W + W − production at the LHC: fiducial cross sections and distributions in NNLO QCD

Journal of High Energy Physics, Aug 2016

We consider QCD radiative corrections to W + W − production at the LHC and present the first fully differential predictions for this process at next-to-next-to-leading order (NNLO) in perturbation theory. Our computation consistently includes the leptonic decays of the W bosons, taking into account spin correlations, off-shell effects and non-resonant contributions. Detailed predictions are presented for the different-flavour channel \( pp\to {\mu}^{+}{e}^{-}{\nu}_{\mu }{\overline{\nu}}_e+X \) at \( \sqrt{s}=8 \) and 13 TeV. In particular, we discuss fiducial cross sections and distributions in the presence of standard selection cuts used in experimental W + W − and H → W + W − analyses at the LHC. The inclusive W + W − cross section receives large NNLO corrections, and, due to the presence of a jet veto, typical fiducial cuts have a sizeable influence on the behaviour of the perturbative expansion. The availability of differential NNLO predictions, both for inclusive and fiducial observables, will play an important role in the rich physics programme that is based on precision studies of W + W − signatures at the LHC.

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W + W − production at the LHC: fiducial cross sections and distributions in NNLO QCD

Received: May production at the LHC: ducial cross sections and distributions in NNLO QCD Massimiliano Grazzini 0 1 3 6 Stefan Kallweit 0 1 3 4 5 Stefano Pozzorini 0 1 3 5 6 Dirk Rathlev 0 1 2 3 Marius Wiesemann 0 1 3 6 Zurich 0 1 3 Switzerland 0 1 3 0 D-22607 Hamburg , Germany 1 University of California , Santa Barbara, CA 93106 , U.S.A 2 Theory Group, Deutsches Elektronen-Synchrotron , DESY 3 Johannes Gutenberg University , D-55099 Mainz , Germany 4 PRISMA Cluster of Excellence, Institute of Physics 5 Kavli Institute for Theoretical Physics 6 Physik-Institut, Universitat Zurich 7 (Higgs cuts) @ 8 TeV We consider QCD radiative corrections to W +W QCD Phenomenology - W W and H ! W +W e + X at p production at the LHC and present the rst fully di erential predictions for this process at next-to-next-to-leading order (NNLO) in perturbation theory. Our computation consistently includes the leptonic decays of the W bosons, taking into account spin correlations, o -shell e ects and nonresonant contributions. Detailed predictions are presented for the di erent- avour channel s = 8 and 13 TeV. In particular, we discuss ducial cross sections and distributions in the presence of standard selection cuts used in experimental W +W analyses at the LHC. The inclusive W +W cross section receives large NNLO corrections, and, due to the presence of a jet veto, typical ducial cuts have a sizeable in uence on the behaviour of the perturbative expansion. The availability of di erential NNLO predictions, both for inclusive and ducial observables, will play an important role in the rich physics programme that is based on precision studies of W +W signatures at the LHC. 1 Introduction 2 Description of the calculation 3 Results 4 Summary 1 Introduction 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 contamination through single-top and tt production The production of W -boson pairs is one of the most important electroweak (EW) processes at hadron colliders. Experimental studies of W +W production play a central role in precision tests of the gauge symmetry structure of EW interactions and of the mechanism of EW symmetry breaking. The W +W cross section has been measured at the Tevatron [ 1, 2 ] and at the LHC, both at 7 TeV [3, 4] and 8 TeV [5{8]. The dynamics of W -pair production is of great interest, not only in the context of precision tests of the Standard Model, but also in searches of physics beyond the Standard Model (BSM). Any small anomaly in the production rate or in the shape of distributions could be a signal of new physics. In particular, due to the high sensitivity to modi cations of the Standard Model trilinear gauge couplings, W +W measurements are a powerful tool for indirect BSM searches via anomalous couplings [3, 4, 6, 8, 9]. Thanks to the increasing reach in transverse momentum, Run 2 of the LHC will considerably tighten the present bounds on anomalous couplings. Final states with W -boson pairs are widely studied also in the context of direct BSM searches [10]. In Higgs-boson studies [11{16], W +W production plays an important role as irreducible background in the H ! W +W channel. Such measurements are mostly based on nal states with two leptons and two neutrinos, which provide a clean experimental a consequence, it is not possible to extract the irreducible W +W signature, but do not allow for a full reconstruction of the H ! W +W resonance. As background from data with a simple side-band approach. Thus, the availability of precise theory predictions { 1 { for the W +W background is essential for the sensitivity to H ! W +W and to any BSM particle that decays into W -boson pairs. In the context of Higgs studies, the o shell treatment of W -boson decays is of great relevance, both for the description of the signal region below the W +W threshold, and for indirect determinations of the Higgs-boson width through signal-background interference e ects at high invariant The accurate description of the jet activity is another critical aspect of Higgs measurements, and of W +W measurements in general. Such analyses typically rely on a rather strict jet veto, which suppresses the severe signal contamination due to the tt background, but induces potentially large logarithms that challenge the reliability of xed-order predictions in perturbation theory. All these requirements, combined with the ever increasing accuracy of experimental measurements, call for continuous improvements in the theoretical description of W +W production. Next-to-leading order (NLO) QCD predictions for W +W production at hadron colliders have been available for a long time, both for the case of stable W -bosons [20, 21] and with spin-correlated decays of vector bosons into leptons [22{25]. Recently, also the NLO EW corrections have been computed [26{28]. Their impact on inclusive cross sections hardly exceeds a few percent, but can be strongly enhanced up to several tens of percent at transverse momenta of about 1 TeV. Given the sizeable impact of O( S) corrections, the calculation of higher-order QCD e ects is indispensable in order to reach high precision. The simplest ingredient of pp ! + X at O( S2) is given by the loop-induced gluon-fusion contribution. Due to the strong enhancement of the gluon luminosity, the gg channel was generally regarded as the H ! W +W masses [17{19]. dominant source of NNLO QCD corrections to pp ! W +W tions for gg ! W +W contributions at LO are known also for gg ! W +W at LO have been widely studied [25, 29{32], and squared quark-loop + X in the literature. Predicg [33, 34]. Two-loop helicity amplithe NLO QCD corrections to gg ! tudes for gg ! V V 0 became available in refs. [35, 36], and have been used to compute [37], including all partonic processes with external gluons, while the ones with external quarks are still unknown to date. Calculations at NLO QCD for W +W production in association with one [38{41] and two [42, 43] jets are also important ingredients of inclusive W +W production at NNLO QCD and beyond. The merging of NLO QCD predictions for pp ! W +W + 0; 1 jets1 has been presented in ref. [45]. This merged calculation also consistently includes squared quark-loop + 0; 1 jets in all gluon- and quark-induced channels. contributions to pp ! W +W First NNLO QCD predictions for the inclusive W +W cross section became available in ref. [46]. This calculation was based on two-loop scattering amplitudes for on-shell production, while two-loop helicity amplitudes are now available for all vectorboson pair production processes, including o -shell leptonic decays [47, 48]. In the energy range from 7 to 14 TeV, NNLO corrections shift the NLO predictions for the total cross section by about 9% to 12% [46], which is around three times as large as the gg ! W +W contribution alone. Thus, contrary to what was widely expected, gluon-gluon fusion is not the dominant source of radiative corrections beyond NLO. Moreover, the relatively 1See also [44] for a combination of xed-order NLO predictions for W +W +0,1 jet production. { 2 { large size of NNLO e ects turned out to alleviate the tension that was observed between earlier experimental measurements [5, 7] and NLO QCD predictions supplemented with the loop-induced gluon fusion contribution [25]. In fact, NNLO QCD predictions are in good agreement with the latest measurements of the W +W cross section [6, 8]. Besides perturbative calculations for the inclusive cross section, the modelling of the jet-veto e ciency is another theoretical ingredient that plays a critical role in the comparison of data with Standard Model predictions. In particular, it was pointed out that a possible underestimate of the jet-veto e ciency through the Powheg Monte Carlo [49], which