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NLP threshold corrections to W+jet production

Sourav Pal, Satyajit Seth

TL;DR

The work addresses NLP threshold logarithms in $W+$jet production, introducing a unified helicity-amplitude framework that combines shifted spinors for next-to-soft gluon emissions with soft-quark operators for quark emissions. By computing NLP-LL contributions across all partonic channels and helicity configurations, the authors validate a universal NLP structure that connects to the known results for $H+$jet through mass substitutions and total-sum coherence. The findings demonstrate that the NLP corrections arising from soft radiation exhibit a process-independent form when summed over relevant helicities and polarizations, reinforcing the universality of NLP logarithms in color-singlet plus jet production. The results have practical impact for precision predictions and resummation in jet-associated colorless final states, enabling more reliable inclusion of NLP effects in both polarized and unpolarized cross sections.

Abstract

We perform a detailed computation of the helicity-dependent next-to-leading power leading logarithms in W+jet production, originating from next-to-soft gluon radiation and soft (anti-)quark emissions. These contributions are systematically captured via helicity-sensitive spinor shifts and soft quark operators. The resulting expressions exhibit full agreement with a recently proposed universal structure of NLP corrections for processes involving the production of an arbitrary massive colourless final state in association with a jet.

NLP threshold corrections to W+jet production

TL;DR

The work addresses NLP threshold logarithms in jet production, introducing a unified helicity-amplitude framework that combines shifted spinors for next-to-soft gluon emissions with soft-quark operators for quark emissions. By computing NLP-LL contributions across all partonic channels and helicity configurations, the authors validate a universal NLP structure that connects to the known results for jet through mass substitutions and total-sum coherence. The findings demonstrate that the NLP corrections arising from soft radiation exhibit a process-independent form when summed over relevant helicities and polarizations, reinforcing the universality of NLP logarithms in color-singlet plus jet production. The results have practical impact for precision predictions and resummation in jet-associated colorless final states, enabling more reliable inclusion of NLP effects in both polarized and unpolarized cross sections.

Abstract

We perform a detailed computation of the helicity-dependent next-to-leading power leading logarithms in W+jet production, originating from next-to-soft gluon radiation and soft (anti-)quark emissions. These contributions are systematically captured via helicity-sensitive spinor shifts and soft quark operators. The resulting expressions exhibit full agreement with a recently proposed universal structure of NLP corrections for processes involving the production of an arbitrary massive colourless final state in association with a jet.

Paper Structure

This paper contains 9 sections, 47 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Soft and next-to-soft emissions
  3. Amplitudes for $W$+1 jet production
  4. NLP threshold logarithms
  5. $q\bar{q}^{\prime}$ initiated process
  6. $gg$ initiated process
  7. $qg$ initiated process
  8. $\bar{q}^{\prime} g$ initiated process
  9. Summary

Figures (5)

  • Figure 1: Plot of $\bar{\mathcal{C}}_{LL}$ for $\bar{q}^{\prime} q$ initiated process with next-to-soft gluon radiation, evaluated at various phase space points. The vertical red line indicates $\bar{\mathcal{C}}_{LL}$ obtained by summing the contributions from eqs. \ref{['eq:qaqggWp']} and \ref{['eq:qaqggWm']}, while the blue line represents the corresponding result for $H$ plus one jet production.
  • Figure 2: Panel (a) shows the $\bar{\mathcal{C}}_{LL}$ result for radiation of a final state soft quark for $Q=q$, as given in eq. \ref{['eq:xsec4qq']}. Panel (a) displays the corresponding result for radiation of a final state soft anti-quark in the case $Q=q^\prime$, as given in eq. \ref{['eq:diff-flav']}. In both panels, the vertical red lines indicate the results for the $W\!+\!jet$ process, while the vertical blue lines correspond to the results for $H\!+\!jet$ production.
  • Figure 3: The $\bar{\mathcal{C}}_{LL}$ coefficients arising from final state soft quark and anti-quark radiation are shown in panels (a) and (b), corresponding to eqs. \ref{['eq:ggcllsq']} and \ref{['eq:ggcllsaq2']} respectively. In both plots, the vertical red lines represent the results for the $W\!+\!jet$ process, while the vertical blue lines denote the corresponding results for $H\!+\!jet$ production. The result associated with eq. \ref{['eq:ggcllsaq1']} is not displayed explicitly, as its behaviour can be readily inferred from the results presented in these two panels.
  • Figure 4: Panel (a) shows the $\bar{\mathcal{C}}_{LL}$ result for the radiation of an unpolarised next-to-soft gluon, obtained by summing both the expressions in eq.\ref{['eq:qgsg']}. Panel (b) presents the $\bar{\mathcal{C}}_{LL}$ for soft final state quark emission, including two helicity configurations, which yield identical expressions as given in eq. \ref{['eq:qgsaq']}. In both panels, the vertical red lines indicate the results for the $W\!+\!jet$ process. The vertical blue line in panel (a) corresponds to the full result for $H\!+\!jet$ production. In contrast, the blue line in panel (b) shows only the partial $H\!+\!jet$ contribution proportional to $C_F$, as additional contribution arises in this case due to the non-vanishing $ggHg\to0$ non-radiative process.
  • Figure 5: Panel (a) shows the $\bar{\mathcal{C}}_{LL}$ result for next-to-soft gluon radiation, with both polarisation states summed, while panel (b) presents the $\bar{\mathcal{C}}_{LL}$ for radiation of a soft final state anti-quark. In both panels, the vertical red lines indicate the results for the $W\!+\!jet$ process, corresponding to eq. \ref{['eq:aqgsg']} (sum of both contributions) in panel (a), and eq. \ref{['eq:aqgsq']} in panel (b). The vertical blue line in panel (a) denotes the result for $H\!+\!jet$ production. In panel (b), the blue line represents the partial $H\!+\!jet$ result proportional to $C_F$, as the $C_A$ contribution is absent in the $W\!+\!jet$ case due to the absence of non-radiative $ggWg\to0$ process at leading order.