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, † Nigel Glover,4, ‡ Alexander Huss,5, § 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¯ı/B(xB, b)S(b)H(Q ) through a virtual photon, for which all relevant matrix 0 + (i ↔ ¯ı) + 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 fractions√ are orem [40, 41]. The challenge is that all individual sub- y √fixed 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 hA|ψi(0, b , b) ψi(0)|Ai . 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 † † tion only to NNLO. The genuine N LO infrared singu- S(b) = hΩ|T{Yn¯ Yn(0, 0, b)}T{Yn Yn¯ (0)}|Ωi , (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. the unresolved contribution. Cancellation of these large 3 logarithms occurs when resolved and unresolved contri- For N LO accuracy, we need the third order correc- butions are combined. tions to the perturbative beam function Ii/j(x, b), soft function and hard function. The hard function has been A major advantage of qT -subtraction is that the struc- known to three loops for some time [58–60]. The calcu- ture of perturbation theory in the unresolved region is lation of the beam and soft function is less straightfor- well understood from the development of qT resumma- ward, due to the presence of rapidity divergences [61], tion [46–49]. This allows to write the unresolved contri- which only disappear in physical cross sections. Various butions in a factorised form to all orders in perturbation approaches for rapidity regularization have been adopted theory, in terms of a hard function H, beam functions B in the literature to obtain the beam and soft function at 3

NNLO [62–69]. At N3LO, the scale dependence of per- * turbative beam and soft functions are completely ﬁxed SCET+NNLOJET pp to (NLO to N3LO) s = 13 TeV 400 by renormalisation group (RG) evolution in SCET, see 300 e.g. [70, 71]. The initial conditions of this RG evolution ] 200 3 b f form the genuine N LO contributions, and require cal- [

) 100 culation to this order in SCET. Very recently, this was * , T 0 q accomplished in a series of works for the soft function [72] ( PDF4LHC15 nnlo n l 100 d and the beam functions [73–75], using the rapidity reg- / F = R = 100 GeV

d 200 ulator proposed in [76]. These newly available results SCET N3LO NNLOJET N3LO SCET NNLO NNLOJET NNLO provide the key ingredients for applying q -subtraction 300 T SCET NLO NNLOJET NLO to processes with colorless ﬁnal states at N3LO. The 10 perturbative beam functions are expressed in terms of 5 harmonic polylogarithms [77] up to weight 5, which can 0 be evaluated numerically with standard tools [78]. 5 cut 3 The resolved contribution above the q for N LO 100 101 102

T Non-Singular [fb] Drell-Yan production contains the same ingredients of qT, * [GeV] the NNLO calculation with one extra jet. Fully differential NNLO contributions for Drell-Yan-plus-jet produc- FIG. 1: Perturbative contributions to transverse mo- 3 tion has been computed in [79–81]. The application to mentum distribution of the virtual photon up to αs. 3 N LO qT -subtraction further requires stable fixed-order The upper panel displays the qT -distribution obtained predictions at small qT [82–84], enabling the cancella- from NNLOJET and from expanding SCET to each cut tion of the qT between resolved and unresolved con- order. The bottom panel contains the non-singular tributions to sufficient accuracy. In this letter, we em- remainder (NNLOJET minus SCET). ploy the antenna subtraction method [85–88] to compute cut Drell-Yan production above qT up to NNLO in perturbation theory. The calculation is performed within the obtained by expanding the leading-power factorised pre- parton-level event generator NNLOJET [79, 82], which diction at small qT using (2). Note that although the qT removes the double unresolved infrared divergences from distribution is obtained from virtual photon production 3 tree level, single unresolved and explicit infrared diver- up to O(αs), its infrared divergences are of NNLO type. 5 gences from one-loop level and explicit infrared diver- The highest logarithms at this order are 1/qT ln (Q/qT ). gences from two-loop level with appropriate antenna sub- The singular qT distribution is expected to match be- traction terms. To achieve stable and reliable fixed order tween NNLOJET and SCET, which is a prerequisite for predictions down to the qT ∼ 0.4 GeV region, NNLOJET the qT -subtraction method. This requirement is ful- has been developing dedicated optimizations of its phase filled by the non-singular contribution (NNLOJET minus space generation based on the work in [70]. This ensures SCET) demonstrated in the bottom panel of Fig. 1. Re- sufficient coverage in the multiply unresolved regions re- markably, the agreement starts for qT at about 2 GeV quired for the qT -subtraction. and extends down to 0.32 GeV for each perturbative order. Numerical uncertainties from phase space integra- tions are displayed as error bars. We emphasize that RESULTS the observed agreement is highly non-trivial, and provides a very strong support to the correctness of both Applying the qT -subtraction method described above, the NNLOJET and SCET predictions. we compute Drell-Yan lepton pair production to N3LO In Fig. 2, we display the N3LO QCD corrections to accuracy. For the phenomenological analysis, we restrict the total cross section for Drell-Yan production through ourselves to the production of a di-lepton pair through a a virtual photon, using the qT -subtraction procedure, de- virtual photon only. We take ECM = 13 TeV as center composed into different partonic channels. The cross of mass collision energy and fix the invariant mass of section is shown as a function of the unphysical cut- cut the di-lepton pair at Q = 100 GeV. Central scales of off parameter qT , which separates resolved and un- renormalization (µR) and factorization (µF ) are taken at resolved contributions. Integrated over qT , both the 3 Q, allowing us to compare with the N LO total cross NNLOJET and SCET predictions involve logarithms up 6 cut section results from [14]. We use the central member of to ln (Q/qT ), which become explicit in the SCET cal- PDF4LHC15_nnlo PDFs [89] throughout the calculation. culation. The NNLOJET calculation produces the same cut To establish the cancellation of qT -dependent terms large logarithms but with opposite sign, as well as power cut m n cut between resolved and unresolved contributions, Fig. 1 suppressed logarithms (qT ) ln (Q/qT ), where m ≥ 2 3 displays the qT distribution of virtual photon obtained and n ≤ 6. The physical N LO total cross section con- cut with NNLOJET (used for the resolved contribution) and tribution must not depend on the unphysical cutoff qT , 4