is used to extrapolate the measured cross section from the ducial region to the full phase space, would lead to an arti cial excess in the total cross section [50, 51]. The +e relatively large size of higher-order e ects and the large intrinsic uncertainties of NLO+PS Monte Carlo simulations call for improved theoretical predictions for the jet-veto e ciency. The resummation of logarithms of the jet-veto scale at next-to-next-to-leading logarithmic (NNLL) accuracy was presented in refs. [ 52, 53 ]. Being matched to the pp ! W +W + X cross sections at NLO, these predictions cannot describe the vetoing of hard jets beyond LO accuracy. In order to reach higher theoretical accuracy, NNLL resummation needs to be matched to di erential NNLO calculations. Such NNLL+NNLO predictions have been presented in ref. [ 54 ] for the distribution in the transverse momentum of the W +W system, and could be used to obtain accurate predictions for the jet-veto e ciency through a reweighting of Monte Carlo samples, along the lines of refs. [50, 55]. In this paper we present, for the rst time, fully di erential predictions for W +W production with leptonic decays at NNLO. More precisely, the full process that leads to a nal state with two leptons and two neutrinos is considered, including all relevant o shell and interference e ects in the complex-mass scheme [56]. The calculation is carried out with Matrix,2 a new tool that is based on the Munich Monte Carlo program3 interfaced with the OpenLoops generator of one-loop scattering amplitudes [57, 58], and includes an automated implementation of the qT -subtraction [59] and -resummation [60] formalisms. This widely automated framework has already been used, in combination with the two-loop scattering amplitudes of refs. [48, 61], for the calculations of Z tons and two neutrinos, but in this paper we will focus on the di erent- avour signature e. The impact of QCD corrections on cross sections and distributions will be studied both at inclusive level and in presence of typical experimental selection cuts for W +W measurements and H ! W +W studies. The presented NNLO results for ducial cross sections and for the e ciencies of the corresponding acceptance cuts provide rst insights into acceptance e ciencies and jet-veto e ects at NNLO. 2Matrix is the abbreviation of \Munich Automates qT subtraction and Resummation to Integrate X-sections", by M. Grazzini, S. Kallweit, D. Rathlev and M. Wiesemann. In preparation. 3Munich is the abbreviation of \MUlti-chaNnel Integrator at Swiss (CH) precision" an automated parton level NLO generator by S. Kallweit. In preparation. { 3 { As pointed out in ref. [46], radiative QCD corrections resulting from real bottomquark emissions lead to a severe contamination of W -pair production through top-quark resonances in the W +W b and W +W bb channels. The enhancement of the W +W cross section that results from the opening of the tt channel at NNLO can exceed a factor of ve. It is thus clear that a careful subtraction of tt and single-top contributions is indispensable in order to ensure a decent convergence of the perturbative series. To this end, we adopt a top-free de nition of the W +W cross section based on a complete bottom-quark veto in the four- avour scheme. The uncertainty related with this prescription will be assessed by means of an alternative top-subtraction approach based on the top-quark-width dependence of the W +W cross section in the ve- avour scheme [46]. The manuscript is organized as follows. In section 2 we describe technical aspects of the computation, including the subtraction of resonant top-quark contributions (section 2.1), qT subtraction (section 2.2), the Matrix framework (section 2.3), and the stability of (N)NLO predictions based on qT subtraction (section 2.4). Section 3 describes our numere + X: we present the input parameters (section 3.1), cross sections and distributions without acceptance cuts (section 3.2) and with cuts correspondsignal (section 3.3) and Higgs analyses (section 3.4). The main results are ical results for pp ! +e ing to W +W summarized in section 4. 2 Description of the calculation We study the process pp ! l+l0 l l0 + X; (2.1) including all resonant and non-resonant Feynman diagrams that contribute to the production of two charged leptons and two neutrinos. Depending on the avour of the nal-state leptons, the generic reaction in eq. (2.1) can involve di erent combinations of vector-boson resonances. The di erent- avour nal state l+l0 l l0 is generated, as shown in gure 1 for the qq process at LO, (a) via resonant t-channel W +W production with subsequent W + ! l+ l and W ! l0 l0 decays; (b) via s-channel production in Z( )= ! W W ( ) topologies through a triple-gaugeboson vertex with subsequent W + ! l+ l and W ! l0 l0 decays, where either both W bosons, or the Z boson and one of the W bosons can become simultaneously (c) via Z= production with a subsequent decay Z= ! l lW ! ll0 l l0 . Note that kinematics again allows for a resonant W boson in the decay chain of a resonant Z resonant; boson. Additionally, in the case of equal lepton avours, l = l0, o -shell ZZ production diagrams are involved, as shown in gure 2, where the l+l l l nal state is generated (d) via resonant t-channel ZZ production with Z ! l+l and Z ! l l decays; { 4 { avour case (l 6= l0) and in the same- avour case (l = l0). (e) via further Z ! 4 leptons topologies, Z= ! llZ ! ll l l or Z ! l lZ ! ll l l. Any double-resonant con gurations are kinematically suppressed or excluded by phase-space cuts. process as W +W avour channel pp ! +e production though. Note that the appearance of infrared (IR) divergent equal lepton avours would prevent a fully inclusive phase-space integration. ! l+l splittings in the case of Our calculation is performed in the complex-mass scheme [56], and besides resonances, it includes also contributions from o -shell EW bosons and all relevant interferences; no resonance approximation is applied. Our implementation can deal with any combination of leptonic avours, l; l0 2 fe; ; g. However, in this paper we will focus on the di erente + X. For the sake of brevity, we will often denote this The NNLO computation requires the following scattering amplitudes at O( S2): tree amplitudes for qq ! l+l0 l l0 gg, qq(0) ! l+l0 l l0 q(00)q(000), and crossing-related processes; one-loop amplitudes for qq ! l+l0 l l0 g, and crossing-related processes; squared one-loop amplitudes for qq ! l+l0 l l0 and gg ! l+l0 l l0 ; two-loop amplitudes for qq ! l+l0 l l0 . All required tree-level and one-loop amplitudes are obtained from the OpenLoops generator [57, 58], which implements a fast numerical recursion for the calculation of NLO scattering amplitudes within the Standard Model. For the numerically stable evaluation of tensor integrals we employ the Collier library [67{69], which is based on the Denner-Dittmaier reduction techniques [70, 71] and the scalar integrals of ref. [72]. For the two-loop helicity { 5 { amplitudes we rely on a public C++ library [73] that implements the results of ref. [48], and for the numerical evaluation of the relevant multiple polylogarithms we use the implementation [74] in the GiNaC [75] library. The contribution of the massive-quark loops is neglected in the two-loop amplitudes, but accounted for anywhere else, in particular in the loop-induced gg channel. Based on the size of two-loop contributions with a massless-quark loop, we estimate that the impact of the neglected diagrams with massive-quark loops will be well below the per mille level. 2.1 contamination through single-top and tt production The theoretical description of W +W production at higher orders in QCD is complicated by a subtle interplay with top-production processes, which originates from real-emission channels with nal-state bottom quarks [38, 45, 46]. In the ve- avour scheme (5FS), where bottom quarks are included in the parton-distribution functions and the bottomquark mass is set to zero, the presence of real bottom-quark emission is essential to cancel collinear singularities that arise from g ! bb splittings in the virtual corrections. At the same time, the occurrence of W b pairs in the real-emission matrix elements induces W b resonances that lead to a severe contamination of W +W production. The problem starts with the NLO cross section, which receives a single-resonant tW ! W +W b contribution of about 30% (60%) at 7 (14) TeV. At NNLO, the appearance of doubleresonant tt ! W +W bb production channels enhances the W +W cross section by about a factor of four (eight) [46]. Such single-top and tt contributions arise through the couplings of W bosons to external bottom quarks and enter at the same orders in and S as (N)NLO QCD contributions from light quarks. Their huge impact jeopardises the convergence of the perturbative expansion. Thus, precise theoretical predictions for W +W production require a consistent prescription to subtract the top contamination. In principle, resonant top contributions can be suppressed by imposing a b-jet veto, similarly as in experimental analyses. However, for