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JHEP03(2018)136

Published for SISSA by Springer

Received: January 28, 2018

Accepted: March 19, 2018

Published: March 22, 2018

Three-loop massive form factors: complete

light-fermion corrections for the vector current

Roman N. Lee,

Alexander V. Smirnov,

Vladimir A. Smirnov

c,d

and Matthias Steinhauser

Budker Institute of Nuclear Physics,

630090 Novosibirsk, Russia

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-Gaed e Straße 1 , D-76128 Karlsruhe, Germany

E-mail:

roman.n.lee@gmail.com, asmirnov80@gmail.com,

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

Abstract: We compute the three-loop QCD corrections to the m assi ve quark-anti-quark-

photon form factors F

and F

involving a closed loop of massless fermions. This subset

is gauge invariant and contains both planar and non-planar contributions. We perform

the reduction using FIRE and compute the master integrals with the help of diﬀerential

equations. Our analytic results can be expressed in te rm s of Goncharov polylogarithms.

We provide analytic results for all master integrals which are not present in the l ar ge- N

calculation considered in refs. [

1, 2].

Keywords: NLO Computations, QCD Phenomenology

ArXiv ePrint: 1801.08151

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP03(2018)136

JHEP03(2018)136

Contents

1 Introduction

2 Notation, renormalization and infrared structure 2

3 One- and two-loop form factors 3

4 Three-loop form factor 5

5 Analytical and numerical results 8

5.1 Form factor s in the static limit 8

5.2 Form factor s at high energies 10

5.3 Form factor s and threshold cross section 13

5.4 Numerical results 15

6 Conclusions and outlook 17

1 Introduction

In the absence of striking experimental signals which hint to phy s i cs beyond the Standard

Model it is of utmost importance to increase the precision of the theoretical predictions.

A subsequent detailed comparison to pr e c i se measurements will help to uncover deviations

and will provide hints for the construction of beyond-the-Standard-Model theories.

Quark and gluon form factors play a special role in the context of precision calculations.

On the one hand t he y are suﬃciently simple which allows to comput e them to high or de r

in perturbation theory. On the other hand they enter as building bl ocks into a variety of

physical cros s secti ons and decay rates, most prominently into Higgs boson production and

decay, the Drell Yan production of leptons and the production of massive quark s . Form

factors also c onst i tu te an ideal playground to study the infrared p r operties of a quantum

ﬁeld theory, in particular of QCD. As far as massle ss form factors are concerned the state-

of-the-art is four loops where diﬀerent groups have contributed to partial resu l ts [

3–8].

Massive quark form factors are known to two loops [

9] including O(ǫ) [2, 10] and O(ǫ

)

terms [

11, 12]. Three-loop corr ec t i ons in the l arge -N

limit for the vector current form

factor have been computed in ref. [

2]. In this paper we extend these considerations and

compute the complete contributions (i.e. all colour factors) from the diagrams involving

a closed massless quark loop. This well-deﬁned and gauge invariant subset contains for

the ﬁrst time non-planar contributions which we study in detail. Furthermore, new planar

master integrals have to be evaluated which are not present in the large-N

result. As a

by-product of our calculation we obtain the two-loop form factor including order ǫ

terms.

We do not consider singlet diagrams where the external photon couples to a clos ed massless

– 1 –

JHEP03(2018)136

quark loop which is connected via gluons to the ﬁnal-state massive quarks. Such diagrams

form agai n a separate gauge i nvariant subset which requires the computation of diﬀerent

integral families. Let us mention that all-order corrections to the massive form factor in

the large-β

limit have been considered in ref. [

13].

The remainde r of the paper is structured as follows: in the next section we introduce

the notation and discuss the ultraviolet and infrared divergences. One- and two-loop

results are presented in section

3. The three-loop calculation is descri bed in section 4,

in particular the calculation of the master integrals. Section

5 contains a discussion of

the three -l oop form factor. We provide both numerical results and analytic expressions

in various kinematic al limits. In section

6 we sum mar i ze our resul t s and comment on the

perspective for the full result.

