Next-to-leading-order electroweak corrections to the production of four charged leptons at the LHC

Journal of High Energy Physics, Jan 2017

We present a state-of-the-art calculation of the next-to-leading-order electroweak corrections to ZZ production, including the leptonic decays of the Z bosons into μ + μ −e+e− or μ + μ − μ + μ − final states. We use complete leading-order and next-to-leading-order matrix elements for four-lepton production, including contributions of virtual photons and all off-shell effects of Z bosons, where the finite Z-boson width is taken into account using the complex-mass scheme. The matrix elements are implemented into Monte Carlo programs allowing for the evaluation of arbitrary differential distributions. We present integrated and differential cross sections for the LHC at 13 TeV both for an inclusive setup where only lepton identification cuts are applied, and for a setup motivated by Higgs-boson analyses in the four-lepton decay channel. The electroweak corrections are divided into photonic and purely weak contributions. The former show the well-known pronounced tails near kinematical thresholds and resonances; the latter are generically at the level of ∼ −5% and reach several −10% in the high-energy tails of distributions. Comparing the results for μ + μ −e+e− and μ + μ − μ + μ − final states, we find significant differences mainly in distributions that are sensitive to the μ + μ − pairing in the μ + μ − μ + μ − final state. Differences between μ + μ −e+e− and μ + μ − μ + μ − channels due to interferences of equal-flavour leptons in the final state can reach up to 10% in off-shell-sensitive regions. Contributions induced by incoming photons, i.e. photon-photon and quark-photon channels, are included, but turn out to be phenomenologically unimportant.

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Next-to-leading-order electroweak corrections to the production of four charged leptons at the LHC

Received: November Next-to-leading-order electroweak corrections to the production of four charged leptons at the LHC Benedikt Biedermann 0 1 2 4 5 6 Ansgar Denner 0 1 2 4 5 6 Stefan Dittmaier 0 1 2 4 5 6 Lars Hofer 0 1 2 3 4 5 6 0 08028 Barcelona , Spain 1 79104 Freiburg , Germany 2 97074 Wu ̈rzburg , Germany 3 Institut de Ci`encies del Cosmo (ICCUB) 4 Open Access , c The Authors 5 72076 Tu ̈bingen , Germany 6 [45] B. Biedermann, A. Denner , S. Dittmaier, L. Hofer and B. J ̈ager, Electroweak corrections to and Barbara J¨agerd aInstitut fu¨r Theoretische Physik und Astrophysik, Julius-Maximilians-Universita¨t Wu¨rzburg, Departament de F´ısica Qu`antica i Astrof´ısica (FQA), Universitat de Barcelona (UB), , , bPhysikalisches Institut; Albert-Ludwigs-Universita¨t Freiburg - Abstract: We present a state-of-the-art calculation of the next-to-leading-order electroweak corrections to ZZ production, including the leptonic decays of the Z bosons into order matrix elements for four-lepton production, including contributions of virtual photons and all off-shell effects of Z bosons, where the finite Z-boson width is taken into account using the complex-mass scheme. The matrix elements are implemented into Monte Carlo programs allowing for the evaluation of arbitrary differential distributions. integrated and differential cross sections for the LHC at 13 TeV both for an inclusive setup where only lepton identification cuts are applied, and for a setup motivated by Higgsboson analyses in the four-lepton decay channel. The electroweak corrections are divided into photonic and purely weak contributions. The former show the well-known pronounced tails near kinematical thresholds and resonances; the latter are generically at the level of leptons in the final state can reach up to 10% in off-shell-sensitive regions. Contributions induced by incoming photons, i.e. photon-photon and quark-photon channels, are included, but turn out to be phenomenologically unimportant. ArXiv ePrint: 1611.05338 1 Introduction Details of the calculation Partonic channels Virtual corrections Real corrections Phenomenological results 3.1 Input parameters 2.1 2.2 2.3 Numerical implementation and independent checks of the calculation Definition of observables and acceptance cuts Results on integrated cross sections Results on differential cross sections in the inclusive setup Results on differential cross sections in the Higgs-specific setup Introduction The physics programme of the LHC at Run I was particularly successful in the investigation of electroweak (EW) interactions and culminated in the discovery of a Higgs boson, but no evidence for physics beyond the Standard Model (SM) was found. While the community is looking forward to a major discovery at Run II, an important task is the precise measurement of the properties of the Higgs boson and the other particles of the SM. Small deviations from the predictions of the SM in the observed event rates or distributions might reveal signs of new physics. One class of processes particularly relevant for tests of the EW sector of the SM is EW gauge-boson pair production. These reactions allow to measure the triple gauge-boson couplings and to study the EW gauge bosons in more detail. Moreover, they constitute a background to Higgs-boson production with subsequent decay into weak gauge-boson pairs and to searches for new physics such as heavy vector bosons. In the Higgs-signal region below the WW and ZZ production thresholds, off-shell effects of the W and Z bosons are of particular importance. In this paper we focus on the production of Z-boson pairs with subsequent decays to four charged leptons, covering all off-shell domains in phase space. While this channel has the smallest cross section among the vector-boson pair production processes, it is the cleanest, as it leads to final states with four charged leptons that can be well studied experimentally. At Run I both ATLAS and CMS measured the cross section of Z-boson pair production [1–4] using final states with four charged leptons or two charged leptons and two neutrinos. The results of these measurements are in agreement with the predictions of the SM and permitted to derive improved limits on triple gauge-boson couplings between neutral gauge bosons [5–7]. Run II allows to improve these measurements, and first analyses have already been published [8, 9]. Precise measurements call for precise predictions. The next-to-leading order (NLO) QCD corrections to Z-boson pair production were calculated a long time ago for stable Z bosons [10, 11] and including leptonic decays in the narrow-width approximation [12]. Once the one-loop helicity amplitudes were available [13], complete calculations including spin correlations and off-shell effects became possible [14, 15]. Gluon-induced one-loop contributions were evaluated for stable Z bosons [16, 17], including off-shell effects [18, 19], and studied as a background to Higgs-boson searches [20]. NLO QCD corrections were matched to parton showers in various frameworks without [21] and with [22–25] including leptonic decays. In ref. [26], a comprehensive NLO-QCD-based prediction was presented for off-shell weak diboson production as a background to Higgs production. Recently, the next-to-next-to-leading order (NNLO) QCD corrections to Z-pair production have been calculated for the total cross section [27] and including leptonic decays [28]. The NNLO QCD calculation has been combined with next-to-next-to-leading-order resummation of multiple soft-gluon emission [29]. Although formally being beyond NNLO in the pp cross section, even the NLO corrections to the loop-induced gluon-fusion channel were calculated [30–32] because of their particular relevance in Higgs-boson analyses. Besides QCD corrections also EW NLO corrections are important for precise predictions of vector-boson pair production at the LHC. EW corrections typically increase with energy owing to the presence of large Sudakov and other subleading EW logarithms [33– 38] and reach several 10% in the high-energy tails of distributions. In addition, photonic corrections lead to pronounced radiative tails near resonances or kinematical thresholds. Logarithmic EW corrections to gauge-boson pair production at the LHC were studied in ref. [39] and found to reach 30% for Z-pair production for ZZ invariant masses in the TeV range. Later, the complete NLO EW corrections were calculated for stable vector bosons and all pair production processes including photon-induced contributions [40, 41]. The size and in particular