Abstract
A number of important observables exhibit logarithms in their perturbative description that are induced by emissions at widely separated rapidities. These include transverse-momentum (qT) logarithms, logarithms involving heavy-quark or electroweak gauge boson masses, and small-x logarithms. In this paper, we initiate the study of rapidity logarithms, and the associated rapidity divergences, at subleading order in the power expansion. This is accomplished using the soft collinear effective theory (SCET). We discuss the structure of subleading-power rapidity divergences and how to consistently regulate them. We introduce a new pure rapidity regulator and a corresponding overline{mathrm{MS}} -like scheme, which handles rapidity divergences while maintaining the homogeneity of the power expansion. We find that power-law rapidity divergences appear at subleading power, which give rise to derivatives of parton distribution functions. As a concrete example, we consider the qT spectrum for color-singlet production, for which we compute the complete qT2/Q2 suppressed power corrections at mathcal{O}left({alpha}_sright) , including both logarithmic and nonlogarithmic terms. Our results also represent an important first step towards carrying out a resummation of subleading-power rapidity logarithms.
Highlights
Rapidity logarithms can be resummed to all orders in αs using rapidity evolution equations
We have studied in detail the structure and consistent regularization of rapidity divergences at subleading order in the power expansion
We have shown that the η regulator, which in principle can be applied at subleading power, is not homogeneous in the power expansion, which leads to undesirable complications at subleading power
Summary
Rapidity divergences naturally arise in the calculation of observables sensitive to the transverse momentum of soft emissions. Unlike SCETI where the modes are separated in virtuality, in SCETII the modes in the EFT have the same virtuality, but are distinguished by their longitudinal momentum (k+ or k−), or equivalently, their rapidity e2yk = k−/k+ This separation into modes at hierarchical rapidities introduces divergences, which arise when k+/k− → ∞ or k+/k− → 0 [9, 101,102,103,104]. At leading power in the EFT expansion, the structure of rapidity divergences and the associated rapidity renormalization group are well understood and they have been studied to high perturbative orders
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