Vector boson pair production at one loop: analytic results for the process $q \bar{q} \ell \bar\ell \ell^\prime \bar{\ell}^\prime g$

TL;DR

This paper derives compact analytic one-loop amplitude results for vector boson pair production with a jet, incorporating quark mass effects, and introduces advanced mathematical techniques for simplifying complex calculations.

Contribution

It provides novel analytic formulas for one-loop amplitudes involving massive and massless quarks, utilizing advanced algebraic and numerical methods beyond previous massless five-point calculations.

Findings

Analytic expressions for $q ar{q} o Z Z g$ and $W W g$ processes.

Implementation of efficient, fast numerical code for these amplitudes.

Application of algebraic-geometry and number theory techniques to simplify complex integrals.

Abstract

We present compact analytic results for the one-loop amplitude for the process $0 \rightarrow q \bar{q} \ell \bar\ell \ell^\prime \bar{\ell}^\prime g$, relevant for both the production of a pair of $Z$ and $W$-bosons in association with a jet. We focus on the gauge-invariant contribution mediated by a loop of quarks. We explicitly include all effects of the loop-quark mass $m$, appropriate for the production of a pair of $Z$-bosons. In the limit $m \to 0$, our results are also applicable to the production of $W$-boson pairs, mediated by a loop of massless quarks. Implemented in a numerical code, the results are fast. The calculation uses novel advancements in spinor-helicity simplification techniques, for the first time applied beyond five-point massless kinematics. We make use of primary decompositions from algebraic-geometry, which now involve non-radical ideals, and $p$-adic numbers from number theory. We show how to infer whether numerator polynomials belong to symbolic powers of non-radical ideals through numerical evaluations.