Systematic analysis of double-scale evolution

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.