Three-loop massive form factors: complete light-fermion and large-$N_c$ corrections for vector, axial-vector, scalar and pseudo-scalar currents

TL;DR

This work delivers the first complete three-loop QCD corrections to massive-quark form factors for vector, axial-vector, scalar, and pseudoscalar currents in the non-singlet sector, including all massless-fermion loops and the planar (large-$N_c$) non-fermionic part. Analytically, the results are expressed in Goncharov polylogarithms and complemented by precise expansions in static, high-energy, and threshold regimes, enabling accurate phenomenology for heavy-quark production and Higgs-related processes. Rigorous checks—cusp anomalous dimension universality, axial Ward identity, gauge-parameter independence, and master-integral verifications—validate the calculation. Numerical analyses demonstrate robust convergence of the limit expansions and provide a practical, computer-readable dataset for further phenomenological applications and future refinements including singlet and massive fermion-loop contributions.

Abstract

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-$N_c$ limit, where only planar Feynman diagrams contribute.

Figures (6)

• Figure 1: Sample diagrams contributing to the form factors. Solid and curly lines represent quarks and gluons, respectively. The grey blob refers to one of the external currents given in Eq. (\ref{['eq::currents']}). Singlet contributions, as shown in (a), are not considered in this paper.
• Figure 2: 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 the right the complex $x$ plane is shown. The (coloured) wiggled and zigzag lines show the mapping of the various intervals, whereas the straight lines indicate the mapping for special values of $q^2/m^2$ and $x$.
• Figure 3: Real part of the $\epsilon^0$ term of the vector and axial-vector form factors as a function of $x$. Exact results and approximations are shown as solid and dashed lines, respectively. At three-loop order we add the complete light-fermion part for $n_l=5$ and the $N_c^3$ contribution. Short- (blue), medium- (red) and long- (green) dashed lines correspond to the low-energy, high-energy and threshold approximation, respectively
• Figure 4: Same as Fig. \ref{['fig::x_re_va']} but for the the scalar and pseudo-scalar currents.
• Figure 5: $\epsilon^0$ term of the vector and axial-vector form factors as a function of $\phi$. Exact results and approximations are shown as solid and dashed lines, respectively. At three-loop order we add the complete light-fermion part for $n_l=5$ and the $N_c^3$ contribution. Note that for $\phi\in[0,\pi]$ the form factors are real. Short- (blue) and long- (green) dashed lines correspond to the low-energy and threshold approximation, respectively