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The Three Loop Two-Mass Contribution to the Gluon Vacuum Polarization

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Johannes Bluemlein at Deutsches Elektronen-Synchrotron
Johannes Bluemlein
A. De Freitas
A. De Freitas
A. De Freitas
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Carsten Schneider at Johannes Kepler University of Linz
Carsten Schneider
K. Schönwald
K. Schönwald
K. Schönwald
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Abstract

We calculate the two-mass contribution to the 3-loop vacuum polarization of the gluon in Quantum Chromodynamics at virtuality p2=0p^2 = 0 for general masses and also present the analogous result for the photon in Quantum Electrodynamics.
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arXiv:1710.04500v1 [hep-ph] 12 Oct 2017
DESY 17–157
DO–TH 17/26
October 2017
The Three Loop Two-Mass Contribution to the
Gluon Vacuum Polarization
J. Bl¨
umleina, A. De Freitasa, C. Schneiderb, K. Sch¨
onwalda
aDeutsches Elektronen–Synchrotron, DESY,
Platanenallee 6, D-15738 Zeuthen, Germany
bResearch Institute for Symbolic Computation (RISC),
Johannes Kepler University, Altenbergerstraße 69, A–4040, Linz, Austria
Abstract
We calculate the two-mass contribution to the 3-loop vacuum polarization of the gluon in
Quantum Chromodynamics at virtuality p2= 0 for general masses and also present the
analogous result for the photon in Quantum Electrodynamics.
The 3-loop heavy flavor corrections to deep-inelastic scattering at larger virtualities [1–7]
form an important ingredient for the determination of the strong coupling constant αs(M2
Z),
the parton distribution functions and the measurement of the mass of the charm quark mcat
high precision [8, 9]. Starting at 3-loop order the QCD corrections contain also 2-mass contri-
butions in single Feynman diagrams [10–12]. The heavy flavor contributions to deep-inelastic
structure functions in the region of larger virtualities Q2≫m2
Q, with mQthe heavy quark mass,
can be obtained in terms of massive on-shell operator matrix elements (OMEs) [1, 13]. These
quantities also receive massive self-energy insertions, such as the on-shell vacuum polarization
function ˆ
˜
Π(3)(0, m2
1, m2
2, µ2) and fermion self energy ˆ
˜
Σ(0, m2
1, m2
2, µ2). Here m1,2denote the cor-
responding heavy quark masses and µis the renormalization scale. These quantities are of more
general interest, as they appear in various massive higher loop calculations. The expression for
ˆ
˜
Σ(0, m2
1, m2
2, µ2) for general ratios η=m2
1/m2
2has been given in Ref. [10,14]. In the present note,
we calculate the polarization function ˆ
˜
Π(3)(0, m2
1, m2
2, µ2), which is obtained as the 3rd term in
the expansion
ˆ
Πµν
ab,H (p2, m2
1, m2
2, µ2, ε, ˆas) = i(−p2gµν +pµpν)δab
∞
X
k=1
ˆak
sˆ
ΠH(p2, m2
1, m2
2, µ2, ε) (1)
in the limit p2→0. Here the index Hlabels the heavy quark part of the polarization function,
ˆas=g2
s/(4π)2denotes the unrenormalized strong coupling constant, m1,2are the bare heavy
quark masses, and ε=D−4 is the dimensional parameter. In the calculation we refer to the
Feynman rules given in [15]. The corresponding (single mass) expressions ˆ
Π(1,2)
H(0, m2
1, m2
2, µ2)
were calculated in [1, 16–20] for QED and/or QCD.
There are six physical topologies contributing to ˆ
˜
Π(3)(0, m2
1, m2
2, µ2), Eq. (2). The corre-
sponding 3-loop Feynman diagrams have been generated using QGRAF [21] and the color-traces
were calculated using Color [22]. Standard Feynman parameter integration has been applied,
cf. [23], representing one of the integrals using the Mellin-Barnes contour integral [24–28], also
using the package [29]. The sums over the residues have been performed analytically using the
packages Sigma [30, 31], EvaluateMultiSums and SumProduction [32], applying procedures of
HarmonicSums [33–37] also for the limiting processes in case of infinite sums.
