|ei| sin2 θeffW . (2.50)
2.2.1 Validity of the Parton Model
The validity of the parton picture can be justified by considering an impulse approximation
of the scattering process as seen from a certain class of reference frames, in which the
proton momentum is taken to be very large (P∞-frames). Two things happen to the
proton when combining this limit with the Bjorken–limit: The internal interactions of its
partons are time dilated, and it is Lorentz contracted in the direction of the collision. As
the cms energy increases, the parton lifetimes are lengthened and the time it takes the
electron to interact with the proton is shortened. Therefore the condition for the validity
of the parton model is given by, cf. [26, 204],
τint
τlife
≪ 1 . (2.51)
Here τint denotes the interaction time and τlife the average life time of a parton. If (2.51)
holds, the proton will be in a single virtual state characterized by a certain number of
partons during the entire interaction time. This justifies the assumption that parton i
carries a definite momentum fraction zi, 0 ≤ zi ≤ 1, of the proton in the cms. This parton
model is also referred to as collinear parton model, since the proton is assumed to consist
out of a stream of partons with parallel momenta. Further
∑
i zi = 1 holds. In order
to derive the fraction of times in (2.51), one aligns the coordinate system parallel to the
proton’s momentum. Thus one obtains in the limit P 23 ≫M2, [205],
P =
(√
P 23 +M2; 0, 0, P3
)
≈
(
P3 +
M2
2 · P3
; 0, 0, P3
)
. (2.52)
20
The photon momentum can be parametrized by
q = (q0; q3, ~q⊥) , (2.53)
where ~q⊥ denotes its transverse momentum with respect to the proton. By choosing
the cms of the initial states as reference and requiring that νM and q2 approach a limit
independent of P3 as P3 →∞, one finds for the characteristic interaction time scale, using
an (approximate) time–energy uncertainty relation,
τint ≃
1
q0
=
4P3x
Q2(1− x) . (2.54)
The life time of the individual partons is estimated accordingly to be inversely proportional
to the energy fluctuations of the partons around the average energy E
τlife ≃
1∑
i Ei − E
. (2.55)
Here Ei denote the energies of the individual partons. After introducing the two-
momentum ~k⊥i of the partons perpendicular to the direction of motion of the proton
as given in (2.52), a simple calculation yields, cf. [205],
τint
τlife
=
2x
Q2(1− x)
(
∑
i
(m2i + k2⊥i)
zi
−M2
)
, (2.56)
where mi denotes the mass of the i-th parton. This expression is independent of P3. The
above procedure allows therefore to estimate the probability of deeply inelastic scattering
to occur independently of the large momentum of the proton. Accordingly, we consider
now the case of two partons with momentum fractions x and 1−x and equal perpendicular
momentum, neglecting all masses. One obtains
τint
τlife
≈ 2k
2
⊥
Q2(1− x)2 . (2.57)
This example leads to the conclusion, that deeply inelastic scattering probes single partons
if the virtuality of the photon is much larger than the transverse momenta squared of the
partons and Bjorken-x is neither close to one nor zero. In the latter case, xP3 would be
the large momentum to be considered. If one does not neglect the quark masses, one has
to adjust this picture, as will be described in Section 3.3.
2.3 The Light–Cone Expansion
In quantum field theory one usually considers time-ordered products, denoted by T, rather
than a commutator as it appears in the hadronic tensor in Eq. (2.24). The hadronic tensor
can be expressed as the imaginary part of the forward Compton amplitude for virtual
gauge boson–nucleon scattering, Tµν(q, P ). The optical theorem, depicted graphically in
Figure 3, yields
Wµν(q, P ) =
1
π Im Tµν(q, P ) , (2.58)
21
where the Compton amplitude is given by, cf. [189],
Tµν(q, P ) = i
∫
d4ξ exp(iqξ)〈P | TJµ(ξ)Jν(0) | P 〉 . (2.59)
∑
F
F
2
= 1π Im
Figure 3: Schematic picture of the optical theorem.
By applying the same invariance and conservation conditions as for the hadronic ten-
sor, the Compton amplitude can be expressed in the unpolarized case by two amplitudes
TL(x,Q2) and T2(x,Q2). It is then given by
Tµν(q, P ) =
1
2x
(
gµν +
qµqν
Q2
)
TL(x,Q2)
+
2x
Q2
(
PµPν +
qµPν + qνPµ
2x −
Q2
4x2 gµν
)
T2(x,Q2) . (2.60)
Using translation invariance, one can show that (2.59) is crossing symmetric under
q → −q, cf. [195,206],
Tµν(q, P ) = Tµν(−q, P ) , (2.61)
with q → −q being equivalent to ν, x→ (−ν), (−x). The corresponding relations for the
amplitudes are then obtained by considering (2.60)
T(2,L)(x,Q2) = T(2,L)(−x,Q2) . (2.62)
By (2.58) these amplitudes relate to the structure functions FL and F2 as
F(2,L)(x,Q2) =
1
π Im T(2,L)(x,Q
2) . (2.63)
Another general property of the Compton amplitude is that TL and T2 are real analytic
functions of x at fixed Q2, cf. [50], i.e.
T(2,L)(x∗, Q2) = T ∗(2,L)(x,Q2) . (2.64)
Using this description one can perform the LCE, [42], or the cut–vertex method in the
time–like case, [207–209], respectively, and derive general properties of the moments of
the structure functions as will be shown in the subsequent Section. A technical aspect
22
which has been proved very useful is to work in Mellin space rather than in x–space. The
Nth Mellin moment of a function f is defined through the integral
M[f ](N) ≡
∫ 1
0
dz zN−1f(z) . (2.65)
This transform diagonalizes the Mellin–convolution f ⊗ g of two functions f, g
[f ⊗ g](z) =
∫ 1
0
dz1
∫ 1
0
dz2 δ(z − z1z2) f(z1)g(z2) . (2.66)
The convolution (2.66) decomposes into a simple product of the Mellin-transforms of the
two functions,
M[f ⊗ g](N) = M[f ](N)M[g](N) . (2.67)
In Eqs. (2.65, 2.67), N is taken to be an integer. However, later on one may perform an
analytic continuation to arbitrary complex values of N , [210]. Note that it is enough to
know all even or odd integer moments – as is the case for inclusive DIS – of the functions
f, g to perform an analytic continuation to arbitrary complex values N ∈ C, [211]. Then
Eq. (2.66) can be obtained from the relation for the moments, (2.67), by an inverse Mellin–
transform. Hence in this case the z– and N–space description are equivalent, which we
will frequently use later on.
2.3.1 Light–Cone Dominance
It can be shown that in the Bjorken limit, Q2 →∞, ν →∞, x fixed, the hadronic tensor
is dominated by its contribution near the light–cone, i.e. by the values of the integrand in
(2.24) at ξ2 ≈ 0, cf. [42]. This can be understood by considering the infinite momentum
frame, see Section 2.2.1,
P = (P3; 0, 0, P3) , (2.68)
q =
( ν
2P3
;
√
Q2, 0, −ν
2P3
)
, (2.69)
P3 ≈
√
ν →∞ . (2.70)
According to the Riemann–Lebesgue theorem, the integral in (2.24) is dominated by the
region where q.ξ ≈ 0 due to the rapidly oscillating exponential exp(iq.ξ), [37]. One can
now rewrite the dot product as, cf. [190],
q.ξ = 1
2
(q0 − q3)(ξ0 + ξ3) + 1
2
(q0 + q3)(ξ0 − ξ3)− q1ξ1 , (2.71)
and infer that the condition q.ξ ≈ 0 in the Bjorken-limit is equivalent to
ξ0 ± ξ3 ∝ 1√ν , ξ
1 ∝ 1√ν , (2.72)
which results in
ξ2 ≈ 0 , (2.73)
23
called light–cone dominance: for DIS in the Bjorken-limit the dominant contribution to
the hadronic tensor Wµν(q, P ) and the Compton Amplitude comes from the region where
ξ2 ≈ 0.
This property allows to apply the LCE of the current–current correlation in Eq. (2.24)
and for the time ordered product in Eq. (2.59), respectively. In the latter case it reads
for scalar currents, cf. [42],
lim
ξ2→0
TJ(ξ), J(0) ∝
∑
i,N,τ
CNi,τ (ξ2, µ2)ξµ1 ...ξµNOµ1...µNi,τ (0, µ2) . (2.74)
The Oi,τ (ξ, µ2) are local operators which are finite as ξ2 → 0. The singularities which
appear for the product of two operators as their arguments become equal are shifted
to the c-number coefficients CNi,τ (ξ2, µ2), the Wilson coefficients, and can therefore be
treated separately. In Eq. (2.74), µ2 is the factorization scale describing at which point
the separation between the perturbative and non–perturbative contributions takes place.
The summation index i runs over the set of allowed operators in the model, while the
sum over N extends to infinity. Dimensional analysis shows that the degree of divergence
of the functions CNi,τ as ξ2 → 0 is given by
CNi,τ (ξ2, µ2) ∝
(
1
ξ2
)−τ/2+dJ
. (2.75)
Here, dJ denotes the canonical dimension of the current J(ξ) and τ is the twist of the
local operator Oµ1..µNi,τ (ξ, µ2), which is defined by, cf. [43],
τ ≡ DO −N . (2.76)
DO is the canonical (mass) dimension of Oµ1..µNi,τ (ξ, µ2) and N is called its spin. From
(2.75) one can infer that the most singular coefficients are those related to the operators
of lowest twist, i.e. in the case of the LCE of the electromagnetic current (2.17), twist
τ = 2. The contributions due to higher twist operators are suppressed by factors of
(µ2/Q2)k, with µ a typical hadronic mass scale of O(1 GeV). In a wide range of phase–
space it is thus sufficient to consider the leading twist contributions only, which we will
do in the following and omit the index τ .
2.3.2 A Simple Example
In this Section, we consider a simple example of the LCE applied to the Compton am-
plitude and its relation to the hadronic tensor, neglecting all Lorentz–indices and model
dependence, cf. Ref. [38, 106]. The generalization to the case of QCD is straightforward
and hence we will already make some physical arguments which apply in both cases. The
scalar expressions corresponding to the hadronic tensor and the Compton amplitude are
given by
W (x,Q2) = 1
2π
∫
d4ξ exp(iqξ)〈P | [J(ξ), J(0)] | P 〉 , (2.77)
T (x,Q2) = i
∫
d4ξ exp(iqξ)〈P | TJ(ξ)J(0) | P 〉 . (2.78)
24
Eq. (2.78) can be evaluated in the limit ξ2 → 0 for twist τ = 2 by using the LCE given
in Eq. (2.74), where for brevity only one local operator is considered. The coefficient
functions in momentum space are defined as
∫
exp(iq.ξ)ξµ1 ..ξµNC
N
(ξ2, µ2) ≡ −i
( 2
−q2
)N
qµ1 ...qµNCN
(Q2
µ2
)
. (2.79)
The nucleon states act on the composite operators only and the corresponding matrix
elements can be expressed as
〈P | Oµ1...µN (0, µ2) | P 〉 = AN
(P 2
µ2
)
P µ1 ...P µN + trace terms. (2.80)
The trace terms in the above equation can be neglected, because due to dimensional
counting they would give contributions of the order 1/Q2, 1/ν and hence are irrelevant
in the Bjorken–limit. Thus the Compton amplitude reads, cf. e.g. [38,54],
T (x′, Q2) = 2
∑
N=0,2,4,..
CN
(Q2
µ2
)
AN
(P 2
µ2
)
x′N , x′ = 1x (2.81)
In (2.81) only the even moments contribute. This is a consequence of crossing symmetry,
Eq. (2.62), and holds as well in the general case of unpolarized DIS for single photon
exchange. In other cases the projection is onto the odd moments. Depending on the type
of the observable the series may start at different initial values, cf. e.g. [195,196]. The sum
in Eq. (2.81) is convergent in the unphysical region x ≥ 1 and an analytic continuation
to the physical region 0 ≤ x ≤ 1 has to be performed. Here, one of the assumptions is
that scattering amplitudes are analytic in the complex plane except at values of kinematic
variables allowing intermediate states to be on mass–shell. This general feature has been
proved to all orders in perturbation theory, [212,213]. In QCD, it is justified on grounds of
the parton model. When ν ≥ Q2/2M , i.e. 0 ≤ x ≤ 1, the virtual photon-proton system
can produce a physical hadronic intermediate state, so the T(2,L)(x,Q2) and T (x,Q2),
respectively, have cuts along the positive (negative) real x-axis starting from 1(−1) and
poles at ν = Q2/2M (x = 1,−1). The discontinuity along the cut is then just given
by (2.58). The Compton amplitude can be further analyzed by applying (subtracted)
dispersion relations, cf. [195, 196]. Equivalently, one can divide both sides of Eq. (2.81)
by x′m and integrate along the path shown in Figure 4, cf. [38, 56].
1−1
Figure 4: Integration contour in the complex x′-plane.
25
For the left–hand side of (2.81) one obtains
1
2πi
∮
dx′T (x
′, Q2)
x′m =
2
π
∫ ∞
1
dx′
x′m ImT (x
′, Q2) = 2
∫ 1
0
dx xm−2W (x,Q2) , (2.82)
where the optical theorem, (2.58), and crossing symmetry, (2.62) have been used. The
right–hand side of (2.81) yields
1
πi
∑
N=0,2,4,..
CN
(Q2
µ2
)
AN
(P 2
µ2
)∮
dx′ x′N−m = 2Cm−1
(Q2
µ2
)
Am−1
(P 2
µ2
)
. (2.83)
Thus from Eqs. (2.82) and (2.83) one obtains for the moments of the scalar hadronic
tensor defined in Eq. (2.77)
∫ 1
0
dx xN−1W (x,Q2) = CN
(Q2
µ2
)
AN
(P 2
µ2
)
. (2.84)
2.3.3 The Light–Cone Expansion applied to DIS
In order to derive the moment–decomposition of the structure functions one essentially
has to go through the same steps as in the previous Section. The LCE of the physical
forward Compton amplitude (2.59) at the level of twist τ = 2 in the Bjorken–limit is
given by, cf. [48, 106],
Tµν(q, P ) →
∑
i,N
{ [
Q2gµµ1gνµ2 + gµµ1qνqµ2 + gνµ2qµqµ1 − gµνqµ1qµ2
]
Ci,2
(
N, Q
2
µ2
)
+
[
gµν +
qµqµ
Q2
]
qµ1qµ2Ci,L
(
N, Q
2
µ2
)}
qµ3 ...qµN
( 2
Q2
)N
〈P | Oµ1...µNi (µ2) | P 〉 .
(2.85)
Additionally to Section 2.3.2, the index i runs over the allowed operators which emerge
from the expansion of the product of two electromagnetic currents, Eq. (2.17). The
possible twist–2 operators are given by 7, [208],
ONSq,r;µ1,... ,µN = i
N−1S[ψγµ1Dµ2 . . . DµN
λr
2
ψ]− trace terms , (2.86)
OSq;µ1,... ,µN = i
N−1S[ψγµ1Dµ2 . . . DµNψ]− trace terms , (2.87)
OSg;µ1,... ,µN = 2i
N−2SSp[F aµ1αDµ2 . . . DµN−1F
α,a
µN ]− trace terms . (2.88)
Here, S denotes the symmetrization operator of the Lorentz indices µ1, . . . , µN . λr is the
flavor matrix of SU(nf ) with nf light flavors, ψ denotes the quark field, F aµν the gluon
field–strength tensor, and Dµ the covariant derivative. The indices q, g represent the
quark– and gluon–operator, respectively. Sp in (2.88) is the color–trace and a the color
index in the adjoint representation, cf. Appendix A. The quark–fields carry color indices
in the fundamental representation, which have been suppressed. The classification of the
7Here we consider only the spin–averaged case for single photon exchange. Other operators contribute
for parity–violating processes, in the polarized case and for transversity, cf. Sections 8 and 9.
26
composite operators (2.86–2.88) in terms of flavor singlet (S) and non-singlet (NS) refers
to their symmetry properties with respect to the flavor group SU(nf ). The operator
in Eq. (2.86) belongs to the adjoint representation of SU(nf ), whereas the operators in
Eqs. (2.87, 2.88) are singlets under SU(nf ). Neglecting the trace terms, one rewrites the
matrix element of the composite operators in terms of its Lorentz structure and the scalar
operator matrix elements, cf. [54, 190],
〈P | Oµ1...µNi | P 〉 = Ai
(
N, P
2
µ2
)
P µ1 ...P µN . (2.89)
Eq. (2.85) then becomes
Tµν(q, P ) = 2
∑
i,N
{
2x
Q2
[
PµPν +
Pµqν + Pνqµ
2x −
Q2
4x2 gµν
]
Ci,2
(
N, Q
2
µ2
)
+
1
2x
[
gµν +
qµqµ
Q2
]
Ci,L
(
N, Q
2
µ2
)} 1
xN−1Ai
(
N, P
2
µ2
)
. (2.90)
Comparing Eq. (2.90) with the general Lorentz structure expected for the Compton ampli-
tude, Eq. (2.60), the relations of the scalar forward amplitudes to the Wilson coefficients
and nucleon matrix elements can be read off
T(2,L)(x,Q2) = 2
∑
i,N
1
xN−1Ci,(2,L)
(
N, Q
2
µ2
)
Ai
(
N, P
2
µ2
)
. (2.91)
Eq. (2.91) is of the same type as Eq. (2.81) and one thus obtains for the moments of the
structure functions
F(2,L)(N,Q2) = M[F(2,L)(x,Q2)](N) (2.92)
=
∑
i
Ci,(2,L)
(Q2
µ2 , N
)
Ai
(P 2
µ2 , N
)
. (2.93)
The above equations have already been written in Mellin space, which we will always do
from now on, if not indicated otherwise. Eqs. (2.91, 2.93), together with the general struc-
ture of the Compton amplitude, Eqs. (2.60, 2.90), and the hadronic tensor, Eq. (2.27), are
the basic equations for theoretical or phenomenological analysis of DIS in the kinematic
regions where higher twist effects can be safely disregarded. Note that the generalization
of these equations to electroweak or polarized interactions is straightforward by includ-
ing additional operators and Wilson coefficients. In order to interpret Eqs. (2.91, 2.93),
one uses the fact that the Wilson coefficients Ci,(2,L) are independent of the proton state.
This is obvious since the wave function of the proton only enters into the definition of the
operator matrix elements, cf. Eq. (2.89). In order to calculate the Wilson coefficients,
the proton state has therefore to be replaced by a suitably chosen quark or gluon state i
with momentum p. The corresponding partonic tensor is denoted byW iµν(q, p), cf. below
Eq. (2.40), with scalar amplitudes F i(2,L)(τ,Q2). Here τ is the partonic scaling variable
defined in Eq. (2.38). The LCE of the electromagnetic current does not change and the
replacement only affects the operator matrix elements. The forward Compton amplitude
for photon–quark (gluon) scattering corresponding to W iµν(q, p) can be calculated order
27
by order in perturbation theory, provided the scale Q2 is large enough for the strong
coupling constant to be small. In the same manner, the contributing operator matrix
elements with external partons may be evaluated. Finally, one can read off the Wilson
coefficients from the partonic equivalent of Eq. (2.91) 8. By identifying the nucleon OMEs
(2.89) with the PDFs, one obtains the QCD improved parton model. At LO it coincides
with the naive parton model, which we described in Section 2.2, as can be inferred from
the discussion below Eq. (2.41). The improved parton model states that in the Bjorken
limit at the level of twist τ = 2 the unpolarized nucleon structure functions Fi(x,Q2)
are obtained in Mellin space as products of the universal parton densities fi(N,µ2) with
process–dependent Wilson coefficients Ci,(2,L)(N,Q2/µ2)
F(2,L)(N,Q2) =
∑
i
Ci,(2,L)
(
N, Q
2
µ2
)
fi(N,µ2) (2.94)
to all orders in perturbation theory. This property is also formulated in the factorization
theorems, [203], cf. also [51], where it is essential that an inclusive, infrared–safe cross sec-
tion is considered, [214]. We have not yet dealt with the question of how renormalization
is being performed. However, we have already introduced the scale µ2 into the right–hand
side of Eq. (2.94). This scale is called factorization scale. It describes a mass scale at
which the separation of the structure functions into the perturbative hard scattering coef-
ficients Ci,(2,L) and the non–perturbative parton densities fi can be performed. This choice
is arbitrary at large enough scales and the physical structure functions do not depend on
it. This independence is used in turn to establish the corresponding renormalization group
equation, [215,216], which describes the scale–evolution of the Wilson coefficients, parton
densities and structure functions w.r.t. to µ2 and Q2, cf. Refs. [37, 48, 54, 217–219] and
Section 2.4.
These evolution equations then predict scaling violations and are used to analyze
experimental data in order to unfold the twist–2 parton distributions at some scale Q20,
together with the QCD–scale ΛQCD, cf. [217,220,221].
Before finishing this Section, we describe the quantities appearing in Eq. (2.94) in
detail. Starting from the operators defined in Eqs. (2.86)–(2.88), three types of parton
densities are expected. Since the question how heavy quarks are treated in this framework
will be discussed in Section 3, we write the following equations for nf light flavors in
massless QCD. The gluon density is denoted by G(nf , N, µ2) and multiplies the gluonic
Wilson coefficients Cg,(2,L)(nf , N,Q2/µ2), which describe the interaction of a gluon with
a photon and emerge for the first time at O(αs). Each quark and its anti–quark have a
parton density, denoted by fk(k)(nf , N, µ2). These are grouped together into the flavor
singlet combination Σ(nf , N, µ2) and a non–singlet combination ∆k(nf , N, µ2) as follows
Σ(nf , N, µ2) =
nf∑
l=1
[
fl(nf , N, µ2) + fl¯(nf , N, µ2)
]
, (2.95)
∆k(nf , N, µ2) = fk(nf , N, µ2) + fk¯(nf , N, µ2)−
1
nf
Σ(nf , N, µ2) . (2.96)
8Due to the optical theorem, one may also obtain the Wilson coefficients by calculating the inclusive
hard scattering cross sections of a virtual photon with a quark(gluon) using the standard Feynman–rules
and phase–space kinematics.
28
The distributions multiply the quarkonic Wilson coefficients CS,NSq,(2,L)(nf , N,Q2/µ2), which
describe the hard scattering of a photon with a light quark. The complete factorization
formula for the structure functions is then given by
F(2,L)(nf , N,Q2) =
1
nf
nf∑
k=1
e2k
[
Σ(nf , N, µ2)CSq,(2,L)
(
nf , N,
Q2
µ2
)
+ G(nf , N, µ2)CSg,(2,L)
(
nf , N,
Q2
µ2
)
+ nf∆k(nf , N, µ2)CNSq,(2,L)
(
nf , N,
Q2
µ2
)]
.
(2.97)
Note, that one usually splits the quarkonic S contributions into a NS and pure–singlet
(PS) part via S = PS + NS. The perturbative expansions of the Wilson coefficients read
CSg,(2,L)
(
nf , N,
Q2
µ2
)
=
∞∑
i=1
aisC
(i),S
g,(2,L)
(
nf , N,
Q2
µ2
)
, (2.98)
CPSq,(2,L)
(
nf , N,
Q2
µ2
)
=
∞∑
i=2
aisC
(i),PS
q,(2,L)
(
nf , N,
Q2
µ2
)
, (2.99)
CNSq,(2,L)
(
nf , N,
Q2
µ2
)
= δ2 +
∞∑
i=1
aisC
(i),NS
q,(2,L)
(
nf , N,
Q2
µ2
)
, (2.100)
where as ≡ αs/(4π) and
δ2 = 1 for F2 and δ2 = 0 for FL . (2.101)
These terms are at present known up to O(a3s). The O(as) terms have been calculated
in Refs. [105–107] and the O(a2s) contributions by various groups in Refs. [108,109]. The
O(a3s) terms have first been calculated for fixed moments in Refs. [110–112, 114] and the
complete result for all N has been obtained in Refs. [115] 9.
2.4 RGE–improved Parton Model and Anomalous Dimensions
In the following, we present a derivation of the RGEs for the Wilson coefficients, and
subsequently, the evolution equations for the parton densities. When calculating scat-
tering cross sections in quantum field theories, they usually contain divergences of dif-
ferent origin. The infrared and collinear singularities are connected to the limit of soft–
and collinear radiation, respectively. Due to the Bloch–Nordsieck theorem, [223], it is
known that the infrared divergences cancel between virtual and bremsstrahlung contribu-
tions. The structure functions are inclusive quantities. Therefore, all final state collinear
(mass) singularities cancel as well, which is formulated in the Lee–Kinoshita–Nauenberg
theorem, [224]. Thus in case of the Wilson coefficients, only the initial state collinear
divergences of the external light partons and the ultraviolet divergences remain. The
9Recently, the O(a3s) Wilson coefficient for the structure function xF3(x,Q
2) was calculated in
Ref. [222].
29
latter are connected to the large scale behavior and are renormalized by a redefinition of
the parameters of the theory, as the coupling constant, the masses, the fields, and the
composite operators, [225, 226]. This introduces a renormalization scale µr, which forms
the subtraction point for renormalization. The scale which appears in the factorization
formulas (2.94, 2.97) is denoted by µf and called factorization scale, cf. [51, 203]. Its
origin lies in the arbitrariness of the point at which short– and long–distance effects are
separated and is connected to the redefinition of the bare parton densities by absorbing
the initial state collinear singularities of the Wilson coefficients into them. Note, that
one usually adopts dimensional regularization to regularize the infinities in perturbative
calculations, cf. Section 4, which causes another scale µ to appear. It is associated to
the mass dimension of the coupling constant in D 6= 4 dimensions. In principle all these
three scales have to be treated separately, but we will set them equal in the subsequent
analysis, µ = µr = µf .
The renormalization group equations are obtained using the argument that all these
scales are arbitrary and therefore physical quantities do not alter when changing these
scales, [215,216,225,226]. One therefore defines the total derivative w.r.t. to µ2
D(µ2) ≡ µ2 ∂∂µ2 + β(as(µ
2))
∂
∂as(µ2)
− γm(as(µ2))m(µ2)
∂
∂m(µ2) . (2.102)
Here the β–function and the anomalous dimension of the mass, γm, are given by
β(as(µ2)) ≡ µ2
∂as(µ2)
∂µ2 , (2.103)
γm(as(µ2))) ≡ −
µ2
m(µ2)
∂m(µ2)
∂µ2 , (2.104)
cf. Sections 4.3, 4.4. The derivatives have to be performed keeping the bare quantities
aˆs, mˆ fixed. Additionally, we work in Feynman–gauge and therefore the gauge–parameter
is not present in Eq. (2.102). In the following we will consider only one mass m. The
composite operators (2.86)–(2.88) are renormalized introducing operator Z–factors
ONSq,r;µ1,...,µN = Z
NS(µ2)OˆNSq,r;µ1,...,µN , (2.105)
OSi;µ1,...,µN = Z
S
ij(µ2)OˆSj;µ1,...,µN , i = q, g , (2.106)
where in the singlet case mixing occurs since these operators carry the same quantum
numbers. The anomalous dimensions of the operators are defined by
γNSqq = µZ−1,NS(µ2)
∂
∂µZ
NS(µ2) , (2.107)
γSij = µZ−1,Sil (µ2)
∂
∂µZ
S
lj(µ2) . (2.108)
We begin by considering the partonic structure functions calculated with external fields l.
Here we would like to point out that we calculate matrix elements of currents, operators,
etc. and not vacuum expectation values of time–ordered products with the external fields
included. The anomalous dimensions of the latter therefore do not contribute, [190], and
they are parts of the anomalous dimensions of the composite operators, respectively. The
RGE reads
D(µ2)F l(2,L)(N,Q2) = 0 . (2.109)
30
On the partonic level, Eq. (2.93) takes the form
F l(2,L)(N,Q2) =
∑
j
Cj,(2,L)
(
N, Q
2
µ2
)
〈l | Oj(µ2) | l〉 . (2.110)
From the operator renormalization constants of the Oi, Eqs. (2.105, 2.106), the following
RGE is derived for the matrix elements, [48],
∑
j
(
D(µ2) δij +
1
2
γS,NSij
)
〈l | Oj(µ2) | l〉 = 0 , (2.111)
where we write the S and NS case in one equation for brevity and we remind the reader
that in the latter case, i, j, l = q only. Combining Eqs. (2.109, 2.110, 2.111), one can
determine the RGE for the Wilson coefficients. It reads
∑
i
(
D(µ2) δij −
1
2
γS,NSij
)
Ci,(2,L)
(
N, Q
2
µ2
)
= 0 . (2.112)
The structure functions, which are observables, obey the same RGE as on the partonic
level
D(µ2)F(2,L)(N,Q2) = µ2
d
dµ2F(2,L)(N,Q
2) = 0 . (2.113)
Using the factorization of the structure functions into Wilson coefficients and parton
densities, Eqs. (2.94, 2.97), together with the RGE derived for the Wilson coefficients in
Eq. (2.112), one obtains from the above formula the QCD evolution equations for the
parton densities, cf. e.g. [37,48,54,217–219],
d
d lnµ2f
S,NS
i (nf , N, µ2) = −
1
2
∑
j
γS,NSij fS,NSj (nf , N, µ2) . (2.114)
Eq. (2.114) describes the change of the parton densities w.r.t. the scale µ. In the more
familiar matrix notation, these equations read
d
d lnµ2
(
Σ(nf , N, µ2)
G(nf , N, µ2)
)
= −1
2
(
γqq γqg
γgq γgg
)(
Σ(nf , N, µ2)
G(nf , N, µ2)
)
, (2.115)
d
d lnµ2∆k(nf , N, µ
2) = −1
2
γNSqq ∆k(nf , N, µ2) , (2.116)
where we have used the definition for the parton densities in Eqs. (2.95, 2.96). The anoma-
lous dimensions in the above equations can be calculated order by order in perturbation
theory. At LO, [46], and NLO, [119–123], they have been known for a long time. The
NNLO anomalous dimension were calculated first for fixed moments in Refs. [111,112,114]
and the complete result for all moments has been obtained in Refs. [124, 125] 10. As de-
scribed, the PDFs are non–perturbative quantities and have to be extracted at a certain
scale from experimental data using the factorization relation (2.94). If the scale µ2 is large
enough to apply perturbation theory, the evolution equations can be used to calculate the
10Note that from our convention in Eqs. (2.107, 2.108) follows a relative factor 2 between the anomalous
dimensions considered in this work compared to Refs. [124,125].
31
PDFs at another perturbative scale, which provides a detailed QCD test comparing to
precision data. There are similar evolution equations for the structure functions and Wil-
son coefficients, cf. e.g. [37, 48, 54, 217–219]. Different groups analyze the evolution of
the parton distribution functions based on precision data from deep–inelastic scattering
experiments and other hard scattering cross sections. Analyzes were performed by the
Dortmund group, [96,102,227–233], by Alekhin et. al., [97,234], Blu¨mlein et. al., [93,98],
the MSTW–, [235], QTEQ–, [236], and the NNPDF–collaborations, [237]. The PDFs de-
termined in this way can e.g. be used as input data for the pp collisions at the LHC,
since they are universal quantities and only relate to the structure of the proton and not
to the particular kind of scattering events considered. Apart from performing precision
analyzes of the PDFs, one can also use the evolution equations to determine as more
precisely, [93,96–98,102,232–235].
The evolution equations (2.114, 2.115, 2.116) are written for moments only. The
representation in x–space is obtained by using (2.65, 2.66, 2.67) and is usually expressed
in terms of the splitting functions Pij(x), [117]. At the level of twist–2 the latter are
connected to the anomalous dimensions by the Mellin–transform
γij(N) = −M[Pij](N) . (2.117)
The behavior of parton distribution functions in the small x region attracted special inter-
est due to possibly new dynamical contributions, such as Glauber–model based screening
corrections, [238], and the so-called BFKL contributions, a ‘leading singularity’ resum-
mation in the anomalous dimensions for all orders in the strong coupling constant, [239].
For both effects there is no evidence yet in the data both for F2(x,Q2) and FL(x,Q2),
beyond the known perturbative contributions to O(a3s). This does not exclude that at
even smaller values of x contributions of this kind will be found. The BFKL contributions
were investigated on the basis of a consistent renormalization group treatment, together
with the fixed order contributions in Refs. [240, 241]. One main characteristic, compar-
ing with the fixed order case, is that several sub-leading series, which are unknown, are
required to stabilize the results, see also [242]. This aspect also has to be studied within
the framework of recent approaches, [243].
32
3 Heavy Quark Production in DIS
In the Standard Model, the charm, bottom and top quark are treated as heavy quarks, all
of which have a mass larger than the QCD–scale ΛQCD(nf = 4) ≈ 240 MeV, [93, 96–98,
232, 233, 244]. The up, down and strange quark are usually treated as massless. Because
of confinement, the quarks can only be observed via the asymptotic states baryons and
mesons, in which they are contained. In the following, we concentrate on the inclusive
production of one species of a heavy quark, denoted by Q(Q), with mass m. In the
case of HERA kinematics, Q = c. This is justified to a certain extent by the observation
that bottom quark contributions to DIS structure functions are much smaller, cf. [86] 11.
Since the ratio m2c/m2b ≈ 1/10 is not small, there are regions in which both masses
are potentially important. The description of these effects is beyond the scope of the
formalism outlined below and of comparable order as the m2c/Q2 corrections. Top–quark
production in l±N scattering is usually treated as a semi–inclusive process, cf. [246,247].
Charmed mesons are more abundantly produced at HERA than baryons. D–mesons
are bound states of charm and lighter quarks, e.g. Du = uc, Dd = dc etc. Furthermore
also cc resonances contribute, such as J/Ψ, by the observation of which charm was dis-
covered, [57, 58]. The charm contributions to the structure functions are determined in
experiment by tagging charm quarks in the final state, e.g. through the D–meson decay
channel D∗ → D0πs → Kππs. In the case of DIS, the measured visible cross section is
then extrapolated to the full inclusive phase space using theoretical models if structure
functions are considered, [86,91,248–250].
Within the approach of this thesis, the main objective for studying heavy quark pro-
duction in DIS is to provide a framework allowing for more precise measurements of as
and of the parton densities and for a better description of the structure functions F cc2 ,
F bb2 . The current world data for the nucleon structure functions F p,d2 (x,Q2) reached the
precision of a few per cent over a wide kinematic region. Both the measurements of the
heavy flavor contributions to the deep-inelastic structure functions, cf. [85, 86, 249], and
numerical studies, [89, 251, 252], based on the leading, [101, 102], and next-to-leading or-
der heavy flavor Wilson coefficients, [103], show that the scaling violations of the light
and the heavy contributions to (2.97) exhibit a different behavior over a wide range of
Q2. This is both due to the logarithmic contributions lnk(Q2/m2) and power corrections
∝ (m2/Q2)k, k ≥ 1. Moreover, in the region of smaller values of x the heavy flavor contri-
butions amount to 20–40%. Therefore, the precision measurement of the QCD parameter
ΛQCD, [92,93,96–99,102,232,233,244], and of the parton distribution functions in deeply
inelastic scattering requires the analysis at the level of the O(a3s) corrections to control
the theory-errors at the level of the experimental accuracy and below, [92,96–99].
The precise value of ΛQCD, a fundamental parameter of the Standard Model, is of
central importance for the quantitative understanding of all strongly interacting processes.
Moreover, the possible unification of the gauge forces, [253], depends crucially on its value.
In recent non–singlet analyzes, [93, 96, 98], errors for as(M2Z) of O(1.5 %) were obtained,
partially extending the analysis effectively to N3LO. In the flavor singlet case the so
far unknown 3–loop heavy flavor Wilson coefficients do yet prevent a consistent 3–loop
analysis, [94, 95, 100]. Due to the large statistics in the lower x region, one may hope to
eventually improve the accuracy of as(M2Z) beyond the above value.
11Likewise, for even higher scales the b–quark could be considered as the heavy quark with u, d, s, c
being effectively massless, cf. e.g. [245].
33
Of similar importance is the detailed knowledge of the PDFs for all hadron-induced
processes, notably for the interpretation of all scattering cross sections measured at the
TEVATRON and the LHC. For example, the process of Higgs-boson production at the
LHC, cf. e.g. [254], depends on the gluon density and its accuracy is widely determined
by this distribution.
In Section 3.1, we describe the general framework of electroproduction of heavy quarks
in DIS within the fixed–flavor–number–scheme (FFNS), treating only the light quarks
and the gluon as constituents of the nucleon. In the following Section, 3.2, we out-
line the method, which we use to extract all but the power suppressed contributions
∝ (m2/Q2)k, k ≥ 1 of the heavy flavor Wilson coefficients, [126]. The latter are equiva-
lent to the Wilson coefficients introduced in Section 2.3.3, including heavy quarks. Finally,
in Section 3.3 we comment on the possibility to define heavy quark parton densities within
a variable–flavor–number–scheme (VFNS), [129].
3.1 Electroproduction of Heavy Quarks
We study electroproduction of heavy quarks in unpolarized DIS via single photon ex-
change, cf. [101, 102, 255], at sufficiently large virtualities Q2, Q2 ≥ 5GeV 12. Here, one
can distinguish two possible production mechanisms for heavy quarks: extrinsic produc-
tion and intrinsic heavy quark excitation. In the latter case, one introduces a heavy quark
state in the nucleon wave function, i.e. the heavy quark is treated at the same level as
the light quarks in the factorization of the structure functions, cf. Eqs. (2.97)–(2.100).
The LO contribution is then given by the flavor excitation process shown in Figure 5,
γ∗ +Q(Q)→ Q(Q) . (3.1)
P
}
PF
γ∗
Q(Q)
l
l′
Figure 5: LO intrinsic heavy quark production.
Several experimental and theoretical studies suggest that the intrinsic contribution to
the heavy flavor cross section is of the order of 1% or smaller, [90, 91], and we will not
consider it any further.
12One may however, also consider photoproduction of heavy quarks in ep collisions where Q2 ≈ 0,
which is a widely hadronic process, cf. [256, 257], and especially important for the production of heavy
quark resonances, as e.g. the J/Ψ.
34
In extrinsic heavy flavor production, the heavy quarks are produced as final states
in virtual gauge boson scattering off massless partons. This description is also referred
to as the fixed flavor number scheme. At higher orders, one has to make the distinction
between whether one considers the complete inclusive structure functions or only those
heavy quark contributions, which can be determined in experiments by tagging the final
state heavy quarks. In the former case, virtual corrections containing heavy quark loops
have to be included into the theoretical calculation as well, cf. also Section 5.1.
We consider only twist-2 parton densities in the Bjorken limit. Therefore no transverse
momentum effects in the initial parton distributions will be allowed, since these contribu-
tions are related, in the kinematic sense, to higher twist operators. From the conditions
for the validity of the parton model, Eqs. (2.51, 2.56), it follows that in the region of not
too small nor too large values of the Bjorken variable x, the partonic description holds for
massless partons. Evidently, iff Q2(1−x)2/m2 ≫/ 1 no partonic description for a potential
heavy quark distribution can be obtained. The question under which circumstances one
may introduce a heavy flavor parton density will be further discussed in Section 3.3. In
a general kinematic region the parton densities in Eq. (2.97) are enforced to be massless
and the heavy quark mass effects are contained in the inclusive Wilson coefficients. These
are calculable perturbatively and denoted by
CS,PS,NSi,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
. (3.2)
The argument nf + 1 denotes the presence of nf light and one heavy flavor. τ is the
partonic scaling variable defined in Eq. (2.38) and we will present some of the following
equations in x–space rather than in Mellin space.
One may identify the massless flavor contributions in Eq. (3.2) and separate the Wilson
coefficients into a purely light part Ci,(2,L), cf. Eq. (2.97), and a heavy part
CS,PS,NSi,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
= CS,PS,NSi,(2,L)
(
τ, nf ,
Q2
µ2
)
+HS,PSi,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
+ LS,PS,NSi,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
.
(3.3)
Here, we denote the heavy flavor Wilson coefficients by Li,j and Hi,j, respectively, de-
pending on whether the photon couples to a light (L) or heavy (H) quark line. From
this it follows that the light flavor Wilson coefficients Ci,j depend on nf light flavors only,
whereas Hi,j and Li,j may contain light flavors in addition to the heavy quark, indicated
by the argument nf + 1. The perturbative series of the heavy flavor Wilson coefficients
read
HSg,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
=
∞∑
i=1
aisH
(i),S
g,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
, (3.4)
HPSq,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
=
∞∑
i=2
aisH
(i),PS
q,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
, (3.5)
LSg,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
=
∞∑
i=2
aisL
(i),S
g,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
, (3.6)
35
LSq,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
=
∞∑
i=2
aisL
(i),S
q,(2,L)
(
τ, nf + 1,
Q2
µ2 ,
m2
µ2
)
. (3.7)
Note that we have not yet specified a scheme for treating as, but one has to use the same
scheme when combining the above terms with the light flavor Wilson coefficients. At LO,
only the term Hg,(2,L) contributes via the photon–gluon fusion process shown in Figure 6,
γ∗ + g → Q+Q . (3.8)
The LO Wilson coefficients corresponding to this process are given by, [101,102,255] 13,
P
}
PF
γ∗
g
Q
Q
l
l′
Figure 6: LO extrinsic heavy quark production.
H(1)g,2
(
τ, m
2
Q2
)
= 8TF
{
v
[
−1
2
+ 4τ(1− τ) + 2m
2
Q2 τ(τ − 1)
]
+
[
−1
2
+ τ − τ 2 + 2m
2
Q2 τ(3τ − 1) + 4
m4
Q4 τ
2
]
ln
(
1− v
1 + v
)}
, (3.9)
H(1)g,L
(
τ, m
2
Q2
)
= 16TF
[
τ(1− τ)v + 2m
2
Q2 τ
2 ln
(
1− v
1 + v
)]
. (3.10)
The cms velocity v of the produced heavy quark pair is given by
v =
√
1− 4m
2τ
Q2(1− τ) . (3.11)
The LO heavy flavor contributions to the structure functions are then
FQQ(2,L)(x,Q2,m2) = e2Qas
∫ 1
ax
dz
z H
(1)
g,(2,L)
(x
z ,
m2
Q2
)
G(nf , z, Q2) , a = 1 + 4m2/Q2 , (3.12)
where the integration boundaries follow from the kinematics of the process. Here eQ
denotes the electric charge of the heavy quark.
13Eqs. (16), (17) in Ref. [130] contain misprints
36
At O(a2s), the terms HPSq,(2,L) and LSq,(2,L) contribute as well. They result from the
process
γ∗ + q(q)→ q(q) +X , (3.13)
where X = Q+Q in case of extrinsic heavy flavor production. The latter is of phenomeno-
logical relevance if the heavy quarks are detected in the final states, e.g. via the produced
Dc–mesons in case Q = c. For a complete inclusive analysis summing over all final states,
both light and heavy, one has to include radiative corrections containing virtual heavy
quark contributions as well. The term LSq,(2,L) can be split into a NS and a PS piece via
LSq,(2,L) = LNSq,(2,L) + LPSq,(2,L), (3.14)
where the PS–term emerges for the first time at O(a3s) and the NS–term at O(a2s), re-
spectively. Finally, LSg,(2,L) contributes for the first time at O(a3s) in case of heavy quarks
in the final state but there is a O(a2s) term involving radiative corrections, which will be
commented on in Section 5.1. The terms H(2)g,(2,L), H
(2),PS
q,(2,L) and L
(2),NS
q,(2,L) have been calcu-
lated in x–space in the complete kinematic range in semi-analytic form in Refs. [103] 14,
considering heavy quarks in the final states only.
The heavy quark contribution to the structure functions F(2,L)(x,Q2) for one heavy
quark of mass m and nf light flavors is then given by, cf. [129] and Eq. (2.97),
FQQ(2,L)(x, nf + 1, Q2,m2) =
nf∑
k=1
e2k
{
LNSq,(2,L)
(
x, nf + 1,
Q2
m2 ,
m2
µ2
)
⊗
[
fk(x, µ2, nf ) + fk(x, µ2, nf )
]
+
1
nf
LPSq,(2,L)
(
x, nf + 1,
Q2
m2 ,
m2
µ2
)
⊗ Σ(x, µ2, nf )
+
1
nf
LSg,(2,L)
(
x, nf + 1,
Q2
m2 ,
m2
µ2
)
⊗G(x, µ2, nf )
}
+ e2Q
[
HPSq,(2,L)
(
x, nf + 1,
Q2
m2 ,
m2
µ2
)
⊗ Σ(x, µ2, nf )
+HSg,(2,L)
(
x, nf + 1,
Q2
m2 ,
m2
µ2
)
⊗G(x, µ2, nf )
]
, (3.15)
where the integration boundaries of the Mellin–convolutions follow from phase space kine-
matics, cf. Eq. (3.12).
3.2 Asymptotic Heavy Quark Coefficient Functions
An important part of the kinematic region in case of heavy flavor production in DIS
is located at larger values of Q2, cf. e.g. [247, 258]. As has been shown in Ref. [126],
cf. also [129, 259], the heavy flavor Wilson coefficients Hi,j, Li,j factorize in the limit
Q2 ≫ m2 into massive operator matrix elements Aki and the massless Wilson coefficients
14A precise representation in Mellin space was given in [116].
37
Ci,j, if one heavy quark flavor and nf light flavors are considered. The massive OMEs
are process independent quantities and contain all the mass dependence except for the
power corrections ∝ (m2/Q2)k, k ≥ 1. The process dependence is given by the light
flavor Wilson coefficients only. This allows the analytic calculation of the NLO heavy
flavor Wilson coefficients, [126, 128]. Comparing these asymptotic expressions with the
exact LO and NLO results obtained in Refs. [101, 102] and [103], respectively, one finds
that this approximation becomes valid in case of FQQ2 for Q2/m2 >∼ 10. These scales are
sufficiently low and match with the region analyzed in deeply inelastic scattering for
precision measurements. In case of FQQL , this approximation is only valid for Q2/m2 >∼ 800,
[126]. For the latter case, the 3–loop corrections were calculated in Ref. [127]. This
difference is due to the emergence of terms ∝ (m2/Q2) ln(m2/Q2), which vanish only
slowly in the limit Q2/m2 →∞.
In order to derive the factorization formula, one considers the inclusive Wilson coef-
ficients CS,PS,NSi,j , which have been defined in Eq. (3.2). After applying the LCE to the
partonic tensor, or the forward Compton amplitude, corresponding to the respective Wil-
son coefficients, one arrives at the factorization relation, cf. Eq. (2.93),
CS,PS,NS,asympj,(2,L)
(
N,nf + 1,
Q2
µ2 ,
m2
µ2
)
=
∑
i
AS,PS,NSij
(
N,nf + 1,
m2
µ2
)
CS,PS,NSi,(2,L)
(
N,nf + 1,
Q2
µ2
)
+O
(m2
Q2
)
. (3.16)
Here µ refers to the factorization scale between the heavy and light contributions in Cj,i and
’asymp’ denotes the limit Q2 ≫ m2. The Ci,j are precisely the light Wilson coefficients,
cf. Eqs. (2.97)–(2.99), taken at nf + 1 flavors. This can be inferred from the fact that in
the LCE, Eq. (2.74), the Wilson coefficients describe the singularities for very large values
of Q2, which can not depend on the presence of a quark mass. The mass dependence is
given by the OMEs Aij, cf. Eqs. (2.80,2.89), between partonic states. Eq. (3.16) accounts
for all mass effects but corrections which are power suppressed, (m2/Q2)k, k ≥ 1. This
factorization is only valid if the heavy quark coefficient functions are defined in such a
way that all radiative corrections containing heavy quark loops are included. Otherwise,
(3.16), would not show the correct asymptotic Q2–behavior, [129].
An equivalent way of describing Eq. (3.16) is obtained by considering the calcula-
tion of the massless Wilson coefficients. Here, the initial state collinear singularities are
given by evaluating the massless OMEs between off–shell partons, leading to transition
functions Γij. The Γij are given in terms of the anomalous dimensions of the twist–2
operators and transfer the initial state singularities to the bare parton–densities due to
mass factorization, cf. e.g. [126, 129]. In the case at hand, something similar happens:
The initial state collinear singularities are transferred to the parton densities except for
those which are regulated by the quark mass and described by the OMEs. Instead of
absorbing these terms into the parton densities as well, they are used to reconstruct the
asymptotic behavior of the heavy flavor Wilson coefficients. Here,
AS,NSij
(
N,nf + 1,
m2
µ2
)
= 〈j|OS,NSi |j〉 = δij +
∞∑
i=1
aisA
(i),S,NS
ij (3.17)
are the operator matrix elements of the local twist–2 operators being defined in Eqs.
(2.86)–(2.88) between on–shell partonic states |j〉, j = q, g. As usual, the S contribution
38
can be split into a NS and PS part via
ASqq = ANSqq + APSqq . (3.18)
Due to the on–shell condition, all contributions but the O(a0s) terms vanish 15 if no heavy
quark is present in the virtual loops. This is due to the fact that integrals without scale
vanish in dimensional regularization, cf. Section 4.1. Hence only those terms with a mass
remain and these are referred to as massive OMEs. The calculation of these massive
OMEs is the main objective of this thesis. In case of the gluon operator, (2.88), the
contributing terms are denoted by Agq,Q and Agg,Q, where the perturbative series of the
former starts at O(a2s) and the one of the latter at O(a0s) 16. For the quark operator,
one distinguishes whether the operator couples to a heavy or light quark. In the NS–
case, the operator by definition couples to the light quark. Thus there is only one term,
ANSqq,Q, which contributes at O(a0s). In the S and PS–case, two OMEs can be distinguished,
{APSqq,Q, ASqg,Q} and {APSQq, ASQg}, where in the former case the operator couples to a light
quark and in the latter case to a heavy quark. The terms Aqi,Q emerge for the first time
at O(a3s), APSQq at O(a2s) and ASQg at O(as).
In this work we refer only to the even moments, cf. Section 2.3. In the non–singlet
case we will obtain, however, besides the NS+ contributions for the even moments also
the NS− terms, which correspond to the odd moments.
Eq. (3.16) can now be split into its parts by considering the different nf–terms. We
adopt the following notation for a function f(nf )
f˜(nf ) ≡
f(nf )
nf
. (3.19)
This is necessary in order to separate the different types of contributions in Eq. (3.15),
weighted by the electric charges of the light and heavy flavors, respectively. Since we
concentrate on only the heavy flavor part, we define as well for later use
fˆ(nf ) ≡ f(nf + 1)− f(nf ) , (3.20)
with ˆ˜f(nf ) ≡ ̂[f˜(nf )]. The following Eqs. (3.21)–(3.25) are the same as Eqs. (2.31)–(2.35)
in Ref. [129]. We present these terms here again, however, since Ref. [129] contains a
few inconsistencies regarding the f˜–description. Contrary to the latter reference, the
argument corresponding to the number of flavors stands for all flavors, light or heavy.
The separation for the NS–term is given by
CNSq,(2,L)
(
N,nf ,
Q2
µ2
)
+ LNSq,(2,L)
(
N,nf + 1,
Q2
µ2 ,
m2
µ2
)
=
ANSqq,Q
(
N,nf + 1,
m2
µ2
)
CNSq,(2,L)
(
N,nf + 1,
Q2
µ2
)
. (3.21)
Here and in the following, we omit the index ”asymp” to denote the asymptotic heavy
flavor Wilson coefficients, since no confusion is to be expected. For the remaining terms,
15In Ref. [111] use was made of this fact to calculate the massless Wilson coefficients without having
to calculate the massless OMEs.
16The O(a0s) term of Agg does not contain a heavy quark, but still remains in Eq. (3.16) because no
loops have to be calculated.
39
we suppress for brevity the arguments N , Q2/µ2 and m2/µ2, all of which can be inferred
from Eqs. (3.3, 3.16). Additionally, we will suppress from now on the index S and label
only the NS and PS terms explicitly. The contributions to Li,j read
CPSq,(2,L)(nf ) + LPSq,(2,L)(nf + 1) =
[
ANSqq,Q(nf + 1) + APSqq,Q(nf + 1) +APSQq(nf + 1)
]
×nf C˜PSq,(2,L)(nf + 1) + APSqq,Q(nf + 1)CNSq,(2,L)(nf + 1)
+Agq,Q(nf + 1)nf C˜g,(2,L)(nf + 1) ,
(3.22)
Cg,(2,L)(nf ) + Lg,(2,L)(nf + 1) = Agg,Q(nf + 1)nf C˜g,(2,L)(nf + 1)
+Aqg,Q(nf + 1)CNSq,(2,L)(nf + 1)
+
[
Aqg,Q(nf + 1) + AQg(nf + 1)
]
nf C˜PSq,(2,L)(nf + 1) .
(3.23)
The terms Hi,j are given by
HPSq,(2,L)(nf + 1) = APSQq(nf + 1)
[
CNSq,(2,L)(nf + 1) + C˜PSq,(2,L)(nf + 1)
]
+
[
ANSqq,Q(nf + 1) + APSqq,Q(nf + 1)
]
C˜PSq,(2,L)(nf + 1)
+Agq,Q(nf + 1)C˜g,(2,L)(nf + 1) , (3.24)
Hg,(2,L)(nf + 1) = Agg,Q(nf + 1)C˜g,(2,L)(nf + 1) + Aqg,Q(nf + 1)C˜PSq,(2,L)(nf + 1)
+AQg(nf + 1)
[
CNSq,(2,L)(nf + 1) + C˜PSq,(2,L)(nf + 1)
]
. (3.25)
Expanding the above equations up to O(a3s), we obtain, using Eqs. (3.19, 3.20), the heavy
flavor Wilson coefficients in the asymptotic limit :
LNSq,(2,L)(nf + 1) = a2s
[
A(2),NSqq,Q (nf + 1) δ2 + Cˆ
(2),NS
q,(2,L)(nf )
]
+ a3s
[
A(3),NSqq,Q (nf + 1) δ2 + A
(2),NS
qq,Q (nf + 1)C
(1),NS
q,(2,L)(nf + 1)
+Cˆ(3),NSq,(2,L)(nf )
]
, (3.26)
LPSq,(2,L)(nf + 1) = a3s
[
A(3),PSqq,Q (nf + 1) δ2 + A
(2)
gq,Q(nf ) nf C˜
(1)
g,(2,L)(nf + 1)
+nf ˆ˜C
(3),PS
q,(2,L)(nf )
]
, (3.27)
LSg,(2,L)(nf + 1) = a2sA
(1)
gg,Q(nf + 1)nf C˜
(1)
g,(2,L)(nf + 1)
+ a3s
[
A(3)qg,Q(nf + 1) δ2 + A
(1)
gg,Q(nf + 1) nf C˜
(2)
g,(2,L)(nf + 1)
+A(2)gg,Q(nf + 1) nf C˜
(1)
g,(2,L)(nf + 1)
+ A(1)Qg(nf + 1) nf C˜
(2),PS
q,(2,L)(nf + 1) + nf
ˆ˜C
(3)
g,(2,L)(nf )
]
, (3.28)
HPSq,(2,L)(nf + 1) = a2s
[
A(2),PSQq (nf + 1) δ2 + C˜
(2),PS
q,(2,L)(nf + 1)
]
+ a3s
[
A(3),PSQq (nf + 1) δ2 + C˜
(3),PS
q,(2,L)(nf + 1)
40
+A(2)gq,Q(nf + 1) C˜
(1)
g,(2,L)(nf + 1) + A
(2),PS
Qq (nf + 1) C
(1),NS
q,(2,L)(nf + 1)
]
, (3.29)
HSg,(2,L)(nf + 1) = as
[
A(1)Qg(nf + 1) δ2 + C˜
(1)
g,(2,L)(nf + 1)
]
+ a2s
[
A(2)Qg(nf + 1) δ2 + A
(1)
Qg(nf + 1) C
(1),NS
q,(2,L)(nf + 1)
+ A(1)gg,Q(nf + 1) C˜
(1)
g,(2,L)(nf + 1) + C˜
(2)
g,(2,L)(nf + 1)
]
+ a3s
[
A(3)Qg(nf + 1) δ2 + A
(2)
Qg(nf + 1) C
(1),NS
q,(2,L)(nf + 1)
+ A(2)gg,Q(nf + 1) C˜
(1)
g,(2,L)(nf + 1)
+ A(1)Qg(nf + 1)
{
C(2),NSq,(2,L)(nf + 1) + C˜
(2),PS
q,(2,L)(nf + 1)
}
+ A(1)gg,Q(nf + 1) C˜
(2)
g,(2,L)(nf + 1) + C˜
(3)
g,(2,L)(nf + 1)
]
. (3.30)
Note that δ2 has been defined in Eq. (2.101). The above equations include radiative
corrections due to heavy quark loops to the Wilson coefficients. Therefore, in order to
compare e.g. with the calculation in Refs. [103], these terms still have to be subtracted.
Since the light flavor Wilson coefficients were calculated in the MS–scheme, the same
scheme has to be used for the massive OMEs. It should also be thoroughly used for
renormalization to derive consistent results in QCD analyzes of deep-inelastic scattering
data and to be able to compare to other analyzes directly. This means that one has to take
special attendance of which scheme for the definition of as was used. In Section 4.4 we will
describe a scheme for as, to which one is naturally led in the course of renormalization.
We refer to this scheme as MOM–scheme and present the transformation formula to the
MS as well. How this affects the asymptotic heavy flavor Wilson coefficients is described
in Section 5.1, where we compare Eqs. (3.26)–(3.30) to those presented in Ref. [126].
3.3 Heavy Quark Parton Densities
The FFNS forms a general starting point to describe and to calculate the heavy fla-
vor contributions to the DIS structure functions. Approaching higher values of Q2, one
may think of the heavy quark becoming effectively light and thus acquiring an own par-
ton density. Different variable flavor scheme treatments were considered in the past, cf.
e.g. [260]. Here we follow [129] to obtain a description in complete accordance with the
renormalization group in the MS–scheme. In the kinematic region in which the factor-
ization relation (3.16) holds, one may redefine the results obtained in the FFNS, which
allows for a partonic description at the level of (nf + 1) flavors.
In the strict sense, only massless particles can be interpreted as partons in hard scatter-
ing processes since the lifetime of these quantum-fluctuations off the hadronic background
τlife ∝ 1/(k2⊥ +m2Q) has to be large against the interaction time τint ∝ 1/Q2 in the infinite
momentum frame, [204], cf. also Section 2.2.1. In the massive case, τlife is necessarily
finite and there exists a larger scale Q20 below which any partonic description fails. From
this it follows, that the heavy quark effects are genuinely described by the process depen-
dent Wilson coefficients. Since parton-densities are process independent quantities, only
process independent pieces out of the Wilson coefficients can be used to define them for
heavy quarks at all. Clearly this is impossible in the region close to threshold but requires
Q2/m2Q = r ≫ 1, with r >∼ 10 in case of F2(x,Q2). For FL(x,Q2) the corresponding ratio
41
even turns out to be r >∼ 800, [126, 127, 252]. Heavy flavor parton distributions can thus
be constructed only for scales µ2 ≫ m2Q. This is done under the further assumption that
for the other heavy flavors the masses mQi form a hierarchy m2Q1 ≪ m2Q2 ≪ etc. Their
use in observables is restricted to a region, in which the power corrections can be safely
neglected. This range may strongly depend on the observable considered as the examples
of F2 and FL show. Also in case of the structure functions associated to transverse virtual
gauge boson polarizations, like F2(x,Q2), the factorization (3.16) only occurs far above
threshold, Q2thr ∼ 4m2Qx/(1− x), and at even larger scales for FL(x,Q2).