a b-jet veto with typical pT values of 20 30 GeV, the top contamination remains as large as about 10% [46], while in the limit of a vanishing b-jet veto pT 's the NLO and NNLO W +W singularities associated with massless bottom quarks in the 5FS. cross sections su er from collinear To circumvent this problem, throughout this paper we use the four- avour scheme (4FS), where the bottom mass renders all partonic subprocesses with bottom quarks in the nal state separately nite. In this scheme, the contamination from tt and single-top production is easily avoided by omitting bottom-quark emission subprocesses. However, this prescription generates logarithms of the bottom mass that could have a non-negligible impact on the W +W cross section. In order to assess the related uncertainty, results in the 4FS are compared against a second calculation in the 5FS. In that case, the contributions that are free from top resonances are isolated with a gauge-invariant approach that exploits the scaling behaviour of the cross sections in the limit of a vanishing topquark width [46]. The idea is that double-resonant (single-resonant) contributions depend quadratically (linearly) on 1= t, while top-free W +W contributions are not enhanced at small t . Exploiting this scaling property, the tt, tW and (top-free) W +W components in the 5FS are separated from each other through a numerical t based on multiple { 6 { high-statistics evaluations of the cross section for increasingly small values of t. The subtracted result in the 5FS can then be understood as a theoretical prediction of the genuine cross section and directly compared to the 4FS result. The di erence should be regarded as an ambiguity in the de nition of a top-free W +W cross section and includes, among other contributions, the quantum interference between W +W production (plus unresolved bottom quarks) and tt or single-top production. This ambiguity was shown to be around 1% 2% for the inclusive W +W cross section at NNLO [46], and turns out to be of the same size or even smaller in presence of a jet veto (see section 3). 2.2 The qT -subtraction formalism The implementation of the various IR-divergent amplitudes into a numerical code that provides nite NNLO predictions for physical observables is a highly non-trivial task. In particular, the numerical computations need to be arranged in a way that guarantees the cancellation of IR singularities across subprocesses with di erent parton multiplicities. To this end various methods have been developed. They can be classi ed in two broad categories. In the rst one, the NNLO calculation is organized so as to cancel IR singularities of both NLO and NNLO type at the same time. The formalisms of antenna subtraction [76{ 79], colourful subtraction [80{82] and Stripper [83{85] belong to this category. Antenna subtraction and colourful subtraction can be considered as extensions of the NLO subtraction methods of refs. [86{89] to NNLO. Stripper, instead, is a combination of the FKS subtraction method [86] with numerical techniques based on sector decomposition [90, 91]. The methods in the second category start from an NLO calculation with one additional parton (jet) in the nal state and devise suitable subtractions to make the cross section nite in the region in which the additional parton (jet) leads to further divergences. The qT subtraction method [59] as well as N -jettiness subtraction [92{94], and the Born-projection method of ref. [95] belong to this class. The qT -subtraction formalism [59] has been conceived in order to deal with the production of any colourless4 high-mass system F at hadron colliders. This method has already been applied in several NNLO calculations [46, 59, 62{66, 97{100], and we have employed it also to obtain the results presented in this paper. In the qT -subtraction framework, the pp ! F + X cross section at (N)NLO can be written as d (FN)NLO = H(N)NLO F h d LFO + d (FN+)jLeOt d (CNT)NLO : i (2.2) The term d (FN+)jLeOt represents the cross section for the production of the system F plus one jet at (N)LO accuracy and can be evaluated with any available NLO subtraction formalism. The counterterm d (CNT)NLO guarantees the cancellation of the remaining IR divergences of the F +jet cross section. It is obtained via xed-order expansion from the resummation formula for logarithmically enhanced contributions at small transverse momenta [60]. The practical implementation of the contributions in the square bracket in eq. (2.2) is described in more detail in section 2.3. 4The extension to heavy-quark production has been discussed in ref. [96]. { 7 { process and compensates5 for the subtraction of d (CNT)NLO. It is obtained from the (N)NLO truncation of the process-dependent perturbative function H F = 1 + S H F(1) + S 2 H F(2) + : : : : (2.3) The NLO calculation of d F requires the knowledge of HF(1), and the NNLO calculation also requires HF(2). The general structure of HF(1) has been known for a long time [101]. Exploiting the explicit results of HF(2) for Higgs [102] and vector-boson [103] production, the result of ref. [101] has been extended to the calculation of the NNLO coe cient HF(2) [104]. These results have been con rmed through an independent calculation in the framework of Soft-Collinear E ective Theory [105, 106]. The counterterm d (CNT)NLO only depends on H(N)LO, i.e. for an NNLO computation it requires only HF(1) as input, which F can be derived from the one-loop amplitudes for the Born subprocesses. 2.3 Organization of the calculation in MATRIX Our calculation of W +W production is based on Matrix, a widely automated program for NNLO calculations at hadron colliders. This new tool is based on qT subtraction, and is thus applicable to any process with a colourless high-mass nal state, provided that the two-loop amplitudes for the Born subprocess are available. Moreover, besides xed-order calculations, it supports also the resummation of logarithmically enhanced terms at NNLL accuracy (see ref. [ 54 ], and ref. [107] for more details). Matrix is based on Munich, a general-purpose Monte Carlo program that includes a fully automated implementation of the Catani-Seymour dipole subtraction method [88, 89], an e cient phase-space integration, as well as an interface to the one-loop generator OpenLoops [57, 58] to obtain all required (spin- and colour-correlated) tree-level and one-loop amplitudes. Munich takes care of the bookkeeping of all relevant partonic subprocesses. For each subprocess it automatically generates adequate phase-space parameterizations based on the resonance structure of the underlying (squared) tree-level Feynman diagrams. These parameterizations are combined using a multi-channel approach to simultaneously atten the resonance structure of the amplitudes, and thus guarantee a fast convergence of the numerical integration. Several improvements like an adaptive weight-optimization procedure are implemented as well. Supplementing the fully automated NLO framework of Munich with a generic implementation of the qT -subtraction and -resummation techniques, Matrix achieves NNLL+NNLO accuracy in a way that limits the additionally introduced dependence on F the process to the two-loop amplitudes that enter HNNLO in eq. (2.2). All other processdependent information entering the various ingredients in eq. (2.2) are expressed in terms of NLO quantities already available within Munich+OpenLoops. 5More precisely, while the behaviour of d (CNT)NLO for qT ! 0 is dictated by the singular structure of d (FN+)jLeOt, its non-divergent part in the same limit is to some extent arbitrary, and its choice determines the F explicit form of H(N)NLO. { 8 { All NNLO contributions with vanishing total transverse momentum qT of the nalF state system F are collected in the coe cient HNNLO. The remaining part of the NNLO cross section, namely the di erence in the square bracket in eq. (2.2), is formally nite in the limit qT ! 0, but each term separately exhibits logarithmic divergences in this limit. Since the subtraction is non-local, a technical cut on qT is introduced in order to render both terms separately nite. In this way, the qT -subtraction method works very similarly to a phase-space slicing method. In practice, it turns out to be more convenient to use a cut, rcut, on the dimensionless quantity r = qT =M , where M denotes the invariant mass of the nal-state system F . The counterterm d (CNT)NLO cancels all divergent terms from the real-emission contributions at small qT , implying that the rcut dependence of their di erence should become numerically negligible for su ciently small values of rcut. In practice, as both the counterterm and the real-emission contribution grow arbitrarily large for rcut ! 0, the statistical accuracy of the Monte Carlo integration degrades, preventing one from pushing rcut too low. In general, the absence of any strong residual rcut dependence provides a stringent check on the correctness of the computation since any signi cant mismatch between the contributions would result in a divergent cross section in the limit rcut ! 