Relating transverse-momentum-dependent and collinear factorization theorems in a generalized formalism

Abstract

We construct an improved implementation for combining transverse-momentum-dependent (TMD) factorization and collinear factorization. TMD factorization is suitable for low transverse momentum physics, while collinear factorization is suitable for high transverse momenta and for a cross section integrated over transverse momentum. The result is a modified version of the standard W + Y prescription traditionally used in the Collins-Soper-Sterman (CSS) formalism and related approaches. We further argue that questions regarding the shape and Q-dependence of the cross sections at lower Q are largely governed by the matching to the Y -term.

Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production

“…In this work, we focus on the simplest ones, i.e., the unpolarized TMD PDF f q 1 (x, k 2 ⊥ ) and the unpolarized TMD FF D q→h 1 (z, P 2 ⊥ ), where z is the fractional energy carried by the detected hadron h, k ⊥ is the transverse momentum of the parton with respect to the parent hadron, and P ⊥ is the transverse momentum of the produced hadron with respect to the parent parton. Despite their simplicity, the phenomenology of these unpolarized TMDs presents several challenges [22]: the choice of a functional form for the nonperturbative components of TMDs, the inclusion of a possible dependence on partonic flavor [23], the implementation of TMD evolution [4,24], the matching to fixed-order calculations in collinear factorization [25].…”

Twist decomposition of Drell-Yan structure functions: phenomenological implications

“…The overall normalization, however, cancels in the analysis of the relative twist content so the small discrepancy of this parameter does not prohibit using this model in the twist analysis. Note that in the q T integrated cross sections the effects of the CSS resummation cancel [48].…”

“…Furthermore, it was shown by Collins, Sterman and Soper (CSS) [47] that in the relevant small q T region these corrections may be resummed or parameterized by a universal non-perturbative transverse momentum dependent parton distribution at very small q T . It was proven that the contribution of the resummed corrections of this type cancel after q T integrations [48], but it is essential for the correct description of the Drell-Yan q T -dependent cross section at small q T . Recently the problem of joint resummation of small x effects and the transverse momentum logarithms was addressed [49,50] providing important results for analyses of the q T dependent DY distribution at small x.…”

Evolution of sum rules and positivity constraints

“…For the TMD evolution the asymptotic nullification of the moment of TMD pdf much discussed here [7] means the violation of positivity for large Q T . One may understand that as a reversal of scale arrow at Q T ∼ Q which is also supported by the sign change of the corresponding log.…”

“…One may understand that as a reversal of scale arrow at Q T ∼ Q which is also supported by the sign change of the corresponding log. The substitution of W by Y term [7] may be interpreted as a restoration of the direction of scale arrow.…”