In order to maintain the process independence of the parton distributions, we define
them for (nf + 1) flavors from the light flavor parton distribution functions for nf flavors
together with the massive operator matrix elements. The following set of parton densities
is obtained in Mellin–space, [129] :
fk(nf + 1, N, µ2,m2) + fk(nf + 1, N, µ2,m2) =
ANSqq,Q
(
N,nf + 1,
µ2
m2
)
·
[
fk(nf , N, µ2) + fk(nf , N, µ2)
]
+
1
nf
APSqq,Q
(
N,nf + 1,
µ2
m2
)
· Σ(nf , N, µ2)
+
1
nf
Aqg,Q
(
N,nf + 1,
µ2
m2
)
·G(nf , N, µ2), (3.31)
fQ(nf + 1, N, µ2,m2) + fQ(nf + 1, N, µ2,m2) =
APSQq
(
N,nf + 1,
µ2
m2
)
· Σ(nf , N, µ2)
+AQg
(
N,nf + 1,
µ2
m2
)
·G(nf , N, µ2) . (3.32)
The flavor singlet, non–singlet and gluon densities for (nf + 1) flavors are given by
Σ(nf + 1, N, µ2,m2) =
[
ANSqq,Q
(
N,nf + 1,
µ2
m2
)
+ APSqq,Q
(
N,nf + 1,
µ2
m2
)
+APSQq
(
N,nf + 1,
µ2
m2
)]
· Σ(nf , N, µ2)
+
[
Aqg,Q
(
N,nf + 1,
µ2
m2
)
+ AQg
(
N,nf + 1,
µ2
m2
)]
·G(nf , N, µ2) ,
(3.33)
∆k(nf + 1, N, µ2,m2) = fk(nf + 1, N, µ2,m2) + fk(nf + 1, N, µ2,m2)
− 1nf + 1
Σ(nf + 1, N, µ2,m2) , (3.34)
G(nf + 1, N, µ2,m2) = Agq,Q
(
N,nf + 1,
µ2
m2
)
· Σ(nf , N, µ2)
+Agg,Q
(
N,nf + 1,
µ2
m2
)
·G(nf , N, µ2) . (3.35)
Note, that the new parton densities depend on the renormalized heavy quark mass m2 =
m2(a2s(µ2)). As will be outlined in Sections 4, 5, the corresponding relations for the
42
operator matrix elements depend on the mass–renormalization scheme. This has to be
taken into account in QCD-analyzes, in particular, m2 cannot be chosen constant. The
quarkonic and gluonic operators obtained in the light–cone expansion can be normalized
arbitrarily. It is, however, convenient to chose the relative factor such, that the non-
perturbative nucleon-state expectation values, Σ(nf , N, µ2) and G(nf , N, µ2), obey
Σ(nf , N = 2, µ2) +G(nf , N = 2, µ2) = 1 (3.36)
due to 4-momentum conservation. As a consequence, the OMEs fulfill the relations, [129],
ANSqq,Q(N = 2) +APSqq,Q(N = 2) + APSQq(N = 2) + Agq,Q(N = 2) = 1 , (3.37)
Aqg,Q(N = 2) +AQg(N = 2) + Agg,Q(N = 2) = 1 . (3.38)
The above scenario can be easily followed up to 2-loop order. Also here diagrams con-
tribute which carry two different heavy quark flavors. At this level, the additional heavy
degree of freedom may be absorbed into the coupling constant and thus decoupled tem-
porarily. Beginning with 3-loop order the situation becomes more involved since there
are graphs in which two different heavy quark flavors occur in nested topologies, i.e., the
corresponding diagrams depend on the ratio ρ = m2c/m2b yielding power corrections in ρ.
There is no strong hierarchy between these two masses. The above picture, leading to
heavy flavor parton distributions whenever Q2 ≫ m2 will not hold anymore, since, in case
of the two-flavor graphs, one cannot decide immediately whether they belong to the c– or
the b–quark distribution. Hence, the partonic description can only be maintained within
a certain approximation by assuming ρ≪ 1.
Conversely, one may extend the kinematic regime for deep-inelastic scattering to define
the distribution functions (3.31)–(3.35) upon knowing the power corrections which occur
in the heavy flavor Wilson coefficients Hi,j = Hi,j, Li,j. This is the case for 2-loop order.
We separate
Hi,j
(
x, Q
2
m2 ,
m2
µ2
)
= Hasympi,j
(
x, Q
2
m2 ,
m2
µ2
)
+ Hpoweri,j
(
x, Q
2
m2 ,
m2
µ2
)
, (3.39)
where Hasympi,j (x,Q2/m2,m2/µ2) denotes the part of the Wilson coefficient given in
Eq. (3.16). If one accounts for Hpoweri,j (x,Q2/m2,m2/µ2) in the fixed flavor number scheme,
Eqs. (3.31)–(3.35) are still valid, but they do not necessarily yield the dominant contri-
butions in the region closer to threshold. There, the kinematics of heavy quarks is by far
not collinear, which is the main reason that a partonic description has to fail. Moreover,
relation Eq. (2.51) may be violated. In any case, it is not possible to use the partonic
description (3.31)–(3.35) alone for other hard processes in a kinematic domain with sig-
nificant power corrections.
For processes in the high p⊥ region at the LHC, in which condition (2.51) is fulfilled and
the characteristic scale µ2 obeys µ2 ≫ m2, one may use heavy flavor parton distributions
by proceeding as follows. In the region Q2 >∼ 10 m2 the heavy flavor contributions to the
F2(x,Q2)–world data are very well described by the asymptotic representation in the
FFNS. For large scales one can then form a variable flavor representation including one
heavy flavor distribution, [129]. This process can be iterated towards the next heavier
flavor, provided the universal representation holds and all power corrections can be safely
neglected. One has to take special care of the fact, that the matching scale in the coupling
constant, at which the transition nf → nf + 1 is to be performed, often differs rather
significantly from m, cf. [261].
43
4 Renormalization of Composite Operator Matrix
Elements
Before renormalizing the massive OMEs, they have to be calculated applying a suitable
regularization scheme, for which we apply dimensional regularization in D = 4 + ε di-
mensions, see Section 4.1. The unrenormalized massive OMEs are then denoted by a
double–hat and are expanded into a perturbative series in the bare coupling constant
aˆs 17 via
ˆˆAij
(mˆ2
µ2 , ε, N
)
= δij +
∞∑
l=1
aˆls
ˆˆA
(l)
ij
(mˆ2
µ2 , ε, N
)
= δij +
∞∑
l=1
aˆls
(mˆ2
µ2
)lε/2 ˆˆA
(l)
ij
(
mˆ2 = µ2, ε, N
)
. (4.1)
The OMEs in Eq. (4.1) depend on ε, the Mellin–Parameter N , the bare mass mˆ and the
renormalization scale µ = µR. Also the factorization scale µF will be identified with µ in
the following. Note that in the last line of (4.1), the dependence on the ratio of the mass
and the renormalization scale was made explicit for each order in aˆs. The possible values
of the indices ij have been described in Section 3.2, below Eq. (3.17).
The factorization between the massive OMEs and the massless Wilson coefficients
(3.16) requires the external legs of the operator matrix elements to be on–shell,
p2 = 0 , (4.2)
where p denotes the external momentum. Unlike in the massless case, where the scale
of the OMEs is set by an off–shell momentum −p2 < 0, in our framework the internal
heavy quark mass yields the scale. In the former case, one observes a mixing of the
physical OMEs with non–gauge invariant (NGI) operators, cf. [123,262], and contributions
originating in the violation of the equations of motion (EOM). Terms of this kind do not
contribute in the present case, as will be discussed in Section 4.2.
Renormalizing the OMEs then consists of four steps. First, mass and charge renormal-
ization have to performed. The former is done in the on–mass–shell–scheme and described
in Section 4.3. For the latter, we present the final result in the MS–scheme, but in an
intermediate step, we adopt an on–shell subtraction scheme (MOM–scheme) for the gluon
propagator, cf. Section 4.4. This is necessary to maintain condition (4.2), i.e., to keep the
external massless partons on–shell. Note, that there are other, differing MOM–schemes
used in the literature, cf. e.g. [263].
After mass and coupling constant renormalization, we denote the OMEs with a single
hat, Aˆij. The remaining singularities are then connected to the composite operators and
the particle kinematics of the corresponding Feynman–diagrams. One can distinguish
between ultraviolet (UV) and collinear (C) divergences. In Section 4.5, we describe how
the former are renormalized via the operator Z–factors. The UV–finite OMEs are de-
noted by a bar, A¯ij. Finally, the C–divergences are removed via mass factorization, cf.
Section 4.6. The renormalized OMEs are then denoted by Aij. Section 4.7 contains the
general structure of the massive OMEs up to O(a3s) in terms of renormalization constants
and lower order contributions.
17We would like to remind the reader of the definition of the hat–symbol for a function f, Eq. (3.20),
which is not to be confused with the hat–symbol denoting unrenormalized quantities
44
4.1 Regularization Scheme
When evaluating momentum integrals of Feynman diagrams in D = 4 dimensions, one
encounters singularities, which have to be regularized. A convenient method is to apply
D-dimensional regularization, [264, 265]. The dimensionality of space–time is analyti-
cally continued to values D 6= 4, for which the corresponding integrals converge. After
performing a Wick rotation, integrals in Euclidean space of the form
∫ dDk
(2π)D
(k2)r
(k2 +R2)m =
1
(4π)D/2
Γ(r +D/2)Γ(m− r −D/2)
Γ(D/2)Γ(m) (R
2)r+D/2−m (4.3)
are obtained. Note that within dimensional regularization, this integral vanishes if R = 0,
i.e. if it does not contain a scale, [226]. The properties of the Γ–function in the complex
plane are well known, see Appendix C. Therefore one can analytically continue the right-
hand side of Eq. (4.3) from integer values of D to arbitrary complex values. In order
to recover the physical space-time dimension, we set D = 4 + ε. The singularities can
now be isolated by expanding the Γ–functions into Laurent-series around ε = 0. Note
that this method regularizes both UV- and C- singularities and one could in principle
distinguish their origins by a label, εUV , εC , but we treat all singularities by a common
parameter ε in the following. Additionally, all other quantities have to be considered in D
dimensions. This applies for the metric tensor gµν and the Clifford-Algebra of γ–matrices,
see Appendix A. Also the bare coupling constant gˆs, which is dimensionless in D = 4,
has to be continued to D dimensions. Due to this it acquires the dimension of mass,
gˆs,D = µ−ε/2gˆs , (4.4)
which is described by a scale µ corresponding to the renormalization scale in Eq. (4.1).
From now on, Eq. (4.4) is understood to have been applied and we set
gˆ2s
(4π)2 = aˆs . (4.5)
Dimensional regularization has the advantage, unlike the Pauli–Villars regularization,
[266], that it obeys all physical requirements such as Lorentz-invariance, gauge invariance
and unitarity, [264,267]. Hence it is suitable to be applied in perturbative calculations in
quantum field theory including Yang–Mills fields.
Using dimensional regularization, the poles of the unrenormalized results appear as
terms 1/εi, where in the calculations in this thesis i can run from 1 to the number of
loops. In order to remove these singularities, one has to perform renormalization and
mass factorization. To do this, a suitable scheme has to be chosen. The most commonly
used schemes in perturbation theory are the MS-scheme, [268], and the MS-scheme, [106],
to which we will refer in the following.
In the MS-scheme only the pole terms in ε are subtracted. More generally, the MS-
scheme makes use of the observation that 1/ε–poles always appear in combination with
the spherical factor
Sε ≡ exp
[ε
2
(γE − ln(4π))
]
, (4.6)
which may be bracketed out for each loop order. Here γE denotes the Euler-Mascheroni
constant
γE ≡ limN→∞
( N∑
k=1
1
k − ln(N)
)
≈ 0.577215664901 . . . . (4.7)
45
By subtracting the poles in the form Sε/ε in the MS-scheme, no terms containing
lnk(4π), γkE will appear in the renormalized result, simplifying the expression. This is
due to the fact that for a k–loop calculation, one will always obtain the overall term
Γ(1− k ε2)
(4π) kε2
= Skε exp
( ∞∑
i=2
ζi
i
(kε
2
)i)
, (4.8)
with ζi being Riemann’s ζ–values, cf. Appendix C. In the following, we will always
assume that the MS-scheme is applied and set Sε ≡ 1.
4.2 Projectors
We consider the expectation values of the local operators (2.86)–(2.88) between partonic
states j
Gij,Q = 〈j | Oi | j〉Q . (4.9)
Here, i, j = q, g and the subscript Q denotes the presence of one heavy quark. In case of
massless QCD, one has to take the external parton j of momentum p off–shell, p2 < 0,
which implies that the OMEs derived from Eq. (4.9) are not gauge invariant. As has been
outlined in Ref. [269], they acquire unphysical parts which are due to the breakdown
of the equations of motion (EOM) and the mixing with additional non–gauge–invariant
(NGI) operators. The EOM terms may be dealt with by applying a suitable projection
operator to eliminate them, [269]. The NGI terms are more difficult to deal with, since
they affect the renormalization constants and one has to consider additional ghost– and
alien– OMEs, see [123,262,269,270] for details.
In the case of massive OMEs, these difficulties do not occur. The external particles
are massless and taken to be on–shell. Hence the equations of motion are not violated.
Additionally, the OMEs remain gauge invariant quantities, since the external states are
physical and therefore no mixing with NGI–operators occurs, [123,226,269,270].
The computation of the Green’s functions will reveal trace terms which do not contribute
since the local operators are traceless and symmetric under the Lorentz group. It is
convenient to project these terms out from the beginning by contracting with an external
source term
JN ≡ ∆µ1 ...∆µN . (4.10)
Here ∆µ is a light-like vector, ∆2 = 0. In this way, the Feynman–rules for composite
operators can be derived, cf. Appendix B. In addition, one has to amputate the external
field. Note that we nonetheless choose to renormalize the mass and the coupling multi-
plicative and include self–energy insertions containing massive lines on external legs into
our calculation. The Green’s functions in momentum space corresponding to the OMEs
with external gluons are then given by
ǫµ(p)GabQ,µνǫν(p) = ǫµ(p)JN〈Aaµ(p) | OQ;µ1...µN | Abν(p)〉ǫν(p) , (4.11)
ǫµ(p)Gabq,Q,µνǫν(p) = ǫµ(p)JN〈Aaµ(p) | Oq;µ1...µN | Abν(p)〉Qǫν(p) , (4.12)
ǫµ(p)Gabg,Q,µνǫν(p) = ǫµ(p)JN〈Aaµ(p) | Og;µ1...µN | Abν(p)〉Qǫν(p) . (4.13)
In Eqs. (4.11-4.13), Aaµ denote the external gluon fields with color index a, Lorentz index
µ and momentum p. The polarization vector of the external gluon is given by ǫµ(p). Note
46
that in Eq. (4.11), the operator couples to the heavy quark. In Eqs. (4.12, 4.13) it couples
to a light quark or gluon, respectively, with the heavy quark still being present in virtual
loops.
In the flavor non–singlet case, there is only one term which reads
u(p, s)Gij,NSq,Q λru(p, s) = JN〈Ψi(p) | ONSq,r;µ1...µN | Ψ
j(p)〉Q , (4.14)
with u(p, s), u(p, s) being the bi–spinors of the external massless quark and anti–quark,
respectively. The remaining Green’s functions with an outer quark are given by
u(p, s)GijQu(p, s) = JN〈Ψi(p) | OQ,µ1...µN | Ψj(p)〉 , (4.15)
u(p, s)Gijq,Qu(p, s) = JN〈Ψi(p) | Oq,µ1...µN | Ψj(p)〉Q , (4.16)
u(p, s)Gijg,Qu(p, s) = JN〈Ψi(p) | Og,µ1...µN | Ψj(p)〉Q . (4.17)
Note that in the quarkonic case the fields Ψ, Ψ with color indices i, j stand for the external
light quarks only. Further, we remind that the S– contributions are split up according to
Eq. (3.18), which is of relevance for Eq. (4.16).
The above tensors have the general form, cf. [126,269],
GˆabQ,µν =
ˆˆAQg
(mˆ2
µ2 , ε, N
)
δab(∆ · p)N
[
− gµν +
pµ∆ν +∆µpν
∆ · p
]
, (4.18)
Gˆabl,Q,µν =
ˆˆAlg,Q
(mˆ2
µ2 , ε, N
)
δab(∆ · p)N
[
− gµν +
pµ∆ν +∆µpν
∆ · p
]
, l = g, q , (4.19)
GˆijQ =
ˆˆA
PS
Qq
(mˆ2
µ2 , ε, N
)
δij(∆ · p)N−1/∆ , (4.20)
Gˆij,rl,Q =
ˆˆA
r
lq,Q
(mˆ2
µ2 , ε, N
)
δij(∆ · p)N−1/∆ , l = g, q , r = S, NS, PS . (4.21)
Here, we have denoted the Green’s function with a hat to signify that the above equations
are written on the unrenormalized level. In order to simplify the evaluation, it is useful to
define projection operators which, applied to the Green’s function, yield the corresponding
OME. For outer gluons, one defines
P (1)g Gˆabl,(Q),µν ≡ −
δab
N2c − 1
gµν
D − 2(∆ · p)
−NGˆabl,(Q),µν , (4.22)
P (2)g Gˆabl,(Q),µν ≡
δab
N2c − 1
1
D − 2(∆ · p)
−N
(
−gµν + p
µ∆ν + pν∆µ
∆ · p
)
Gˆabl,(Q),µν . (4.23)
The difference between the gluonic projectors, Eq. (4.22) and Eq. (4.23), can be traced
back to the fact that in the former case, the summation over indices µ, ν includes un-
physical transverse gluon states. These have to be compensated by adding diagrams with
external ghost lines, which is not the case when using the physical projector in Eq. (4.23).
In the case of external quarks there is only one projector which reads
PqGˆijl,(Q) ≡
δij
Nc
(∆ · p)−N 1
4
Tr[ /p Gˆijl,(Q)] . (4.24)
47
In Eqs. (4.22)–(4.24), Nc denotes the number of colors, cf. Appendix A. The unrenor-
malized OMEs are then obtained by
ˆˆAlg
(mˆ2
µ2 , ε, N
)
= P (1,2)g Gˆabl,(Q),µν , (4.25)
ˆˆAlq
(mˆ2
µ2 , ε, N
)
= PqGˆijl,(Q) . (4.26)
The advantage of these projection operators is that one does not have to resort to compli-
cated tensorial reduction. In perturbation theory, the expressions in Eqs. (4.25, 4.26) can
then be evaluated order by order in the coupling constant by applying the Feynman-rules
given in Appendix B.
4.3 Renormalization of the Mass
In a first step, we perform mass renormalization. There are two traditional schemes for
mass renormalization: the on–shell–scheme and the MS–scheme. In the following, we will
apply the on–shell–scheme, defining the renormalized mass m as the pole of the quark
propagator. The differences to the MS–scheme will be discussed in Section 5. The bare
mass in Eq. (4.1) is replaced by the renormalized on–shell mass m via
mˆ = Zmm = m
[
1 + aˆs
(m2
µ2
)ε/2
δm1 + aˆ2s
(m2
µ2
)ε
δm2
]
+O(aˆ3s) . (4.27)
The constants in the above equation are given by 18
δm1 = CF
[6
ε − 4 +
(
4 +
3
4
ζ2
)
ε
]
(4.28)
≡ δm
(−1)
1
ε + δm
(0)
1 + δm
(1)
1 ε , (4.29)
δm2 = CF
{
1
ε2
(
18CF − 22CA + 8TF (nf +Nh)
)
+
1
ε
(
−45
2
CF +
91
2
CA
−14TF (nf +Nh)
)
+ CF
(199
8
− 51
2
ζ2 + 48 ln(2)ζ2 − 12ζ3
)
+CA
(
−605
8
+
5
2
ζ2 − 24 ln(2)ζ2 + 6ζ3
)
+TF
[
nf
(45
2
+ 10ζ2
)
+Nh
(69
2
− 14ζ2
)]}
(4.30)
≡ δm
(−2)
2
ε2 +
δm(−1)2
ε + δm
(0)
2 . (4.31)
Eq. (4.28) is easily obtained. In Eq. (4.30), nf denotes the number of light flavors and
Nh the number of heavy flavors, which we will set equal to Nh = 1 from now on. The
pole contributions were given in Refs. [271, 272], and the constant term was derived in
Refs. [273], cf. also [274]. In Eqs. (4.29, 4.31), we have defined the expansion coefficients
18Note that there is a misprint in the double–pole term of Eq. (28) in Ref. [137].
48
in ε of the corresponding quantities. After mass renormalization, the OMEs read up to
O(aˆ3s)
ˆˆAij
(m2
µ2 , ε, N
)
= δij + aˆs ˆˆA
(1)
ij
(m2
µ2 , ε, N
)
+aˆ2s
[
ˆˆA
(2)
ij
(m2
µ2 , ε, N
)
+ δm1
(m2
µ2
)ε/2md
dm
ˆˆA
(1)
ij
(m2
µ2 , ε, N
)]
+aˆ3s
[
ˆˆA
(3)
ij
(m2
µ2 , ε, N
)
+ δm1
(m2
µ2
)ε/2md
dm
ˆˆA
(2)
ij
(m2
µ2 , ε, N
)
+δm2
(m2
µ2
)εmd
dm
ˆˆA
(1)
ij
(m2
µ2 , ε, N
)
+
δm21
2
(m2
µ2
)εm2d2
dm2
ˆˆA
(1)
ij
(m2
µ2 , ε, N
)]
.
(4.32)
4.4 Renormalization of the Coupling
Next, we consider charge renormalization. At this point it becomes important to define
in which scheme the strong coupling constant is renormalized, cf. Section 3.2. We briefly
summarize the main steps in the massless case for nf flavors in the MS–scheme. The bare
coupling constant aˆs is expressed by the renormalized coupling aMSs via
aˆs = ZMSg
2
(ε, nf )aMSs (µ2)
= aMSs (µ2)
[
1 + δaMSs,1 (nf )aMSs (µ2) + δaMSs,2 (nf )aMSs
2
(µ2)
]
+O(aMSs
3
) . (4.33)
The coefficients in Eq. (4.33) are, [39–41,275] and [276],
δaMSs,1 (nf ) =
2
εβ0(nf ) , (4.34)
δaMSs,2 (nf ) =
4
ε2β
2
0(nf ) +
1
εβ1(nf ) , (4.35)
with
β0(nf ) =
11
3
CA −
4
3
TFnf , (4.36)
β1(nf ) =
34
3
C2A − 4
(
5
3
CA + CF
)
TFnf . (4.37)
From the above equations, one can determine the β–function, Eq. (2.103), which describes
the running of the strong coupling constant and leads to asymptotic freedom in case of
QCD, [39,40]. It can be calculated using the fact that the bare strong coupling constant
does not depend on the renormalization scale µ. Using Eq. (4.4), one obtains
0 =
daˆs,D
d lnµ2 =
d
d lnµ2 aˆsµ
−ε =
d
d lnµ2as(µ
2)Z2g (ε, nf , µ2)µ−ε , (4.38)
=⇒ β = ε
2
as(µ2)− 2as(µ2)
d
d lnµ2 lnZg(ε, nf , µ
2) . (4.39)
49
Note that in Eq. (4.39) we have not specified a scheme yet and kept a possible µ–
dependence for Zg, which is not present in case of the MS–scheme. From (4.39), one
can calculate the expansion coefficients of the β–function. Combining it with the re-
sult for ZMSg in Eqs. (4.34, 4.35), one obtains in the MS-scheme for nf light flavors,
cf. [39–41,275,276],
βMS(nf ) = −β0(nf )aMSs
2
− β1(nf )aMSs
3
+O(aMSs
4
) . (4.40)
Additionally, it follows
das(µ2)
d ln(µ2) =
1
2
εas(µ2)−
∞∑
k=0
βkak+2s (µ2) . (4.41)
The factorization relation (3.16) strictly requires that the external massless particles are
on–shell. Massive loop corrections to the gluon propagator violate this condition, which
has to be enforced subtracting the corresponding corrections. These can be uniquely
absorbed into the strong coupling constant applying the background field method, [277], to
maintain the Slavnov-Taylor identities of QCD. We thus determine the coupling constant
renormalization in the MS-scheme as far as the light flavors and the gluon are concerned.
In addition, we make the choice that the heavy quark decouples in the running coupling
constant as(µ2) for µ2 < m2 and thus from the renormalized OMEs. This implies the
requirement that ΠH(0,m2) = 0, where ΠH(p2,m2) is the contribution to the gluon self-
energy due to the heavy quark loops, [126]. Since this condition introduces higher order
terms in ε into Zg, we left the MS–scheme. This new scheme is a MOM–scheme. After
mass renormalization in the on–shell–scheme via Eq. (4.27), we obtain for the heavy quark
contributions to the gluon self–energy in the background field formalism
ΠˆµνH,ab,BF(p2,m2, µ2, ε, aˆs) = i(−p2gµν + pµpν)δabΠˆH,BF(p2,m2, µ2, ε, aˆs) ,
ΠˆH,BF(0,m2, µ2, ε, aˆs) = aˆs
2β0,Q
ε
(m2
µ2
)ε/2
exp
( ∞∑
i=2
ζi
i
(ε
2
)i)
+aˆ2s
(m2
µ2
)ε
[
1
ε
(
−20
3
TFCA − 4TFCF
)
− 32
9
TFCA + 15TFCF
+ε
(
−86
27
TFCA −
31
4
TFCF −
5
3
ζ2TFCA − ζ2TFCF
)]
, (4.42)
with
β0,Q = βˆ0(nf ) = −
4
3
TF . (4.43)
Note that Eq. (4.42) holds only up to order O(ε), although we have partially included
higher orders in ε in order to keep the expressions shorter. We have used the Feynman–
rules of the background field formalism as given in Ref. [190]. In the following, we define
f(ε) ≡
(m2
µ2
)ε/2
exp
( ∞∑
i=2
ζi
i
(ε
2
)i)
. (4.44)
50
The renormalization constant of the background field ZA is related to Zg via
ZA = Z−2g . (4.45)
The light flavor contributions to ZA, ZA,l, can thus be determined by combining
Eqs. (4.34, 4.35, 4.45). The heavy flavor part follows from the condition
ΠH,BF(0,m2) + ZA,H ≡ 0 , (4.46)
which ensures that the on–shell gluon remains strictly massless. Thus we newly define
the renormalization constant of the strong coupling with nf light and one heavy flavor as
ZMOMg (ε, nf + 1, µ2,m2) ≡
1
(ZA,l + ZA,H)1/2
(4.47)
and obtain
ZMOMg 2(ε, nf + 1, µ2,m2) = 1 + aMOMs (µ2)
[2
ε (β0(nf ) + β0,Qf(ε))
]
+aMOMs 2(µ2)
[β1(nf )
ε +
4
ε2 (β0(nf ) + β0,Qf(ε))
2
+
1
ε
(m2
µ2
)ε(
β1,Q + εβ(1)1,Q + ε2β
(2)
1,Q
)]
+O(aMOMs 3) , (4.48)
with
β1,Q = βˆ1(nf ) = −4
(
5
3
CA + CF
)
TF , (4.49)
β(1)1,Q = −
32
9
TFCA + 15TFCF , (4.50)
β(2)1,Q = −
86
27
TFCA −
31
4
TFCF − ζ2
(
5
3
TFCA + TFCF
)
. (4.51)
The coefficients corresponding to Eq. (4.33) then read in the MOM–scheme
δaMOMs,1 =
[2β0(nf )
ε +
2β0,Q
ε f(ε)
]
, (4.52)
δaMOMs,2 =
[β1(nf )
ε +
(2β0(nf )
ε +
2β0,Q
ε f(ε)
)2
+
1
ε
(m2
µ2
)ε(
β1,Q + εβ(1)1,Q + ε2β
(2)
1,Q
)]
+O(ε2) . (4.53)
Since the MS–scheme is commonly used, we transform our results back from the MOM–
description into the MS–scheme, in order to be able to compare to other analyzes. This is
achieved by observing that the bare coupling does not change under this transformation
and one obtains the condition
ZMSg
2
(ε, nf + 1)aMSs (µ2) = ZMOMg 2(ε, nf + 1, µ2,m2)aMOMs (µ2) . (4.54)
The following relations hold :
aMOMs = aMSs − β0,Q ln
(m2
µ2
)
aMSs
2
+
[
β20,Q ln2
(m2
µ2
)
− β1,Q ln
(m2
µ2
)
− β(1)1,Q
]
aMSs
3
+O(aMSs
4
) , (4.55)
51
or,
aMSs = aMOMs + aMOMs 2
(
δaMOMs,1 − δaMSs,1 (nf + 1)
)
+ aMOMs 3
(
δaMOMs,2 − δaMSs,2 (nf + 1)
−2δaMSs,1 (nf + 1)
[
δaMOMs,1 − δaMSs,1 (nf + 1)
])
+O(aMOMs 4) , (4.56)
vice versa. Eq. (4.56) is valid to all orders in ε. Here, aMSs = aMSs (nf + 1). Applying
the on–shell–scheme for mass renormalization and the described MOM–scheme for the
renormalization of the coupling, one obtains as general formula for mass and coupling
constant renormalization up to O(aMOMs 3)
Aˆij = δij + aMOMs
ˆˆA
(1)
ij + aMOMs 2
[
ˆˆA
(2)
ij + δm1
(m2
µ2
)ε/2
m ddm
ˆˆA
(1)
ij + δaMOMs,1
ˆˆA
(1)
ij
]
+aMOMs 3
[
ˆˆA
(3)
ij + δaMOMs,2
ˆˆA
(1)
ij + 2δaMOMs,1
(
ˆˆA
(2)
ij + δm1
(m2
µ2
)ε/2
m ddm
ˆˆA
(1)
ij
)
+δm1
(m2
µ2
)ε/2
m ddm
ˆˆA
(2)
ij + δm2
(m2
µ2
)ε
m ddm
ˆˆA
(1)
ij
+
δm21
2
(m2
µ2
)ε
m2 d
2
dm2
ˆˆA
(1)
ij
]
, (4.57)
where we have suppressed the dependence on m, ε and N in the arguments 19.
4.5 Operator Renormalization
The renormalization of the UV singularities of the composite operators is being performed
introducing the corresponding Zij-factors, which have been defined in Eqs. (2.105, 2.106).
We consider first only nf massless flavors, cf. [269], and do then include subsequently one
heavy quark. In the former case, renormalization proceeds in the MS–scheme via
ANSqq
(−p2
µ2 , a
MS
s , nf , N
)
= Z−1,NSqq (aMSs , nf , ε, N)AˆNSqq
(−p2
µ2 , a
MS
s , nf , ε, N
)
, (4.58)
Aij
(−p2
µ2 , a
MS
s , nf , N
)
= Z−1il (aMSs , nf , ε, N)Aˆlj
(−p2
µ2 , a
MS
s , nf , ε, N
)
, i, j = q, g,
(4.59)
with p a space–like momentum. As is well known, operator mixing occurs in the singlet
case, Eq. (4.59). As mentioned before, we neglected all terms being associated to EOM
and NGI parts, since they do not contribute in the renormalization of the massive on–shell
operator matrix elements. The NS and PS contributions are separated via
Z−1qq = Z−1,PSqq + Z−1,NSqq , (4.60)
Aqq = APSqq + ANSqq . (4.61)
19Here we corrected a typographical error in [137], Eq. (48).
52
The anomalous dimensions γij of the operators are defined in Eqs. (2.107, 2.108) and can
be expanded in a perturbative series as follows
γS,PS,NSij (aMSs , nf , N) =
∞∑
l=1
aMSs
l
γ(l),S,PS,NSij (nf , N) . (4.62)
Here, the PS contribution starts at O(a2s). In the following, we do not write the depen-
dence on the Mellin–variable N for the OMEs, the operator Z–factors and the anomalous
dimensions explicitly. Further, we will suppress the dependence on ε for unrenormalized
quantities and Z–factors. From Eqs. (2.107, 2.108), one can determine the relation be-
tween the anomalous dimensions and the Z–factors order by order in perturbation theory.
In the general case, one finds up to O(aMSs
3
)
Zij(aMSs , nf ) = δij + aMSs
γ(0)ij
ε + a
MS
s
2
{
1
ε2
(1
2
γ(0)il γ
(0)
lj + β0γ
(0)
ij
)
+
1
2εγ
(1)
ij
}
+aMSs
3
{
1
ε3
(1
6
γ(0)il γ
(0)
lk γ
(0)
kj + β0γ
(0)
il γ
(0)
lj +
4
3
β20γ
(0)
ij
)
+
1
ε2
(1
6
γ(1)il γ
(0)
lj +
1
3
γ(0)il γ
(1)
lj +
2
3
β0γ(1)ij +
2
3
β1γ(0)ij
)
+
γ(2)ij
3ε
}
. (4.63)
The NS and PS Z–factors are given by 20
ZNSqq (aMSs , nf ) = 1 + aMSs
γ(0),NSqq
ε + a
MS
s
2
{
1
ε2
(1
2
γ(0),NSqq
2
+ β0γ(0),NSqq
)
+
1
2εγ
(1),NS
qq
}
+aMSs
3
{
1
ε3
(1
6
γ(0),NSqq
3
+ β0γ(0),NSqq
2
+
4
3
β20γ(0),NSqq
)
+
1
ε2
(1
2
γ(0),NSqq γ(1),NSqq +
2
3
β0γ(1),NSqq +
2
3
β1γ(0),NSqq
)
+
1
3εγ
(2),NS
qq
}
,
(4.64)
ZPSqq (aMSs , nf ) = aMSs
2
{
1
2ε2γ
(0)
qg γ(0)gq +
1
2εγ
(1),PS
qq
}
+ aMSs
3
{
1
ε3
(1
3
γ(0)qq γ(0)qg γ(0)gq
+
1
6
γ(0)qg γ(0)gg γ(0)gq + β0γ(0)qg γ(0)gq
)
+
1
ε2
(1
3
γ(0)qg γ(1)gq
+
1
6
γ(1)qg γ(0)gq +
1
2
γ(0)qq γ(1),PSqq +
2
3
β0γ(1),PSqq
)
+
γ(2),PSqq
3ε
}
. (4.65)
All quantities in Eqs. (4.63)–(4.65) refer to nf light flavors and renormalize the massless
off–shell OMEs given in Eqs. (4.58, 4.59).
In the next step, we consider an additional heavy quark with mass m. We keep
the external momentum artificially off–shell for the moment, in order to deal with the
UV–singularities only. For the additional massive quark, one has to account for the
20In Eq. (4.65) we corrected typographical errors contained in Eq. (34), [137].
53
renormalization of the coupling constant we defined in Eqs. (4.52, 4.53). The Z–factors
including one massive quark are then obtained by taking Eqs. (4.63)–(4.65) at (nf + 1)
flavors and performing the scheme transformation given in (4.56). The emergence of
δaMOMs,k in Zij is due to the finite mass effects and cancels singularities which emerge for
real radiation and virtual processes at p2 → 0. Thus one obtains up to O(aMOMs 3)
Z−1ij (aMOMs , nf + 1, µ2) = δij − aMOMs
γ(0)ij
ε + a
MOM
s
2
[
1
ε
(
−1
2
γ(1)ij − δaMOMs,1 γ
(0)
ij
)
+
1
ε2
(1
2
γ(0)il γ
(0)
lj + β0γ
(0)
ij
)]
+ aMOMs 3
[
1
ε
(
−1
3
γ(2)ij − δaMOMs,1 γ
(1)
ij
−δaMOMs,2 γ
(0)
ij
)
+
1
ε2
(4
3
β0γ(1)ij + 2δaMOMs,1 β0γ
(0)
ij +
1
3
β1γ(0)ij
+δaMOMs,1 γ
(0)
il γ
(0)
lj +
1
3
γ(1)il γ
(0)
lj +
1
6
γ(0)il γ
(1)
lj
)
+
1
ε3
(
−4
3
β20γ
(0)
ij
−β0γ(0)il γ
(0)
lj −
1
6
γ(0)il γ
(0)
lk γ
(0)
kj
)]
, (4.66)
and
Z−1,NSqq (aMOMs , nf + 1, µ2) = 1− aMOMs
γ(0),NSqq
ε + a
MOM
s
2
[
1
ε
(
−1
2
γ(1),NSqq − δaMOMs,1 γ(0),NSqq
)
+
1
ε2
(
β0γ(0),NSqq +
1
2
γ(0),NSqq
2
)]
+ aMOMs 3
[
1
ε
(
−1
3
γ(2),NSqq
−δaMOMs,1 γ(1),NSqq − δaMOMs,2 γ(0),NSqq
)
+
1
ε2
(4
3
β0γ(1),NSqq
+2δaMOMs,1 β0γ(0),NSqq +
1
3
β1γ(0),NSqq +
1
2
γ(0),NSqq γ(1),NSqq
+δaMOMs,1 γ(0),NSqq
2
)
+
1
ε3
(
−4
3
β20γ(0),NSqq
−β0γ(0),NSqq
2 − 1
6
γ(0),NSqq
3
)]
, (4.67)
Z−1,PSqq (aMOMs , nf + 1, µ2) = aMOMs 2
[
1
ε
(
−1
2
γ(1),PSqq
)
+
1
ε2
(1
2
γ(0)qg γ(0)gq
)]
+aMOMs 3
[
1
ε
(
−1
3
γ(2),PSqq − δaMOMs,1 γ(1),PSqq
)
+
1
ε2
(1
6
γ(0)qg γ(1)gq
+
1
3
γ(0)gq γ(1)qg +
1
2
γ(0)qq γ(1),PSqq +
4
3
β0γ(1),PSqq + δaMOMs,1 γ(0)qg γ(0)gq
)
+
1
ε3
(
−1
3
γ(0)qg γ(0)gq γ(0)qq −
1
6
γ(0)gq γ(0)qg γ(0)gg − β0γ(0)qg γ(0)gq
)]
.
(4.68)
The above equations are given for nf + 1 flavors. One re-derives the expressions for nf
light flavors by setting (nf + 1) =: nf and δaMOMs = δaMSs . As a next step, we split the
54
OMEs into a part involving only light flavors and the heavy flavor part
Aˆij(p2,m2, µ2, aMOMs , nf + 1) = Aˆij
(−p2
µ2 , a
MS
s , nf
)
+AˆQij(p2,m2, µ2, aMOMs , nf + 1) . (4.69)
In (4.69, 4.70), the light flavor part depends on aMSs , since the prescription adopted for
coupling constant renormalization only applies to the massive part. AˆQij denotes any
massive OME we consider. The correct UV–renormalization prescription for the massive
contribution is obtained by subtracting from Eq. (4.69) the terms applying to the light
part only :
A¯Qij(p2,m2, µ2, aMOMs , nf + 1) = Z−1il (aMOMs , nf + 1, µ2)Aˆ
Q
ij(p2,m2, µ2, aMOMs , nf + 1)
+Z−1il (aMOMs , nf + 1, µ2)Aˆij
(−p2
µ2 , a
MS
s , nf
)
−Z−1il (aMSs , nf , µ2)Aˆij
(−p2
µ2 , a
MS
s , nf
)
, (4.70)
where
Z−1ij = δij +
∞∑
k=1
aksZ
−1,(k)
ij . (4.71)
In the limit p2 = 0, integrals without a scale vanish within dimensional regularization.
Hence for the light flavor OMEs only the term δij remains and one obtains the UV–finite
massive OMEs after expanding in as
A¯Qij
(m2
µ2 , a
MOM
s , nf + 1
)
= aMOMs
(
Aˆ(1),Qij
(m2
µ2
)
+ Z−1,(1)ij (nf + 1, µ2)− Z
−1,(1)
ij (nf )
)
+aMOMs 2
(
Aˆ(2),Qij
(m2
µ2
)
+ Z−1,(2)ij (nf + 1, µ2)− Z
−1,(2)
ij (nf )
+ Z−1,(1)ik (nf + 1, µ2)Aˆ
(1),Q
kj
(m2
µ2
))
+aMOMs 3
(
Aˆ(3),Qij
(m2
µ2
)
+ Z−1,(3)ij (nf + 1, µ2)− Z
−1,(3)
ij (nf )
+ Z−1,(1)ik (nf + 1, µ2)Aˆ
(2),Q
kj
(m2
µ2
)
+ Z−1,(2)ik (nf + 1, µ2)Aˆ
(1),Q
kj
(m2
µ2
))
. (4.72)
The Z–factors at nf + 1 flavors refer to Eqs. (4.66)–(4.68), whereas those at nf flavors
correspond to the massless case.
4.6 Mass Factorization
Finally, we have to remove the collinear singularities contained in A¯ij, which emerge in
the limit p2 = 0. They are absorbed into the parton distribution functions and are not
55
present in case of the off–shell massless OMEs. As a generic renormalization formula,
generalizing Eqs. (4.58, 4.59), one finds
Aij = Z−1il AˆlkΓ−1kj . (4.73)
The renormalized operator matrix elements are obtained by
AQij
(m2
µ2 , a
MOM
s , nf + 1
)
= A¯Qil
(m2
µ2 , a
MOM
s , nf + 1
)
Γ−1lj . (4.74)
If all quarks were massless, the identity, [126],
Γij = Z−1ij . (4.75)
would hold. However, due to the presence of a heavy quark Q, the transition functions
Γ(nf ) refer only to massless sub-graphs. Hence the Γ–factors contribute up to O(a2s) only
and do not involve the special scheme adopted for the renormalization of the coupling.
Due to Eq. (4.75), they can be read off from Eqs. (4.63)–(4.65). The renormalized
operator matrix elements are then given by:
AQij
(m2
µ2 , a
MOM
s , nf + 1
)
=
aMOMs
(
Aˆ(1),Qij
(m2
µ2
)
+ Z−1,(1)ij (nf + 1)− Z
−1,(1)
ij (nf )
)
+aMOMs 2
(
Aˆ(2),Qij
(m2
µ2
)
+ Z−1,(2)ij (nf + 1)− Z
−1,(2)
ij (nf ) + Z
−1,(1)
ik (nf + 1)Aˆ
(1),Q
kj
(m2
µ2
)
+
[
Aˆ(1),Qil
(m2
µ2
)
+ Z−1,(1)il (nf + 1)− Z
−1,(1)
il (nf )
]
Γ−1,(1)lj (nf )
)
+aMOMs 3
(
Aˆ(3),Qij
(m2
µ2
)
+ Z−1,(3)ij (nf + 1)− Z
−1,(3)
ij (nf ) + Z
−1,(1)
ik (nf + 1)Aˆ
(2),Q
kj
(m2
µ2
)
+ Z−1,(2)ik (nf + 1)Aˆ
(1),Q
kj
(m2
µ2
)
+
[
Aˆ(1),Qil
(m2
µ2
)
+ Z−1,(1)il (nf + 1)
− Z−1,(1)il (nf )
]
Γ−1,(2)lj (nf ) +
[
Aˆ(2),Qil
(m2
µ2
)
+ Z−1,(2)il (nf + 1)− Z
−1,(2)
il (nf )
+ Z−1,(1)ik (nf + 1)Aˆ
(1),Q
kl
(m2
µ2
)]
Γ−1,(1)lj (nf )
)
+O(aMOMs 4) . (4.76)
From (4.76) it is obvious that the renormalization of AQij to O(a3s) requires the 1–loop
terms up to O(ε2) and the 2–loop terms up to O(ε), cf. [126, 128–130, 137]. These terms
are calculated in Section 6. Finally, we transform the coupling constant back into the
MS–scheme by using Eq. (4.55). We do not give the explicit formula here, but present
the individual renormalized OMEs after this transformation in the next Section as per-
turbative series in aMSs ,
AQij
(m2
µ2 , a
MS
s , nf + 1
)
= δij + aMSs A
Q,(1)
ij
(m2
µ2 , nf + 1
)
+ aMSs
2
AQ,(2)ij
(m2
µ2 , nf + 1
)
+aMSs
3
AQ,(3)ij
(m2
µ2 , nf + 1
)
+O(aMSs
4
) . (4.77)
56
As stated in Section 3, one has to use the same scheme when combining the massive OMEs
with the massless Wilson coefficients in the factorization formula (3.16). The effects
of the transformation between the MOM– and MS–scheme are discussed in Section 5.
The subscript Q was introduced in this Section to make the distinction between the
massless and massive OMEs explicit and will be dropped from now on, since no confusion
is expected. Comparing Eqs. (4.76) and (4.77), one notices that the term δij is not
present in the former because it was subtracted together with the light flavor contributions.
However, as one infers from Eq. (3.16) and the discussion below, this term is necessary
when calculating the massive Wilson coefficients in the asymptotic limit and we therefore
have re–introduced it into Eq. (4.77).
4.7 General Structure of the Massive Operator Matrix Ele-
ments
In the following, we present the general structure of the unrenormalized and renormalized
massive operator matrix elements for the specific partonic channels. The former are
expressed as a Laurent–series in ε via
ˆˆA
(l)
ij
(mˆ2
µ2 , ε, N
)
=
(mˆ2
µ2
)lε/2 ∞∑
k=0
a(l,k)ij
εl−k . (4.78)
Additionally, we set
a(l,l)ij ≡ a
(l)
ij , a
(l,l+1)
ij ≡ a
(l)
ij , etc. . (4.79)
The pole terms can all be expressed by known renormalization constants and lower order
contributions to the massive OMEs, which provides us with a strong check on our calcu-
lation. In particular, the complete NLO anomalous dimensions, as well as their TF–terms
at NNLO, contribute at O(a3s). The moments of the O(ε0)–terms of the unrenormalized
OMEs at the 3–loop level, a(3)ij , are a new result of this thesis and will be calculated in
Section 7, cf. [134]. The O(ε) terms at the 2–loop level, a(2)ij , contribute to the non–
logarithmic part of the renormalized 3–loop OMEs and are calculated for general values
of N in Section 6, cf. [130, 137]. The pole terms and the O(ε0) terms, a(2)ij , at 2–loop
have been calculated for the first time in Refs. [126, 129]. The terms involving the quark
operator, (2.86, 2.87), were confirmed in [128] and the terms involving the gluon oper-
ator (2.88) by the present work, cf. [130]. In order to keep up with the notation used
in [126, 129], we define the 2–loop terms a(2)ij , a
(2)
ij after performing mass renormalization
in the on–shell–scheme. This we do not apply for the 3–loop terms. We choose to calcu-
late one–particle reducible diagrams and therefore have to include external self–energies
containing massive quarks into our calculation. Before presenting the operator matrix
elements up to three loops, we first summarize the necessary self–energy contributions in
the next Section. The remaining Sections, (4.7.2)–(4.7.6), contain the general structure
of the unrenormalized and renormalized massive OMEs up to 3–loops. In these Sections,
we always proceed as follows: From Eqs. (4.57, 4.76), one predicts the pole terms of the
respective unrenormalized OMEs by demanding that these terms have to cancel through
renormalization. The unrenormalized expressions are then renormalized in the MOM–
scheme. Finally, Eq. (4.55) is applied and the renormalized massive OMEs are presented
in the MS–scheme.
57
4.7.1 Self–energy contributions
The gluon and quark self-energy contributions due to heavy quark lines are given by
Πˆabµν(p2, mˆ2, µ2, aˆs) = iδab
[
−gµνp2 + pµpν
]
Πˆ(p2, mˆ2, µ2, aˆs) , (4.80)
Πˆ(p2, mˆ2, µ2, aˆs) =
∞∑
k=1
aˆksΠˆ(k)(p2, mˆ2, µ2). (4.81)
Σˆij(p2, mˆ2, µ2, aˆs) = i δij /p Σˆ(p2, mˆ2, µ2, aˆs) , (4.82)
Σˆ(p2, mˆ2, µ2, aˆs) =
∞∑
k=2
aˆksΣˆ(k)(p2, mˆ2, µ2) . (4.83)
Note, that the quark self–energy contributions start at 2–loop order. These self–energies
are easily calculated using MATAD, [164], cf. Section 7. The expansion coefficients for
p2 = 0 of Eqs. (4.82, 4.83) are needed for the calculation of the gluonic and quarkonic
OMEs, respectively. The contributions to the gluon vacuum polarization for general gauge
parameter ξ are
Πˆ(1)
(
0, mˆ
2
µ2
)
= TF
(mˆ2
µ2
)ε/2
(
− 8
3ε exp
( ∞∑
i=2
ζi
i
(ε
2
)i)
)
, (4.84)
Πˆ(2)
(
0, mˆ
2
µ2
)
= TF
(mˆ2
µ2
)ε
(
− 4ε2CA +
1
ε
{
−12CF + 5CA
}
+ CA
(13
12
− ζ2
)
− 13
3
CF
+ε
{
CA
(169
144
+
5
4
ζ2 −
ζ3
3
)
+ CF
(
−35
12
− 3ζ2
)})
+O(ε2) , (4.85)
Πˆ(3)
(
0, mˆ
2
µ2
)
= TF
(mˆ2
µ2
)3ε/2
(
1
ε3
{
−32
9
TFCA
(
2nf + 1
)
+ C2A
(164
9
+
4
3
ξ
)}
+
1
ε2
{
80
27
(
CA − 6CF
)
nfTF +
8
27
(
35CA − 48CF
)
TF +
C2A
27
(
−781
+63ξ
)
+
712
9
CACF
}
+
1
ε
{
4
27
(
CA(−101− 18ζ2)− 62CF
)
nfTF
+
2
27
(
CA(−37− 18ζ2)− 80CF
)
TF + C2A
(
−12ζ3 +
41
6
ζ2 +
3181
108
+
ζ2
2
ξ
+
137
36
ξ
)
+ CACF
(
16ζ3 −
1570
27
)
+
272
3
C2F
}
+ nfTF
{
CA
(56
9
ζ3 +
10
9
ζ2
−3203
243
)
+ CF
(
−20
3
ζ2 −
1942
81
)}
+ TF
{
CA
(
−295
18
ζ3 +
35
9
ζ2 +
6361
486
)
+CF
(
−7ζ3 −
16
3
ζ2 −
218
81
)}
+ C2A
{
4B4 − 27ζ4 +
1969
72
ζ3 −
781
72
ζ2
+
42799
3888
− 7
6
ζ3ξ +
7
8
ζ2ξ +
3577
432
ξ
}
+ CACF
{
−8B4 + 36ζ4 −
1957
12
ζ3
58
+
89
3
ζ2 +
10633
81
}
+ C2F
{
95
3
ζ3 +
274
9
})
+O(ε) , (4.86)
and for the quark self–energy,
Σˆ(2)(0, mˆ
2
µ2 ) = TFCF
(mˆ2
µ2
)ε{2
ε +
5
6
+
[
89
72
+
ζ2
2
]
ε
}
+O(ε2) , (4.87)
Σˆ(3)(0, mˆ
2
µ2 ) = TFCF
(mˆ2
µ2
)3ε/2
(
8
3ε3CA{1− ξ}+
1
ε2
{32
9
TF (nf + 2)− CA
(40
9
+ 4ξ
)
−8
3
CF
}
+
1
ε
{
40
27
TF (nf + 2) + CA
{
ζ2 +
454
27
− ζ2ξ −
70
9
ξ
}
− 26CF
}
+nfTF
{4
3
ζ2 +
674
81
}
+ TF
{8
3
ζ2 +
604
81
}
+ CA
{17
3
ζ3 −
5
3
ζ2 +
1879
162
+
7
3
ζ3ξ −
3
2
ζ2ξ −
407
27
ξ
}
+ CF
{
−8ζ3 − ζ2 −
335
18
})
+O(ε) , (4.88)
see also [263,278]. In Eq. (4.86) the constant
B4 = −4ζ2 ln2(2) +
2
3
ln4(2)− 13
2
ζ4 + 16Li4
(1
2
)
≈ −1.762800093... (4.89)
appears due to genuine massive effects, cf. [279–282].
4.7.2 ANSqq,Q
The lowest non–trivial NS–contribution is of O(a2s),
ANSqq,Q = 1 + a2sA
(2),NS
qq,Q + a3sA
(3),NS
qq,Q +O(a4s) . (4.90)
The expansion coefficients are obtained in the MOM–scheme from the bare quantities,
using Eqs. (4.57, 4.76). After mass– and coupling constant renormalization, the OMEs
are given by
A(2),NS,MOMqq,Q = Aˆ
(2),NS,MOM
qq,Q + Z−1,(2),NSqq (nf + 1)− Z−1,(2),NSqq (nf ) , (4.91)
A(3),NS,MOMqq,Q = Aˆ
(3),NS,MOM
qq,Q + Z−1,(3),NSqq (nf + 1)− Z−1,(3),NSqq (nf )
+Z−1,(1),NSqq (nf + 1)Aˆ
(2),NS,MOM
qq,Q +
[
Aˆ(2),NS,MOMqq,Q
+Z−1,(2),NSqq (nf + 1)− Z−1,(2),NSqq (nf )
]
Γ−1,(1)qq (nf ) . (4.92)
From (4.57, 4.76, 4.91, 4.92), one predicts the pole terms of the unrenormalized OME. At
second and third order they read
ˆˆA
(2),NS
qq,Q =
(mˆ2
µ2
)ε
(
β0,Qγ(0)qq
ε2 +
γˆ(1),NSqq
2ε + a
(2),NS
qq,Q + a
(2),NS
qq,Q ε
)
, (4.93)
ˆˆA
(3),NS
qq,Q =
(mˆ2
µ2
)3ε/2
{
−4γ
(0)
qq β0,Q
3ε3
(
β0 + 2β0,Q
)
+
1
ε2
(
2γ(1),NSqq β0,Q
3
− 4γˆ
(1),NS
qq
3
[
β0 + β0,Q
]
+
2β1,Qγ(0)qq
3
− 2δm(−1)1 β0,Qγ(0)qq
)
+
1
ε
(
γˆ(2),NSqq
3
− 4a(2),NSqq,Q
[
β0 + β0,Q
]
+ β(1)1,Qγ(0)qq
59
+
γ(0)qq β0β0,Qζ2
2
− 2δm(0)1 β0,Qγ(0)qq − δm
(−1)
1 γˆ(1),NSqq
)
+ a(3),NSqq,Q
}
. (4.94)
Note, that we have already used the general structure of the unrenormalized lower order
OME in the evaluation of the O(aˆ3s) term, as we will always do in the following. Using
Eqs. (4.57, 4.91, 4.92), one can renormalize the above expressions. In addition, we finally
transform back to the MS–scheme using Eq. (4.55). Thus one obtains the renormalized
expansion coefficients of Eq. (4.90)
A(2),NS,MSqq,Q =
β0,Qγ(0)qq
4
ln2
(m2
µ2
)
+
γˆ(1),NSqq
2
ln
(m2
µ2
)
+ a(2),NSqq,Q −
β0,Qγ(0)qq
4
ζ2 , (4.95)
A(3),NS,MSqq,Q = −
γ(0)qq β0,Q
6
(
β0 + 2β0,Q
)
ln3
(m2
µ2
)
+
1
4
{
2γ(1),NSqq β0,Q − 2γˆ(1),NSqq
(
β0 + β0,Q
)
+β1,Qγ(0)qq
}
ln2
(m2
µ2
)
+
1
2
{
γˆ(2),NSqq −
(
4a(2),NSqq,Q − ζ2β0,Qγ(0)qq
)
(β0 + β0,Q)
+γ(0)qq β
(1)
1,Q
}
ln
(m2
µ2
)
+ 4a(2),NSqq,Q (β0 + β0,Q)− γ(0)qq β
(2)
1,Q −
γ(0)qq β0β0,Qζ3
6
−γ
(1),NS
qq β0,Qζ2
4
+ 2δm(1)1 β0,Qγ(0)qq + δm
(0)
1 γˆ(1),NSqq + 2δm
(−1)
1 a
(2),NS
qq,Q
+a(3),NSqq,Q . (4.96)
Note that in the NS–case, one is generically provided with even and odd moments due to
a Ward–identity relating the results in the polarized and unpolarized case. The former
refer to the anomalous dimensions γNS,+qq and the latter to γNS,−qq , respectively, as given in
Eqs. (3.5, 3.7) and Eqs. (3.6, 3.8) in Ref. [124]. The relations above also apply to other
twist–2 non–singlet massive OMEs, as to transversity, for which the 2- and 3–loop heavy
flavor corrections are given in Section 9, cf. also [160].
4.7.3 APSQq and APSqq,Q
There are two different PS–contributions, cf. the discussion below Eq. 3.18,
APSQq = a2sA
(2),PS
Qq + a3sA
(3),PS
Qq +O(a4s) , (4.97)
APSqq,Q = a3sA
(3),PS
qq,Q +O(a4s) . (4.98)
Separating these contributions is not straightforward, since the generic renormalization
formula for operator renormalization and mass factorization, Eq. (4.76), applies to the
sum of these terms only. At O(a2s), this problem does not occur and renormalization
proceeds in the MOM–scheme via
A(2),PS,MOMQq = Aˆ
(2),PS,MOM
Qq + Z−1,(2),PSqq (nf + 1)− Z−1,(2),PSqq (nf )
+
[
Aˆ(1),MOMQg + Z−1,(1)qg (nf + 1)− Z−1,(1)qg (nf )
]
Γ−1,(1)gq (nf ) . (4.99)
The unrenormalized expression is given by
ˆˆA
(2),PS
Qq =
(mˆ2
µ2
)ε
(
− γˆ
(0)
qg γ(0)gq
2ε2 +
γˆ(1),PSqq
2ε + a
(2),PS
Qq + a
(2),PS
Qq ε
)
. (4.100)
60
The renormalized result in the MS–scheme reads
A(2),PS,MSQq = −
γˆ(0)qg γ(0)gq
8
ln2
(m2
µ2
)
+
γˆ(1),PSqq
2
ln
(m2
µ2
)
+ a(2),PSQq +
γˆ(0)qg γ(0)gq
8
ζ2 . (4.101)
The corresponding renormalization relation at third order is given by
A(3),PS,MOMQq + A
(3),PS,MOM
qq,Q = Aˆ
(3),PS,MOM
Qq + Aˆ
(3),PS,MOM
qq,Q + Z−1,(3),PSqq (nf + 1)
− Z−1,(3),PSqq (nf ) + Z−1,(1)qq (nf + 1)Aˆ
(2),PS,MOM
Qq + Z−1,(1)qg (nf + 1)Aˆ
(2),MOM
gq,Q
+
[
Aˆ(1),MOMQg + Z−1,(1)qg (nf + 1)− Z−1,(1)qg (nf )
]
Γ−1,(2)gq (nf ) +
[
Aˆ(2),PS,MOMQq
+ Z−1,(2),PSqq (nf + 1)− Z−1,(2),PSqq (nf )
]
Γ−1,(1)qq (nf ) +
[
Aˆ(2),MOMQg + Z−1,(2)qg (nf + 1)
− Z−1,(2)qg (nf ) + Z−1,(1)qq (nf + 1)A
(1),MOM
Qg + Z−1,(1)qg (nf + 1)A
(1),MOM
gg,Q
]
Γ−1,(1)gq (nf ) .
