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We present the first complete calculation for the quark and gluon $N$-jettiness ($Tau_N$) beam functions at next-to-next-to-next-to-leading order (N$^3$LO) in perturbative QCD. Our calculation is based on an expansion of the differential Higgs boson and Drell-Yan production cross sections about their collinear limit. This method allows us to employ cutting edge techniques for the computation of cross sections to extract the universal building blocks in question. The class of functions appearing in the matching coefficents for all channels includes iterated integrals with non-rational kernels, thus going beyond the one of harmonic polylogarithms. Our results are a key step in extending the $Tau_N$ subtraction methods to N$^3$LO, and to resum $Tau_N$ distributions at N$^3$LL$^prime$ accuracy both for quark as well as for gluon initiated processes.
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$N$-jettiness subtractions provide a general approach for performing fully-differential next-to-next-to-leading order (NNLO) calculations. Since they are based on the physical resolution variable $N$-jettiness, $mathcal{T}_N$, subleading power corrections in $tau=mathcal{T}_N/Q$, with $Q$ a hard interaction scale, can also be systematically computed. We study the structure of power corrections for $0$-jettiness, $mathcal{T}_0$, for the $ggto H$ process. Using the soft-collinear effective theory we analytically compute the leading power corrections $alpha_s tau lntau$ and $alpha_s^2 tau ln^3tau$ (finding partial agreement with a previous result in the literature), and perform a detailed numerical study of the power corrections in the $gg$, $gq$, and $qbar q$ channels. This includes a numerical extraction of the $alpha_stau$ and $alpha_s^2 tau ln^2tau$ corrections, and a study of the dependence on the $mathcal{T}_0$ definition. Including such power suppressed logarithms significantly reduces the size of missing power corrections, and hence improves the numerical efficiency of the subtraction method. Having a more detailed understanding of the power corrections for both $qbar q$ and $gg$ initiated processes also provides insight into their universality, and hence their behavior in more complicated processes where they have not yet been analytically calculated.
We compute the unpolarized quark and gluon transverse-momentum dependent fragmentation functions (TMDFFs) at next-to-next-to-next-to-leading order (N$^3$LO) in perturbative QCD. The calculation is based on a relation between the TMDFF and the limit of the semi-inclusive deep inelastic scattering cross section where all final-state radiation becomes collinear to the detected hadron. The required cross section is obtained by analytically continuing our recent computation of the Drell-Yan and Higgs boson production cross section at N$^3$LO expanded around the limit of all final-state radiation becoming collinear to one of the initial states. Our results agree with a recent independent calculation by Luo et al.
We present the Higgs boson production cross section at Hadron colliders in the gluon fusion production mode through N3LO in perturbative QCD. Specifically, we work in an effective theory where the top quark is assumed to be infinitely heavy and all other quarks are considered to be massless. Our result is the first exact formula for a partonic hadron collider cross section at N3LO in perturbative QCD. Furthermore, this result represents the first analytic computation of a hadron collider cross section involving elliptic integrals. We derive numerical predictions for the Higgs boson cross section at the LHC. Previously this result was approximated by an expansion of the cross section around the production threshold of the Higgs boson and we compare our findings. Finally, we study the impact of our new result on the state of the art prediction for the Higgs boson cross section at the LHC.
We derive the leading-power singular terms at three loops for both $q_T$ and 0-jettiness, $cal{T}_0$, for generic color-singlet processes. Our results provide the complete set of differential subtraction terms for $q_T$ and $cal{T}_0$ subtractions at N$^3$LO, which are an important ingredient for matching N$^3$LO calculations with parton showers. We obtain the full three-loop structure of the relevant beam and soft functions, which are necessary ingredients for the resummation of $q_T$ and $cal{T}_0$ at N$^3$LL$$ and N$^4$LL order, and which constitute important building blocks in other contexts as well. The nonlogarithmic boundary coefficients of the beam functions, which contribute to the integrated subtraction terms, are not yet fully known at three loops. By exploiting consistency relations between different factorization limits, we derive results for the $q_T$ and $cal{T}_0$ beam function coefficients at N$^3$LO in the $zto 1$ threshold limit, and we also estimate the size of the unknown terms beyond threshold.
We present several key steps towards the computation of differential Higgs boson cross sections at N$^3$LO in perturbative QCD. Specifically, we work in the framework of Higgs-differential cross sections that allows to compute precise predictions for realistic LHC observables. We demonstrate how to perform an expansion of the analytic N$^3$LO coefficient functions around the production threshold of the Higgs boson. Our framework allows us to compute to arbitrarily high order in the threshold expansion and we explicitly obtain the first two expansion coefficients in analytic form. Furthermore, we assess the phenomenological viability of threshold expansions for differential distributions. In addition, we report on an interesting obstacle for the computation of N$^3$LO corrections with LHAPDF parton distribution functions and our solution. We provide files containing the analytic expressions for the partonic cross sections together with the arXiv submission.
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