Rapidity renormalized TMD soft and beam functions at two loops

Thomas Lübbert,Maximilian Stahlhofen,Joel Oredsson

doi:10.1007/jhep03(2016)168

Abstract

We compute the transverse momentum dependent (TMD) soft function for the production of a color-neutral final state at the LHC within the rapidity renormalization group (RRG) framework to next-to-next-to-leading order (NNLO). We use this result to extract the universal renormalized TMD beam functions (aka TMDPDFs) in the same scheme and at the same order from known results in another scheme. We derive recurrence relations for the logarithmic structure of the soft and beam functions, which we use to cross check our calculation. We also explicitly confirm the non-Abelian exponentiation of the TMD soft function in the RRG framework at two loops. Our results provide the ingredients for resummed predictions of pT-differential cross sections at NNLL' in the RRG formalism. The RRG provides a systematic framework to resum large (rapidity) logarithms through (R)RG evolution and to assess the associated perturbative uncertainties.

Highlights

Sufficiently high order provide an accurate description of the p⊥-spectrum
Our results provide the ingredients for resummed predictions of p⊥-differential cross sections at NNLL in the renormalization group (RRG) formalism
We briefly review the renormalization of the beam and soft functions according to ref. [10] and predict their logarithmic structure based on the (R)RG equations (RGEs)

Summary

Introduction

In the region where p2⊥ M 2, the distribution is peaked and the resummation of (Sudakov) logarithms ∝ lnn(p2⊥/M 2) is crucial for reliable theoretical predictions. The traditional approach to resum such logarithms in QCD is based on the factorization theorem devised by Collins, Soper, and Sterman (CSS) in ref. In this framework the resummation has reached the next-to-next-to-leading-logarithmic (NNLL) level [2,3,4,5,6,7,8,9]. Throughout this paper we will focus on the perturbatively accessible resummation region of the p⊥-spectrum, i.e. where Λ2QCD p2⊥ M 2. The leading order SCET factorization theorem generically has the schematic form dσ dp2⊥ = H × S ⊗ Ba ⊗ Bb (p⊥)

Objectives

Results

Conclusion