(4.102)
Taking into account the structure of the UV– and collinear singularities of the contributing
Feynman–diagrams, these two contributions can be separated. For the bare quantities we
obtain
ˆˆA
(3),PS
Qq =
(mˆ2
µ2
)3ε/2
[
γˆ(0)qg γ(0)gq
6ε3
(
γ(0)gg − γ(0)qq + 6β0 + 16β0,Q
)
+
1
ε2
(
−4γˆ
(1),PS
qq
3
[
β0 + β0,Q
]
−γ
(0)
gq γˆ(1)qg
3
+
γˆ(0)qg
6
[
2γˆ(1)gq − γ(1)gq
]
+ δm(−1)1 γˆ(0)qg γ(0)gq
)
+
1
ε
(
γˆ(2),PSqq
3
− nf
ˆ˜γ(2),PSqq
3
+γˆ(0)qg a
(2)
gq,Q − γ(0)gq a
(2)
Qg − 4(β0 + β0,Q)a
(2),PS
Qq −
γˆ(0)qg γ(0)gq ζ2
16
[
γ(0)gg − γ(0)qq + 6β0
]
+δm(0)1 γˆ(0)qg γ(0)gq − δm
(−1)
1 γˆ(1),PSqq
)
+ a(3),PSQq
]
, (4.103)
ˆˆA
(3),PS
qq,Q = nf
(mˆ2
µ2
)3ε/2
[
2γˆ(0)qg γ(0)gq β0,Q
3ε3 +
1
3ε2
(
2γˆ(1),PSqq β0,Q + γˆ(0)qg γˆ(1)gq
)
+
1
ε
(
ˆ˜γ(2),PSqq
3
+ γˆ(0)qg a
(2)
gq,Q −
γˆ(0)qg γ(0)gq β0,Qζ2
4
)
+
a(3),PSqq,Q
nf
]
. (4.104)
The renormalized terms in the MS–scheme are given by
A(3),PS,MSQq =
γˆ(0)qg γ(0)gq
48
{
γ(0)gg − γ(0)qq + 6β0 + 16β0,Q
}
ln3
(m2
µ2
)
+
1
8
{
−4γˆ(1),PSqq
(
β0 + β0,Q
)
+γˆ(0)qg
(
γˆ(1)gq − γ(1)gq
)
− γ(0)gq γˆ(1)qg
}
ln2
(m2
µ2
)
+
1
16
{
8γˆ(2),PSqq − 8nf ˆ˜γ
(2),PS
qq
−32a(2),PSQq (β0 + β0,Q) + 8γˆ(0)qg a
(2)
gq,Q − 8γ(0)gq a
(2)
Qg − γˆ(0)qg γ(0)gq ζ2
(
γ(0)gg − γ(0)qq
+6β0 + 8β0,Q
)}
ln
(m2
µ2
)
+ 4(β0 + β0,Q)a(2),PSQq + γ(0)gq a
(2)
Qg − γˆ(0)qg a
(2)
gq,Q
+
γ(0)gq γˆ(0)qg ζ3
48
(
γ(0)gg − γ(0)qq + 6β0
)
+
γˆ(0)qg γ(1)gq ζ2
16
− δm(1)1 γˆ(0)qg γ(0)gq + δm
(0)
1 γˆ(1),PSqq
61
+2δm(−1)1 a
(2),PS
Qq + a
(3),PS
Qq . (4.105)
A(3),PS,MSqq,Q = nf
{
γ(0)gq γˆ(0)qg β0,Q
12
ln3
(m2
µ2
)
+
1
8
(
4γˆ(1),PSqq β0,Q + γˆ(0)qg γˆ(1)gq
)
ln2
(m2
µ2
)
+
1
4
(
2ˆ˜γ(2),PSqq + γˆ(0)qg
{
2a(2)gq,Q − γ(0)gq β0,Qζ2
})
ln
(m2
µ2
)
−γˆ(0)qg a
(2)
gq,Q +
γ(0)gq γˆ(0)qg β0,Qζ3
12
− γˆ
(1),PS
qq β0,Qζ2
4
}
+ a(3),PSqq,Q . (4.106)
4.7.4 AQg and Aqg,Q
The OME AQg is the most complex expression. As in the PS–case, there are two different
contributions
AQg = asA(1)Qg + a2sA
(2)
Qg + a3sA
(3)
Qg +O(a4s) . (4.107)
Aqg,Q = a3sA
(3)
qg,Q +O(a4s) . (4.108)
In the MOM–scheme the 1– and 2–loop contributions obey the following relations
A(1),MOMQg = Aˆ
(1),MOM
Qg + Z−1,(1)qg (nf + 1)− Z−1,(1)qg (nf ) , (4.109)
A(2),MOMQg = Aˆ
(2),MOM
Qg + Z−1,(2)qg (nf + 1)− Z−1,(2)qg (nf ) + Z−1,(1)qg (nf + 1)Aˆ
(1),MOM
gg,Q
+Z−1,(1)qq (nf + 1)Aˆ
(1),MOM
Qg +
[
Aˆ(1),MOMQg + Z−1,(1)qg (nf + 1)
−Z−1,(1)qg (nf )
]
Γ−1,(1)gg (nf ) . (4.110)
The unrenormalized terms are given by
ˆˆA
(1)
Qg =
(mˆ2
µ2
)ε/2 γˆ(0)qg
ε exp
( ∞∑
i=2
ζi
i
(ε
2
)i)
, (4.111)
ˆˆA
(2)
Qg =
(mˆ2
µ2
)ε
[
− γˆ
(0)
qg
2ε2
(
γ(0)gg − γ(0)qq + 2β0 + 4β0,Q
)
+
γˆ(1)qg − 2δm(−1)1 γˆ
(0)
qg
2ε + a
(2)
Qg
−δm(0)1 γˆ(0)qg −
γˆ(0)qg β0,Qζ2
2
+ ε
(
a(2)Qg − δm
(1)
1 γˆ(0)qg −
γˆ(0)qg β0,Qζ2
12
)]
. (4.112)
Note that we have already made the one–particle reducible contributions to Eq. (4.112)
explicit, which are given by the LO–term multiplied with the 1–loop gluon–self energy,
cf. Eq. (4.84). Furthermore, Eq. (4.112) already contains terms in the O(ε0) and O(ε)
expressions which result from mass renormalization. At this stage of the renormalization
procedure they should not be present, however, we have included them here in order to
have the same notation as in Refs. [126,129] at the 2–loop level. The renormalized terms
then become in the MS–scheme
A(1),MSQg =
γˆ(0)qg
2
ln
(m2
µ2
)
, (4.113)
A(2),MSQg = −
γˆ(0)qg
8
[
γ(0)gg − γ(0)qq + 2β0 + 4β0,Q
]
ln2
(m2
µ2
)
+
γˆ(1)qg
2
ln
(m2
µ2
)
62
+a(2)Qg +
γˆ(0)qg ζ2
8
(
γ(0)gg − γ(0)qq + 2β0
)
. (4.114)
The generic renormalization relation at the 3–loop level is given by
A(3),MOMQg + A
(3),MOM
qg,Q = Aˆ
(3),MOM
Qg + Aˆ
(3),MOM
qg,Q + Z−1,(3)qg (nf + 1)− Z−1,(3)qg (nf )
+ Z−1,(2)qg (nf + 1)Aˆ
(1),MOM
gg,Q + Z−1,(1)qg (nf + 1)Aˆ
(2),MOM
gg,Q + Z−1,(2)qq (nf + 1)Aˆ
(1),MOM
Qg
+ Z−1,(1)qq (nf + 1)Aˆ
(2),MOM
Qg +
[
Aˆ(1),MOMQg + Z−1,(1)qg (nf + 1)
− Z−1,(1)qg (nf )
]
Γ−1,(2)gg (nf ) +
[
Aˆ(2),MOMQg + Z−1,(2)qg (nf + 1)− Z−1,(2)qg (nf )
+ Z−1,(1)qq (nf + 1)A
(1),MOM
Qg + Z−1,(1)qg (nf + 1)A
(1),MOM
gg,Q
]
Γ−1,(1)gg (nf )
+
[
Aˆ(2),PS,MOMQq + Z−1,(2),PSqq (nf + 1)− Z−1,(2),PSqq (nf )
]
Γ−1,(1)qg (nf )
+
[
Aˆ(2),NS,MOMqq,Q + Z−1,(2),NSqq (nf + 1)− Z−1,(2),NSqq (nf )
]
Γ−1,(1)qg (nf ) . (4.115)
Similar to the PS–case, the different contributions can be separated and one obtains the
following unrenormalized results
ˆˆA
(3)
Qg =
(mˆ2
µ2
)3ε/2
[
γˆ(0)qg
6ε3
(
(nf + 1)γ(0)gq γˆ(0)qg + γ(0)qq
[
γ(0)qq − 2γ(0)gg − 6β0 − 8β0,Q
]
+ 8β20
+28β0,Qβ0 + 24β20,Q + γ(0)gg
[
γ(0)gg + 6β0 + 14β0,Q
])
+
1
6ε2
(
γˆ(1)qg
[
2γ(0)qq − 2γ(0)gg
−8β0 − 10β0,Q
]
+ γˆ(0)qg
[
γˆ(1),PSqq {1− 2nf}+ γ(1),NSqq + γˆ(1),NSqq + 2γˆ(1)gg − γ(1)gg − 2β1
−2β1,Q
]
+ 6δm(−1)1 γˆ(0)qg
[
γ(0)gg − γ(0)qq + 3β0 + 5β0,Q
])
+
1
ε
(
γˆ(2)qg
3
− nf
ˆ˜γ(2)qg
3
+γˆ(0)qg
[
a(2)gg,Q − nfa
(2),PS
Qq
]
+ a(2)Qg
[
γ(0)qq − γ(0)gg − 4β0 − 4β0,Q
]
+
γˆ(0)qg ζ2
16
[
γ(0)gg
{
2γ(0)qq
−γ(0)gg − 6β0 + 2β0,Q
}
− (nf + 1)γ(0)gq γˆ(0)qg + γ(0)qq
{
−γ(0)qq + 6β0
}
− 8β20
+4β0,Qβ0 + 24β20,Q
]
+
δm(−1)1
2
[
−2γˆ(1)qg + 3δm
(−1)
1 γˆ(0)qg + 2δm
(0)
1 γˆ(0)qg
]
+δm(0)1 γˆ(0)qg
[
γ(0)gg − γ(0)qq + 2β0 + 4β0,Q
]
− δm(−1)2 γˆ(0)qg
)
+ a(3)Qg
]
. (4.116)
ˆˆA
(3)
qg,Q = nf
(mˆ2
µ2
)3ε/2
[
γˆ(0)qg
6ε3
(
γ(0)gq γˆ(0)qg + 2β0,Q
[
γ(0)gg − γ(0)qq + 2β0
])
+
1
ε2
(
γˆ(0)qg
6
[
2γˆ(1)gg
+γˆ(1),PSqq − 2γˆ(1),NSqq + 4β1,Q
]
+
γˆ(1)qg β0,Q
3
)
+
1
ε
(
ˆ˜γ(2)qg
3
+ γˆ(0)qg
[
a(2)gg,Q − a
(2),NS
qq,Q
+β(1)1,Q
]
− γˆ
(0)
qg ζ2
16
[
γ(0)gq γˆ(0)qg + 2β0,Q
{
γ(0)gg − γ(0)qq + 2β0
}])
+
a(3)qg,Q
nf
]
. (4.117)
63
The renormalized expressions are
A(3),MSQg =
γˆ(0)qg
48
{
(nf + 1)γ(0)gq γˆ(0)qg + γ(0)gg
(
γ(0)gg − 2γ(0)qq + 6β0 + 14β0,Q
)
+ γ(0)qq
(
γ(0)qq
−6β0 − 8β0,Q
)
+ 8β20 + 28β0,Qβ0 + 24β20,Q
}
ln3
(m2
µ2
)
+
1
8
{
γˆ(1)qg
(
γ(0)qq − γ(0)gg
−4β0 − 6β0,Q
)
+ γˆ(0)qg
(
γˆ(1)gg − γ(1)gg + (1− nf )γˆ(1),PSqq + γ(1),NSqq + γˆ(1),NSqq − 2β1
−2β1,Q
)}
ln2
(m2
µ2
)
+
{
γˆ(2)qg
2
− nf
ˆ˜γ(2)qg
2
+
a(2)Qg
2
(
γ(0)qq − γ(0)gg − 4β0 − 4β0,Q
)
+
γˆ(0)qg
2
(
a(2)gg,Q − nfa
(2),PS
Qq
)
+
γˆ(0)qg ζ2
16
(
−(nf + 1)γ(0)gq γˆ(0)qg + γ(0)gg
[
2γ(0)qq − γ(0)gg − 6β0
−6β0,Q
]
− 4β0[2β0 + 3β0,Q] + γ(0)qq
[
−γ(0)qq + 6β0 + 4β0,Q
])}
ln
(m2
µ2
)
+ a(2)Qg
(
γ(0)gg
−γ(0)qq + 4β0 + 4β0,Q
)
+ γˆ(0)qg
(
nfa(2),PSQq − a
(2)
gg,Q
)
+
γˆ(0)qg ζ3
48
(
(nf + 1)γ(0)gq γˆ(0)qg
+γ(0)gg
[
γ(0)gg − 2γ(0)qq + 6β0 − 2β0,Q
]
+ γ(0)qq
[
γ(0)qq − 6β0
]
+ 8β20 − 4β0β0,Q
−24β20,Q
)
+
γˆ(1)qg β0,Qζ2
8
+
γˆ(0)qg ζ2
16
(
γ(1)gg − γˆ(1),NSqq − γ(1),NSqq − γˆ(1),PSqq + 2β1
+2β1,Q
)
+
δm(−1)1
8
(
16a(2)Qg + γˆ(0)qg
[
−24δm(0)1 − 8δm
(1)
1 − ζ2β0 − 9ζ2β0,Q
])
+
δm(0)1
2
(
2γˆ(1)qg − δm
(0)
1 γˆ(0)qg
)
+ δm(1)1 γˆ(0)qg
(
γ(0)qq − γ(0)gg − 2β0 − 4β0,Q
)
+δm(0)2 γˆ(0)qg + a
(3)
Qg . (4.118)
A(3),MSqg,Q = nf
[
γˆ(0)qg
48
{
γ(0)gq γˆ(0)qg + 2β0,Q
(
γ(0)gg − γ(0)qq + 2β0
)}
ln3
(m2
µ2
)
+
1
8
{
2γˆ(1)qg β0,Q
+γˆ(0)qg
(
γˆ(1),PSqq − γˆ(1),NSqq + γˆ(1)gg + 2β1,Q
)}
ln2
(m2
µ2
)
+
1
2
{
ˆ˜γ(2)qg + γˆ(0)qg
(
a(2)gg,Q
−a(2),NSqq,Q + β
(1)
1,Q
)
− γˆ
(0)
qg
8
ζ2
(
γ(0)gq γˆ(0)qg + 2β0,Q
[
γ(0)gg − γ(0)qq + 2β0
])}
ln
(m2
µ2
)
+γˆ(0)qg
(
a(2),NSqq,Q − a
(2)
gg,Q − β
(2)
1,Q
)
+
γˆ(0)qg
48
ζ3
(
γ(0)gq γˆ(0)qg + 2β0,Q
[
γ(0)gg − γ(0)qq + 2β0
])
− ζ2
16
(
γˆ(0)qg γˆ(1),PSqq + 2γˆ(1)qg β0,Q
)
+
a(3)qg,Q
nf
]
. (4.119)
4.7.5 Agq,Q
The gq–contributions start at O(a2s),
Agq,Q = a2sA
(2)
gq,Q + a3sA
(3)
gq,Q +O(a4s) . (4.120)
64
The renormalization formulas in the MOM–scheme read
A(2),MOMgq,Q = Aˆ
(2),MOM
gq,Q + Z−1,(2)gq (nf + 1)− Z−1,(2)gq (nf )
+
(
Aˆ(1),MOMgg,Q + Z−1,(1)gg (nf + 1)− Z−1,(1)gg (nf )
)
Γ−1,(1)gq , (4.121)
A(3),MOMgq,Q = Aˆ
(3),MOM
gq,Q + Z−1,(3)gq (nf + 1)− Z−1,(3)gq (nf ) + Z−1,(1)gg (nf + 1)Aˆ
(2),MOM
gq,Q
+Z−1,(1)gq (nf + 1)Aˆ(2),MOMqq +
[
Aˆ(1),MOMgg,Q + Z−1,(1)gg (nf + 1)
−Z−1,(1)gg (nf )
]
Γ−1,(2)gq (nf ) +
[
Aˆ(2),MOMgq,Q + Z−1,(2)gq (nf + 1)
−Z−1,(2)gq (nf )
]
Γ−1,(1)qq (nf ) +
[
Aˆ(2),MOMgg,Q + Z−1,(2)gg (nf + 1)
−Z−1,(2)gg (nf ) + Z−1,(1)gg (nf + 1)Aˆ
(1),MOM
gg,Q
+Z−1,(1)gq (nf + 1)Aˆ
(1),MOM
Qg
]
Γ−1,(1)gq (nf ) , (4.122)
while the unrenormalized expressions are
ˆˆA
(2)
gq,Q =
(mˆ2
µ2
)ε
[
2β0,Q
ε2 γ
(0)
gq +
γˆ(1)gq
2ε + a
(2)
gq,Q + a
(2)
gq,Qε
]
, (4.123)
ˆˆA
(3)
gq,Q =
(mˆ2
µ2
)3ε/2
{
−γ
(0)
gq
3ε3
(
γ(0)gq γˆ(0)qg +
[
γ(0)qq − γ(0)gg + 10β0 + 24β0,Q
]
β0,Q
)
+
1
ε2
(
γ(1)gq β0,Q +
γˆ(1)gq
3
[
γ(0)gg − γ(0)qq − 4β0 − 6β0,Q
]
+
γ(0)gq
3
[
γˆ(1),NSqq + γˆ(1),PSqq − γˆ(1)gg
+2β1,Q
]
− 4δm(−1)1 β0,Qγ(0)gq
)
+
1
ε
(
γˆ(2)gq
3
+ a(2)gq,Q
[
γ(0)gg − γ(0)qq − 6β0,Q − 4β0
]
+γ(0)gq
[
a(2),NSqq,Q + a
(2),PS
Qq − a
(2)
gg,Q
]
+ γ(0)gq β
(1)
1,Q +
γ(0)gq ζ2
8
[
γ(0)gq γˆ(0)qg + β0,Q(γ(0)qq
−γ(0)gg + 10β0)
]
− δm(−1)1 γˆ(1)gq − 4δm
(0)
1 β0,Qγ(0)gq
)
+ a(3)gq,Q
}
. (4.124)
The contributions to the renormalized operator matrix element are given by
A(2),MSgq,Q =
β0,Qγ(0)gq
2
ln2
(m2
µ2
)
+
γˆ(1)gq
2
ln
(m2
µ2
)
+ a(2)gq,Q −
β0,Qγ(0)gq
2
ζ2 , (4.125)
A(3),MSgq,Q = −
γ(0)gq
24
{
γ(0)gq γˆ(0)qg +
(
γ(0)qq − γ(0)gg + 10β0 + 24β0,Q
)
β0,Q
}
ln3
(m2
µ2
)
+
1
8
{
6γ(1)gq β0,Q + γˆ(1)gq
(
γ(0)gg − γ(0)qq − 4β0 − 6β0,Q
)
+ γ(0)gq
(
γˆ(1),NSqq + γˆ(1),PSqq
−γˆ(1)gg + 2β1,Q
)}
ln2
(m2
µ2
)
+
1
8
{
4γˆ(2)gq + 4a
(2)
gq,Q
(
γ(0)gg − γ(0)qq − 4β0
−6β0,Q
)
+ 4γ(0)gq
(
a(2),NSqq,Q + a
(2),PS
Qq − a
(2)
gg,Q + β
(1)
1,Q
)
+ γ(0)gq ζ2
(
γ(0)gq γˆ(0)qg +
[
γ(0)qq
65
−γ(0)gg + 12β0,Q + 10β0
]
β0,Q
)}
ln
(m2
µ2
)
+ a(2)gq,Q
(
γ(0)qq − γ(0)gg + 4β0 + 6β0,Q
)
+γ(0)gq
(
a(2)gg,Q − a
(2),PS
Qq − a
(2),NS
qq,Q
)
− γ(0)gq β
(2)
1,Q −
γ(0)gq ζ3
24
(
γ(0)gq γˆ(0)qg +
[
γ(0)qq − γ(0)gg
+10β0
]
β0,Q
)
− 3γ
(1)
gq β0,Qζ2
8
+ 2δm(−1)1 a
(2)
gq,Q + δm
(0)
1 γˆ(1)gq + 4δm
(1)
1 β0,Qγ(0)gq + a
(3)
gq,Q .
(4.126)
4.7.6 Agg,Q
The gg–contributions start at O(a0s),
Agg,Q = 1 + asA(1)gg,Q + a2sA
(2)
gg,Q + a3sA
(3)
gg,Q +O(a4s) . (4.127)
The corresponding renormalization formulas read in the MOM–scheme
A(1),MOMgg,Q = Aˆ
(1),MOM
gg,Q + Z−1,(1)gg (nf + 1)− Z−1,(1)gg (nf ) , (4.128)
A(2),MOMgg,Q = Aˆ
(2),MOM
gg,Q + Z−1,(2)gg (nf + 1)− Z−1,(2)gg (nf )
+Z−1,(1)gg (nf + 1)Aˆ
(1),MOM
gg,Q + Z−1,(1)gq (nf + 1)Aˆ
(1),MOM
Qg
+
[
Aˆ(1),MOMgg,Q + Z−1,(1)gg (nf + 1)− Z−1,(1)gg (nf )
]
Γ−1,(1)gg (nf ) , (4.129)
A(3),MOMgg,Q = Aˆ
(3),MOM
gg,Q + Z−1,(3)gg (nf + 1)− Z−1,(3)gg (nf ) + Z−1,(2)gg (nf + 1)Aˆ
(1),MOM
gg,Q
+Z−1,(1)gg (nf + 1)Aˆ
(2),MOM
gg,Q + Z−1,(2)gq (nf + 1)Aˆ
(1),MOM
Qg
+Z−1,(1)gq (nf + 1)Aˆ
(2),MOM
Qg +
[
Aˆ(1),MOMgg,Q + Z−1,(1)gg (nf + 1)
−Z−1,(1)gg (nf )
]
Γ−1,(2)gg (nf ) +
[
Aˆ(2),MOMgg,Q + Z−1,(2)gg (nf + 1)
−Z−1,(2)gg (nf ) + Z−1,(1)gq (nf + 1)A
(1),MOM
Qg
+Z−1,(1)gg (nf + 1)A
(1),MOM
gg,Q
]
Γ−1,(1)gg (nf )
+
[
Aˆ(2),MOMgq,Q + Z−1,(2)gq (nf + 1)− Z−1,(2)gq (nf )
]
Γ−1,(1)qg (nf ) . (4.130)
The general structure of the unrenormalized 1–loop result is then given by
ˆˆA
(1)
gg,Q =
(mˆ2
µ2
)ε/2
(
γˆ(0)gg
ε + a
(1)
gg,Q + εa
(1)
gg,Q + ε2a
(1)
gg,Q
)
. (4.131)
One obtains
ˆˆA
(1)
gg,Q =
(mˆ2
µ2
)ε/2(
−2β0,Qε
)
exp
( ∞∑
i=2
ζi
i
(ε
2
)i)
. (4.132)
Using Eq. (4.132), the 2–loop term is given by
ˆˆA
(2)
gg,Q =
(mˆ2
µ2
)ε
[
1
2ε2
{
γ(0)gq γˆ(0)qg + 2β0,Q
(
γ(0)gg + 2β0 + 4β0,Q
)}
+
γˆ(1)gg + 4δm(−1)1 β0,Q
2ε
+a(2)gg,Q + 2δm
(0)
1 β0,Q + β20,Qζ2 + ε
[
a(2)gg,Q + 2δm
(1)
1 β0,Q +
β20,Qζ3
6
]]
. (4.133)
66
Again, we have made explicit one–particle reducible contributions and terms stemming
from mass renormalization in order to refer to the notation of Refs. [126, 129], cf. the
discussion below (4.112). The 3–loop contribution becomes
ˆˆA
(3)
gg,Q =
(mˆ2
µ2
)3ε/2
[
1
ε3
(
−γ
(0)
gq γˆ(0)qg
6
[
γ(0)gg − γ(0)qq + 6β0 + 4nfβ0,Q + 10β0,Q
]
−2γ
(0)
gg β0,Q
3
[
2β0 + 7β0,Q
]
− 4β0,Q
3
[
2β20 + 7β0,Qβ0 + 6β20,Q
])
+
1
ε2
(
γˆ(0)qg
6
[
γ(1)gq − (2nf − 1)γˆ(1)gq
]
+
γ(0)gq γˆ(1)qg
3
− γˆ
(1)
gg
3
[
4β0 + 7β0,Q
]
+
2β0,Q
3
[
γ(1)gg + β1 + β1,Q
]
+
2γ(0)gg β1,Q
3
+ δm(−1)1
[
−γˆ(0)qg γ(0)gq − 2β0,Qγ(0)gg
−10β20,Q − 6β0,Qβ0
])
+
1
ε
(
γˆ(2)gg
3
− 2(2β0 + 3β0,Q)a(2)gg,Q − nf γˆ(0)qg a
(2)
gq,Q
+γ(0)gq a
(2)
Qg + β
(1)
1,Qγ(0)gg +
γ(0)gq γˆ(0)qg ζ2
16
[
γ(0)gg − γ(0)qq + 2(2nf + 1)β0,Q + 6β0
]
+
β0,Qζ2
4
[
γ(0)gg {2β0 − β0,Q}+ 4β20 − 2β0,Qβ0 − 12β20,Q
]
+δm(−1)1
[
−3δm(−1)1 β0,Q − 2δm
(0)
1 β0,Q − γˆ(1)gg
]
+ δm(0)1
[
−γˆ(0)qg γ(0)gq
−2γ(0)gg β0,Q − 4β0,Qβ0 − 8β20,Q
]
+ 2δm(−1)2 β0,Q
)
+ a(3)gg,Q
]
. (4.134)
The renormalized results are
A(1),MSgg,Q = −β0,Q ln
(m2
µ2
)
, (4.135)
A(2),MSgg,Q =
1
8
{
2β0,Q
(
γ(0)gg + 2β0
)
+ γ(0)gq γˆ(0)qg + 8β20,Q
}
ln2
(m2
µ2
)
+
γˆ(1)gg
2
ln
(m2
µ2
)
−ζ2
8
[
2β0,Q
(
γ(0)gg + 2β0
)
+ γ(0)gq γˆ(0)qg
]
+ a(2)gg,Q , (4.136)
A(3),MSgg,Q =
1
48
{
γ(0)gq γˆ(0)qg
(
γ(0)qq − γ(0)gg − 6β0 − 4nfβ0,Q − 10β0,Q
)
− 4
(
γ(0)gg
[
2β0 + 7β0,Q
]
+4β20 + 14β0,Qβ0 + 12β20,Q
)
β0,Q
}
ln3
(m2
µ2
)
+
1
8
{
γˆ(0)qg
(
γ(1)gq + (1− nf )γˆ(1)gq
)
+γ(0)gq γˆ(1)qg + 4γ(1)gg β0,Q − 4γˆ(1)gg [β0 + 2β0,Q] + 4[β1 + β1,Q]β0,Q
+2γ(0)gg β1,Q
}
ln2
(m2
µ2
)
+
1
16
{
8γˆ(2)gg − 8nfa
(2)
gq,Qγˆ(0)qg − 16a
(2)
gg,Q(2β0 + 3β0,Q)
+8γ(0)gq a
(2)
Qg + 8γ(0)gg β
(1)
1,Q + γ(0)gq γˆ(0)qg ζ2
(
γ(0)gg − γ(0)qq + 6β0 + 4nfβ0,Q + 6β0,Q
)
+4β0,Qζ2
(
γ(0)gg + 2β0
)(
2β0 + 3β0,Q
)}
ln
(m2
µ2
)
+ 2(2β0 + 3β0,Q)a(2)gg,Q
67
+nf γˆ(0)qg a
(2)
gq,Q − γ(0)gq a
(2)
Qg − β
(2)
1,Qγ(0)gg +
γ(0)gq γˆ(0)qg ζ3
48
(
γ(0)qq − γ(0)gg − 2[2nf + 1]β0,Q
−6β0
)
+
β0,Qζ3
12
(
[β0,Q − 2β0]γ(0)gg + 2[β0 + 6β0,Q]β0,Q − 4β20
)
− γˆ
(0)
qg ζ2
16
(
γ(1)gq + γˆ(1)gq
)
+
β0,Qζ2
8
(
γˆ(1)gg − 2γ(1)gg − 2β1 − 2β1,Q
)
+
δm(−1)1
4
(
8a(2)gg,Q
+24δm(0)1 β0,Q + 8δm
(1)
1 β0,Q + ζ2β0,Qβ0 + 9ζ2β20,Q
)
+ δm(0)1
(
β0,Qδm(0)1 + γˆ(1)gg
)
+δm(1)1
(
γˆ(0)qg γ(0)gq + 2β0,Qγ(0)gg + 4β0,Qβ0 + 8β20,Q
)
− 2δm(0)2 β0,Q + a
(3)
gg,Q . (4.137)
68
5 Representation in Different Renormalization
Schemes
As outlined in Section 4, there are different obvious possibilities to choose a scheme for the
renormalization of the mass and the coupling constant. Concerning the coupling constant,
we intermediately worked in a MOM–scheme, which derives from the condition that the
external gluon lines have to be kept on–shell. In the end, we transformed back to the
MS–description via. Eq. (4.55), since this is the commonly used renormalization scheme.
If masses are involved, it is useful to renormalize them in the on–mass–shell–scheme, as
it was done in the previous Section. In this scheme, one defines the renormalized mass m
as the pole of the quark propagator. In this Section, we present the relations required to
transform the renormalized results from Section 4.7 into the different, related schemes. In
Section 5.1, we show how these scheme transformations affect the NLO results. Denoting
the MS–mass by m, there are in addition to the {aMS, m}–scheme adopted in Section 4.7
the following schemes
{
aMOMs , m
}
,
{
aMOMs , m
}
,
{
aMSs , m
}
. (5.1)
In case of mass renormalization in the MS–scheme, Eq. (4.27) becomes
mˆ = ZMSm m = m
[
1 + aˆsδm1 + aˆ2sδm2
]
+O(aˆ3s) . (5.2)
The corresponding coefficients read, [271],
δm1 =
6
εCF ≡
δm(−1)1
ε , (5.3)
δm2 =
CF
ε2 (18CF − 22CA + 8TF (nf + 1)) +
CF
ε
(
3
2
CF +
97
6
CA −
10
3
TF (nf + 1)
)
≡ δm
(−2)
2
ε2 +
δm(−1)2
ε . (5.4)
One notices that the following relations hold between the expansion coefficients in ε of
the on–shell– and MS–terms
δm(−1)1 = δm
(−1)
1 , (5.5)
δm(−2)2 = δm
(−2)
2 , (5.6)
δm(−1)2 = δm
(−1)
2 − δm
(−1)
1 δm
(0)
1 + 2δm
(0)
1 (β0 + β0,Q) . (5.7)
One has to be careful, since the choice of this scheme also affects the renormalization
constant of the coupling in the MOM–scheme. This is due to the fact that in Eq. (4.42)
mass renormalization had been performed in the on–shell–scheme. Going through the
same steps as in Eqs. (4.42)–(4.47), but using the MS–mass, we obtain for Zg in the
MOM–scheme.
ZMOMg 2(ε, nf + 1, µ2,m2) = 1 + aMOMs (µ2)
[2
ε (β0(nf ) + β0,Qf(ε))
]
+aMOMs 2(µ2)
[β1(nf )
ε +
4
ε2 (β0(nf ) + β0,Qf(ε))
2 +
2β0,Q
ε δm
(−1)
1 f(ε)
+
1
ε
(m2
µ2
)ε(
β1,Q + εβ
(1)
1,Q + ε2β
(2)
1,Q
)]
+O(aMOMs 3) , (5.8)
69
where in the term f(ε), cf. Eq. (4.44), the MS–mass has to be used. The coefficients
differing from the on–shell–scheme in the above equation are given by, cf. Eqs. (4.50, 4.51)
β1,Q = β1,Q − 2β0,Qδm(−1)1 , (5.9)
β(1)1,Q = β
(1)
1,Q − 2β0,Qδm
(0)
1 , (5.10)
β(2)1,Q = β
(2)
1,Q −
β0,Q
4
(
8δm(1)1 + δm
(−1)
1 ζ2
)
. (5.11)
The transformation formulas between the different schemes follow from the condition that
the unrenormalized terms are equal.
In order to transform from the {aMSs , m}–scheme to the {aMOMs , m}–scheme, the
inverse of Eq. (4.55)
aMSs (m2) = aMOMs
[
1 + β0,Q ln
(m2
µ2
)
aMOMs
+
{
β20,Q ln2
(m2
µ2
)
+ β1,Q ln
(m2
µ2
)
+ β(1)1,Q
}
aMOMs 2
]
(5.12)
is used. For the transformation to the {aMSs , m}–scheme one obtains
m(aMSs ) = m(aMSs )
(
1 +
{
−δm
(−1)
1
2
ln
(m2
µ2
)
− δm(0)1
}
aMSs
+
{
δm(−1)1
8
[
2β0 + 2β0,Q + δm(−1)1
]
ln2
(m2
µ2
)
+
1
2
[
−δm(0)1
(
2β0 + 2β0,Q
−3δm(−1)1
)
+ δm(−1)1
2 − 2δm(−1)2
]
ln
(m2
µ2
)
+ δm(1)1
[
δm(−1)1 − 2β0 − 2β0,Q
]
+δm(0)1
[
δm(−1)1 + δm
(0)
1
]
− δm(0)2
}
aMSs
2
)
. (5.13)
Finally, the transformation to the {aMOMs , m} is achieved via
aMSs (m2) = aMOMs
[
1 + β0,Q ln
(m2
µ2
)
aMOMs +
{
β20,Q ln2
(m2
µ2
)
+
(
β1,Q − β0,Qδm(−1)1
)
ln
(m2
µ2
)
+ β(1)1,Q − 2δm
(0)
1 β0,Q
}
aMOMs 2
]
, (5.14)
and
m(aMSs ) = m(aMOMs )
(
1 +
{
−δm
(−1)
1
2
ln
(m2
µ2
)
− δm(0)1
}
aMOMs
+
{
δm(−1)1
8
[
2β0 − 2β0,Q + δm(−1)1
]
ln2
(m2
µ2
)
+
1
2
[
−δm(0)1
(
2β0 + 4β0,Q
−3δm(−1)1
)
+ δm(−1)1
2 − 2δm(−1)2
]
ln
(m2
µ2
)
+ δm(1)1
[
δm(−1)1 − 2β0 − 2β0,Q
]
+δm(0)1
[
δm(−1)1 + δm
(0)
1
]
− δm(0)2
}
aMOMs 2
)
. (5.15)
70
The expressions for the OMEs in different schemes are then obtained by inserting the
relations (5.12)–(5.15) into the general expression (4.77) and expanding in the coupling
constant.
5.1 Scheme Dependence at NLO
Finally, we would like to comment on how the factorization formulas for the heavy flavor
Wilson coefficients, (3.26)–(3.30), have to be applied to obtain a complete description.
Here, the renormalization of the coupling constant has to be carried out in the same way
for all quantities contributing. The general factorization formula (3.16) holds only for
completely inclusive quantities, including radiative corrections containing heavy quark
loops, [129].
One has to distinguish one-particle irreducible and reducible diagrams, which both
contribute in the calculation. We would like to remind the reader of the background of
this aspect. If one evaluates the heavy-quark Wilson coefficients, diagrams of the type
shown in Figure 7 may appear as well. Diagram (a) contains a virtual heavy quark loop
correction to the gluon propagator in the initial state and contributes to the terms Lg,i
and Hg,i, respectively, depending on whether a light or heavy quark pair is produced in
the final state. Diagrams (b), (c) contribute to LNSq,i and contain radiative corrections to
the gluon propagator due to heavy quarks as well. The latter diagrams contribute to
F(2,L)(x,Q2) in the inclusive case, but are absent in the semi–inclusive QQ–production
cross section. The same holds for diagram (a) if a qq–pair is produced. In Refs. [103], the
Q(Q)
Q(q)
Q(q)
(a)
q(q)
Q(Q)
(b)
q(q)
Q(Q)
(c)
Figure 7: O(a2s) virtual heavy quark corrections.
coupling constant was renormalized in the MOM–scheme by absorbing the contributions
of diagram (a) into the coupling constant, as a consequence of which the term Lg,i appears
for the first time at O(a3s). This can be made explicit by considering the complete gluonic
Wilson coefficient up to O(a2s), including one heavy quark, cf. Eqs. (3.28, 3.30),
Cg,(2,L)(nf ) + Lg,(2,L)(nf + 1) +Hg,(2,L)(nf + 1) = aMSs
[
A(1),MSQg δ2 + C
(1)
g,(2,L)(nf + 1)
]
+aMSs
2[
A(2),MSQg δ2 + A
(1),MS
Qg C
(1),NS
q,(2,L)(nf + 1) +A
(1),MS
gg,Q C
(1)
g,(2,L)(nf + 1)
+C(2)g,(2,L)(nf + 1)
]
. (5.16)
71
The above equation is given in the MS–scheme, and the structure of the OMEs can be
inferred from Eqs. (4.113, 4.114). Here, diagram (a) gives a contribution, correspond-
ing exactly to the color factor T 2F . The transformation to the MOM–scheme for as, cf.
Eqs. (4.55, 4.56), yields
Cg,(2,L)(nf ) + Lg,(2,L)(nf + 1) +Hg,(2,L)(nf + 1) = aMOMs
[
A(1),MSQg δ2 + C
(1)
g,(2,L)(nf + 1)
]
+aMOMs 2
[
A(2),MSQg δ2 + β0,Q ln
(m2
µ2
)
A(1),MSQg δ2 + A
(1),MS
Qg C
(1),NS
q,(2,L)(nf + 1)
+A(1),MSgg,Q C
(1)
g,(2,L)(nf + 1) + β0,Q ln
(m2
µ2
)
C(1)g,(2,L)(nf + 1) + C
(2)
g,(2,L)(nf + 1)
]
. (5.17)
By using the general structure of the renormalized OMEs, Eqs. (4.113, 4.114, 4.135), one
notices that all contributions due to diagram (a) cancel in the MOM–scheme, i.e., the
color factor T 2F does not occur at the 2–loop level. Thus the factorization formula reads
Cg,(2,L)(nf ) + Lg,(2,L)(nf + 1) +Hg,(2,L)(nf + 1) =
aMOMs
[
A(1),MOMQg δ2 + C
(1)
g,(2,L)(nf + 1)
]
+aMOMs 2
[
A(2),MOMQg δ2 + A
(1),MOM
Qg C
(1),NS
q,(2,L)(nf + 1) + C
(2)
g,(2,L)(nf + 1)
]
. (5.18)
Splitting up Eq. (5.18) into Hg,i and Lg,i, one observes that Lg,i vanishes at O(a2s) and
the term Hg,i is the one calculated in Ref. [126]. This is the asymptotic expression of the
gluonic heavy flavor Wilson coefficient as calculated in Refs. [103]. Note that the observed
cancellation was due to the fact that the term A(1)gg,Q receives only contributions from the
heavy quark loops of the gluon–self energy, which also enters into the definition of the
MOM–scheme. It is not clear whether this can be achieved at the 3–loop level as well, i.e.,
transforming the general inclusive factorization formula (3.16) in such a way that only
the contributions due to heavy flavors in the final state remain. Therefore one should
use these asymptotic expressions only for completely inclusive analyzes, where heavy and
light flavors are treated together. This approach has also been adopted in Ref. [129] for
the renormalization of the massive OMEs, which was performed in the MS–scheme and
not in the MOM–scheme, as previously in Ref. [126]. The radiative corrections in the
NS–case can be treated in the same manner. Here the scheme transformation affects only
the light Wilson coefficients and not the OMEs at the 2–loop level. In the MS–scheme,
one obtains the following asymptotic expression up to O(a2s) from Eqs. (3.21, 3.26).
CNSq,(2,L)(nf ) + LNSq,(2,L)(nf + 1) =
1 + aMSs C
(1),NS
q,(2,L)(nf + 1) + aMSs
2[
A(2),NS,MSqq,Q (nf + 1) δ2 + C
(2),NS
q,(2,L)(nf + 1)
]
. (5.19)
Transformation to the MOM–scheme yields
CNSq,(2,L)(nf ) + LNSq,(2,L)(nf + 1) =
1 + aMOMs C
(1),NS
q,(2,L)(nf + 1) + aMOMs
2
[
A(2),NS,MOMqq,Q (nf + 1) δ2
+ β0,Q ln
(m2
µ2
)
C(1),NSq,(2,L)(nf + 1) + C
(2),NS
q,(2,L)(nf + 1)
]
. (5.20)
72
Note that A(2),NSqq,Q , Eq. (4.95), is not affected by this scheme transformation. As is obvious
from Figure 7, the logarithmic term in Eq. (5.20) can therefore only be attributed to the
massless Wilson coefficient. Separating the light from the heavy part one obtains
L(2),NS,MOMq,(2,L) (nf + 1) =
A(2),NS,MOMqq,Q (nf + 1) δ2 + β0,Q ln
(m2
µ2
)
C(1),NSq,(2,L)(nf + 1) + Cˆ
(2),NS
q,(2,L)(nf ) . (5.21)
This provides the same results as Eqs. (4.23)–(4.29) of Ref. [126]. These are the asymptotic
expressions of the NS heavy flavor Wilson coefficients from Refs. [103], where only the
case of QQ–production in the final state has been considered. Hence the logarithmic term
in Eq. (5.21) just cancels the contributions due to diagrams (b), (c) in Figure 7.
73
6 Calculation of the Massive Operator Matrix Ele-
ments up to O(a2sε)
The quarkonic 2–loop massive OMEs A(2)Qg, A
(2),PS
Qq and A
(2)
qq,Q have been calculated for the
first time in Ref. [126] to construct asymptotic expressions for the NLO heavy flavor Wil-
son Coefficients in the limit Q2 ≫ m2, cf. Section 3.2. The corresponding gluonic OMEs
A(2)gg,Q and A
(2)
gq,Q were calculated in Ref. [129], where they were used within a VFNS de-
scription of heavy flavors in high–energy scattering processes, see Section 3.3. In these
calculations, the integration–by–parts technique, [283], has been applied to reduce the
number of propagators occurring in the momentum integrals. Subsequently, the integrals
were calculated in z–space, which led to a variety of multiple integrals of logarithms, par-
tially with complicated arguments. The final results were given in terms of polylogarithms
and Nielsen–integrals, see Appendix C.4. The quarkonic terms have been confirmed in
Ref. [128], cf. also [284], where a different approach was followed. The calculation was
performed in Mellin–N space and by avoiding the integration–by–parts technique. Using
representations in terms of generalized hypergeometric functions, the integrals could be
expressed in terms of multiple finite and infinite sums with one free parameter, N . The
advantage of this approach is that the evaluation of these sums can be automatized us-
ing various techniques, simplifying the calculation. The final result is then obtained in
Mellin–space in terms of nested harmonic sums or Z–sums, cf. [142, 143] and Appendix
C.4. An additional simplification was found since the final result, e.g., for A(2)Qg can be
expressed in terms of two basic harmonic sums only, using algebraic, [146], and structural
relations, [147, 148], between them. This is another example of an observation which
has been made for many different single scale quantities in high–energy physics, namely
that the Mellin–space representation is better suited to the problem than the z–space
representation.
As has been outlined in Section 4, the O(ε)–terms of the unrenormalized 2–loop mas-
sive OMEs are needed in the renormalization of the 3–loop contributions. In this Section,
we calculate these terms based on the approach advocated in Ref. [128], which is a new
result, [130, 137]. Additionally, we re–calculate the gluonic OMEs up to the constant
term in ε for the first time, cf. [129,130]. Example diagrams for each OME are shown in
Figure 8.
In Section 6.1, we explain how the integrals are obtained in terms of finite and infinite
sums using representations in terms of generalized hypergeometric functions, cf. [285,286]
and Appendix C.2. For the calculation of these sums we mainly used the MATHEMATICA–
based program Sigma, [153, 154], which is discussed in Section 6.2. The results are pre-
sented in Section 6.3. Additionally, we make several remarks on the MOM–scheme, which
has to be adopted intermediately for the renormalization of the coupling constant, cf.
Section 4.4. In Section 6.4, different checks of the results are presented.
6.1 Representation in Terms of Hypergeometric Functions
All diagrams contributing to the massive OMEs are shown in Figures 1–4 in Ref. [126] and
in Figures 3,4 in Ref. [129], respectively. They represent 2–point functions with on–shell
external momentum p, p2 = 0. They are expressed in two parameters, the heavy quark
mass m and the Mellin–parameter N . Since the mass can be factored out of the integrals,
the problem effectively contains a single scale. The parameter N represents the spin of
74
(Qg) (gq,Q) (gg,Q) (NS) (Qq, PS)
Figure 8: Examples for 2–loop diagrams contributing to the massive OMEs. Thick lines:
heavy quarks, curly lines: gluons, full lines: quarks.
the composite operators, (2.86)–(2.88), and enters the calculation via the Feynman–rules
for these objects, cf. Appendix B.
Since the external momentum does not appear in the final result, the corresponding
scalar integrals reduce to massive tadpoles if one sets N = 0. In order to explain our
method, we consider first the massive 2–loop tadpole shown in Figure 9, from which all
OMEs can be derived at this order, by attaching 2 outer legs and inserting the composite
operator in all possible ways, i.e., both on the lines and on the vertices.
ν3ν2ν1
Figure 9: Basic 2–loop massive tadpole
In Figure 9, the wavy line is massless and the full lines are massive. Here νi labels
the power of the propagator. We adopt the convention νi...j ≡ νi + ... + νj etc. The
corresponding dimensionless momentum integral reads in Minkowski–space
I1 =
∫ ∫ dDk1dDk2
(4π)4D
(4π)4(−1)ν123−1(m2)ν123−D
(k21 −m2)ν1(k21 − k22)ν2(k22 −m2)ν3
, (6.1)
where we have attached a factor (4π)4(−1)ν123−1 for convenience. Using standard
Feynman–parametrization and Eq. (4.3) for momentum integration, one obtains the fol-
lowing Feynman–parameter integral
I1 = Γ
[
ν123 − 4− ε
ν1, ν2, ν3
]∫∫ 1
0
dxdy x
1+ε/2−ν2(1− x)ν23−3−ε/2yν3−1(1− y)ν12−3−ε/2
(4π)ε(1− xy)ν123−4−ε ,
(6.2)
which belongs to the class of the hypergeometric function 3F2 with argument z = 1, see
75
Appendix C.2. Applying Eq. (C.19), one obtains
I1 = S2ε exp
( ∞∑
i=2
ζi
i ε
i
)
Γ
[
ν123 − 4− ε, 2 + ε/2− ν2, ν23 − 2− ε/2, ν12 − 2− ε/2
1− ε, ν1, ν2, ν3, ν123 − 2− ε/2
]
× 3F2
[
ν123 − 4− ε, 2 + ε/2− ν2, ν3
ν3, ν123 − 2− ε/2
; 1
]
, (6.3)
where we have used Eq. (4.8). The term ν3 in the argument of the 3F2 cancels between
nominator and denominator and thus one can use Gauss’s theorem, Eq. (C.16), to write
the result in terms of Γ–functions
I1 = Γ
[
ν123 − 4− ε, 2 + ε/2− ν2, ν12 − 2− ε/2, ν23 − 2− ε/2
1− ε, 2 + ε/2, ν1, ν3, ν123 + ν2 − 4− ε
]
S2ε exp
( ∞∑
i=2
ζi
i ε
i
)
.
(6.4)
This calculation is of course trivial and Eq. (6.4) can be easily checked using MATAD,
cf. Ref. [164] and Section 7.2. Next, let us consider the case of arbitrary moments in
presence of the complete numerator structure. Since the final result contains the factor
(∆.p)N , one cannot set p to zero anymore. This increases the number of propagators and
hence the number of Feynman–parameters in Eq. (6.2). Additionally, the terms (∆.q)A
in the integral lead to polynomials in the Feynman–parameters to a symbolic power in
the integral, which can not be integrated trivially. Hence neither Eq. (C.19) nor Gauss’s
theorem can be applied anymore in the general case.
However, the structure of the integral in Eq. (6.2) does not change. For any diagram
deriving from the 2–loop tadpole, a general integral of the type
I2 = C2
∫∫ 1
0
dxdy x
a(1− x)byc(1− y)d
(1− xy)e
∫ 1
0
dz1...
∫ 1
0
dzi P
(
x, y, z1 . . . zi, N
)
(6.5)
is obtained. Here P is a rational function of x, y and possibly more parameters z1...zi.
N denotes the Mellin–parameter and occurs in some exponents. Note that operator
insertions with more than two legs give rise to additional finite sums in P, see Appendix
B. For fixed values of N , one can expand P and the integral I2 turns into a finite sum
over integrals of the type I1. The terms νi in these integrals might have been shifted by
integers, but after expanding in ε, the one–fold infinite sum can be performed, e.g., using
the FORM–based code Summer, [143].
To illustrate the sophistication occurring once one keeps the complete dependence on
N in an example, we consider the scalar integral contributing to A(2)Qg shown in Figure 8.
After momentum integration, it reads
I3 =
(∆p)N−2Γ(1− ε)
(4π)4+ε(m2)1−ε
∫∫∫∫
dudzdydx(1− u)
−ε/2z−ε/2(1− z)ε/2−1
(1− u+ uz)1−ε(x− y)[(
zyu+ x(1− zu)
)N−1
−
(
(1− u)x+ uy
)N−1
]
, (6.6)
where we have performed the finite sum already, which stems from the operator insertion.
Here and below, the Feynman-parameter integrals are carried out over the respective
76
unit-cube. This integral is of the type of Eq. (6.5) and the term x− y in the denominator
cancels for fixed values of N . Due to the operator insertion on an internal vertex, it
is one of the more involved integrals in the 2–loop case. For almost all other integrals,
all but two parameters can be integrated automatically, leaving only a single infinite
sum of the type of Eq. (6.3) with N appearing in the parameters of the hypergeometric
function, cf. e.g. [128, 284, 287]. In order to render this example calculable, suitable
variable transformations, as, e.g., given in Ref. [270], are applied, [128, 284]. Thus one
arrives at the following double sum
I3 =
S2ε (∆p)N−2
(4π)4(m2)1−ε exp
{ ∞∑
l=2
ζl
l ε
l
}
2π
N sin(π2 ε)
N∑
j=1
{(N
j
)
(−1)j + δj,N
}
×
{
Γ(j)Γ(j + 1− ε2)
Γ(j + 2− ε)Γ(j + 1 + ε2)
− B(1−
ε
2 , 1 + j)
j 3F2
[
1− ε, ε2 , j + 1
1, j + 2− ε2
; 1
]}
=
S2ε (∆p)N−2
(4π)4(m2)1−ε
{
I(0)3 + I
(1)
3 ε+O(ε2)
}
. (6.7)
Note that in our approach no expansion in ε is needed until a sum–representation of
the kind of Eq. (6.7) is obtained. Having performed the momentum integrations, the
expressions of almost all diagrams were given in terms of single generalized hypergeometric
series 3F2 at z = 1, with possibly additional finite summations. These infinite sums could
then be safely expanded in ε, leading to different kinds of sums depending on the Mellin–
parameter N . The summands are typically products of harmonic sums with different
arguments, weighted by summation parameters and contain hypergeometric terms 21,
like binomials or Beta–function factors B(N, i), cf. Eq. (C.9). Here i is a summation–
index. In the most difficult cases, double sums as in Eq. (6.7) or even triple sums were
obtained, which had to be treated accordingly. In general, these sums can be expressed in
terms of nested harmonic sums and ζ–values. Note that sums containing Beta–functions
with different arguments, e.g. B(i, i), B(N + i, i), usually do not lead to harmonic
sums in the final result. Some of these sums can be performed by the existing packages
[143, 149, 150]. However, there exists so far no automatic computer program to calculate
sums which contain Beta–function factors of the type B(N, i) and single harmonic sums
in the summand. These sums can be calculated applying analytic methods, as integral
representations, and general summation methods, as encoded in the Sigma package [151–
154]. In the next Section, we will present details on this.
Before finishing this Section, we give the result in terms of harmonic sums for the
double sum in Eq. (6.7) applying these summation methods. The O(ε0) of Eq. (6.7) is
needed for the constant term a(2)Qg, cf. Refs. [128,287]. The linear term in ε reads
I(1)3 =
1
N
[
−2S2,1 + 2S3 +
4N + 1
N S2 −
S21
N −
4
NS1
]
, (6.8)
where we adopt the notation to take harmonic sums at argument N , if not stated other-
wise.
21f(k) is hypergeometric in k iff f(k + 1)/f(k) = g(k) for some fixed rational function g(k).
77
6.2 Difference Equations and Infinite Summation
Single scale quantities in renormalizable quantum field theories are most simply repre-
sented in terms of nested harmonic sums, cf. [142, 143] and Appendix C.4, which holds
at least up to 3–loop order for massless Yang–Mills theories and for a wide class of
different processes. This includes the anomalous dimensions and massless Wilson co-
efficients for unpolarized and polarized space- and time-like processes to 3–loop order,
the Wilson coefficients for the Drell-Yan process and pseudoscalar and scalar Higgs–
boson production in hadron scattering in the heavy quark mass limit, as well as the
soft- and virtual corrections to Bhabha scattering in the on–mass–shell–scheme to 2–loop
order, cf. [95, 115, 124, 125, 138, 144, 145]. The corresponding Feynman–parameter inte-
grals are such that nested harmonic sums appear in a natural way, working in Mellin
space, [147, 148]. Single scale massive quantities at 2 loops, like the unpolarized and
polarized heavy-flavor Wilson coefficients in the region Q2 ≫ m2 as considered in this
thesis, belong also to this class, [126–128, 157, 160, 165, 287–289]. Finite harmonic sums
obey algebraic, cf. [146], and structural relations, [147], which can be used to obtain
simplified expressions and both shorten the calculations and yield compact final results.
These representations have to be mapped to momentum-fraction space to use the respec-
tive quantities in experimental analyzes. This is obtained by an Mellin inverse transform
which requires the analytic continuation of the harmonic sums w.r.t. the Mellin index
N ∈ C, [147,148,210].
Calculating the massive OMEs in Mellin space, new types of infinite sums occur if
compared to massless calculations. In the latter case, summation algorithms as Sum-
mer, [143], Nestedsums, [149], and Xsummer, [150], may be used to calculate the respec-
tive sums. Summer and Xsummer are based on FORM, while Nestedsums is based on
GiNaC, [290]. The new sums which emerge in [128,137,157,287–289] can be calculated in
different ways. In Ref. [128, 157], we chose analytic methods and in the former reference
all sums are given which are needed to calculate the constant term of the massive OMEs.
Few of these sums can be calculated using general theorems, as Gauss’ theorem, (C.16),
Dixon’s theorem, [285], or summation tables in the literature, cf. [143,291].
In order to calculate the gluonic OMEs as well as the O(ε)–terms, many new sums
had to be evaluated. For this we adopted a more systematic technique based on difference
equations, which are the discrete equivalent of differential equations, cf. [292]. This is a
promising approach, since it allowed us to obtain all sums needed automatically and it
may be applied to entirely different single–scale processes as well. It is based on applying
general summation algorithms in computer algebra. A first method is Gosper’s telescoping
algorithm, [293], for hypergeometric terms. For practical applications, Zeilberger’s exten-
sion of Gosper’s algorithm to creative telescoping, [294, 295], can be considered as the
breakthrough in symbolic summation. The recent summation package Sigma, [151–154],
written in MATHEMATICA opens up completely new possibilities in symbolic summation.
Based on Karr’s ΠΣ-difference fields, [296], and further refinements, [151,297], the pack-
age contains summation algorithms, [298], that allow to solve not only hypergeometric
sums, like Gosper’s and Zeilberger’s algorithms, but also sums involving indefinite nested
sums. In this algebraic setting, one can represent completely algorithmically indefinite
nested sums and products without introducing any algebraic relations between them.
Note that this general class of expressions covers as special cases the harmonic sums or
generalized nested harmonic sums, cf. [155, 299–301]. Given such an optimal representa-
tion, by introducing as less sums as possible, various summation principles are available
78
in Sigma. In this work, we applied the following strategy which has been generalized from
the hypergeometric case, [295,302], to the ΠΣ-field setting.
1. Given a definite sum that involves an extra parameter N , we compute a recurrence
relation in N that is fulfilled by the input sum. The underlying difference field
algorithms exploit Zeilberger’s creative telescoping principle, [295,302].
2. Then we solve the derived recurrence in terms of the so-called d’Alembertian solu-
tions, [295, 302]. Since this class covers the harmonic sums, we find all solutions in
terms of harmonic sums.
3. Taking the initial values of the original input sum, we can combine the solutions
found from step 2 in order to arrive at a closed representation in terms of harmonic
sums.
In the following, we give some examples on how Sigma works. A few typical sums we had
to calculate are listed in Appendix D and a complete set of sums needed to calculate the
2–Loop OMEs up to O(ε) can be found in Appendix B of Refs. [128, 137]. Note that in
this calculation also more well-known sums are occurring which can, e.g., be easily solved
using Summer.
6.2.1 The Sigma-Approach
As a first example we consider the sum
T1(N) ≡
∞∑
i=1
B(N, i)
i+N + 2S1(i)S1(N + i) . (6.9)
We treat the upper bound of the sum as a finite integer, i.e., we consider the truncated
version
T1(a,N) ≡
a∑
i=1
B(N, i)
i+N + 2S1(i)S1(N + i),
for a ∈ N. Given this sum as input, we apply Sigma’s creative telescoping algorithm and
find a recurrence for T1(a,N) of the form
c0(N)T (a,N) + . . . cd(N)T (a,N + d) = q(a,N) (6.10)
with order d = 4. Here, the ci(N) and q(a,N) are known functions of N and a. Finally,
we perform the limit a→∞ and we end up at the recurrence
−N(N + 1)(N + 2)2
{
4N5 + 68N4 + 455N3 + 1494N2 + 2402N + 1510
}
T1(N)
− (N + 1)(N + 2)(N + 3)
{
16N5 + 260N4 + 1660N3 + 5188N2 + 7912N + 4699
}
× T1(N + 1) + (N + 2)(N + 4)(2N + 5)
{
4N6 + 74N5 + 542N4 + 1978N3 + 3680N2
+ 3103N + 767
}
T1(N + 2) + (N + 4)(N + 5)
{
16N6 + 276N5 + 1928N4 + 6968N3
+ 13716N2 + 13929N + 5707
}
T1(N + 3)− (N + 4)(N + 5)2(N + 6)
{
4N5 + 48N4
+ 223N3 + 497N2 + 527N + 211
}
T1(N + 4) = P1(N) + P2(N)S1(N)
79
where
P1(N) =
(
32N18 + 1232N17 + 21512N16 + 223472N15 + 1514464N14 + 6806114N13
+ 18666770N12 + 15297623N11 − 116877645N10 − 641458913N9 − 1826931522N8
− 3507205291N7 − 4825457477N6 − 4839106893N5 − 3535231014N4
− 1860247616N3 − 684064448N2 − 160164480N − 17395200
)
/(
N3(N + 1)3(N + 2)3(N + 3)2(N + 4)(N + 5)
)
and
P2(N) = −4
(
(4N14 + 150N13 + 2610N12 + 27717N11 + 199197N10 + 1017704N9
+ 3786588N8 + 10355813N7 + 20779613N6 + 30225025N5 + 31132328N4
+ 21872237N3 + 9912442N2 + 2672360N + 362400
)
/(
N2(N + 1)2(N + 2)2(N + 3)(N + 4)(N + 5)
)
.
In the next step, we apply Sigma’s recurrence solver to the computed recurrence and find
the four linearly independent solutions
h1(N) =
1
N + 2 , h2(N) =
(−1)N
N(N + 1)(N + 2) ,
h3(N) =
S1(N)
N + 2 , h4(N) = (−1)
N
(
1 + (N + 1)S1(N)
)
N(N + 1)2(N + 2) ,
of the homogeneous version of the recurrence and the particular solution
p(N) = 2(−1)
N
N(N + 1)(N + 2)
[
2S−2,1(N)− 3S−3(N)− 2S−2(N)S1(N)− ζ2S1(N)
−ζ3 −
2S−2(N) + ζ2
N + 1
]
− 2S3(N)− ζ3N + 2 −
S2(N)− ζ2
N + 2 S1(N)
+
2 + 7N + 7N2 + 5N3 +N4
N3(N + 1)3(N + 2) S1(N) + 2
2 + 7N + 9N2 + 4N3 +N4
N4(N + 1)3(N + 2)
of the recurrence itself. Finally, we look for constants c1, . . . , c4 such that
T1(N) = c1 h1(N) + c2 h2(N) + c3 h3(N) + c4 h4(N) + p(N) .
The calculation of the necessary initial values for N = 0, 1, 2, 3 does not pose a problem
for Sigma and we conclude that c1 = c2 = c3 = c4 = 0. Hence the final result reads
T1(N) =
2(−1)N
N(N + 1)(N + 2)
[
2S−2,1(N)− 3S−3(N)− 2S−2(N)S1(N)− ζ2S1(N)
−ζ3 −
2S−2(N) + ζ2
N + 1
]
− 2S3(N)− ζ3N + 2 −
S2(N)− ζ2
N + 2 S1(N)
+
2 + 7N + 7N2 + 5N3 +N4
N3(N + 1)3(N + 2) S1(N) + 2
2 + 7N + 9N2 + 4N3 +N4
N4(N + 1)3(N + 2) .
(6.11)
80
Using more refined algorithms of Sigma, see e.g. [303], even a first order difference equation
can be obtained
(N + 2)T1(N)− (N + 3)T1(N + 1)
= 2
(−1)N
N(N + 2)
(
− 3N + 4
(N + 1)(N + 2)
(
ζ2 + 2S−2(N)
)
− 2ζ3 − 2S−3(N)− 2ζ2S1(N)
−4S1,−2(N)
)
+
N6 + 8N5 + 31N4 + 66N3 + 88N2 + 64N + 16
N3(N + 1)2(N + 2)3 S1(N)
+
S2(N)− ζ2
N + 1 + 2
N5 + 5N4 + 21N3 + 38N2 + 28N + 8
N4(N + 1)2(N + 2)2 . (6.12)
However, in deriving Eq. (6.12), use had to be made of further sums of less complexity,
which had to be calculated separately. As above, we can easily solve the recurrence and
obtain again the result (6.11). Here and in the following we applied various algebraic
relations between harmonic sums to obtain a simplification of our results, cf. [146].