the non-uniform effect on the shapes of distributions were confirmed. Leptonic vector-boson decays were first included in NLO EW calculations in the form of a consistent expansion about the resonances for W-pair production [42], and in an approximate variant via the Herwig++ Monte Carlo generator for WW, WZ, and ZZ production [43]. Most recently, NLO EW calculations based on full 2 → 4 particle amplitudes, including all off-shell effects, have been presented for W-pair [44] and Z-pair production [45] for four-lepton final states of different fermion generations (i.e. without identical particle effects or WW/ZZ interferences). For Z-pair production, the off-shell effects include also the contributions of virtual photons that cannot be separated from the Z-pair signal, but only suppressed by using appropriate invariant-mass cuts. Note that these full off-shell calculations are essential to safely assess the EW corrections below the WW and ZZ thresholds, ¯ q e− ¯ q e− e− boson analyses. Moreover, a detailed comparison of the full four-lepton calculation [44] to the double-pole approximation for W-boson pairs [42] revealed limitations of the latter approach for transverse-momentum distributions of the leptons in the high-energy domain where new-physics signals are searched for. In ref. [45] we have presented some selected results for the NLO EW corrections to offshell ZZ production in a scenario relevant for Higgs-boson studies. In this paper we provide more detailed phenomenological studies in various phase-space regions relevant for LHC including interference effects from identical final-state leptons. We follow the same concepts and strategies as in refs. [44, 45], i.e. finite-width effects of the Z bosons are consistently included using the complex-mass scheme [46–48], so that we obtain NLO EW precision everywhere in phase space. We also include photon-induced partonic processes originating The paper is organized as follows: some details on the calculational methods are presented in section 2. Phenomenological results for two different experimental setups are discussed in section 3. Our conclusions are given in section 4. Details of the calculation Partonic channels +X receive contributions from the quark-antiquark annihilation channels q¯q/qq¯ → µ +µ −e+e−, µ +µ −µ +µ −, channels in the following, are shown in figures 1(a) and 1(b). Note that all LO diagrams involve Z-boson and photon exchange only. There are LO channels with two photons in the initial state as well, γγ → µ +µ −e+e−, µ +µ −µ +µ −, with corresponding diagrams shown in figure 1(c). Due to their small numerical impact, LO cross section and do not include higher-order corrections to these processes. The NLO EW corrections comprise virtual and real contributions of the q¯q channels, q¯q/qq¯ → µ +µ −e+e− (+γ) , µ +µ −µ +µ − and the real photon-induced contributions with one (anti)quark and one photon in the γq¯/q¯γ → µ +µ −e+e− q¯, µ +µ −µ +µ − Virtual corrections The one-loop virtual corrections to the q¯q channels are computed including the full set of Feynman diagrams. We employ the complex-mass scheme for the proper handling of unstable internal particles [46–48]. This approach allows the simultaneous treatment of phase-space regions above, near, and below the Z resonances within a single framework, leading to NLO accuracy both in resonant and non-resonant regions. Sample diagrams for the virtual EW corrections are shown in figure 2. A first set of diagrams is obtained by exchanging Z bosons or photons in all possible ways between the fermion lines of the tree-level diagrams in figure 1: diagram types (a) and (b) of figure 2 would also be present in narrow-width or pole approximations for the Z bosons and contain separate corrections to the production and the decay of the Z boson. Diagrams (c) and (d) feature correlations between the initial and final states or between different Z-boson decays and are only present in a full off-shell calculation. The sample diagrams (e)–(i) cannot be obtained by naive vector-boson insertions between fermion lines. They involve, for example, closed fermion loops (e) or the exchange of W or Higgs bosons. In our calculation, we perform a gauge-invariant decomposition of the full NLO EW virtual photonic part is defined as the set of all diagrams with at least one photon in the loop coupling to the fermion lines. The weak contribution is then the set of all remaining one-loop diagrams, including also self-energy insertions and vertex corrections induced by closed fermion loops. The contributions to the renormalization constants have to be split accordingly. This splitting is possible, because the LO process with four charged leptons in the final state does not involve charged-current interactions, i.e. there is no W-boson exchange at tree level. The vector-boson insertions between fermion lines exemplarily shown in the diagrams (a)–(d) of figure 2 thus exhaust all generic possibilities how a photon appears in a loop propagator and can systematically be used to construct the virtual photonic contribution. Figure 3 shows the decomposition of the eight diagrams represented by figure 2(d) into the purely weak part with only Z bosons in the loop coupling to fermion lines (upper row) and the photonic part with one or two photons in the loop (lower row). ¯ q ¯ q ¯ q e− e− e− ¯ q ¯ q ¯ q photon and Z-boson insertions between the fermion lines of the tree-level diagrams in figure 1(a). The remaining diagrams may involve couplings (f)–(i) or corrections to vertices (e) that are not present at LO. Note that the criterion for the splitting considers only the vector bosons in the loop, while it does not refer to the tree-level part of the diagram. The contributions to the field renormalization constants of the fermions are decomposed in an analogous manner. Since only loops with internal photons lead to soft and collinear divergences, the purely weak contribution is infrared (IR) finite. The full and finite photonic corrections to the q¯q channels, on the other hand, comprise the virtual photonic corrections plus the real photon emission described in the next section. In general, the decomposition of an amplitude into gauge-invariant parts requires great caution. The gauge-invariant isolation of photonic corrections in processes that proceed in LO via neutral-current interactions is only possible because the LO amplitude can be group with the photon as massless gauge boson and U(1)Z to an Abelian gauge theory with the Z boson as gauge boson. The corrections within this theory, which is a consistent and e− e− e− ¯ q ¯ q ¯ q e− e− e− ¯ q e− ¯ q ¯ q contributions for the diagram type shown in figure 2(d). renormalizable field theory on its own (see e.g. ref. [49] or ref. [50], chapter 12.9), form a gauge-invariant subset of the full SM corrections to our neutral-current process. Since theory, the photonic contributions form a gauge-invariant subset of the corrections (which could even be further decomposed into subsets defined by the global charge factors of the fermions linked by the photon). Note that the subset of closed fermion loops could be isolated as another gauge-invariant part of the EW correction, since each fermion generation (in the absence of generation mixing) delivers a gauge-invariant subset of diagrams. For the process at hand, we, however, prefer to keep the closed fermion loops in the weak Real corrections The real corrections to the q¯q channels include all possible ways of photon radiation off the initial-state quarks or off the final-state leptons, schematically depicted in figure 4. The phase-space integrations over the squared real-emission amplitudes diverge if the radiated photon becomes soft or collinear to one of the external fermions. For IR-safe (i.e. softand collinear-safe) observables, however, the collinear final-state singularities and the soft singularities cancel exactly the corresponding soft and collinear divergences from the virtual corrections after integration over the phase space. The collinear initial-state singularities do not fully cancel between real and virtual corrections; the remnants are absorbed into the parton distribution functions (PDFs) via factorization, in complete analogy to the usual procedure applied in QCD. We employ the dipole subtraction formalism for the numerical integration of the real corrections. In detail, we apply two different variants based on dimensional regularization [51] and mass regularization [52], respectively. The results obtained with the