Fixed Order σpp→γ∗ (fb) * SCET+NNLOJET pp to s = 13 TeV +34.06 LO 339.62−37.48 5 +10.84 NLO 391.25−16.62 0 +3.06 NNLO 390.09−4.11 ]

b 3 +2.64

f 5 [ N LO 382.08−3.09 from [14] y l

n 3

o 10 N LO only qT -subtraction Results from [14] O L 3

N 15 qg −15.32(32) −15.29

d qT sub. total Inclusive qg qq¯ + qQ¯ +5.08(11) +4.97 20 Inclusive total qT sub. qq + qQ PDF4LHC15 nnlo Inclusive qq + qQ qT sub. qq + qQ 7-point scale variation gg +2.17(6) +2.12 25 Inclusive qq + qQ qT sub. gg Inclusive gg F = R = 100 GeV qT sub. qg qq + qQ +0.09(13) +0.17 30 2 Total −7.98(36) −8.03

1 TABLE I: Inclusive cross sections with up to N3LO 0 Ratio to QCD corrections to Drell-Yan production through inclusive total 100 101 102 3 qcut [GeV] a virtual photon. N LO results are from the qT - T cut subtraction method (qT = 0.63 GeV) and from the analytic calculation in [14]. Cross sections at central FIG. 2: Inclusive N3LO QCD corrections to total scale of Q = 100 GeV are presented together with cross section for Drell-Yan production through a vir- 7-point scale variation. Numerical integration errors tual photon. In the bottom panel we plot the ratio to from q -subtraction are indicated in brackets. the analytic calculation in [14]. T

* cut SCET+NNLOJET pp to s = 13 TeV therefore it is important to choose a suﬃciently small qT 110.0 to suppress such power corrections. LO NNLO 107.5 NLO N3LO Fig. 2 demonstrates the SCET+NNLOJET predictions 105.0

cut ]