6.2.2 Alternative Approaches
As a second example we consider the sum
T2(N) ≡
∞∑
i=1
S21(i+N)
i2 , (6.13)
which does not contain a Beta–function. In a first attempt, we proceed as in the first
example T1(N). The naive application of Sigma yields a fifth order difference equation,
which is clearly too complex for this sum. However, similar to the situation T1(N), Sigma
can reduce it to a third order relation which reads
T2(N)(N + 1)2 − T2(N + 1)(3N2 + 10N + 9)
+T2(N + 2)(3N2 + 14N + 17)− T2(N + 3)(N + 3)2
=
6N5 + 48N4 + 143N3 + 186N2 + 81N − 12
(N + 1)2(N + 2)3(N + 3)2 − 2
2N2 + 7N + 7
(N + 1)(N + 2)2(N + 3)S1(N)
+
−2N6 − 24N5 − 116N4 − 288N3 − 386N2 − 264N − 72
(N + 1)2(N + 2)3(N + 3)2 ζ2 . (6.14)
Solving this recurrence relation in terms of harmonic sums gives a closed form, see (6.20)
below. Still (6.14) represents a rather involved way to solve the problem. It is of advantage
to map the numerator S21(i+N) into a linear representation, which can be achieved using
Euler’s relation
S2a(N) = 2Sa,a(N)− S2a(N), a > 0 . (6.15)
This is realized in Summer by the basis–command for general–type harmonic sums,
T2(N) =
∞∑
i=1
2S1,1(i+N)− S2(i+N)
i2 . (6.16)
81
As outlined in Ref. [143], sums of this type can be evaluated by considering the difference
D2(j) = T2(j)− T2(j − 1) = 2
∞∑
i=1
S1(j + i)
i −
∞∑
i=1
1
i2(j + i)2 . (6.17)
The solution is then obtained by summing (6.17) to
T2(N) =
N∑
j=1
D2(j) + T2(0) . (6.18)
The sums in Eq. (6.17) are now calculable trivially or are of less complexity than the
original sum. In the case considered here, only the first sum on the left hand side is not
trivial. However, after partial fractioning, one can repeat the same procedure, resulting
into another difference equation, which is now easily solved. Thus using this technique, the
solution of Eq. (6.13) can be obtained by summing two first order difference equations
or solving a second order one. The above procedure is well known and some of the
summation–algorithms of Summer are based on it. As a consequence, infinite sums with
an arbitrary number of harmonic sums with the same argument can be performed using
this package. Note that sums containing harmonic sums with different arguments, see e.g
Eq. (6.21), can in principle be summed automatically using the same approach. However,
this feature is not yet built into Summer. A third way to obtain the sum (6.13) consists
of using integral representations for harmonic sums, [142]. One finds
T2(N) = 2
∞∑
i=1
∫ 1
0
dxx
i+N
i2
( ln(1− x)
1− x
)
+
−
∞∑
i=1
(∫ 1
0
dxx
i+N
i2
ln(x)
1− x +
ζ2
i2
)
= 2M
[( ln(1− x)
1− x
)
+
Li2(x)
]
(N + 1)
−
(
M
[ ln(x)
1− xLi2(x)
]
(N + 1) + ζ22
)
. (6.19)
Here the Mellin–transform is defined in Eq. (2.65). Eq. (6.19) can then be easily calculated
since the corresponding Mellin–transforms are well–known, [142]. Either of these three
methods above lead to
T2(N) =
17
10
ζ22 + 4S1(N)ζ3 + S21(N)ζ2 − S2(N)ζ2 − 2S1(N)S2,1(N)− S2,2(N) . (6.20)
As a third example we would like to evaluate the sum
T3(N) =
∞∑
i=1
S21(i+N)S1(i)
i . (6.21)
Note that (6.21) is divergent. In order to treat this divergence, the symbol σ1, cf.
Eq. (C.35), is used. The application of Sigma to this sum yields a fourth order difference
equation
(N + 1)2(N + 2)T2(N)− (N + 2)
(
4N2 + 15N + 15
)
T2(N + 1)
+(2N + 5)
(
3N2 + 15N + 20
)
T2(N + 2)− (N + 3)
(
4N2 + 25N + 40
)
T2(N + 3)
+(N + 3)(N + 4)2T2(N + 4)
=
6N5 + 73N4 + 329N3 + 684N2 + 645N + 215
(N + 1)2(N + 2)2(N + 3)2 +
6N2 + 19N + 9
(N + 1)(N + 2)(N + 3)S1(N) ,
(6.22)
82
which can be solved. As in the foregoing example the better way to calculate the sum is
to first change S21(i+N) into a linear basis representation
T3(N) =
∞∑
i=1
2S1,1(i+N)− S2(i+N)
i S1(i) . (6.23)
One may now calculate T3(N) using telescoping for the difference
D3(j) = T3(j)− T3(j − 1) = 2
∞∑
i=1
S1(i+ j)S1(i)
i(i+ j) −
∞∑
i=1
S1(i)
i(i+ j)2 , (6.24)
with
T3(N) =
N∑
j=1
D2(j) + T3(0) . (6.25)
One finally obtains
T3(N) =
σ41
4
+
43
20
ζ22 + 5S1(N)ζ3 +
3S21(N)− S2(N)
2
ζ2 − 2S1(N)S2,1(N)
+S21(N)S2(N) + S1(N)S3(N)−
S22(N)
4
+
S41(N)
4
. (6.26)
6.3 Results
For the singlet contributions, we leave out an overall factor
1 + (−1)N
2
(6.27)
in the following. This factor emerges naturally in our calculation and is due to the fact that
in the light–cone expansion, only even values of N contribute to F2 and FL, cf. Section 2.3.
Additionally, we do not choose a linear representation in terms of harmonic sums as was
done in Refs. [115,124,125], since these are non–minimal w.r.t. to the corresponding quasi–
shuffle algebra, [304]. Due to this a much smaller number of harmonic sums contributes.
Remainder terms can be expressed in polynomials Pi(N). Single harmonic sums with
negative index are expressed in terms of the function β(N + 1), cf. Appendix C.4. For
completeness, we also give all pole terms and the constant terms of the quarkonic OMEs.
The latter have been obtained before in Refs. [126,128]. The pole terms can be expressed
via the LO–, [46], and the fermionic parts of the NLO, [119–123], anomalous dimensions
and the 1–loop β–function, [39–41,275].
We first consider the matrix element A(2)Qg, which is the most complex of the 2–loop
OMEs. For the calculation we used the projector given in Eq. (4.22) and therefore have
to include diagrams with external ghost lines as well. The 1–loop result is straightforward
to calculate and has already been given in Eqs. (4.111, 4.113). As explained in Section 4,
we perform the calculation accounting for 1–particle reducible diagrams. Hence the 1–
loop massive gluon self–energy term, Eq. (4.84), contributes. The unrenormalized 2–loop
OME is then given in terms of 1–particle irreducible and reducible contributions by
ˆˆA
(2)
Qg =
ˆˆA
(2),irr
Qg −
ˆˆA
(1)
QgΠˆ
(1)
(
0, mˆ
2
µ2
)
. (6.28)
83
Using the techniques described in the previous Sections, the pole–terms predicted by
renormalization in Eq. (4.112) are obtained, which have been given in Refs. [126, 128]
before. Here, the contributing 1–loop anomalous dimensions are
γ(0)qq = 4CF
{
2S1 −
3N2 + 3N + 2
2N(N + 1)
}
, (6.29)
γˆ(0)qg = −8TF
N2 +N + 2
N(N + 1)(N + 2) , (6.30)
γ(0)gg = 8CA
{
S1 −
2(N2 +N + 1)
(N − 1)N(N + 1)(N + 2)
}
− 2β0 , (6.31)
and the 2–loop contribution reads
γˆ(1)qg = 8CFTF
{
2
N2 +N + 2
N(N + 1)(N + 2)
[
S2 − S21
]
+
4
N2S1 −
P1
N3(N + 1)3(N + 2)
}
+16CATF
{
N2 +N + 2
N(N + 1)(N + 2)
[
S2 + S21 − 2β′ − ζ2
]
− 4(2N + 3)S1
(N + 1)2(N + 2)2 −
P2
(N − 1)N3(N + 1)3(N + 2)3
}
, (6.32)
P1 = 5N6 + 15N5 + 36N4 + 51N3 + 25N2 + 8N + 4 ,
P2 = N9 + 6N8 + 15N7 + 25N6 + 36N5 + 85N4 + 128N3
+104N2 + 64N + 16 . (6.33)
These terms agree with the literature and provide a strong check on the calculation. The
constant term in ε in Eq. (4.112) is determined after mass renormalization, [126,128,284].
a(2)Qg = TFCF
{
4(N2 +N + 2)
3N(N + 1)(N + 2)
(
4S3 − 3S2S1 − S31 − 6S1ζ2
)
+ 4
3N + 2
N2(N + 2)S
2
1
+4
N4 + 17N3 + 17N2 − 5N − 2
N2(N + 1)2(N + 2) S2 + 2
(3N2 + 3N + 2)(N2 +N + 2)
N2(N + 1)2(N + 2) ζ2
+4
N4 −N3 − 20N2 − 10N − 4
N2(N + 1)2(N + 2) S1 +
2P3
N4(N + 1)4(N + 2)
}
+TFCA
{
2(N2 +N + 2)
3N(N + 1)(N + 2)
(
−24S−2,1 + 6β′′ + 16S3 − 24β′S1 + 18S2S1 + 2S31
−9ζ3
)
− 16 N
2 −N − 4
(N + 1)2(N + 2)2β
′ − 47N
5 + 21N4 + 13N3 + 21N2 + 18N + 16
(N − 1)N2(N + 1)2(N + 2)2 S2
−4N
3 + 8N2 + 11N + 2
N(N + 1)2(N + 2)2 S
2
1 − 8
N4 − 2N3 + 5N2 + 2N + 2
(N − 1)N2(N + 1)2(N + 2)ζ2
− 4P4N(N + 1)3(N + 2)3S1 +
4P5
(N − 1)N4(N + 1)4(N + 2)4
}
, (6.34)
84
where the polynomials in Eq. (6.34) are given by
P3 = 12N8 + 52N7 + 132N6 + 216N5 + 191N4 + 54N3 − 25N2
−20N − 4 , (6.35)
P4 = N6 + 8N5 + 23N4 + 54N3 + 94N2 + 72N + 8 , (6.36)
P5 = 2N12 + 20N11 + 86N10 + 192N9 + 199N8 −N7 − 297N6 − 495N5
−514N4 − 488N3 − 416N2 − 176N − 32 . (6.37)
The newly calculated O(ε) contribution to A(2)Qg, [137], reads after mass renormalization
a(2)Qg = TFCF
{
N2 +N + 2
N(N + 1)(N + 2)
(
16S2,1,1 − 8S3,1 − 8S2,1S1 + 3S4 −
4
3
S3S1 −
1
2
S22
−S2S21 −
1
6
S41 + 2ζ2S2 − 2ζ2S21 −
8
3
ζ3S1
)
− 8 N
2 − 3N − 2
N2(N + 1)(N + 2)S2,1
+
2
3
3N + 2
N2(N + 2)S
3
1 +
2
3
3N4 + 48N3 + 43N2 − 22N − 8
N2(N + 1)2(N + 2) S3 + 2
3N + 2
N2(N + 2)S2S1
+4
S1
N2 ζ2 +
2
3
(N2 +N + 2)(3N2 + 3N + 2)
N2(N + 1)2(N + 2) ζ3 +
P6
N3(N + 1)3(N + 2)S2
+
N4 − 5N3 − 32N2 − 18N − 4
N2(N + 1)2(N + 2) S
2
1 − 2
2N5 − 2N4 − 11N3 − 19N2 − 44N − 12
N2(N + 1)3(N + 2) S1
−5N
6 + 15N5 + 36N4 + 51N3 + 25N2 + 8N + 4
N3(N + 1)3(N + 2) ζ2 −
P7
N5(N + 1)5(N + 2)
}
+TFCA
{
N2 +N + 2
N(N + 1)(N + 2)
(
16S−2,1,1 − 4S2,1,1 − 8S−3,1 − 8S−2,2 − 4S3,1 −
2
3
β′′′
+9S4 − 16S−2,1S1 +
40
3
S1S3 + 4β′′S1 − 8β′S2 +
1
2
S22 − 8β′S21 + 5S21S2 +
1
6
S41
−10
3
S1ζ3 − 2S2ζ2 − 2S21ζ2 − 4β′ζ2 −
17
5
ζ22
)
− 8 N
2 +N − 1
(N + 1)2(N + 2)2 ζ2S1
+
4(N2 −N − 4)
(N + 1)2(N + 2)2
(
−4S−2,1 + β′′ − 4β′S1
)
− 2
3
N3 + 8N2 + 11N + 2
N(N + 1)2(N + 2)2 S
3
1
−16
3
N5 + 10N4 + 9N3 + 3N2 + 7N + 6
(N − 1)N2(N + 1)2(N + 2)2 S3 + 8
N4 + 2N3 + 7N2 + 22N + 20
(N + 1)3(N + 2)3 β
′
+2
3N3 − 12N2 − 27N − 2
N(N + 1)2(N + 2)2 S2S1 −
2
3
9N5 − 10N4 − 11N3 + 68N2 + 24N + 16
(N − 1)N2(N + 1)2(N + 2)2 ζ3
− P8S2
(N − 1)N3(N + 1)3(N + 2)3 −
P10S21
N(N + 1)3(N + 2)3 +
2P11S1
N(N + 1)4(N + 2)4
− 2P9ζ2
(N − 1)N3(N + 1)3(N + 2)2 −
2P12
(N − 1)N5(N + 1)5(N + 2)5
}
, (6.38)
with the polynomials
P6 = 3N6 + 30N5 + 15N4 − 64N3 − 56N2 − 20N − 8 , (6.39)
85
P7 = 24N10 + 136N9 + 395N8 + 704N7 + 739N6 + 407N5 + 87N4
+27N3 + 45N2 + 24N + 4 , (6.40)
P8 = N9 + 21N8 + 85N7 + 105N6 + 42N5 + 290N4 + 600N3 + 456N2
+256N + 64 , (6.41)
P9 = (N3 + 3N2 + 12N + 4)(N5 −N4 + 5N2 +N + 2) , (6.42)
P10 = N6 + 6N5 + 7N4 + 4N3 + 18N2 + 16N − 8 , (6.43)
P11 = 2N8 + 22N7 + 117N6 + 386N5 + 759N4 + 810N3 + 396N2
+72N + 32 , (6.44)
P12 = 4N15 + 50N14 + 267N13 + 765N12 + 1183N11 + 682N10 − 826N9
−1858N8 − 1116N7 + 457N6 + 1500N5 + 2268N4 + 2400N3
+1392N2 + 448N + 64 . (6.45)
Note that the terms ∝ ζ3 in Eq. (6.34) and ∝ ζ22 in Eq. (6.38) are only due to the
representation using the β(k)–functions and are absent in representations using harmonic
sums. The results for the individual diagrams contributing to A(2)Qg can be found up to
O(ε0) in Ref. [128] and at O(ε) in Ref. [137].
Since harmonic sums appear in a wide variety of applications, it is interesting to study
the pattern in which they emerge. In Table 1, we list the harmonic sums contributing to
each individual diagram 22. The β–function and their derivatives can be traced back to
Table 1: Complexity of the results for the individual diagrams contributing to A(2)Qg
Diagram S1 S2 S3 S4 S−2 S−3 S−4 S2,1 S−2,1 S−2,2 S3,1 S−3,1 S2,1,1 S−2,1,1
A + +
B + + + + + + +
C + +
D + + + +
E + + + +
F + + + + + +
G + + + +
H + + + +
I + + + + + + + + + + + + + +
J + +
K + +
L + + + + + + +
M + +
N + + + + + + + + + + + + + +
O + + + + + + +
P + + + + + + +
S + +
T + +
the single non–alternating harmonic sums, allowing for half-integer arguments, cf. [142]
and Appendix C.4. Therefore, all single harmonic sums form an equivalence class being
represented by the sum S1, from which the other single harmonic sums are easily derived
through differentiation and half-integer relations Additionally, we have already made use
of the algebraic relations, [146], between harmonic sums in deriving Eqs. (6.34, 6.38).
Moreover, the sums S−2,2 and S3,1 obey structural relations to other harmonic sums,
i.e., they lie in corresponding equivalence classes and may be obtained by either rational
argument relations and/or differentiation w.r.t. N . Reference to these equivalence classes
is useful since the representation of these sums for N ǫ C needs not to be derived newly,
22Cf. Ref. [126] for the labeling of the diagrams.
86
except of straightforward differentiations. All functions involved are meromorphic, with
poles at the non–negative integers. Thus the O(ε0)–term depends on two basic functions
only, S1 and S−2,1 23. This has to be compared to the z–space representation used in
Ref. [126], in which 48 different functions were needed. As shown in [142], various of
these functions have Mellin transforms containing triple sums, which do not occur in
our approach even on the level of individual diagrams. Thus the method applied here
allowed to compactify the representation of the heavy flavor matrix elements and Wilson
coefficients significantly.
The O(ε)–term consists of 6 basic functions only, which are given by
{S1, S2, S3, S4, S−2, S−3, S−4}, S2,1, S−2,1, S−3,1, S2,1,1, S−2,1,1 , (6.46)
S−2,2 : depends on S−2,1, S−3,1
S3,1 : depends on S2,1 .
The absence of harmonic sums containing {−1} as index was noted before for all other
classes of space– and time–like anomalous dimensions and Wilson coefficients, including
those for other hard processes having been calculated so far, cf. [95,144,145]. This can not
be seen if one applies the z–space representation or the linear representation in Mellin–
space, [109].
Analytic continuation, e.g., for S−2,1 proceeds via the equality,
M
[
Li2(x)
1 + x
]
(N + 1)− ζ2β(N + 1) = (−1)N+1
[
S−2,1(N) +
5
8
ζ3
]
(6.47)
with similar representations for the remaining sums, [142] 24.
As discussed in [127], the result for a(2)Qg agrees with that in z–space given in Ref. [126].
However, there is a difference concerning the complete renormalized expression for A(2)Qg.
This is due to the scheme–dependence for the renormalization of the coupling constant,
which has been described in Sections 4.4, 5.1 and emerges for the first time at O(a2s).
Comparing Eq. (4.114) for the renormalized result in the MS–scheme for the coupling
constant with the transformation formula to the MOM–scheme, Eq. (5.12), this difference
is given by
A(2),MSQg = A
(2),MOM
Qg − β0,Q
γˆ(0)qg
2
ln2
(m2
µ2
)
. (6.48)
As an example, the second moment of the massive OME up to 2–loops reads in the
MS–scheme for coupling constant renormalization
AMSQg = aMSs
{
−4
3
TF ln
(m2
µ2
)}
+ aMSs
2
{
TF
[22
9
CA −
16
9
CF −
16
9
TF
]
ln2
(m2
µ2
)
+ TF
[
−70
27
CA −
148
27
CF
]
ln
(m2
µ2
)
− 7
9
CATF +
1352
81
CFTF
}
, (6.49)
23The associated Mellin transform to this sum has been discussed in Ref. [121] first.
24Note that the argument of the Mellin-transform in Eq. (36), Ref. [127], should read (N + 1).
87
and in the MOM–scheme
AMOMQg = aMOMs
{
−4
3
TF ln
(m2
µ2
)}
+ aMOMs 2
{
TF
[22
9
CA −
16
9
CF
]
ln2
(m2
µ2
)
+ TF
[
−70
27
CA −
148
27
CF
]
ln
(m2
µ2
)
− 7
9
CATF +
1352
81
CFTF
}
. (6.50)
As one infers from the above formulas, this difference affects at the 2–loop level only
the double logarithmic term and stems from the treatment of the 1–particle–reducible
contributions. In Ref. [126], these contributions were absorbed into the coupling constant,
applying the MOM–scheme. This was motivated by the need to eliminate the virtual
contributions due to heavier quarks (b, t) and was also extended to the charm–quark,
thus adopting the same renormalization scheme as has been used in Refs. [103] for the
exact calculation of the heavy flavor contributions to the Wilson coefficients. Contrary,
in Ref. [129], the MS–description was applied and the strong coupling constant depends
on nf + 1 flavors, cf. the discussion in Section 5.1.
The remaining massive OMEs are less complex than the term A(2)Qg and depend only
on single harmonic sums, i.e. on only one basic function, S1. In the PS–case, the LO and
NLO anomalous dimensions
γ(0)gq = −4CF
N2 +N + 2
(N − 1)N(N + 1) , (6.51)
γˆ(1),PSqq = −16CFTF
5N5 + 32N4 + 49N3 + 38N2 + 28N + 8
(N − 1)N3(N + 1)3(N + 2)2 (6.52)
contribute. The pole–terms are given by Eq. (4.100) and we obtain for the higher order
terms in ε
a(2),PSQq = CFTF
{
− 4(N
2 +N + 2)2 (2S2 + ζ2)
(N − 1)N2(N + 1)2(N + 2) +
4P13
(N − 1)N4(N + 1)4(N + 2)3
}
,
(6.53)
P13 = N10 + 8N9 + 29N8 + 49N7 − 11N6 − 131N5 − 161N4
−160N3 − 168N2 − 80N − 16 , (6.54)
a(2),PSQq = CFTF
{
−2(5N
3 + 7N2 + 4N + 4)(N2 + 5N + 2)
(N − 1)N3(N + 1)3(N + 2)2 (2S2 + ζ2)
− 4(N
2 +N + 2)2 (3S3 + ζ3)
3(N − 1)N2(N + 1)2(N + 2) +
2P14
(N − 1)N5(N + 1)5(N + 2)4
}
, (6.55)
P14 = 5N11 + 62N10 + 252N9 + 374N8 − 400N6 + 38N7 − 473N5
−682N4 − 904N3 − 592N2 − 208N − 32 . (6.56)
Since the PS–OME emerges for the first time at O(a2s), there is no difference between its
representation in the MOM– and the MS–scheme. The renormalized OME A(2)PSQq is given
88
in Eq. (4.101) and the second moment reads
APS,MSQq = aMSs
2
{
−16
9
ln2
(m2
µ2
)
− 80
27
ln
(m2
µ2
)
− 4
}
CFTF +O(aMSs
3
) . (6.57)
The flavor non-singlet NLO anomalous dimension is given by
γˆ(1),NSqq =
4CFTF
3
{
8S2 −
40
3
S1 +
3N4 + 6N3 + 47N2 + 20N − 12
3N2(N + 1)2
}
. (6.58)
The unrenormalized OME is obtained from the 1–particle irreducible graphs and the
contributions of heavy quark loops to the quark self–energy. The latter is given at O(aˆ2s)
in Eq. (4.87). One obtains
ˆˆA
(2),NS
qq,Q =
ˆˆA
(2),NS,irred
qq,Q − Σˆ(2)(0,
mˆ2
µ2 ) . (6.59)
Our result is of the structure given in Eq. (4.93) and the higher order terms in ε read
a(2),NSqq,Q =
CFTF
3
{
−8S3 − 8ζ2S1 +
40
3
S2 + 2
3N2 + 3N + 2
N(N + 1) ζ2 −
224
9
S1
+
219N6 + 657N5 + 1193N4 + 763N3 − 40N2 − 48N + 72
18N3(N + 1)3
}
, (6.60)
a(2),NSqq,Q =
CFTF
3
{
4S4 + 4S2ζ2 −
8
3
S1ζ3 +
112
9
S2 +
3N4 + 6N3 + 47N2 + 20N − 12
6N2(N + 1)2 ζ2
−20
3
S1ζ2 −
20
3
S3 −
656
27
S1 + 2
3N2 + 3N + 2
3N(N + 1) ζ3 +
P15
216N4(N + 1)4
}
, (6.61)
P15 = 1551N8 + 6204N7 + 15338N6 + 17868N5 + 8319N4
+944N3 + 528N2 − 144N − 432 . (6.62)
The anomalous dimensions in Eqs. (6.51, 6.52, 6.58) agree with the literature.
Eqs. (6.53, 6.60), cf. Ref. [128], were first given in Ref. [126] and agree with the re-
sults presented there. Eqs. (6.55, 6.61), [137], are new results of this thesis. As in the PS
case, the NS OME emerges for the first time at O(a2s). The corresponding renormalized
OME A(2),NSqq,Q is given in Eq. (4.95) and the second moment reads
ANS,MSqq,Q = aMSs
2
{
−16
9
ln2
(m2
µ2
)
− 128
27
ln
(m2
µ2
)
− 128
27
}
CFTF +O(aMSs
3
) . (6.63)
Note that the first moment of the NS–OME vanishes, even on the unrenormalized level
up to O(ε). This provides a check on the results in Eqs. (6.60, 6.61), because this is
required by fermion number conservation.
At this point an additional comment on the difference between the MOM and the
MS–scheme is in order. The MOM–scheme was applied in Ref. [126] for two different
89
purposes. The first one is described below Eq. (6.50). It was introduced to absorb the
contributions of one–particle reducible diagrams and heavier quarks into the definition
of the coupling constant. However, in case of A(2)Qg, renormalization in the MOM–scheme
and the scheme transformation from the MOM–scheme to the MS–scheme accidentally
commute. This means, that one could apply Eq. (4.110) in the MS–scheme, i.e., set
δaMOMs,1 = δaMSs,1 (nf + 1) (6.64)
from the start and obtain Eq. (4.114) for the renormalized result. This is not the case
for A(2),NSqq,Q . As mentioned earlier, the scheme transformation does not have an effect on
this term at 2–loop order. This means that Eq. (4.91) should yield the same renormalized
result in the MOM– and in the MS–scheme. However, in the latter case, the difference of
Z–factors does not contain the mass. Thus a term
∝ 1ε ln
(m2
µ2
)
, (6.65)
which stems from the expansion of the unrenormalized result in Eq. (4.93), can not be
subtracted. The reason for this is the following. As pointed out in Ref. [126], the term
Aˆ(2),NSqq,Q is only UV–divergent. However, this is only the case if one imposes the condition
that the heavy quark contributions to the gluon self–energy vanishes for on–shell momen-
tum of the gluon. This is exactly the condition we imposed for renormalization in the
MOM–scheme, cf. Section 4.4. Hence in this case, the additional divergences absorbed
into the coupling are of the collinear type, contrary to the term in A(2)Qg. By applying the
transformation back to the MS–scheme, we treat these two different terms in a concise
way. This is especially important at the three–loop level, since in this case both effects
are observed for all OMEs and the renormalization would not be possible if not applying
the MOM–scheme first.
Let us now turn to the gluonic OMEs A(2)gg,Q, A
(2)
gq,Q, which are not needed for the
asymptotic 2–loop heavy flavor Wilson coefficients. They contribute, however, in the
VFNS–description of heavy flavor parton densities, cf. Ref. [129] and Section 3.3. The
1–loop term A(1)gg,Q has already been given in Eqs. (4.132, 4.135). In case of A
(2)
gg,Q, the
part
γˆ(1)gg = 8CFTF
N8 + 4N7 + 8N6 + 6N5 − 3N4 − 22N3 − 10N2 − 8N − 8
(N − 1)N3(N + 1)3(N + 2)
+
32CATF
9
{
−5S1 +
3N6 + 9N5 + 22N4 + 29N3 + 41N2 + 28N + 6
(N − 1)N2(N + 1)2(N + 2)
}
(6.66)
of the 2–loop anomalous dimension is additionally needed. As for A(2)Qg, the massive parts
of the gluon self–energy contribute, Eqs. (4.84, 4.85). The unrenormalized OME at the
2–loop level is then given in terms of reducible and irreducible contributions via
ˆˆA
(2)
gg,Q =
ˆˆA
(2),irred
gg,Q −
ˆˆA
(1)
gg,QΠˆ
(1)
(
0, mˆ
2
µ2
)
− Πˆ(2)
(
0, mˆ
2
µ2
)
. (6.67)
90
In the unrenormalized result, we observe the same pole structure as predicted in
Eq. (4.133). The constant and O(ε) contributions a(2)gg,Q and a
(2)
gg,Q are
a(2)gg,Q = TFCA
{
−8
3
ζ2S1 +
16(N2 +N + 1)ζ2
3(N − 1)N(N + 1)(N + 2) − 4
56N + 47
27(N + 1)S1
+
2P16
27(N − 1)N3(N + 1)3(N + 2)
}
+TFCF
{
4(N2 +N + 2)2ζ2
(N − 1)N2(N + 1)2(N + 2) −
P17
(N − 1)N4(N + 1)4(N + 2)
}
, (6.68)
a(2)gg,Q = TFCA
{
−8
9
ζ3S1 −
20
9
ζ2S1 +
16(N2 +N + 1)
9(N − 1)N(N + 1)(N + 2)ζ3 +
2N + 1
3(N + 1)S2
− S
2
1
3(N + 1) − 2
328N4 + 256N3 − 247N2 − 175N + 54
81(N − 1)N(N + 1)2 S1
+
4P18ζ2
9(N − 1)N2(N + 1)2(N + 2) +
P19
81(N − 1)N4(N + 1)4(N + 2)
}
+TFCF
{
4(N2 +N + 2)2ζ3
3(N − 1)N2(N + 1)2(N + 2) +
P20ζ2
(N − 1)N3(N + 1)3(N + 2)
+
P21
4(N − 1)N5(N + 1)5(N + 2)
}
, (6.69)
P16 = 15N8 + 60N7 + 572N6 + 1470N5 + 2135N4
+1794N3 + 722N2 − 24N − 72 , (6.70)
P17 = 15N10 + 75N9 + 112N8 + 14N7 − 61N6 + 107N5 + 170N4 + 36N3
−36N2 − 32N − 16 , (6.71)
P18 = 3N6 + 9N5 + 22N4 + 29N3 + 41N2 + 28N + 6 , (6.72)
P19 = 3N10 + 15N9 + 3316N8 + 12778N7 + 22951N6 + 23815N5 + 14212N4
+3556N3 − 30N2 + 288N + 216 , (6.73)
P20 = N8 + 4N7 + 8N6 + 6N5 − 3N4 − 22N3 − 10N2 − 8N − 8 , (6.74)
P21 = 31N12 + 186N11 + 435N10 + 438N9 − 123N8 − 1170N7 − 1527N6
−654N5 + 88N4 − 136N2 − 96N − 32 . (6.75)
We agree with the result for a(2)gg,Q given in [129], which is presented in Eq. (6.68). The
new term a(2)gg,Q, Eq. (6.69), contributes to all OMEs A
(3)
ij through renormalization. The
renormalized OME is then given by Eq. (4.136). Since this OME already emerges at LO,
the O(a2s) term changes replacing the MOM– by the MS–scheme. The second moment in
the MS–scheme reads
AMSgg,Q = aMSs
{
4
3
TF ln
(m2
µ2
)}
+ aMSs
2
{
TF
[
−22
9
CA +
16
9
CF +
16
9
TF
]
ln2
(m2
µ2
)
91
+ TF
[70
27
CA +
148
27
CF
]
ln
(m2
µ2
)
+
7
9
CATF −
1352
81
CFTF
}
+O(aMSs
3
) . (6.76)
In the MOM–scheme it is given by
AMOMgg,Q = aMOMs
{
4
3
TF ln
(m2
µ2
)}
+ aMOMs 2
{
TF
[
−22
9
CA +
16
9
CF
]
ln2
(m2
µ2
)
+ TF
[70
27
CA +
148
27
CF
]
ln
(m2
µ2
)
+
7
9
CATF −
1352
81
CFTF
}
+O(aMOMs 3) .(6.77)
The difference between the schemes reads
A(2),MSgg,Q = A
(2),MOM
gg,Q + β20,Q ln2
(m2
µ2
)
. (6.78)
The need for applying intermediately the MOM–scheme for renormalization becomes ob-
vious again for the term A(2)gg,Q. As in the NS–case, renormalization in the MS–scheme
for the coupling constant does not cancel all singularities. The remaining term is A(2)gq,Q,
which emerges for the first time at O(a2s) and the same result is obtained in the MS– and
MOM–schemes. The corresponding NLO anomalous dimension is given by
γˆ(1)gq =
32CFTF
3
{
− (N
2 +N + 2)S1
(N − 1)N(N + 1) +
8N3 + 13N2 + 27N + 16
3(N − 1)N(N + 1)2
}
. (6.79)
Again, we obtain the pole terms as predicted in Eq. (4.123). The constant and O(ε)
contributions a(2)gq,Q and a
(2)
gq,Q then read
a(2)gq,Q = TFCF
{
4
3
N2 +N + 2
(N − 1)N(N + 1)
(
2ζ2 + S2 + S21
)
−8
9
8N3 + 13N2 + 27N + 16
(N − 1)N(N + 1)2 S1 +
8
27
P22
(N − 1)N(N + 1)3
}
, (6.80)
a(2)gq,Q = TFCF
{
2
9
N2 +N + 2
(N − 1)N(N + 1)
(
−2S3 − 3S2S1 − S31 + 4ζ3 − 6ζ2S1
)
+
2
9
8N3 + 13N2 + 27N + 16
(N − 1)N(N + 1)2
(
2ζ2 + S2 + S21
)
− 4
27
P22S1
(N − 1)N(N + 1)3
+
4
81
P23
(N − 1)N(N + 1)4
}
, (6.81)
with
P22 = 43N4 + 105N3 + 224N2 + 230N + 86 (6.82)
P23 = 248N5 + 863N4 + 1927N3 + 2582N2 + 1820N + 496 . (6.83)
The second moment of the renormalized result, cf. Eq. (4.125), reads
AMSgq,Q = aMSs
2
{
32
9
ln2
(m2
µ2
)
+
208
27
ln
(m2
µ2
)
+
236
27
}
CFTF +O(aMSs
3
) . (6.84)
92
We agree with the result for a(2)gq,Q given in [129], which is presented in (6.80).
Let us summarize so far. In this Section, we newly calculated the O(ε) terms of the
2–loop massive OMEs. We additionally recalculated for the first time the terms a(2)gg,Q,
Eq. (6.68), and a(2)gq,Q, Eq. (6.80), which were given in Ref. [129] and find full agreement.
For completeness, we showed as well the terms a(2),NSqq,Q , a
(2),PS
Qq and a
(2)
Qg, which have been
calculated for the first time in Ref. [126] and were recalculated in Refs. [128, 284]. The
latter terms contribute to the heavy flavor Wilson coefficients in deeply inelastic scattering
to the non power-suppressed contributions at O(a2s). In the renormalization of the heavy
flavor Wilson coefficients to 3–loop order, all these terms contribute together with lower
order single pole terms. The O(a2sε) contributions form parts of the constant terms of the
3–loop heavy flavor unpolarized operator matrix elements needed to describe the 3–loop
heavy flavor Wilson coefficients in the region Q2 ≫ m2.
The mathematical structure of our results is as follows. The terms a(2)ij can be ex-
pressed in terms of polynomials of the basic nested harmonic sums up to weight w = 4
and derivatives thereof. They belong to the complexity-class of the general two-loop
Wilson coefficients or hard scattering cross sections in massless QED and QCD and are
described by six basic functions and their derivatives in Mellin space. Their analytic con-
tinuation to complex values of N is known in explicit form. The package Sigma, [151–154],
proved to be a useful tool to solve the sums occurring in the present problem and was
extended accordingly by its author.
6.4 Checks on the Calculation
There are several checks which we can use for our results. First of all, the terms up
to O(ε0) have been calculated in Refs. [126, 129] and we agree with all unrenormalized
results. As described in Sections 6.1, 6.2, we keep the complete ε–dependence until we
expand the summand of the finite or infinite sums, which serves as a consistency check
on the O(ε) results.
Another test is provided by the sum rules in Eqs. (3.37, 3.38) for N = 2, which are
fulfilled by the renormalized OMEs presented here and in Refs. [126, 129]. These rules
are obeyed regardless of the renormalization scheme. We observe that they hold on the
unrenormalized level as well, even up to O(ε).
For the term A(2)Qg, we evaluated fixed moments of N for the contributing unrenormal-
ized diagrams using the Mellin0-Barnes method, [305–307], cf. also Appendix C.3. Here,
we used an extension of a method developed for massless propagators in Ref. [308] to
massive on–shell operator matrix elements, [158, 287, 289]. The Mellin–Barnes integrals
are then evaluated numerically using the package MB, [309]. Using this method, we cal-
culated the even moments N = 2, 4, 6, 8 and agree with the corresponding fixed moments
of our all–N result 25.
For the first moment of the Abelian part of the unrenormalized term ˆˆA
(2)
Qg, there exists
even another check. After analytic continuation from the even values of N to N ǫ C is
performed, one may consider the limit N → 1. In this procedure the term (1+ (−1)N)/2
equals to 1. At O(a2s) the terms ∝ TFCA contain 1/z contributions in momentum fraction
space and their first moment diverges. For the other contributions to the unrenormalized
25In Table 2 of Ref. [137], the moments N = 2 and N = 6 for the more difficult two–loop diagrams are
presented.
93
operator matrix element, after mass renormalization to 2–loop order, the first moment
is related to the Abelian part of the transverse contribution to the gluon propagator
ΠV (p2,m2)|p2=0, except the term ∝ T 2F which results from wave function renormalization.
This was shown in [126] up to the constant term in ε. One obtains
ΠˆV (p2,m2) = aˆsTF Πˆ(1)V (p2,m2) + aˆ2sCFTF Πˆ
(2)
V (p2,m2) +O(aˆ3s) , (6.85)
with
lim
p2→0
Πˆ(1)V (p2,m2) =
1
2
ˆˆA
(1),N=1
Qg (6.86)
lim
p2→0
Πˆ(2)V (p2,m2) =
1
2
ˆˆA
(2),N=1
Qg |CF . (6.87)
Here, we extend the relation to the linear terms in ε. For the first moment the double
pole contributions in ε vanish in Eq. (6.87). We compare with the corresponding QED–
expression for the photon–propagator, ΠV,(k)T , which has been obtained in Ref. [310]. Due
to the transition from QED to QCD, the relative color factor at the 2–loop level has to
be adjusted to 1/4 = 1/(CFCA). After asymptotic expansion in m2/p2, the comparison
can be performed up to the linear term in ε. One obtains
lim
p2→0
1
p2 Πˆ
V,(1)
T (p2,m2) =
1
2TF
ˆˆA
(1),N=1
Qg = −
(m2
µ2
)ε/2 [ 8
3ε +
ε
3
ζ2
]
(6.88)
lim
p2→0
1
p2 Πˆ
V,(2)
T (p2,m2) =
1
2TFCF
ˆˆA
(2),N=1
Qg |CF =
(m2
µ2
)ε [
−4ε + 15−
(
31
4
+ ζ2
)
ε
]
.
(6.89)
Additionally, we notice that the renormalized results do not anymore contain ζ2–terms.
The renormalized terms in Eqs. (4.95, 4.101, 4.114, 4.125, 4.136) contain expressions pro-
portional to ζ2 in the non–logarithmic contributions, which just cancel the corresponding
ζ2–terms in a(2)ij , cf. Eqs. (6.34, 6.53, 6.60, 6.68, 6.80). For explicit examples of this
cancellation, one may compare the second moments of the renormalized OMEs presented
in Eqs. (6.49, 6.50, 6.57, 6.63, 6.76, 6.77, 6.84). The latter provides no stringent test,
but is in accordance with general observations made in higher loop calculations, namely
that even ζ–values cancel for massless calculations in even dimensions in the renormalized
results if presented in the MS–scheme, [311]. In the present work, this observation holds
for the ζ2–terms in a single–scale massive calculation as well.
The most powerful test is provided by the FORM–based program MATAD, [164], which
we used to calculate fixed moments of the 2–loop OMEs up to O(ε). The setup is the same
as in the 3–loop case and is explained in the next Section. At the 2–loop level we worked
in general Rξ–gauges and explicitly observe the cancellation of the gauge parameter. For
the terms A(2)Qg, A
(2)
gg we used both projection operators given in Eqs. (4.22, 4.23), which
serves as another consistency check. In the singlet case, we calculated the even moments
N = 2, 4, ..., 12 and found full agreement with the results presented in this Section up to
O(ε). The same holds in the non–singlet case, where we calculated the odd moments as
well, N = 1, 2, 3, ..., 12.
94
7 Calculation of Moments at O(a3s)
In this Chapter, we describe the computation of the 3–loop corrections to the massive
operator matrix elements in detail, cf. [134]. Typical Feynman diagrams contributing
for the different processes are shown in Figure 10, where ⊗ denotes the corresponding
composite operator insertions, cf. Appendix B. The generation of these diagrams with
the FORTRAN–based program QGRAF, [161], is described in Section 7.1 along with the
subsequent steps to prepare the input for the FORM–based program MATAD, [164]. The
latter allows the calculation of massive tadpole integrals in D dimensions up to three loops
and relies on the MINCER algorithm, [312, 313]. The use of MATAD and the projection
onto fixed moments are explained in Section 7.2. Finally, we present our results for the
fixed moments of the 3–loop OMEs and the fermionic contributions to the anomalous
dimensions in Section 7.3. The calculation is mainly performed using FORM programs
while in a few cases codes have also been written in MAPLE.
(NS) (PSH) (PSl) (qgH) (qgl) (gq) (gg) ghost
Figure 10: Examples for 3–loop diagrams contributing to the massive operator matrix ele-
ments: NS - non–singlet, PSH,l - pure–singlet, singlet qgH,l, gq, gg and ghost contributions.
Here the coupling of the gauge boson to a heavy or light fermion line is labeled by H and l,
respectively. Thick lines: heavy quarks, curly lines: gluons, full lines: quarks, dashed lines:
ghosts.
7.1 Generation of Diagrams
QGRAF is a quite general program to generate Feynman diagrams and allows to spec-
ify various kinds of particles and interactions. Our main issue is to generate diagrams
which contain composite operator insertions, cf. (2.86)–(2.88) and Appendix B, as spe-
cial vertices. To give an example, let us consider the contributions to A(1)Qg. Within the
light–cone expansion, Section 2.3, this term derives from the Born diagrams squared of
the photon–gluon fusion process shown in Figure 11, cf. Section 3.1 and Figure 6.
Figure 11: Diagrams contributing to H(1)g,(2,L) via the optical theorem. Wavy lines: photons;
curly lines: gluons; full lines: quarks.
95
After expanding these diagrams with respect to the virtuality of the photon, the mass
effects are given by the diagrams in Figure 12. These are obtained by contracting the
lines between the external photons.
Figure 12: Diagrams contributing to A(1)Qg.
Thus, one may think of the operator insertion as being coupled to two external particles,
an incoming and an outgoing one, which carry the same momentum. Therefore, one
defines in the model file of QGRAF vertices which resemble the operator insertions in this
manner, using a scalar field φ, which shall not propagate in order to ensure that there is
only one of these vertices for each diagram. For the quarkonic operators, one defines the
vertices
φ+ φ+ q + q + n g , 0 ≤ n ≤ 3 , (7.1)
which is illustrated in Figure 13.
....
︸ ︷︷ ︸
n
=⇒
....
︸ ︷︷ ︸
n
φ , p2 φ , p2
Figure 13: Generation of the operator insertion.
The same procedure can be used for the purely gluonic interactions and one defines in
this case
φ+ φ+ n g , 0 ≤ n ≤ 4 . (7.2)
The Green’s functions we have to consider and their relation to the respective OMEs
were given in Eqs. (4.18)–(4.21). The number of diagrams we obtain contributing to each
OME is shown in Table 7.1.
Term # Term # Term # Term #
A(3)Qg 1358 A
(3)
qg,Q 140 A
(3),PS
Qq 125 A
(3),PS
qq,Q 8
A(3),NSqq,Q 129 A
(3)
gq,Q 89 A
(3)
gg,Q 886
Table 2: Number of diagrams contributing to the 3–loop massive OMEs.
96
The next step consists in rewriting the output provided by QGRAF in such a way, that the
Feynman rules given in Appendix B can be inserted. Thus, one has to introduce Lorentz
and color indices and align the fermion lines. Additionally, the integration momenta
have to be written in such a way that MATAD can handle them. For the latter step, all
information on the types of particles, the operator insertion and the external momentum
are irrelevant, leading to only two basic topologies to be considered at the 2–loop level,
which are shown in Figure 14.
p1
p3
p2
p1 p2
Figure 14: 2–Loop topologies for MATAD, indicating labeling of momenta.
Note, that in the case at hand the topology on the right–hand side of Figure 14 always
yields zero after integration. At the 3–loop level, the master topology is given in Figure 15.
p2 p6
p4
p3
p5
p1
Figure 15: Master 3–loop topology for MATAD, indicating labeling of momenta.
From this topology, five types of diagrams are derived by shrinking various lines. These
diagrams are shown in Figure 16. Finally the projectors given in Eqs. (4.22, 4.24) are
applied to project onto the scalar massive OMEs. We only use the physical projector (4.23)
as a check for lower moments, since it causes a significant increase of the computation
time. To calculate the color factor of each diagram, we use the program provided in
Ref. [163] and for the calculation of fermion traces we use FORM. Up to this point,
all operations have been performed for general values of Mellin N and the dimensional
parameter ε. The integrals do not contain any Lorentz or color indices anymore. In order
to use MATAD, one now has to assign to N a specific value. Additionally, the unphysical
momentum ∆ has to be replaced by a suitable projector, which we define in the following
Section.
97
p2 p1
p3
p5 p6 p2 p1
p3
p3
p6
p2 p6
p3
p3
p1 p5 p2
p6
p3
p5
p2
p3
p3
p6
Figure 16: Additional 3–loop topologies for MATAD.
7.2 Calculation of Fixed 3–Loop Moments Using MATAD
We consider integrals of the type
Il(p,m, n1 . . . nj) ≡
∫ dDk1
(2π)D . . .
∫ dDkl
(2π)D (∆.q1)
n1 . . . (∆.qj)njf(k1 . . . kl, p,m) .
(7.3)
Here p denotes the external momentum, p2 = 0, m is the heavy quark mass, and ∆ is a
light–like vector, ∆2 = 0. The momenta qi are given by any linear combination of the
loop momenta ki and external momentum p. The exponents ni are integers or possibly
sums of integers, see the Feynman rules in Appendix B. Their sum is given by
j∑
i=1
ni = N . (7.4)
The function f in Eq. (7.3) contains propagators, of which at least one is massive, dot-
products of its arguments and powers of m. If one sets N = 0, (7.3) is given by
Il(p,m, 0 . . . 0) = Il(m) =
∫ dDk1
(2π)D . . .
∫ dDkl
(2π)D f(k1 . . . kl,m) . (7.5)
From p2 = 0 it follows, that the result can not depend on p anymore. The above integral
is a massive tadpole integral and thus of the type MATAD can process. Additionally,
MATAD can calculate the integral up to a given order as a power series in p2/m2. Let us
return to the general integral given in Eq. (7.3). One notes, that for fixed moments of
N , each integral of this type splits up into one or more integrals of the same type with
the ni having fixed integer values. At this point, it is useful to recall that the auxiliary
vector ∆ has only been introduced to get rid of the trace terms of the expectation values
of the composite operators and has no physical significance. By undoing the contraction
98
with ∆, these trace terms appear again. Consider as an example
Il(p,m, 2, 1) =
∫ dDk1
(2π)D . . .
∫ dDkl
(2π)D (∆.q1)
2(∆.q2)f(k1 . . . kl, p,m) (7.6)
= ∆µ1∆µ2∆µ3
∫ dDk1
(2π)D . . .
∫ dDkl
(2π)D q1,µ1q1,µ2q2,µ3f(k1 . . . kl, p,m) .
(7.7)
One notices that the way of distributing the indices in Eq. (7.7) is somewhat arbitrary,
since after the contraction with the totally symmetric tensor ∆µ1∆µ2∆µ3 only the com-
pletely symmetric part of the corresponding tensor integral contributes. This is made
explicit by distributing the indices among the qi in all possible ways and dividing by the
number of permutations one has used. Thus Eq. (7.7) is written as
Il(p,m, 2, 1) = ∆µ1∆µ2∆µ3
1
3
∫ dDk1
(2π)D . . .
∫ dDkl
(2π)D (q1,µ2q1,µ3q2,µ1 + q1,µ1q1,µ3q2,µ2
+q1,µ1q1,µ2q2,µ3)f(k1 . . . kl, p,m) . (7.8)
Generally speaking, the symmetrization of the tensor resulting from
j∏
i=1
(∆.q1)ni (7.9)
can be achieved by shuffling indices, [142, 143, 146, 155, 301, 314], and dividing by the
number of terms. The shuffle product is given by
C

(k1, . . . , k1)︸ ︷︷ ︸
n1
⊔⊔ (k2, . . . , k2)︸ ︷︷ ︸
n2
⊔⊔ . . . ⊔⊔ (kI , . . . , kI)︸ ︷︷ ︸
nI

 , (7.10)
where C is the normalization constant
C =
( N
n1, . . . , nI
)−1
. (7.11)
As an example, the symmetrization of
q1,µ1q1,µ2q2,µ3 (7.12)
can be inferred from Eq. (7.8). After undoing the contraction with ∆ in (7.3) and shuffling
the indices, one may make the following ansatz for the result of this integral, which follows
from the necessity of complete symmetry in the Lorentz indices
R{µ1...µN} ≡
[N/2]+1∑
j=1
Aj
(j−1∏
k=1
g{µ2kµ2k−1
)( N∏
l=2j−1
pµl}
)
. (7.13)
In the above equation, [ ] denotes the Gauss–bracket and {} symmetrization with respect
to the indices enclosed and dividing by the number of terms, as outlined above. The first
99
few terms are then given by
R0 ≡ 1 , (7.14)
R{µ1} = A1pµ1 , (7.15)
R{µ1µ2} = A1pµ1pµ2 + A2gµ1µ2 , (7.16)
R{µ1µ2µ3} = A1pµ1pµ2pµ3 + A2g{µ1µ2pµ3} . (7.17)
The scalars Aj have in general different mass dimensions. By contracting again with ∆,
all trace terms vanish and one obtains
Il(p,m, n1 . . . nj) = ∆µ1 . . .∆µNR{µ1...µN} (7.18)
= A1(∆.p)N (7.19)
and thus the coefficient A1 in Eq. (7.13) gives the desired result. To obtain it, one
constructs a different projector, which is made up only of the external momentum p and
the metric tensor. By making a general ansatz for this projector, applying it to Eq. (7.13)
and demanding that the result shall be equal to A1, the coefficients of the different Lorentz
structures can be determined. The projector reads
Πµ1...µN = F (N)
[N/2]+1∑
i=1
C(i, N)
([N/2]−i+1∏
l=1
gµ2l−1µ2l
p2
)( N∏
k=2[N/2]−2i+3
pµk
p2
)
. (7.20)
For the overall pre-factors F (N) and the coefficients C(i, N), one has to distinguish be-
tween even and odd values of N ,
Codd(k,N) = (−1)N/2+k+1/22
2k−N/2−3/2Γ(N + 1)Γ(D/2 +N/2 + k − 3/2)
Γ(N/2− k + 3/2)Γ(2k)Γ(D/2 +N/2− 1/2) ,
(7.21)
F odd(N) = 2
3/2−N/2Γ(D/2 + 1/2)
(D − 1)Γ(N/2 +D/2− 1) , (7.22)
Ceven(k,N) = (−1)N/2+k+1 2
2k−N/2−2Γ(N + 1)Γ(D/2 +N/2− 2 + k)
Γ(N/2− k + 2)Γ(2k − 1)Γ(D/2 +N/2− 1) , (7.23)
F even(N) = 2
1−N/2Γ(D/2 + 1/2)
(D − 1)Γ(N/2 +D/2− 1/2) . (7.24)
The projector obeys the normalization condition
Πµ1...µNRµ1...µN = A1 , (7.25)
which implies
Πµ1...µNpµ1 . . . pµN = 1 . (7.26)
(7.27)
As an example for the above procedure, we consider the case N = 3,
Πµ1µ2µ3 =
1
D − 1
(
−3gµ1µ2pµ3p4 + (D + 2)
pµ1pµ2pµ3
p6
)
. (7.28)
100
Applying this term to (7.8) yields
Il(p,m, 2, 1) =
1
(D − 1)p6
∫ dDk1
(2π)D . . .
∫ dDkl
(2π)D
(
−2p2q1.q2p.q1
−p2q21p.q2 + (D + 2)(q1.p)2q2.p
)
f(k1 . . . kl, p,m) . (7.29)
Up to 3–loop integrals of the type (7.29) can be calculated by MATAD as a Taylor series
in p2/m2. It is important to keep p artificially off–shell until the end of the calculation.
By construction, the overall result will not contain any term ∝ 1/p2, since the integral
one starts with is free of such terms. Thus, at the end, these terms have to cancel. The
remaining constant term in p2 is the desired result.
The above projectors are similar to the harmonic projectors used in the MINCER–
program, cf. [313, 315]. These are, however, applied to the virtual forward Compton–
amplitude to determine the anomalous dimensions and the moments of the massless Wil-
son coefficients up to 3–loop order.
The calculation was performed in Feynman gauge in general. Part of the calculation
was carried out keeping the gauge parameter in Rξ–gauges, in particular for the moments
N = 2, 4 in the singlet case and for N = 1, 2, 3, 4 in the non–singlet case, yielding
agreement with the results being obtained using Feynman–gauge. In addition, for the
moments N = 2, 4 in the terms with external gluons, we applied the physical projector
in Eq. (4.23), which serves as another verification of our results. The computation of the
more complicated diagrams was performed on various 32/64 Gb machines using FORM
and for part of the calculation TFORM, [316], was used. The complete calculation required
about 250 CPU days.
7.3 Results
We calculated the unrenormalized operator matrix elements treating the 1PI-contributions
explicitly. They contribute to A(3)Qg, A
(3)
gg,Q and A
(3),NS
qq,Q . One obtains the following represen-
tations
ˆˆA
(3)
Qg =
ˆˆA
(3),irr
Qg −
ˆˆA
(2),irr
Qg Πˆ
(1)
(
0, mˆ
2
µ2
)
− ˆˆA
(1)
QgΠˆ
(2)
(
0, mˆ
2
µ2
)
+ ˆˆA
(1)
QgΠˆ
(1)
(
0, mˆ
2
µ2
)
Πˆ(1)
(
0, mˆ
2
µ2
)
, (7.30)
ˆˆA
(3)
gg,Q =
ˆˆA
(3),irr
gg,Q − Πˆ(3)
(
0, mˆ
2
µ2
)
− ˆˆA
(2),irr
gg,Q Πˆ
(1)
(
0, mˆ
2
µ2
)
−2 ˆˆA
(1)
gg,QΠˆ
(2)
(
0, mˆ
2
µ2
)
+ ˆˆA
(1)
gg,QΠˆ
(1)
(
0, mˆ
2
µ2
)
Πˆ(1)
(
0, mˆ
2
µ2
)
, (7.31)
ˆˆA
(3),NS
qq,Q =
ˆˆA
(3),NS,irr
qq,Q − Σˆ(3)
(
0, mˆ
2
µ2
)
. (7.32)
The self-energies are given in Eqs. (4.84, 4.85, 4.86, 4.88). The calculation of the one-
particle irreducible 3–loop contributions is performed as described in the previous Sec-
tion 26. The amount of moments, which could be calculated, depended on the available
26Partial results of the calculation were presented in [131,132].
101
computer resources w.r.t. memory and computational time, as well as the possible par-
allelization using TFORM. Increasing the Mellin moment from N → N + 2 demands
both a factor of 6–8 larger memory and CPU time. We have calculated the even mo-
ments N = 2, . . . , 10 for A(3)Qg, A
(3)
gg,Q, and A
(3)
qg,Q, for A
(3),PS
Qq up to N = 12, and for
A(3),NSqq,Q , A
(3),PS
qq,Q , A
(3)
gq,Q up to N = 14. In the NS–case, we also calculated the odd moments
N = 1, . . . , 13, which correspond to the NS−–terms.
(i) Anomalous Dimensions :
The pole terms of the unrenormalized OMEs emerging in the calculation agree with
the general structure we presented in Eqs. (4.94, 4.103, 4.104, 4.116, 4.117, 4.124, 4.134).
Using lower order renormalization coefficients and the constant terms of the 2–loop results,
[126,128–130], allows to determine the fixed moments of the 2–loop anomalous dimensions
and the contributions ∝ TF of the 3–loop anomalous dimensions, cf. Appendix E. All
our results agree with the results of Refs. [111, 112, 124, 125, 317, 318]. The anomalous
dimensions γ(2)qg and γ(2),PSqq are obtained completely. The present calculation is fully inde-
pendent both in the algorithms and codes compared to Refs. [111, 112, 124, 125, 318] and
thus provides a stringent check on these results.
(ii) The constant terms a(3)ij (N):
The constant terms at O(a3s), cf. Eqs. (4.94, 4.103, 4.104, 4.116, 4.117, 4.124, 4.134),
are the new contributions to the non–logarithmic part of the 3–loop massive operator
matrix elements, which can not be constructed by other renormalization constants cal-
culated previously. They are given in Appendix F. All other contributions to the heavy
flavor Wilson coefficients in the region Q2 ≫ m2 are known for general values of N , cf.
Sections 4.7 and 6. The functions a(3)ij (N) still contain coefficients ∝ ζ2 and we will see
below, under which circumstances these terms will contribute to the heavy flavor contri-
butions to the deep–inelastic structure functions. The constant B4, (4.89), emerges as in
other massive single–scale calculations, [279–282].
(iii) Moments of the Constant Terms of the 3–loop Massive OMEs
The logarithmic terms of the renormalized 3–loop massive OMEs are determined by
known renormalization constants and lower order contributions to the massive OMEs.
They can be inferred from Eqs. (4.96, 4.105, 4.106, 4.118, 4.119, 4.126, 4.137). In the
following, we consider as examples the non–logarithmic contributions to the second mo-
ments of the renormalized massive OMEs. We refer to coupling constant renormalization
in the MS–scheme and compare the results performing the mass renormalization in the
on–shell–scheme (m) and the MS–scheme (m), cf. Section 5. For the matrix elements
with external gluons, we obtain :
A(3),MSQg (µ2 = m2, 2) = TFC2A
(
174055
4374
− 88
9
B4 + 72ζ4 −
29431
324
ζ3
)
+TFCFCA
(
−18002
729
+
208
9
B4 − 104ζ4 +
2186
9
ζ3 −
64
3
ζ2 + 64ζ2 ln(2)
)
102
+TFC2F
(
−8879
729
− 64
9
B4 + 32ζ4 −
701
81
ζ3 + 80ζ2 − 128ζ2 ln(2)
)
+T 2FCA
(
−21586
2187
+
3605
162
ζ3
)
+ T 2FCF
(
−55672
729
+
889
81
ζ3 +
128
3
ζ2
)
+nfT 2FCA
(
−7054
2187
− 704
81
ζ3
)
+ nfT 2FCF
(
−22526
729
+
1024
81
ζ3 −
64
3
ζ2
)
. (7.33)
A(3),MSQg (µ2 = m2, 2) = TFC2A
(
174055
4374
− 88
9
B4 + 72ζ4 −
29431
324
ζ3
)
+TFCFCA
(
−123113
729
+
208
9
B4 − 104ζ4 +
2330
9
ζ3
)
+ TFC2F
(
−8042
729
− 64
9
B4
+32ζ4 −
3293
81
ζ3
)
+ T 2FCA
(
−21586
2187
+
3605
162
ζ3
)
+ T 2FCF
(
−9340
729
+
889
81
ζ3
)
+nfT 2FCA
(
−7054
2187
− 704
81
ζ3
)
+ nfT 2FCF
(
478
729
+
1024
81
ζ3
)
. (7.34)
A(3),MSqg,Q (µ2 = m2, 2) = nfT 2FCA
(
64280
2187
− 704
81
ζ3
)
+ nfT 2FCF
(
−7382
729
+
1024
81
ζ3
)
.
