Two-Loop Formfactors in Theories with Mass Gap and Z-Boson Production

TL;DR

The paper develops a method to compute two-loop form factors with mass gaps for theories relevant to Z-boson production, combining infrared factorization, large-mass expansions, and a basis of generalized polylogarithms. It first solves the Abelian $U(1)\times U(1)$ case to obtain a finite non-factorizable remainder $\phi(z)$, then extends to the full electroweak theory, identifying Abelian and non-Abelian two-loop pieces $\phi_A$ and $\phi_{NA}$, the latter involving novel generalized polylogarithms. The results yield analytic expressions and numerical implementations for off-shell $q^2$ and Sudakov regimes, providing essential ingredients for mixed $O(\alpha_{\rm weak}\alpha_s)$ corrections to $Z$ production and decay. The work advances precision predictions for collider phenomenology by clarifying the infrared structure and high-energy behaviour of the relevant two-loop amplitudes.

Abstract

The two-loop formfactor both for a U(1)xU(1) and a SU(2)xU(1) gauge theory with massive and massless gauge bosons respectively is evaluated at arbitrary momentum transfer q^2. The asymptotic behaviour for q^2->\infinity is compared to a recent calculation of Sudakov logarithms. The result is an important ingredient for the calculation of radiative corrections to Z-boson production at hadron and lepton colliders.

Figures (2)

• Figure 1: Diagrams, contributing to the vertex $Zq\bar{q}$ (a) and (b). The two-loop diagrams are obtained by attaching one virtual gluon in all possible ways. The case (b) represents nonabelian part. That gives contribution $\phi_{\rm NA}(z)$ in the text. Diagram (a) with $W$ exchange also contributes to $\phi_{\rm NA}(z)$.
• Figure 2: Non-factorizable two-loop correction to the abelian formfactor in the euclidean regime ($z=q^2/m^2$). The solid line represents the exact result, the dashed line the Sudakov approximation and the dash-dotted line includes the power suppressed terms.