Factorization connecting continuum & lattice TMDs

Abstract

Transverse-momentum-dependent parton distribution functions (TMDs) can be studied from first principles by a perturbative matching onto lattice-calculable quantities: so-called lattice TMDs, which are a class of equal-time correlators that includes quasi-TMDs and TMDs in the Lorentz-invariant approach. We introduce a general correlator that includes as special cases these two Lattice TMDs and continuum TMDs, like the Collins scheme. Then, to facilitate the derivation of a factorization relation between lattice and continuum TMDs, we construct a new scheme, the Large Rapidity (LR) scheme, intermediate between the Collins and quasi-TMDs. The LR and Collins schemes differ only by an order of limits, and can be matched onto one another by a multiplicative kernel. We show that this same matching also holds between quasi and Collins TMDs, which enables us to prove a factorization relation between these quantities to all orders in αs. Our results imply that there is no mixing between various quark flavors or gluons when matching Collins and quasi TMDs, making the lattice calculation of individual flavors and gluon TMDs easier than anticipated. We cross-check these results explicitly at one loop and discuss implications for other physical-to-lattice scheme factorizations.

Nonperturbative Determination of Collins-Soper Kernel from Quasi Transverse-Momentum Dependent Wave Functions

“…With the help of soft function, the infrared contributions in the subtracted quasi TMDWFs can be properly accounted for such that the infrared structures for the quasi TMDWFs and light-cone ones are matched. This implies a multiplicative factorization theorem in the framework of LaMET [25][26][27][28]39]:…”

One-loop matching for gluon lattice TMDs

“…Here we focus on gluons, taking R = g for the adjoint representation. Beam functions involve staple-shaped Wilson [39] and (b) is adapted from ref. [48].…”

“…Quasi-TMDs [26][27][28][29][30][31][32][33][34][35]39]. The lattice quasi-TMD for gluons takes the form…”

“…The leadingpower factorization formula connecting quasi-TMDs to the continuum has been proposed in Refs. [28,29,31,32], and recently derived to all orders in α s in [39]. First lattice results have been presented for various components of quasi-TMDs: the beam function and Collins-Soper kernel [40][41][42][43][44][45], and soft function [43,44].…”

“…At leading power, the quasi-TMD (lattice) to Collins TMD (continuum) factorization formula reads [39]:…”