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Borel resummation of transverse momentum distributions

Marco Bonvini, Stefano Forte, Giovanni Ridolfi

TL;DR

The paper addresses the divergence arising in the perturbative expansion of transverse-momentum resummation when transforming from impact-parameter space back to q_T-space due to the Landau pole. It introduces a Borel resummation framework, incorporating a higher-twist term to regularize the inversion and yield an asymptotic, well-defined result. Numerical comparisons show the Borel prescription is stable, with small resummation ambiguities for large Q^2 and modest q_T, and superior matching behavior relative to some existing prescriptions. This provides a reliable, implementable method for precise predictions of transverse momentum distributions in hadronic collisions, with clear ambiguity estimates via the twist parameter.

Abstract

We present a new prescription for the resummation of contributions due to soft gluon emission to the trasverse momentum distribution of processes such as Drell-Yan production in hadronic collisions. We show that familiar difficulties in obtaining resummed results as a function of transverse momentum starting from impact-parameter space resummation are related to the divergence of the perturbative expansion of the momentum-space result. We construct a resummed expression by Borel resummation of this divergent series, removing the divergence in the Borel inversion through the inclusion of a suitable higher twist term. The ensuing resummation prescription is free of numerical instabilities, is stable upon the inclusion of subleading terms, and the original divergent perturbative series is asymptotic to it. We compare our results to those obtained using alternative prescriptions, and discuss the ambiguities related to the resummation procedure.

Borel resummation of transverse momentum distributions

TL;DR

The paper addresses the divergence arising in the perturbative expansion of transverse-momentum resummation when transforming from impact-parameter space back to q_T-space due to the Landau pole. It introduces a Borel resummation framework, incorporating a higher-twist term to regularize the inversion and yield an asymptotic, well-defined result. Numerical comparisons show the Borel prescription is stable, with small resummation ambiguities for large Q^2 and modest q_T, and superior matching behavior relative to some existing prescriptions. This provides a reliable, implementable method for precise predictions of transverse momentum distributions in hadronic collisions, with clear ambiguity estimates via the twist parameter.

Abstract

We present a new prescription for the resummation of contributions due to soft gluon emission to the trasverse momentum distribution of processes such as Drell-Yan production in hadronic collisions. We show that familiar difficulties in obtaining resummed results as a function of transverse momentum starting from impact-parameter space resummation are related to the divergence of the perturbative expansion of the momentum-space result. We construct a resummed expression by Borel resummation of this divergent series, removing the divergence in the Borel inversion through the inclusion of a suitable higher twist term. The ensuing resummation prescription is free of numerical instabilities, is stable upon the inclusion of subleading terms, and the original divergent perturbative series is asymptotic to it. We compare our results to those obtained using alternative prescriptions, and discuss the ambiguities related to the resummation procedure.

Paper Structure

This paper contains 6 sections, 67 equations, 3 figures.

Table of Contents

  1. Transverse momentum resummation
  2. The need for a resummation prescription
  3. The Borel prescription
  4. Comparison of resummation prescriptions
  5. Summary
  6. Appendix

Figures (3)

  • Figure 1: The integration contour $H$ in eq. (\ref{['eq:tappeto']}).
  • Figure 2: The NLL partonic resummed Drell-Yan transverse momentum distribution computed with various resummation prescriptions with $Q^2=10^4$ GeV$^2$ and in a narrow (left) and wide (right) range of $q_{ T}$.
  • Figure 3: Dependence of the results shown in fig. \ref{['fig:sigma']} for the Borel prescription on the parameter $C$. The vertical line at $q_{ T}=156$ MeV indicates the position of the Landau pole.