The $n_f^2$ contributions to fermionic four-loop form factors

TL;DR

This work resolves the complete $n_f^2$ four-loop contributions to massless fermionic form factors $F_q$ (photon-quark) and $F_b$ (Higgs-quark) by focusing on two non-planar integral families. It combines diagram generation, IBP reduction with FIRE/LiteRed, and a canonical $\\epsilon$-form differential-equation approach to obtain analytic results for all non-planar master integrals up to weight 8, including a two-scale setup with $q_2^2=x q^2$. The authors extract the cusp and collinear anomalous dimensions up to $\gamma_{ m cusp}^3$ and $\gamma_q^3$ for the $n_f^2$ sector, verify the universal IR structure of $\log(F_x)$, and provide the finite parts of the four-loop form factors (with data and master integrals available in progdata). The Higgs-fermion case serves as a cross-check with fewer integrals and confirms parity with the photon case, while extending the three-loop Higgs form factor to $\epsilon^2$, thus furnishing essential ingredients for future higher-order, $n_f$-dependent analyses.

Abstract

We compute the four-loop contributions to the photon quark and Higgs quark form factors involving two closed fermion loops. We present analytical results for all non-planar master integrals of the two non-planar integral families which enter our calculation.

Figures (2)

• Figure 1: Sample Feynman diagrams contributing to $F_q$ and $F_b$ at the four-loop order and containing two closed fermion loops. The gray box indicates either a scalar or a vector coupling of the fermions to the external current. Solid and curly lines represent quarks and gluons, respectively. All particles are massless.
• Figure 2: Graphs associated with the non-planar families 7 and 786 needed for the $n_f^2$ contribution to $F_q$ and $F_b$. The numbers $n$ next to the lines correspond to the indices of the propagators, i.e. to the $n^{\rm th}$ integer argument of the functions representing the integrals. In addition to the 12 propagators we have for each family six linear independent numerator factors. However, the corresponding indices are always zero for our master integrals.