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

Abstract

We present compact analytic results for the one-loop amplitude for the process $0\to q\bar{q}\ell \bar{\ell }{\ell }^{\prime }{\bar{\ell }}^{\prime }g$, 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.

Comments: 40 pages, 2 figures, attached python test scripts (one working from TeX source) also fortran code give access to the results of the paper in a machine-readable form