0. To monitor the rcut dependence without the need of repeated CPU-intensive runs, Matrix allows for simultaneous cross-section evaluations at variable rcut values. The numerical information on the rcut dependence of the cross section can be used to quantify the uncertainty due to nite rcut values (see section 2.4). 2.4 Stability of qT subtraction for +e e production +e In the following we investigate the stability of the qT subtraction approach for pp ! e + X. To this end, in gure 3 we plot the NLO and NNLO cross sections as functions of the qT -subtraction cut, rcut, which acts on the dimensionless variable r = pT; +e e =m +e e . Validation plots are presented at 8 TeV both for the fully inclusive cross section (see section 3.2) and for the most exclusive case we have investigated, i.e. the cross section in presence of standard ducial cuts for Higgs background analyses (see section 3.4). All considered scenarios at 8 and 13 TeV lead essentially to the same conclusions. At NLO the rcut-independent cross section obtained with Catani-Seymour subtraction perturbative correction at NnLO is is used as a reference for the validation of the qT -subtraction result. The comparison of the NLO cross sections in the left panels of gure 3 demonstrates that qT subtraction reaches about half-permille accuracy already at the moderate value of rcut = 1%, where we can, however, still resolve a di erence, which is slightly larger than the respective numerical uncertainties, with respect to the rcut-independent result achieved using Catani-Seymour subtraction. This di erence is due to the power-suppressed contributions that are left after the cancellation of the logarithmic singularity at small rcut. Going to even smaller values of rcut, we observe a perfect convergence within statistical uncertainties towards the Catani-Seymour-subtracted result in the limit rcut ! 0. The expected behaviour of the (n)(rcut) = (n) + f (n)(rcut) n = 1; 2; : : : (2.4) { 9 { +0.70 σ/σNNLO − 1[%] +0.70 σ/σNNLO − 1[%] HJEP08(216)4 +0.08 +0.06 +0.04 +0.02 0 −0.02 −0.04 −0.06 −0.08 −0.10 +0.08 +0.06 +0.04 +0.02 0 −0.02 −0.04 −0.06 −0.08 −0.10 σNNLO σNqTNLO(r) σNNLO σNqTNLO(r) σNCLSO σNqTLO(r) σNCLSO σNqTLO(r) +0.60 +0.50 +0.40 +0.30 +0.20 +0.10 −0.10 −0.20 −0.30 −0.40 −0.50 −0.60 −0.70 0 +0.60 +0.50 +0.40 +0.30 +0.20 +0.10 −0.10 −0.20 −0.30 −0.40 −0.50 −0.60 −0.70 0 rcut, for both NLO (left plots) and NNLO (right plots) results in the inclusive phase space (upper plots) and with Higgs cuts (lower plots). NLO results are normalized to the rcut-independent NLO cross section computed with Catani-Seymour subtraction, and the NNLO results are normalized to their values at rcut ! 0, with a conservative extrapolation-error indicated by the blue bands. where (n) is the rcut-independent result and the function f (n)(rcut) has the general form 2n 1 k=0 f (n)(rcut) = rc2ut X ak;n lnk(rcut) + : : : (2.5) At NNLO, where an rcut-independent control result is not available, we observe no signi cant, i.e. beyond the numerical uncertainties, rcut dependence below about rcut = 1%; we thus use the nite-rcut results to extrapolate to rcut = 0, taking into account the breakdown of predictivity for very low rcut values, and conservatively assign an additional numerical error to our results due to this extrapolation.6 This procedure allows us to control all NNLO predictions to inclusive and ducial cross sections presented in section 3 well below the level of two per mille. The increasing error bars indicate that arbitrarily low rcut values cannot be tested as the contributions cancelling in the limit are separately divergent. Based on the observation that no signi cant rcut dependence is found below rcut = 1%, the value rcut = 0:25% was adopted for the calculation of the di erential observables presented in section 3. We have checked that the total rates for that value are fully consistent within numerical uncertainties with our extrapolated results and that a smaller value rcut = 0:1% leads to distributions in full statistical agreement, thus con rming the robustness of our results also at the di erential level. 6In the NNLO calculation the O( S) contributions are evaluated by using Catani-Seymour subtraction. We present numerical results for the di erent- avour process pp ! +e e + X at s = 8 TeV and 13 TeV. Cross sections and distributions are studied both in the inclusive phase space and in presence of typical selection cuts for W +W Di erent- avour nal states provide the highest sensitivity both in W +W ments and Higgs studies. We note that, due to the charge asymmetry of W +W in proton-proton collisions and the di erences in the muon and electron acceptance cuts (in particular regarding the rapidity cuts), the two di erent- avour channels, analyses. measureproduction +e e and e , do not yield identical cross sections. However, we have checked that the absolute di erences are not resolved on the level of our statistical errors. Thus (N)NLO predictions and K-factors for +e e production can be safely applied also to pp ! e+ e + X. Input parameters, PDFs and selection cuts Results in this paper are based on the EW input parameters G = 1:1663787 10 5 GeV 2 mW = 80:385 GeV and mZ = 91:1876 GeV. The other couplings in the EW sector are derived in the G -scheme, where cos w = mW =mZ and complex-mass scheme, the physical gauge-boson masses and the weak mixing angle are = p 2G m2W sin2 w= . In the replaced by V = q m2V i V mV and cos ^w = W = Z , while for the above real-valued expression is used. For the vector-boson widths we employ W = 2:085 GeV and Z = 2:4952 GeV [108], and for the heavy quarks we set mb = 4:92 GeV and mt = 172:5 GeV. These input parameters result in a branching fraction BR(W ! l l) = 0:1090040 for each massless lepton generation, i.e. l = e; . Contributions from resonant Higgs bosons and continuum are fully supported in our implementation. production as EW signal or as background their interference with the W +W However, since this study is focused on W +W to H ! W +W , Higgs contributions have been decoupled by taking the mH ! 1 limit. To compute hadronic cross sections, we use NNPDF3.0 parton-distribution functions (PDFs) [109], and, unless stated otherwise, we work in the 4FS, while removing all contributions with nal-state bottom quarks in order to avoid any contamination from top-quark resonances. In the NNPDF framework, 4FS PDFs are derived from the standard variableavour-number PDF set with (5F)(MZ ) = 0:118 via appropriate backward and forward s evolution with ve and four active avours, respectively. The resulting values of the strong coupling (4F)(MZ ) at LO, NLO and NNLO are 0.1136, 0.1123 and 0.1123, respectively. s Predictions at NnLO are obtained using PDFs at the corresponding perturbative order and the evolution of S at (n + 1)-loop order, as provided by the PDF set. The central values of the factorization and renormalization scales are set to F = R = mW . Scale uncertainties are estimated by varying F and R in the range 0:5 mW F ; R 2 mW with the restriction 0:5 F = R In the following subsections we investigate +e e production in the inclusive phase space (section 3.2) and in presence of typical selection cuts that are designed for measurements of W +W production (section 3.3) and for H ! W +W studies (section 3.4) at the LHC. The detailed list of cuts is speci ed in table 1. Besides the requirement of two charged leptons within a certain transverse-momentum and rapidity region, they involve cut variable W +W cuts pT;l1 pT;l2 jy j jyej pmiss T T pmiss,rel pT;ll mll Rll ll ll; Njets lepton de nition leptonic cuts > 25 GeV > 20 GeV anti-kT jets with R = 0:4, pT;j > 25 GeV, jyj j < 4:5 momentum of the pmiss T closest lepton; sin j j, where ll; momenta, pT;ll, and pTmiss. signal measurements (central column) and H ! W +W studies (right column). The hardest and second hardest lepton are denoted as l1 and l2, respectively. The missing transverse momentum, pmiss, is identi ed with the total transverse T pair, while the relative missing transverse momentum pmiss,rel is de ned as T is the azimuthal separation between pTmiss and the momentum of the is the azimuthal angle between the vectorial sum of the leptons' transverse additional restrictions on the missing transverse momentum (pTmiss = pT; ), the transverse momentum (pT;ll) and invariant mass (mll) of the dilepton system, the combined rapidity-azimuth ( Rll) and azimuthal ( ll) separation of the charged leptons, as well as on the relative missing transverse momentum (pTmiss,rel) and the azimuthal angle between pT;ll, and pTmiss ( ll; ), as de ned in table 1. Moreover, the W +W and Higgs selection criteria involve a veto against anti-kT jets [110] with R = 0:4, pT > 25 GeV and jyj < 4:5. 