Di-Lepton Rapidity Distribution in Drell-Yan Production to Third Order in QCD

KA-TP-17-2021, ZU-TH 33/21, CERN-TH-2021-110, IPPP/21/13 Di-lepton Rapidity Distribution in Drell-Yan Production to Third Order in QCD Xuan Chen,1, 2, 3, ∗ Thomas Gehrmann,1, y Nigel Glover,4, z Alexander Huss,5, x Tong-Zhi Yang,1, { and Hua Xing Zhu6, ∗∗ 1Physik-Institut, UniversitätZürich,Winterthurerstrasse 190, CH-8057 Zürich,Switzerland 2Institute for Theoretical Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany 3Institute for Astroparticle Physics, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany 4Institute for Particle Physics Phenomenology, Durham University, Durham, DH1 3LE, UK 5Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland 6Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou, 310027, China We compute for the first time the lepton-pair rapidity distribution in the photon-mediated Drell- Yan process to next-to-next-to-next-to-leading order (N3LO) in QCD. The calculation is based on the qT -subtraction method, suitably extended to this order for quark-antiquark initiated Born processes. Our results display sizeable QCD corrections at N3LO over the full rapidity region and provide a fully independent confirmation of the recent results for the total Drell-Yan cross section at this order. INTRODUCTION dictions. For Higgs production, distributions that are fully differential in the decay products are now available at N3LO [21] using the analytic results for the inclusive Precision physics is becoming increasingly important cross section and rapidity distribution of the Higgs bo- for the CERN Large Hadron Collider (LHC) physics pro- son at N3LO as input [22{24]. Unfortunately, the same gram, in particular in view of the absence of striking sig- approach does not work well for the Drell-Yan process, nals for beyond the Standard Model (SM) phenomena. as the threshold expansion used for the analytic calcula- Among the most important precision processes at the tion of the rapidity distribution does not converge well LHC is Drell-Yan lepton-pair production through neu- for quark-induced processes. tral current Z/γ∗ or charged current W ± exchanges. It In this Letter we present for the first time the di-lepton plays a central role in the extraction of SM parameters rapidity distribution for Drell-Yan production at N3LO, and as input to the determination of parton distribution computed using the q -subtraction method at this or- functions (PDFs). The Drell-Yan process is also highly T der. We focus on the contribution from virtual pho- important in new physics searches, both as background ton production alone, neglecting the contribution from Z to direct signals, as well as an indirect probe of dynamics boson exchange and from virtual photon-Z interference. beyond the collider energy. While the remaining contributions are important, the vir- The Drell-Yan process further plays a special role in tual photon contributions are sufficiently representative the development of modern precision theory calculations to gain knowledge about the size of QCD corrections at in particle physics. It was the first hadron collider pro- this order [14], and most importantly sufficient for illus- cess for which next-to-leading order (NLO) QCD cor- trating the subtraction of infrared singularities at N3LO. rections were calculated [1, 2], due to its simplicity in This is also the first time that the qT -subtraction method kinematics on one hand, as well as its phenomenologi- is being applied at the N3LO order without any input cal importance on the other hand. The large perturba- from a total inclusive cross section, thereby establishing tive corrections observed at the NLO also sparked the the validity and practicality of the method at this or- interest in soft gluon resummation [3, 4], which subse- der, as well as validating independently the recent N3LO arXiv:2107.09085v1 [hep-ph] 19 Jul 2021 quently developed into a field of its own. The first calcu- result [14] for the inclusive Drell-Yan coefficient function. lation of inclusive next-to-next-to-leading order (NNLO) QCD corrections to a hadron collider process was also performed for Drell-Yan process [5, 6], followed by the QT-SUBTRACTION AT N3LO first NNLO rapidity distributions [7, 8], and then fully differential distributions [9{13]. Very recently, next-to- 3 3 The N LO corrections in QCD receive contributions next-to-next-to-leading order (N LO) QCD corrections from four types of parton-level sub-processes, each cor- have been computed for the inclusive Drell-Yan process recting the underlying Born-level process: triple real ra- with an off-shell photon [14], and for charged current diation at tree-level, double real radiation