The automation of next-to-leading order electroweak calculations

TL;DR

This paper develops and validates the automation of next-to-leading order electroweak calculations within a mixed coupling expansion (QCD+EW) using MadGraph5_aMC@NLO. It extends the FKS subtraction framework to handle mixed couplings and to observables defined via fragmentation functions, and it implements the complex-mass scheme to consistently treat unstable particles at NLO EW accuracy. The authors present detailed methodological extensions, provide thorough cross-checks, and deliver illustrative 13 TeV LHC results for a wide set of processes, highlighting that subleading terms can defy naive coupling-counting expectations. They also release an MG5_aMC@NLO version capable of mixed-coupling computations and discuss current limitations and future improvements. Overall, the work significantly advances automated, high-precision collider predictions by integrating mixed QCD+EW effects, fragmentation, and a robust CM framework into a single framework.

Abstract

We present the key features relevant to the automated computation of all the leading- and next-to-leading order contributions to short-distance cross sections in a mixed-coupling expansion, with special emphasis on the first subleading NLO term in the QCD+EW scenario, commonly referred to as NLO EW corrections. We discuss, in particular, the FKS subtraction in the context of a mixed-coupling expansion; the extension of the FKS subtraction to processes that include final-state tagged particles, defined by means of fragmentation functions; and some properties of the complex mass scheme. We combine the present paper with the release of a new version of MadGraph5_aMC@NLO, capable of dealing with mixed-coupling expansions. We use the code to obtain illustrative inclusive and differential results for the 13-TeV LHC.

Figures (16)

• Figure 1: Fermion (left panel) and photon-$W$ (right panel) bubble graphs relevant to the $W$ self-energy one-loop correction, that contribute to $\Sigma_{{\rm U},T}^{f_1f_2}$ (see eq. (\ref{['SigffW']})) and to $\Sigma_{{\rm U},T}^{\gamma W}$ (see eq. (\ref{['SiggaWW']})), respectively.
• Figure 2: Examples of trajectories, for a function with a branch cut on the negative real axis. The arrow indicates the direction one moves in from $\bar{\gamma}=0$ to $\bar{\gamma}=\bar{\Gamma}$. See the text for details.
• Figure 3: Trajectories of eq. (\ref{['sampletrj']}) for the five configurations of table \ref{['tab:trj']}. In the left panel, trajectories relevant to configurations A and E are multiplied by the numerical factors reported in the labels in order for them to fit into the layout, and to improve visibility; note that $\Re(T(\bar{\gamma}=0))>0$ for configuration E.
• Figure 4: Results for the real (left panel) and imaginary (right panel) part of the UV-finite contribution to the r.h.s. eq. (\ref{['B0mm']}) (i.e. the non-divergent component of $B_0/(i\pi^2)$), as a function of $\bar{\Gamma}$. The other widths and the masses are chosen according to configuration E in table \ref{['tab:trj']}. See the text for details.
• Figure 5: Representative virtual (panels (a) and (b)) and real (panel (c)) contributions to the $q\bar{q}\to q^\prime\bar{q}^\prime g$ partonic cross section. The figure highlights the coupling-constant combinations relevant to each contribution. See the text for details.