3.2 Analysis of inclusive +e e production in absence of acceptance cuts. Predictions for the total inclusive cross section at LO, NLO and NNLO are listed in table 2. The NLO cross section computed with NNLO PDFs, denoted by NLO0, and NLO0 supplemented with the loop-induced gluon-fusion contribution (NLO0+gg) are provided as well. At p s = 8 (13) TeV the NLO corrections increase the LO cross section by 47% (55%), and the NNLO corrections result in a further sizeable shift of +11% (+14%) with respect to NLO. The total NNLO correction can be understood as the sum of three contributions that can be read o table 2: evaluating the cross section up to O( S) with NNLO PDFs increases the NLO result by about 2% (3%). The loop-induced gluon-fusion channel, which used to be considered the dominant part of the NNLO corrections, further raises the cross inclusive [fb] = NLO p s LO respect to NLO. The quoted uncertainties correspond to scale variations as described in the text, and the numerical integration errors on the previous digit(s) are stated in brackets; for the NNLO results, the latter include the uncertainty due the rcut extrapolation (see section 2.4). section by only 3% (4%), while the genuine O( S2) corrections to the qq channel7 amount to about +6% (+7%). Neglecting PDF e ects, we nd that the loop-induced gg contribution corresponds to only 37% (38%) of the total O( S2) e ect, i.e. of NNLO remaining 63% (62%) being due to genuine NNLO corrections. NLO0 , with the These results are in line with the inclusive on-shell predictions of ref. [46], where the relative weight of the gg contribution was found to be 35% (36%), and the small di erence is due to the chosen PDFs. We also nd by up to about 2% larger NNLO corrections than stated in ref. [46], which can also be attributed to the chosen PDF sets. Indeed, repeating the on-shell calculation of ref. [46] using the input parameters of section 3.1 (with W = Z = 0), we nd that the relative corrections agree on the level of the statistical error when the same PDF sets are applied. Moreover, comparing the results of table 2 with this on-shell calculation allows us to quantify the size of o -shell e ects, which turn out to reduce the on-shell result by about 2% with a very mild dependence (at the permille level) on the perturbative order and the collider energy. The results for the two considered collider energies con rm that the size of relative corrections slightly increases with the centre-of-mass energy, as in the on-shell case. We add a few comments on the theoretical uncertainties of the above results. As is well known, scale variations do not give a reliable estimate of the size of missing higherorder contributions at the rst orders of the perturbative expansion. In fact, LO and NLO predictions are not consistent within scale uncertainties, and the same conclusion can be drawn by comparing NLO or NLO0+gg predictions with their respective scale uncertainties to the central NNLO result. This can be explained by the fact that the qg (as well as qg) and gg (as well as qq(0), qq(0) and qq0) channels open up only at NLO and NNLO, respectively. Since the NNLO is the rst order where all the partonic channels are contributing, the NNLO scale dependence should provide a realistic estimate of the 7Here and in what follows, all NNLO corrections that do not stem from the loop-induced gg ! W +W channel are denoted as genuine O( S2 ) corrections or NNLO corrections to the qq channel. Besides qqinduced partonic processes, they actually contain also gq and gq channels with one extra nal-state parton as well as gg, qq(0), qq(0) and qq0 channels with two extra nal-state partons. dσ/dmWW [fb/GeV] µ+e- νµν‾ e(inclusive)@LHC 8 TeV dσ/dmWW [fb/GeV] µ+e- νµν‾ e(inclusive)@LHC 13 TeV acceptance cuts are applied. Absolute LO (black, dotted), NLO (red, dashed) and NNLO (blue, solid) predictions at p s = 8 TeV (left) and p s = 13 TeV (right) are plotted in the upper frames. The lower frames display NLO0+gg (green, dot-dashed) and NNLO predictions normalized to NLO. The bands illustrate the scale dependence of the NLO and NNLO predictions. In the case of ratios, scale variations are applied only to the numerator, while the NLO prediction in the denominator corresponds to the central scale. uncertainty from missing higher-order corrections. The loop-induced gluon-gluon channel, which contributes only at its leading order at O( S2) and thus could receive large relative corrections, was not expected to break this picture due to its overall smallness already in ref. [46]. That conclusion is supported by the recent calculation of the NLO corrections to the loop-induced gg channel [37]. In gures 4{7 we present distributions that characterize the kinematics of the reconstructed W bosons.8 Absolute predictions at the various perturbative orders are complemented by ratio plots that illustrate the relative di erences with respect to NLO. In order to assess the importance of genuine NNLO corrections, full NNLO results are compared to NLO0+gg predictions in the ratio plots. In gure 4 we show the distribution in the total invariant mass, mW +W = m +e e . This observable features the characteristic threshold behaviour around 2 mW , with a rather long tail and a steeply falling cross section in the o -shell region below threshold. Although suppressed by two orders of magnitude, the Z-boson resonance that originates from topologies of type (b) and (c) in gure 1 is clearly visible at m +e e = mZ . Radiative QCD e ects turn out to be largely insensitive to the EW dynamics that governs o -shell W -boson 8The various kinematic variables are de ned in terms of the o -shell W -boson momenta, pW + = p + +p and pW = pe + p e . R T T A A i i w w u u d d o o pT,WW [GeV] (b) applied. Absolute predictions and relative corrections as in gure 4. decays and dictates the shape of the m +e e distribution. In fact, the NNLO= NLO ratio is rather at, and shape distortions do not exceed about 5%, apart from the strongly suppressed region far below the 2 mW threshold. The distribution in the transverse momentum of the W +W pair, shown in gure 5, vanishes at LO. Thus, at non-zero transverse momenta NLO (NNLO) results are formally only LO (NLO) accurate. Moreover, the loop-induced gg channel contributes only at pT;W W = 0. The relative NNLO corrections are consistent with the results discussed in ref. [ 54 ]: they are large and exceed the estimated scale uncertainties in the small and intermediate transverse-momentum regions, while the NLO and NNLO uncertainty bands overlap at large transverse momenta. At very low pT , the xed-order NNLO calculation diverges, but NNLL+NNLO resummation [ 54 ] can provide accurate predictions also in that region. In gures 6 and 7 the transverse-momentum distributions of the harder W boson, pT;W1 , and the softer W boson, pT;W2 , are depicted. The rst eye-catching feature is the large NLO/LO correction in case of the harder W boson, which grows with pT and leads to an enhancement by a factor of ve at pT 500 GeV, whereas such large corrections are absent for the softer W boson. This feature is due to the fact that the phase-space region with at least one hard W boson is dominantly populated by events with the NLO jet recoiling against this W boson, while the other W boson is relatively soft. The LO-like nature of this dominant contribution for moderate and large values of pT;W1 is re ected by the large NLO scale band. The phase-space region where the softer W boson has moderate or high transverse momentum as well is naturally dominated by topologies with the two W bosons recoiling against each other. Such topologies are present already at LO, and thus do not result in exceptionally large corrections. Both for the leading and subleading W 10-1 10-2 10-3 1.4 1.3 1.2 1.1 1 0.9 0.8 0 101 100 10-1 10-2 10-3 1.4 1.3 1.2 1.1 1 0.9 0.8 0 pT,W2 [GeV] NLO NNLO NLO'+gg LO NLO NNLO NLO'+gg 101 X RIT 100 dσ/dpT,W1 [fb/GeV] µ+e- νµν‾ e(inclusive)@LHC 8 TeV dσ/dpT,W1 [fb/GeV] µ+e- νµν‾ e(inclusive)@LHC 13 TeV 100 200 300 400 500 0 100 200 300 400 500 LO NLO NNLO NLO'+gg LO NLO NNLO X I R T h t d e c o r p I R t d e c o r p acceptance cuts are applied. Absolute predictions and relative corrections as in gure 4. dσ/dpT,W2 [fb/GeV] µ+e- νµν‾ e(inclusive)@LHC 8 TeV dσ/dpT,W2 [fb/GeV] µ+e- νµν‾ e(inclusive)@LHC 13 TeV 100 200 300 400 500 0 100 200 300 400 500 acceptance cuts are applied. Absolute predictions and relative corrections as in gure 4. pT,W2 [GeV] (b) A NLO'+gg boson, the NNLO corrections tend to exceed the NLO scale band at moderate transversemomentum values. For all distributions discussed so far, we nd qualitatively the same e ects at 8 and 13 TeV, essentially only di ering by the larger overall size of the NNLO corrections at the higher collider energy. Contributing only about one third of the total NNLO correction, the NLO0+gg approximation does not provide a reliable description of the full NNLO result. Moreover, in general the loop-induced gluon-gluon channel alone cannot reproduce the correct shapes of the full NNLO correction. In this section we investigate the behaviour of radiative corrections in presence of acceptance cuts used in W +W measurements. The full set of cuts is summarized in table 1 and is inspired by the W +W analysis of ref. [6].9 Besides various restrictions on the leptonic degrees of freedom and the missing transverse momentum, this analysis implements a jet veto. Predictions for ducial cross sections at di erent perturbative orders are