Factorization connecting continuum and lattice TMDs

TL;DR

This paper develops a new theoretical framework connecting continuum and lattice TMDs, enabling more precise lattice calculations of parton distributions by establishing a universal factorization scheme and proving all-order matching relations.

Contribution

Introduces the Large Rapidity (LR) scheme for connecting continuum and lattice TMDs, and proves all-order factorization and matching relations between quasi, Collins, and continuum TMDs.

Findings

LR scheme is intermediate between Collins and quasi-TMDs

Matching between schemes is multiplicative and scheme-independent

No flavor or gluon mixing occurs in the matching process

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 $\alpha_s$. 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.