two e− e− ¯ q e− ¯ q e− ¯ q e− approaches are in perfect agreement. The underlying idea is to add and subtract auxiliary terms |Msub|2 at the integrand level which pointwise mimic the universal singularity structure of the squared real matrix elements |Mreal|2 and which, on the other hand, can be integrated analytically in a process-independent way. In the dipole subtraction approach, the subtraction function is constructed from “emitter-spectator pairs”, where the “emitter” is the particle whose collinear splitting leads to an IR singularity and the “spectator” is the particle balancing momentum and charge conservation in the emission process. + 2 Re MBornMvirt where 2 Re[MBornMvirt] denotes the interference of the Born amplitude MBorn with the one-loop amplitude Mvirt. The last term represents the (IR-divergent) factorization contribution from the PDF redefinition, which takes the form of a convolution over the momentum fraction x quantifying the momentum loss via collinear parton/photon emission. particle real phase space, which includes the bremsstrahlung photon, is decomposed into integration over [dk] involves an integration over the variable x which controls the momentum loss from initial-state radiation. Splitting off the soft singularities developing in this integration of the subtraction terms over x, decomposes the integral R1[dk]|Msub|2 into analytically combined with the virtual corrections, the latter with the factorization contribution, to produce individually IR-finite terms which may be integrated separately in a fully numerical way: dΦ |Mreal|2Θ(p1, . . . , pn, pγ ) − |Msub|2Θ(p˜1, . . . , p˜n) ∗ dxh|Msub(x)|2fin + δ(1 − x)2 Re MBornMvirt finiΘ(p˜1, . . . , p˜n) . ∗ Here, |Msub(x)|2fin and 2 Re[MBornMvirt]fin are the finite parts resulting from this splitting of the x integration into continuum and endpoint parts. The momenta p˜i of the reduced nparticle phase space in the first integral are related to the momenta pi of the (n+1)-particle The integration over x is a remainder from the factorized one-particle phase space and the phase-space cuts applied to the particle momenta {p1, . . . }, possibly after applying a recombination procedure which is discussed in the next paragraphs. Quark-induced channels — the collinear-safe setup. Observables that are collinear safe with respect to final-state radiation — as described in ref. [52] — are constructed by applying an appropriate procedure for recombining radiated photons with nearly collinear final-state leptons. In collinear regions, a photon-lepton pair with photon and lepton moAny phase-space cut or any evaluation of an observable is performed for the recombined momenta, while the matrix elements themselves are evaluated with the original kinematics. Both the local and the integrated subtraction terms and the virtual corrections are cut in edge-of-phase-space effects which do not spoil integrability). Since collinear lepton-photon configurations are treated fully inclusively within some collinear cone defined by the photon recombination, the conditions for the KLN theorem are fulfilled, guaranteeing the cancellation of the collinear mass singularity. The formation of such a quasi-particle is close to the experimental concept of “dressed leptons”, as e.g. described by the ATLAS collaboration in ref. [53]. Quark-induced channels — the collinear-unsafe setup. It is not a priori necessary that observables sensitive to photon radiation off a final-state lepton are defined in a collinear-safe way. The reason is that photons and charged leptons may be detected in geometrically separated places, i.e. the photons in the electromagnetic calorimeter and the muons in the muon chamber. This allows the measurement of an arbitrarily collinear photon emission off a muon. In the absence of photon recombination, the lepton masses serve as a physical cutoff for collinear singularities. On the computational side, this simply forbids the recombination of a muon with momentum pµ and a photon with momentum do in a collinear-safe setup. In the case of photon emission off electrons, the detection of the two particles takes place in the electromagnetic calorimeter. The finite resolution of the detector then defines a natural “cone size” for the recombination of the lepton-photon pair to a single quasi-particle. In our collinear-unsafe setup, we exclude the muons from recombination, while the electrons are recombined with photons like in the collinear-safe In ref. [54], the dipole subtraction formalism was extended to collinear-unsafe observables. As in the collinear-safe case, the starting point of the formalism is eq. (2.5), the fundamental difference being that without recombination of a lepton-photon pair, some observables may now be sensitive to the individual lepton and photon momenta within the collinear region. While this is obvious for the real-emission matrix element, this is also required from the subtraction terms in order to guarantee the local subtraction of the singularities. To this end, the reduced n-point kinematics of the local subtraction terms (which are integrated over the (n + 1)-particle phase space) is a posteriori extended to an effective (n + 1)-particle configuration with a resolved collinear lepton-photon pair with Here p˜i denotes the momentum of a particular final-state emitter in the reduced kinematics, lepton-photon pair is denoted by z; it is constructed from kinematical invariants of the (n + 1)-particle phase space. The local subtraction terms are evaluated in the same way as in the collinear-safe case. However, any collinear-unsafe contribution is now cut with respect to the unrecombined (n + 1)-particle phase space: dΦ |Mreal|2Θ(p1, . . . , pn, pγ) − |Msub|2Θ(p˜1, . . . , p˜n; z)i Z dΦ˜ Z 1dxZ 1dzh|Msub(x, z)|2fin +δ(1−x)δ(1−z)2 Re MBornMvirt finiΘ(p˜1, . . . , p˜n; z) . ∗ The schematic shorthand notation emission. Note that the one-particle phase-space integral [dk] in the integrated subtraction terms in eq. (2.5) is modified, because the z dependence is required in the re-added subtraction contribution in order to allow for cuts on the bare lepton momentum also in the collinear region. This is in contrast to the collinear-safe case where z could be integrated in a process-independent way [52]. Splitting off the soft- and collinear-singular contributions in the z-integration, the subtraction terms can be separated into an inclusive part for the collinear-safe case plus extra terms for the collinear-unsafe case. The detailed form of |Msub(x, z)|2fin can be found in ref. [54] and is not repeated here. For collinear-unsafe observables, the integration over z is not inclusive, so that the conditions for the KLN theorem are not fulfilled. Hence, the collinear singularities from the virtual corrections do not entirely cancel against those from the real corrections. Using and modify the cross section, which often leads to significant shape distortions of differential distributions. Since partonic scattering energies at the LHC are much larger than the muon mass, all terms suppressed by factors of mµ can be safely neglected. From a practical point of view this means that all kinematics is evaluated with exactly massless muons and the relicts from collinearly-sensitive observables remain in the finite but possibly large ¯ q e− ¯ q e− Quark-photon-induced channels. nels with one photon and one (anti)quark in the initial state. Since an external soft quark does not lead to a singularity and since there is no collinear divergence when the final-state quark becomes collinear to one of the final-state leptons, the matrix elements exhibit only collinear initial-state singularities. As illustrated in figure 5, they can be grouped into two classes: first, the incoming photon splits into a quark-antiquark pair with the final-state (anti)quark becoming collinear to the incoming photon, or second, the incoming (anti)quark splits into a photon-(anti)quark pair with the final- and initial-state (anti)quark becoming collinear. With the dipole subtraction method each of the two collinear singularities may be locally subtracted with a single dipole whose functional form is given in ref. [54]. The singularities in the two corresponding integrated subtraction terms cancel against the collinear counterterm from the PDFs, which can, e.g., be found in ref. [55]. Numerical implementation and independent checks of the calculation We have performed a complete calculation of all contributions using the publicly available matrix element generator Recola [56] for the evaluation of the virtual corrections and for all tree-level amplitudes at Born level and at the level of the real corrections. The phasespace integration has been carried out with a multi-channel Monte Carlo integrator with an implementation of the dipole-subtraction formalism [51, 52, 54, 57] for collinear-safe and collinear-unsafe observables. The calculation has been cross-checked both at the level of phase-space points and differential cross sections with two other independent implementations. The one-loop matrix elements of the equal-flavour case have been checked against amplitudes from the Mathematica package Pole [58], which employs FeynArts [59, 60] and FormCalc [61]. The one-loop matrix elements of the mixed-flavour case have been checked against a calculation based on diagrammatic methods like those developed for four-fermion production in electron-positron collisions [47, 62], starting from a generation of amplitudes with FeynArts [59, 60] and further algebraic processing with in-house Mathematica routines. In all three calculational approaches, the one-loop integrals are evaluated with the tensor-integral library Collier [63] containing two independent implementations of the tensor and scalar integrals. Collier employs the numerical reduction schemes of refs. [64, 65] for one-loop tensor integrals and the explicit results of one-loop scalar integrals of refs. [66–68] for complex masses. The phase-space integration in all three approaches is carried out with independent multi-channel Monte Carlo integrators which are further developments of the ones described in refs. [69–71]. For all differential and total cross sections obtained with the different implementations we find agreement within the statistical uncertainty of the Monte Carlo integration. Phenomenological results Input parameters In the numerical analysis presented below, we consider the LHC at a centre-of-mass (CM) energy of 13 TeV and choose the following input parameters. For the values of the on-shell masses and widths of the gauge bosons we use Gµ = 1.16637 × 10−5 GeV−2 1 − MZ2 MZos = 91.1876 GeV, M Wos = 80.385 GeV, In the complex-mass scheme, the on-shell masses and widths need to be converted to pole quantities according to the relations [72] MV = V = W, Z . The complex weak mixing angle used in the complex-mass scheme is derived from the ratio EW parameters, we refer to ref. [47]. Since the Higgs boson and the top quark do not appear as internal resonances in our calculation, their widths are set equal to zero. For the corresponding masses we choose the values MH = 125 GeV, mt = 173 GeV. particles with zero mass throughout the calculation. In the computation of collinear-unsafe observables, the physical muon mass appears as a regulator with numerical value mµ = 105.6583715 MeV. Note that this non-zero value for the muon mass is only kept in otherwise divergent logarithms from photon radiation off muons, while everywhere else the muons are strictly treated as massless particles. scale into the LO cross section and thus avoids mass singularities in the charge renormalization (see refs. [73, 74] and the discussion in the “EW dictionary” in ref. [75]). Moreover, is only used as coupling parameter in the relative photonic corrections, i.e. the NLO conwe do not consider QCD corrections in this paper and work in the on-shell renormalization scheme [47, 73], our cross-section predictions do not depend on the renormalization scale µ ren. The dependence of the relative NLO EW corrections on the factorization scale µ fact is need to investigate residual scale dependences or alternative scale choices. As PDFs we use the NNPDF23 nlo as 0118 qed set [76].1 Throughout our calculation of EW corrections, we employ the deep-inelastic-scattering (DIS) factorization scheme, following the arguments given in ref. [78]. The corresponding finite terms for the EW corrections to be included in the subtraction formalism can be found in ref. [55]. Definition of observables and acceptance cuts In the following we define two different event selections: an “inclusive” and a “Higgsspecific” setup. The former uses typical lepton-identification cuts without any further selection criteria; the latter is motivated by specific criteria designed for Higgs-boson analyses by ATLAS [79] and CMS [80]. Inclusive setup. In the collinear-safe case, photons emerging in the real-emission contributions are recombined with the closest charged lepton (cf. section 2.3) if their separation in the rapidity-azimuthal-angle plane obeys ΔRℓi,γ = q(yℓi − yγ )2 + (Δφℓiγ )2 < 0.2 . with larger rapidities as lost in the beam pipe. In the collinear-unsafe case, photons are recombined only with electrons/positrons, while no recombination with muons/antimuons 1In this calculation we take the photon density of NNPDF23 nlo as 0118 qed in spite of its larger uncertainty compared to the LUXqed photon distribution [77], in order to rely on one consistent PDF set. This leptons are thus labelled as ℓ1±, and subleading leptons as ℓ2±. is performed. For observables of the equal-flavour-lepton final state, the leading lepton pair As default setup, we consider a minimal set of selection cuts inspired by the ATLAS analysis [1]. For each charged lepton ℓi, we restrict transverse momentum pT,ℓi and rapidity yℓi according to pT,ℓi > pT,min = 15 GeV, Any pair of charged leptons (ℓi, ℓj ) is required to be well separated in the rapidity-azimuthal-angle plane, Higgs-specific setup. For the Higgs-specific setup, motivated by the ATLAS and CMS analyses [79, 80], we replace the cuts of eq. (3.9) by the less restrictive criteria pT,ℓi > pT,min = 6 GeV, retain the cut (3.10), and complement them with additional invariant-mass cuts on the charged leptons. For the mixed-flavour final state, we require for the two same-flavour 40 GeV < M + − < 120 GeV, 12 GeV < M + − < 120 GeV, or further away from the nominal Z-boson mass, respectively. For the same-flavour final state, we apply the cuts of eq. (3.12) after selecting the leading and subleading lepton pairs (ℓ1+, ℓ1 ) and (ℓ2+, ℓ2−) in the same way as described above. The invariant mass M4ℓ of the − four-lepton system is subjected to the cut M4ℓ > 100 GeV, which is independent of the flavour of the final-state leptons. In both setups we treat the additional (anti)quark in the final state of the photoninduced contributions in a fully inclusive way, i.e. we do not apply any jet veto. Finally, we note that the employed single-particle lepton identification cuts are chosen to be equal for all charged leptons. Since the lepton pairing in the equal-flavour final state is flavour indefinal state within the collinear-safe setup. In the following, [4µ ] denotes the equal-flavour final state, while [2µ 2e] denotes the mixed-flavour final state. Whenever possible, we have compared the results of our full off-shell calculation with the available results for on-shell Z-boson pair production [40, 41]. Since our calculation sets phase-space cuts on the charged leptons, a direct comparison is not possible for most observables. Moreover, the calculations for stable Z bosons do not take into account corrections to the Z-boson decays. Finally, there are differences in the setup: the values in incl. [2µ 2e] Higgs [2µ 2e] 13.8598(3) −4.32 −4.32 −3.59 −3.42 −0.93 −0.94 −0.04 −0.09 −1.68 −2.43 −0.28 −0.66 −0.09 −0.14 corrections δi = σi/σqL¯qO for the LHC at √ (“incl.”) with the cuts of eqs. (3.9)–(3.10) and the Higgs-specific setup (“Higgs”) with the selection cuts of eqs. (3.10)–(3.13), respectively. the literature are given for a CM energy of √ PDFs. Nevertheless, the relative EW corrections to those observables that are less sensitive to off-shell effects and corrections to the Z-boson decays can still be directly compared. This holds in particular for the Sudakov enhancement at large transverse momentum of the on-shell Z boson or the corresponding charged final-state lepton pair. Results on integrated cross sections given for the inclusive selection cuts of eqs. (3.9)–(3.10) and the Higgs-specific selection cuts of eqs. (3.10)–(3.13). Together with the LO cross sections, the NLO EW corrections . For the photonic corrections, we further distinguish between the collinear-safe phot,safe, where the bremsstrahlung photon is recombined with any charged lepton, phot,unsafe, where the muons are excluded from recombination, EW/weak/phot to the q¯q channels are practically independent with state-of-the-art QCD predictions. EW/weak/phot well suited for a combination We