reported in very di erently as compared to the inclusive case. The NLO corrections with respect to LO amount to only about +4% (+1%) at 8 (13) TeV. Neglecting the +2% (+3%) shift due to the PDFs, the NNLO corrections amount to +5% (+7%). Their positive impact is, however, entirely due to the loop-induced gluon-fusion contribution, which is not a ected by the jet veto. In fact, comparing the NNLO and NLO0+gg predictions we see that the genuine O( S2) corrections are negative and amount to roughly 1% ( 2%). The reduction of the impact of radiative corrections when a jet veto is applied is a well-known feature in perturbative QCD calculations [111]. A stringent veto on the radiation recoiling against the W +W system tends to unbalance the cancellation between positive real and negative virtual contributions, possibly leading to large logarithmic terms. The resummation of such logarithms has been the subject of intense theoretical studies, especially in the important case of Higgs-boson production [112{115], and it has been recently addressed also for W +W production [ 52, 53 ]. In the case at hand, the moderate size of radiative e ects beyond NLO suggests that, similarly as for Higgs production, xedorder NNLO predictions should provide a fairly reliable description of jet-vetoed ducial cross sections and distributions. The reduced impact of radiative e ects in the presence of a jet veto is often accompanied by a reduction of scale uncertainties in xed-order perturbative calculations. Comparing the results in table 3 with those in table 2 we indeed see that the size of the NNLO scale uncertainty is reduced when cuts, particularly the jet veto, are applied. Such a small scale dependence should be interpreted with caution as it tends to underestimate the true uncertainty due to missing higher-order perturbative contributions. The e ect of radiative corrections on the e ciency of W +W ducial cuts, = ducial= inclusive ; (3.1) 9We do not apply any lepton-isolation criteria with respect to hadronic activity. 1:3% p s LO NLO NLO' di erences with respect to NLO. Scale uncertainties and errors as in table 2. acceptance cuts at di erent perturbative orders and relative di erences with respect to NLO. Scale uncertainties and errors as in table 2. is illustrated in table 4, where numerator and denominator are evaluated at the same perturbative order and both with R = F = mW . Due to the negative impact of the newly computed NNLO corrections on the ducial cross section (see table 3) and their positive impact on the inclusive cross section (see table 2), the NNLO corrections on the cut e ciency are quite signi cant. In particular, at p s = 8 (13) TeV the full NNLO prediction lies about 6% (9%) below the NLO0+gg result. The uncertainties quoted in table 4 are obtained by varying R and F in a fully correlated way in the numerator and denominator of eq. (3.1). Clearly, there is a large correlation at LO, which results in a particularly small uncertainty. At NNLO the uncertainties are comparable to those of the ducial cross sections. As discussed in section 2.1, our default 4FS predictions are compared to an alternative obtain NLO = 165:7(3) fb and NNLO = 181:9(4) fb at p top-subtracted computation in the 5FS, in order to assess the uncertainty related to the prescription for the subtraction of the top contamination. Without top subtraction we s = 8 TeV in the 5FS. Due to the jet veto, these ducial cross sections feature a moderate top contamination of about 8% at NLO and 12% at NNLO. Removing the top contributions, we nd NLO = 153:4(4) fb and NNLO = 162:5(3) fb, which agree with the 4FS results within 1%. At p s = 13 TeV, the top contamination in the 5FS is somewhat larger and amounts to 12% (17%) at NLO (NNLO). The top-subtracted ducial cross sections, NLO = 238:3(6) fb and NNLO = 265(2) fb, on the other hand, are again in agreement with the 4FS results at the 1% LO NLO NNLO 200 dσ/dΔϕll [fb] 0.5 1 2 2.5 3 0.5 1 2 2.5 3 R R T i w w d d e e c c u u d d o o r r p p X I M h 1.5 Δϕll (b) applied. Absolute predictions and relative corrections as in gure 4. Di erential distributions in presence of W +W ducial cuts are presented in gures 8{15. We rst consider, in gure 8, the distribution in the azimuthal separation of the charged leptons, full NNLO result at small ll. The NLO0+gg approximation is in good agreement with ll, but in the peak region the di erence exceeds 5%, and the NLO0+gg result lies outside the NNLO uncertainty band. The di erence signi cantly increases in the large ll region, where the cross section is strongly suppressed though. The uncertainty bands of the NLO and NNLO predictions do not overlap. This feature is common to all distributions that are considered in the following. It is primarily caused by the loop-induced gg contribution, which enters only at NNLO and is not accounted for by the NLO scale variations. Ignoring the gluon-induced component, we observe a good perturbative convergence, apart from some peculiar phase-space corners. In gure 9 we study the cross section as a function of the azimuthal separation ll; between the transverse momentum of the dilepton pair (pT;ll) and the missing transverse ll; = at LO, the (N)NLO calculation is only (N)LO < . The NNLO corrections have a dramatic impact on the shape of momentum (pTmiss). Since accurate at ll; to O(10) in the region the distribution: the NNLO= NLO K-factor grows with decreasing ll; and reaches up ll; . 1, where the cross section is suppressed by more than three orders of magnitude. This huge e ect results from the interplay of the jet veto with the cuts on the pT 's of the individual leptons and on pTmiss. At small momenta pT;ll and pTmiss must be balanced by recoiling QCD partons. However, at NLO the emitted parton can deliver a sizeable recoil only in the region that is not subject to the ll; the transverse jet veto, i.e. in the strongly suppressed rapidity range jyj j > 4:5. At NNLO, the presence of a second parton relaxes this restriction to some extent, thereby reducing the suppression 1.5 Δϕll,νν (b) ton system and the missing transverse momentum. W +W cuts are applied. Absolute predictions and relative corrections as in gure 4. dσ/dmll [fb/GeV] µ+e- νµν‾ e(WW-cuts )@LHC 8 TeV dσ/dmll [fb/GeV] µ+e- νµν‾ e(WW-cuts )@LHC 13 TeV LO NLO NNLO X I R R T LO NLO NNLO LO (b) 104 dσ/dΔϕll,νν [fb] 101 100 10-1 10-2 101 100 100 10-1 10-2 1.4 1.3 1.2 1.1 1 0.9 0 I R T u d r p M h c u d predictions and relative corrections as in gure 4. HJEP08(216)4 103 RIT 102 1.4 1.3 1.2 1.1 1 0.9 d d o o LO NLO NNLO T t i d e c u r p NLO'+gg 10-2 10-3 10-4 1.5 1.4 mTATLAS [GeV] NLO0+gg predictions at to the PDFs. transverse mass. W +W cuts are applied. Absolute predictions and relative corrections as in gure 4. by about one order of magnitude. The loop-induced gg contribution does not involve any QCD radiation and contributes only at ll; = . As a consequence, the NLO and ll; are almost identical, apart from minor di erences due The invariant-mass distribution of the dilepton pair is presented in gure 10. On the one hand, if one takes into account NNLO scale variations, the NLO0+gg result is by and large consistent with the NNLO prediction. On the other hand, the shapes of the NLO0+gg and NNLO distributions feature non-negligible di erences, which range from +5% at low masses to 5% in the high-mass tail. Nevertheless, NLO0+gg provides a reasonable approximation of the full NNLO result, in particular regarding the normalization. The distribution in the W +W transverse mass, mTATLAS = q ET;l1 + ET;l2 + pTmiss 2 pT;l1 + pT;l2 + pTmiss 2 ; (3.2) is displayed in gure 11. Also in this case, apart from the strongly suppressed region of small mTATLAS, the NLO0+gg approximation is in quite good agreement with the full NNLO prediction. In gures 12 and 13 we show results for the pT distributions of the leading and subleading lepton, respectively. In both cases the impact of NNLO corrections grows with pT . This is driven by the gluon-induced contribution, which overshoots the complete NNLO result in the small-pT region and behaves in the opposite way as pT becomes large. In the case of the subleading lepton, the genuine NNLO corrections are as large as O(10%) around pT;l2 = 200 GeV. Overall, there is a visible di erence in shape between NLO0+gg and NNLO for both the leading and subleading lepton transverse-momentum distributions. 100 10-1 10-2 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0 101 100 10-1 10-2 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0 LO NLO NNLO I R T i w d r p I R t i c u d 101 dσ/dpT,l1 [fb/GeV] µ+e- νµν‾ e(WW-cuts )@LHC 13 TeV pT,l2 [GeV] 100 (a) LO NLO NNLO NLO'+gg LO NLO NNLO RT 100 I i dw 10-1 (b) cuts are applied. Absolute w d d e e o o r r p p predictions and relative corrections as in gure 4. dσ/dpT,l2 [fb/GeV] µ+e- νµν‾ e(WW-cuts )@LHC 8 TeV dσ/dpT,l2 [fb/GeV] µ+e- νµν‾ e(WW-cuts )@LHC 13 TeV 50 150 200 0 50 150 200 (b) cuts are applied. Absolute predictions and relative corrections as in gure 4. 100 10-1 10-2 10-3 10-1 10-2 10-3 1.8 1.6 1.4 1.2 1 0.8 0 LO NLO NNLO NLO'+gg LO NLO NNLO X 100 I R R T predictions and relative corrections as in gure 4. 