  The artificial separation of a full-theory mode into distinct collinear and
soft modes in SCET 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 NLL and NNLL
calculations are improved.
