Quark and gluon two-loop beam functions for leading-jet $p_T$ and slicing at NNLO

We compute the complete set of two-loop beam functions for the transverse momentum distribution of the leading jet produced in association with an arbitrary colour-singlet system. Our results constitute the last missing ingredient for the calculation of the jet-vetoed cross section at small veto scales at the next-to-next-to-leading order, as well as an important ingredient for its resummation to next-to-next-to-next-to-leading logarithmic order. Our calculation is performed in the soft-collinear effective theory framework with a suitable regularisation of the rapidity divergences occurring in the phase-space integrals. We discuss the occurrence of soft-collinear mixing terms that might violate the factorisation theorem, and demonstrate that they vanish at two loops in the exponential rapidity regularisation scheme when performing a multipole expansion of the measurement function. As in our recent computation of the two-loop soft function, we present the results as a Laurent expansion in the jet radius $R$. We provide analytic expressions for all flavour channels in $x$ space with the exception of a set of $R$-independent non-logarithmic terms that are given as numerical grids. We also perform a fully numerical calculation with exact $R$ dependence, and find that it agrees with our analytic expansion at the permyriad level or better. Our calculation allows us to define a next-to-next-to-leading order slicing method using the leading-jet $p_T$ as a slicing variable. As a check of our results, we carry out a calculation of the Higgs and $Z$ boson total production cross sections at the next-to-next-to-leading order in QCD.