Reemergence of rapidity-scale uncertainty in soft-collinear effective theory

Abstract

The artificial separation of a full-theory mode into distinct collinear and soft modes in soft-collinear effective theory leads to divergent integrals over rapidity, which are not present in the full theory. Rapidity divergence introduces an additional scale into the problem, giving rise to its own renormalization group with respect to this new scale. Two contradicting claims exist in the literature concerning rapidity scale uncertainty. One camp has shown that the results of perturbative calculations depend on the precise choice of rapidity scale. The other has derived an all-order factorization formula with no dependence on rapidity scale, by using a form of analytic regulator to regulate rapidity divergences. We deliver a simple resolution to this controversy by deriving an alternative form of the all-order factorization formula with an analytic regulator that, despite being formally rapidity scale independent, reveals how rapidity scale dependence arises when it is truncated at a finite order in perturbation theory. With our results, one can continue to take advantage of the technical ease and simplicity of the analytic regulator approach while correctly taking into account rapidity scale dependence. As an application, we update our earlier study of $WW$ production with jet veto by including rapidity scale uncertainty. While the central values of the predictions are unchanged, the scale uncertainties are increased and consistency between the next-to-leading logarithm and next-to-next-to-leading logarithm calculations is improved.