Abstract
In this work, we perform a lattice QCD study of the intrinsic, rapidity-independent soft function within the framework of large momentum effective theory. The computation is carried out using a gauge ensemble of N_{f}=2+1+1 clover-improved twisted mass fermion. After applying an appropriate renormalization procedure and the removal of significant higher-twist contamination, we obtain the intrinsic soft function that is comparable to the one-loop perturbative result at large external momentum. The determination of the nonperturbative soft function from first principles is crucial to sharpen our understanding of the processes with small transverse momentum such as the Drell-Yan production and the semi-inclusive deep inelastic scattering. Additionally, we calculate the Collins-Soper evolution kernel using the quasi-transverse-momentum-dependent wave function as input.
Highlights
Introduction.—Understanding the structure of matter within the framework of quantum chromodynamics (QCD) is one of the central goals of hadron and nuclear physics
These functions cannot be obtained from totally inclusive processes, as we need an observable in the final state carrying information on k⊥, obtained, e.g., by measuring the transverse momentum Q⊥ of a lepton pair produced in a Drell-Yan process
As in the case of collinear PDFs, the extraction of TMD parton distribution functions (TMDPDFs) from the measured Drell-Yan or semi-inclusive deep inelastic cross sections is possible, thanks to factorization theorems, which isolate the nonperturbative physics into suitable definitions of TMDPDFs [7,8,9,10,11]
Summary
We perform a calculation of the soft function using a different fermionic discretization, namely, the twisted mass fermion. Two current operators u Γu and d Γd are inserted at the same time slice, but with a spatial separation b⊥ that is perpendicular to the momentum direction. FΓðb⊥; PzÞ can be factorized into the quasi-TMD wave function (quasiTMDWF) Φ and the intrinsic soft function Sðb⊥; μÞ [14,17] at large Pz through. Up to OðαsÞ corrections, the hard kernel takes a simple form, denoted here as H0Γ It can be obtained from a Fierz identity that H0Γ 1⁄4 1=ð2NcÞ for Γ 1⁄4 I; γ⊥; γ5γ⊥ and −1=ð2NcÞ for Γ 1⁄4 γ5 with Nc 1⁄4 3 the number of colors. Using H0Γ as an input, one can further simplify the expression (2) as
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