being independent on q for values below 1 GeV. In b T f 102.5 [ |

fact, for all partonic channels except qg, the cross section * 100.0 y predictions become flat and therefore reliable already at | d cut / 97.5 q ∼ 5 GeV. It is only the qg channel that requires a PDF4LHC15 nnlo T d 7-point scale variation cut 95.0 much smaller qT , indicating more sizeable power cor- F = R = 100 GeV rections than in other channels. A more detailed under- 92.5 standing of this feature could become useful when apply- 90.0 1.02 cut cut cut qT = 0.75 GeV qT = 1.0 GeV qT = 1.5 GeV ing qT -subtraction to more complicated final states. Also shown in the upper panel of Fig. 2 in dashed 1.00 0.98 NNLO lines are the inclusive predictions from [14], decomposed Ratio to 0.96 into different partonic channels. We observe an excellent 0.0 0.5 1.0 1.5 2.0 2.5 3.0 |y * | agreement at small-qT region with a detailed compari- son given in Tab. I. This agreement provides a fully independent confirmation of the analytic calculation [14], FIG. 3: Di-lepton rapidity distribution from LO to 3 and lends strong support to the correctness for our qT - N LO. The colored bands represent theory uncer- subtraction-based calculation. In the bottom panel of tainties from scale variations. The bottom panel is Fig. 2, we plot the ratio between different partonic chan- the ratio of the N3LO prediction to NNLO, with dif- 3 cut nels to the total inclusive N LO corrections. We ob- ferent cutoff qT . serve large cancellation between qg channel (blue) and qq¯ channel (orange). While the inclusive N3LO correction is about −8 fb, the qg channel alone can be as large dictions of increasing perturbative orders up to N3LO as −15.3 fb. Similar cancellations between qg and qq¯ are displayed. We estimate the theory uncertainty band channel can already be observed at NLO and NNLO. on our predictions by independently varying µR and µF The numerical smallness of the NNLO corrections (and around 100 GeV with factors of 1/2 and 2 while elimi- of its associated scale uncertainty) is due to these cancel- nating the two extreme combinations (7-point scale vari- lations, which may potentially lead to an underestimate ation). With large QCD corrections from LO to NLO, of theory uncertainties at NNLO. the NNLO corrections are only modest and come with In Fig. 3 we show for the first time the N3LO pre- scale uncertainties that are significantly reduced [5, 7, 8]. dictions for the Drell-Yan di-lepton rapidity distribution, However, as has been observed for the total cross sec- which constitutes the main new result of this letter. Pre- tion, the smallness of NNLO corrections is due to cancel- 5 lations between the qg and qq¯ channels. Indeed, Fig. 3 over a direct calculation in qT -subtraction, which poten- 3 cut shows clearly that the N LO correction is large compared tially suffers from large power corrections in qT once with NNLO, and that the NNLO scale uncertainty band fiducial cuts are applied [93, 94]. fails to overlap with N3LO over the full rapidity range. For total Drell-Yan cross section, our results are in ex- It should however be noted that the uncertainties from cellent agreement with a previous calculation [14]. We PDFs, especially from the missing N3LO effects in their found that N3LO corrections are significant over the full evolution, can be at the per-cent level [14], which high- rapidity region. They are largely rapidity-independent, lights the necessity for a consistent PDF evolution and indicating only very small corrections to distributions extraction at N3LO in the future. that are normalized to the total cross section. Moreover, In the bottom panel of Fig. 3 we show the ratio perturbative uncertainties estimated from scale variation of N3LO rapidity distribution to the previously known do not overlap between NNLO and N3LO, indicating an NNLO result [7, 8]. As can be seen, the corrections are underestimate of perturbative uncertainties at NNLO. about −2% of the NNLO results, and are flat over a To apply our results in precision phenomenology, one large rapidity range. There is minimal overlap between needs to supplement them by contributions from Z boson the scale uncertainty bands only at large yγ∗ . To test exchange and Z-photon interference. They give rise to 3 cut the numerical stability at N LO, three values of qT are new subprocesses which are infrared finite in the small cut 3 examined in the bottom panel. We observe the qT de- qT limit, and therefore one can apply the N LO qT - pendence to be smaller than the numerical error, which subtraction method without further modification. It will cut justifies to use predictions with qT = 1 GeV in the top also be important to combine the QCD results with elec- 3 panel. Since the N LO corrections are largely rapidity- troweak corrections and mixed electroweak-QCD correc- independent, their effect will cancel out in the normalized tions. We leave these studies to future work. rapidity distribution, which can thus be expected to be Acknowledgements. The authors would like to thank described theoretically to sub-per-cent accuracy, thereby Claude Duhr, Falko Dulat and Bernhard Mistlberger for meeting the precision requirements of the experimental discussions and Julien Baglio for providing detailed re- measurements for normalized distributions in the Drell- sults for the predictions in [14]. This research was sup- Yan process [90, 91]. ported in part by the Swiss National Science Foundation (SNF) under contract 200020-175595 and by the Swiss National Supercomputing Centre (CSCS) under project CONCLUSION AND OUTLOOK ID UZH10. H.X.Z. was supported by the National Sci- ence Foundation of China (NSFC) under contract No. In this letter we calculated for the first time the 11975200. E.W.N.G. was supported by the U.K. STFC di-lepton rapidity distribution for Drell-Yan production through grant ST/P001246/1. through virtual photon exchange to third order in perturbative QCD. We employed the qT -subtraction method at N3LO, by combining results from NNLO Drell-Yan production at large qT and leading-power factorised pre- ∗ dictions from SCET at small q . 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