  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.