101 dσ/dpTmiss [fb/GeV] 101 dσ/dpTmiss [fb/GeV] µ+e- νµν‾ e(WW-cuts )@LHC 13 TeV 101 dσ/dpT,ll [fb/GeV] µ+e- νµν‾ e(WW-cuts )@LHC 13 TeV (b) cuts are applied. Absolute R R t i i w w u u d d o o LO NLO NNLO NLO'+gg LO NLO NNLO X I d e c r p X I d e c r p 50 150 200 0 50 150 200 (b) cuts are applied. Absolute predictions and relative corrections as in gure 4. s = 13 TeV in presence of W +W The pT distribution of the dilepton pair is displayed in gure 14. This observable has a kinematical boundary at LO, where the requirement pTmiss > 20 GeV implies that pT;ll > 20 GeV. The region pT;ll < 20 GeV starts to be populated at NLO, but each perturbative higher-order contribution (beyond LO) produces integrable logarithmic singularities leading to perturbative instabilities at the boundary [116]. This becomes particularly evident in the d NNLO=d NLO ratio. The loop-induced gg contribution, having Born-like kinematics, does not contribute to the region pT;ll < 20 GeV. In contrast, NNLO corrections are huge, and the formal accuracy of NNLO predictions is only NLO in that region. In the region of high pT;ll we observe signi cant NNLO corrections, and the NLO0+gg approximation works rather well. Similar features are observed in the pmiss distribution, displayed in gure 15, but without the perturbative instability at pTmiss = 20 GeV, as the cut on pmiss is explicit. T In general, radiative corrections behave in a rather similar way at p s = 8 TeV and T cuts. Comparing the NLO0+gg approximation with the full NNLO prediction, we nd that the overall normalization is typically reproduced quite well, while genuine NNLO corrections can lead to signi cant shape di erences of up to 10%. It does not come as a surprise that in kinematic regions that imply the presence of QCD radiation, loop-induced gg contributions cannot provide a reasonable approximation of the full NNLO correction. e production with Higgs selection cuts In this section we repeat our study of radiative corrections in presence of cuts that are designed for H ! W +W studies at the LHC. In this case, W +W production plays the role of irreducible background, and more stringent cuts are applied in order to minimize its impact on the H ! W +W corresponds to the H ! W +W of cuts similar to the ones used in W +W suppression of on-shell W +W pT;ll, mll, ll and ll; . signal. The precise list of cuts is speci ed in table 1 and analysis of ref. [12].10 This selection implements a series signal measurements, including a jet veto. The production is achieved through additional restrictions on In table 5 we report predictions for ducial cross sections at di erent perturbative orders. The corresponding acceptance e ciencies, computed as in section 3.3, are presented in table 6. It turns out that Higgs cuts suppress the impact of QCD radiative e ects in a similar way as W +W cuts. At 8 (13) TeV the NLO and NNLO corrections amount to +5% (+3%) and to +9% (+13%), respectively. The latter consist of a positive +3% shift due to NNLO PDFs, a sizeable loop-induced gg component of +9% (+13%), and a rather small genuine O( S2) contribution of 2% ( 4%). at p We compare the 4FS predictions against the top-subtracted calculation in the 5FS: s = 8 (13) TeV the latter yields NLO = 48:7 (3) fb ( NLO = 73:4 (2) fb) and NNLO = 53:0 (5) fb ( NNLO = 83:1 (5) fb), which corresponds to a 1% 2% agreement with the 4FS results. The size of the subtracted top contamination in the 5FS is slightly smaller than what was found for W +W cuts. It amounts to 5% (9%) at NLO and 6% (11%) at NNLO. 10In our analysis, we require jy j < 2:4 as for W +W signal cuts, in contrast to jy j < 2:5 in the ATLAS analysis. Moreover, we do not apply any lepton-isolation criteria with respect to hadronic activity. HJEP08(216)4 s LO NLO NLO' ducial cuts at di erent perturbative orders and relative di erences with respect to NLO. Scale uncertainties and errors as in table 2. = ducial(H cuts)= inclusive = NLO 1 s LO 8 TeV with respect to NLO. Scale uncertainties and errors as in table 2. Similarly to the case of W +W impact on the acceptance e ciency: at p cuts, genuine O( S2) corrections have a signi cant s = 8 (13) TeV the NNLO prediction lies roughly 8% (10%) below the NLO0+gg result, which exceeds the respective scale uncertainties. While the relative size of higher-order e ects on the Higgs-cut e ciency is almost identical to the one found for W +W selection cuts, the absolute size of the acceptance e ciencies is much smaller. In the case of Higgs cuts it is almost a factor of three lower, primarily due to the stringent cut on the invariant mass of the dilepton system. Di erential distributions with Higgs cuts applied are presented in gures 16{23. In general, they behave in a similar way as for the case of W +W cuts discussed in section 3.3. However, a few observables are quite sensitive to the additional cuts that are applied in the Higgs analysis. Most notably, the distribution in the azimuthal separation of the charged leptons in gure 16 exhibits a completely di erent shape as compared to gure 8. In corrections with respect to the NLO distribution at p particular, it features an approximate plateau in the region 0:4 ll 1:2. The NNLO s = 8 (13) TeV range from about +13% (+18%) at small ll to roughly +2% (+5%) at separations close to the ducial cut. The loop-induced gg component provides a good approximation of the complete NNLO result for small separations, but in the large ll region it overshoots the complete NNLO result by about 5% (7%). In the case of W +W ll; distribution, displayed in gure 17, we observe that, similarly to the gure 9), also Higgs cuts lead to huge NNLO corrections at 40 dσ/dΔϕll [fb] I I R R T A M M h t t i i w w d e e c u u d d o r r Absolute predictions and relative corrections as in gure 4. dσ/dΔϕll,νν [fb] µ+e- νµν‾ e(H-cuts )@LHC 8 TeV dσ/dΔϕll,νν [fb] µ+e- νµν‾ e(H-cuts )@LHC 13 TeV LO NLO NNLO NLO'+gg 60 55 50 45 40 35 30 25 1.2 I TRA 101 M h 100 LO NLO NNLO NLO'+gg 103 dσ/dσNLO 102 101 100 Δϕll Δϕll,νν (b) 35 30 25 20 15 1.2 10-1 10-2 10-3 10-4 10-5 102 101 100 t t NLO'+gg R T i w d e c u d o r p 1.6 dilepton system and the missing transverse momentum. Higgs cuts are applied. Absolute predictions and relative corrections as in gure 4. 1.6 dσ/dmll [fb/GeV] 2.5 dσ/dmll [fb/GeV] AT 1.5 M h t i i w w d d d o r r p LO NLO NNLO T A c u NLO'+gg (b) tions and relative corrections as in gure 4. T small ll; . As discussed in section 3.3, this behaviour is due to the fact that at small the leptonic and pmiss cuts require the presence of a sizeable QCD recoil, which is, however, strongly suppressed by the jet veto at NLO. In the Higgs analysis, this suppression mechanism becomes even more powerful due to the additional cut pT;ll > 30 GeV, which forbids the two leptons to recoil against each other. This leads to the kink at ll; = 2:2 in the NLO distribution and to the explosion of NNLO corrections below and slightly above this threshold. The invariant mass of the dilepton system, shown in gure 18, is restricted to the region 10 GeV mll 55 GeV. The peak of the distribution is around mll = 38 GeV, and the NNLO= NLO K-factor is essentially at. Also the NLO0+gg curve has a very similar shape so that the radiative corrections precisely match those on the ducial rates. The distribution in mTATLAS is presented in gure 19. As compared to the W +W analysis (see gure 11), we observe that the tail of the distribution drops signi cantly faster when Higgs cuts are applied. Moreover, in the high-mTATLAS region the size of the loopinduced gg corrections relative to NLO and, hence, the size of the full NNLO correction, is much larger than in the W +W 40% (60%) of the NLO cross section at p when W +W cuts are applied. analysis. The NNLO corrections amount up to about s = 8 (13) TeV, while they hardly exceed 15% The distributions in the lepton pT 's, depicted in gures 20 and 21, behave in a similar way as in gures 12 and 13, apart from a steeper drop-o in the tail and slightly larger corrections. The shape of the pT;l2 distribution can be qualitatively explained as follows. As pT;l2 becomes large, since the dilepton invariant mass is constrained to be smaller than 55 GeV (see table 1), the total transverse momentum of the dilepton system increases. Such large dilepton pT has to be balanced by the missing transverse momentum and by the i w w d d o o X I R t d c u r HJEP08(216)4 100 10-1 10-2 10-3 1.8 mTATLAS [GeV] mTATLAS [GeV] tions and relative corrections as in gure 4. transverse mass. Higgs cuts are applied. Absolute predicrecoiling QCD radiation, whose pT must increase accordingly. At pT;l2 50 GeV the jet of the impact of radiative corrections. This e ect is particularly visible at p veto starts to suppress QCD radiation harder than 25 GeV, thereby leading to a reduction s = 13 TeV, where the available energy is larger. W +W T of W +W For the distributions in the pT of the dilepton pair (see gure 22) and in pmiss (see T gure 23), we also nd a similar behaviour as in the case where W +W cuts are applied. We note, however, that the perturbative instability observed in the pT;ll distribution with gure 14) is removed by the explicit cut pT;ll > 30 GeV in the Higgs analysis. Accordingly, the pT;ll cut implicitly vetoes events with pmiss < 30 GeV at Born T level, which leads to a perturbative instability in the pTmiss distribution, particularly visible in the NNLO= NLO ratio. In fact, it is evident from gure 23 that the phase-space region pmiss < 30 GeV is lled only upon inclusion of higher-order corrections. Similarly to the case cuts, the behaviour of radiative e ects is rather insensitive to the collider energy. Comparing NLO0+gg and full NNLO predictions, in spite of the fairly good agreement at the level of ducial cross section, we observe that the genuine O( S2) corrections lead to signi cant shape distortions at the 10% level. 