Three-loop massive form factors: complete light-fermion corrections for the vector current

TL;DR

This work delivers the first complete three-loop QCD corrections to the massive quark form factors F1 and F2 arising from light-fermion loops, including non-planar diagrams. By combining FIRE-based IBP reduction with differential equations, the authors obtain analytic results in Goncharov polylogarithms and provide new master integrals beyond the large-Nc limit. They analyze the static, high-energy, and threshold limits and connect the form factors to cross-section corrections in e+e− annihilation, while validating against known two- and three-loop results and cusp anomalous dimension calculations. The results furnish essential inputs for precision predictions in heavy-quark production and decay, and establish a framework for incorporating remaining non-fermionic contributions and potential elliptic structures in future work.

Abstract

We compute the three-loop QCD corrections to the massive quark-anti-quark-photon form factors $F_1$ and $F_2$ 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 terms of Goncharov polylogarithms. We provide analytic results for all master integrals which are not present in the large-$N_c$ calculation considered in Refs. [1,2].

Figures (7)

• Figure 1: Illustration of the variable transformation between $q^2/m^2$ and $x$ as given in Eq. (\ref{['eq::trans_x_q']}). The left graph represents the $q^2/m^2$ plane and on right the complex $x$ plane is shown. The straight lines indicate the mapping for special values of $q^2/m^2$ and $x$.
• Figure 2: Sample diagrams contributing to $F_1$ and $F_2$ at one and two loops. Solid, curly and wavy lines represent quarks, gluons and photons, respectively.
• Figure 3: The two one-loop master integrals are shown in (a) and (b). One of the 17 master two-loop integrals is shown in (c) and the remaining 16 master integrals are obtained from (d) as described in the text. Solid and dashed internal lines correspond to massive and massless scalar propagators. Thin external lines are on the mass shell and thick external lines carry the (off-shell) momentum $q$.
• Figure 4: Sample diagrams contributing to $F_1$ and $F_2$ at three-loop order. Solid, curly and wavy lines represent quarks, gluons and photons, respectively. In our calculation we only consider contributions with at least one closed massless quark loop.
• Figure 5: New three-loop integral families needed for the fermionic contributions to the three-loop vertex corrections. Solid and dashed lines represent massive and massless lines, respectively. Thin external lines are on the mass shell and thick external lines carry the off-shell momentum $q$. For convenience we keep our internal numeration of the integral families, which is shown below the Feynman diagrams.