  The zero-jettiness beam functions describe collinear emissions from initial
state legs and appear in the factorisation theorem for cross sections in the
limit of small zero-jettiness. They are an important building block for slicing
schemes for colour-singlet production at hadron colliders. We report on our
ongoing calculation of this quantity at next-to-next-to-next-to-leading order
(N$^3$LO) in QCD, highlighting in particular the aspects of partial fraction
relations and the calculation of master integrals.