4 We have presented the rst fully di erential calculation of the NNLO QCD corrections to W +W production with decays at the LHC. O -shell e ects and spin correlations, as well as all possible topologies that lead to a nal state with two charged leptons and two neutrinos are consistently taken into account in the complex-mass scheme. At higher orders (b) tions and relative corrections as in gure 4. 101 dσ/dpT,l2 [fb/GeV] µ+e- νµν‾ e(H-cuts )@LHC 8 TeV 101 dσ/dpT,l2 [fb/GeV] µ+e- νµν‾ e(H-cuts )@LHC 13 TeV 100 10-1 10-2 10-3 10-4 1.8 10-1 10-2 10-3 10-4 1.8 1.6 1.4 1.2 1 LO NLO NNLO LO NLO NNLO R T A i e c p i w e c u r p I I h h d d d d HJEP08(216)4 I I h h d d d d o o LO NLO NNLO LO NLO NNLO R MAT 10-1 wit 10-2 uce 10-3 R MAT 10-1 wit 10-2 euc 10-3 predictions and relative corrections as in gure 4. NLO'+gg pT,l2 [GeV] 40 (b) pT,ll [GeV] (b) predictions and relative corrections as in gure 4. dσ/dpTmiss [fb/GeV] µ+e- νµν‾ e(H-cuts )@LHC 13 TeV I R R h h t t i i c c u u d d predictions and relative corrections as in gure 4. 10-1 10-2 10-3 1.6 1.4 1.2 1 100 10-1 10-2 10-3 1.8 1.6 1.4 1.2 1 0.8 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 pmiss [GeV] T (a) (b) I I R R h h t t i i c c u u d d HJEP08(216)4 1.8 dσ/dσNLO LO NLO NNLO T A M w r p T A M w d e o r p LO NLO NNLO LO NLO NNLO 100 in QCD perturbation theory, the inclusive W +W cross section is plagued by a huge contamination from top-quark production processes, and the subtraction of top contributions is mandatory for a perturbatively stable de nition of the W +W rate. In our calculation, any top contamination is avoided by excluding partonic channels with nal-state bottom quarks in the 4FS, where the bottom-quark mass renders such contributions separately nite. In order to quantify the sensitivity of the top-free W +W cross section on the details of the top-subtraction prescription, our default predictions in the 4FS have been compared to an alternative calculation in the 5FS. In the latter case a numerical extrapolation in the narrow top-width limit is used to separate contributions that involve top resonances from genuine W +W production and its interference with tW and tt production. The comparison of 4FS and 5FS predictions for inclusive and ducial cross sections indicates that the dependence on the top-subtraction prescription is at the 1% Numerical predictions at p di erent- avour channel pp ! s = 8 and 13 TeV have been discussed in detail for the e + X. As compared to the case of on-shell W +W production [46], the inclusion of leptonic decays leads to a reduction of the total cross section that corresponds to the e ect of leptonic branching ratios plus an additional correction of about 2% due to o -shell e ects. The in uence of o -shell W -boson decays on the behaviour of (N)NLO QCD corrections is negligible. In fact, apart from minor di erences due to the employed PDFs, we nd that the relative impact of QCD corrections on the total cross sections is the same as for on-shell W +W production [46]. At p s = 8 (13) TeV, ignoring the shift of +2% (+3%) due to the di erence between NNLO and NLO PDFs, the overall NNLO correction is as large as +9% (+11%), while the loop-induced gluon-gluon contribution amounts to only +3% (+4%); i.e., contrary to what was generally expected in the literature, the NNLO corrections are dominated by genuine NNLO contributions to the qq channel, and the loop-induced gg contribution plays only a subdominant role. The complete calculation of NNLO QCD corrections allows us to provide a rst realistic estimate of theoretical uncertainties through scale variations: as is well-known, uncertainties from missing higher-order contributions obtained through scale variations are completely unreliable at LO and still largely underestimated at NLO. This is due to the fact that the qg (as well as qg) and gg (as well as qq(0), qq(0) and qq0) partonic channels do not contribute at LO and NLO, respectively. In fact, NNLO is the rst order at which all partonic channels contribute. Thus NNLO scale variations, which are at the level of 2% 3% for the inclusive cross sections, can be regarded as a reasonable estimate of the theoretical uncertainty due to the truncation of the perturbative series. This is supported by the moderate impact of the recently computed NLO corrections to the loop-induced gg contribution [37]. Imposing a jet veto has a strong in uence on the size of NNLO corrections and on the relative importance of NNLO contributions from the qq channel and the loop-induced gg channel. This was studied in detail for the case of standard ducial cuts used in W +W and H ! W +W analyses by the LHC experiments. As a result of the jet veto, such cuts signi cantly suppress all (N)NLO contributions that involve QCD radiation, thereby enhancing the relative importance of the loop-induced gg channel at NNLO. More precisely, depending on the analysis and the collider energy, ducial cuts lift the loop-induced gg contribution up to 6% 13% with respect to NLO, whereas the genuine NNLO corrections to the qq channel are negative and range between 1% and 4%, while the NLO corrections vary between +1% and +5%. The reduction of the impact of radiative corrections is accompanied by a reduction of scale uncertainties, which, for the NNLO sections, are at the 1% 2% level. This is a typical side-e ect of jet vetoes, and scale uncertainties are likely to underestimate unknown higher-order e ects in this situation. As a result of the di erent behaviour of radiative corrections to the inclusive and ducial cross sections, their ratios, which determine the e ciencies of acceptance cuts, turn out to be quite sensitive to higher-order e ects. More explicitly, the overall NNLO corrections to the cut e ciency are small and range between 3% and 1%. However, they arise from a positive shift between +3% and +9% due to the loop-induced gg channel, and a negative shift between 6% and 10% from genuine NNLO corrections to the qq channel. The NLO prediction supplemented by the loop-induced gg channel, i.e. the \best" prediction before the complete NNLO corrections were known, would thus lead to a signi cant overestimation of the e ciency, by up to about 10%. Similarly to the case of ducial cross sections, the scale uncertainties of cut e ciencies are at the 1% level, and further studies are needed in order to estimate unknown higher-order e ects in a fully realistic way. This, in particular, involves a more accurate modelling of the jet veto, which is left for future work. Our analysis of di erential distributions demonstrates that, in absence of ducial cuts, genuine NNLO corrections to the qq channel can lead to signi cant modi cations in the shapes of observables that are sensitive to QCD radiation, such as the transverse momentum of the leading W boson or of the W +W system. On the other hand, in presence of ducial cuts, NLO predictions supplemented with the loop-induced gg contribution yield a reasonably good description of the shape of di erential observables, such as dilepton invariant masses and single-lepton transverse momenta. We nd, however, that even for standard W +W and Higgs selection cuts, which include a jet veto, genuine NNLO corrections tend to distort such distributions by up to about 10%. In phase-space regions that imply the presence of QCD radiation, loop-induced gg contributions cannot approximate the shapes of full NNLO corrections. The predictions presented in this paper have been obtained with Matrix, a widely automated and exible framework that supports NNLO calculations for all processes of the class pp ! l+l0 l l0 + X, including in particular also the channels with equal lepton avours, l = l0. More generally, Matrix is able to address fully exclusive NNLO computations for all diboson production processes at hadron colliders. Acknowledgments We thank A. Denner, S. Dittmaier and L. Hofer for providing us with the one-loop tensorintegral library Collier well before publication, and we are grateful to P. Maierhofer and J. Lindert for advice on technical aspects of OpenLoops. 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Massimiliano Grazzini, Stefan Kallweit, Stefano Pozzorini. W + W − production at the LHC: fiducial cross sections and distributions in NNLO QCD, Journal of High Energy Physics, 2016, 140, DOI: 10.1007/JHEP08(2016)140