The gluon beam function at two loops

The virtuality-dependent beam function is a universal ingredient in the resummation for observables probing the virtuality of incoming partons, including N-jettiness and beam thrust. We compute the gluon beam function at two-loop order. Together with our previous results for the two-loop quark beam function, this completes the full set of virtuality-dependent beam functions at next-to-next-to-leading order (NNLO). Our results are required to account for all collinear initial-state radiation effects on the N-jettiness event shape through (NLL)-L-3 order. We present numerical results for both the quark and gluon beam functions up to NNLO and (NLL)-L-3 order. Numerically, the NNLO matching corrections are important. They reduce the residual matching scale dependence in the resummed beam function by about a factor of two.