Two-loop beam and soft functions for rapidity-dependent jet vetoes

Jet vetoes play an important role in many analyses at the LHC. Traditionally, jet vetoes have been imposed using a restriction on the transverse momentum p_Tj of jets. Alternatively, one can also consider jet observables for which p_Tj is weighted by a smooth function of the jet rapidity y_j that vanishes as |y_j| → ∞. Such observables are useful as they provide a natural way to impose a tight veto on central jets but a looser one at forward rapidities. We consider two such rapidity-dependent jet veto observables, T_Bj and T_Cj, and compute the required beam and dijet soft functions for the jet-vetoed color-singlet production cross section at two loops. At this order, clustering effects from the jet algorithm become important. The dominant contributions are computed fully analytically while corrections that are subleading in the limit of small jet radii are expressed in terms of finite numerical integrals. Our results enable the full NNLL′ resummation and are an important step towards N³LL resummation for cross sections with a T_Bj or T_Cj jet veto.