Three-loop massive form factors: complete light-fermion and large-
corrections for vector, axial-vector, scalar and pseudo-scalar currents
We compute the three-loop QCD corrections to the massive quark form factors with external vector, axial-vector, scalar and pseudo-scalar currents. All corrections with closed loops of massless fermions are included. The non-fermionic part is computed in the large- limit, where only planar Feynman diagrams contribute.
Vertex form factors with external fermions play a crucial role in a number of phenomenologically interesting processes. Among them are the massive fermion production in electron positron collisions, and in particular the forward-backward asymmetry, where form factor contributions induced by vector and axial-vector currents are needed. Furthermore, building blocks to the decay rates of scalar and pseudo-scalar Higgs bosons are provided by the corresponding form factors. Last but not least, form factors constitute important toys, which help to investigate the structure of high-order quantum corrections.
In this work we consider vertex form factors where the external current is of vector, axial-vector, scalar or pseudo-scalar type. They are given by
where for convenience the heavy quark mass has been introduced in the scalar and pseudo-scalar currents such that no additional overall (ultraviolet) renormalization constants have to be introduced (as for the vector and axial-vector111In this paper we do not consider Feynman diagrams which contribute to the axial anomaly. cases) . If or are used to compute properties of the Higgs boson there is a one-to-one relation of to the corresponding Yukawa coupling.
We consider the three-point functions of the currents in Eq. (1) and a quark-anti-quark pair. The corresponding vertex functions can be decomposed into scalar form factors which are defined as
with incoming momentum , outgoing momentum and being the outgoing momentum at . The external quarks are on-shell, i.e., and we have . We note that in all cases the colour structure is a simple Kronecker delta in the fundamental colour indices of the external quarks and not written out explicitly.
For later convenience we define the perturbative expansion of the scalar form factors as
with and .
The two-loop corrections to the vector current contributions and have been computed for the first time in Ref.  (see also Ref.  for the fermionic contributions) and have been cross checked by several groups [4, 5, 6, 7, 8]. In some cases higher order terms in have been added. Two-loop axial-vector, scalar and pseudo-scalar contributions have been computed in Refs. [9, 10, 11] and recently been confirmed in Ref.  where and terms have been added. Three-loop corrections are only known for well-defined subsets of the vector form factor: The large- limit has been computed in Ref.  using the master integrals of . This involves only planar integrals. The complete (planar and non-planar) light-fermion contributions to and have been obtained in Ref. . In this reference also the results of the relevant master integrals are given. Let us mention that all-order corrections to the massive vector form factor in the large- limit have been considered in Ref. .
For the three-point functions one in general distinguishes singlet and non-singlet contributions. The former includes a closed fermion loop which contains the coupling to the external current. It is connected to the fermions in the final state via gluons as is shown in Fig. 1(a). In case the external current contains singlet contributions need special attention since the anti-commuting definition for can not be used. Instead prescriptions like the one introduced in Ref.  have to be applied.
If the external current does not contain the singlet contributions can be treated along the same lines as the non-singlet part. However, in contrast to the latter the singlet contributions have massless cuts which requires modifications of the technique described in [12, 5, 8] to compute the master integrals. Thus, in this paper we restrict ourselves to non-singlet contributions (cf. Figs. 1(b)–(l)), i.e., the external current couples to the fermions in the final state. At three loops we compute the complete light-fermion contributions and consider the large- expansion of the remaining part. At one- and two-loop order all colour factors are computed and agreement with the literature  is found.
The techniques and the setup of the programs, which are used to obtain the results of this paper, are straightforward extensions of the works [5, 8] and thus we refrain from repeating in detail the technical descriptions. Note, however, that in contrast to Ref.  we do not define a “super family”, which includes the eight relevant planar families as sub-cases. Rather, we generated the input files for FIRE  from scratch and computed separate tables for each individual family. Let us mention that for the reduction to master integrals and the minimization of the latter it is useful to combine FIRE  with LiteRed [16, 17], which provides important symmetry information. In fact, for the most complicated integral family the reduction took about a day of CPU time on a computer with 18 cores, even for general gauge parameter.
