${\mathcal{O}(α_s^2 α)}$ corrections to quark form factor

TL;DR

The study delivers a complete analytic evaluation of the non-singlet three-loop mixed QCD–EW corrections to quark form factors at ${\mathcal{O}}(\alpha_s^2\alpha)$, addressing the complications from massive vector bosons through advanced IBP reduction and differential-equation techniques. It establishes UV renormalization and universal IR subtraction within a background-field gauge, and constructs finite remainders expressed in terms of Harmonic Polylogarithms and Generalized Polylogarithms, including new master integrals for triple-vector-boson topologies. The work provides detailed computational methodology (25 integral families expanded to 61, localized IBP reduction) and delivers OS numerical results for the finite parts at representative kinematics, alongside ancillary files with full analytic expressions. These results are essential building blocks for precise predictions of Drell–Yan processes at hadron colliders, enabling improved constraints on EW–QCD interplay and setting the stage for future inclusion of singlet contributions. Overall, the paper advances high-precision SM predictions by pushing the frontier of mixed QCD–EW corrections at three loops.

Abstract

We present the analytic results for the non-singlet contributions to the three-loop mixed strong-electroweak ${\mathcal{O}}(α_s^2α)$ virtual corrections to the quark form factors. The primary challenge of this computation arises from the presence of massive vector bosons within the loops. This significantly increases the complexity of the integration-by-parts reduction of the scalar integrals and complicates their evaluation via the method of differential equations. To obtain the physical results, we perform the appropriate ultraviolet renormalization and subtract the universal infrared divergences. The resulting finite remainders are expressed in terms of Harmonic Polylogarithms and Generalized Polylogarithms.

Figures (4)

• Figure 1: The figure illustrates the transformation between $x$ and $y$ . The left and right panel displays the $x$-plane and the complex $y$-plane, respectively. Colored lines show the mapping of intervals. In the $y$-plane, straight and wiggly lines represent two distinct roots.
• Figure 2: The figure illustrates the individual contributions of each color factor to the coefficients $F_{Z,1}^{(2,1),\text{fin}}$ and $F_{Z,2}^{(2,1),\text{fin}}$ over a range of center-of-mass energies, $50$ GeV $\leq \sqrt{s} \leq 150$ GeV.
• Figure 3: The figure illustrates the individual contributions of each color factor to the coefficients $F_{W,1}^{(2,1),\text{fin}}$ and $F_{W,2}^{(2,1),\text{fin}}$ over a range of center-of-mass energies, $50$ GeV $\leq \sqrt{s} \leq 150$ GeV.
• Figure 4: The figure illustrates the $C_F n_F$ contributions to the coefficients $F_{W,3}^{(2,1),\text{fin}}$ over a range of center-of-mass energies, $50$ GeV $\leq \sqrt{s} \leq 150$ GeV.