Towards pp -> VVjj at NLO QCD: Bosonic contributions to triple vector boson production plus jet

TL;DR

The paper addresses the challenge of delivering precise NLO QCD predictions for $pp \to VVjj+X$ by computing bosonic one-loop corrections to the related $pp \to VVVV+X$ and $pp \to VVVj+X$ topologies. It introduces a Mathematica/FeynCalc framework that automates tensor reduction up to hexagon rank-5, and establishes infrared factorization alongside Ward-identity gauge checks to ensure numerical stability. The work provides master equations for tensor coefficients, benchmarks hexagon contributions, and demonstrates competitive CPU performance with stability strategies including LU decomposition and targeted quadruple-precision rescues. These results enable reliable multi-vector-boson plus jet predictions at NLO QCD and pave the way for applying the approach to other $2 \to 4$ processes relevant to LHC phenomenology.

Abstract

In this work, some of the NLO QCD corrections for pp -> VVjj + X are presented. A program in Mathematica based on the structure of FeynCalc which automatically simplifies a set of amplitudes up to the hexagon level of rank 5 has been created for this purpose. We focus on two different topologies. The first involves all the virtual contributions needed for quadruple electroweak vector boson production, i.e. pp -> VVVV + X. In the second, the remaining "bosonic" corrections to electroweak triple vector boson production with an additional jet (pp -> VVV j + X) are computed. We show the factorization formula of the infrared divergences of the bosonic contributions for VVVV and VVVj production with V=(W,Z,gamma). Stability issues associated with the evaluation of the hexagons up to rank 5 are studied. The CPU time of the FORTRAN subroutines rounds the 2 milliseconds and seems to be competitive with other more sophisticated methods. Additionally, in Appendix A the master equations to obtain the tensor coefficients up to the hexagon level in the external momenta convention are presented including the ones needed for small Gram determinants.

Figures (16)

• Figure 1: "One loop fermion" contributions (left) and "bosonic" contributions (right).
• Figure 2: Different "bosonic" topologies. $V_i$ stands for vector bosons emitted from the quark line.
• Figure 3: "bosonic" one loop QED-like topologies appearing in the calculation of the virtual contributions for $pp \to VVjj +X$ production, with V$\in ($W $^\pm$,Z,$\gamma)$.
• Figure 4: Virtual corrections for a fermion line with two vector bosons attached, $V_1(k_1)$ and $V_2(k_2)$ in a given permutation. The sum of these graphs defines ${\cal M }_{V_1V_2,\tau}$ in Eq. (\ref{['boxline']}).
• Figure 5: Virtual corrections for a fermion line with three vector bosons attached in a given permutation. The sum of these graphs defines ${\cal M }_{V_1V_2V_3,\tau}$ in Eq. (\ref{['penline']}).