Systematic analysis of double-scale evolution

TL;DR

This paper analyzes the complex two-dimensional evolution of double-scale distributions like TMDs, introduces an optimal universal non-perturbative function, and discusses how perturbative truncation affects evolution uncertainties.

Contribution

It formulates the $ta$-prescription for double-scale evolution, defines an optimal universal TMD distribution, and examines the impact of perturbation truncation on evolution ambiguities.

Findings

The two-dimensional structure of TMD evolution is characterized.

Perturbation truncation introduces ambiguities and uncertainties.

Proposed solutions reduce scale variation uncertainties.

Abstract

Often the factorization of differential cross sections results in the definition of fundamental hadronic functions/distributions which have a double-scale evolution, as provided by a pair of coupled equations. Typically, the two scales are the renormalization and rapidity scales. The two-dimensional structure of their evolution is the object of the present study . In order to be more specific, we consider the case of the transverse momentum dependent distributions (TMD). Nonetheless, most of our findings can be used with other double-scale parton distributions. On the basis of the two-dimensional structure of TMD evolution, we formulate the general statement of the $\zeta$-prescription introduced in \cite{Scimemi:2017etj}, and we define an optimal TMD distribution, which is a scaleless model-independent universal non-perturbative function. A significant part of this work is devoted to the study of the effects of truncation of perturbation theory on the double-scale evolution. We show that within truncated perturbation theory the solution of evolution equations is ambiguous and this fact generates extra uncertainties within the resummed cross-section. The alternatives to bypass this issue are discussed. We discuss the effects of the scale variation effects and show that these uncertainties reduce within the non-ambiguous solutions that we propose.