  Finding an efficient and compelling regularization of soft and collinear
degrees of freedom at the same invariant mass scale, but separated in rapidity
is a persistent problem in high-energy factorization. In the course of a
calculation, one encounters divergences unregulated by dimensional
regularization, often called rapidity divergences. Once regulated, a general
framework exists for their renormalization, the rapidity renormalization group
(RRG), leading to fully resummed calculations of transverse momentum (to the
jet axis) sensitive quantities. We examine how this regularization can be
implemented via a multi-differential factorization of the soft-collinear
phase-space, leading to an (in principle) alternative non-perturbative
regularization of rapidity divergences. As an example, we examine the
fully-differential factorization of a color singlet's momentum spectrum in a
hadron-hadron collision at threshold. We show how this factorization acts as a
mother theory to both traditional threshold and transverse momentum
resummation, recovering the classical results for both resummations. Examining
the refactorization of the transverse momentum beam functions in the threshold
region, we show that one can directly calculate the rapidity renormalized
function, while shedding light on the structure of joint resummation. Finally,
we show how using modern bootstrap techniques, the transverse momentum spectrum
is determined by an expansion about the threshold factorization, leading to a
viable higher loop scheme for calculating the relevant anomalous dimensions for
the transverse momentum spectrum.