(7.35)
A(3),MSgg,Q (µ2 = m2, 2) = TFC2A
(
−174055
4374
+
88
9
B4 − 72ζ4 +
29431
324
ζ3
)
+TFCFCA
(
18002
729
− 208
9
B4 + 104ζ4 −
2186
9
ζ3 +
64
3
ζ2 − 64ζ2 ln(2)
)
+TFC2F
(
8879
729
+
64
9
B4 − 32ζ4 +
701
81
ζ3 − 80ζ2 + 128ζ2 ln(2)
)
+T 2FCA
(
21586
2187
− 3605
162
ζ3
)
+ T 2FCF
(
55672
729
− 889
81
ζ3 −
128
3
ζ2
)
+nfT 2FCA
(
−57226
2187
+
1408
81
ζ3
)
+ nfT 2FCF
(
29908
729
− 2048
81
ζ3 +
64
3
ζ2
)
. (7.36)
A(3),MSgg,Q (µ2 = m2, 2) = TFC2A
(
−174055
4374
+
88
9
B4 − 72ζ4 +
29431
324
ζ3
)
+TFCFCA
(
123113
729
− 208
9
B4 + 104ζ4 −
2330
9
ζ3
)
+ TFC2F
(
8042
729
+
64
9
B4
−32ζ4 +
3293
81
ζ3
)
+ T 2FCA
(
21586
2187
− 3605
162
ζ3
)
+ T 2FCF
(
9340
729
− 889
81
ζ3
)
103
+nfT 2FCA
(
−57226
2187
+
1408
81
ζ3
)
+ nfT 2FCF
(
6904
729
− 2048
81
ζ3
)
. (7.37)
Comparing the operator matrix elements in case of the on–shell–scheme and MS–
scheme, one notices that the terms ln(2)ζ2 and ζ2 are absent in the latter. The ζ2 terms,
which contribute to a(3)ij , are canceled by other contributions through renormalization.
Although the present process is massive, this observation resembles the known result that
ζ2–terms do not contribute in space–like massless higher order calculations in even dimen-
sions, [311]. This behavior is found for all calculated moments. The occurring ζ4–terms
may partly cancel with those in the 3–loop light Wilson coefficients, [115]. Note, that
Eq. (7.35) is not sensitive to mass renormalization due to the structure of the contribut-
ing diagrams.
An additional check is provided by the sum rule (3.38), which is fulfilled in all renor-
malization schemes and also on the unrenormalized level.
Unlike the operator matrix elements with external gluons, the second moments of the
quarkonic OMEs emerge for the first time at O(a2s). To 3–loop order, the renormalized
quarkonic OMEs do not contain terms ∝ ζ2. Due to their simpler structure, mass renor-
malization in the on–shell–scheme does not give rise to terms ∝ ζ2, ln(2)ζ2. Only the
rational contribution in the color factor ∝ TFC2F turns out to be different compared to
the on–mass–shell–scheme and APS,(3)qq,Q , (7.40), is not affected at all. This holds again for
all moments we calculated. The non–logarithmic contributions are given by
A(3),PS,MSQq (µ2 = m2, 2) = TFCFCA
(
830
2187
+
64
9
B4 − 64ζ4 +
1280
27
ζ3
)
+TFC2F
(
95638
729
− 128
9
B4 + 64ζ4 −
9536
81
ζ3
)
+ T 2FCF
(
53144
2187
− 3584
81
ζ3
)
+nfT 2FCF
(
−34312
2187
+
1024
81
ζ3
)
. (7.38)
A(3),PS,MSQq (µ2 = m2, 2) = TFCFCA
(
830
2187
+
64
9
B4 − 64ζ4 +
1280
27
ζ3
)
+TFC2F
(
78358
729
− 128
9
B4 + 64ζ4 −
9536
81
ζ3
)
+ T 2FCF
(
53144
2187
− 3584
81
ζ3
)
+nfT 2FCF
(
−34312
2187
+
1024
81
ζ3
)
. (7.39)
A(3),PS,MSqq,Q (µ2 = m2, 2) = nfT 2FCF
(
−52168
2187
+
1024
81
ζ3
)
. (7.40)
A(3),NS,MSqq,Q (µ2 = m2, 2) = TFCFCA
(
−101944
2187
+
64
9
B4 − 64ζ4 +
4456
81
ζ3
)
+TFC2F
(
283964
2187
− 128
9
B4 + 64ζ4 −
848
9
ζ3
)
104
+T 2FCF
(
25024
2187
− 1792
81
ζ3
)
+ nfT 2FCF
(
−46336
2187
+
1024
81
ζ3
)
. (7.41)
A(3),NS,MSqq,Q (µ2 = m2, 2) = TFCFCA
(
−101944
2187
+
64
9
B4 − 64ζ4 +
4456
81
ζ3
)
+TFC2F
(
201020
2187
− 128
9
B4 + 64ζ4 −
848
9
ζ3
)
+T 2FCF
(
25024
2187
− 1792
81
ζ3
)
+ nfT 2FCF
(
−46336
2187
+
1024
81
ζ3
)
. (7.42)
A(3),MSgq,Q (µ2 = m2, 2) = TFCFCA
(
101114
2187
− 128
9
B4 + 128ζ4 −
8296
81
ζ3
)
+TFC2F
(
−570878
2187
+
256
9
B4 − 128ζ4 +
17168
81
ζ3
)
+T 2FCF
(
−26056
729
+
1792
27
ζ3
)
+ nfT 2FCF
(
44272
729
− 1024
27
ζ3
)
. (7.43)
A(3),MSgq,Q (µ2 = m2, 2) = TFCFCA
(
101114
2187
− 128
9
B4 + 128ζ4 −
8296
81
ζ3
)
+TFC2F
(
−436094
2187
+
256
9
B4 − 128ζ4 +
17168
81
ζ3
)
+T 2FCF
(
−26056
729
+
1792
27
ζ3
)
+ nfT 2FCF
(
44272
729
− 1024
27
ζ3
)
. (7.44)
Finally, the sum rule (3.38) holds on the unrenormalized level, as well as for the renor-
malized expressions in all schemes considered.
FORM–codes for the constant terms a(3)ij , Appendix F, and the corresponding moments
of the renormalized massive operator matrix elements, both for the mass renormalization
carried out in the on–shell– and MS–scheme, are attached to Ref. [134] and can be obtained
upon request. Phenomenological studies of the 3–loop heavy flavor Wilson coefficients in
the region Q2 ≫ m2 will be given elsewhere, [319].
105
8 Heavy Flavor Corrections to Polarized Deep-
Inelastic Scattering
The composition of the proton spin in terms of partonic degrees of freedom has attracted
much interest after the initial experimental finding, [320], that the polarization of the
three light quarks alone does not add to the required value 1/2. Subsequently, the po-
larized proton structure functions have been measured in great detail by various exper-
iments, [321] 27. To determine the different contributions to the nucleon spin, both the
flavor dependence as well as the contributions due to gluons and angular excitations at
virtualities Q2 in the perturbative region have to be studied in more detail in the future.
As the nucleon spin contributions are related to the first moments of the respective dis-
tribution functions, it is desirable to measure to very small values of x at high energies,
cf. [177,179,325].
A detailed treatment of the flavor structure requires the inclusion of heavy flavor. As
in the unpolarized case, this contribution is driven by the gluon and sea–quark densities.
Exclusive data on charm–quark pair production in polarized deep–inelastic scattering are
available only in the region of very low photon virtualities at present, [326]. However,
the inclusive measurement of the structure functions g1(x,Q2) and g2(x,Q2) contains the
heavy flavor contributions for hadronic masses W 2 ≥ (2m+M)2.
The polarized heavy flavor Wilson coefficients are known to first order in the whole
kinematic range, [327–329]. In these references, numerical illustrations for the LO con-
tributions were given as well, cf. also [202]. The polarized parton densities have been
extracted from deep-inelastic scattering data in [330–333]. Unlike the case for photo-
production, [334], the NLO Wilson coefficients have not been calculated for the whole
kinematic domain, but only in the region Q2 ≫ m2, [165], applying the same technique as
described in Section 3.2. As outlined in the same Section, the heavy flavor contributions
to the structure function F2(x,Q2) are very well described by the asymptotic represen-
tation for Q2/m2 >∼ 10, i.e., Q2 >∼ 22.5GeV2, in case of charm. A similar approximation
should hold in case of the polarized structure function g1(x,Q2).
In this chapter, we re-calculate for the first time the heavy flavor contributions to the
longitudinally polarized structure function g1(x,Q2) to O(a2s) in the asymptotic region
Q2 ≫ m2, [165]. The corresponding contributions to the structure function g2(x,Q2)
can be obtained by using the Wandzura–Wilczek relation, [335], at the level of twist–2
operators, as has been shown in Refs. [195,200–202] within the covariant parton model.
In the polarized case, the twist-2 heavy flavor Wilson coefficients factorize in the limit
Q2 ≫ m2 in the same way as in the unpolarized case, cf. Section 3.2 and [165]. The
corresponding light flavor Wilson coefficients were obtained in Ref. [336]. We proceed by
calculating the 2–loop polarized massive quarkonic OMEs, as has been done in Ref. [165].
Additionally, we newly calculate the O(ε) terms of these objects, which will be needed to
evaluate the O(a3s) corrections, cf. Section 4.
The calculation is performed in the same way as described in Section 6 and we therefore
only discuss aspects that are specific to the polarized case. The notation for the heavy
flavor Wilson coefficients is the same as in Eq. (3.2) and below, except that the index
(2, L) has to be replaced by (g1, g2). The polarized massive operator matrix elements are
denoted by ∆Aij and obey the same relations as in Sections 3 and 4, if one replaces the
27For theoretical surveys see [322–324].
106
anomalous dimensions, cf. Eq. (2.107, 2.108), by their polarized counterparts, ∆γij.
The asymptotic heavy flavor corrections for polarized deeply inelastic scattering to
O(a2s), [165], were calculated in a specific scheme for the treatment of γ5 in dimensional
regularization. This was done in order to use the same scheme as has been applied in the
calculation of the massless Wilson coefficients in [336]. Here, we refer to the version prior
to an Erratum submitted in 2007, which connected the calculation to the MS–scheme. In
this chapter we would like to compare to the results given in Ref. [165], which requires to
apply the conventions used there.
In Section 8.1, we summarize main relations such as the differential cross sections
for polarized deeply inelastic scattering and the leading order heavy flavor corrections.
We give a brief outline on the representation of the asymptotic heavy flavor corrections
at NLO. In Sections (8.2.1)–(8.2.3), the contributions to the operator matrix elements
∆A(2),NSqq,Q , ∆A
(2)
Qg and ∆A
PS,(2)
Qq are calculated up to the linear terms in ε.
8.1 Polarized Scattering Cross Sections
We consider the process of deeply inelastic longitudinally polarized charged lepton scat-
tering off longitudinally (L) or transversely (T) polarized nucleons in case of single photon
exchange 28. The differential scattering cross section is given by
d3σ
dxdydθ =
yα2
Q4 L
µνWµν , (8.1)
cf. [201,323]. Here, θ is the azimuthal angle of the final state lepton. One may define an
asymmetry between the differential cross sections for opposite nucleon polarization
A(x, y, θ)L,T =
d3σ→L,T
dxdydθ −
d3σ←L,T
dxdydθ , (8.2)
which projects onto the asymmetric part of both the leptonic and hadronic tensors, LAµν
and WAµν . The hadronic tensor is then expressed by two nucleon structure functions
WAµν = iεµνλσ
[qλSσ
P.q g1(x,Q
2) +
qλ(P.qSσ − S.qP σ)
(P.q)2 g2(x,Q
2)
]
. (8.3)
Here S denotes the nucleon’s spin vector
SL = (0, 0, 0,M)
ST = M(0, cos(θ¯), sin(θ¯), 0) , (8.4)
with θ¯ a fixed angle in the plane transverse to the nucleon beam. εµνλσ is the Levi–Civita
symbol. The asymmetries A(x, y, θ)L,T read
A(x, y)L = 4λ
α2
Q2
[(
2− y − 2xyM
2
s
)
g1(x,Q2) + 4
yxM2
s g2(x,Q
2)
]
, (8.5)
A(x, y, θ¯, θ)T = −8λ
α2
Q2
√
M2
s
√
x
y
[
1− y − xyM
2
S
]
cos(θ¯ − θ)[yg1(x,Q2)
+2g2(x,Q2)] , (8.6)
28For the basic kinematics of DIS, see Section 2.1.
107
where λ is the degree of polarization. In case of A(x, y)L, the azimuthal angle was inte-
grated out, since the differential cross section depends on it only through phase space.
The twist–2 heavy flavor contributions to the structure function g1(x,Q2) are cal-
culated using the collinear parton model. This is not possible in case of the structure
function g2(x,Q2). As has been shown in Ref. [202], the Wandzura–Wilczek relation
holds for the gluonic heavy flavor contributions as well
gτ=22 (x,Q2) = −gτ=21 (x,Q2) +
∫ 1
x
dz
z g
τ=2
1 (z,Q2) , (8.7)
from which g2(x,Q2) can be calculated for twist τ = 2. At leading order the heavy flavor
corrections are known for the whole kinematic region, [327–329],
gQQ1 (x,Q2,m2) = 4e2Qas
∫ 1
ax
dz
z H
(1)
g,g1
(x
z ,
m2
Q2
)
∆G(z, nf , Q2) (8.8)
and are of the same structure as in the unpolarized case, cf. Eq. (3.12). Here, ∆G is the
polarized gluon density. The LO heavy flavor Wilson coefficient then reads
H(1)g,g1
(
τ, m
2
Q2
)
= 4TF
[
v(3− 4τ) + (1− 2τ) ln
(
1− v
1 + v
)]
. (8.9)
The support of H(1)g,g1 (τ,m2/Q2) is τ ǫ [0, 1/a]. As is well known, its first moment vanishes
∫ 1/a
0
dτH(1)g1
(
τ, m
2
Q2
)
= 0 , (8.10)
which has a phenomenological implication on the heavy flavor contributions to polarized
structure functions, resulting in an oscillatory profile, [202]. The unpolarized heavy flavor
Wilson coefficients, [103,126,128,135,136,287,289], do not obey a relation like (8.10) but
exhibit a rising behavior towards smaller values of x.
At asymptotic values Q2 ≫ m2 one obtains
H(1),asg,g1
(
τ, m
2
Q2
)
= 4TF
[
(3− 4τ)− (1− 2τ) ln
(Q2
m2
1− τ
τ
)]
. (8.11)
The factor in front of the logarithmic term ln(Q2/m2) in (8.11) is the leading order
splitting function ∆Pqg(τ), [117,337] 29,
∆Pqg(τ) = 8TF
[
τ 2 − (1− τ)2
]
= 8TF [2τ − 1] . (8.12)
The sum–rule (8.10) also holds in the asymptotic case extending the range of integration
to τ ǫ [0, 1],
∫ 1
0
dτH(1),asg,g1
(
τ, m
2
Q2
)
= 0 . (8.13)
29Early calculations of the leading order polarized singlet splitting functions in Refs. [337] still contained
some errors.
108
8.2 Polarized Massive Operator Matrix Elements
The asymptotic heavy flavor Wilson coefficients obey the same factorization relations in
the limit Q2 ≫ m2 as in the unpolarized case, Eqs. (3.21)–(3.25), if one replaces all
quantities by their polarized counterparts.
The corresponding polarized twist–2 composite operators, cf. Eqs. (2.86)–(2.88), are
given by
ONSq,r;µ1,... ,µN = i
N−1S[ψγ5γµ1Dµ2 . . . DµN
λr
2
ψ]− trace terms , (8.14)
OSq;µ1,... ,µN = i
N−1S[ψγ5γµ1Dµ2 . . . DµNψ]− trace terms , (8.15)
OSg;µ1,... ,µN = 2i
N−2SSp[
1
2
εµ1αβγF aβγDµ2 . . . DµN−1F µNα,a ]− trace terms . (8.16)
The Feynman rules needed are given in Appendix B. The polarized anomalous dimensions
of these operators are defined in the same way as in Eqs. (2.107, 2.108), as is the case for
the polarized massive OMEs, cf. Eq. (3.17) and below.
In the subsequent investigation, we will follow Ref. [165] and calculate the quarkonic
heavy quark contributions to O(a2s). The diagrams contributing to the corresponding
massive OMEs are the same as in the unpolarized case and are shown in Figures 1–4
in Ref. [126]. The formal factorization relations for the heavy flavor Wilson coefficients
can be inferred from Eqs. (3.26, 3.29, 3.30). Here, we perform the calculation in the
MOM–scheme, cf. Section 5.1, to account for heavy quarks in the final state only. The
same scheme has been adopted in Ref. [165]. Identifying µ2 = Q2, the heavy flavor Wilson
coefficients in the limit Q2 ≫ m2 become, [165],
H(1)g,g1
(Q2
m2 , N
)
= −1
2
∆γˆ(0)qg ln
(Q2
m2
)
+ cˆ(1)g,g1 , (8.17)
H(2)g,g1
(Q2
m2 , N
)
=
{
1
8
∆γˆ(0)qg
[
∆γ(0)qq −∆γ(0)gg − 2β0
]
ln2
(Q2
m2
)
−1
2
[
∆γˆ(1)qg +∆γˆ(0)qg c(1)q,g1
]
ln
(Q2
m2
)
+
[
∆γ(0)gg −∆γ(0)qq + 2β0
] ∆γˆ(0)qg ζ2
8
+ cˆ(2)g,g1 +∆a
(2)
Qg
}
, (8.18)
H(2),PSq,g1
(Q2
m2 , N
)
=
{
−1
8
∆γˆ(0)qg ∆γ(0)gq ln2
(Q2
m2
)
− 1
2
∆γˆ(1),PSqq ln
(Q2
m2
)
+
∆γˆ(0)qg ∆γ(0)gq
8
ζ2 + cˆ(2),PSq,g1 +∆a
(2),PS
Qq
}
, (8.19)
L(2),NSq,g1
(Q2
m2 , N
)
=
{
1
4
β0,Q∆γ(0)qq ln2
(Q2
m2
)
−
[
1
2
∆γˆ(1),NSqq + β0,Qc(1)q,g1
]
ln
(Q2
m2
)
−1
4
β0,Qζ2∆γ(0)qq + cˆ
(2),NS
q,g1,Q +∆a
(2),NS
qq,Q
}
. (8.20)
c(k)i,g1 are the kth order non–logarithmic terms of the polarized coefficient functions. As has
been described in [165], the relations (8.18)–(8.20) hold if one uses the same scheme for
109
the description of γ5 in dimensional regularization for the massive OMEs and the light
flavor Wilson coefficients. This is the case for the massive OMEs as calculated in [165],
to which we refer, and the light flavor Wilson coefficients as calculated in Ref. [336].
8.2.1 ∆A(2),NSqq,Q
The non–singlet operator matrix element ∆A(2),NSqq,Q has to be the same as in the unpolarized
case due to the Ward–Takahashi identity, [338]. Since it is obtained as zero–momentum
insertion on a graph for the transition 〈p| → |p〉, one may write it equivalently in terms
of the momentum derivative of the self–energy. The latter is independent of the operator
insertion and yields therefore the same in case of /∆(∆.p)N−1 and /∆γ5(∆.p)N−1. Hence,
∆A(2),NSqq,Q reads, cf. Eq. (4.95),
∆A(2),NSqq,Q
(
N, m
2
µ2
)
= A(2),NSqq,Q
(
N, m
2
µ2
)
=
β0,Qγ(0)qq
4
ln2
(m2
µ2
)
+
γˆ(1),NSqq
2
ln
(m2
µ2
)
+ a(2),NSqq,Q −
γ(0)qq
4
β0,Qζ2 ,
(8.21)
where the constant term in ε of the unrenormalized result, Eq. (4.93), is given in Eq. (6.60)
and the O(ε)–term in Eq. (6.61).
8.2.2 ∆A(2)Qg
To calculate the OME ∆AQg up to O(a2s), the Dirac-matrix γ5 is represented in D = 4+ε
dimensions via, [165,264,339],
/∆γ5 = i
6
εµνρσ∆µγνγργσ . (8.22)
The Levi–Civita symbol will be contracted later with a second Levi–Civita symbol emerg-
ing in the general expression for the Green’s function, cf. Eq. (4.18),
∆GˆabQ,µν = ∆
ˆˆAQg
(mˆ2
µ2 , ε, N
)
δab(∆ · p)N−1εµναβ∆αpβ , (8.23)
by using the following relation in D–dimensions, [194],
εµνρσεαλτγ = −Det
[
gβω
]
, (8.24)
β = α, λ, τ, γ , ω = µ, ν, ρ, σ .
In particular, anti–symmetry relations of the Levi-Civita tensor or the relation γ25 = 1,
holding in four dimensions, are not used. The projector for the gluonic OME then reads
∆ ˆˆAQg =
δab
N2c − 1
1
(D − 2)(D − 3)(∆.p)
−N−1εµνρσ∆GˆabQ,µν∆ρpσ . (8.25)
In the following, we will present the results for the operator matrix element using the above
prescription for γ5. This representation allows a direct comparison to Ref. [165] despite the
110
fact that in this scheme even some of the anomalous dimensions are not those of the MS–
scheme. We will discuss operator matrix elements for which only mass renormalization
was carried out, cf. Section 4.3. Due to the crossing relations of the forward Compton
amplitude corresponding to the polarized case, only odd moments contribute. Therefore
the overall factor
1
2
[
1− (−1)N
]
, N ∈ N, (8.26)
is implied in the following. To obtain the results in x–space the analytic continuation
to complex values of N can be performed starting from the odd integers. The O(as)
calculation is straightforward
∆ ˆˆA
(1)
Qg =
(m2
µ2
)ε/2 [1
ε +
ζ2
8
ε2 + ζ3
24
ε
]
∆γˆ(0)qg +O(ε3) (8.27)
=
(m2
µ2
)ε/2 [1
ε∆γˆ
(0)
qg +∆a
(1)
Qg + ε∆a
(1)
Qg + ε2∆a
(1)
Qg
]
+O(ε3) . (8.28)
The matrix element contains the leading order anomalous dimension ∆γˆ(0)qg ,
∆A(1)Qg =
1
2
∆γˆ(0)qg ln
(m2
µ2
)
, (8.29)
where
∆γˆ(0)qg = −8TF
N − 1
N(N + 1) . (8.30)
The leading order polarized Wilson coefficient c(1)g,g1 reads, [329,336,340],
c(1)g,g1 = −4TF
N − 1
N(N + 1)
[
S1 +
N − 1
N
]
. (8.31)
The Mellin transform of Eq. (8.11) then yields the same expression as one obtains from
Eq. (8.17)
H(1),asg,g1
(
N, m
2
Q2
)
=
[
−1
2
∆γˆ(0)qg ln
(Q2
m2
)
+ c(1)g,g1
]
, (8.32)
for which the proportionality
H(1),asg,g1
(
N, m
2
Q2
)
∝ (N − 1) (8.33)
holds, leading to a vanishing first moment.
At the 2–loop level, we express the operator matrix element ∆ ˆˆA
(2)
Qg, after mass renor-
malization, in terms of anomalous dimensions, cf. [126,128,135,136,287,289], by
∆ ˆˆA
(2)
Qg =
(m2
µ2
)ε
[
∆γˆ(0)qg
2ε2
{
∆γ(0)qq −∆γ(0)gg − 2β0
}
+
∆γˆ′(1)qg
ε +∆a
′(2)
Qg +∆a
′(2)
Qg ε
]
−2εβ0,Q
(m2
µ2
)ε/2
(
1 +
ε2
8
ζ2 +
ε3
24
ζ3
)
∆ ˆˆA
(1)
Qg +O(ε2) , (8.34)
111
The remaining LO anomalous dimensions are
∆γ(0)qq = −CF
(
−8S1 + 2
3N2 + 3N + 2
N(N + 1)
)
, (8.35)
∆γ(0)gg = −CA
(
−8S1 + 2
11N2 + 11N + 24
3N(N + 1)
)
+
8
3
TFnf . (8.36)
The renormalized expression in the MOM–scheme is given by
∆A′(2),MOMQg =
∆γˆ(0)qg
8
[
∆γ(0)qq −∆γ(0)gg − 2β0
]
ln2
(m2
µ2
)
+
γˆ′(1)qg
2
ln
(m2
µ2
)
+
(
∆γ(0)gg −∆γ(0)qq + 2β0
) γˆ(0)qg
8
ζ2 +∆a′(2)Qg . (8.37)
The LO anomalous dimensions which enter the double pole term in Eq. (8.34) and the
ln2(m2/µ2) term in Eq. (8.37), respectively, are scheme–independent. This is not the
case for the remaining terms, which depend on the particular scheme we adopted in
Eqs. (8.24, 8.22) and are therefore denoted by a prime. The NLO anomalous dimension
we obtain is given by
∆γˆ′(1)qg = −TFCF
(
−16 N − 1N(N + 1)S2 + 16
N − 1
N(N + 1)S
2
1 − 32
N − 1
N2(N + 1)S1
+8
(N − 1)(5N4 + 10N3 + 8N2 + 7N + 2)
N3(N + 1)3
)
+TFCA
(
32
N − 1
N(N + 1)β
′ + 16
N − 1
N(N + 1)S2 + 16
N − 1
N(N + 1)S
2
1
−16 N − 1N(N + 1)ζ2 −
64S1
N(N + 1)2 − 16
N5 +N4 − 4N3 + 3N2 − 7N − 2
N3(N + 1)3
)
.
(8.38)
It differs from the result in the MS–scheme, [341,342], by a finite renormalization. This is
due to the fact that we contracted the Levi–Civita symbols in D dimensions. The correct
NLO splitting function is obtained by
∆γˆ(1)qg = ∆γˆ′(1)qg + 64TFCF
N − 1
N2(N + 1)2 . (8.39)
In an earlier version of Ref. [336], ∆γˆ′(1)qg was used as the anomalous dimension departing
from the MS scheme. Therefore, in Ref. [165] the finite renormalization (8.39), as the
corresponding one for c(2)g,g1 , [336], was not used for the calculation of ∆A(2)Qg. For the
112
higher order terms in ε in Eq. (8.34) we obtain
∆a′(2)Qg = −TFCF
{
4(N − 1)
3N(N + 1)
(
−4S3 + S31 + 3S1S2 + 6S1ζ2
)
− 4(3N
2 + 3N − 2)S21
N2(N + 1)(N + 2)
−4N
4 + 17N3 + 43N2 + 33N + 2
N2(N + 1)2(N + 2) S2 − 2
(N − 1)(3N2 + 3N + 2)
N2(N + 1)2 ζ2
−4N
3 − 2N2 − 22N − 36
N2(N + 1)(N + 2) S1 +
2P1
N4(N + 1)4(N + 2)
}
−TFCA
{
4
N − 1
3N(N + 1)
(
12M
[
Li2(x)
1 + x
]
(N + 1) + 3β′′ − 8S3 − S31 − 9S1S2
−12S1β′ − 12βζ2 − 3ζ3
)
− 16 N − 1N(N + 1)2β
′ + 4
N2 + 4N + 5
N(N + 1)2(N + 2)S
2
1
+4
7N3 + 24N2 + 15N − 16
N2(N + 1)2(N + 2) S2 + 8
(N − 1)(N + 2)
N2(N + 1)2 ζ2
+4
N4 + 4N3 −N2 − 10N + 2
N(N + 1)3(N + 2) S1 −
4P2
N4(N + 1)4(N + 2)
}
, (8.40)
∆a′(2)Qg = TFCF
{
N − 1
N(N + 1)
(
16S2,1,1 − 8S3,1 − 8S2,1S1 + 3S4 −
4
3
S3S1 −
1
2
S22 −
1
6
S41
−8
3
S1ζ3 − S2S21 + 2S2ζ2 − 2S21ζ2
)
− 8S2,1N2 +
3N2 + 3N − 2
N2(N + 1)(N + 2)
(
2S2S1 +
2
3
S31
)
+2
3N4 + 48N3 + 123N2 + 98N + 8
3N2(N + 1)2(2 +N) S3 +
4(N − 1)
N2(N + 1)S1ζ2
+
2
3
(N − 1)(3N2 + 3N + 2)
N2(N + 1)2 ζ3 +
P3S2
N3(N + 1)3(N + 2) +
N3 − 6N2 − 22N − 36
N2(N + 1)(N + 2) S
2
1
+
P4ζ2
N3(N + 1)3 − 2
2N4 − 4N3 − 3N2 + 20N + 12
N2(N + 1)2(N + 2) S1 +
P5
N5(N + 1)5(N + 2)
}
+TFCA
{
N − 1
N(N + 1)
(
16S−2,1,1 − 4S2,1,1 − 8S−3,1 − 8S−2,2 − 4S3,1 +
2
3
β′′′
−16S−2,1S1 − 4β′′S1 + 8β′S2 + 8β′S21 + 9S4 +
40
3
S3S1 +
1
2
S22 + 5S2S21 +
1
6
S41
+4ζ2β′ − 2ζ2S2 − 2ζ2S21 −
10
3
S1ζ3 −
17
5
ζ22
)
− N − 1N(N + 1)2
(
16S−2,1 + 4β′′ − 16β′S1
)
−16
3
N3 + 7N2 + 8N − 6
N2(N + 1)2(N + 2) S3 +
2(3N2 − 13)S2S1
N(N + 1)2(N + 2) −
2(N2 + 4N + 5)
3N(N + 1)2(N + 2)S
3
1
− 8ζ2S1
(N + 1)2 −
2
3
(N − 1)(9N + 8)
N2(N + 1)2 ζ3 −
8(N2 + 3)
N(N + 1)3β
′ − P6S2N3(N + 1)3(N + 2)
−N
4 + 2N3 − 5N2 − 12N + 2
N(N + 1)3(N + 2) S
2
1 −
2P7ζ2
N3(N + 1)3 +
2P8S1
N(N + 1)4(N + 2)
− 2P9N5(N + 1)5(N + 2)
}
, (8.41)
113
with the polynomials
P1 = 4N8 + 12N7 + 4N6 − 32N5 − 55N4 − 30N3 − 3N2 − 8N − 4 , (8.42)
P2 = 2N8 + 10N7 + 22N6 + 36N5 + 29N4 + 4N3 + 33N2 + 12N + 4 , (8.43)
P3 = 3N6 + 30N5 + 107N4 + 124N3 + 48N2 + 20N + 8 , (8.44)
P4 = (N − 1)(7N4 + 14N3 + 4N2 − 7N − 2) , (8.45)
P5 = 8N10 + 24N9 − 11N8 − 160N7 − 311N6 − 275N5 − 111N4 − 7N3
+11N2 + 12N + 4 , (8.46)
P6 = N6 + 18N5 + 63N4 + 84N3 + 30N2 − 64N − 16 , (8.47)
P7 = N5 −N4 − 4N3 − 3N2 − 7N − 2 , (8.48)
P8 = 2N5 + 10N4 + 29N3 + 64N2 + 67N + 8 , (8.49)
P9 = 4N10 + 22N9 + 45N8 + 36N7 − 11N6 − 15N5 + 25N4 − 41N3
−21N2 − 16N − 4 . (8.50)
The Mellin–transform in Eq. (8.40) is given in Eq. (6.47) in terms of harmonic sums. As
a check, we calculated several lower moments (N = 1 . . . 9) of each individual diagram
contributing to A(2)Qg 30 using the Mellin–Barnes method, [287,309]. In Table 3, we present
the numerical results we obtain for the moments N = 3, 7 of the individual diagrams.
We agree with the results obtained for general values of N . The contributions from the
individual diagrams are given in [159]. Our results up to O(ε0), Eqs. (8.34, 8.40), agree
with the results presented in [165], which we thereby confirm for the first time. Eq. (8.41)
is a new result.
In this calculation extensive, use was made of the representation of the Feynman-
parameter integrals in terms of generalized hypergeometric functions, cf. Section 6. The
infinite sums, which occur in the polarized calculation, are widely the same as in the
unpolarized case, [128,135,136,287,289]. The structure of the result for the higher order
terms in ε is completely the same as in the unpolarized case as well, see Eq. (6.34) and
the following discussion. Especially, the structural relations between the finite harmonic
sums, [138, 144, 147, 148], allow to express ∆a′(2)Qg by only two basic Mellin transforms,
S1 and S−2,1. This has to be compared to the 24 functions needed in Ref. [165] to
express the constant term in z–space. Thus we reached a more compact representation.
∆a′(2)Qg depends on the six sums S1(N), S±2,1(N), S−3,1(N), S±2,1,1(N), after applying the
structural relations. The O(ε0) term has the same complexity as the 2–loop anomalous
dimensions, whereas the complexity of the O(ε) term corresponds to the level observed for
2–loop Wilson coefficients and other hard scattering processes which depend on a single
scale, cf. [95, 145].
8.2.3 ∆A(2),PSQq
The operator matrix element ∆A(2),PSQq is obtained from the diagrams shown in Figure 3 of
Ref. [126]. In this calculation, we did not adopt any specific scheme for γ5, but calculated
the corresponding integrals without performing any traces or (anti)commuting γ5.
30These are shown in Figure 2 of Ref. [126].
114
order 1/ε2 1/ε 1 ε ε2
A N = 3 −0.22222 0.06481 −0.13343 −0.15367 −0.06208
N = 7 −0.03061 0.00409 −0.01669 −0.01900 −0.00639
B N = 3 4.44444 −1.07407 4.45579 0.515535 3.13754
N = 7 5.46122 0.74491 6.09646 2.97092 5.35587
C N = 3 1.33333 −8.14444 0.13303 −6.55515 −2.64601
N = 7 0.85714 −5.12329 0.14342 −4.10768 −1.59526
D N = 3 2.66666 −0.02222 2.19940 1.03927 1.69331
N = 7 1.71428 0.85340 1.78773 1.56227 1.80130
E N = 3 −2.66667 5 −2.27719 4.89957 0.73208
N = 7 −1.71429 2.97857 −1.3471 2.83548 0.44608
F N = 3 0 0.77777 −5.80092 −2.63560 −6.57334
N = 7 0 1.40105 −3.54227 −0.78565 −3.72466
L N = 3 −9.33333 0.25000 −8.83933 −3.25228 −6.84460
N = 7 −6.73878 −1.86855 −7.09938 −4.56051 −6.501
M N = 3 −0.22222 0.71296 −0.41198 0.69938 −0.11618
N = 7 −0.03061 0.11324 −0.05861 0.11969 −0.01207
N N = 3 −2.22222 1.26851 −1.37562 0.69748 −0.36030
N = 7 −3.19184 −0.50674 −3.39832 −1.7667 −2.97339
Table 3: Numerical values for moments of individual diagrams of ∆ ˆˆA
(2)
Qg.
The result can then be represented in terms of three bi–spinor structures
C1(ε) =
1
∆.pTr
{
/∆ /pγµγνγ5
}
/∆γµγν =
24
3 + ε/∆γ5 (8.51)
C2(ε) = Tr
{
/∆γµγνγργ5
}
γµγνγρ = 24/∆γ5 (8.52)
C3(ε) =
1
m2Tr
{
/∆ /pγµγνγ5
}
/pγµγν . (8.53)
These are placed between the states 〈p| . . . |p〉, with
/p|p〉 = m0|p〉 (8.54)
and m0 the light quark mass. Therefore, the contribution due to C3(ε) vanishes in the
limit m0 → 0. The results for C1,2(ε) in the r.h.s. of Eqs. (8.51, 8.52) can be obtained by
115
applying the projector
−3
2(D − 1)(D − 2)(D − 3)Tr[ /pγ5 Ci] (8.55)
and performing the trace in D–dimensions using relations (8.22, 8.24). Note that the
result in 4–dimensions is recovered by setting ε = 0. One obtains from the truncated
2–loop Green’s function ∆ Gij,(2)Qq
∆ Gij,(2)Qq = ∆
ˆˆA
(2),PS
Qq /∆γ5δij(∆.p)N−1 , (8.56)
the following result for the massive OME
∆ ˆˆA
(2),PS
Qq /∆γ5 =
(m2
µ2
)ε
C(ε)8(N + 2)
{
− 1ε2
2(N − 1)
N2(N + 1)2 +
1
ε
N3 + 2N + 1
N3(N + 1)3
−(N − 1)(ζ2 + 2S2)
2N2(N + 1)2 −
4N3 − 4N2 − 3N − 1
2N4(N + 1)4 + ε
[(N3 + 2N + 1)(ζ2 + 2S2)
4N3(N + 1)3
−(N − 1)(ζ3 + 3S3)
6N2(N + 1)2 +
N5 − 7N4 + 6N3 + 7N2 + 4N + 1
4N5(N + 1)5
]}
+O(ε2) , (8.57)
where
C(ε) = C1(ε) · (N − 1) + C2(ε)
8(N + 2) = /∆γ5
3(N + 2 + ε)
(N + 2)(3 + ε)
= 1 +
N − 1
3(N + 2)
(
−ε+ ε
2
3
− ε
3
9
)
+O(ε4) . (8.58)
Comparing our result, Eq. (8.57), to the result obtained in [165], one notices that there
the factor C(ε) was calculated in 4–dimensions, i.e. C(ε) = 1. Therefore, we do the same
and obtain
∆ ˆˆA
(2),PS
Qq = S2ε
(m2
µ2
)ε
[
−∆γˆ
(0)
qg ∆γ(0)gq
2ε2 +
∆γˆ(1),PSqq
2ε +∆a
(2),PS
Qq +∆a
(2),PS
Qq ε
]
.
(8.59)
with
∆γ(0)gq = −4CF
N + 2
N(N + 1) , (8.60)
∆γˆ(1),PSqq = 16TFCF
(N + 2)(N3 + 2N + 1)
N3(N + 1)3 , (8.61)
∆a(2),PSQq = −
(N − 1)(ζ2 + 2S2)
2N2(N + 1)2 −
4N3 − 4N2 − 3N − 1
2N4(N + 1)4 , (8.62)
∆a(2),PSQq = −
(N − 1)(ζ3 + 3S3)
6N2(N + 1)2
(N3 + 2N + 1)(ζ2 + 2S2)
4N3(N + 1)3
+
N5 − 7N4 + 6N3 + 7N2 + 4N + 1
4N5(N + 1)5 . (8.63)
116
Here, we agree up to O(ε0) with Ref. [165] and Eq. (8.63) is a new result. Note, that
Eq. (8.61) is already the MS anomalous dimension as obtained in Refs. [341,342]. There-
fore any additional scheme dependence due to γ5 can only be contained in the higher
order terms in ε. As a comparison the anomalous dimension ∆γˆ′(1),PSqq which is obtained
by calculating C(ε) in D dimensions, is related to the MS one by
∆γˆ(1),PSqq = ∆γˆ′(1),PSqq − TFCF
16(N − 1)2
3N2(N + 1)2 . (8.64)
The renormalized result becomes
∆A(2),PSQq = −
∆γˆ(0)qg ∆γ(0)gq
8
ln2
(m2
µ2
)
+
∆γˆ(1),PSqq
2
ln
(m2
µ2
)
+∆a(2),PSQq +
∆γˆ(0)qg ∆γ(0)gq
8
ζ2 .
(8.65)
The results in this Section constitute a partial step towards the calculation of the asymp-
totic heavy flavor contributions at O(a2s) in the MS–scheme, thereby going beyond the
results of Ref. [165]. The same holds for the O(a2sε)–terms, which we calculated for the
first time, using the same description for γ5 as has been done in [165]. The correct finite
renormalization to transform to the MS–scheme remains to be worked out and will be
presented elsewhere, [159].
117
9 Heavy Flavor Contributions to Transversity
The transversity distribution ∆Tf(x,Q2) is one of the three possible quarkonic twist-2
parton distributions besides the unpolarized and the longitudinally polarized quark dis-
tribution, f(x,Q2) and ∆f(x,Q2), respectively. Unlike the latter distribution functions,
it cannot be measured in inclusive deeply inelastic scattering in case of massless partons
since it is chirally odd. However, it can be extracted from semi–inclusive deep-inelastic
scattering (SIDIS) studying isolated meson production, [343, 344], and in the Drell-Yan
process, [344–346], off transversely polarized targets 31. Measurements of the transversity
distribution in different polarized hard scattering processes are currently performed or in
preparation, [347]. In the past, phenomenological models for the transversity distribu-
tion were developed based on bag-like models, chiral models, light–cone models, spectator
models, and non-perturbative QCD calculations, cf. Section 8 of Ref. [166]. The main
characteristics of the transversity distributions are that they vanish by some power law
both at small and large values of Bjorken–x and exhibit a shifted bell-like shape. Recent
attempts to extract the distribution out of data were made in Refs. [348]. The moments
of the transversity distribution can be measured in lattice simulations, which help to
constrain it ab initio, where first results were given in Refs. [192,349]. From these investi-
gations there is evidence, that the up-quark distribution is positive while the down-quark
distribution is negative, with first moments between {0.85 . . . 1.0} and {−0.20 . . .−0.24},
respectively. This is in qualitative agreement with phenomenological fits.
Some of the processes which have been proposed to measure transversity contain k⊥−
and higher twist effects, cf. [166]. We will limit our considerations to the class of purely
twist–2 contributions, for which the formalism to calculate the heavy flavor corrections
is established, cf. Section 3. As for the unpolarized flavor non–singlet contributions, we
apply the factorization relation of the heavy flavor Wilson coefficient (3.21) in the region
Q2 ≫ m2.
As physical processes one may consider the SIDIS process lN → l′h+X off transversely
polarized targets in which the transverse momentum of the produced final state hadron
h is integrated. The differential scattering cross section in case of single photon exchange
reads
d3σ
dxdydz =
4πα2
xyQ2
∑
i=q,q
e2ix
{
1
2
[
1 + (1− y)2
]
Fi(x,Q2)D˜i(z,Q2)
−(1− y)|S⊥||Sh⊥| cos (φS + φSh)∆TFi(x,Q2)∆T D˜i(z,Q2)
}
. (9.1)
Here, in addition to the Bjorken variables x and y, the fragmentation variable z occurs.
S⊥ and Sh⊥ are the transverse spin vectors of the incoming nucleon N and the measured
hadron h. The angles φS,Sh are measured in the plane perpendicular to the γ∗N (z–)
axis between the x-axis and the respective vector. The transversity distribution can be
obtained from Eq. (9.1) for a transversely polarized hadron h by measuring its polarization.
31For a review see Ref. [166].
118
The functions Fi, D˜i,∆TFi,∆T D˜i are given by
Fi(x,Q2) = Ci(x,Q2)⊗ fi(x,Q2) (9.2)
D˜i(z,Q2) = C˜i(z,Q2)⊗Di(z,Q2) (9.3)
∆TFi(x,Q2) = ∆TCi(x,Q2)⊗∆Tfi(x,Q2) (9.4)
∆T D˜i(z,Q2) = ∆T C˜i(z,Q2)⊗∆TDi(z,Q2) . (9.5)
Here, Di,∆TDi are the fragmentation functions and C˜i, ∆TCi, ∆T C˜i are the corresponding
space- and time-like Wilson coefficients. The functions Ci are the Wilson coefficients as
have been considered in the unpolarized case, cf. Sections 2 and 3. The Wilson coefficient
for transversity, ∆TCi(x,Q2), contains light– and heavy flavor contributions, cf. Eq. (3.3),
∆TCi(x,Q2) = ∆TCi(x,Q2) + ∆THi(x,Q2) . (9.6)
∆TCi denotes the light flavor transversity Wilson coefficient and ∆THi(x,Q2) the heavy
flavor part. We dropped arguments of the type nf , m2, µ2 for brevity, since they can all
be inferred from the discussion in Section 3.
Eq. (9.1) holds for spin–1/2 hadrons in the final state, but the transversity distribution
may also be measured in the leptoproduction process of spin–1 hadrons, [350]. In this
case, the Ph⊥-integrated Born cross section reads
d3σ
dxdydz =
4πα2
xyQ2 sin (φS + φSLT ) |S⊥||SLT |(1− y)
∑
i=q,q
e2ix∆TFi(x,Q2)Ĥi,1,LT (z,Q2) .
(9.7)
Here, the polarization state of a spin–1 particle is described by a tensor with five inde-
pendent components, [351]. φLT denotes the azimuthal angle of ~SLT , with
|SLT | =
√
(SxLT )
2 + (SyLT )
2 . (9.8)
Ĥa,1,LT (z,Q2) is a T - and chirally odd twist-2 fragmentation function at vanishing k⊥.
The process (9.7) has the advantage that the transverse polarization of the produced
hadron can be measured from its decay products.
The transversity distribution can also be measured in the transversely polarized Drell–
Yan process, see Refs. [352–354]. However, the SIDIS processes have the advantage that
in high luminosity experiments the heavy flavor contributions can be tagged like in deep-
inelastic scattering. This is not the case for the Drell-Yan process, where the heavy flavor
effects appear as inclusive radiative corrections in the Wilson coefficients only.
The same argument as in Section 3.2 can be applied to obtain the heavy flavor Wilson
coefficients for transversity in the asymptotic limit Q2 ≫ m2. Since transversity is a NS
quantity, the relation is the same as in the unpolarized NS case and reads up to O(a3s),
cf. Eq. (3.26),
∆THAsymq (nf + 1) = a2s
[
∆TA(2),NSqq,Q (nf + 1) + ∆T Cˆ(2)q (nf )
]
+ a3s
[
∆TA(3),NSqq,Q (nf + 1) + ∆TA
(2),NS
qq,Q (nf + 1)∆TC(1)q (nf + 1)
+∆T Cˆ(3)q (nf )
]
. (9.9)
119
The operator matrix elements ∆TA(2,3),NSqq,Q are – as in the unpolarized case – universal
and account for all mass contributions but power corrections. The respective asymptotic
heavy flavor Wilson coefficients are obtained in combination with the light flavor process–
dependent Wilson coefficients 32. In the following, we will concentrate on the calculation
of the massive operator matrix elements. The twist–2 local operator in case of transversity
has a different Lorentz–structure compared to Eqs. (2.86)–(2.88) and is given by
OTR,NSq,r;µ,µ1,... ,µN = i
N−1S[ψσµµ1Dµ2 . . . DµN λr
2
ψ]− trace terms , (9.10)
with σνµ = (i/2) [γνγµ − γµγν ] and the definition of the massive operator matrix element
is the same as in Section 3.2. Since (9.10) denotes a twist–2 flavor non–singlet operator,
it does not mix with other operators. After multiplying with the external source JN ,
cf. Eq. (4.10) and below, the Green’s function in momentum space corresponding to the
transversity operator between quarkonic states is given by
u(p, s)Gij,TR,NSµ,q,Q λru(p, s) = JN〈Ψi(p) | OTR,NSq,r;µ,µ1,... ,µN | Ψ
j(p)〉Q . (9.11)
It relates to the unrenormalized transversity OME via
Gˆij,TR,NSµ,q,Q = δij(∆ · p)N−1
(
∆ρσµρ∆T ˆˆA
NS
qq,Q
(mˆ2
µ2 , ε, N
)
+ c1∆µ + c2pµ + c3γµp/
+c4∆/ p/∆µ + c5∆/ p/pµ
)
. (9.12)
The Feynman rules for the operators multiplied with the external source are given in
Appendix B. The projection onto the massive OME is found to be
∆T
ˆˆA
NS
qq,Q
(mˆ2
µ2 , ε, N
)
= − iδ
ij
4Nc(∆.p)2(D − 2)
{
Tr[∆/ p/ pµGˆij,TR,NSµ,q,Q ]−∆.pTr[pµGˆij,TR,NSµ,q,Q ]
+i∆.pTr[σµρpρGˆij,TR,NSµ,q,Q ]
}
. (9.13)
Renormalization for transversity proceeds in the same manner as in the NS–case. The
structure of the unrenormalized expressions at the 2– and 3–loop level are the same as
shown in Eqs. (4.93, 4.94), if one inserts the respective transversity anomalous dimensions.
The expansion coefficients of the renormalized OME then read up to O(a3s) in the MS–
scheme, cf. Eqs. (4.95, 4.96),
∆TA(2),NS,MSqq,Q =
β0,Qγ(0),TRqq
4
ln2
(m2
µ2
)
+
γˆ(1),TRqq
2
ln
(m2
µ2
)
+ a(2),TRqq,Q −
β0,Qγ(0),TRqq
4
ζ2 ,
(9.14)
∆TA(3),NS,MSqq,Q = −
γ(0),TRqq β0,Q
6
(
β0 + 2β0,Q
)
ln3
(m2
µ2
)
+
1
4
{
2γ(1),TRqq β0,Q
32Apparently, the light flavor Wilson coefficients for SIDIS were not yet calculated even at O(as),
although this calculation and the corresponding soft-exponentiation should be straightforward. For the
transversely polarized Drell-Yan process the O(as) light flavor Wilson coefficients were given in [353] and
higher order terms due to soft resummation were derived in [354].
120
−2γˆ(1),TRqq
(
β0 + β0,Q
)
+ β1,Qγ(0),TRqq
}
ln2
(m2
µ2
)
+
1
2
{
γˆ(2),TRqq
−
(
4a(2),TRqq,Q − ζ2β0,Qγ(0),TRqq
)
(β0 + β0,Q) + γ(0),TRqq β
(1)
1,Q
}
ln
(m2
µ2
)
+4a(2),TRqq,Q (β0 + β0,Q)− γ(0)qq β
(2)
1,Q −
γ(0),TRqq β0β0,Qζ3
6
− γ
(1),TR
qq β0,Qζ2
4
+2δm(1)1 β0,Qγ(0),TRqq + δm
(0)
1 γˆ(1),TRqq + 2δm
(−1)
1 a
(2),TR
qq,Q + a
(3),TR
qq,Q . (9.15)
Here, γ(k),TRqq , {k = 0, 1, 2}, denote the transversity quark anomalous dimensions at
O(ak+1s ) and a
(2,3),TR
qq,Q , a
(2),TR
qq,Q are the constant and O(ε) terms of the massive operator
matrix element at 2– and 3–loop order, respectively, cf. the discussion in Section 4. At
LO the transversity anomalous dimension was calculated in [345,355,356] 33, and at NLO
in [353,358] 34. At three-loop order the moments N = 1 . . . 8 are known, [360].
The 2–loop calculation for allN proceeds in the same way as described in Section 6. We
also calculated the unprojected Green’s function to check the projector (9.13). Fixed mo-
ments at the 2– and 3–loop level were calculated using MATAD as described in Section 7.
From the pole terms of the unrenormalized 2–loop OMEs, the leading and next-to-leading
order anomalous dimensions γ(0),TRqq and γˆ(1),TRqq can be determined. We obtain
γ(0),TRqq = 2CF [−3 + 4S1] , (9.16)
and
γˆ(1),TRqq =
32
9
CFTF
[
3S2 − 5S1 +
3
8
]
, (9.17)
confirming earlier results, [353,358]. The finite and O(ε) contributions are given by
a(2),TRqq,Q = CFTF
{
−8
3
S3 +
40
9
S2 −
[
224
27
+
8
3
ζ2
]
S1 + 2ζ2 +
(24 + 73N + 73N2)
18N (N + 1)
}
,
(9.18)
a(2),TRqq,Q = CFTF
{
−
[
656
81
+
20
9
ζ2 +
8
9
ζ3
]
S1 +
[
112
27
+
4
3
ζ2
]
S2 −
20
9
S3
+
4
3
S4 +
1
6
ζ2 +
2
3
ζ3 +
(−144− 48N + 757N2 + 1034N3 + 517N4)
216N2 (N + 1)2
}
.
(9.19)
The renormalized 2–loop massive OME (9.14) reads
∆TA(2),NS,MSqq,Q = CFTF
{[
−8
3
S1 + 2
]
ln2
(m2
µ2
)
+
[
−80
9
S1 +
2
3
+
16
3
S2
]
ln
(m2
µ2
)
−8
3
S3 +
40
9
S2 −
224
27
S1 +
24 + 73N + 73N2
18N (N + 1)
}
. (9.20)
33The small x limit of the LO anomalous dimension was calculated in [357].
34For calculations in the non-forward case, see [356,359].
121
Using MATAD, we calculated the moments N = 1 . . . 13 at O(a2s) and O(a3s). At the
2–loop level, we find complete agreement with the results presented in Eqs. (9.16)–
(9.19). At O(a3s), we also obtain γˆ
(2),TR
qq , which can be compared to the TF -terms in
the calculation [360] for N = 1...8. This is the first re-calculation of these terms and
we find agreement. For the moments N = 9...13 this contribution to the transversity
anomalous dimension is calculated for the first time. We list the anomalous dimensions
in Appendix G. There, also the constant contributions a(3),TRqq,Q are given for N = 1 . . . 13,
which is a new result. Furthermore, we obtain in the 3–loop calculation the moments
N = 1...13 of the complete 2–loop anomalous dimensions. These are in accordance with
Refs. [353,358].
Finally, we show as examples the first moments of the MS–renormalized O(a3s) massive
transversity OME. Unlike the case for the vector current, the first moment does not vanish,
since there is no conservation law to enforce this.
∆T A(3),NS,MSqq,Q (1) = CFTF
{(16
27
TF (1− nf ) +
44
27
CA
)
ln3
(m2
µ2
)
+
(104
27
TF
−106
9
CA +
32
3
CF
)
ln2
(m2
µ2
)
+
[
−604
81
nfTF −
4
3
TF +
(
−2233
81
− 16ζ3
)
CA
+
(
16ζ3 +
233
9
)
CF
]
ln
(m2
µ2
)
+
(
−6556
729
+
128
27
ζ3
)
TFnf
+
(2746
729
− 224
27
ζ3
)
TF +
(8
3
B4 +
437
27
ζ3 − 24ζ4 −
34135
729
)
CA
+
(
−16
3
B4 + 24ζ4 −
278
9
ζ3 +
7511
81
)
CF
}
, (9.21)
∆T A(3),NS,MSqq,Q (2) = CFTF
{(16
9
TF (1− nf ) +
44
9
CA
)
ln3
(m2
µ2
)
+
(
−34
3
CA
+8TF
)
ln2
(m2
µ2
)
+
[
−196
9
nfTF −
92
27
TF +
(
−48ζ3 −
73
9
)
CA +
(
48ζ3
+15
)
CF
]
ln
(m2
µ2
)
+
(128
9
ζ3 −
1988
81
)
TFnf +
(338
27
− 224
9
ζ3
)
TF +
(
−56
−72ζ4 + 8B4 +
533
9
ζ3
)
CA +
(
−16B4 +
4133
27
+ 72ζ4 −
310
3
ζ3
)
CF
}
. (9.22)
The structure of the result and the contributing numbers are the same as in the unpo-
larized case, cf. Eq. (7.41). We checked the moments N = 1 . . . 4 keeping the complete
dependence on the gauge–parameter ξ and find that it cancels in the final result. Again,
we observe that the massive OMEs do not depend on ζ2, cf. Section 7.3.
Since the light flavor Wilson coefficients for the processes from which the transversity
distribution can be extracted are not known to 2– and 3–loop order, phenomenological
studies on the effect of the heavy flavor contributions cannot yet be performed. However,
the results of this Section can be used in comparisons with upcoming lattice simulations
with (2+1+1)-dynamical fermions including the charm quark. More details on this cal-
culation are given in [160].
122
10 First Steps Towards a Calculation of A(3)ij for all
Moments.
In Section 7, we described how the various massive OMEs are calculated for fixed integer
values of the Mellin variable N at 3–loop order using MATAD. The ultimate goal is to
calculate these quantities for general values of N . So far no massive single scale calculation
at O(a3s) has been performed. In the following we would like to present some first results
and a general method, which may be of use in later work calculating the general N–
dependence of the massive OMEs A(3)ij .
In Section 10.1, we solve, as an example, a 3–loop ladder graph contributing to A(3)Qg
for general values of N by direct integration, avoiding the integration–by–parts method.
In Section 10.2, Ref. [140,141], we discuss a general algorithm which allows to determine
from a sufficiently large but finite number of moments for a recurrent quantity its general
N–dependence. This algorithm has been successfully applied in [141] to reconstruct the
3–loop anomalous dimensions, [124, 125], and massless 3–loop Wilson coefficients, [115],
from their moments. These are the largest single scale quantities known at the moment
and are well suited to demonstrate the power of this formalism. Similarly, one may apply
this method to new problems of smaller size which emerge in course of the calculation of
the OMEs A(3)ij for general values of N .
10.1 Results for all–N Using Generalized Hypergeometric
Functions
In Section 6.1, we showed that there is only one basic 2–loop massive tadpole which
needs to be considered. From it, all diagrams contributing to the massive 2–loop OMEs
can be derived by attaching external quark–, gluon– and ghost–lines, respectively, and
including one operator insertion according to the Feynman rules given in Appendix B.
The corresponding parameter–integrals are then all of the same structure, Eq. (6.5). If
one knows a method to calculate the basic topology for arbitrary integer powers of the
propagators, the calculation of the 2–loop OMEs is straightforward for fixed values of N .
In the general case, we arrived at infinite sums containing the parameter N . To calculate
these sums, additional tools are needed, e.g. the program Sigma, cf. Section 6.2.
(a) (b) (c) (d)
Figure 17: Basic 3–loop topologies. Straight lines: quarks, curly lines: gluons.
The gluon loop in (a) can also be replaced by a ghost loop.
123
We would like to follow the same approach in the 3–loop case. Here, five basic topologies
need to considered, which are shown in Figures 17, 18. Diagram (a) and (b) – if one of
the quark loops corresponds to a massless quark – can be reduced to 2–loop integrals,
because the massless loop can be integrated trivially. For the remaining terms, this is not
the case. Diagrams (c) and (d) are the most complex topologies, the former giving rise to
the number B4, Eq. (4.89), whereas the latter yields single ζ–values up to weight 4, cf.
e.g. [279]. Diagram (b) – if both quarks are massive – and (e) are ladder topologies and
of less complexity. Let us, as an example, consider diagram (e).
ν4 ν2
ν5
ν1
ν3
(e)
Figure 18: 3–loop ladder graph
Our notation is the same as in Section 6.1. The scalar D–dimensional integral corre-
sponding to diagram (e) reads for arbitrary exponents of the propagators
Te =
∫∫∫ dDqdDkdDl
(2π)3D
i(−1)ν12345(m2)ν12345−3D/2(4π)3D/2
(k2)ν1((k − l)2 −m2)ν2(l2 −m2)ν3((q − l)2 −m2)ν4(q2)ν5 .
(10.1)
Again, we have attached suitable normalization factors for convenience. After loop–by–
loop integration of the momenta k, q, l (in this order) using Feynman–parameters, one
obtains after a few steps the following parameter integral
Te = Γ
[
ν12345 − 6− 3ε/2
ν1, ν2, ν3, ν4, ν5
]
∫ 1
0
dw1 . . .
∫ 1
0
dw4 θ(1− w1 − w2)
w−3−ε/2+ν121 w
−3−ε/2+ν45
2 (1− w1 − w2)ν3−1
(1 + w1
w3
1− w3
+ w2
w4
1− w4
)ν12345−6−3ε/2
×w1+ε/2−ν13 (1− w3)1+ε/2−ν2w
1+ε/2−ν5
4 (1− w4)1+ε/2−ν4 . (10.2)
The θ–function enforces w1 + w2 ≤ 1. In order to perform the {w1, w2} integration, one
considers
I =
∫ 1
0
dw1
∫ 1
0
dw2 θ(1− w1 − w2)wb−11 wb
′−1
2 (1− w1 − w2)c−b−b
′−1(1− w1x− w2y)−a .
(10.3)
124
The parameters a, b, b′, c shall be such that this integral is convergent. It can be expressed
in terms of the Appell function F1 via, [285] 35,
I = Γ
[
b, b′, c− b− b′
c
] ∞∑
m,n=0
(a)m+n(b)n(b′)m
(1)m(1)n(c)m+n
xnym (10.4)
= Γ
[
b, b′, c− b− b′
c
]
F1
[
a; b, b′; c;x, y
]
. (10.5)
The parameters x, y correspond to w3/(1−w3) and w4/(1−w4) in Eq. (10.2), respectively.
