A Toolbox for q_T and 0-Jettiness Subtractions at N³LO

We derive the leading-power singular terms at three loops for both q _T and 0-jettiness, T, for generic color-singlet processes. Our results provide the complete set of differential subtraction terms for q _T and T subtractions at N ³LO , which are an important ingredient for matching N ³LO calculations with parton showers. We obtain the full three-loop structure of the relevant beam and soft functions, which are necessary ingredients for the resummation of q _T and T at N ³LL ^′ and N ⁴LL order, and which constitute important building blocks in other contexts as well. The nonlogarithmic boundary coefficients of the beam functions, which contribute to the integrated subtraction terms, are not yet fully known at three loops. By exploiting consistency relations between different factorization limits, we derive results for the q _T and T beam function coefficients at N ³LO in the z→ 1 threshold limit, and we also estimate the size of the unknown terms beyond threshold.