Resummation of Transverse Momentum Distributions in Distribution Space

Differential spectra in observables that resolve additional soft or collinear QCD emissions exhibit Sudakov double logarithms in the form of logarithmic plus distributions. Important examples are the total transverse momentum $q_T$ in color-singlet production, N-jettiness (with thrust or beam thrust as special cases), but also jet mass and more complicated jet substructure observables. The all-order logarithmic structure of such distributions is often fully encoded in differential equations, so-called (renormalization group) evolution equations. We introduce a well-defined technique of distributional scale setting, which allows one to treat logarithmic plus distributions like ordinary logarithms when solving these differential equations. In particular, this allows one (through canonical scale choices) to minimize logarithmic contributions in the boundary terms of the solution, and to obtain the full distributional logarithmic structure from the solution's evolution kernel directly in distribution space. We apply this technique to the $q_T$ distribution, where the two-dimensional nature of convolutions leads to additional difficulties (compared to one-dimensional cases like thrust), and for which the resummation in distribution (or momentum) space has been a long-standing open question. For the first time, we show how to perform the RG evolution fully in momentum space, thereby directly resumming the logarithms $[\ln^n(q_T^2/Q^2)/q_T^2]_+$ appearing in the physical $q_T$ distribution. The resummation accuracy is then solely determined by the perturbative expansion of the associated anomalous dimensions.