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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.
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In this paper we calculate analytically the perturbative matching coefficients for unpolarized quark and gluon Transverse-Momentum-Dependent (TMD) Parton Distribution Functions (PDFs) and Fragmentation Functions (FFs) through Next-to-Next-to-Next-to-Leading Order (N$^3$LO) in QCD. The N$^3$LO TMD PDFs are calculated by solving a system of differential equation of Feynman and phase space integrals. The TMD FFs are obtained by analytic continuation from space-like quantities to time-like quantities, taking into account the probability interpretation of TMD PDFs and FFs properly. The coefficient functions for TMD FFs exhibit double logarithmic enhancement at small momentum fraction $z$. We resum such logarithmic terms to the third order in the expansion of $alpha_s$. Our results constitute important ingredients for precision determination of TMD PDFs and FFs in current and future experiments.
We compute the quark and gluon transverse momentum dependent parton distribution 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 corresponding perturbative matching kernels for all channels are expressed in terms of simple harmonic polylogarithms up to weight five. As a byproduct, we confirm a previous computation of the soft function for transverse momentum factorization at N$^3$LO. Our results are the last missing ingredient to extend the $q_T$ subtraction methods to N$^3$LO and to obtain resummed $q_T$ spectra at N$^3$LL$^prime$ accuracy both for gluon as well as for quark initiated processes.
We study the transverse-momentum spectrum of quarkonium production from single light-parton fragmentation mechanism. In the case of semi-inclusive deep inelastic scattering, we observe that there are two possible initiating processes, namely photon-gluon fusion and light-quark fragmentation. For the second case we derive the factorization theorem, which involves a new hadronic quantity: the quarkonium transverse-momentum-dependent fragmentation functions in NRQCD. We calculate their matching onto the non-perturbative long distance matrix elements at the lowest order in the strong-coupling constant (${mathcal O}(alpha_s^2)$). Focusing on the case of the electron-ion collider, we make a comparative phenomenological study of the two production mechanisms and find the regions of the phase space where one is dominant over the other.
We present the analytic formula for the Energy-Energy Correlation (EEC) in electron-positron annihilation computed in perturbative QCD to next-to-next-to-next-to-leading order (N$^3$LO) in the back-to-back limit. In particular, we consider the EEC arising from the annihilation of an electron-positron pair into a virtual photon as well as a Higgs boson and their subsequent inclusive decay into hadrons. Our computation is based on a factorization theorem of the EEC formulated within Soft-Collinear Effective Theory (SCET) for the back-to-back limit. We obtain the last missing ingredient for our computation - the jet function - from a recent calculation of the transverse-momentum dependent fragmentation function (TMDFF) at N$^3$LO. We combine the newly obtained N$^3$LO jet function with the well known hard and soft function to predict the EEC in the back-to-back limit. The leading transcendental contribution of our analytic formula agrees with previously obtained results in $mathcal{N} = 4$ supersymmetric Yang-Mills theory. We obtain the $N=2$ Mellin moment of the bulk region of the EEC using momentum sum rules. Finally, we obtain the first resummation of the EEC in the back-to-back limit at N$^3$LL$^prime$ accuracy, resulting in a factor of $sim 4$ reduction of uncertainties in the peak region compared to N$^3$LL predictions.
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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