2 Notation, renormalization and infrared structure

Let us deﬁne the form factors we are going to c onsi d er . Starting point is the photon-quark-

anti-quark vertex which we introduce as

µ,ij

, q

) = δ

¯u(q

)Γ

, q

)v(q

) , (2.1)

where i and j are (fundamental) colour indices and ¯u(q

) and v(q

) are the spinors of the

quark and anti-quark, respecti vely, with incoming momentum q

and out goi ng momentum

. The external quarks are on-shell, i.e., we have q

= q

= m

. The form factors are

deﬁned as prefactors of the Lorentz decomposition of the vertex function Γ

, q

) which

is introduced as

, q

) = Q



)γ

−

)σ

µν



, (2.2)

with q = q

− q

being the outgoing momentum of the photon and σ

µν

= i[γ

, γ

]/2.

is the charge of the c onsi de r e d quark. For on-sh el l renormalized form factors we have

(0) = 1 and F

(0) = (g − 2)/2 where g is the gyromagnetic ratio of the quark (or lepton

in the case of QED). For later convenience we deﬁne the perturbative expan si on of F

and

n≥0

(n)



(µ)

4π



, (2.3)

with F

(0)

= 1 and F

(0)

= 0.

To obtain the ren orm al i ze d form factors we use the

MS scheme for the strong coupling

constant and the on-shel l scheme for the heavy quark mass and wave function of the

external quarks. In all cases the counterterm contributions are simply obtained by re-

scaling the bare parameters with the corresponding renormalization constants, Z

, Z

and Z

. The latter is needed to three loops whereas two-loop corrections for Z

and Z

are suﬃcient to obtain re nor mal i z e d three-loop r es ul t s for F

and F

. Note that higher

order ǫ coeﬃcients are needed for the on-shell r e nor mal i z ati on constants since the one- and

two-loop form factors develop 1/ǫ and 1/ǫ

poles, respect i vely.

– 2 –

JHEP03(2018)136

After renormaliz ati on of the ultr aviolet divergences the for m factors still contain in-

frared poles which are connec te d to the cusp anomalous dimens i on, Γ

cusp

[

14–16]. We

adapt the notation from ref. [

2] and write

F = ZF

, (2.4)

where the factor Z, which is deﬁned in the

MS scheme and thus onl y contains poles in ǫ,

absorbs the infrared divergences and F

is ﬁnite. The coeﬃcients of the poles of Z are

determined by the QCD beta function and Γ

cusp

. In fact, the 1/ǫ

pole of the α

term of

Z is proportional to the n- l oop correction to Γ

cusp

(see, e.g., re f. [

2].

)

A dedicated calculation of Γ

cusp

to three loops has been performed in refs. [

14, 16–18].

An independent cross check of the large-N

result has been provided in ref. [

2]. In this

work we reproduce all n

terms at th re e -l oop order by extracting Γ

cusp

from the pole part

of the form factors.

For the practical computation of the master integrals, for the di sc uss i on of the various

kinematic limits and also for the numerical evaluation it is convenient to introduce the

dimensionless variable

= −

(1 − x)

, (2.5)

which maps the complex q

plane into the unit circle. The low-energy (q

→ 0), h i gh-

energy (q

→ ∞) and threshold (q

→ 4m

) limits correspond to x → 1, x → 0 and

x → −1, respectively. Furthermore, as can be se en i n ﬁgu re

1, the interval q

< 0 is

mapped to x ∈ (0, 1) and q

∈ [0, 4m

] t o the upper se mi - ci r c l e . For these values of x the

form factors have to be re al -valued since the corresponding Feynman diagrams do not have

cuts. Thi s is diﬀerent for the region q

> 4m

, which corresponds to x ∈ (−1, 0), where

the form factors are complex-valued.

For the threshold limit it is also convenient to introduce the velocity of the produce d

quarks

β =

1 −

, (2.6)

which is related to x via

x =

2β

1 + β

− 1 . (2.7)

3 One- and two-loop form factors

Let us in the following brieﬂy outline the main steps of the two-loop calculation. Sample

Feynman diagrams contributing to F

and F

can be found in ﬁgure

2. After generating

the amplitudes we ﬁnd it convenient to deﬁne one integral family at one and four integral

Note that t h ere is a typo in the second equation of eq. (12) in ref. [

2]: a factor “2” is miss in g in front

of Γ

(1)

cusp

inside the round brackets. The corrected equation reads z

2,2

= Γ

(1)

cusp

(β

+ 2Γ

(1)

cusp

)/16.

– 3 –

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