Hence the integral over these variables would yield a divergent sum. Therefore one uses
the following analytic continuation relation for F1, [285],
F1[a; b, b′; c;
x
x− 1 ,
y
y − 1] = (1− x)
b(1− y)b′F1[c− a; b, b′; c;x, y] . (10.6)
Finally one arrives at an infinite double sum
Te = Γ
[
−2− ε/2 + ν12,−2− ε/2 + ν45,−6− 3ε/2 + ν12345
ν2, ν4,−4− ε+ ν12345
]
×
∞∑
m,n=0
Γ
[
2 +m+ ε/2− ν1, 2 + n+ ε/2− ν5
1 +m, 1 + n, 2 +m+ ε/2, 2 + n+ ε/2
]
×(2 + ε/2)n+m(−2− ε/2 + ν12)m(−2− ε/2 + ν45)n
(−4− ε+ ν12345)n+m
. (10.7)
Here, we have adopted the notation for the Γ–function defined in (C.8) and (a)b is
Pochhammer’s symbol, Eq. (C.14). As one expects, Eq. (10.7) is symmetric w.r.t. ex-
changes of the indices {ν1, ν2} ↔ {ν4, ν5}. For any values of νi of the type νi = ai + biε,
with ai ∈ N, bi ∈ C, the Laurent–series in ε of Eq. (10.7) can be calculated using e.g.
Summer, [143]. We have checked Eq. (10.7) for various values of the νi using MATAD, cf.
Section 7.2.
Next, let us consider the diagram shown in Figure 19, which contributes to A(3)Qg and
derives from diagram (e). We treat the case where all exponents of the propagators are
equal to one.
p→ → p
q
q − p
k
k − p
q − l
k − l
l
l − p
Figure 19: Example 3–loop graph
35Note that Eq. (8.2.2) of Ref. [285] contains typos.
125
Including the factor i(m2)2−3ε/2(4π)3D/2 and integrating q, k, l (in this order), we obtain
Iex = Γ(2− 3ε/2)
∫ 1
0
dwi θ(1− w1 − w2)
w−ε/21 w
−ε/2
2 (1− w1 − w2)
(1 + w1
1− w3
w3
+ w2
1− w4
w4
)2−3ε/2
×w−1+ε/23 (1− w3)ε/2w
−1+ε/2
4 (1− w4)ε/2
×(1− w5w1 − w6w2 − (1− w1 − w2)w7)N , (10.8)
where all parameters w1 . . . w7 have to be integrated from 0 . . . 1. As in the 2–loop case,
(6.5), one observes that the integral–kernel given by the corresponding massive tadpole
integral (10.2) is multiplied with a polynomial containing various integration parameters
to the power N . The same holds true for the remaining 3–loop diagrams. Hence, if a
general sum representation for the corresponding tadpoles integrals is known and one
knows how to evaluate the corresponding sums, at least fixed moments of the 3–loop
massive OMEs can be calculated right away. The presence of the polynomial to the
power N (which may also involve a finite sum, cf. the Feynman–rules in Appendix B,)
complicates the calculation further. One has to find a suitable way to deal with this
situation, which depends on the integral considered. For Iex, we split it up into several
finite sums, rendering the integrals calculable in the same way as for Te. We obtain
Iex =
−Γ(2− 3ε/2)
(N + 1)(N + 2)(N + 3)
∞∑
m,n=0
{
N+2∑
t=1
(
3 +N
t
)
(t− ε/2)m(2 +N + ε/2)n+m(3− t+N − ε/2)n
(4 +N − ε)n+m
×Γ
[
t, t− ε/2, 1 +m+ ε/2, 1 + n+ ε/2, 3− t+N, 3− t+N − ε/2
4 +N − ε, 1 +m, 1 + n, 1 + t+m+ ε/2, 4− t+ n+N + ε/2
]
−
N+3∑
s=1
s−1∑
r=1
(s
r
)(
3 +N
s
)
(−1)s (r − ε/2)m(−1 + s+ ε/2)n+m(s− r − ε/2)n
(1 + s− ε)n+m
×Γ
[
r, r − ε/2, s− r, 1 +m+ ε/2, 1 + n+ ε/2, s− r − ε/2
1 +m, 1 + n, 1 + r +m+ ε/2, 1 + s− r + n+ ε/2, 1 + s− ε
]}
.
(10.9)
After expanding in ε, the summation can be performed using Sigma and the summation
techniques explained in Section 6.2. The result reads
Iex = −
4(N + 1)S1 + 4
(N + 1)2(N + 2)ζ3 +
2S2,1,1
(N + 2)(N + 3) +
1
(N + 1)(N + 2)(N + 3)
{
−2(3N + 5)S3,1 −
S14
4
+
4(N + 1)S1 − 4N
N + 1 S2,1 + 2
(
(2N + 3)S1 +
5N + 6
N + 1
)
S3
+
9 + 4N
4
S22 +
(
2
7N + 11
(N + 1)(N + 2) +
5N
N + 1S1 −
5
2
S12
)
S2 +
2(3N + 5)S12
(N + 1)(N + 2)
+
N
N + 1S1
3 +
4(2N + 3)S1
(N + 1)2(N + 2) −
(2N + 3)S4
2
+ 8
2N + 3
(N + 1)3(N + 2)
}
+O(ε) , (10.10)
126
which agrees with the fixed moments N = 1 . . . 10 obtained using MATAD, cf. Section
7.2.
We have shown that in principle one can be apply similar techniques as on the 2–loop
level, Section 6.1, to calculate the massive 3–loop OMEs considering only the five basic
topologies. In this approach the integration-by-parts method is not used. We have given
the necessary formulas for one non–trivial topology (e) and showed for on of the cases
there how the calculation proceeds keeping the all–N dependence. In order to obtain
complete results for the massive OMEs, suitable integral representations for diagrams
(b), (c) and (d) of Figure 17 have to be derived first. This will allow for a calculation of
fixed moments not relying on MATAD. Next, an automatization of the step from (10.8)
to (10.9) has to be found in order to obtain sums which can be handled e.g. by Sigma.
The latter step is not trivial, since it depends on the respective diagram and the flow of
the outer momentum p through it.
10.2 Reconstructing General–N Relations
from a Finite Number of Mellin–Moments
Higher order calculations in Quantum Field Theories easily become tedious due to the
large number of terms emerging and the sophisticated form of the contributing Feynman
parameter integrals. This applies already to zero scale and single scale quantities. Even
more this is the case for problems containing at least two scales. While in the latter case
the mathematical structure of the solution of the Feynman integrals is widely unknown,
it is explored to a certain extent for zero- and single scale quantities. Zero scale quantities
emerge as the expansion coefficients of the running couplings and masses, as fixed moments
of splitting functions, etc.. They can be expressed by rational numbers and certain special
numbers as multiple zeta-values (MZVs), [155,156] and related quantities.
Single scale quantities depend on a scale z which may be given as a ratio of Lorentz
invariants s′/s in the respective physical problem. One may perform a Mellin transform
over z, Eq. (2.65). All subsequent calculations are then carried out in Mellin space and
one assumes N ∈ N, N > 0. By this transformation, the problem at hand becomes
discrete and one may seek a description in terms of difference equations, [292]. Zero
scale problems are obtained from single scale problems treating N as a fixed integer or
considering the limit N →∞.
A main question concerning zero scale quantities is: Do the corresponding Feynman
integrals always lead to MZVs? In the lower orders this is the case. However, starting
at some order, even for single-mass problems, other special numbers will occur, [281,
361]. Since one has to known the respective basis completely, this makes it difficult to
use methods like PSLQ, [362], to determine the analytic structure of the corresponding
terms even if one may calculate them numerically at high enough precision. Zero scale
problems are much easier to calculate than single scale problems. In some analogy to the
determination of the analytic structure in zero scale problems through integer relations
over a known basis (PSLQ) one may think of an automated reconstruction of the all–N
relation out of a finite number of Mellin moments given in analytic form. This is possible
for recurrent quantities. At least up to 3-loop order, presumably even to higher orders,
single scale quantities belong to this class. Here we report on a general algorithm for this
purpose, which we applied to the problem being currently the most sophisticated one: the
anomalous dimensions and massless Wilson coefficients to 3–loop order for unpolarized
127
DIS, [115,124,125]. Details of our calculation are given in Refs. [140,141].
10.2.1 Single Scale Feynman Integrals as Recurrent Quantities
For a large variety of massless problems single scale Feynman integrals can be represented
as polynomials in the ring formed by the nested harmonic sums, cf. Appendix C.4, and
the MZVs ζa1,...,al , which we set equal to the σ–values defined in Eq. (C.35). Rational
functions in N and harmonic sums obey recurrence relations. Thus, due to closure proper-
ties, [363,364], also any polynomial expression in such terms is a solution of a recurrence.
Consider as an example the recursion
F (N + 1)− F (N) = sign(a)
N+1
(N + 1)|a| . (10.11)
It is solved by the harmonic sum Sa(N). Corresponding difference equations hold for
harmonic sums of deeper nestedness. Feynman integrals can often be decomposed into a
combination containing terms of the form
∫ 1
0
dz z
N−1 − 1
1− z H~a(z),
∫ 1
0
dz (−z)
N−1 − 1
1 + z H~a(z) , (10.12)
with H~a(z) being a harmonic polylogarithm, [314]. This structure also leads to recur-
rences, [365]. Therefore, it is very likely that single scale Feynman diagrams do always
obey difference equations.
10.2.2 Establishing and Solving Recurrences
Suppose we are given a finite array of rational numbers,
q1, q2, . . . , qK ,
which are the first terms of an infinite sequence F (N), i.e., F (1) = q1, F (2) = q2, etc. Let
us assume that F (N) represents a physical quantity and satisfies a recurrence of type
l∑
k=0
( d∑
i=0
ci,kN i
)
F (N + k) = 0 , (10.13)
which we would like to deduce from the given numbers qm. In a strict sense, this is not
possible without knowing how the sequence continues for N > K. One thing we can do is
to determine the recurrence equations satisfied by the data we are given. Any recurrence
for F (N) must certainly be among those.
To find the recurrence equations of F (N) valid for the first terms, the simplest way
to proceed is by making an ansatz with undetermined coefficients. Let us fix an order l ∈
N and a degree d ∈ N and consider the generic recurrence (10.13), where the ci,k are
unknown. For each specific choice N = 1, 2, . . . , K − l, we can evaluate the ansatz,
because we know all the values of F (N+k) in this range, and we obtain a system of K− l
homogeneous linear equations for (l + 1)(d+ 1) unknowns ci,j.
If K − l > (l + 1)(d + 1), this system is under-determined and is thus guaranteed
to have nontrivial solutions. All these solutions will be valid recurrences for F (N) for
128
N = 1, . . . , K − l, but they will most typically fail to hold beyond. If, on the other hand,
K− l ≤ (l+1)(d+1), then the system is overdetermined and nontrivial solutions are not
to be expected. But at least recurrence equations valid for all N , if there are any, must
appear among the solutions. We therefore expect in this case that the solution set will
precisely consist of the recurrences of F (N) of order l and degree d valid for all N .
As an example, let us consider the contribution to the gluon splitting function ∝ CA
at leading order, P (0)gg (N). The first 20 terms, starting with N = 3, of the sequence F (N)
are
14
5 , 215 , 18135 , 8314 , 4129630 , 31945 , 261863465 , 184212310 , 75232790090 , 712038190 , 81163790090 , 12891113860 , 293211293063060 ,
2508266
255255 , 29288626129099070 , 7045513684684 , 61125926958198140 , 1561447145860 , 4862237357446185740 , 98880845589237148 .
Making an ansatz for a recurrence of order 3 with polynomial coefficients of degree 3 leads
to an overdetermined homogeneous linear system with 16 unknowns and 17 equations.
Despite of being overdetermined and dense, this system has two linearly independent
solutions. Using bounds for the absolute value of determinants depending on the size of
a matrix and the bit size of its coefficients, one can very roughly estimate the probability
for this to happen “by coincidence” to about 10−65. And in fact, it did not happen by
coincidence. The solutions to the system correspond to the two recurrence equations
(7N3 + 113N2 + 494N + 592)F (N)− (12N3 + 233N2 + 1289N + 2156)F (N + 1)
+ (3N3 + 118N2 + 1021N + 2476)F (N + 2) + (2N3 + 2N2 − 226N − 912)F (N + 3)
= 0 (10.14)
and
(4N3 + 64N2 + 278N + 332)F (N)− (7N3 + 134N2 + 735N + 1222)F (N + 1)
+ (2N3 + 71N2 + 595N + 1418)F (N + 2) + (N3 −N2 − 138N − 528)F (N + 3)
= 0, (10.15)
which both are valid for all N ≥ 1. If we had found that the linear system did not
have a nontrivial solution, then we could have concluded that the sequence F (N) would
definitely (i.e. without any uncertainty) not satisfy a recurrence of order 3 and degree 3.
It might then still have satisfied recurrences with larger order or degree, but more terms
of the sequence had to be known for detecting those.
The method of determining (potential) recurrence equations for sequences as just de-
scribed is not new. It is known to the experimental mathematics community as automated
guessing and is frequently applied in the study of combinatorial sequences. Standard
software packages for generating functions such as gfun [363] for MAPLE or Generating-
Functions.m [364] for MATHEMATICA provide functions which take as input a finite array
of numbers, thought of as the first terms of some infinite sequence, and produce as output
recurrence equations that are, with high probability, satisfied by the infinite sequence.
These packages apply the method described above more or less literally, and this is
perfectly sufficient for small examples. But if thousands of terms of a sequence are needed,
there is no way to solve the linear systems using rational number arithmetic. Even worse,
already for medium sized problems from our collection, the size of the linear system
exceeds by far typical memory capacities of 16–64Gb. Let us consider as an example the
129
difference equation associated to the contribution of the color factor C3F for the 3-loop
Wilson coefficient C(3)2,q in unpolarized deeply inelastic scattering. 11 Tb of memory would
be required to establish (10.13) in a naive way. Therefore refined methods have to be
applied. We use arithmetic in finite fields together with Chinese remaindering, [366], which
reduces the storage requirements to a few Gb of memory. The linear system approximately
minimizes for l ≈ d. If one finds more than one recurrence the different recurrences are
joined to reduce l to a minimal value. It seems to be a general phenomenon that the
recurrence of minimal order is that with the smallest integer coefficients, cf. also [367].
For even larger problems than those dealt with in the present analysis, a series of further
technical improvements may be carried out, [368].
For the solution of the recurrence low orders are clearly preferred. It is solved in depth-
optimal ΠΣ fields, [152, 153, 296, 369, 370]; here we apply advanced symbolic summation
methods as: efficient recurrence solvers and refined telescoping algorithms. They are
available in the summation package Sigma, [153,154].
The solutions are found as linear combinations of rational terms in N combined with
functions, which cannot be further reduced in the ΠΣ fields. In the present application
they turn out to be nested harmonic sums, cf. Appendix C.4. Other or higher order
applications may lead to sums of different type as well, which are uniquely found by the
present algorithm.
10.2.3 Determination of the 3-Loop Anomalous Dimensions
and Wilson Coefficients
We apply the method to determine the unpolarized anomalous dimensions and massless
Wilson coefficients to 3–loop order. Here we apply the above method to the contributions
stemming from a single color/ζi-factor. These are 186 terms. As input we use the respec-
tive Mellin moments, which were calculated by a MAPLE–code based on the harmonic
sum representation calculated in Refs. [115, 124, 125]. We need very high moments and
calculate the input recursively. As an example, let us illustrate the size of the moments
for the C3F -contribution to the Wilson coefficient C
(3)
2,q . The highest moment required is
N = 5114. It cannot be calculated simply using Summer, [143], and we used a recursive
algorithm in MAPLE for it.
The corresponding difference equations (10.13) are determined by a recurrence finder.
Furthermore, the order of the difference equation is reduced to the smallest value possible.
The difference equations are then solved order by order using the summation package
Sigma. For the C3F -term in C
(3)
q,2 , the recurrence was established after 20.7 days of CPU
time. Here 4h were required for the modular prediction of the dimension of the system, 5.8
days were spent on solving modular linear systems, and 11 days for the modular operator
GCDs. The Chinese remainder method and rational reconstruction took 3.8 days. 140
word size primes were needed. As output one obtains a recurrence of 31 Mb, which is
of order 35 and degree 938, with a largest integer of 1227 digits. The recurrence was
solved by Sigma after 5.9 days. We reached a compactification from 289 harmonic sums
needed in [115,124,125] to 58 harmonic sums. The determination of the 3–loop anomalous
dimensions is a much smaller problem. Here the computation takes only about 18 h for
the complete result.
For the three most complicated cases, establishing and solving of the difference equa-
tions took 3 + 1 weeks each, requiring ≤ 10Gb on a 2 GHz processor. This led to an
130
overall computation time of about sixteen weeks.
In the final representation, we account for algebraic reduction, [146]. For this task
we used the package HarmonicSums, [371], which complements the functionality of Sigma.
One observes that different color factor contributions lead to the same, or nearly the
same, amount of sums for a given quantity. This points to the fact that the amount
of sums contributing, after the algebraic reduction has been carried out, is governed by
topology rather than the field- and color structures involved. The linear harmonic sum
representations used in [115,124,125] require many more sums than in the representation
reached by the present analysis. A further reduction can be obtained using the structural
relations, which leads to maximally 35 different sums up to the level of the 3–loop Wilson
coefficients, [147, 148, 365]. It is not unlikely that the present method can be applied to
single scale problems in even higher orders. As has been found before in [126–128, 137,
138,145,259,365,372,373], representing a large number of 2- and 3-loop processes in terms
of harmonic sums, the basis elements emerging are always the same.
In practice no method does yet exist to calculate such a high number of moments ab
initio as required for the determination of the all–N formulas in the 3–loop case. On
the other hand, a proof of existence has been delivered of a quite general and powerful
automatic difference-equation solver, standing rather demanding tests. It opens up good
prospects for the development of even more powerful methods, which can be applied in
establishing and solving difference equations for single scale quantities such as the classes
of Feynman–parameter integrals contributing to the massive operator matrix elements for
general values of N .
131
11 Conclusions
In this thesis, we extended the description of the contributions of a single heavy quark
to the unpolarized Wilson coefficients CS,PS,NS(q,g),2 to O(a3s). In upcoming precision analyzes
of deep–inelastic data, this will allow more precise determinations of parton distribution
functions and of the strong coupling constant. We applied a factorization relation for the
complete inclusive heavy flavor Wilson coefficients, which holds in the limit Q2 ≫ 10m2
in case of F2(x,Q2), [126], at the level of twist–2. It relates the asymptotic heavy flavor
Wilson coefficients to a convolution of the corresponding light flavor Wilson coefficients,
which are known up to O(a3s), [115], and describe all process dependence, with the mas-
sive operator matrix elements. The latter are process independent quantities and describe
all mass–dependent contributions but the power–suppressed terms ((m2/Q2)k, k ≥ 1).
They are obtained from the unpolarized twist–2 local composite operators stemming from
the light–cone expansion of the electromagnetic current between on–shell partonic states,
including virtual heavy quark lines. The first calculation of fixed moments of all 3–loop
massive OMEs is the main result of this thesis.
In Section 3.2, we applied the factorization formula at the O(a3s)–level. It holds for
the inclusive heavy flavor Wilson coefficients, including radiative corrections due to heavy
quark loops. In order to describe the production of heavy quarks in the final states only,
further assumptions have to be made. This description succeeded at the 2–loop level in
Ref. [126] because of the possible comparison with the exact calculation in Refs. [103]
and since the contributing virtual heavy flavor corrections are easily identified, cf. Sec-
tion 5.1. At O(a3s) this is not possible anymore and only the inclusive description should
be used, as has been done in Ref. [129] in order to derive heavy flavor parton densities.
These are obtained as convolutions of the light flavor densities with the massive OMEs,
cf. Section 3.3.
In Section 4, we derived and presented in detail the renormalization of the massive
operator matrix elements up to O(a3s). This led to an intermediary representation in a
defined MOM–scheme to maintain the partonic description required for the factorization
of the heavy flavor Wilson coefficients into OMEs and the light flavor Wilson coefficients.
Finally, we applied the MS–scheme for coupling constant renormalization in order to refer
to the inclusive heavy flavor Wilson coefficients and to be able to combine our results
with the light flavor Wilson coefficients, which have been calculated in the same scheme.
For mass renormalization we chose the on–mass–shell–scheme and provided in Section 5
all necessary formulas to transform between the MOM– and the on–shell–scheme, respec-
tively, and the MS–scheme.
For renormalization at O(a3s), all O(a2s) massive OMEs AQg, APSQq, ANSqq,Q, Agg,Q, Agq,Q
are needed up to O(ε) in dimensional regularization. In Section 6, we newly calculated
all the corresponding O(ε) contributions in Mellin space for general values of N . This
involved a first re–calculation of the complete terms A(2)gg,Q and A
(2)
gq,Q, in which we agree
with the literature, [129]. We made use of the representation of the Feynman–parameter
integrals in terms of generalized hypergeometric functions. The O(ε)–expansion led to
new infinite sums which had to be solved by analytic and algebraic methods. The results
can be expressed in terms of polynomials of the basic nested harmonic sums up to weight
132
w = 4 and derivatives thereof. They belong to the complexity-class of the general two-
loop Wilson coefficients or hard scattering cross sections in massless QED and QCD and
are described by six basic functions and their derivatives in Mellin space. The package
Sigma, [151–154], proved to be a useful tool to solve the sums occurring in the present
problem, leading to extensions of this code by the author.
The main part of the thesis was the calculation of fixed moments of all 3–loop massive
operator matrix elements AQg, Aqg,Q, APSQq, APSqq,Q, ANSqq,Q, Agq,Q, Agg,Q, cf. Section 7.
These are needed to describe the asymptotic heavy flavor Wilson coefficients at O(a3s) and
to derive massive quark–distributions at the same level, [129]. We developed computer
algebra codes which allow based on QGRAF, [161], the automatic generation of 3–loop
Feynman diagrams with local operator insertions. These were then projected onto massive
tadpole diagrams for fixed values of the Mellin variable N . For the final calculation of
the diagrams, use was made of the FORM–code MATAD, [164]. The representation of the
massive OMEs is available for general values ofN in analytic form, apart from the constant
terms a(3)ij of the unrenormalized 3–loop OMEs. This is achieved by combining our general
expressions for the renormalized results, the all–N results up to O(a2sε) and results given
in the literature. A number of fixed Mellin moments of the terms a(3)ij were calculated,
reaching up to N = 10, 12, 14, depending on the complexity of the corresponding operator
matrix element. The computation required about 250 CPU days on 32/64 Gb–machines.
Through the renormalization of the massive OMEs, the corresponding moments of
the complete 2-loop anomalous dimensions and the TF–terms of the 3–loop anomalous
dimensions were obtained, as were the moments of the complete anomalous dimensions
γ(2),PSqq and γ(2)qg , in which we agree with the literature. This provides a first independent
check of the moments of the fermionic contributions to the 3–loop anomalous dimensions,
which have been obtained in Refs. [111,112].
In Section 8, we presented results on the effects of heavy quarks in polarized deep–
inelastic scattering, using essentially the same description as in the unpolarized case. We
worked in the scheme for γ5 in dimensional regularization used in Ref. [165] and could
confirm the results given there for the 2–loop massive OMEs ∆APSQq and ∆AQg. Addition-
ally, we newly presented the O(ε) contributions of these terms.
We calculated the 2–loop massive OMEs of transversity for all–N and the 3–loop terms
for the moments N = 1, . . . , 13 in Section 9. This calculation is not yet of phenomenolog-
ical use, since the corresponding light flavor Wilson coefficients have not been calculated
so far. However, these results could be obtained by making only minor changes to the
computer programs written for the unpolarized case. We confirmed for the first time the
moments N = 1, . . . , 8 of the fermionic contributions to the 3–loop transversity anoma-
lous dimension obtained in Refs. [360]. Our results can, however, be used in comparison
with lattice calculations.
Several steps were undertaken towards an all–N calculation of the massive OMEs.
Four non–trivial 3–loop massive topologies contribute. We presented in an example a
first all–N result for a ladder–topology in Section 10.1.
In Section 10.2, we described a general algorithm to calculate the exact expression for
single scale quantities from a finite (suitably large) number of moments, which are zero
133
scale quantities. The latter are much more easily calculable than single scale quantities.
We applied the method to the anomalous dimensions and massless Wilson coefficients
up to 3–loop order, [115, 124, 125]. Solving 3–loop problems in this way directly is not
possible at present, since the number of required moments is too large for the methods
available. Yet this method constitutes a proof of principle and may find application in
medium–sized problems in the future.
134
A Conventions
We use natural units
~ = 1 , c = 1 , ε0 = 1 , (A.1)
where ~ denotes Planck’s constant, c the vacuum speed of light and ε0 the permittivity
of vacuum. The electromagnetic fine–structure constant α is given by
α = α′(µ2 = 0) = e
2
4πε0~c
=
e2
4π ≈
1
137.03599911(46) . (A.2)
In this convention, energies and momenta are given in the same units, electron volt (eV).
The space–time dimension is taken to be D = 4 + ε and the metric tensor gµν in
Minkowski–space is defined as
g00 = 1 , gii = −1 , i = 1 . . . D − 1 , gij = 0 , i 6= j . (A.3)
Einstein’s summation convention is used, i.e.
xµyµ :=
D−1∑
µ=0
xµyµ . (A.4)
Bold–faced symbols represent (D − 1)–dimensional spatial vectors:
x = (x0,x) . (A.5)
If not stated otherwise, Greek indices refer to the D–component space–time vector and
Latin ones to the D−1 spatial components only. The dot product of two vectors is defined
by
p.q = p0q0 −
D−1∑
i=1
piqi . (A.6)
The γ–matrices γµ are taken to be of dimension D and fulfill the anti–commutation
relation
{γµ, γν} = 2gµν . (A.7)
It follows that
γµγµ = D (A.8)
Tr (γµγν) = 4gµν (A.9)
Tr (γµγνγαγβ) = 4[gµνgαβ + gµβgνα − gµαgνβ] . (A.10)
The slash–symbol for a D-momentum p is defined by
/p := γµpµ . (A.11)
The conjugate of a bi–spinor u of a particle is given by
u = u†γ0 , (A.12)
137
where † denotes Hermitian and ∗ complex conjugation, respectively. The bi–spinors u
and v fulfill the free Dirac–equation,
(/p−m)u(p) = 0 , u(p)(/p−m) = 0 (A.13)
(/p+m)v(p) = 0 , v(p)(/p+m) = 0 . (A.14)
Bi–spinors and polarization vectors are normalized to
∑
σ
u(p, σ)u(p, σ) = /p+m (A.15)
∑
σ
v(p, σ)v(p, σ) = /p−m (A.16)
∑
λ
ǫµ(k, λ)ǫν(k, λ) = −gµν , (A.17)
where λ and σ represent the spin.
The commonly used caret “ˆ” to signify an operator, e.g. Oˆ, is omitted if confusion
is not to be expected.
The gauge symmetry group of QCD is the Lie–Group SU(3)c. We consider the general
case of SU(Nc). The non–commutative generators are denoted by ta, where a runs from
1 to N2c − 1. The generators can be represented by Hermitian, traceless matrices, [38].
The structure constants fabc and dabc of SU(Nc) are defined via the commutation and
anti–commutation relations of its generators, [190],
[ta, tb] = ifabctc (A.18)
{ta, tb} = dabctc + 1Nc
δab . (A.19)
The indices of the color matrices, in a certain representation, are denoted by i, j, k, l, ...
The color invariants most commonly encountered are
δabCA = facdf bcd (A.20)
δijCF = tailtalj (A.21)
δabTF = taiktbki . (A.22)
These constants evaluate to
CA = Nc , CF =
N2c − 1
2Nc
, TF =
1
2
. (A.23)
At higher loops, more color–invariants emerge. At 3–loop order, one additionally obtains
dabddabc = (N2c − 1)(N2c − 4)/Nc . (A.24)
In case of SU(3)c, CA = 3 , CF = 4/3 , dabcdabc = 40/3 holds.
138
B Feynman Rules
For the QCD Feynman rules, Figure 20, we follow Ref. [190], cf. also Refs. [374]. D–
dimensional momenta are denoted by pi and Lorentz-indices by Greek letters. Color
indices are a, b, ... and i, j are indices of the color matrices. Solid lines represent fermions,
wavy lines gluons and dashed lines ghosts. Arrows denote the direction of the momenta.
A factor (−1) has to be included for each closed fermion– or ghost loop.
ji
µ, a
igsγµtaji
ρ, c, p3
↓
ν, b, p2
↓
µ, a, p1
↑
−gsfabc[(p1 − p2)ρgµν + (p2 − p3)µgνρ + (p3 − p1)νgµρ]
c, pb
µ, a
−gsfabcpµ
ρ, cν, b
σ, dµ, a
−ig2s
∑
e
{
fabef cde[gµρgνσ − gµσgνρ]
+facef bde[gµνgρσ − gµσgνρ]
+fadef cbe[gµρgνσ − gµνgρσ]
}
i p j
i
p/−m+i0δij
a, µ p b, ν
i
p2+i0(−gµν + ξpµpν/(p2 + i0))δab
a p b
i
p2+i0δab
Figure 20: Feynman rules of QCD.
139
The Feynman rules for the quarkonic composite operators are given in Figure 21. Up
to O(g2) they can be found in Ref. [119] and also in [341]. Note that the O(g) term in
the former reference contains a typographical error. We have checked these terms and
agree up to normalization factors, which may be due to other conventions being applied
there. We newly derived the rule with three external gluons. The terms γ± refer to the
unpolarized (+) and polarized (−) case, respectively. Gluon momenta are taken to be
incoming.
p, jp, i
δij/∆γ±(∆ · p)N−1 , N ≥ 1
p2, jp1, i
µ, a
gtaji∆
µ/∆γ±
∑N−2
j=0 (∆ · p1)j(∆ · p2)N−j−2 , N ≥ 2
p2, jp1, i
p3, µ, a p4, ν, b
g2∆µ∆ν/∆γ±
∑N−3
j=0
∑N−2
l=j+1(∆p2)
j(∆p1)N−l−2[
(tatb)ji(∆p1 +∆p4)l−j−1 + (tbta)ji(∆p1 +∆p3)l−j−1
]
,
N ≥ 3
p2, jp1, i
p3, µ, a p4, ν, b p5, ρ, c
g3∆µ∆ν∆ρ/∆γ±
∑N−4
j=0
∑N−3
l=j+1
∑N−2
m=l+1(∆.p2)
j(∆.p1)N−m−2[
(tatbtc)ji(∆.p4 +∆.p5 +∆.p1)l−j−1(∆.p5 +∆.p1)m−l−1
+(tatctb)ji(∆.p4 +∆.p5 +∆.p1)l−j−1(∆.p4 +∆.p1)m−l−1
+(tbtatc)ji(∆.p3 +∆.p5 +∆.p1)l−j−1(∆.p5 +∆.p1)m−l−1
+(tbtcta)ji(∆.p3 +∆.p5 +∆.p1)l−j−1(∆.p3 +∆.p1)m−l−1
+(tctatb)ji(∆.p3 +∆.p4 +∆.p1)l−j−1(∆.p4 +∆.p1)m−l−1
+(tctbta)ji(∆.p3 +∆.p4 +∆.p1)l−j−1(∆.p3 +∆.p1)m−l−1
]
,
N ≥ 4
γ+ = 1 , γ− = γ5 . For transversity, one has to replace: /∆γ± → σµν∆ν .
Figure 21: Feynman rules for quarkonic composite operators. ∆ denotes a light-like 4-vector,
∆2 = 0; N is a suitably large positive integer.
140
The Feynman rules for the unpolarized gluonic composite operators are given in Figure
22. Up to O(g2), they can be found in Refs. [120] and [123]. We have checked these terms
and agree up to O(g0). At O(g), we agree with [120], but not with [123]. At O(g2), we
do not agree with either of these results, which even differ from each other36.
p, µ, ap, ν, b 1+(−1)N
2 δ
ab(∆ · p)N−2
[
gµν(∆ · p)2 − (∆µpν +∆νpµ)∆ · p + p2∆µ∆ν
]
, N ≥ 2
p1, µ, a
→
p2, ν, b
↑
p3, λ, c
←
−ig 1+(−1)N2 fabc
(
[
(∆νgλµ −∆λgµν)∆ · p1 +∆µ(p1,ν∆λ − p1,λ∆ν)
]
(∆ · p1)N−2
+∆λ
[
∆ · p1p2,µ∆ν +∆ · p2p1,ν∆µ −∆ · p1∆ · p2gµν − p1 · p2∆µ∆ν
]
×
∑N−3
j=0 (−∆ · p1)j(∆ · p2)N−3−j
+
{
p1→p2→p3→p1
µ→ν→λ→µ
}
+
{
p1→p3→p2→p1
µ→λ→ν→µ
})
, N ≥ 2
p1, µ, a
→
p2, ν, b
↑
p3, λ, c
↑
p4, σ, d
←
g2 1+(−1)
N
2
(
fabef cdeOµνλσ(p1, p2, p3, p4)
+facef bdeOµλνσ(p1, p3, p2, p4) + fadef bceOµσνλ(p1, p4, p2, p3)
)
,
Oµνλσ(p1, p2, p3, p4) = ∆ν∆λ
{
−gµσ(∆ · p3 +∆ · p4)N−2
+[p4,µ∆σ −∆ · p4gµσ]
∑N−3
i=0 (∆ · p3 +∆ · p4)i(∆ · p4)N−3−i
−[p1,σ∆µ −∆ · p1gµσ]
∑N−3
i=0 (−∆ · p1)i(∆ · p3 +∆ · p4)N−3−i
+[∆ · p1∆ · p4gµσ + p1 · p4∆µ∆σ −∆ · p4p1,σ∆µ −∆ · p1p4,µ∆σ]
×∑N−4i=0
∑i
j=0(−∆ · p1)N−4−i(∆ · p3 +∆ · p4)i−j(∆ · p4)j
}
−
{
p1↔p2
µ↔ν
}
−
{
p3↔p4
λ↔σ
}
+
{
p1↔p2, p3↔p4
µ↔ν, λ↔σ
}
, N ≥ 2
Figure 22: Feynman rules for gluonic composite operators. ∆ denotes a light-like 4-vector,
∆2 = 0; N is an integer.
36We would like to thank J. Smith for the possibility to compare with their FORM–code used in
Refs. [126,165,269,375], to which we agree.
141
C Special Functions
In the following we summarize for convenience some relations for special functions which
occur repeatedly in quantum field theory and are used in this thesis.
C.1 The Γ–function
The Γ-function, cf. [376,377], is analytic in the whole complex plane except at single poles
at the non-positive integers. Its inverse is given by Euler’s infinite product
1
Γ(z) = z exp(γEz)
∞∏
i=1
[(
1 +
z
i
)
exp(−z/i)
]
. (C.1)
The residues of the Γ-function at its poles are given by
Res[Γ(z)]z=−N =
(−1)N
N ! , N ∈ N ∪ 0 . (C.2)
In case of Re(z) > 0, it can be expressed by Euler’s integral
Γ(z) =
∫ ∞
0
exp(−t)tz−1dt , (C.3)
from which one infers the well known functional equation of the Γ-function
Γ(z + 1) = zΓ(z) , (C.4)
which is used for analytic continuation. Around z = 1, the following series expansion is
obtained
Γ(1− ε) = exp(εγE) exp
{ ∞∑
i=2
ζi
εi
i
}
, (C.5)
|ε| < 1 . (C.6)
Here and in (C.1), γE denotes the Euler-Mascheroni constant, see Eq. (4.7). In (C.5)
Riemann’s ζ–function is given by
ζk =
∞∑
i=1
1
ik , 2 ≤ k ∈ N . (C.7)
A shorthand notation for rational functions of Γ–functions is
Γ
[
a1, ..., ai
b1, ..., bj
]
:=
Γ(a1)...Γ(ai)
Γ(b1)...Γ(bj)
. (C.8)
Functions closely related to the Γ-function are the function ψ(x), the Beta-function
B(A,C) and the function β(x).
The Beta-function can be defined by Eq. (C.8)
B(A,C) = Γ
[
A,C
A+ C
]
. (C.9)
142
If Re(A),Re(C) > 0, the following integral representation is valid
B(A,C) =
∫ 1
0
dx xA−1(1− x)C−1 . (C.10)
For arbitrary values of A and C, (C.10) can be continued analytically using Eqs.
(C.1, C.9). Its expansion around singularities can be performed via Eqs. (C.2, C.5).
The ψ-function and β(x) are defined as derivatives of the Γ-function via
ψ(x) = 1
Γ(x)
d
dxΓ(x) . (C.11)
β(x) = 1
2
[
ψ
(x+ 1
2
)
− ψ
(x
2
)]
. (C.12)
C.2 The Generalized Hypergeometric Function
The generalized hypergeometric function PFQ is defined by, cf. [285,286],
PFQ
[
a1, ..., aP
b1, ..., bQ
; z
]
=
∞∑
i=0
(a1)i...(aP )i
(b1)i...(bQ)i
zi
Γ(i+ 1) . (C.13)
Here (c)n is Pochhammer’s symbol
(c)n =
Γ(c+ n)
Γ(c) , (C.14)
for which the following relation holds
(N + 1)−i =
(−1)i
(−N)i
, N ∈ N . (C.15)
In (C.13), there are P numerator parameters a1...aP , Q denominator parameters b1...bQ
and one variable z, all of which may be real or complex. Additionally, the denominator
parameters must not be negative integers, since in that case (C.13) is not defined. The
generalized hypergeometric series PFQ are evaluated at a certain value of z, which in this
thesis is always z = 1 for the final integrals.
Gauss was the first to study this kind of functions, introducing the Gauss function 2F1,
and proved the theorem, cf. [285],
2F1[a, b; c; 1] = Γ
[
c, c− a− b
c− a, c− b
]
, Re(c− a− b) > 0 (C.16)
which is called Gauss’ theorem. An integral representation for the Gauss function is given
by the integral, cf. [285],
2F1
[
a, b+ 1
c+ b+ 2; z
]
= Γ
[
c+ b+ 2
c+ 1, b+ 1
]∫ 1
0
dx xb(1− x)c(1− zx)−a , (C.17)
provided that the conditions
|z| < 1 , Re(c+ 1), Re(b+ 1) > 0 , (C.18)
143
are obeyed. Applying Eq. (C.17) recursively, one obtains the following integral represen-
tation for a general P+1FP–function
P+1FP
[
a0, a1, . . . , aP
b1, . . . , bP
; z
]
= Γ
[
b1, . . . , bP
a1, . . . , aP , b1 − a1, . . . , bP − aP
]
×
∫ 1
0
dx1 . . .
∫ 1
0
dxPxa1−11 (1− x1)b1−a1−1 . . . xaP−1P (1− xP )bP−aP−1(1− zx1 . . . xP )−a0 ,
(C.19)
under similar conditions as in Eq. (C.18).
C.3 Mellin–Barnes Integrals
For the Gauss function, there exists a representation in terms of a complex contour integral
over Γ-functions. It is given by, cf. [285],
2F1
[
a, b
c ; z
]
=
Γ(c)
2πiΓ(a)Γ(b)
∫ i∞+α
−i∞+α
Γ(a+ s)Γ(b+ s)Γ(−s)
Γ(c+ s) (−z)
sds , (C.20)
under the conditions
|z| < 1 , | arg(−z)| < π . (C.21)
(C.20) only holds if one chooses the integration contour in the complex plane and the
positive constant α in such a way that the poles of the Γ-functions containing (+s) are
separated from those arising from the Γ-functions containing (−s) and closes the contour
to the right.
Setting b = 1, c = 1 in (C.20) one obtains
1F0[a; z] =
1
(1− z)a , (C.22)
which yields the Mellin-Barnes transformation, cf. [305,307,378],
1
(X + Y )λ =
1
2πiΓ(λ)
∫ +i∞+α
−i∞+α
dsΓ(λ+ s)Γ(−s) Y
s
Xλ+s . (C.23)
The contour has to be chosen as in (C.20) and the conditions 0 < α < Re(λ) ,
| arg(Y/X)| < π have to be fulfilled.
C.4 Harmonic Sums and Nielsen–Integrals
Expanding the Γ–function in ε, its logarithmic derivatives, the ψ(k)-functions, emerge. In
many applications of perturbative QCD and QED, harmonic sums occur, cf. [142, 143],
which can be considered as generalization of the ψ-function and the β-function. These
are defined by
Sa1,... ,am(N) =
N∑
n1=1
n1∑
n2=1
. . .
nm−1∑
nm=1
(sign(a1))n1
n|a1|1
(sign(a2))n2
n|a2|2
. . . (sign(am))
nm
n|am|m
,
N ∈ N, ∀ l al ∈ Z \ 0 , (C.24)
S∅ = 1 . (C.25)
144
We adopt the convention
Sa1,... ,am ≡ Sa1,... ,am(N) , (C.26)
i.e. harmonic sums are taken at argument (N), if no argument is indicated. Related
quantities are the Z–sums defined by
Zm1,... ,mk(N) =
∑
N≥i1>i2...>ik>0
∏k
l=1[sign(ml)]il
i|ml|l
. (C.27)
The depth d and the weight w of a harmonic sum are given by
d := m , (C.28)
w :=
m∑
i=1
|ai| . (C.29)
Harmonic sums of depth d = 1 are referred to as single harmonic sums. The complete
set of algebraic relations connecting harmonic sums to other harmonic sums of the same
or lower weight is known [146], see also [143] for an implementation in FORM. Thus the
number of independent harmonic sums can be reduced significantly, e.g., for w = 3 the 18
possible harmonic sums can be expressed algebraically in terms of 8 basic harmonic sums
only. One introduces a product for the harmonic sums, the shuffle product ⊔⊔ , cf. [146].
For the product of a single and a general finite harmonic sum it is given by
Sa1(N)⊔⊔Sb1,... ,bm(N) = Sa1,b1,... ,bm(N) + Sb1,a1,b2,... ,bm(N) + . . .+ Sb1,b2,... ,bm,a1(N) .
(C.30)
For sums Sa1,... ,an(N) and Sb1,... ,bm(N) of arbitrary depth, the shuffle product is then the
sum of all harmonic sums of depth m+ n in the index set of which ai occurs left of aj for
i < j, likewise for bk and bl for k < l. Note that the shuffle product is symmetric.
One can show that the following relation holds, cf. [146],
Sa1(N) · Sb1,... ,bm(N) = Sa1(N)⊔⊔Sb1,... ,bm(N)
−Sa1∧b1,b2,... ,bm(N)− . . .− Sb1,b2,... ,a1∧bm(N) , (C.31)
where the ∧ symbol is defined as
a ∧ b = sign(a)sign(b) (|a|+ |b|) . (C.32)
Due to the additional terms containing wedges (∧) between indices, harmonic sums form
a quasi–shuffle algebra, [304]. By summing (C.31) over permutations, one obtains the
symmetric algebraic relations between harmonic sums. At depth 2 and 3 these read, [142],
Sm,n + Sn,m = SmSn + Sm∧n , (C.33)∑
perm{l,m,n}
Sl,m,n = SlSmSn +
∑
inv perm{l,m,n}
SlSm∧n + 2 Sl∧m∧n , (C.34)
which we used extensively to simplify our expressions. In (C.33, C.34), “perm” denotes
all permutations and “inv perm” invariant ones.
145
The limit N →∞ of finite harmonic sums exists only if a1 6= 1 in (C.24). Additionally,
one defines all σ-values symbolically as
σkl,... ,k1 = limN→∞Sa1,... ,al(N) . (C.35)
The finite σ-values are related to multiple ζ-values, [142, 143, 155, 379, 380], Eq. (C.7).
Further we define the symbol
σ0 :=
∞∑
i=1
1 . (C.36)
It is useful to include these σ-values into the algebra, since they allow to treat parts
of sums individually, accounting for the respective divergences, cf. also [143]. These
divergent pieces cancel in the end if the overall sum is finite.
The relation of single harmonic sums with positive or negative indices to the ψ(k)–
functions is then given by
S1(N) = ψ(N + 1) + γE , (C.37)
Sa(N) =
(−1)a−1
Γ(a) ψ
(a−1)(N + 1) + ζa , k ≥ 2 , (C.38)
S−1(N) = (−1)Nβ(N + 1)− ln(2) , (C.39)
S−a(N) = −
(−1)N+a
Γ(a) β
(a−1)(N + 1)−
(
1− 21−a
)
ζa , k ≥ 2 . (C.40)
Thus single harmonic sums can be analytically continued to complex values of N by
these relations. At higher depths, harmonic sums can be expressed in terms of Mellin–
transforms of polylogarithms and the more general Nielsen-integrals, [291, 381,382]. The
latter are defined by
Sn,p(z) =
(−1)n+p−1
(n− 1)!p!
∫ 1
0
dx
x log
n−1(x) logp(1− zx) (C.41)
and fulfill the relation
dSn,p(x)
d log(x) = Sn−1,p(x) . (C.42)
If p = 1, one obtains the polylogarithms
Lin(x) = Sn−1,1(x) , (C.43)
where
Li0(x) =
x
1− x . (C.44)
These functions do not suffice for arbitrary harmonic sums, in which case the harmonic
polylogarithms have to be considered, [314]. The latter functions obey a direct shuffle
algebra, cf. [146, 155]. The representation in terms of Mellin–transforms then allows an
analytic continuation of arbitrary harmonic sums to complex N , cf. [210, 211]. Equiva-
lently, one may express harmonic sums by factorial series, [377, 383], up to polynomials
of S1(N) and harmonic sums of lower degree, and use this representation for the analytic
continuation to N ∈ C, cf. [121,147].
146
D Finite and Infinite Sums
In this appendix, we list some examples for infinite sums which were needed in the present
analysis and are newly calculated. The calculation was done using the Sigma–package as
explained in Section 6.2. A complete set of sums contributing to the calculation of the
2–loop massive OMEs can be found in Appendix B of Refs. [128,137].
∞∑
i=1
B(N − 2, i)
(i+N)3 = (−1)
N 4S1,−2 + 2S−3 + 2ζ2S1 + 2ζ3 − 6S−2 − 3ζ2
(N − 2)(N − 1)N
+
1
(N − 2)(N − 1)N2 . (D.1)
∞∑
i=1
B(N − 2, i)
(i+N)2 S1(i+N − 2) =
(−1)N+1
(N − 2)(N − 1)N
(
8S1,−2 − 4S−3 − 4S1S−2 − 2ζ3
+2ζ2S1 − 10S−2 − 5ζ2
)
+
N2 − 3N + 3
(N − 2)(N − 1)2N2S1
− N
3 − 5N + 3
(N − 2)(N − 1)3N3 . (D.2)
∞∑
i=1
B(N, i)
i+N + 2S1(i)S1(N + i) =
(−1)N
N(N + 1)(N + 2)
(
4S−2,1 − 6S−3 − 4S−2S1 − 2ζ3
−2ζ2S1 − 2
ζ2
(N + 1) − 4
S−2
(N + 1)
)
+
−2S3 − S1S2 + ζ2S1 + 2ζ3
N + 2
+
2 + 7N + 7N2 + 5N3 +N4
N3(N + 1)3(N + 2) S1
+2
2 + 7N + 9N2 + 4N3 +N4
N4(N + 1)3(N + 2) . (D.3)
∞∑
i=1
S1(i+N)S21(i)
i+N =
σ41
4
− 3ζ
2
2
4
+
( 2
N − 2S1
)
ζ3 +
(S1
N −
S21
2
− S2
2
)
ζ2 +
S31
N
−S
4
1
4
+ S21
(
− 1N2 −
3S2
2
)
− S2N2 −
S22
4
− S2,1N
+S1
(
3
S2
N + S2,1 − 2S3
)
+ 2
S3
N + S3,1 − S4 . (D.4)
∞∑
i=1
(
S1(i+N)− S1(i)
)3
= −3
2
S21 − S31 −
1
2
S2 + 3NS2,1 −NS3 +Nζ3 . (D.5)
147
∞∑
k=1
B(k + ε/2, N + 1)
N + k = (−1)
N
[
2S−2 + ζ2
]
+
ε
2
(−1)N
[
−ζ3 + ζ2S1 + 2S1,−2 − 2S−2,1
]
+
ε2
4
(−1)N
[
2
5
ζ22 − ζ3S1 + ζ2S1,1
+2
{
S1,1,−2 + S−2,1,1 − S1,−2,1
}]
+ ε3(−1)N
[
−ζ5
8
+
S1
20
ζ22 −
S1,1
8
ζ3 +
S1,1,1
8
ζ2
+
S1,−2,1,1 + S1,1,1,−2 − S−2,1,1,1 − S1,1,−2,1
4
]
+O(ε4) . (D.6)
An example for a double infinite sum we encountered is given by
N
∞∑
i,j=1
S1(i)S1(i+ j +N)
i(i+ j)(j +N) = 4S2,1,1 − 2S3,1 + S1
(
−3S2,1 +
4S3
3
)
− S4
2
−S22 + S21S2 +
S41
6
+ 6S1ζ3 + ζ2
(
2S21 + S2
)
. (D.7)
A detailed description of the method to calculate this sum can be found in Appendix B
of Ref. [137].
148
E Moments of the Fermionic Contributions to the
3–Loop Anomalous Dimensions
The pole terms of the unrenormalized OMEs in our calculation agree with the general
structure we presented in Eqs. (4.94, 4.103, 4.104, 4.116, 4.117, 4.124, 4.134). Using the
lower order renormalization coefficients and the constant terms of the 2–loop results, Eqs.
(6.34, 6.53, 6.60, 6.68, 6.80), allows to determine the TF–terms of the 3–loop anomalous
dimensions for fixed values of N . All our results agree with the results of Refs. [111,112,
124,125,317,318]. Note that in this way we obtain the complete expressions for the terms
γ(2)qg and γ(2),PSqq , since they always involve an overall factor TF . For them we obtain
(i) γˆ(2)qg :
γˆ(2)qg (2) = TF
[
(1 + 2nf )TF
(8464
243
CA −
1384
243
CF
)
+
ζ3
3
(
−416CACF + 288C2A
+128C2F
)
− 7178
81
C2A +
556
9
CACF −
8620
243
C2F
]
, (E.1)
γˆ(2)qg (4) = TF
[
(1 + 2nf )TF
(4481539
303750
CA +
9613841
3037500
CF
)
+
ζ3
25
(
2832C2A − 3876CACF
+1044C2F
)
− 295110931
3037500
C2A +
278546497
2025000
CACF −
757117001
12150000
C2F
]
, (E.2)
γˆ(2)qg (6) = TF
[
(1 + 2nf )TF
(86617163
11668860
CA +
1539874183
340341750
CF
)
+
ζ3
735
(
69864C2A
−94664CACF + 24800C2F
)
− 58595443051
653456160
C2A +
1199181909343
8168202000
CACF
−2933980223981
40841010000
C2F
]
, (E.3)
γˆ(2)qg (8) = TF
[
(1 + 2nf )TF
(10379424541
2755620000
CA +
7903297846481
1620304560000
CF
)
+ζ3
(128042
1575
C2A −
515201
4725
CACF +
749
27
C2F
)
− 24648658224523
289340100000
C2A
+
4896295442015177
32406091200000
CACF −
4374484944665803
56710659600000
C2F
]
, (E.4)
γˆ(2)qg (10) = TF
[
(1 + 2nf )TF
(1669885489
988267500
CA +
1584713325754369
323600780868750
CF
)
+ζ3
(1935952
27225
C2A −
2573584
27225
CACF +
70848
3025
C2F
)
− 21025430857658971
255684567600000
C2A
149
+
926990216580622991
6040547909550000
CACF −
1091980048536213833
13591232796487500
C2F
]
. (E.5)
(ii) γˆ(2),PSqq :
γˆ(2),PSqq (2) = TFCF
[
−(1 + 2nf )TF
5024
243
+
256
3
(
CF − CA
)
ζ3 +
10136
243
CA
−14728
243
CF
]
, (E.6)
γˆ(2),PSqq (4) = TFCF
[
−(1 + 2nf )TF
618673
151875
+
968
75
(
CF − CA
)
ζ3 +
2485097
506250
CA
−2217031
675000
CF
]
, (E.7)
γˆ(2),PSqq (6) = TFCF
[
−(1 + 2nf )TF
126223052
72930375
+
3872
735
(
CF − CA
)
ζ3
+
1988624681
4084101000
CA +
11602048711
10210252500
CF
]
, (E.8)
γˆ(2),PSqq (8) = TFCF
[
−(1 + 2nf )TF
13131081443
13502538000
+
2738
945
(
CF − CA
)
ζ3
−343248329803
648121824000
CA +
39929737384469
22684263840000
CF
]
, (E.9)
γˆ(2),PSqq (10) = TFCF
[
−(1 + 2nf )TF
265847305072
420260754375
+
50176
27225
(
CF − CA
)
ζ3
−1028766412107043
1294403123475000
CA +
839864254987192
485401171303125
CF
]
, (E.10)
γˆ(2),PSqq (12) = TFCF
[
−(1 + 2nf )TF
2566080055386457
5703275664286200
+
49928
39039
(
CF − CA
)
ζ3
−69697489543846494691
83039693672007072000
CA +
86033255402443256197
54806197823524667520
CF
]
. (E.11)
For the remaining terms, only the projection onto the color factor TF can be obtained :
(iii) γˆ(2),NS,+qq :
γˆ(2),NS,+qq (2) = TFCF
[
−(1 + 2nf )TF
1792
243
+
256
3
(
CF − CA
)
ζ3 −
12512
243
CA
150
−13648
243
CF
]
, (E.12)
γˆ(2),NS,+qq (4) = TFCF
[
−(1 + 2nf )TF
384277
30375
+
2512
15
(
CF − CA
)
ζ3
−8802581
121500
CA −
165237563
1215000
CF
]
, (E.13)
γˆ(2),NS,+qq (6) = TFCF
[
−(1 + 2nf )TF
160695142
10418625
+
22688
105
(
CF − CA
)
ζ3
−13978373
171500
CA −
44644018231
243101250
CF
]
, (E.14)
γˆ(2),NS,+qq (8) = TFCF
[
−(1 + 2nf )TF
38920977797
2250423000
+
79064
315
(
CF − CA
)
ζ3
−1578915745223
18003384000
CA −
91675209372043
420078960000
CF
]
, (E.15)
γˆ(2),NS,+qq (10) = TFCF
[
−(1 + 2nf )TF
27995901056887
1497656506500
+
192880
693
(
CF − CA
)
ζ3
−9007773127403
97250422500
CA −
75522073210471127
307518802668000
CF
]
, (E.16)
γˆ(2),NS,+qq (12) = TFCF
[
−(1 + 2nf )TF
65155853387858071
3290351344780500
+
13549568
45045
(
CF − CA
)
ζ3
−25478252190337435009
263228107582440000
CA −
35346062280941906036867
131745667845011220000
CF
]
,
(E.17)
γˆ(2),NS,+qq (14) = TFCF
[
−(1 + 2nf )TF
68167166257767019
3290351344780500
+
2881936
9009
(
CF − CA
)
ζ3
−92531316363319241549
921298376538540000
CA −
37908544797975614512733
131745667845011220000
CF
]
.
(E.18)
(iv) γˆ(2),NS,−qq :
γˆ(2),NS,−qq (1) = 0 , (E.19)
γˆ(2),NS,−qq (3) = TFCF
[
−(1 + 2nf )TF
2569
243
+
400
3
(
CF − CA
)
ζ3 −
62249
972
CA
151
−203627
1944
CF
]
, (E.20)
γˆ(2),NS,−qq (5) = TFCF
[
−(1 + 2nf )TF
431242
30375
+
2912
15
(
CF − CA
)
ζ3
−38587
500
CA −
5494973
33750
CF
]
, (E.21)
γˆ(2),NS,−qq (7) = TFCF
[
−(1 + 2nf )TF
1369936511
83349000
+
8216
35
(
CF − CA
)
ζ3
−2257057261
26671680
CA −
3150205788689
15558480000
CF
]
, (E.22)
γˆ(2),NS,−qq (9) = TFCF
[
−(1 + 2nf )TF
20297329837
1125211500
+
16720
63
(
CF − CA
)
ζ3
−126810403414
1406514375
CA −
1630263834317
7001316000
CF
]
, (E.23)
γˆ(2),NS,−qq (11) = TFCF
[
−(1 + 2nf )TF
28869611542843
1497656506500
+
1005056
3465
(
CF − CA
)
ζ3
−1031510572686647
10892047320000
CA −
1188145134622636787
4612782040020000
CF
]
, (E.24)
γˆ(2),NS,−qq (13) = TFCF
[
−(1 + 2nf )TF
66727681292862571
3290351344780500
+
13995728
45045
(
CF − CA
)
ζ3
−90849626920977361109
921298376538540000
CA −
36688336888519925613757
131745667845011220000
CF
]
.