first analyse the inclusive scenario. The major contribution to the corrections stems The photon-induced contribution matters only at the permille level, justifying that we neglect higher-order corrections to this channel. Besides the small photon flux in the proton, yet another reason for the strong suppression is that channels with two photons in the initial state involve at most one resonant Z boson, as illustrated by the sample diagram of figure 1(c). We count such kinematical topologies as background topologies to the dominant contribution with two possibly resonant Z-boson propagators as they appear another order of magnitude and are thus entirely negligible in the integrated cross section. Summing up all contributions, we find for both the [2µ 2e] and the [4µ ] final state the in a slightly different setup. The differences can be attributed to the off-shell effects, including also additional virtual photon exchange, and to differences in phase-space cuts and in the employed numerical setup (cf. comments at the end of section 3.2). Note that on-shell approximation [41]. Comparing the collinear-unsafe photonic corrections with the collinear-safe case, we tively. In the collinear-unsafe case, final-state radiation off muons is enhanced through the bined with the muons, there are systematically more events with a large energy loss in one of the muon momenta induced by final-state radiation. For this reason, less events pass the event-selection in the collinear-unsafe case, leading to more negative corrections. This “acceptance correction” scales with the number of leptons treated in a collinear-unsafe way, The amplitudes for the equal-flavour final state can be obtained from the differentflavour amplitudes by antisymmetrization with respect to a pair of equal final-state leptons. two different vector bosons. These interference terms lead to a deviation from a naive rescaling of the different-flavour cross section by a symmetry factor of two. From the integrated LO cross section. Comparing this number with the total relative correction of section in the inclusive setup is a “LO effect” in the sense that the relative corrections do not modify this behaviour. The reason for the smallness of the interferences is that the LO cross section is dominated by contributions with two resonant Z-boson propagators. In the interference terms, however, in at least one diagram both Z-boson propagators are off shell. with at most one resonant Z boson. We now turn to the Higgs-specific setup. Despite the additional cuts of eqs. (3.12)– (3.13), the cross sections for this scenario are larger than for the previously considered inclusive setup. This feature is due to the less severe cut of 6 GeV imposed on the transverse momenta of the charged leptons in the Higgs-specific setup, as compared to a cut of 15 GeV resonances due to the coupling of the W boson to the photon. in the inclusive setup. Moreover, we observe a percent-level deviation at LO from the naive 0.973. This reflects, on the one hand, an expected enhancement of the interference terms since, by construction, the whole scenario is more sensitive to the off-shell effects of the vector bosons. On the other hand, the additional invariant-mass cuts in eq. (3.12) depend in the equal-flavour case on the chosen lepton-pairing algorithm, while they do not in the mixed-flavour case. A quantitative statement on the size of the pure interference effects would thus only be possible if the same lepton pairing was applied also in the mixed-flavour case. The same arguments apply also to the corrections in table 1, i.e. the differences between the equal-flavour and the mixed-flavour final state are due to interferences and event selection. As a general pattern, we observe that the bulk of corrections to the total contribute only at the sub-percent level. Results on differential cross sections in the inclusive setup Invariant-mass and transverse-momentum distributions. In order to illustrate the impact of EW corrections on differential observables, we present several results on distris = 13 TeV. We choose the collinear-safe setup as default and provide selected results within the collinear-unsafe case subsequently. Figure 6 shows the four-lepton invariant-mass distribution for the unequal-flavour [2µ 2e] and the equal-flavour [4µ ] final states. The left-hand side resolves the off-shell region with its threshold and resonance structures with a fine histogram binning, while the panels on the right-hand side show the whole range from the off-shell region over the resonances and thresholds up to the TeV regime in coarse-grained resolution. The absolute predictions of the LO and NLO distributions of both the [2µ 2e] and [4µ ] final states in the upper panels follow the characteristic pattern of Z-boson pair producZ boson in the s-channel according to the sample diagram in figure 1(b), the threshical cut of eq. (3.9) on the lepton transverse momentum. For M4ℓ & MZ + 2pT,min, the cross section is dominated by events with one resonant Z boson (→ ℓ1 ℓ1 ) and the ℓ2 ℓ2 + − + − plane. Since E + + E − > 2pT,min = 30 GeV is necessary for the event to pass the cuts, ℓ ℓ the transition rate, since no resonance enhancement is present anymore in the diagram type illustrated in figure 1(a). The panels directly below the absolute predictions for the cross sections show the relative EW corrections to the q¯q channels in the collinear-safe setup, comparing the purely shell region below the single Z resonance, we observe that the relative EW corrections of the mixed-flavour final state and the equal-flavour final state are equal over the whole √s = 13 TeV 10−4 []% 10 −10 −20 ] 2 [ δ 1 −1 100 200 300 400 500 600 700 800 900 1000 unequal-flavour [2µ 2e] and the equal-flavour [4µ ] final states in the inclusive setup. The panels at the bottom show the ratio of the [2µ 2e] and [4µ ] final states. invariant-mass spectrum. This confirms at the level of differential distributions that the interference effect is mainly a LO effect, in accordance with what we have already seen for the integrated cross section. The four-lepton invariant mass in the inclusive setup is well suited to study the relative size of the interferences, as this observable does not depend on the lepton pairing. We show LO and NLO curves are, as expected, almost equal. The size of the interference effect equal-flavour final state there. In the region MZ + 2pT,min . M4ℓ . 2MZ, where only one √s = 13 TeV lepton pair can be resonant, the interference effect amounts to 2%. Above the ZZ threshold, the ratio is equal to one up to fractions of a percent, since in this region of phase space the doubly-resonant contribution dominates over any non-resonant interference effect. For higher invariant masses M4ℓ, the overlap of the two resonance pairs becomes smaller and smaller in phase space, so that the ratio asymptotically approaches one. We inspect the EW corrections in more detail. In the high-invariant-mass region, the correction is entirely dominated by the purely weak contribution and reaches about regime of ZZ production where all Mandelstam variables (s, t, u) of the 2 → 2 particle process would have to be large. Instead, Z-pair production at high energies is dominated by forward/backward-produced Z bosons, where t and u are small. At the ZZ production threshold, the weak corrections change their sign and reach up to 5% below. Note that this non-trivial sign change makes it impossible to approximate the full NLO EW results by a global rescaling factor. At the Z peak, the weak corrections are extremely suppressed. LO cross section falls off steeply. Final-state radiation of a real photon, however, may shift the value of the measured invariant mass to smaller values. Since the LO cross section is small in this phase-space region, the relative correction due to the bremsstrahlung photon becomes large. The structure of the radiative tails follows precisely the thresholds and another one below the Z resonance. For completeness, we also show the photon-induced corrections in a separate panel nel has only a single Z resonance according to the diagram in figure 1(c), it is strongest suppressed with respect to the q¯q LO cross section above the ZZ threshold and near the s-channel resonance at MZ. In the non-resonant region below MZ + 2pT,min it reaches up to 1%. Since there the LO cross section is small anyway, the overall impact remains small, in agreement with the result for the integrated cross section. Differences between the equal-flavour