at one loop, Drell-Yan production [15]. With such level of accuracy in single real radiation at two loops and purely virtual three- perturbative QCD, mixed electroweak-QCD corrections, loop corrections. At this order, only very few collider derived recently [16{20], become equally important. processes were computed so far, including inclusive and For many phenomenological applications of Drell-Yan differential Higgs production from gluon fusion [21{25], production, it is more desirable to have differential pre- inclusive Drell-Yan production [14], inclusive Higgs pro- 2 duction from b quark annihilation [26], vector boson fu- for the incoming particle beams, and a soft function S: sion Higgs production [27], di-Higgs production [28], in- 4 Born Z 2 clusive deep inelastic scattering [29] and jet production d σγ∗ X σi d b iq ·b = e T d2q dQ2dy E2 (2π)2 in deep-inelastic scattering [30, 31]. T i CM 2 In this letter we focus on the Drell-Yan production × Bi=A(xA; b)B¯{=B(xB; b)S(b)H(Q ) through a virtual photon, for which all relevant matrix 0 + (i $ ¯{) + O(qT ) : (2) elements have been available for some time [32{39]. Af- ter subtraction of universal initial-state collinear singu- Born 2 2 2 where σi = 4πQi αem=(3NcQ ), Qi is the electric larities, perturbative predictions for infrared safe observ- charge, αem is the fine structure constant of QED, ECM is ables are finite due to the Kinoshita-Lee-Nauenberg the- the center of mass energy. The momentum fractionsp are orem [40, 41]. The challenge is that all individual sub- y pfixed by the final-state kinematics as xA = τe , xB = processes with different multiplicities are separately in- −y 2 2 2 τe , with τ = (Q + qT )=ECM. In contrast to the frared divergent. In particular the infrared divergences in 0 leading-power terms [50{52], the power corrections O(qT ) sub-processes with real radiations reside in phase space are far less well understood but their contribution can integrals, which makes them difficult to handle. An im- cut be suppressed by choosing a sufficiently small qT value. portant part of the NLO and NNLO revolution in the The factorisation structure in Eq. (2) is most transparent past decades has been the development of convenient and in Soft-Collinear Effective Theory (SCET) [53{57], which efficient algorithms for handling these infrared singular- also provides a convenient framework for the calculation ities from real emissions. Two among these methods of the unresolved contribution beyond NNLO. (Projection-to-Born [42] and qT -subtraction [43]) have The hard function H is simply the electromagnetic 3 been extended to be applied in specifc N LO calcula- quark form factor. The beam function Bi=A(xA; b) en- tions [21, 25, 31]. codes initial-state collinear radiation. For a high energy hadron A moving in the light-cone direction nµ = The q -subtraction method [11, 12, 43, 44] was ini- T (1; 0; 0; 1) with four momentum P µ, the beam function tially developed for processes with color-less final states. A can be written in light-cone gauge and coordinates as The key idea is that the most singular phase space con- figurations are associated with the small qT region of − P + + Z db − A γ ixb 2 − the color-less system, and can be isolated by an artifi- Bi=A(x; b) = e hAj i(0; b ; b) i(0)jAi : cial q cut. The extension of the q -subtraction method 4π 2 T T (3) to N3LO has been outlined for gluon-induced [25] and This beam function is a priori a non-perturbative ma- quark-induced processes [45]. For Drell-Yan production trix element, which can be expressed in terms of per- at N3LO, the double differential cross section in di-lepton turbatively calculable Wilson coefficients I and parton invariant mass squared Q2 and di-lepton rapidity y is di- i=j distribution functions fj=A using a light-cone operator vided into the unresolved (resolved) part, in which qT is cut product expansion: bounded by qT from above (below), X Z 1 dξ Bi=A(x; b) = Ii=j(ξ; b)fj=A(x/ξ) : (4) cut x ξ 2 Z qT 4 Z 4 j d σγ∗ 2 d σγ∗ 2 d σγ∗ 2 = d qT 2 2 + d qT 2 2 : dQ dy d q dQ dy cut d q dQ dy 0 T qT T The soft function describes multiple soft gluon radiation (1) with a constraint on the total qT . It is given by the The resolved contribution can be regarded as Drell-Yan vacuum matrix element plus jet production, therefore requiring infrared subtrac- 3 tr y y tion only to NNLO. The genuine N LO infrared singu- S(b) = hΩjTfYn¯ Yn(0; 0; b)gTfYn Yn¯ (0)gjΩi ; (5) larities cancel within the unresolved contribution. While Nc the singularities themselves are canceled, they give rise R 0 m where Y (x) = P exp(ig ds A(x + sn)) is a path- to large logarithms, ln qcut=Q, both in the resolved and n −∞ T ordered semi-infinite light-like Wilson line.

Load more