(E.25)
(v) γˆ(2)gg :
γˆ(2)gg (2) = TF
[
(1 + 2nf )TF
(
−8464
243
CA +
1384
243
CF
)
+
ζ3
3
(
−288C2A + 416CACF
−128C2F
)
+
7178
81
C2A −
556
9
CACF +
8620
243
C2F
]
, (E.26)
γˆ(2)gg (4) = TF
[
(1 + 2nf )TF
(
−757861
30375
CA −
979774
151875
CF
)
+
ζ3
25
(
−6264C2A + 6528CACF
152
−264C2F
)
+
53797499
607500
C2A −
235535117
1012500
CACF +
2557151
759375
C2F
]
, (E.27)
γˆ(2)gg (6) = TF
[
(1 + 2nf )TF
(
−52781896
2083725
CA −
560828662
72930375
CF
)
+ ζ3
(
−75168
245
C2A
+
229024
735
CACF −
704
147
C2F
)
+
9763460989
116688600
C2A −
9691228129
32672808
CACF
−11024749151
10210252500
C2F
]
, (E.28)
γˆ(2)gg (8) = TF
[
(1 + 2nf )TF
(
−420970849
16074450
CA −
6990254812
843908625
CF
)
+ζ3
(
−325174
945
C2A +
327764
945
CACF −
74
27
C2F
)
+
2080130771161
25719120000
C2A
−220111823810087
648121824000
CACF −
14058417959723
5671065960000
C2F
]
, (E.29)
γˆ(2)gg (10) = TF
[
(1 + 2nf )TF
(
−2752314359
101881395
CA −
3631303571944
420260754375
CF
)
+ζ3
(
−70985968
190575
C2A +
71324656
190575
CACF −
5376
3025
C2F
)
+
43228502203851731
549140719050000
C2A −
3374081335517123191
9060821864325000
CFCA
−3009386129483453
970802342606250
C2F
]
. (E.30)
(vi) γˆ(2)gq :
γˆ(2)gq (2) = TFCF
[
(1 + 2nf )TF
2272
81
+
512
3
(
CA − CF
)
ζ3 +
88
9
CA +
28376
243
CF
]
,
(E.31)
γˆ(2)gq (4) = TFCF
[
(1 + 2nf )TF
109462
10125
+
704
15
(
CA − CF
)
ζ3 −
799
12150
CA
+
14606684
759375
CF
]
, (E.32)
γˆ(2)gq (6) = TFCF
[
(1 + 2nf )TF
22667672
3472875
+
2816
105
(
CA − CF
)
ζ3 −
253841107
145860750
CA
153
+
20157323311
2552563125
CF
]
, (E.33)
γˆ(2)gq (8) = TFCF
[
(1 + 2nf )TF
339184373
75014100
+
1184
63
(
CA − CF
)
ζ3
−3105820553
1687817250
CA +
8498139408671
2268426384000
CF
]
, (E.34)
γˆ(2)gq (10) = TFCF
[
(1 + 2nf )TF
1218139408
363862125
+
7168
495
(
CA − CF
)
ζ3
−18846629176433
11767301122500
CA +
529979902254031
323600780868750
CF
]
, (E.35)
γˆ(2)gq (12) = TFCF
[
(1 + 2nf )TF
13454024393417
5222779912350
+
5056
429
(
CA − CF
)
ζ3
−64190493078139789
48885219979596000
CA +
1401404001326440151
3495293228541114000
CF
]
, (E.36)
γˆ(2)gq (14) = TFCF
[
(1 + 2nf )TF
19285002274
9495963477
+
13568
1365
(
CA − CF
)
ζ3
−37115284124613269
35434552943790000
CA −
40163401444446690479
104797690331258925000
CF
]
. (E.37)
154
F The O(ε0) Contributions to ˆˆA
(3)
ij
Finally, we present all moments we calculated. We only give the constant term in ε of
the unrenormalized result, cf. Eqs. (4.94, 4.103, 4.104, 4.116, 4.117, 4.124, 4.134). These
terms have to be inserted into the general results on the renormalized level, cf. Eqs. (4.96,
4.105, 4.106, 4.118, 4.119, 4.126, 4.137). We obtain
(i) a(3),PSQq :
a(3),PSQq (2) = TFCFCA
(
117290
2187
+
64
9
B4 − 64ζ4 +
1456
27
ζ3 +
224
81
ζ2
)
+TFC2F
(
42458
243
− 128
9
B4 + 64ζ4 −
9664
81
ζ3 +
704
27
ζ2
)
+T 2FCF
(
−36880
2187
− 4096
81
ζ3 −
736
81
ζ2
)
+nfT 2FCF
(
−76408
2187
+
896
81
ζ3 −
112
81
ζ2
)
, (F.1)
a(3),PSQq (4) = TFCFCA
(
23115644813
1458000000
+
242
225
B4 −
242
25
ζ4 +
1403
180
ζ3 +
283481
270000
ζ2
)
+TFC2F
(
−181635821459
8748000000
− 484
225
B4 +
242
25
ζ4 +
577729
40500
ζ3
+
4587077
1620000
ζ2
)
+ T 2FCF
(
−2879939
5467500
− 15488
2025
ζ3 −
1118
2025
ζ2
)
+nfT 2FCF
(
−474827503
109350000
+
3388
2025
ζ3 −
851
20250
ζ2
)
, (F.2)
a(3),PSQq (6) = TFCFCA
(
111932846538053
10291934520000
+
968
2205
B4 −
968
245
ζ4 +
2451517
1852200
ζ3
+
5638039
7779240
ζ2
)
+ TFC2F
(
−238736626635539
5145967260000
− 1936
2205
B4 +
968
245
ζ4
+
19628197
555660
ζ3 +
8325229
10804500
ζ2
)
+ T 2FCF
(
146092097
1093955625
− 61952
19845
ζ3
− 7592
99225
ζ2
)
+ nfT 2FCF
(
−82616977
45378900
+
1936
2835
ζ3 −
16778
694575
ζ2
)
, (F.3)
a(3),PSQq (8) = TFCFCA
(
314805694173451777
32665339929600000
+
1369
5670
B4 −
1369
630
ζ4
155
−202221853
137168640
ζ3 +
1888099001
3429216000
ζ2
)
+ TFC2F
(
−25652839216168097959
457314759014400000
−1369
2835
B4 +
1369
630
ζ4 +
2154827491
48988800
ζ3 +
12144008761
48009024000
ζ2
)
+T 2FCF
(
48402207241
272211166080
− 43808
25515
ζ3 +
1229
142884
ζ2
)
+nfT 2FCF
(
−16194572439593
15122842560000
+
1369
3645
ζ3 −
343781
14288400
ζ2
)
, (F.4)
a(3),PSQq (10) = TFCFCA
(
989015303211567766373
107642563748181000000
+
12544
81675
B4 −
12544
9075
ζ4
−1305489421
431244000
ζ3 +
2903694979
6670805625
ζ2
)
+ TFC2F
(
−4936013830140976263563
80731922811135750000
−25088
81675
B4 +
12544
9075
ζ4 +
94499430133
1940598000
ζ3 +
282148432
4002483375
ζ2
)
+T 2FCF
(
430570223624411
2780024890190625
− 802816
735075
ζ3 +
319072
11026125
ζ2
)
+nfT 2FCF
(
−454721266324013
624087220246875
+
175616
735075
ζ3 −
547424
24257475
ζ2
)
, (F.5)
a(3),PSQq (12) = TFCFCA
(
968307050156826905398206547
107727062441920086477312000
+
12482
117117
B4 −
12482
13013
ζ4
−64839185833913
16206444334080
ζ3 +
489403711559293
1382612282251200
ζ2
)
+TFC2F
(
−190211298439834685159055148289
2962494217152802378126080000
− 24964
117117
B4 +
12482
13013
ζ4
+
418408135384633
8103222167040
ζ3 −
72904483229177
15208735104763200
ζ2
)
+T 2FCF
(
1727596215111011341
13550982978344011200
− 798848
1054053
ζ3 +
11471393
347837490
ζ2
)
+nfT 2FCF
(
− 6621557709293056160177
12331394510293050192000
+
24964
150579
ζ3
− 1291174013
63306423180
ζ2
)
. (F.6)
156
(ii) a(3),PSqq,Q :
a(3),PSqq,Q (2) = nfT 2FCF
(
−100096
2187
+
896
81
ζ3 −
256
81
ζ2
)
, (F.7)
a(3),PSqq,Q (4) = nfT 2FCF
(
−118992563
21870000
+
3388
2025
ζ3 −
4739
20250
ζ2
)
, (F.8)
a(3),PSqq,Q (6) = nfT 2FCF
(
−17732294117
10210252500
+
1936
2835
ζ3 −
9794
694575
ζ2
)
, (F.9)
a(3),PSqq,Q (8) = nfT 2FCF
(
−20110404913057
27221116608000
+
1369
3645
ζ3 +
135077
4762800
ζ2
)
, (F.10)
a(3),PSqq,Q (10) = nfT 2FCF
(
−308802524517334
873722108345625
+
175616
735075
ζ3 +
4492016
121287375
ζ2
)
, (F.11)
a(3),PSqq,Q (12) = nfT 2FCF
(
− 6724380801633998071
38535607844665781850
+
24964
150579
ζ3 +
583767694
15826605795
ζ2
)
,
(F.12)
a(3),PSqq,Q (14) = nfT 2FCF
(
− 616164615443256347333
7545433703850642600000
+
22472
184275
ζ3
+
189601441
5533778250
ζ2
)
.
(F.13)
(iii) a(3)Qg :
a(3)Qg(2) = TFC2A
(
170227
4374
− 88
9
B4 + 72ζ4 −
31367
324
ζ3 +
1076
81
ζ2
)
+TFCFCA
(
−154643
729
+
208
9
B4 − 104ζ4 +
7166
27
ζ3 − 54ζ2
)
+TFC2F
(
−15574
243
− 64
9
B4 + 32ζ4 −
3421
81
ζ3 +
704
27
ζ2
)
+ T 2FCA
(
−20542
2187
+
4837
162
ζ3 −
670
81
ζ2
)
+ T 2FCF
(
11696
729
+
569
81
ζ3 +
256
9
ζ2
)
− 64
27
T 3F ζ3
+nfT 2FCA
(
−6706
2187
− 616
81
ζ3 −
250
81
ζ2
)
+ nfT 2FCF
(
158
243
+
896
81
ζ3 +
40
9
ζ2
)
,
(F.14)
157
a(3)Qg(4) = TFC2A
(
−425013969083
2916000000
− 559
50
B4 +
2124
25
ζ4 −
352717109
5184000
ζ3
−4403923
270000
ζ2
)
+ TFCFCA
(
−95898493099
874800000
+
646
25
B4 −
2907
25
ζ4
+
172472027
864000
ζ3 −
923197
40500
ζ2
)
+ TFC2F
(
−87901205453
699840000
− 174
25
B4
+
783
25
ζ4 +
937829
12960
ζ3 +
62019319
3240000
ζ2
)
+ T 2FCA
(
960227179
29160000
+
1873781
51840
ζ3
+
120721
13500
ζ2
)
+ T 2FCF
(
−1337115617
874800000
+
73861
324000
ζ3 +
8879111
810000
ζ2
)
−176
135
T 3F ζ3 + nfT 2FCA
(
947836283
72900000
− 18172
2025
ζ3 −
11369
13500
ζ2
)
+nfT 2FCF
(
8164734347
4374000000
+
130207
20250
ζ3 +
1694939
810000
ζ2
)
, (F.15)
a(3)Qg(6) = TFC2A
(
−48989733311629681
263473523712000
− 2938
315
B4 +
17466
245
ζ4 −
748603616077
11379916800
ζ3
−93013721
3457440
ζ2
)
+ TFCFCA
(
712876107019
55319040000
+
47332
2205
B4 −
23666
245
ζ4
+
276158927731
1896652800
ζ3 +
4846249
11113200
ζ2
)
+ TFC2F
(
−38739867811364113
137225793600000
−2480
441
B4 +
1240
49
ζ4 +
148514798653
711244800
ζ3 +
4298936309
388962000
ζ2
)
+T 2FCA
(
706058069789557
18819537408000
+
3393002903
116121600
ζ3 +
6117389
555660
ζ2
)
+T 2FCF
(
−447496496568703
54890317440000
− 666922481
284497920
ζ3 +
49571129
9724050
ζ2
)
−176
189
T 3F ζ3 + nfT 2FCA
(
12648331693
735138180
− 4433
567
ζ3 +
23311
111132
ζ2
)
+nfT 2FCF
(
−8963002169173
1715322420000
+
111848
19845
ζ3 +
11873563
19448100
ζ2
)
, (F.16)
a(3)Qg(8) = TFC2A
(
−358497428780844484961
2389236291993600000
− 899327
113400
B4 +
64021
1050
ζ4
158
−12321174818444641
112368549888000
ζ3 −
19581298057
612360000
ζ2
)
+TFCFCA
(
941315502886297276939
8362327021977600000
+
515201
28350
B4 −
515201
6300
ζ4
+
5580970944338269
56184274944000
ζ3 +
495290785657
34292160000
ζ2
)
+TFC2F
(
−23928053971795796451443
36585180721152000000
− 749
162
B4 +
749
36
ζ4
+
719875828314061
1404606873600
ζ3 +
2484799653079
480090240000
ζ2
)
+ T 2FCA
(
156313300657148129
4147979673600000
+
58802880439
2388787200
ζ3 +
46224083
4082400
ζ2
)
+ T 2FCF
(
−986505627362913047
87107573145600000
−185046016777
50164531200
ζ3 +
7527074663
3429216000
ζ2
)
− 296
405
T 3F ζ3
+nfT 2FCA
(
24718362393463
1322697600000
− 125356
18225
ζ3 +
2118187
2916000
ζ2
)
+nfT 2FCF
(
−291376419801571603
32665339929600000
+
887741
174960
ζ3 −
139731073
1143072000
ζ2
)
, (F.17)
a(3)Qg(10) = TFC2A
(
6830363463566924692253659
685850575063965696000000
− 563692
81675
B4
+
483988
9075
ζ4 −
103652031822049723
415451499724800
ζ3 −
20114890664357
581101290000
ζ2
)
+TFCFCA
(
872201479486471797889957487
2992802509370032128000000
+
1286792
81675
B4
−643396
9075
ζ4 −
761897167477437907
33236119977984000
ζ3 +
15455008277
660342375
ζ2
)
+TFC2F
(
−247930147349635960148869654541
148143724213816590336000000
− 11808
3025
B4
+
53136
3025
ζ4 +
9636017147214304991
7122025709568000
ζ3 +
14699237127551
15689734830000
ζ2
)
159
+T 2FCA
(
23231189758106199645229
633397356480430080000
+
123553074914173
5755172290560
ζ3 +
4206955789
377338500
ζ2
)
+T 2FCF
(
−18319931182630444611912149
1410892611560158003200000
− 502987059528463
113048027136000
ζ3
+
24683221051
46695639375
ζ2
)
− 896
1485
T 3F ζ3 + nfT 2FCA
(
297277185134077151
15532837481700000
−1505896
245025
ζ3 +
189965849
188669250
ζ2
)
+ nfT 2FCF
(
−1178560772273339822317
107642563748181000000
+
62292104
13476375
ζ3 −
49652772817
93391278750
ζ2
)
. (F.18)
(iv) a(3)qg,Q :
a(3)qg,Q(2) = nfT 2FCA
(
83204
2187
− 616
81
ζ3 +
290
81
ζ2
)
+nfT 2FCF
(
−5000
243
+
896
81
ζ3 −
4
3
ζ2
)
, (F.19)
a(3)qg,Q(4) = nfT 2FCA
(
835586311
14580000
− 18172
2025
ζ3 +
71899
13500
ζ2
)
+nfT 2FCF
(
−21270478523
874800000
+
130207
20250
ζ3 −
1401259
810000
ζ2
)
, (F.20)
a(3)qg,Q(6) = nfT 2FCA
(
277835781053
5881105440
− 4433
567
ζ3 +
2368823
555660
ζ2
)
+nfT 2FCF
(
−36123762156197
1715322420000
+
111848
19845
ζ3 −
26095211
19448100
ζ2
)
, (F.21)
a(3)qg,Q(8) = nfT 2FCA
(
157327027056457
3968092800000
− 125356
18225
ζ3 +
7917377
2268000
ζ2
)
+nfT 2FCF
(
−201046808090490443
10888446643200000
+
887741
174960
ζ3
−3712611349
3429216000
ζ2
)
, (F.22)
a(3)qg,Q(10) = nfT 2FCA
(
6542127929072987
191763425700000
− 1505896
245025
ζ3 +
1109186999
377338500
ζ2
)
160
+nfT 2FCF
(
−353813854966442889041
21528512749636200000
+
62292104
13476375
ζ3 −
83961181063
93391278750
ζ2
)
.
(F.23)
(v) a(3)gq,Q :
a(3)gq,Q(2) = TFCFCA
(
−126034
2187
− 128
9
B4 + 128ζ4 −
9176
81
ζ3 −
160
81
ζ2
)
+TFC2F
(
−741578
2187
+
256
9
B4 − 128ζ4 +
17296
81
ζ3 −
4496
81
ζ2
)
+T 2FCF
(
21872
729
+
2048
27
ζ3 +
416
27
ζ2
)
+ nfT 2FCF
(
92200
729
− 896
27
ζ3
+
208
27
ζ2
)
, (F.24)
a(3)gq,Q(4) = TFCFCA
(
−5501493631
218700000
− 176
45
B4 +
176
5
ζ4 −
8258
405
ζ3
+
13229
8100
ζ2
)
+ TFC2F
(
−12907539571
145800000
+
352
45
B4 −
176
5
ζ4
+
132232
2025
ζ3 −
398243
27000
ζ2
)
+T 2FCF
(
1914197
911250
+
2816
135
ζ3 +
1252
675
ζ2
)
+nfT 2FCF
(
50305997
1822500
− 1232
135
ζ3 +
626
675
ζ2
)
, (F.25)
a(3)gq,Q(6) = TFCFCA
(
−384762916141
24504606000
− 704
315
B4 +
704
35
ζ4 −
240092
19845
ζ3
+
403931
463050
ζ2
)
+ TFC2F
(
−40601579774533
918922725000
+
1408
315
B4 −
704
35
ζ4
+
27512264
694575
ζ3 −
24558841
3472875
ζ2
)
+ T 2FCF
(
−279734446
364651875
+
11264
945
ζ3 +
8816
33075
ζ2
)
+ nfT 2FCF
(
4894696577
364651875
− 704
135
ζ3
161
+
4408
33075
ζ2
)
, (F.26)
a(3)gq,Q(8) = TFCFCA
(
−10318865954633473
816633498240000
− 296
189
B4 +
296
21
ζ4 −
1561762
178605
ζ3
+
30677543
85730400
ζ2
)
+ TFC2F
(
−305405135103422947
11432868975360000
+
592
189
B4 −
296
21
ζ4
+
124296743
4286520
ζ3 −
4826251837
1200225600
ζ2
)
+ T 2FCF
(
−864658160833
567106596000
+
4736
567
ζ3 −
12613
59535
ζ2
)
+ nfT 2FCF
(
9330164983967
1134213192000
− 296
81
ζ3
− 12613
119070
ζ2
)
, (F.27)
a(3)gq,Q(10) = TFCFCA
(
−1453920909405842897
130475834846280000
− 1792
1485
B4 +
1792
165
ζ4 −
1016096
147015
ζ3
+
871711
26952750
ζ2
)
+ TFC2F
(
−11703382372448370173
667205973645750000
+
3584
1485
B4
−1792
165
ζ4 +
62282416
2695275
ζ3 −
6202346032
2547034875
ζ2
)
+ T 2FCF
(
−1346754066466
756469357875
+
28672
4455
ζ3 −
297472
735075
ζ2
)
+ nfT 2FCF
(
4251185859247
756469357875
− 12544
4455
ζ3
−148736
735075
ζ2
)
, (F.28)
a(3)gq,Q(12) = TFCFCA
(
−1515875996003174876943331
147976734123516602304000
− 1264
1287
B4 +
1264
143
ζ4
−999900989
173918745
ζ3 −
693594486209
3798385390800
ζ2
)
+TFC2F
(
−48679935129017185612582919
4069360188396706563360000
+
2528
1287
B4 −
1264
143
ζ4
+
43693776149
2260943685
ζ3 −
2486481253717
1671289571952
ζ2
)
+T 2FCF
(
−2105210836073143063
1129248581528667600
+
20224
3861
ζ3 −
28514494
57972915
ζ2
)
+nfT 2FCF
(
9228836319135394697
2258497163057335200
− 8848
3861
ζ3 −
14257247
57972915
ζ2
)
, (F.29)162
a(3)gq,Q(14) = TFCFCA
(
−1918253569538142572718209
199199449781656964640000
− 3392
4095
B4
+
3392
455
ζ4 −
2735193382
553377825
ζ3 −
1689839813797
5113211103000
ζ2
)
+TFC2F
(
−143797180510035170802620917
17429951855894984406000000
+
6784
4095
B4
−3392
455
ζ4 +
12917466836
774728955
ζ3 −
4139063104013
4747981738500
ζ2
)
+T 2FCF
(
−337392441268078561
179653183425015300
+
54272
12285
ζ3 −
98112488
184459275
ζ2
)
+nfT 2FCF
(
222188365726202803
71861273370006120
− 3392
1755
ζ3 −
49056244
184459275
ζ2
)
. (F.30)
(vi) a(3)gg,Q :
a(3)gg,Q(2) = TFC2A
(
−170227
4374
+
88
9
B4 − 72ζ4 +
31367
324
ζ3 −
1076
81
ζ2
)
+TFCFCA
(
154643
729
− 208
9
B4 + 104ζ4 −
7166
27
ζ3 + 54ζ2
)
+TFC2F
(
15574
243
+
64
9
B4 − 32ζ4 +
3421
81
ζ3 −
704
27
ζ2
)
+T 2FCA
(
20542
2187
− 4837
162
ζ3 +
670
81
ζ2
)
+ T 2FCF
(
−11696
729
− 569
81
ζ3
−256
9
ζ2
)
+
64
27
T 3F ζ3 + nfT 2FCA
(
−76498
2187
+
1232
81
ζ3 −
40
81
ζ2
)
+nfT 2FCF
(
538
27
− 1792
81
ζ3 −
28
9
ζ2
)
, (F.31)
a(3)gg,Q(4) = TFC2A
(
29043652079
291600000
+
533
25
B4 −
4698
25
ζ4 +
610035727
2592000
ζ3
+
92341
6750
ζ2
)
+ TFCFCA
(
272542528639
874800000
− 1088
25
B4 +
4896
25
ζ4
−3642403
17280
ζ3 +
73274237
810000
ζ2
)
+ TFC2F
(
41753961371
1749600000
163
+
44
25
B4 −
198
25
ζ4 +
2676077
64800
ζ3 −
4587077
1620000
ζ2
)
+ T 2FCA
(
−1192238291
14580000
−2134741
25920
ζ3 −
16091
675
ζ2
)
+ T 2FCF
(
−785934527
43740000
− 32071
8100
ζ3
−226583
8100
ζ2
)
+
64
27
T 3F ζ3 + nfT 2FCA
(
−271955197
1822500
+
13216
405
ζ3
−6526
675
ζ2
)
+ nfT 2FCF
(
−465904519
27337500
− 6776
2025
ζ3 −
61352
10125
ζ2
)
, (F.32)
a(3)gg,Q(6) = TFC2A
(
37541473421359
448084224000
+
56816
2205
B4 −
56376
245
ζ4 +
926445489353
2844979200
ζ3
+
11108521
555660
ζ2
)
+ TFCFCA
(
18181142251969309
54890317440000
− 114512
2205
B4 +
57256
245
ζ4
−12335744909
67737600
ζ3 +
94031857
864360
ζ2
)
+ TFC2F
(
16053159907363
635304600000
+
352
441
B4
−176
49
ζ4 +
3378458681
88905600
ζ3 −
8325229
10804500
ζ2
)
+ T 2FCA
(
−670098465769
6001128000
−25725061
259200
ζ3 −
96697
2835
ζ2
)
+ T 2FCF
(
−8892517283287
490092120000
− 12688649
2540160
ζ3
−2205188
77175
ζ2
)
+
64
27
T 3F ζ3 + nfT 2FCA
(
−245918019913
1312746750
+
3224
81
ζ3
−250094
19845
ζ2
)
+ nfT 2FCF
(
−71886272797
3403417500
− 3872
2835
ζ3 −
496022
77175
ζ2
)
, (F.33)
a(3)gg,Q(8) = TFC2A
(
512903304712347607
18665908531200000
+
108823
3780
B4 −
162587
630
ζ4
+
2735007975361
6502809600
ζ3 +
180224911
7654500
ζ2
)
+TFCFCA
(
13489584043443319991
43553786572800000
− 163882
2835
B4 +
81941
315
ζ4
−3504113623243
25082265600
ζ3 +
414844703639
3429216000
ζ2
)
+TFC2F
(
5990127272073225467
228657379507200000
+
37
81
B4 −
37
18
ζ4 +
3222019505879
87787929600
ζ3
164
−12144008761
48009024000
ζ2
)
+ T 2FCA
(
−16278325750483243
124439390208000
−871607413
7962624
ζ3 −
591287
14580
ζ2
)
+ T 2FCF
(
−7458367007740639
408316749120000
−291343229
52254720
ζ3 −
2473768763
85730400
ζ2
)
+
64
27
T 3F ζ3
+nfT 2FCA
(
−102747532985051
486091368000
+
54208
1215
ζ3 −
737087
51030
ζ2
)
+nfT 2FCF
(
−1145917332616927
51039593640000
− 2738
3645
ζ3 −
70128089
10716300
ζ2
)
, (F.34)
a(3)gg,Q(10) = TFC2A
(
−15434483462331661005275759
327337774462347264000000
+
17788828
571725
B4
−17746492
63525
ζ4 +
269094476549521109
519314374656000
ζ3 +
1444408720649
55468759500
ζ2
)
+TFCFCA
(
207095356146239371087405921
771581896946961408000000
− 35662328
571725
B4
+
17831164
63525
ζ4 −
3288460968359099
37093883904000
ζ3 +
6078270984602
46695639375
ζ2
)
+TFC2F
(
553777925867720521493231
20667372239650752000000
+
896
3025
B4 −
4032
3025
ζ4
+
7140954579599
198717235200
ζ3 −
282148432
4002483375
ζ2
)
+T 2FCA
(
−63059843481895502807
433789788579840000
− 85188238297
729907200
ζ3 −
33330316
735075
ζ2
)
+T 2FCF
(
−655690580559958774157
35787657557836800000
− 71350574183
12043468800
ζ3 −
3517889264
121287375
ζ2
)
+
64
27
T 3F ζ3 + nfT 2FCA
(
−6069333056458984
26476427525625
+
215128
4455
ζ3 −
81362132
5145525
ζ2
)
+nfT 2FCF
(
−100698363899844296
4368610541728125
− 351232
735075
ζ3 −
799867252
121287375
ζ2
)
. (F.35)
165
(vii) a(3),NSqq,Q :
a(3),NSqq,Q (1) = 0 , (F.36)
a(3),NSqq,Q (2) = TFCFCA
(
8744
2187
+
64
9
B4 − 64ζ4 +
4808
81
ζ3 −
64
81
ζ2
)
+TFC2F
(
359456
2187
− 128
9
B4 + 64ζ4 −
848
9
ζ3 +
2384
81
ζ2
)
+T 2FCF
(
−28736
2187
− 2048
81
ζ3 −
512
81
ζ2
)
+ nfT 2FCF
(
−100096
2187
+
896
81
ζ3 −
256
81
ζ2
)
, (F.37)
a(3),NSqq,Q (3) = TFCFCA
(
522443
34992
+
100
9
B4 − 100ζ4 +
15637
162
ζ3 +
175
162
ζ2
)
+TFC2F
(
35091701
139968
− 200
9
B4 + 100ζ4 −
1315
9
ζ3 +
29035
648
ζ2
)
+T 2FCF
(
−188747
8748
− 3200
81
ζ3 −
830
81
ζ2
)
+nfT 2FCF
(
−1271507
17496
+
1400
81
ζ3 −
415
81
ζ2
)
, (F.38)
a(3),NSqq,Q (4) = TFCFCA
(
419369407
21870000
+
628
45
B4 −
628
5
ζ4 +
515597
4050
ζ3 +
10703
4050
ζ2
)
+TFC2F
(
137067007129
437400000
− 1256
45
B4 +
628
5
ζ4 −
41131
225
ζ3
+
4526303
81000
ζ2
)
+ T 2FCF
(
−151928299
5467500
− 20096
405
ζ3 −
26542
2025
ζ2
)
+nfT 2FCF
(
−1006358899
10935000
+
8792
405
ζ3 −
13271
2025
ζ2
)
, (F.39)
a(3),NSqq,Q (5) = TFCFCA
(
816716669
43740000
+
728
45
B4 −
728
5
ζ4 +
12569
81
ζ3 +
16103
4050
ζ2
)
+TFC2F
(
13213297537
36450000
− 1456
45
B4 +
728
5
ζ4 −
142678
675
ζ3
+
48391
750
ζ2
)
+ T 2FCF
(
−9943403
303750
− 23296
405
ζ3 −
31132
2025
ζ2
)
(F.40)
166
+nfT 2FCF
(
−195474809
1822500
+
10192
405
ζ3 −
15566
2025
ζ2
)
, (F.41)
a(3),NSqq,Q (6) = TFCFCA
(
1541550898907
105019740000
+
5672
315
B4 −
5672
35
ζ4
+
720065
3969
ζ3 +
1016543
198450
ζ2
)
+ TFC2F
(
186569400917
463050000
−11344
315
B4 +
5672
35
ζ4 −
7766854
33075
ζ3 +
55284811
771750
ζ2
)
+T 2FCF
(
−26884517771
729303750
− 181504
2835
ζ3 −
1712476
99225
ζ2
)
+nfT 2FCF
(
−524427335513
4375822500
+
11344
405
ζ3 −
856238
99225
ζ2
)
, (F.42)
a(3),NSqq,Q (7) = TFCFCA
(
5307760084631
672126336000
+
2054
105
B4 −
6162
35
ζ4
+
781237
3780
ζ3 +
19460531
3175200
ζ2
)
+ TFC2F
(
4900454072126579
11202105600000
−4108
105
B4 +
6162
35
ζ4 −
8425379
33075
ζ3 +
1918429937
24696000
ζ2
)
+T 2FCF
(
−8488157192423
210039480000
− 65728
945
ζ3 −
3745727
198450
ζ2
)
+nfT 2FCF
(
−54861581223623
420078960000
+
4108
135
ζ3 −
3745727
396900
ζ2
)
, (F.43)
a(3),NSqq,Q (8) = TFCFCA
(
−37259291367883
38887309440000
+
19766
945
B4 −
19766
105
ζ4
+
1573589
6804
ζ3 +
200739467
28576800
ζ2
)
+ TFC2F
(
3817101976847353531
8166334982400000
−39532
945
B4 +
19766
105
ζ4 −
80980811
297675
ζ3 +
497748102211
6001128000
ζ2
)
+T 2FCF
(
−740566685766263
17013197880000
− 632512
8505
ζ3 −
36241943
1786050
ζ2
)
+nfT 2FCF
(
−4763338626853463
34026395760000
+
39532
1215
ζ3 −
36241943
3572100
ζ2
)
, (F.44)
167
a(3),NSqq,Q (9) = TFCFCA
(
−3952556872585211
340263957600000
+
4180
189
B4 −
4180
21
ζ4
+
21723277
85050
ζ3 +
559512437
71442000
ζ2
)
+ TFC2F
(
1008729211999128667
2041583745600000
−8360
189
B4 +
4180
21
ζ4 −
85539428
297675
ζ3 +
131421660271
1500282000
ζ2
)
+T 2FCF
(
−393938732805271
8506598940000
− 133760
1701
ζ3 −
19247947
893025
ζ2
)
+nfT 2FCF
(
−2523586499054071
17013197880000
+
8360
243
ζ3 −
19247947
1786050
ζ2
)
, (F.45)
a(3),NSqq,Q (10) = TFCFCA
(
−10710275715721975271
452891327565600000
+
48220
2079
B4
−48220
231
ζ4 +
2873636069
10291050
ζ3 +
961673201
112266000
ζ2
)
+TFC2F
(
170291990048723954490137
328799103812625600000
− 96440
2079
B4
+
48220
231
ζ4 −
10844970868
36018675
ζ3 +
183261101886701
1996875342000
ζ2
)
+T 2FCF
(
−6080478350275977191
124545115080540000
− 1543040
18711
ζ3
−2451995507
108056025
ζ2
)
+ nfT 2FCF
(
−38817494524177585991
249090230161080000
+
96440
2673
ζ3 −
2451995507
216112050
ζ2
)
, (F.46)
a(3),NSqq,Q (11) = TFCFCA
(
−22309979286641292041
603855103420800000
+
251264
10395
B4
−251264
1155
ζ4 +
283300123
935550
ζ3 +
1210188619
130977000
ζ2
)
+TFC2F
(
177435748292579058982241
328799103812625600000
− 502528
10395
B4
+
251264
1155
ζ4 −
451739191
1440747
ζ3 +
47705202493793
499218835500
ζ2
)
168
+T 2FCF
(
−6365809346912279423
124545115080540000
− 8040448
93555
ζ3
−512808781
21611205
ζ2
)
+ nfT 2FCF
(
−40517373495580091423
249090230161080000
+
502528
13365
ζ3 −
512808781
43222410
ζ2
)
, (F.47)
a(3),NSqq,Q (12) = TFCFCA
(
−126207343604156227942043
2463815086971638400000
+
3387392
135135
B4
−3387392
15015
ζ4 +
51577729507
158107950
ζ3 +
2401246832561
243486243000
ζ2
)
+TFC2F
(
68296027149155250557867961293
122080805651901196900800000
− 6774784
135135
B4
+
3387392
15015
ζ4 −
79117185295
243486243
ζ3 +
108605787257580461
1096783781593500
ζ2
)
+T 2FCF
(
−189306988923316881320303
3557133031815302940000
− 108396544
1216215
ζ3
−90143221429
3652293645
ζ2
)
+ nfT 2FCF
(
−1201733391177720469772303
7114266063630605880000
+
6774784
173745
ζ3 −
90143221429
7304587290
ζ2
)
, (F.48)
a(3),NSqq,Q (13) = TFCFCA
(
−12032123246389873565503373
181090408892415422400000
+
3498932
135135
B4
−3498932
15015
ζ4 +
2288723461
6548850
ζ3 +
106764723181157
10226422206000
ζ2
)
+TFC2F
(
10076195142551036234891679659
17440115093128742414400000
− 6997864
135135
B4
+
3498932
15015
ζ4 −
81672622894
243486243
ζ3 +
448416864235277759
4387135126374000
ζ2
)
+T 2FCF
(
−196243066652040382535303
3557133031815302940000
− 111965824
1216215
ζ3
−93360116539
3652293645
ζ2
)
+ nfT 2FCF
(
−1242840812874342588467303
7114266063630605880000
169
+
6997864
173745
ζ3 −
93360116539
7304587290
ζ2
)
, (F.49)
a(3),NSqq,Q (14) = TFCFCA
(
−994774587614536873023863
12072693926161028160000
+
720484
27027
B4
−720484
3003
ζ4 +
6345068237
17027010
ζ3 +
37428569944327
3408807402000
ζ2
)
+TFC2F
(
72598193631729215117875463981
122080805651901196900800000
− 1440968
27027
B4
+
720484
3003
ζ4 −
2101051892878
6087156075
ζ3 +
461388998135343407
4387135126374000
ζ2
)
+T 2FCF
(
−40540032063650894708251
711426606363060588000
− 23055488
243243
ζ3
−481761665447
18261468225
ζ2
)
+ nfT 2FCF
(
−256205552272074402170491
1422853212726121176000
+
1440968
34749
ζ3 −
481761665447
36522936450
ζ2
)
. (F.50)
170
G 3–loop Moments for Transversity
We obtain the following fixed moments of the fermionic contributions to the 3–loop
transversity anomalous dimension γ(2),TRqq (N)
γˆ(2),TRqq (1) = CFTF
[
−8
3
TF (1 + 2nf )−
2008
27
CA
+
196
9
CF + 32(CF − CA)ζ3
]
, (G.1)
γˆ(2),TRqq (2) = CFTF
[
−184
27
TF (1 + 2nf )−
2084
27
CA
−60CF + 96(CF − CA)ζ3
]
, (G.2)
γˆ(2),TRqq (3) = CFTF
[
−2408
243
TF (1 + 2nf )−
19450
243
CA
−25276
243
CF +
416
3
(CF − CA)ζ3
]
, (G.3)
γˆ(2),TRqq (4) = CFTF
[
−14722
1215
TF (1 + 2nf )−
199723
2430
CA
−66443
486
CF +
512
3
(CF − CA)ζ3
]
,
(G.4)
γˆ(2),TRqq (5) = CFTF
[
−418594
30375
TF (1 + 2nf )−
5113951
60750
CA
−49495163
303750
CF +
2944
15
(CF − CA)ζ3
]
, (G.5)
γˆ(2),TRqq (6) = CFTF
[
−3209758
212625
TF (1 + 2nf )−
3682664
42525
CA
−18622301
101250
CF +
1088
5
(CF − CA)ζ3
]
, (G.6)
γˆ(2),TRqq (7) = CFTF
[
−168501142
10418625
TF (1 + 2nf )−
1844723441
20837250
CA
−49282560541
243101250
CF +
8256
35
(CF − CA)ζ3
]
(G.7)
γˆ(2),TRqq (8) = CFTF
[
−711801943
41674500
TF (1 + 2nf )−
6056338297
66679200
CA
171
−849420853541
3889620000
CF +
8816
35
(CF − CA)ζ3
]
(G.8)
These moments (N = 1..8) agree with the corresponding terms obtained in [360]. The
newly calculated moments read
γˆ(2),TRqq (9) = CFTF
[
−20096458061
1125211500
TF (1 + 2nf )−
119131812533
1285956000
CA
−24479706761047
105019740000
CF +
83824
315
(CF − CA)ζ3
]
(G.9)
γˆ(2),TRqq (10) = CFTF
[
−229508848783
12377326500
TF (1 + 2nf )−
4264058299021
45008460000
CA
−25800817445759
105019740000
CF +
87856
315
(CF − CA)ζ3
]
(G.10)
γˆ(2),TRqq (11) = CFTF
[
−28677274464343
1497656506500
TF (1 + 2nf )−
75010870835743
778003380000
CA
−396383896707569599
1537594013340000
CF +
1006736
3465
(CF − CA)ζ3
]
(G.11)
γˆ(2),TRqq (12) = CFTF
[
−383379490933459
19469534584500
TF (1 + 2nf )
−38283693844132279
389390691690000
CA
−1237841854306528417
4612782040020000
CF +
1043696
3465
(CF − CA)ζ3
]
(G.12)
γˆ(2),TRqq (13) = CFTF
[
−66409807459266571
3290351344780500
TF (1 + 2nf )
−6571493644375020121
65807026895610000
CA
−36713319015407141570017
131745667845011220000
CF +
14011568
45045
(CF − CA)ζ3
]
. (G.13)
The fixed moments of the constant terms a(3),TRqq,Q (N) of the unrenormalized OME, see Eq.
(9.15), are given by
a(3),TRqq,Q (1) = TFCFCA
(
−26441
1458
+
8
3
B4 − 24ζ4 +
481
27
ζ3 −
61
27
ζ2
)
+TFC2F
(
15715
162
− 16
3
B4 + 24ζ4 −
278
9
ζ3 +
49
3
ζ2
)
172
+T 2FCF
(
−6548
729
− 256
27
ζ3 −
104
27
ζ2
)
+nfT 2FCF
(
−15850
729
+
112
27
ζ3 −
52
27
ζ2
)
, (G.14)
a(3),TRqq,Q (2) = TFCFCA
(
1043
162
+ 8B4 − 72ζ4 +
577
9
ζ3 +
ζ2
3
)
+TFC2F
(
10255
54
− 16B4 + 72ζ4 −
310
3
ζ3 + 33ζ2
)
+T 2FCF
(
−1388
81
− 256
9
ζ3 − 8ζ2
)
+nfT 2FCF
(
−4390
81
+
112
9
ζ3 − 4ζ2
)
, (G.15)
a(3),TRqq,Q (3) = TFCFCA
(
327967
21870
+
104
9
B4 − 104ζ4 +
40001
405
ζ3 +
121
81
ζ2
)
+TFC2F
(
1170943
4374
− 208
9
B4 + 104ζ4 −
1354
9
ζ3 +
3821
81
ζ2
)
+T 2FCF
(
−52096
2187
− 3328
81
ζ3 −
904
81
ζ2
)
+nfT 2FCF
(
−168704
2187
+
1456
81
ζ3 −
452
81
ζ2
)
, (G.16)
a(3),TRqq,Q (4) = TFCFCA
(
4400353
218700
+
128
9
B4 − 128ζ4 +
52112
405
ζ3 +
250
81
ζ2
)
+TFC2F
(
56375659
174960
− 256
9
B4 + 128ζ4 −
556
3
ζ3 +
4616
81
ζ2
)
+T 2FCF
(
−3195707
109350
− 4096
81
ζ3 −
1108
81
ζ2
)
+nfT 2FCF
(
−20731907
218700
+
1792
81
ζ3 −
554
81
ζ2
)
, (G.17)
a(3),TRqq,Q (5) = TFCFCA
(
1436867309
76545000
+
736
45
B4 −
736
5
ζ4 +
442628
2835
ζ3 +
8488
2025
ζ2
)
+TFC2F
(
40410914719
109350000
− 1472
45
B4 +
736
5
ζ4 −
47932
225
ζ3 +
662674
10125
ζ2
)
173
+T 2FCF
(
−92220539
2733750
− 23552
405
ζ3 −
31924
2025
ζ2
)
+nfT 2FCF
(
−596707139
5467500
+
10304
405
ζ3 −
15962
2025
ζ2
)
, (G.18)
a(3),TRqq,Q (6) = TFCFCA
(
807041747
53581500
+
272
15
B4 −
816
5
ζ4 +
172138
945
ζ3
+
10837
2025
ζ2
)
+ TFC2F
(
14845987993
36450000
− 544
15
B4 +
816
5
ζ4 −
159296
675
ζ3
+
81181
1125
ζ2
)
+ T 2FCF
(
−5036315611
133953750
− 8704
135
ζ3 −
35524
2025
ζ2
)
+nfT 2FCF
(
−32472719011
267907500
+
3808
135
ζ3 −
17762
2025
ζ2
)
, (G.19)
a(3),TRqq,Q (7) = TFCFCA
(
413587780793
52509870000
+
688
35
B4 −
6192
35
ζ4 +
27982
135
ζ3
+
620686
99225
ζ2
)
+ TFC2F
(
12873570421651
29172150000
− 1376
35
B4 +
6192
35
ζ4
−8454104
33075
ζ3 +
90495089
1157625
ζ2
)
+ T 2FCF
(
−268946573689
6563733750
− 22016
315
ζ3
−1894276
99225
ζ2
)
+ nfT 2FCF
(
−1727972700289
13127467500
+
1376
45
ζ3 −
947138
99225
ζ2
)
, (G.20)
a(3),TRqq,Q (8) = TFCFCA
(
− 91321974347
112021056000
+
2204
105
B4 −
6612
35
ζ4 +
87613
378
ζ3
+
11372923
1587600
ζ2
)
+ TFC2F
(
1316283829306051
2800526400000
− 4408
105
B4 +
6612
35
ζ4
−9020054
33075
ζ3 +
171321401
2058000
ζ2
)
+ T 2FCF
(
−4618094363399
105019740000
− 70528
945
ζ3
−2030251
99225
ζ2
)
+ nfT 2FCF
(
−29573247248999
210039480000
+
4408
135
ζ3 −
2030251
198450
ζ2
)
,
(G.21)
a(3),TRqq,Q (9) = TFCFCA
(
−17524721583739067
1497161413440000
+
20956
945
B4 −
20956
105
ζ4
+
9574759
37422
ζ3 +
16154189
2041200
ζ2
)
+ TFC2F
(
1013649109952401819
2041583745600000
− 41912
945
B4
174
+
20956
105
ζ4 −
85698286
297675
ζ3 +
131876277049
1500282000
ζ2
)
+T 2FCF
(
−397003835114519
8506598940000
− 670592
8505
ζ3 −
19369859
893025
ζ2
)
+nfT 2FCF
(
−2534665670688119
17013197880000
+
41912
1215
ζ3 −
19369859
1786050
ζ2
)
, (G.22)
a(3),TRqq,Q (10) = TFCFCA
(
−176834434840947469
7485807067200000
+
21964
945
B4 −
21964
105
ζ4
+
261607183
935550
ζ3 +
618627019
71442000
ζ2
)
+ TFC2F
(
11669499797141374121
22457421201600000
−43928
945
B4 +
21964
105
ζ4 −
3590290
11907
ζ3 +
137983320397
1500282000
ζ2
)
+T 2FCF
(
−50558522757917663
1029298471740000
− 702848
8505
ζ3 −
4072951
178605
ζ2
)
+nfT 2FCF
(
−321908083399769663
2058596943480000
+
43928
1215
ζ3 −
4072951
357210
ζ2
)
, (G.23)
a(3),TRqq,Q (11) = TFCFCA
(
−436508000489627050837
11775174516705600000
+
251684
10395
B4 −
251684
1155
ζ4
+
3687221539
12162150
ζ3 +
149112401
16038000
ζ2
)
+ TFC2F
(
177979311179110818909401
328799103812625600000
−503368
10395
B4 +
251684
1155
ζ4 −
452259130
1440747
ζ3 +
191230589104127
1996875342000
ζ2
)
+T 2FCF
(
−6396997235105384423
124545115080540000
− 8053888
93555
ζ3 −
514841791
21611205
ζ2
)
+nfT 2FCF
(
−40628987857774916423
249090230161080000
+
503368
13365
ζ3 −
514841791
43222410
ζ2
)
, (G.24)
a(3),TRqq,Q (12) = TFCFCA
(
−245210883820358086333
4783664647411650000
+
260924
10395
B4 −
260924
1155
ζ4
+
3971470819
12162150
ζ3 +
85827712409
8644482000
ζ2
)
+ TFC2F
(
2396383721714622551610173
4274388349564132800000
−521848
10395
B4 +
260924
1155
ζ4 −
468587596
1440747
ζ3 +
198011292882437
1996875342000
ζ2
)
175
+T 2FCF
(
−1124652164258976877487
21048124448611260000
− 8349568
93555
ζ3
−535118971
21611205
ζ2
)
+ nfT 2FCF
(
−7126865031281296825487
42096248897222520000
+
521848
13365
ζ3 −
535118971
43222410
ζ2
)
, (G.25)
a(3),TRqq,Q (13) = TFCFCA
(
−430633219615523278883051
6467514603300550800000
+
3502892
135135
B4
−3502892
15015
ζ4 +
327241423
935550
ζ3 +
15314434459241
1460917458000
ζ2
)
+TFC2F
(
70680445585608577308861582893
122080805651901196900800000
− 7005784
135135
B4
+
3502892
15015
ζ4 −
81735983092
243486243
ζ3 +
449066258795623169
4387135126374000
ζ2
)
+T 2FCF
(
−196897887865971730295303
3557133031815302940000
− 112092544
1216215
ζ3
−93611152819
3652293645
ζ2
)
+ nfT 2FCF
(
−1245167831299024242467303
7114266063630605880000
+
7005784
173745
ζ3 −
93611152819
7304587290
ζ2
)
. (G.26)
176
References
[1] M. Gell-Mann, Phys. Lett. 8, 214 (1964).
[2] G. Zweig, An SU(3) model for the strong interaction symmetry and its breaking,
CERN-TH-401, 412 (1964).
[3] M. Gell-Mann and Y. Neemam, The eightfold way: a review with a collection of
reprints, (Benjamin Press, New York, 1964), 317 p.
[4] J. J. J. Kokkedee, The Quark Model, (Benjamin Press, New York, 1969), 239 p.
[5] F. E. Close, An Introduction to Quarks and Partons, (Academic Press, London,
1979), 481 p.
[6] V. E. Barnes et al., Phys. Rev. Lett. 12, 204 (1964).
[7] F. Gu¨rsey and L. A. Radicati, Phys. Rev. Lett. 13, 173 (1964).
[8] M. A. B. Beg, B. W. Lee, and A. Pais, Phys. Rev. Lett. 13, 514 (1964).
[9] B. Sakita, Phys. Rev. Lett. 13, 643 (1964).
[10] W. Pauli, Phys. Rev. 58, 716 (1940).
[11] O. W. Greenberg, Phys. Rev. Lett. 13, 598 (1964).
[12] Y. Nambu, A Systematics Of Hadrons In Subnuclear Physics, in: Preludes in The-
oretical Physics, eds. A. De-Shalit, H. Fehsbach and L. van Hove (North-Holland,
Amsterdam, 1966), pp. 133.
[13] M. Y. Han and Y. Nambu, Phys. Rev. 139, B1006 (1965).
[14] H. Fritzsch and M. Gell-Mann, Current algebra: Quarks and what else?, Proceedings
of 16th International Conference on High-Energy Physics, Batavia, Illinois, 6-13
Sep Vol. 2, J.D. Jackson, A. Roberts, R. Donaldson, eds., pp. 135 (1972), hep-
ph/0208010.
[15] L. W. Mo and C. Peck, 8-GeV/c Spectrometer, SLAC-TN-65-029 (1965).
R. E. Taylor, Nucleon form–factors above 6-GeV, in: Proc. of the Int. Symp. on
Electron and Photon Interactions at High Energies, SLAC, September 5–9, 1967,
(SLAC, Stanford CA, 1967), SLAC-PUB-0372, pp. 78.
[16] S. D. Drell and J. D. Walecka, Ann. Phys. 28, 18 (1964).
E. Derman, Phys. Rev. D19, 133 (1979).
[17] J. D. Bjorken, Phys. Rev. 179, 1547 (1969).
[18] D. H. Coward et al., Phys. Rev. Lett. 20, 292 (1968).
W. K. H. Panofsky, Low q2 electrodynamics, elastic and inelastic electron (and muon)
scattering, Proc. 14th International Conference on High-Energy Physics, Vienna,
1968, J. Prentki and J. Steinberger, eds., (CERN, Geneva, 1968), pp. 23.
179
E. D. Bloom et al., Phys. Rev. Lett. 23, 930 (1969).
M. Breidenbach et al., Phys. Rev. Lett. 23, 935 (1969).
[19] H. W. Kendall, Rev. Mod. Phys. 63, 597 (1991).
R. E. Taylor, Rev. Mod. Phys. 63, 573 (1991).
J. I. Friedman, Rev. Mod. Phys. 63, 615 (1991).
[20] R. P. Feynman, The behavior of hadron collisions at extreme energies, Proc. of 3rd
International Conference on High Energy Collisions, Stony Brook, 1969, C.N. Yang,
J.A. Cole, M. Good, R. Hwa, and J. Lee-Franzini, eds., (Gordon and Breach, New
York, 1970), pp. 237.
[21] R. P. Feynman, Phys. Rev. Lett. 23, 1415 (1969).
[22] R. P. Feynman, Photon-hadron interactions, (Benjamin Press, Reading, 1972), 282 p.
[23] C. G. Callan and D. J. Gross, Phys. Rev. Lett. 22, 156 (1969).
[24] T. D. Lee, S. Weinberg, and B. Zumino, Phys. Rev. Lett. 18, 1029 (1967).
[25] J. Sakurai, Vector-Meson dominance - present status and future prospects, Proc. 4th
International Symposium on Electron and Photon Interactions at High Energies,
Liverpool, 1969, (Daresbury Laboratory, 1969), eds. D.W. Braben and R.E. Rand,
pp. 91.
J. J. Sakurai, Phys. Rev. Lett. 22, 981 (1969).
Y.-S. Tsai, SLAC-PUB-0600 (1969).
H. Fraas and D. Schildknecht, Nucl. Phys. B14, 543 (1969).
[26] J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975 (1969).
[27] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).
[28] S. L. Glashow, Nucl. Phys. 22, 579 (1961).
[29] A. Salam and J. C. Ward, Phys. Lett. 13, 168 (1964).
A. Salam, Weak and Electromagnetic Interactions, Proc. of the 8th Nobel Symposium,
Go¨teborg, Sweden, 19–25 May 1968, ed. N. Svartholm, (Almqvist and Wiskell,
Stockholm, 1968), pp. 367.
[30] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B50, 318 (1972).
[31] J. C. Taylor, Nucl. Phys. B33, 436 (1971).
A. A. Slavnov, Theor. Math. Phys. 10, 99 (1972).
B. W. Lee and J. Zinn-Justin, Phys. Rev.D5, 3121 (1972); Phys. Rev.D7, 1049
(1973).
[32] J. S. Bell and R. Jackiw, Nuovo Cim. A60, 47 (1969).
S. L. Adler, Phys. Rev. 177, 2426 (1969).
180
[33] R. A. Bertlmann, Anomalies in quantum field theory, (Clarendon, Oxford, 1996),
566 p.
[34] G. ’t Hooft, Nucl. Phys. B33, 173 (1971).
[35] C.-N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954).
[36] H. Fritzsch, M. Gell-Mann, and H. Leutwyler, Phys. Lett. B47, 365 (1973).
[37] E. Reya, Phys. Rept. 69, 195 (1981).
[38] T. Muta, Foundations of Quantum Chromodynamics, World Sci. Lect. Notes Phys.
57, (World Scientific, Singapore, 1998), 2nd edition.
[39] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973).
[40] H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).
[41] G. t’Hooft, (1972), unpublished.
[42] K. G. Wilson, Phys. Rev. 179, 1499 (1969).
W. Zimmermann, Lect. on Elementary Particle Physics and Quantum Field Theory,
Brandeis Summer Inst., Vol. 1, (MIT Press, Cambridge, 1970), pp. 395.
Y. Frishman, Annals Phys. 66, 373 (1971).
R. A. Brandt and G. Preparata, Nucl. Phys. B27, 541 (1972).
[43] D. J. Gross and S. B. Treiman, Phys. Rev. D4, 1059 (1971).
[44] C. Chang et al., Phys. Rev. Lett. 35, 901 (1975).
Y. Watanabe et al., Phys. Rev. Lett. 35, 898 (1975).
[45] S. Ferrara, R. Gatto, and A. F. Grillo, Springer Tracts Mod. Phys. 67, 1 (1973),
and references therein.
[46] D. J. Gross and F. Wilczek, Phys. Rev. D8, 3633 (1973); D9, 980 (1974).
H. Georgi and H. D. Politzer, Phys. Rev. D9, 416 (1974).
[47] Photon-Hadron Interactions I, International Summer Institute in Theoretical Physics,
DESY, July 12–14, 1971, Springer Tracts in Modern Physics 62, 147 p.
H. D. Politzer, Phys. Rept. 14, 129 (1974).
W. J. Marciano and H. Pagels, Phys. Rept. 36, 137 (1978).
J. R. Ellis and C. T. Sachrajda, NATO Adv. Study Inst. Ser. B Phys. 59, 285
(1980).
[48] A. J. Buras, Rev. Mod. Phys. 52, 199 (1980).
[49] G. Altarelli, Phys. Rept. 81, 1 (1982).
F. Wilczek, Ann. Rev. Nucl. Part. Sci. 32, 177 (1982).
181
[50] R. L. Jaffe, Deep Inelastic Scattering with Application to nuclear targets, Lectures
presented at the Los Alamos School on Quark Nuclear Physics, Los Alamos, NM,
June 10-14, 1985, M.B. Johnson and A. Picklesimer, eds., (Wiley, New York, 1986),
pp. 82.
[51] J. C. Collins and D. E. Soper, Ann. Rev. Nucl. Part. Sci. 37, 383 (1987).
[52] R. K. Ellis, An Introduction to the QCD Parton Model, Lectures given at 1987
Theoretical Advanced Study Inst. in Elementary Particle Physics, Santa Fe, NM,
Jul 5 - Aug 1, 1987, R. Slansky and GeoffreyWest, eds., (World Scientific, Singapore,
1988), pp. 214.
[53] A. H. Mueller, ed., Perturbative Quantum Chromodynamics, (World Scientific, Sin-
gapore, 1989), 614 p.
[54] R. G. Roberts, The Structure of the proton: Deep inelastic scattering, (Cambridge
University Press, Cambridge, 1990), 182 p.
[55] G. Sterman, An Introduction to quantum field theory, (Cambridge University Press,
Cambridge, 1993), 572 p.
R. K. Ellis, W. J. Stirling, and B. R. Webber, QCD and collider physics, Camb.
Monogr. Part. Phys. Nucl. Phys. Cosmol. 8, (Cambridge University Press, Cam-
bridge, 1996), 435 p.
CTEQ collaboration, R. Brock et al., Rev. Mod. Phys. 67, 157 (1995).
J. Blu¨mlein, Surveys High Energ. Phys. 7, 181 (1994).
[56] P. Mulders, Quantum Chromodynamics and Hard Scattering Processes, Lectures,
Dutch Research School for Theoretical Physics, Dalfsen, January 1996, and Dutch
Research School for Subatomic Physics, Beekbergen, February, 1996, (NIKHEF,
Amsterdam, The Netherlands).
[57] SLAC-SP-017 collaboration, J. E. Augustin et al., Phys. Rev. Lett. 33, 1406 (1974).
G. S. Abrams et al., Phys. Rev. Lett. 33, 1453 (1974).
[58] E598 collaboration, J. J. Aubert et al., Phys. Rev. Lett. 33, 1404 (1974).
[59] Z. Maki and M. Nakagawa, Prog. Theor. Phys. 31, 115 (1964).
Y. Hara, Phys. Rev. 134, B701 (1964).
J. D. Bjorken and S. L. Glashow, Phys. Lett. 11, 255 (1964).
[60] S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D2, 1285 (1970).
[61] Particle Data Group, C. Amsler et al., Phys. Lett. B667, 1 (2008).
[62] S. W. Herb et al., Phys. Rev. Lett. 39, 252 (1977).
[63] CDF collaboration, F. Abe et al., Phys. Rev. Lett. 73, 225 (1994), hep-ex/9405005;
Phys. Rev. D50, 2966 (1994); Phys. Rev. Lett. 74, 2626 (1995), hep-ex/9503002.
D0 collaboration, S. Abachi et al., Phys. Rev. Lett. 74, 2632 (1995), hep-
ex/9503003.
182
[64] S. Stein et al., Phys. Rev. D12, 1884 (1975).
W. B. Atwood et al., Phys. Lett. B64, 479 (1976).
A. Bodek et al., Phys. Rev. D20, 1471 (1979).
M. D. Mestayer et al., Phys. Rev. D27, 285 (1983).
[65] EMC collaboration, O. Allkofer et al., Nucl. Instr. Meth. 179, 445 (1981).
EMC collaboration, J. J. Aubert et al., Nucl. Phys. B259, 189 (1985).
[66] D. Bollini et al., Nucl. Instr. Meth. 204, 333 (1983).
BCDMS collaboration, A. C. Benvenuti et al., Nucl. Instr. Meth. A226, 330 (1984);
Phys. Lett. B195, 91 (1987); Phys. Lett. B223, 485 (1989); Phys. Lett. B237, 592
(1990).
[67] NMC collaboration, P. Amaudruz et al., Nucl. Phys. B371, 3 (1992); Phys. Lett.
B295, 159 (1992).
NMC collaboration, M. Arneodo et al., Phys. Lett. B364, 107 (1995), hep-
ph/9509406.
[68] H. L. Anderson et al., Phys. Rev. D20, 2645 (1979).
[69] E665 collaboration, M. R. Adams et al., Nucl. Instrum. Meth A291, 533 (1990);
Phys. Rev. D54, 3006 (1996).
[70] CHARM collaboration, M. Jonker et al., Phys. Lett. B109, 133 (1982).
CHARM collaboration, F. Bergsma et al., Phys. Lett. B123, 269 (1983); Phys.
Lett. B153, 111 (1985).
[71] J. P. Berge et al., Z. Phys. C49, 187 (1991).
[72] Birmingham-CERN-Imperial College-Mu¨nchen(MPI)-Oxford collaboration, G. T.
Jones et al., Z. Phys. C62, 575 (1994).
[73] CCFR collaboration, M. H. Shaevitz et al., Nucl. Phys. Proc. Suppl. 38, 188 (1995).
[74] Aachen-Bonn-CERN-London-Oxford-Saclay collaboration, P. C. Bosetti et al.,
Nucl. Phys. B142, 1 (1978).
J. G. H. de Groot et al., Z. Phys. C1, 143 (1979).
S. M. Heagy et al., Phys. Rev. D23, 1045 (1981).
Gargamelle SPS collaboration, J. G. Morfin et al., Phys. Lett. B104, 235 (1981).
Aachen-Bonn-CERN-Democritos-London-Oxford-Saclay collaboration, P. C.
Bosetti et al., Nucl. Phys. B203, 362 (1982).
H. Abramowicz et al., Z. Phys. C17, 283 (1983).
D. MacFarlane et al., Z. Phys. C26, 1 (1984).
D. Allasia et al., Z. Phys. C28, 321 (1985).
[75] HERA - a proposal for a large electron proton colliding beam facility at DESY, (Ham-
burg, DESY, 1981), DESY HERA 81-10, 292 p.
183
[76] H1 collaboration, I. Abt et al., The H1 detector at HERA, DESY-93-103 (1993),
194 p.
[77] ZEUS collaboration, M. Derrick et al., Phys. Lett. B303, 183 (1993).
[78] HERMES collaboration, K. Ackerstaff et al., Nucl. Instrum. Meth. A417, 230
(1998), hep-ex/9806008.
[79] E. Hartouni et al., HERA-B: An experiment to study CP violation in the B system
using an internal target at the HERA proton ring. Design report, DESY-PRC-95-01
(1995), 491 p.
[80] H1 collaboration, F. D. Aaron et al., Phys. Lett. B665, 139 (2008), hep-
ex/0805.2809.
ZEUS collaboration, Measurement of the Longitudinal Proton Structure Function at
HERA, (2009), hep-ex/0904.1092.
[81] L. W. Whitlow, S. Rock, A. Bodek, E. M. Riordan, and S. Dasu, Phys. Lett. B250,
193 (1990).
S. Dasu et al., Phys. Rev. D49, 5641 (1994).
E140X collaboration, L. H. Tao et al., Z. Phys. C70, 387 (1996).
NMC collaboration, M. Arneodo et al., Nucl. Phys. B483, 3 (1997), hep-
ph/9610231; Nucl. Phys. B487, 3 (1997), hep-ex/9611022.
Y. Liang, M. E. Christy, R. Ent, and C. E. Keppel, Phys. Rev. C73, 065201 (2006),
nucl-ex/0410028.
[82] H1 collaboration, C. Adloff et al., Phys. Lett. B393, 452 (1997), hep-ex/9611017.
[83] M. Klein, On the future measurement of the longitudinal structure function at low x
at HERA, in: Proc. of the 12th Int. Workshop on Deep Inelastic Scattering, DIS
2004, Strebske Pleso, Slovakia, 14–18 April 2004, D. Bruncko, J. Ferencei and P.
Strizˇenec, eds., (Academic Electronic Press, Bratislava, 2004), pp. 309.
[84] J. Feltesse, Measurement of the longitudinal proton structure function at low x
at HERA, in: Proc. of Ringberg Workshop on New Trends in HERA Physics
2005, Ringberg Castle, Tegernsee, Germany, 2-7 Oct. 2005, G. Grindhammer,
B.A. Kniehl, G. Kramer and W. Ochs, eds., (World Scientific, Singapore, 2005),
pp. 370.
[85] H1 and ZEUS collaboration, K. Lipka, Nucl. Phys. Proc. Suppl. 152, 128 (2006).
[86] P. D. Thompson, J. Phys. G34, N177 (2007), hep-ph/0703103.
[87] H. Jung and A. De Roeck, eds., Proceedings of the workshop: HERA and the LHC
workshop series on the implications of HERA for LHC physics, (2006–2008, Hamburg,
Geneve), DESY-PROC-2009-02, March 2009, 794 p. hep-ph/0903.3861.