and the unequal-flavour final states due to interferences are only visible over the whole spectrum at most at the permille level. The large correction near the phasespace boundary is phenomenologically irrelevant as the corresponding LO cross section in this region is very small anyway. Due to the negligible impact of any photon-induced corrections in the inclusive setup, we do not show them separately any more in the following Up to the details in the event selection and the corrections from the Z-boson decays, the four-lepton invariant mass M4ℓ can be compared to the ZZ-invariant-mass distribution obtained in the NLO calculations of refs. [40, 41] with on-shell Z bosons. The relative 10−1 10−2 −µ 10−4 pd10−5 10−6 10−7 −10 ]−20 −40 −50 (2· 1.02 / ] √s = 13 TeV subleading µ +µ − pair √s = 13 TeV subleading µ +µ − pair Mµ +µ − [GeV] pTµ +µ − [GeV] (upper panels), corresponding EW corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the inclusive setup. The equal-flavour case is binned with respect to the subleading lepton pair. stood from the fact that in the vicinity of the Z resonance there are two different types of contributions: corrections to the resonant part of the squared amplitude, and corrections to the interference of the resonant and non-resonant parts of the amplitude. The and thus change sign at the Z resonance. This qualitatively explains the observed sign change of the purely weak corrections which is slightly shifted below the resonance due to the negative offset mentioned above. The corrections are to a large extent equal for the Including also the photonic corrections, we observe in both cases the typical radiative tail due to final-state radiation effects below the Z resonance, similar to what has been observed in the four-lepton invariant mass. In the high-energy spectrum of the steeply falling invariant-mass distribution, shown in figure 8, both photonic and purely weak corrections differ significantly from the mixed-flavour case due to the large differences at LO. At the peak around 45 GeV, the purely weak corrections basically vanish, which is consistent with the four-lepton invariant-mass distribution in figure 6 where the purely weak corrections the [2µ 2e] and the [4µ ] case because the dominant contribution where the leading and the subleading lepton pairs are both close to the resonance is not sensitive to the lepton pairing panel and the discussion for the subleading case above the resonance). The weak corrections stay always below 5% in absolute size. The photonic corrections exhibit an additional radiative tail below the peak around 45 GeV. The radiative tail below the Z resonance is less pronounced due to the missing resonance enhancement by the subleading lepton pair. In figure 8 (right-hand side) we show the distribution in the transverse momentum boson transverse momentum in on-shell calculations. Since the two lepton pairs are back to back at LO, the transverse-momentum distribution depends on the choice of the lepton pair only very weakly. The interference effect of a few percent is only visible for small ZZ production with on-shell Z bosons discussed in refs. [40, 41]. However, as mentioned above, it should be kept in mind that it cannot be expected to find perfect agreement because of the differences in the event selection, which is based on final-state leptons in our calculation, and the absence of corrections to the Z-boson decays in the on-shell s = 14 TeV, which agrees with our result for √ Figure 9 shows the transverse-momentum distribution of the µ +. The left panels compare the leading µ + from the [4µ ] final state with the µ + from the [2µ 2e] final state, the panels in the right column show the corresponding comparison with the subleading µ +. Recall that our ordering of muons into “leading” and “subleading” corresponds to the oras described in section 3.2, but not to the muon pT, which is frequently used as well. We observe again that the observable is very sensitive to the event selection with characteristic differences between leading and subleading leptons. Especially at high transverse momenta pT, the spectrum of the leading muon in [4µ ] is suppressed with respect to the spectrum of pT in [2µ 2e], while the spectrum of the subleading muon in [4µ ] is enhanced. The difference can be traced back to the impact of the ZZ signal and background contributions at large transverse momenta: the leading lepton belongs to the “more resonant” Z boson, and therefore, the contribution is in general dominated by the doubly-resonant signal contributions (cf. figure 1(a)). The main effect of the background contribution in figure 1(b) for large pT arises when the µ + is back-to-back with the three other charged leptons. As already observed for the related process of pp → WW → leptons [44], the impact of background diagrams on the pT spectrum of a single lepton can be as large as the doubly-resonant contribution in the TeV range. Since there is no preselection of the µ + in the [2µ 2e] final state with respect to the resonance, the pTµ + spectrum of [2µ 2e] lies between the spectra of i 10−3 d10−5 10−6 10−7 −10 ]−20 −40 (2· 1.5 / ] √s = 13 TeV rections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the inclusive setup. The left panels compare the leading µ + from the [4µ ] final state with the µ + from the [2µ 2e] final state, while the panels in the right column show the corresponding comparison with the subleading µ +. the leading and subleading muons. This behaviour is also reflected in the size of the purely weak corrections: since the Sudakov enhancement is larger in doubly-resonant contribucase of the subleading µ +. The photonic corrections give an almost constant negative offset the equal-flavour final state is very similar to the mixed-flavour case. For the subleading −1% and −0.5%. Rapidity and angular distributions. The rapidity distributions in figure 10 do not show any significant dependence on the lepton pairing except for a small effect at the percent level in the forward direction with rapidities |yµ + | > 2. The EW corrections are independent of the lepton pairing as well, and their size is almost equal for both final states. The purely photonic corrections give, in good approximation, a constant negative offset of 10−2 i 10−3 d10−5 10−6 10−7 −10 ]−20 −40 (2· 1.5 / ] √s = 13 TeV −1 −2 ]−3 −5 −6 −7 √s = 13 TeV s√usbl=ea1d3inTgeµV+ −1 −2 −5 −6 −7 −2 −1 −2 −1 panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the inclusive setup. The left panels compare the leading µ + from the [4µ ] final state with the µ + from the [2µ 2e] final state, the panels in the right column show the corresponding comparison with the subleading µ +. less negative in the forward direction with about −3%. In figure 11, the distribution in the azimuthal-angle distance between the muons in the proximation independently of the final state and the lepton pairing. This can be explained as follows: the azimuthal-angle-distance distribution is dominated in the whole range by tion receives the largest contribution from doubly-resonant contributions. Moreover, the t-channel nature of the doubly-resonant diagrams favours small transverse momenta of the Z bosons. As can also be seen in figure 9, most of the leptons have pT . MZ/2 as a result of the decay of the Z bosons that move slowly in the transverse plane, i.e. the Z bosons decay almost isotropically in the transverse plane, without a large influence of boost effects (2· 1.5 / √s = 13 TeV leading µ +µ − pair s√usbl=ea1d3inTgeµV+µ − pair sponding EW corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) comparison with the subleading lepton pair. separation cut of eq. (3.10). ]fb 1.4 [ −2 −3 −4 −6 −7 −8 √s = 13 TeV panels), corresponding EW corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the inclusive setup. configuration preferred at low energies. The effect of increasing negative weak corrections tion owing to the absence of collinear enhancements, because collinear final-state radiation does not change the directions of the radiating leptons significantly. The photonic correccase, the leading lepton pair, and the subleading lepton pair, respectively. They decrease for the unequal-flavour [2µ 2e] final state and the corresponding angle defined by the leading the CM system of the four final-state leptons by kℓ1+ℓ1− × kℓ+ |kℓ1+ℓ1− × kℓ1+ ||kℓ1+ℓ1− × kℓ2+ | kℓ1+ℓ1− × kℓ+ × kℓ1+ℓ1− × kℓ+ 1 2 1 − consequence of the lepton-separation cuts in eq. (3.10). These cuts remove collinear lepton configurations where