N$^3$LO Power Corrections for $0$-jettiness Subtractions With Fiducial Cuts

We compute the leading logarithmic power corrections at next-to-next-to-next-to-leading order for $0$-jettiness subtractions for color singlet production. We discuss how to disentangle these power corrections from those arising from the presence of fiducial and isolation cuts by using Projection-to-Born improved slicing. We present the results for Drell-Yan and Higgs production in gluon fusion differential in both the invariant mass and rapidity of the color singlet. Our results include all the channels contributing at leading logarithmic order for these processes, including the off-diagonal channels that receive contributions from soft quark emission. We study the numerical impact of the power corrections for Drell-Yan and Higgs production and find it to become negligible compared to the size of the N$^3$LO corrections only below $\tau_\text{cut} \sim 10^{-5}$. We estimate that in a fully differential calculation at N$^3$LO combining the Projection-to-Born improved slicing method and our results for the leading logarithmic power corrections may allow for keeping the slicing uncertainties under control already with $\tau_\text{cut} \lesssim 10^{-3}$, marking a significant improvement in efficiency for these methods. These results constitute a crucial ingredient for fully differential N$^3$LO calculations based on the $N$-jettiness subtraction scheme.