The equal mass contributions have been calculated in [1], Eq. (4.7), before in case of a general
Rξgauge using MATAD [38]. The 2-mass term is given by
ˆ
˜
Π(3)(0, m2
1, m2
2, µ2) = lim
p2→0
ˆ
ΠH(p2, m2
1, m2
2, µ2, ε)
=−CFT2
F(256
9ε2+64
3εlnm2
1
µ2+ lnm2
2
µ2+5
9−5η−5
η
+−5η
8−5
8η+51
4ln2(η) + 5
2η−5η
2ln(η) + 32ζ2
3
+32 lnm2
1
µ2lnm2
2
µ2+80
9lnm2
1
µ2+80
9lnm2
2
µ2+1246
81
+5η3/2
2+5
2η3/2+3√η
2+3
2√η"1
8ln 1 + √η
1−√ηln2(η)
−Li3(−√η) + Li3(√η)−1
2ln(η) (Li2(√η)−Li2(−√η))#)
2
−CAT2
F(64
9ε3(1 + 2ξ) + 16
3ε2"(1 + 2ξ)lnm2
1
µ2+ lnm2
2
µ2
−35
9#+4
ε"ln2m2
1
µ2+ ln2m2
2
µ2−35
9lnm2
1
µ2−35
9lnm2
2
µ2
+2
3ζ2+37
27 +ξ4
3ln2(η) + 4 lnm2
1
µ2lnm2
2
µ2+4
3ζ2+292
81 #
+2 (1 + ξ)ln3m2
1
µ2+ ln3m2
2
µ2−70
3lnm2
1
µ2lnm2
2
µ2
+2ξln2m2
1
µ2lnm2
2
µ2+ 2ξlnm2
1
µ2ln2m2
2
µ2−2
9(2 + ξ) ln3(η)
+2 (1 + 2ξ)ζ2+37
9+292
27 ξlnm2
1
µ2+ lnm2
2
µ2
+4
3(2 + ξ) ln(1 −η)−η+1
η2
3+5ξ
24−179
18 −43
36ξln2(η)
−1
3(16 + 5ξ)η+1
η−70
9ζ2−8
9ζ3(7 + 2ξ)−3769
243 +262
243ξ
+1
η−η8
3+5
6ξln(η) + 8
3(2 + ξ)Li2(η) ln(η)−Li3(η)
+8 + 5
2ξ1 + η3
3η3/2+10 + 9
2ξ1 + η
√η"1
8ln 1 + √η
1−√ηln2(η)
−Li3(−√η) + Li3(√η)−1
2ln(η) (Li2(√η)−Li2(−√η))#).(2)
Here CA=Nc, CF= (N2
c−1)/(2Nc), TF= 1/2 are the color factors and Nc= 3 in case of QCD.
Using Q2E/Exp [39,40] the first terms of the η-expansion of (2) have been obtained in Ref. [10]
before. The corresponding expansion of (2) agrees with this expression. Eq. (2) is symmetric un-
der the interchange m1⇔m2and depends on the gauge parameter ξat most linearly. Although
not explicitly visible in the representation given above, one can show that ˆ
˜
Π(3)(0, m2
1, m2
2, µ2)
depends only on ηand not on √η.
The corresponding expression in case of Quantum Electrodynamics is found by setting CF=
1, CA= 0 and TF= 1. From [41] the 3-loop QED result can be inferred in principle.
Acknowledgment. We would like to thank P. Marquard for discussions and M. Steinhauser
for providing the codes MATAD 3.0 and Q2E/Exp. Discussions with A. Behring are gratefully
acknowledged. This work was supported in part by the European Commission through PITN-
GA-2012-316704 (HIGGSTOOLS).
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