the decay planes tend to be coplanar. The local minima around different lepton helicities: if the two equally charged final-state leptons do have the same the interference pattern, it is instructive to consider the limit of nearly coplanar Z decays decay planes. In the vicinity of the coplanar configurations this line divides the event plane the matrix elements are antisymmetrized with respect to the exchange of µ 1+ ↔ µ 2+ or µ 1 ↔ µ 2−, a destructive interference is favoured for φ → 0 in [4µ ], leading to the observed − enhancement in the [2µ 2e]/(2[4µ ]) ratio. This effect is not changed by the EW corrections. only uniformly contribute by −1%. Collinear-safe versus collinear-unsafe observables. In figure 13, the different recombination schemes for the muon are illustrated for the four-lepton and two-lepton invariant masses. The recombination procedure only affects the photonic corrections. As a general pattern, all the radiative tails induced by final-state radiation off the charged leptons are strongly enhanced if collinear photons are not recombined with muons. The enhancement is due to the fact that the collinear logarithms are regularized by the muon mass rather than the size of the recombination cone. The effects can be best isolated in the M4ℓ invariant-mass distribution (left panels of figure 13) which is not sensitive to the lepton i 10−2 10−3 10−4 −20 NLO EW collinear unsafe [4µ ] NLO EW collinear unsafe [2µ 2e] 10−1 10−3 10−4 −20 NLO EW collinear safe [2µ 2e] NLO EW collinear unsafe [2µ 2e] Mµ +µ − [GeV] lepton invariant-mass distributions. The upper panels show the absolute distributions and the lower panels the relative EW corrections. pairing. While the absolute prediction is only shown for the collinear-unsafe case for the mixed- and equal-flavour final states, the relative EW corrections are plotted both for the collinear-safe and -unsafe cases. The results illustrate the impact of the number of muons excluded from recombination to the distribution: the maximum of the radiative tail below the ZZ threshold increases from about +30% with full recombination to more than +50% for excluding one muon pair ([2µ 2e]) up to about +70% by excluding both muon pairs ([4µ ]). For [4µ ], the increase is twice as large as for [2µ 2e], since the recombination effect scales with the number of collinear cones that are subject to the changes in the recombination. A similar behaviour is found for the other radiative tails at smaller values of M4ℓ. panels) below the Z resonance where the relative correction increases from almost +60% to +140%. Note that above the resonance the effect from the collinear-unsafe treatment pushes the negative collinear-safe corrections even more negative. Results on differential cross sections in the Higgs-specific setup Invariant-mass and transverse-momentum distributions. The production of Zboson pairs at the LHC is interesting not only per se, as a signal process, but also conmode. In order to assess the impact of this background on Higgs analyses, we impose the Higgs-specific cuts of eqs. (3.12)–(3.11) in addition to the inclusive cut of eq. (3.10). In ref. [45] we already presented some important results of this study, however, restricted to the unequal-flavour final-state [2µ 2e] and ignoring photon-induced channels. In the follow10−1 10−2 −5 ] 2 √s = 13 TeV ℓ dσ dM410−3 10−4 −10 −20 −30 −0.5 −1 4 [2· 0.8 ( / µ]e2 0.7 EW corrections (2nd panels from above), photonic contributions (third panels from above) for the unequal-flavour [2µ 2e] and the equal-flavour [4µ ] final states in the Higgs-specific setup. The lower panels show the ratio of the [2µ 2e] and [4µ ] final states. ing we continue the discussion started there by comparing results for the [2µ 2e] and [4µ ] final states and considering further observables. Figure 14 illustrates the invariant-mass distribution of the four-lepton system at LO and the corresponding NLO EW corrections for both the [2µ 2e] and the [4µ ] final states. In each case, we observe a steep shoulder at the Z-boson pair production threshold at about at smaller invariant masses. Though smaller in magnitude, a similar effect can be observed mass cuts we impose on the charged leptons. Like in the inclusive setup, both the purely weak and the photonic corrections exhibit a sign change at the pair production threshold √s = 13 TeV similar to the inclusive setup with at most permille level differences between the [4µ ] and the [2µ 2e] case. The photonic corrections decrease in absolute size from approximately region below the pair production threshold, the difference between the [4µ ] and the [2µ 2e] cases in the purely weak corrections is below the percent level. The radiative tails in the photonic corrections are up to 5% larger in the mixed-flavour case. In contrast to the inclusive setup, the phase-space cuts of the Higgs-specific setup introduce a dependence on the lepton pairing even in otherwise symmetric observables like the four-lepton invariant mass. The difference seen in the photonic corrections is thus due to both the lepton pairing corrections with respect to the final states are, however, entirely negligible. The significant differences between the [4µ ] and the [2µ 2e] case in the off-shell-sensitive region are, like in the inclusive case, a priori a LO effect. Note that the non-trivial sign change of the photonic corrections leads to significant cancellations between opposite-sign contributions below and above the ZZ threshold resulting in sub-permille effects in the total cross section (cf. table 1), although the individual photonic corrections can be sizable in distributions. We also show the photon-induced contribution to the four-lepton invariant-mass distribution in the third panels from above in figure 14. Above the ZZ production threshold large extent. The overall impact remains at the sub-percent level. We do not show the photon-induced corrections separately in the following plots. final state. Due to the cuts of eq. (3.12), the invariant mass of the leading muon pair in the equal-flavour final state is restricted to the range of 40–120 GeV. This cut leads in the [2µ 2e] final state to a little bump at 40 GeV. Moreover, the local maximum near MZ/2 because the invariant-mass cut M4ℓ > 100 GeV in eq. (3.13) entirely removes the s-channel similar to the results in the inclusive setup (cf. figure 7). The distribution peaks at the final-state radiation effects. The weak corrections, on the other hand, are of the order of 5% and give rise to a change in sign near the Z-boson resonance. Above the resonance the EW corrections are qualitatively similar to the ones in the inclusive setup for both expected, most prominent in the leading lepton pair where the local minimum of the weak corrections at 45 GeV and the entire additional radiative tail of the photonic corrections are removed. While the EW corrections show sizeable deviations between the mixed- and equal-flavour final states, the main differences are LO effects that can be attributed to the cuts and the lepton pairing. √s = 13 TeV leading µ +µ − pair s√usbl=ea1d3inTgeµV+µ − pair Mµ +µ − [GeV] Mµ +µ − [GeV] dle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the Higgs-specific setup. In the left column the equal-flavour case is binned with respect to the leading lepton pair, while the right column shows results for the subleading one. Figure 16 depicts the transverse-momentum distribution of the µ + in the [2µ 2e] final state together with the leading and the subleading µ + of the [4µ ] final state, respectively. We once again remind the reader that the classification of leptons as “leading” or “subleading” refers to the criteria of eq. (3.12), i.e. the leading muon is not necessarily the muon closest to the mass of the Z boson. We find that, in contrast to the inclusive setup illustrated in figure 9, the weak corrections to the transverse momenta are very similar in size and shape for the equal- and the unequal-flavour cases. They become large and negative suppression of background diagrams of the type shown in figure 1(b), which can already be seen from the suppression of the absolute LO cross section at large transverse momenta (cf. figure 9 and related discussion there). The impact of photonic corrections is at the level of one percent for small transverse momenta and even smaller for large ones for both leptonic final states. 