For the form factors it is useful to introduce the following variable
which maps the complex plane into the unit circle, as illustrated in Fig. 2. The low-energy (), high-energy () and threshold () limits correspond to , and , respectively. Furthermore, the interval is mapped to and to the upper semi-circle. Note that for and with the form factors have to be real-valued since the corresponding Feynman diagrams do not have cuts. This is different for the region , which corresponds to , where the form factors are complex-valued. Note that for negative we interpret as .
For the threshold limit (, ) it is convenient to introduce the velocity of the produced quarks
which is related to via
For the analytic three-loop expressions we furthermore define
In the practical calculation it is convenient to apply projectors in order to extract the scalar form factors. We refrain to provide them explicitly but refer to Ref.  where projectors for the four currents in Eq. (1) can be found.
All one- and two-loop Feynman integrals can be expressed as a linear combination of the master integrals discussed in Ref. . Note that our two-loop basis is smaller than the one of Ref.  where 23 non-singlet master integrals are given. After inserting the -expanded results for the master integrals into the expressions for the form factors we obtain the one- and two-loop expressions expanded up to order and , respectively. Our two-loop results agree with . Let us repeat that we do not consider singlet contributions which occur for the first time at two loops. Note that they vanish for the vector current but give non-vanishing contributions for the other three currents.
At three-loop order we have 89 planar master integrals entering the large- expressions and 15 additional master integrals for the complete light-fermion and contributions, only two of the them are non-planar.
To obtain the renormalized form factors we use the scheme for the strong coupling constant and the on-shell 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, , and . The latter is needed to three loops whereas two-loop corrections for and are required. For the scalar and pseudo-scalar form factors also the overall overall factor has to be renormalized (to three-loop order), which we choose to do in the scheme. Note that this is the natural choice if or are used for Higgs boson production or decay since then takes over the role of the Yukawa coupling. The renormalization constants, of course, only contain pole parts. However, for the on-shell quantities also higher order coefficients are needed since the one- and two-loop form factors develop and poles, respectively. Note that in our case the overall renormalization constants of all currents in Eq. (1) are equal to unity.
3 Analytic results
The analytic results for the form factors are expressed in terms of Goncharov polylogarithms (GPLs)  with letters and . They are quite long and we refrain from presenting them in the paper. Rather we collect all relevant expressions in a computer-readable format; the corresponding file can be downloaded from . To fix the notation we provide one-loop results for the six scalar form factors introduced in Eq. (2) up to the constant term in . For they are given by
where the two GPLs can be written as
In Eq. (8) we have the colour factor . At two-loop order one has , , and where , counts the massless quark loops and the quark loops with mass . At three-loop order there are the colour factors , , , and . The latter is obtained by the replacements and in the non- terms and taking only the leading contribution for large . This limit removes in particular all (pure) terms.
In the following three subsections we discuss the analytic structure of the form factors in three important kinematical regions where the external momentum is either small, large or close to the threshold for producing the heavy quarks on-shell. In these limits the expressions are compact and analytic results can be reproduced in this paper. We obtain the expansions of the full result by expanding the GPLs in the respective region. We restrict ourselves to the choice and refer to  for the general results. Subsection 3.4 contains a brief discussion on the infrared structure of the form factors and mentions several checks on our calculation.
3.1 Static limit
After expanding the GPLs in the low-energy limit we obtain the expansion of the form factor up to order . In the following we present the results for the first two terms for the axial-vector, scalar and pseudo-scalar cases. The results for the vector currents can be found in Refs. [12, 8] (see Sections 4.2.1 and 5.2, respectively). Since the interval is mapped to the upper semi-unit circle in the complex plane we use and parametrize our results as a function of which is real-valued. Our results read