Removing phase-space restrictions in factorized cross sections

Factorization in gauge theories holds at the amplitude or amplitude-squared level for states of given soft or collinear momenta. When performing phase-space integrals over such states, one would generally like to avoid putting in explicit cuts to separate soft from collinear momenta. Removing these cuts induces an overcounting of the soft-collinear region and adds new infrared-ultraviolet divergences in the collinear region. In this paper, we first present a regulator-independent subtraction algorithm for removing soft-collinear overlap at the amplitude level which may be useful in pertubative QCD. We then discuss how both the soft-collinear and infrared-ultraviolet overlap can be undone for certain observables in a way which respects factorization. Our discussion clarifies some of the subtleties in phase-space subtractions and includes a proof of the infrared finiteness of a suitably subtracted jet function. These results complete the connection between factorized QCD and Soft-Collinear Effective Theory.