Abstract
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 qT 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 qT 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 [lnn(qT2/Q2)/qT2]+ appearing in the physical qT distribution. The resummation accuracy is then solely determined by the perturbative expansion of the associated anomalous dimensions.
Highlights
An important class of differential observables at colliders are those that resolve additional soft or collinear QCD emissions on top of the underlying hard Born process
As for the one-dimensional case, we find that performing the resummation in Fourier space adds an additional boundary term compared to the momentum-space resummation
After briefly reviewing qT factorization and the relevant associated RG equations in section 3.1, we argue in section 3.2 that the appearing two-dimensional convolutions requires very careful scale setting, which turns out to be the crucial complication of qT resummation in momentum space
Summary
An important class of differential observables at colliders are those that resolve additional soft or collinear QCD emissions on top of the underlying hard Born process. We derive the solution to perform the RG evolution entirely in distribution (momentum) space, allowing for the explicit resummation of all logarithmic contributions [lnn(qT2 /Q2)/qT2 ]+ appearing in the physical qT distribution We show that it intrinsically requires distributional scale setting due to the two-dimensional convolutions appearing for qT , which is not the case for one-dimensional observables such as thrust. An advantage of performing the resummation via the solution of the qT evolution equations is that the solution automatically applies to all orders in resummed perturbation theory, and the resummation accuracy is solely defined through the perturbative accuracy of the anomalous dimensions (as well as matching conditions) This allows one to completely avoid any discussions of how to consistently count explicit logarithms in the cross section (which has been part of the difficulties in previous attempts). In the appendices we collect many relevant and useful definitions and relations for plus distributions
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