[88] ZEUS collaboration, S. Chekanov, Measurement of D± and D0 production in deep
inelastic scattering using a lifetime tag at HERA, (2008), hep-ex/0812.3775.
184
[89] J. Blu¨mlein and S. Riemersma, QCD corrections to FL(x,Q2), (1996), hep-
ph/9609394.
[90] S. J. Brodsky, P. Hoyer, C. Peterson, and N. Sakai, Phys. Lett. B93, 451 (1980).
E. Hoffmann and R. Moore, Z. Phys. C20, 71 (1983).
ZEUS collaboration, M. Derrick et al., Phys. Lett. B349, 225 (1995), hep-
ex/9502002.
B. W. Harris, J. Smith, and R. Vogt, Nucl. Phys. B461, 181 (1996), hep-
ph/9508403.
[91] H1 collaboration, C. Adloff et al., Z. Phys. C72, 593 (1996), hep-ex/9607012.
[92] S. Bethke, J. Phys. G26, R27 (2000), hep-ex/0004021.
S. Bethke, Nucl. Phys. Proc. Suppl. 135, 345 (2004), hep-ex/0407021.
[93] J. Blu¨mlein, H. Bo¨ttcher, and A. Guffanti, Nucl. Phys. Proc. Suppl. 135, 152
(2004), hep-ph/0407089.
[94] S. Alekhin et al., HERA and the LHC - A workshop on the implications of HERA for
LHC physics: Proceedings Part A, B, (2005), hep-ph/0601012, hep-ph/0601013.
[95] M. Dittmar et al., Parton distributions: Summary report for the HERA - LHC workshop,
(2005), hep-ph/0511119.
[96] M. Glu¨ck, E. Reya, and C. Schuck, Nucl. Phys. B754, 178 (2006), hep-ph/0604116.
[97] S. Alekhin, K. Melnikov, and F. Petriello, Phys. Rev. D74, 054033 (2006), hep-
ph/0606237.
[98] J. Blu¨mlein, H. Bo¨ttcher, and A. Guffanti, Nucl. Phys. B774, 182 (2007), hep-
ph/0607200.
[99] J. Blu¨mlein, ΛQCD and αs(M2Z) from DIS Structure Functions, (2007), hep-
ph/0706.2430.
[100] H. Jung et al., What HERA may provide?, (2008), hep-ph/0809.0549.
[101] E. Witten, Nucl. Phys. B104, 445 (1976).
J. Babcock, D. W. Sivers, and S. Wolfram, Phys. Rev. D18, 162 (1978).
M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B136, 157 (1978).
J. P. Leveille and T. J. Weiler, Nucl. Phys. B147, 147 (1979).
[102] M. Glu¨ck, E. Hoffmann, and E. Reya, Z. Phys. C13, 119 (1982).
[103] E. Laenen, S. Riemersma, J. Smith, and W. L. van Neerven, Nucl. Phys. B392,
162 (1993); Nucl. Phys. B392, 229 (1993) .
S. Riemersma, J. Smith, and W. L. van Neerven, Phys. Lett. B347, 143 (1995),
hep-ph/9411431.
185
[104] R. E. Taylor, Inelastic electron-Nucleon Scattering Experiments, Invited paper pre-
sented at Int. Symposium on Lepton and Photon Interactions, Stanford Univ.,
Calif., Aug 21-27, 1975, W.T. Kirk, ed., (SLAC, Stanford CA, 1976), SLAC–PUB–
1729, 29pp.
[105] A. Zee, F. Wilczek, and S. B. Treiman, Phys. Rev. D10, 2881 (1974).
[106] W. A. Bardeen, A. J. Buras, D. W. Duke, and T. Muta, Phys. Rev. D18, 3998
(1978).
[107] W. Furmanski and R. Petronzio, Z. Phys. C11, 293 (1982), and references therein.
[108] D. W. Duke, J. D. Kimel, and G. A. Sowell, Phys. Rev. D25, 71 (1982).
A. Devoto, D. W. Duke, J. D. Kimel, and G. A. Sowell, Phys. Rev. D30, 541 (1984).
D. I. Kazakov and A. V. Kotikov, Nucl. Phys. B307, 721 (1988).
D. I. Kazakov, A. V. Kotikov, G. Parente, O. A. Sampayo, and J. Sanchez Guillen,
Phys. Rev. Lett. 65, 1535 (1990).
J. Sanchez Guillen, J. Miramontes, M. Miramontes, G. Parente, and O. A. Sampayo,
Nucl. Phys. B353, 337 (1991).
W. L. van Neerven and E. B. Zijlstra, Phys. Lett. B272, 127 (1991); Phys. Lett.
B273, 476 (1991); Nucl. Phys. B383, 525 (1992).
D. I. Kazakov and A. V. Kotikov, Phys. Lett. B291, 171 (1992).
S. A. Larin and J. A. M. Vermaseren, Z. Phys. C57, 93 (1993).
[109] S. Moch and J. A. M. Vermaseren, Nucl. Phys. B573, 853 (2000), hep-ph/9912355.
[110] S. A. Larin, T. van Ritbergen, and J. A. M. Vermaseren, Nucl. Phys. B427, 41
(1994).
[111] S. A. Larin, P. Nogueira, T. van Ritbergen, and J. A. M. Vermaseren, Nucl. Phys.
B492, 338 (1997), hep-ph/9605317.
[112] A. Retey and J. A. M. Vermaseren, Nucl. Phys. B604, 281 (2001), hep-ph/0007294.
[113] S. Moch, J. A. M. Vermaseren, and A. Vogt, Phys. Lett. B606, 123 (2005), hep-
ph/0411112.
[114] J. Blu¨mlein and J. A. M. Vermaseren, Phys. Lett. B606, 130 (2005), hep-
ph/0411111.
[115] J. A. M. Vermaseren, A. Vogt, and S. Moch, Nucl. Phys. B724, 3 (2005), hep-
ph/0504242.
[116] S. I. Alekhin and J. Blu¨mlein, Phys. Lett. B594, 299 (2004), hep-ph/0404034.
[117] G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977).
[118] P. A. Baikov and K. G. Chetyrkin, Nucl. Phys. Proc. Suppl. 160, 76 (2006).
186
[119] E. G. Floratos, D. A. Ross, and C. T. Sachrajda, Nucl. Phys. B129, 66 (1977);
[Erratum-ibid.] B139, 545 (1978).
[120] E. G. Floratos, D. A. Ross, and C. T. Sachrajda, Nucl. Phys. B152, 493 (1979).
[121] A. Gonzalez-Arroyo, C. Lopez, and F. J. Yndurain, Nucl. Phys. B153, 161 (1979).
[122] A. Gonzalez-Arroyo and C. Lopez, Nucl. Phys. B166, 429 (1980).
G. Curci, W. Furmanski, and R. Petronzio, Nucl. Phys. B175, 27 (1980).
W. Furmanski and R. Petronzio, Phys. Lett. B97, 437 (1980).
[123] R. Hamberg and W. L. van Neerven, Nucl. Phys. B379, 143 (1992).
[124] S. Moch, J. A. M. Vermaseren, and A. Vogt, Nucl. Phys. B688, 101 (2004), hep-
ph/0403192.
[125] A. Vogt, S. Moch, and J. A. M. Vermaseren, Nucl. Phys. B691, 129 (2004), hep-
ph/0404111.
[126] M. Buza, Y. Matiounine, J. Smith, R. Migneron, and W. L. van Neerven, Nucl.
Phys. B472, 611 (1996), hep-ph/9601302.
[127] J. Blu¨mlein, A. De Freitas, W. L. van Neerven, and S. Klein, Nucl. Phys. B755,
272 (2006), hep-ph/0608024.
[128] I. Bierenbaum, J. Blu¨mlein, and S. Klein, Nucl. Phys. B780, 40 (2007), hep-
ph/0703285.
[129] M. Buza, Y. Matiounine, J. Smith, and W. L. van Neerven, Eur. Phys. J. C1, 301
(1998), hep-ph/9612398.
[130] I. Bierenbaum, J. Blu¨mlein, and S. Klein, Phys. Lett. B672, 401 (2009), hep-
ph/0901.0669.
[131] I. Bierenbaum, J. Blu¨mlein, and S. Klein, Nucl. Phys. Proc. Suppl. 183, 162 (2008),
hep-ph/0806.4613.
[132] I. Bierenbaum, J. Blu¨mlein, and S. Klein, PoS Confinement8, 185 (2008), hep-
ph/0812.2427.
[133] I. Bierenbaum, J. Blu¨mlein, and S. Klein, 2– and 3–loop heavy flavor contributions
to F2(x,Q2), FL(x,Q2) and g1,2(x,Q2) in [87], pp 363.
[134] I. Bierenbaum, J. Blu¨mlein, and S. Klein, Mellin Moments of the O(α3s) Heavy Flavor
Contributions to unpolarized Deep-Inelastic Scattering at Q2 ≫ m2 and Anomalous
Dimensions, Nucl. Phys. B (in print), (2009), hep-ph/0904.3563.
[135] I. Bierenbaum, J. Blu¨mlein, and S. Klein, Acta Phys. Polon. B38, 3543 (2007); Pos
RADCOR 2007, 034 (2007), hep-ph/0710.3348.
[136] I. Bierenbaum, J. Blu¨mlein, and S. Klein, Acta Phys. Polon. B39, 1531 (2008),
hep-ph/0806.0451.
187
[137] I. Bierenbaum, J. Blu¨mlein, S. Klein, and C. Schneider, Nucl. Phys. B803, 1 (2008),
hep-ph/0803.0273.
[138] J. Blu¨mlein and S. Klein, PoS ACAT 2007, 084 (2007), hep-ph/0706.2426.
[139] I. Bierenbaum, J. Blu¨mlein, S. Klein, and C. Schneider, PoS ACAT 2007, 082 (2007),
math-ph/0707.4659.
[140] J. Blu¨mlein, M. Kauers, S. Klein, and C. Schneider, PoS ACAT 2008, 106 (2008),
hep-ph/0902.4095.
[141] J. Blu¨mlein, M. Kauers, S. Klein, and C. Schneider, Determining the closed forms of
the O(a3s) anomalous dimensions and Wilson coefficients from Mellin moments by means
of computer algebra, Comp. Phys. Commun. (in print), (2009), hep-ph/0902.4091.
[142] J. Blu¨mlein and S. Kurth, Phys. Rev. D60, 014018 (1999), hep-ph/9810241.
[143] J. A. M. Vermaseren, Int. J. Mod. Phys. A14, 2037 (1999), hep-ph/9806280.
[144] J. Blu¨mlein, Nucl. Phys. Proc. Suppl. 135, 225 (2004), hep-ph/0407044.
[145] J. Blu¨mlein and V. Ravindran, Nucl. Phys. B716, 128 (2005), hep-ph/0501178.
J. Blu¨mlein and V. Ravindran, Nucl. Phys. B749, 1 (2006), hep-ph/0604019.
[146] J. Blu¨mlein, Comput. Phys. Commun. 159, 19 (2004), hep-ph/0311046.
[147] J. Blu¨mlein, Structural Relations of Harmonic Sums and Mellin Transforms up to
Weight w=5, (2009), hep-ph/0901.3106.
[148] J. Blu¨mlein, Structural Relations of Harmonic Sums and Mellin Transforms at Weight
w=6, in Proc. of the Workshop “Motives, Quantum Field Theory, and Pseudodiffer-
ential Operators, June (2008)”, (Clay Institute, Boston University, 2009), in print,
math-ph/0901.0837.
[149] S. Weinzierl, Comput. Phys. Commun. 145, 357 (2002), math-ph/0201011.
[150] S. Moch and P. Uwer, Comput. Phys. Commun. 174, 759 (2006), math-ph/0508008.
[151] C. Schneider, Ann.Comb., 9 (1) (2005) 75; Proc. ISSAC’05, (2005) pp. 285 (ACM
Press); Proc. FPSAC’07, (2007) 1.
[152] C. Schneider, J. Algebra Appl. 6 (3), 415 (2007).
[153] C. Schneider, J. Diffr. Equations Appl., 11 (9) (2005) 799.
[154] C. Schneider, Se´m. Lothar. Combin. 56 (2007) Article B56b and Habilitation Thesis,
JKU Linz, (2007).
[155] J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, Trans. Am. Math.
Soc. 353, 907 (2001), math/9910045.
[156] J. Blu¨mlein, D. Broadhurst, and J. Vermaseren, The multiple zeta value data mine,
DESY 09–003.
188
[157] I. Bierenbaum, J. Blu¨mlein, and S. Klein, Two-Loop Massive Operator Matrix El-
ements for Polarized and Unpolarized Deep-Inelastic Scattering, in: Proc. of 15th In-
ternational Workshop On Deep-Inelastic Scattering And Related Subjects (DIS2007),
G. Grindhammer, K. Sachs, eds., (16–20 April 2007, Munich), Vol. 2, pp. 821,
hep-ph/0706.2738.
[158] I. Bierenbaum, J. Blu¨mlein, and S. Klein, Two-loop massive operator matrix elements
for polarized and unpolarized deep-inelastic scattering, PoS ACAT 2007, 070 (2007).
[159] I. Bierenbaum, J. Blu¨mlein, and S. Klein, in preparation.
[160] J. Blu¨mlein, S. Klein, and B. To¨dtli, O(α2s) and O(α3s) Heavy Flavor Contributions
to Transversity at Q2 ≫ m2, DESY 09–60 (2009).
[161] P. Nogueira, J. Comput. Phys. 105, 279 (1993).
[162] J. A. M. Vermaseren, New features of FORM, (2000), math-ph/0010025.
[163] T. van Ritbergen, A. N. Schellekens, and J. A. M. Vermaseren, Int. J. Mod. Phys.
A14, 41 (1999), hep-ph/9802376.
[164] M. Steinhauser, Comput. Phys. Commun. 134, 335 (2001), hep-ph/0009029.
[165] M. Buza, Y. Matiounine, J. Smith, and W. L. van Neerven, Nucl. Phys. B485, 420
(1997), hep-ph/9608342.
[166] V. Barone, A. Drago, and P. G. Ratcliffe, Phys. Rept. 359, 1 (2002), hep-
ph/0104283.
[167] J. A. M. Vermaseren, Comput. Phys. Commun. 83, 45 (1994).
[168] W. Albrecht et al., Phys. Lett. B28, 225 (1968); Nucl. Phys. B13, 1 (1969);
Separation of sigma-L and sigma-t in the region of deep inelastic electron - proton
scattering, DESY-69-046 (1969).
[169] R. Clifft and N. Doble, Proposed Design of a High-Energy, High Intensity Muon Beam
for the SPS North Experimental Area, CERN/LAB. II/EA/74-2 (1974).
[170] D. J. Fox et al., Phys. Rev. Lett. 33, 1504 (1974).
[171] T. Sloan, R. Voss, and G. Smadja, Phys. Rept. 162, 45 (1988).
[172] M. Holder et al., Nucl. Instrum. Meth. 151, 69 (1978).
CDHSW collaboration, W. Von Ruden, IEEE Trans. Nucl. Sci. 29, 360 (1982).
[173] G. Harigel, BEBC user’s handbook, (CERN, Geneva, 1977).
[174] W. K. Sakumoto et al., Nucl. Instrum. Meth. A294, 179 (1990).
B. J. King et al., Nucl. Instrum. Meth. A302, 254 (1991).
189
[175] M. Diemoz, F. Ferroni, and E. Longo, Phys. Rept. 130, 293 (1986).
F. Eisele, Rept. Prog. Phys. 49, 233 (1986).
S. R. Mishra and F. Sciulli, Ann. Rev. Nucl. Part. Sci. 39, 259 (1989).
K. Winter, ed., Neutrino physics, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol.
Vol. 1, (Cambridge University Press, Cambridge, 1991), 670 p.
N. Schmitz, Neutrinophysik, (Teubner, Stuttgart, 1997), 478 p.
[176] R. D. Peccei, ed., Proceedings, HERA Workshop, Hamburg, F.R. Germany, October
12-14, 1987. Vol. 1,2, 937 p.
W. Buchmu¨ller and G. Ingelman, eds., Proceedings, Physics at HERA Workshop,
Hamburg, F.R. Germany, October 29-30, 1991. Vol. 1-3, 1566 p.
J. Blu¨mlein and T. Riemann, eds., Deep inelastic scattering. Proceedings, Zeuthen
Workshop on Elementary Particle Theory, Teupitz, Germany, April 6-10, 1992, Nucl.
Phys. Proc. Suppl. B29A, 295 p. (1992).
HERA - The new frontier for QCD. Proceedings, Workshop, Durham, UK, March 21-26,
1993, J. Phys. G19, (1993), pp. 1427.
J. F. Mathiot and J. Tran Thanh Van, eds., Prepared for 6th Rencontres de Blois:
The Heart of the Matter: from Nuclear Interactions to Quark - Gluon Dynamics, Blois,
France, 20-25 Jun 1994, (Ed. Frontieres, Gif–sur–Yvette, 1995), 556 p.
[177] J. Blu¨mlein and W. D. Nowak, eds., Prospects of spin physics at HERA. Proc.,
Workshop, Zeuthen, Germany, August 28-31, 1995, DESY-95-200, (DESY, Hamburg,
1995), 387 p.
[178] G. Ingelman, A. De Roeck, and R. Klanner, eds., Future physics at HERA. Pro-
ceedings, Workshop, Hamburg, Germany, September 25, 1995-May 31, 1996. Vol. 1,2,
DESY-96-235, 1231 p.
[179] J. Blu¨mlein, A. De Roeck, T. Gehrmann, and W. D. Nowak, eds., Deep inelastic
scattering off polarized targets: Theory meets experiment. Physics with polarized protons
at HERA. Proceedings, Workshops, SPIN’97, Zeuthen, Germany, September 1-5,
1997 and Hamburg, Germany, March-September 1997, DESY-97-200.
[180] J. Blu¨mlein, W. D. Nowak, and G. Schnell, eds., Transverse spin physics, Proceedings,
Topical Workshop, Zeuthen, Germany, July 9-11, 2001, DESY-Zeuthen-01-01, Aug
2001. 374 p.
[181] J. B. Kogut and L. Susskind, Phys. Rept. 8, 75 (1973).
T.-M. Yan, Ann. Rev. Nucl. Part. Sci. 26, 199 (1976).
[182] J. Blu¨mlein and M. Klein, Nucl. Instrum. Meth. A329, 112 (1993).
[183] J. Blu¨mlein, Z. Phys. C65, 293 (1995), hep-ph/9403342.
[184] A. Arbuzov, D. Y. Bardin, J. Blu¨mlein, L. Kalinovskaya, and T. Riemann, Comput.
Phys. Commun. 94, 128 (1996), hep-ph/9511434, and references therein.
[185] J. D. Bjorken, Phys. Rev. D1, 1376 (1970).
190
[186] J. Blu¨mlein, M. Klein, T. Naumann, and T. Riemann, Structure functions, quark
distributions and ΛQCD at HERA in Proc. of DESY Theory Workshop on Physics at
HERA (ed. R.D. Peccei), Hamburg, F.R. Germany, Oct 12-14, 1987, Vol 1, 67pp.
[187] A. Kwiatkowski, H. Spiesberger, and H. J. Mo¨hring, Comp. Phys. Commun. 69,
155 (1992), and references therein.
[188] J. Engelen and P. Kooijman, Prog. Part. Nucl. Phys. 41, 1 (1998).
H. Abramowicz and A. Caldwell, Rev. Mod. Phys. 71, 1275 (1999), hep-ex/9903037.
[189] J. Blu¨mlein, B. Geyer, and D. Robaschik, Nucl. Phys. B560, 283 (1999), hep-
ph/9903520.
[190] F. J. Yndurain, The theory of quark and gluon interactions, (Springer, Berlin, 2006),
474 p, 4th edition.
[191] R. Field, Applications of perturbative QCD, (Addison-Wesley, Redwood City, 1989),
366 p.
[192] LHPC collaboration, D. Dolgov et al., Phys. Rev. D66, 034506 (2002), hep-
lat/0201021.
[193] QCDSF collaboration, M. Go¨ckeler et al., PoS LAT 2007, 147 (2007), hep-
lat/0710.2489.
ETM collaboration, R. Baron et al., PoS LAT 2007, 153 (2007), hep-lat/0710.1580.
W. Bietenholz et al., PoS LAT 2008, 149 (2008), hep-lat/0808.3637.
S. N. Syritsyn et al., PoS LAT 2008, 169 (2008), hep-lat/0903.3063.
[194] C. Itzykson and J. Zuber, Quantum Field Theory, (McGraw-Hill, New York, 1980),
705 p.
[195] J. Blu¨mlein and N. Kochelev, Nucl. Phys. B498, 285 (1997), hep-ph/9612318.
[196] J. Blu¨mlein and A. Tkabladze, Nucl. Phys. B553, 427 (1999), hep-ph/9812478.
[197] E. Rutherford, Phil. Mag. 21, 669 (1911).
[198] R. W. Mcallister and R. Hofstadter, Phys. Rev. 102, 851 (1956).
D. N. Olson, H. F. Schopper, and R. R. Wilson, Phys. Rev. Lett. 6, 286 (1961).
R. Hofstadter, Electron scattering and nuclear and nucleon structure. A collection of
reprints with an introduction, (New York, Benjamin, 1963), 690 p.
[199] C. Nash, Nucl. Phys. B31, 419 (1971).
P. V. Landshoff and J. C. Polkinghorne, Phys. Rept. 5, 1 (1972).
[200] J. D. Jackson, G. G. Ross, and R. G. Roberts, Phys. Lett. B226, 159 (1989).
R. G. Roberts and G. G. Ross, Phys. Lett. B373, 235 (1996), hep-ph/9601235.
[201] J. Blu¨mlein and N. Kochelev, Phys. Lett. B381, 296 (1996), hep-ph/9603397.
191
[202] J. Blu¨mlein, V. Ravindran, and W. L. van Neerven, Phys. Rev. D68, 114004 (2003),
hep-ph/0304292.
[203] D. Amati, R. Petronzio, and G. Veneziano, Nucl. Phys. B140, 54 (1978).
S. B. Libby and G. Sterman, Phys. Rev. D18, 3252 (1978).
S. B. Libby and G. Sterman, Phys. Rev. D18, 4737 (1978).
A. H. Mueller, Phys. Rev. D18, 3705 (1978).
J. C. Collins and G. Sterman, Nucl. Phys. B185, 172 (1981).
G. T. Bodwin, Phys. Rev. D31, 2616 (1985); [Erratum-ibid.] D34, 3932 (1986).
J. C. Collins, D. E. Soper, and G. Sterman, Nucl. Phys. B261, 104 (1985).
[204] S. D. Drell and T.-M. Yan, Ann. Phys. 66, 578 (1971).
[205] J. Blu¨mlein, Introduction into QCD, Lecture Notes (1997).
[206] R. Jackiw, Canonical light-cone commutators and their applications in [47], pp 1.
[207] Y. Frishman, Phys. Rept. 13, 1 (1974).
[208] B. Geyer, D. Robaschik, and E. Wieczorek, Fortschr. Phys. 27, 75 (1979).
[209] A. H. Mueller, Phys. Rept. 73, 237 (1981).
[210] J. Blu¨mlein, Comput. Phys. Commun. 133, 76 (2000), hep-ph/0003100.
J. Blu¨mlein and S.-O. Moch, Phys. Lett. B614, 53 (2005), hep-ph/0503188.
[211] E. Carlson, Sur une classe de se´ries de Taylor, PhD Thesis, Uppsala, 1914.
E. Titchmarsh, Theory of Functions, (Oxford University Press, Oxford, 1939), Chapt.
9.5.
[212] L. D. Landau, Nucl. Phys. 13, 181 (1959).
[213] J. D. Bjorken, Experimental tests of quantum electrodynamics and spectral represen-
tations of Green’s functions in perturbation theory, PhD Thesis, RX–1037 (1959).
[214] A. Bassetto, M. Ciafaloni, and G. Marchesini, Phys. Rept. 100, 201 (1983).
[215] E. C. G. Stu¨ckelberg and A. Petermann, Helv. Phys. Acta 24, 317 (1951).
M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300 (1954).
N. N. Bogolyubov and D. V. Shirkov, Introduction to the theory of quantized fields,
(New York, Interscience, 1959), 720 p.
[216] K. Symanzik, Commun. Math. Phys. 18, 227 (1970).
C. G. Callan, Phys. Rev. D2, 1541 (1970).
[217] G. Altarelli, Ann. Rev. Nucl. Part. Sci. 39, 357 (1989).
[218] J. F. Owens and W.-K. Tung, Ann. Rev. Nucl. Part. Sci. 42, 291 (1992).
192
[219] J. Blu¨mlein, On the Theoretical Status of Deep Inelastic Scattering, (1995), hep-
ph/9512272.
[220] D. W. Duke and R. G. Roberts, Phys. Rept. 120, 275 (1985).
[221] S. Bethke and J. E. Pilcher, Ann. Rev. Nucl. Part. Sci. 42, 251 (1992).
[222] S. Moch, J. A. M. Vermaseren, and A. Vogt, Third-order QCD corrections to
the charged-current structure function F3, Nucl. Phys. B (in print), (2009), hep-
ph/0812.4168.
[223] F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937).
D. R. Yennie, S. C. Frautschi, and H. Suura, Ann. Phys. 13, 379 (1961).
[224] T. Kinoshita, J. Math. Phys. 3, 650 (1962).
T. D. Lee and M. Nauenberg, Phys. Rev. 133, B1549 (1964).
[225] A. Peterman, Phys. Rept. 53, 157 (1979).
[226] J. C. Collins, Renormalization, (Cambridge University Press, Cambridge, 1984),
380 p.
[227] M. Glu¨ck, R. M. Godbole, and E. Reya, Z. Phys. C41, 667 (1989).
[228] M. Glu¨ck, E. Reya, and A. Vogt, Z. Phys. C48, 471 (1990).
[229] M. Glu¨ck, E. Reya, and A. Vogt, Z. Phys. C53, 127 (1992).
[230] M. Glu¨ck, E. Reya, and A. Vogt, Z. Phys. C67, 433 (1995).
[231] M. Glu¨ck, E. Reya, and A. Vogt, Eur. Phys. J. C5, 461 (1998), hep-ph/9806404.
[232] M. Glu¨ck, P. Jimenez-Delgado, and E. Reya, Eur. Phys. J. C53, 355 (2008), hep-
ph/0709.0614.
[233] P. Jimenez-Delgado and E. Reya, Dynamical NNLO parton distributions, (2008),
hep-ph/0810.4274.
[234] S. Alekhin, JETP Lett. 82, 628 (2005), hep-ph/0508248.
[235] A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Parton distributions for
the LHC, (2009), hep-ph/0901.0002.
[236] CTEQ collaboration, H. L. Lai et al., Eur. Phys. J. C12, 375 (2000), hep-
ph/9903282.
[237] NNPDF collaboration, R. D. Ball et al., Nucl. Phys. B809, 1 (2009), hep-
ph/0808.1231.
193
[238] L. V. Gribov, E. M. Levin, and M. G. Ryskin, Nucl. Phys. B188, 555 (1981).
A. H. Mueller and J.-W. Qiu, Nucl. Phys. B268, 427 (1986).
J. C. Collins and J. Kwiecinski, Nucl. Phys. B335, 89 (1990).
J. Bartels, G. A. Schuler, and J. Blu¨mlein, Z. Phys. C50, 91 (1991).
M. Altmann, M. Glu¨ck, and E. Reya, Phys. Lett. B285, 359 (1992).
V. Del Duca, An introduction to the perturbative QCD pomeron and to jet physics at
large rapidities, (1995), hep-ph/9503226.
L. N. Lipatov, Phys. Rept. 286, 131 (1997), hep-ph/9610276.
[239] V. S. Fadin, E. A. Kuraev, and L. N. Lipatov, Phys. Lett. B60, 50 (1975).
I. I. Balitsky and L. N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978).
R. Kirschner and L. N. Lipatov, Nucl. Phys. B213, 122 (1983).
J. Bartels, B. I. Ermolaev, and M. G. Ryskin, Z. Phys. C72, 627 (1996), hep-
ph/9603204.
V. S. Fadin and L. N. Lipatov, Phys. Lett. B429, 127 (1998), hep-ph/9802290.
[240] J. Blu¨mlein and A. Vogt, Phys. Lett. B370, 149 (1996), hep-ph/9510410; Acta
Phys. Polon. B27, 1309 (1996), hep-ph/9603450; Phys. Lett. B386, 350 (1996),
hep-ph/9606254; Phys. Rev. D58, 014020 (1998), hep-ph/9712546.
[241] J. Blu¨mlein, V. Ravindran, W. L. van Neerven, and A. Vogt, (1998), hep-
ph/9806368.
[242] J. Blu¨mlein and W. L. van Neerven, Phys. Lett. B450, 412 (1999), hep-ph/9811519.
[243] G. Altarelli, R. D. Ball, and S. Forte, Nucl. Phys. B799, 199 (2008), 0802.0032.
M. Ciafaloni, D. Colferai, G. P. Salam, and A. M. Stasto, JHEP 08, 046 (2007),
0707.1453.
[244] S. Alekhin, Phys. Rev. D68, 014002 (2003), hep-ph/0211096.
[245] A. Chuvakin, J. Smith, and W. L. van Neerven, Phys. Rev. D62, 036004 (2000),
hep-ph/0002011.
[246] G. A. Schuler, Nucl. Phys. B299, 21 (1988).
U. Baur and J. J. van der Bij, Nucl. Phys. B304, 451 (1988).
[247] M. Glu¨ck, R. M. Godbole, and E. Reya, Z. Phys. C38, 441 (1988); [Erratum-ibid.]
C39, 590 (1988).
[248] ZEUS collaboration, J. Breitweg et al., Eur. Phys. J. C12, 35 (2000), hep-
ex/9908012.
H1 collaboration, C. Adloff et al., Phys. Lett. B528, 199 (2002), hep-ex/0108039.
[249] ZEUS collaboration, S. Chekanov et al., Phys. Rev. D69, 012004 (2004), hep-
ex/0308068.
H1 collaboration, A. Aktas et al., Eur. Phys. J. C40, 349 (2005), hep-ex/0411046.
194
[250] H1 collaboration, A. Aktas et al., Eur. Phys. J. C45, 23 (2006), hep-ex/0507081.
[251] E. Eichten, I. Hinchliffe, K. D. Lane, and C. Quigg, Rev. Mod. Phys. 56, 579 (1984).
[252] M. Glu¨ck, E. Reya, and M. Stratmann, Nucl. Phys. B422, 37 (1994).
[253] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32, 438 (1974).
H. Fritzsch and P. Minkowski, Ann. Phys. 93, 193 (1975).
[254] A. Djouadi, Phys. Rept. 457, 1 (2008), hep-ph/0503172; Phys. Rept. 459, 1 (2008),
hep-ph/0503173; and references therein.
[255] M. Glu¨ck and E. Reya, Phys. Lett. B83, 98 (1979).
[256] M. Glu¨ck and E. Reya, Phys. Lett. B79, 453 (1978).
[257] E. L. Berger and D. L. Jones, Phys. Rev. D23, 1521 (1981).
[258] G. Ingelman and G. A. Schuler, Z. Phys. C40, 299 (1988).
[259] F. A. Berends, W. L. van Neerven, and G. J. H. Burgers, Nucl. Phys. B297, 429
(1988); [Erratum-ibid.] B304, 921 (1988).
W. L. van Neerven, Acta Phys. Polon. B28, 2715 (1997), hep-ph/9708452.
[260] M. A. G. Aivazis, J. C. Collins, F. I. Olness, and W.-K. Tung, Phys. Rev. D50,
3102 (1994), hep-ph/9312319.
R. S. Thorne and W. K. Tung, PQCD Formulations with Heavy Quark Masses and
Global Analysis, (2008), hep-ph/0809.0714.
[261] J. Blu¨mlein and W. L. van Neerven, Phys. Lett. B450, 417 (1999), hep-ph/9811351.
[262] J. C. Collins and R. J. Scalise, Phys. Rev. D50, 4117 (1994), hep-ph/9403231.
B. W. Harris and J. Smith, Phys. Rev. D51, 4550 (1995), hep-ph/9409405.
[263] K. G. Chetyrkin, B. A. Kniehl, and M. Steinhauser, Nucl. Phys. B814, 231 (2009),
hep-ph/0812.1337.
[264] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B44, 189 (1972).
[265] J. F. Ashmore, Lett. Nuovo Cim. 4, 289 (1972).
G. M. Cicuta and E. Montaldi, Nuovo Cim. Lett. 4, 329 (1972).
C. G. Bollini and J. J. Giambiagi, Nuovo Cim. B12, 20 (1972).
[266] W. Pauli and F. Villars, Rev. Mod. Phys. 21, 434 (1949).
[267] E. R. Speer, J. Math. Phys. 15, 1 (1974).
[268] G. ’t Hooft, Nucl. Phys. B61, 455 (1973).
[269] Y. Matiounine, J. Smith, and W. L. van Neerven, Phys. Rev. D57, 6701 (1998),
hep-ph/9801224.
195
[270] R. Hamberg, Second order gluonic contributions to physical quantities, PhD Thesis,
Leiden, 1991.
[271] R. Tarrach, Nucl. Phys. B183, 384 (1981).
[272] O. Nachtmann and W. Wetzel, Nucl. Phys. B187, 333 (1981).
[273] N. Gray, D. J. Broadhurst, W. Gra¨fe, and K. Schilcher, Z. Phys. C48, 673 (1990).
D. J. Broadhurst, N. Gray, and K. Schilcher, Z. Phys. C52, 111 (1991).
[274] J. Fleischer, F. Jegerlehner, O. V. Tarasov, and O. L. Veretin, Nucl. Phys. B539,
671 (1999), hep-ph/9803493.
[275] I. B. Khriplovich, Yad. Fiz. 10, 409 (1969).
[276] W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974).
D. R. T. Jones, Nucl. Phys. B75, 531 (1974).
[277] L. F. Abbott, Nucl. Phys. B185, 189 (1981).
A. Rebhan, Z. Phys. C30, 309 (1986).
F. Jegerlehner and O. V. Tarasov, Nucl. Phys. B549, 481 (1999), hep-ph/9809485.
[278] K. G. Chetyrkin and M. Steinhauser, Phys. Rev. Lett. 83, 4001 (1999), hep-
ph/9907509; Nucl. Phys. B573, 617 (2000), hep-ph/9911434.
[279] D. J. Broadhurst, Z. Phys. C54, 599 (1992).
[280] L. Avdeev, J. Fleischer, S. Mikhailov, and O. Tarasov, Phys. Lett. B336, 560
(1994), hep-ph/9406363.
S. Laporta and E. Remiddi, Phys. Lett. B379, 283 (1996), hep-ph/9602417.
[281] D. J. Broadhurst, Eur. Phys. J. C8, 311 (1999), hep-th/9803091.
[282] R. Boughezal, J. B. Tausk, and J. J. van der Bij, Nucl. Phys. B713, 278 (2005),
hep-ph/0410216.
[283] K. G. Chetyrkin, A. L. Kataev, and F. V. Tkachov, Nucl. Phys. B174, 345 (1980).
[284] S. Klein, Diploma Thesis, University of Potsdam (2006).
[285] L. Slater, Generalized Hypergeometric Functions, (Cambridge University Press, Cam-
bridge, 1966), 273 p.
[286] W. Bailey, Generalized Hypergeometric Series, (Cambridge University Press, Cam-
bridge, 1935), 108 p.
G. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics
and its Applications 71, (Cambridge University Press, Cambridge, 2001), 663 p.
[287] I. Bierenbaum, J. Blu¨mlein, and S. Klein, Phys. Lett. B648, 195 (2007), hep-
ph/0702265.
196
[288] I. Bierenbaum, J. Blu¨mlein, and S. Klein, to appear.
[289] I. Bierenbaum, J. Blu¨mlein, and S. Klein, Nucl. Phys. Proc. Suppl. 160, 85 (2006),
hep-ph/0607300.
[290] C. W. Bauer, A. Frink, and R. Kreckel, Introduction to the GiNaC Framework for Sym-
bolic Computation within the C++ Programming Language, (2000), arxiv: 0004015.
[291] A. Devoto and D. W. Duke, Riv. Nuovo Cim. 7N6, 1 (1984).
J. Blu¨mlein and H. Kawamura, Nucl. Phys. B708, 467 (2005), hep-ph/0409289.
J. Blu¨mlein, Collection of Polylog-Integrals, unpublished.
[292] N. No¨rlund, Vorlesungen u¨ber Differenzenrechnung, (Springer, Berlin, 1924), 551 p.
L. Milne-Thomson, The Calculus of finite Differences, (MacMillan, London, 1951),
273 p.
[293] R. Gosper, Proc. Nat. Acad. Sci. USA 75, 40 (1978).
[294] D. Zeilberger, J. Symbolic Comput. 11, 195 (1991).
[295] M. Petkovsˇek, H. S. Wilf, and D. Zeilberger, A = B, (A. K. Peters, Wellesley, MA,
1996).
[296] M. Karr, J. ACM, 28 (1981) 305. .
[297] C. Schneider, J. Symbolic Comput. (2008), doi:10.1016/j.jsc.2008.01.001 .
[298] C. Schneider, Proc. ISSAC’04, (2004) pp. 282 (ACM Press).
[299] A. Goncharov, Math. Res. Lett. 5 (1998) 497 .
[300] M. P. Hoang Ngoc Minh and J. van der Hoeven, Discr. Math. 225 (2000) 217 .
[301] S. Moch, P. Uwer, and S. Weinzierl, J. Math. Phys. 43, 3363 (2002), hep-
ph/0110083.
[302] D. Zeilberger, J. Symbolic Comput. 11 (1991), 195 .
[303] P. Paule and C. Schneider, Adv. in Appl. Math. 31 (2), 359 (2003).
K. Driver, H. Prodinger, C. Schneider, and A. Weideman, Ramanujan Journal 12
(3), 299 (2006).
[304] M. E. Hoffman, J. Algebra 194, 477 (1997).
M. E. Hoffman, Nucl. Phys. Proc. Suppl. 135, 215 (2004), math/0406589.
[305] E. Barnes, Proc. Lond. Math. Soc. (2) 6 (1908) 141 .
E. Barnes, Quart. J. Math. 41 (1910) 136 .
H. Mellin, Math. Ann. 68 (1910) 305 .
197
[306] E. Whittaker and G. Watson, A Course of Modern Analysis, (Cambridge University
Press, Cambridge, 1927; reprinted 1996) 616 p .
E. Titchmarsh, Introduction to the Theory of Fourier Integrals, (Calendron Press,
Oxford, 1937; 2nd Edition 1948) .
[307] R. Paris and D. D. Kaminski, Asymptotics and Mellin-Barnes Integrals, (Cambridge
University Press, Cambridge, 2001), 438 p.
[308] I. Bierenbaum and S. Weinzierl, Eur. Phys. J. C32, 67 (2003), hep-ph/0308311.
[309] M. Czakon, Comput. Phys. Commun. 175, 559 (2006), hep-ph/0511200.
[310] A. Djouadi and P. Gambino, Phys. Rev. D49, 3499 (1994), hep-ph/9309298.
[311] D. Broadhurst, private communication, 2009.
[312] S. G. Gorishnii, S. A. Larin, L. R. Surguladze, and F. V. Tkachov, Comput. Phys.
Commun. 55, 381 (1989).
[313] S. A. Larin, F. V. Tkachov, and J. A. M. Vermaseren, The FORM version of MINCER,
NIKHEF-H-91-18 (1991).
[314] E. Remiddi and J. A. M. Vermaseren, Int. J. Mod. Phys. A15, 725 (2000), hep-
ph/9905237.
[315] J. A. M. Vermaseren, The Form version of MINCER, unpublished.
[316] M. Tentyukov and J. A. M. Vermaseren, The multithreaded version of FORM, (2007),
hep-ph/0702279.
[317] J. A. Gracey, Phys. Lett. B322, 141 (1994), hep-ph/9401214.
[318] S. Moch, J. A. M. Vermaseren, and A. Vogt, Nucl. Phys. B646, 181 (2002), hep-
ph/0209100.
[319] J. Blu¨mlein and S. Klein, in preparation.
[320] M. J. Alguard et al., Phys. Rev. Lett. 37, 1261 (1976); Phys. Rev. Lett. 41, 70
(1978).
G. Baum et al., Phys. Rev. Lett. 51, 1135 (1983).
EMC collaboration, J. Ashman et al., Phys. Lett. B206, 364 (1988); Nucl. Phys.
B328, 1 (1989).
[321] SMC collaboration, B. Adeva et al., Phys. Lett. B302, 533 (1993).
E142 collaboration, P. L. Anthony et al., Phys. Rev. D54, 6620 (1996), hep-
ex/9610007.
HERMES collaboration, K. Ackerstaff et al., Phys. Lett. B404, 383 (1997), hep-
ex/9703005.
E154 collaboration, K. Abe et al., Phys. Rev. Lett. 79, 26 (1997), hep-ex/9705012.
SMC collaboration, B. Adeva et al., Phys. Rev. D58, 112001 (1998).
198
E143 collaboration, K. Abe et al., Phys. Rev. D58, 112003 (1998), hep-ph/9802357.
HERMES collaboration, A. Airapetian et al., Phys. Lett. B442, 484 (1998), hep-
ex/9807015.
E155 collaboration, P. L. Anthony et al., Phys. Lett. B463, 339 (1999), hep-
ex/9904002.
E155 collaboration, P. L. Anthony et al., Phys. Lett. B493, 19 (2000), hep-
ph/0007248.
Jefferson Lab Hall A collaboration, X. Zheng et al., Phys. Rev. Lett. 92, 012004
(2004), nucl-ex/0308011.
HERMES collaboration, A. Airapetian et al., Phys. Rev. D71, 012003 (2005),
hep-ex/0407032.
COMPASS collaboration, E. S. Ageev et al., Phys. Lett. B612, 154 (2005), hep-
ex/0501073.
COMPASS collaboration, E. S. Ageev et al., Phys. Lett. B647, 330 (2007), hep-
ex/0701014.
HERMES collaboration, A. Airapetian et al., Phys. Rev. D75, 012007 (2007),
hep-ex/0609039.
[322] E. Reya, The Spin structure of the nucleon, in: QCD - 20 years later. Proceedings,
Workshop, Aachen, P.M. Zerwas and H.A. Kastrup, eds., Germany, June 9-13, 1992.
(World Scientific, Singapore, 1993), Vol. 1, pp.272.
[323] B. Lampe and E. Reya, Phys. Rept. 332, 1 (2000), hep-ph/9810270.
[324] M. Burkardt, A. Miller, and W. D. Nowak, Spin-polarized high-energy scattering of
charged leptons on nucleons, (2008), hep-ph/0812.2208.
[325] J. Blu¨mlein, On the measurability of the structure function g1(x,Q2) in ep collisions
at HERA, (1995), hep-ph/9508387.
[326] K. Kurek, ∆G from COMPASS, (2006), hep-ex/0607061.
COMPASS collaboration, G. Brona, Measurement of the gluon polarisation at COM-
PASS, (2007), hep-ex/0705.2372.
[327] A. D. Watson, Z. Phys. C12, 123 (1982).
[328] M. Glu¨ck, E. Reya, and W. Vogelsang, Nucl. Phys. B351, 579 (1991).
[329] W. Vogelsang, Z. Phys. C50, 275 (1991).
[330] G. Altarelli, R. D. Ball, S. Forte, and G. Ridolfi, Acta Phys. Polon. B29, 1145
(1998), hep-ph/9803237.
[331] M. Glu¨ck, E. Reya, M. Stratmann, and W. Vogelsang, Phys. Rev. D63, 094005
(2001), hep-ph/0011215.
199
[332] J. Blu¨mlein and H. Bo¨ttcher, Nucl. Phys. B636, 225 (2002), hep-ph/0203155.
M. Hirai, S. Kumano, and N. Saito, Phys. Rev. D74, 014015 (2006), hep-
ph/0603213.
E. Leader, A. V. Sidorov, and D. B. Stamenov, Phys. Rev. D75, 074027 (2007),
hep-ph/0612360.
[333] D. de Florian, R. Sassot, M. Stratmann, and W. Vogelsang, (2009), 0904.3821.
[334] I. Bojak and M. Stratmann, Nucl. Phys. B540, 345 (1999), hep-ph/9807405.
[335] S. Wandzura and F. Wilczek, Phys. Lett. B72, 195 (1977).
[336] E. B. Zijlstra and W. L. van Neerven, Nucl. Phys. B417, 61 (1994); [Erratum-ibid.]
B426, 245 (1994); [Erratum-ibid.] B773, 105 (2007).
[337] H. Ito, Prog. Theor. Phys. 54, 555 (1975).
K. Sasaki, Prog. Theor. Phys. 54, 1816 (1975).
M. A. Ahmed and G. G. Ross, Nucl. Phys. B111, 441 (1976).
[338] J. C. Ward, Phys. Rev. 78, 182 (1950).
Y. Takahashi, Nuovo Cim. 6, 371 (1957).
[339] D. A. Akyeampong and R. Delbourgo, Nuovo Cim. A17, 578 (1973); Nuovo
Cim. A18, 94 (1973); Nuovo Cim. A19, 219 (1974).
P. Breitenlohner and D. Maison, Commun. Math. Phys. 52, 55 (1977).
[340] G. T. Bodwin and J.-W. Qiu, Phys. Rev. D41, 2755 (1990).
[341] R. Mertig and W. L. van Neerven, Z. Phys. C70, 637 (1996), hep-ph/9506451.
[342] W. Vogelsang, Phys. Rev. D54, 2023 (1996), hep-ph/9512218.
W. Vogelsang, Nucl. Phys. B475, 47 (1996), hep-ph/9603366.
[343] J. P. Ralston and D. E. Soper, Nucl. Phys. B152, 109 (1979).
R. L. Jaffe and X.-D. Ji, Phys. Rev. Lett. 67, 552 (1991); Nucl. Phys. B375, 527
(1992).
[344] J. L. Cortes, B. Pire, and J. P. Ralston, Z. Phys. C55, 409 (1992).
[345] X. Artru and M. Mekhfi, Z. Phys. C45, 669 (1990).
[346] J. C. Collins, Nucl. Phys. B396, 161 (1993), hep-ph/9208213.
R. L. Jaffe and X.-D. Ji, Phys. Rev. Lett. 71, 2547 (1993), hep-ph/9307329.
R. D. Tangerman and P. J. Mulders, Polarized twist - three distributions gT and hL
and the role of intrinsic transverse momentum, (1994), hep-ph/9408305.
D. Boer and P. J. Mulders, Phys. Rev. D57, 5780 (1998), hep-ph/9711485.
200
[347] HERMES collaboration, A. Airapetian et al., Phys. Rev. Lett. 94, 012002 (2005),
hep-ex/0408013; JHEP 06, 017 (2008), hep-ex/0803.2367.
COMPASS collaboration, V. Y. Alexakhin et al., Phys. Rev. Lett. 94, 202002
(2005), hep-ex/0503002.
A. Afanasev et al., Transversity and transverse spin in nucleon structure through SIDIS
at Jefferson Lab, (2007), hep-ph/0703288.
COMPASS collaboration, M. Alekseev et al., Phys. Lett. B673, 127 (2009), hep-
ex/0802.2160.
The PANDA collaboration, M. F. M. Lutz, B. Pire, O. Scholten, and R. Timmer-
mans, (2009), hep-ex/0903.3905.
[348] M. Anselmino et al., Phys. Rev. D75, 054032 (2007), hep-ph/0701006.
M. Anselmino et al., Update on transversity and Collins functions from SIDIS and e+
e− data, (2008), hep-ph/0812.4366.
[349] S. Aoki, M. Doui, T. Hatsuda, and Y. Kuramashi, Phys. Rev. D56, 433 (1997),
hep-lat/9608115.
M. Go¨ckeler et al., Nucl. Phys. Proc. Suppl. 53, 315 (1997), hep-lat/9609039.
A. A. Khan et al., Nucl. Phys. Proc. Suppl. 140, 408 (2005), hep-lat/0409161.
QCDSF collaboration, M. Diehl et al., (2005), hep-ph/0511032.
QCDSF collaboration, M. Go¨ckeler et al., Phys. Rev. Lett. 98, 222001 (2007),
hep-lat/0612032.
D. Renner, private communication, 2009.
[350] X.-D. Ji, Phys. Rev. D49, 114 (1994), hep-ph/9307235.
[351] A. Bacchetta and P. J. Mulders, Phys. Rev. D62, 114004 (2000), hep-ph/0007120.
[352] W. Vogelsang and A. Weber, Phys. Rev. D48, 2073 (1993).
[353] W. Vogelsang, Phys. Rev. D57, 1886 (1998), hep-ph/9706511.
[354] H. Shimizu, G. Sterman, W. Vogelsang, and H. Yokoya, Phys. Rev. D71, 114007
(2005), hep-ph/0503270.
[355] F. Baldracchini, N. S. Craigie, V. Roberto, and M. Socolovsky, Fortschr. Phys. 30,
505 (1981).
M. A. Shifman and M. I. Vysotsky, Nucl. Phys. B186, 475 (1981).
A. P. Bukhvostov, G. V. Frolov, L. N. Lipatov, and E. A. Kuraev, Nucl. Phys.
B258, 601 (1985).
A. Mukherjee and D. Chakrabarti, Phys. Lett. B506, 283 (2001), hep-ph/0102003.
[356] J. Blu¨mlein, Eur. Phys. J. C20, 683 (2001), hep-ph/0104099.
[357] R. Kirschner, L. Mankiewicz, A. Schafer, and L. Szymanowski, Z. Phys. C74, 501
(1997), hep-ph/9606267.
201
[358] A. Hayashigaki, Y. Kanazawa, and Y. Koike, Phys. Rev. D56, 7350 (1997), hep-
ph/9707208.
S. Kumano and M. Miyama, Phys. Rev. D56, 2504 (1997), hep-ph/9706420.
[359] A. V. Belitsky and D. Mu¨ller, Phys. Lett. B417, 129 (1998), hep-ph/9709379.
P. Hoodbhoy and X.-D. Ji, Phys. Rev. D58, 054006 (1998), hep-ph/9801369.
A. V. Belitsky, A. Freund, and D. Mu¨ller, Phys. Lett. B493, 341 (2000), hep-
ph/0008005.
[360] J. A. Gracey, Nucl. Phys. B662, 247 (2003), hep-ph/0304113; Nucl. Phys. B667,
242 (2003), hep-ph/0306163; JHEP 10, 040 (2006), hep-ph/0609231; Phys. Lett.
B643, 374 (2006), hep-ph/0611071.
[361] Y. Andre´, Proc. of the Int. Conf. “Motives, Quantum Field Theory, an Pseudo Differ-
ential Operators”, Clay Mathematical Institute, Boston, June, 2008.
F. Brown, Commun. Math. Phys. 287, 925 (2009), arxiv: 0804.1660.
[362] H. Ferguson and D. H. Bailey, A Polynomial Time, Numerically Stable Integer Relation
Algorithm, RNR Techn. Rept. RNR-91-032, 199 (1991).
[363] B. Salvy and P. Zimmermann, ACM Transactions on Mathematical Software (2)
20, 163 (1994).
[364] C. Mallinger, Algorithmic Manipulations and Transformations of Univariate Holonomic
Functions and Sequences, Master Thesis, J. Kepler University, Linz, 1996.
[365] J. Blu¨mlein, Nucl. Phys. Proc. Suppl. 183, 232 (2008), math-ph/0807.0700.
[366] K. Geddes, S. Czapor, and G. Labahn, Algorithms for Computer Algebra, (Kluwer
Academic Publishers, Boston(USA), 1992, 585 p).
J. von zur Gathen and J. Gerhard, Modern Computer Algebra, (Cambridge University
Press, Cambridge, 1999), 754 p.
M. Kauers, Nucl. Phys. Proc. Suppl. 183, 245 (2008).
[367] A. Bostan and M. Kauers, The full counting function for Gessel walks is algebraic,
INRIA-Rocquencourt report, 2009, in preparation.
[368] B. Beckermann and G. Labahn, Numerical Algorithms 3, 45 (1992).
B. Beckermann and G. Labahn, SIAM Journal of Matrix Analysis and Applications
22, 114 (2000).
[369] M. Karr, J. Symbolic Comput., 1 (1985) 303 .
[370] C. Schneider, Small Symbolic Summation in Difference Fields, PhD Thesis, RISC–
Linz, J. Kepler University, Linz, 2001.
C. Schneider, Symbolic summation finds optimal nested sum representations, SFB-
Report 2007-26, SFB F013, J. Kepler University, Linz, 2007.
C. Schneider, J. Symbolic Comput. 43, 611 (2008).
202
C. Schneider, Parameterized telescoping proves algebraic independence of sums, Ann.
Comb., to appear, 2009.
[371] J. Ablinger, A Computer Algebra Toolbox for Harmonic Sums Related to Particle
Physics, Diploma Thesis, J. Kepler University, Linz, 2009.
[372] J. Blu¨mlein and S. Moch, in preparation.
[373] J. Blu¨mlein, A. de Freitas, and W. van Neerven, PoS RADCOR 2007, 005 (2007),
0812.1588.
[374] M. J. G. Veltman, Diagrammatica: The Path to Feynman rules, (Cambridge Univer-
sity Press, Cambridge, 1994), 284 p.
G. ’t Hooft and M. J. G. Veltman, Diagrammar, CERN Yellow Report 73–9 (1973).
[375] Y. Matiounine, J. Smith, and W. L. van Neerven, Phys. Rev. D58, 076002 (1998),
hep-ph/9803439.
[376] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover
Publications Inc., New York, 1972), 1046 p.
[377] N. Nielsen, Handbuch der Theorie der Gammafunktion, (Chelsea Publishing Company,
New York, 1965), 328 p; first published: (Teubner, Leipzig, 1906), 326 p.
[378] V. A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211, 1
(2004).
[379] L. Euler, Novi Comm. Acad. Sci Petropolitanae 1, 140 (1775).
[380] D. Zagier, Proc. First European Congress Math. (Paris) II, 497 (1994).
[381] N. Nielsen, Nova Acta Leopoldina 90, 123 (1909).
[382] K. S. Ko¨lbig, SIAM J. Math. Anal. 17, 1232 (1986).
[383] K. Knopp, Theorie und Anwendung der unendlichen Reihen, (Springer, Berlin, 1947),
583 p.
E. Landau, S.-Ber. Ko¨nigl. Bayerische Akad. Wiss. Mu¨nchen, math.-naturw. Kl.
36, 151 (1906).
203
Acknowledgement
Foremost, I would like to thank Johannes Blu¨mlein for his constant support during the
last years, putting very much time and effort into supervising and teaching me.
Further I would like to thank Prof. Reya. for giving me the opportunity of getting my
Ph.D. at the University of Dortmund.
I am particularly grateful to I. Bierenbaum for her advice and friendship during the last
years and for reading the manuscript.
I would like to thank D. Broadhurst, K. Chetyrkin, J. Kallarackal, M. Kauers, D. Renner,
C. Schneider, J. Smith, F. Stan, M. Steinhauser and J. Vermaseren for useful discussions.
Additionally, I would like to thank M. Steinhauser for help with the use of MATAD, J. Ver-
maseren for help with the use of FORM, and C. Schneider for help with the use of Sigma.
Further thanks go to B. To¨dtli and K. Litten for reading parts of the manuscript.
Finally, I also would like to thank my family, Frank, Katharina, Max and Ute for moral
support.
This work was supported in part by DFG Sonderforschungsbereich Transregio 9, Com-
putergestu¨tzte Theoretische Teilchenphysik, Studienstiftung des Deutschen Volkes, the
European Commission MRTN HEPTOOLS under Contract No. MRTN-CT-2006-035505,
and DESY. I thank both IT groups of DESY providing me access to special facilities to
perform the calculations involved in this thesis.
Diese Dissertation beinhaltet Beitra¨ge zu folgenden Vero¨ffentlichungen:
i) Publikationen: Journale
1. Mellin Moments of the O(α3s) Heavy Flavor Contributions to unpolarized
Deep-Inelastic Scattering at Q2 ≫ m2 and Anomalous Dimensions
I. Bierenbaum, J. Blu¨mlein and S. Klein
Nucl. Phys. B (in print) (2009) [arXiv:hep-ph/0904.3563].
2. Determining the closed forms of the O(a3s) anomalous dimensions and
Wilson coefficients from Mellin moments by means of computer algebra
J. Blu¨mlein, M. Kauers, S. Klein and C. Schneider
Comput. Phys. Commun. (in print) [arXiv:hep-ph/0902.4091].
3. The Gluonic Operator Matrix Elements at O(α2s) for DIS Heavy Flavor
Production
I. Bierenbaum, J. Blu¨mlein and S. Klein
Phys. Lett. B672, 401 (2009) [arXiv:hep-ph/0901.0669].
4. Two–Loop Massive Operator Matrix Elements for Unpolarized Heavy
Flavor Production to O(ǫ)
I. Bierenbaum, J. Blu¨mlein, S. Klein and C. Schneider
Nucl. Phys. B803, 1 (2008) [arXiv:hep-ph/0803.0273].
ii) Beitra¨ge zu Konferenzba¨nden
5. 2– and 3–loop heavy flavor contributions to F2(x,Q2), FL(x,Q2) and
g1,2(x,Q2)
I. Bierenbaum, J. Blu¨mlein and S. Klein
in: Proceedings of the workshop: HERA and the LHC workshop series on the implica-
tions of HERA for LHC physics, H. Jung and A. De Roeck, eds., (2006–2008, Hamburg,
Geneve), DESY-PROC-2009-02, March 2009, pp. 363 [arXiv:hep-ph/0903.3861].
6. From Moments to Functions in Quantum Chromodynamics
J. Blu¨mlein, M. Kauers, S. Klein and C. Schneider
PoS ACAT 2008, 106 (2008) [arXiv:hep-ph/0902.4095].
7. Heavy flavor operator matrix elements at O(a3s)
I. Bierenbaum, J. Blu¨mlein and S. Klein
PoS Confinement8, 185 (2008) [arXiv:hep-ph/0812.2427].
8. First O(α3s ) heavy flavor contributions to deeply inelastic scattering
I. Bierenbaum, J. Blu¨mlein and S. Klein
Nucl. Phys. Proc. Suppl. 183, 162 (2008) [arXiv:hep-ph/0806.4613].
9. Higher order corrections to heavy flavour production in deep inelastic
scattering
I. Bierenbaum, J. Blu¨mlein and S. Klein
Acta Phys. Polon. B39, 1531 (2008) [arXiv:hep-ph/0806.0451].
10. Heavy flavour production in deep–inelastic scattering - Two–loop massive
operator matrix elements and beyond
I. Bierenbaum, J. Blu¨mlein and S. Klein
Acta Phys. Polon. B38, 3543 (2007); PoS RADCOR 2007, 034 (2007) [arXiv:hep-
ph/0710.3348].
11. Difference Equations in Massive Higher Order Calculations
I. Bierenbaum, J. Blu¨mlein, S. Klein and C. Schneider
PoS ACAT 2007, 082 (2007) [arXiv:math-ph/0707.4659].
12. Two-loop massive operator matrix elements for polarized and unpolar-
ized deep-inelastic scattering
I. Bierenbaum, J. Blu¨mlein and S. Klein
in: Proc. of 15th International Workshop On Deep-Inelastic Scattering And Related
Subjects (DIS2007), G. Grindhammer, K. Sachs, eds., (16–20 April 2007, Munich),
Vol. 2, pp. 821 [arXiv:hep-ph/0706.2738].
13. Structural Relations between Harmonic Sums up to w=6
J. Blu¨mlein and S. Klein
PoS ACAT, 084 (2007) [arXiv/hep-ph:0706.2426].
14. Two–loop massive operator matrix elements for polarized and unpolar-
ized deep-inelastic scattering
I. Bierenbaum, J. Blu¨mlein and S. Klein
PoS ACAT, 070 (2007).
Selbststa¨ndigkeitserkla¨rung
Ich erkla¨re an Eides statt, dass außer den in der Dissertation angegebenen Quellen keine
weiteren Hilfsmittel verwendet wurden. Die Arbeit wurde bisher weder zu einer Promotion
noch zu anderen Pru¨fungen verwendet.
Sebastian Klein, Dortmund den 5. Juni 2009.