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An Exclusive Soft Function for Drell-Yan at Next-to-Next-to-Leading Order

Ye Li, Sonny Mantry, Frank Petriello

TL;DR

This work delivers the NNLO calculation of the exclusive soft function for Drell-Yan production in SCET with unintegrated distributions, a key ingredient for NNLL resummation of low-pT spectra. It provides both position-space and hybrid (q^-, q^+, b_perp) formulations, derives the corresponding anomalous dimensions, and demonstrates exponentiation of the soft function with appropriate convolution structure in light-cone momentum. The authors perform multiple checks, including the Belitsky limit, non-abelian exponentiation, and gauge consistency, validating the results. The results pave the way for extending NNLL resummation to unintegrated distribution frameworks and potentially to gluon-initiated processes like Higgs production, while offering insights into the treatment of rapidity divergences via kinematics.

Abstract

We present next-to-next-to-leading-order (NNLO) results for an exclusive soft function that appears in a recently developed factorization theorem for transverse momentum distributions. The factorization theorem, derived using the Soft Collinear Effective Theory, involves both a soft function and unintegrated nucleon distribution functions fully differential in momentum coordinates. The soft function is given by the vacuum matrix element of soft Wilson lines and is also fully differential in all components. We give results and relevant technical details for the NNLO calculation of the soft function, including finite parts, and derive the corresponding anomalous dimension. These results are necessary for achieving low transverse momentum resummation at next-to-next-to-leading-logarithmic accuracy in this effective field theory approach with unintegrated distribution functions.

An Exclusive Soft Function for Drell-Yan at Next-to-Next-to-Leading Order

TL;DR

This work delivers the NNLO calculation of the exclusive soft function for Drell-Yan production in SCET with unintegrated distributions, a key ingredient for NNLL resummation of low-pT spectra. It provides both position-space and hybrid (q^-, q^+, b_perp) formulations, derives the corresponding anomalous dimensions, and demonstrates exponentiation of the soft function with appropriate convolution structure in light-cone momentum. The authors perform multiple checks, including the Belitsky limit, non-abelian exponentiation, and gauge consistency, validating the results. The results pave the way for extending NNLL resummation to unintegrated distribution frameworks and potentially to gluon-initiated processes like Higgs production, while offering insights into the treatment of rapidity divergences via kinematics.

Abstract

We present next-to-next-to-leading-order (NNLO) results for an exclusive soft function that appears in a recently developed factorization theorem for transverse momentum distributions. The factorization theorem, derived using the Soft Collinear Effective Theory, involves both a soft function and unintegrated nucleon distribution functions fully differential in momentum coordinates. The soft function is given by the vacuum matrix element of soft Wilson lines and is also fully differential in all components. We give results and relevant technical details for the NNLO calculation of the soft function, including finite parts, and derive the corresponding anomalous dimension. These results are necessary for achieving low transverse momentum resummation at next-to-next-to-leading-logarithmic accuracy in this effective field theory approach with unintegrated distribution functions.

Paper Structure

This paper contains 10 sections, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and NLO Results
  3. Renormalization
  4. NLO Soft Function
  5. The soft function at NNLO
  6. The bare soft function
  7. Renormalization, exponentiation, and the finite soft function
  8. Conclusions
  9. Auxiliary integrals
  10. Hypergeometric expansions

Figures (1)

  • Figure 1: Example diagrams contributing the the soft function at next-to-next-to-leading order. From the top left diagram and proceeding in a clockwise fashion, the diagrams are respectively contributing to the terms ${\cal S}^{(2)}_{RR,C_F^2}$, ${\cal S}^{(2)}_{RR,C_F CA}$, ${\cal S}^{(2)}_{RV,C_FC_A}$ and ${\cal S}^{(2)}_{RR,C_F N_F}$.