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Next-to-leading order numerical calculations in Coulomb gauge

Michael Kramer, Davison E. Soper

TL;DR

The paper develops a fully numerical framework for next-to-leading order QCD calculations in Coulomb gauge, aiming to enable approximate all-order effects by avoiding unphysical gluon polarizations. It introduces a Coulomb-gauge formalism with careful treatment of two- and three-point subgraphs, including 3D-renormalization substitutions and contour-deformation techniques to render loop integrals numerically tractable. The approach is implemented in a Monte Carlo event generator and validated by computing thrust-related observables, showing gauge-invariant results when all diagram topologies are summed. The work demonstrates the feasibility of accurate NLO predictions in a physical gauge and outlines a path toward incorporating higher-order and showering effects.

Abstract

Calculations of observables in quantum chromodynamics can be performed using a method in which all of the integrations, including integrations over virtual loop momenta, are performed numerically. We use the flexibility inherent in this method in order to perform next-to-leading order calculations for event shape variables in electron-positron annihilation in Coulomb gauge. The use of Coulomb gauge provides the potential to go beyond a purely order alpha_s^2 calculation by including, for instance, renormalon or parton showering effects. We expect that the approximations needed to include such effects at all orders in alpha_s will be simplest in a gauge in which unphysically polarized gluons do not propagate over long distances.

Next-to-leading order numerical calculations in Coulomb gauge

TL;DR

The paper develops a fully numerical framework for next-to-leading order QCD calculations in Coulomb gauge, aiming to enable approximate all-order effects by avoiding unphysical gluon polarizations. It introduces a Coulomb-gauge formalism with careful treatment of two- and three-point subgraphs, including 3D-renormalization substitutions and contour-deformation techniques to render loop integrals numerically tractable. The approach is implemented in a Monte Carlo event generator and validated by computing thrust-related observables, showing gauge-invariant results when all diagram topologies are summed. The work demonstrates the feasibility of accurate NLO predictions in a physical gauge and outlines a path toward incorporating higher-order and showering effects.

Abstract

Calculations of observables in quantum chromodynamics can be performed using a method in which all of the integrations, including integrations over virtual loop momenta, are performed numerically. We use the flexibility inherent in this method in order to perform next-to-leading order calculations for event shape variables in electron-positron annihilation in Coulomb gauge. The use of Coulomb gauge provides the potential to go beyond a purely order alpha_s^2 calculation by including, for instance, renormalon or parton showering effects. We expect that the approximations needed to include such effects at all orders in alpha_s will be simplest in a gauge in which unphysically polarized gluons do not propagate over long distances.

Paper Structure

This paper contains 23 sections, 160 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Calculations by numerical integration
  3. Elliptical coordinates
  4. Structure of graphs with a cut gluon propagator
  5. Real gluon self-energy graph
  6. Virtual gluon self-energy graph
  7. The energy integral
  8. Components needed
  9. Transverse components
  10. Timelike components
  11. The tensor $F_g^{\mu\nu}$
  12. ${W}_g^{\mu\nu}$ and its supplementary terms
  13. Structure of graphs with a cut quark propagator
  14. Real quark self-energy graph
  15. Virtual quark self-energy graph
  16. ...and 8 more sections

Figures (7)

  • Figure 1: Two cuts of one of the Feynman diagrams that contribute to $e^+e^- \to {\it hadrons}$.
  • Figure 2: Cut gluon propagator at the Born level.
  • Figure 3: Cut gluon self-energy diagram.
  • Figure 4: Off-shell virtual gluon self-energy diagram. (This case does not occur in order $\alpha_s^2$ graphs for $e^+ e^- \to hadrons$, but we consider it as an intermediate step toward analyzing the on-shell gluon self-energy diagram. The analogous off-shell virtual quark self-energy diagram does occur in order $\alpha_s^2$ graphs for $e^+ e^- \to hadrons$.)
  • Figure 5: On-shell virtual gluon self-energy diagram.
  • ...and 2 more figures