10−2 i 10−3 d10−5 10−6 10−7 −10 ]−20 −40 −50 / µ]e2 0.9 i 10−3 d10−5 10−6 10−7 −10 ]−20 −40 −50 (2· 1.0 / √s = 13 TeV √s = 13 TeV Transverse-momentum distribution of the µ + (upper panels), corresponding EW corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the Higgsspecific setup. The left panels compare the leading µ + from the [4µ ] final state with the µ + from the [2µ 2e] final state, while the panels in the right column show the corresponding comparison with the subleading µ +. Rapidity and angular distributions. The rapidity distributions of the µ + and the corresponding EW corrections, shown in figure 17, do not change very much when going from the inclusive to the Higgs-specific setup. The only visible changes are the constant offsets in the relative corrections that can already be observed for the integrated cross sections given in table 1. Figure 18 illustrates the distribution in the azimuthal-angle difference of the leading the analogous distributions in the inclusive setup shown in figure 11 is absent in the HiggsHiggs-specific selection cuts applied in the current setup remove such contributions, leaving −1 ]−2 −4 −5 √s = 13 TeV √s = 13 TeV −2 −1 −2 −1 panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the Higgs-specific setup. The left panels compare the leading µ + from the [4µ ] final state with the µ + from the [2µ 2e] final state, the panels in the right column show the corresponding comparison with the subleading µ +. weak corrections on normalization and shape of the azimuthal-angle differences in the Higgs setup is similar in size as in the inclusive setup. Purely photonic corrections are even more suppressed than in the inclusive case. In the Higgs-specific scenario the fraction of events with leading muon pairs close to the Z resonance is enhanced, while the one for subleading muon pairs is reduced (compare figures 7 and 15). As a consequence, the distribution of while the one of the subleading lepton pair is reduced. For small azimuthal-angle differences the situation is reversed. We show in figure 19 the distribution in the angle between the two Z-boson decay planes in the four-lepton CM system in the Higgs-specific setup.4 The distribution as well as the EW corrections closely resemble those of the inclusive setup shown in figure 12. The ratio 4The distribution in the angle between the two Z-boson decay planes shown in figure 3 of ref. [45] is not the lepton momenta in the laboratory system. √s = 13 TeV leading µ +µ − pair √s = 13 TeV subleading µ +µ − pair corresponding EW corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower corresponding comparison with the subleading lepton pair. the Higgs setup, we cannot attribute the deviations of the ratio from one to interference ZZ⋆ → 4 leptons looks qualitatively similar to the distribution of direct ZZ production shown in figure 19, but the distortions by EW corrections are quite different [82, 83]. Conclusions The production of four charged leptons in hadronic collisions at the LHC is an important process class both for the investigation of the interactions between the neutral Standard Model gauge bosons and as background process to searches for new physics and to precision studies of the Higgs boson. In the confrontation of experimental data with theory predic−1 −2 −4 −5 −6 √s = 13 TeV panels), corresponding EW corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the Higgs-search setup. tions precision plays a key role. In this paper we have further improved the theory prediction by calculating the next-to-leading-order electroweak corrections to the production of mediate states. Our results are thus accurate to next-to-leading order in all phase-space regions, no matter whether they are dominated by two, one, or zero resonant Z bosons. Our numerical discussion of the corrections focuses on two different event-selection scenarios, one based on typical lepton-identification criteria only and another one that is specifically designed for Higgs-boson analyses. Since the Higgs-boson mass of about 125 GeV lies below the Z-pair threshold, the flexibility of our calculation, allowing intermediate Z bosons to be far off shell, is essential for the study of four-lepton production as background to the Higgs-boson decay H → ZZ⋆. vestigated further observables and channels with photons in the initial state and included corrections consist of photonic and purely weak contributions displaying rather different features. Photonic corrections can grow very large, to several tens of percent, in particular in distributions where resonances and kinematic shoulders lead to radiative tails. These effects are significantly enhanced when observables within a collinear-unsafe setup are considered. While photonic corrections might be well approximated with QED parton showers, this is not the case for the weak corrections, which are typically of the size of invariant-mass and transverse-momentum distributions. Moreover, the weak corrections below the ZZ threshold distort distributions that are important in Higgs-boson analyses. On the other hand, contributions induced by incoming photons, i.e. photon-photon and quark-photon channels, turn out to be phenomenologically unimportant. Comparing the state to intermediate Z bosons. Interferences in equal-flavour-lepton final states lead to deviations of up to 10% from the mixed-flavour case in off-shell-sensitive phase-space regions. Their effect is, however, in general hidden in the effects of the selection criteria for the lepton pairing. The relative electroweak corrections are widely insensitive to details next-to-leading order roughly in the same way. The full calculation is available in the form of a Monte Carlo program allowing for the evaluation of arbitrary differential cross sections. The best possible predictions for ZZ production processes can be achieved by combining the electroweak corrections of our calculation with the most accurate QCD predictions available to date. Practically, this could be achieved, e.g., by reweighting differential distributions including QCD corrections with electroweak correction factors. In this way, an overall accuracy at the percent level can be achieved for integrated cross sections that are dominated by energy scales up to a few 100 GeV, where the theoretical uncertainty is completely dominated by QCD. We estimate the contribution of missing higher-order electroweak corrections on the integrated cross section to 0.5%. The impact of missing higher-order electroweak corrections grows in the high-energy tails of transverse-momentum and invariant-mass distributions where weak Sudakov (and subleading high-energy) logarithms are known to be large. In this kinematic domain, the size of this uncertainty may be estimated by the square of the relative electroweak correction. The inclusion of the known leading two-loop effects or a resummation of logarithmically enhanced contributions could reduce these theoretical uncertainties. At the same time, multi-photon emission effects could be systematically taken into account by structure functions or parton showers. Such improvements are, however, left to future studies. For upcoming analyses of LHC data, next-to-leading order precision in electroweak corrections is certainly sufficient, and the remaining electroweak uncertainties are negligible compared to the larger uncertainties from missing QCD corrections and from parton distribution functions. Acknowledgments We would like to thank Jochen Meyer for helpful discussions. The work of B.B. and A.D. was supported by the German Federal Ministry for Education and Research (BMBF) under contract no. 05H15WWCA1 and by the German Science Foundation (DFG) under reference number DE 623/2-1. S.D. gratefully acknowledges support from the DFG research training group RTG 2044. The work of L.H. was supported by the grants FPA201346570-C2-1-P and 2014-SGR-104, and partially by the Spanish MINECO under the project MDM-2014-0369 of ICCUB (Unidad de Excelencia “Mar´ıa de Maeztu”). The work of B.J. was supported in part by the Institutional Strategy of the University of Tu¨bingen (DFG, ZUK 63) and in part by the BMBF under contract number 05H2015. The authors acknowledge support by the state of Baden-Wu¨rttemberg through bwHPC and the German Research Foundation (DFG) through grant no. INST 39/963-1 FUGG. Open Access. 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Benedikt Biedermann, Ansgar Denner, Stefan Dittmaier, Lars Hofer, Barbara Jäger. Next-to-leading-order electroweak corrections to the production of four charged leptons at the LHC, Journal of High Energy Physics, 2017, 33, DOI: 10.1007/JHEP01(2017)033