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JHEP01(2017)074

Published for SISSA by Springer

Received: November 30, 2016

Accepted: January 10, 2017

Published: January 17, 2017

Massive three-loop form factor in the planar limit

Johannes Henn,

Alexander V. Smirnov,

Vladimir A. Smirnov

c,d

and

Matthias Steinhauser

PRISMA Cluster of Excellence, Johannes Gutenberg University,

Staudingerweg 9, 55099 Mainz, Germany

Research Computing Center, Moscow State University,

119991 Moscow, Russia

Skobeltsyn Institute of Nuclear Physics of Moscow State University,

119991 Moscow, Russia

Institut f¨ur Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT),

Wolfgang-Gaede Straße 1, 76128 Karlsruhe, Germany

E-mail:

henn@uni-mainz.de, asmirnov80@gmail.com,

smirnov@theory.sinp.msu.ru, matthias.steinhauser@kit.edu

Abstract: We compute the three-loop QCD corrections to the massive quark-anti-quark-

photon form factors F

and F

in the large-N

limit. The analytic results are expressed in

terms of Goncharov polylogarithms. This allows for a straightforward numerical evaluat i on.

We also derive se r i e s expansions, including power suppressed terms, for three k i ne mat i c

regions corresponding to small and large invariant masses of the photon momentum, and

small velocities of the heavy quarks.

Keywords: NLO Computations, QCD Phenomenology

ArXiv ePrint:

1611.07535

Open Access,

 The Authors.

Article funded by SCOAP

doi:

10.1007/JHEP01(2017)074

JHEP01(2017)074

Figure 1. Sample diagrams contributing to F

and F

at one-, two- and thr e e-l oop order. Solid,

curly and wavy lines represent quarks, gluons and photons, respectively. In our calculation the

closed fermion loops only involve massless quarks.

Sample Feynman diagrams can be found in ﬁgure 1. Two-loop QCD corr ec t i ons to the

electric and magnetic form factors for the heavy quark vector current have been computed

for the ﬁrst time in ref. [

1] (axial vector and anomaly contributions have been consi d er e d

in [

2, 3]) where analytic results have been obtained. An independent cross check of t he two-

loop r e sul t s for F

and F

has been performe d in [

4] where also O(ǫ

) and O(ǫ) terms have

been added to the one- and two-loop results, respectively. The results have be e n used to

obtain predictions for the three-loop form factor F

in the high energy limit, by exploiting

evolution equations and the exponentiation of infrared divergences (see also ref. [

5] for

earlier considerations).

In this paper we compute the three-loop form factor in the planar limit, keeping the ex-

act mass dependence. After expanding our exact result for small quark masses we can com-

pare to the high-energy results of [

4] mentioned above, and complete them by determining

the unknown constants in the 1/ǫ and ǫ

part. We furthermore provide power-suppressed

terms.

Massive form factor s have infrared divergences that are well understood. After the

ultraviolet renormalization, all poles in dimensional regularization are given in terms of the

cusp anomalous dimension [

6, 7], and the beta function. The three-loop cusp anomalous

dimension was computed in re fs. [

8, 9]. By verifying the infrared pole structure at the

three-loop order, we provide a ﬁrst independent check of the result of refs. [

8, 9] (in the

planar limit).

In the static limit, the infrared divergences dis appear, and F

and F

are ﬁnite. In

fact, F

vanishes and F

determines the anomalous magnetic moment of a heavy quark

which has been considered at two-loop order in ref. [

10]. A dedicated calculation at three

loops has been performed in ref. [

11] which serves as a welcome check for our ex ac t result

expanded for q

→ 0.

– 2 –

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