NLO QCD+EW predictions for jets including off-shell vector-boson decays and multijet merging
We present next-to-leading order (NLO) predictions including QCD and electroweak (EW) corrections for the production and decay of off-shell electroweak vector bosons in association with up to two jets at the 13 TeV LHC. All possible dilepton final states with zero, one or two charged leptons that can arise from off-shell and bosons or photons are considered. All predictions are obtained using the automated implementation of NLO QCD+EW corrections in the OpenLoops matrix-element generator combined with the Munich and Sherpa Monte Carlo frameworks. Electroweak corrections play an especially important role in the context of BSM searches, due to the presence of large EW Sudakov logarithms at the TeV scale. In this kinematic regime, important observables such as the jet transverse momentum or the total transverse energy are strongly sensitive to multijet emissions. As a result, fixed-order NLO QCD+EW predictions are plagued by huge QCD corrections and poor theoretical precision. To remedy this problem we present an approximate method that allows for a simple and reliable implementation of NLO EW corrections in the M
Keywords:Electroweak radiative corrections, NLO computations, Hadronic colliders
The production of electroweak (EW) vector bosons in association with jets plays a key role in the physics programme of the Large Hadron Collider (LHC). Inclusive and differential measurements of vector-boson plus multijet cross sections [1, 2, 3, 4, 5, 6] can be performed for a wide range of jet multiplicities exploiting various clean final states that arise from the leptonic decays of and bosons or off-shell photons. This offers unique opportunities to test the Standard Model at high precision and to validate fundamental aspects of theoretical simulations at hadron colliders. Associated multijet production () represents also an important background to a large variety of analyses based on signatures with leptons, missing energy and jets. In particular, it is a prominent background in searches for physics beyond the Standard Model (BSM) at the TeV scale. In this context, the availability of precise theoretical predictions for multijet production can play a critical role for the sensitivity to new phenomena and for the interpretation of possible discoveries.
Predictions for multijet production at next-to-leading order (NLO) in QCD [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] are widely available, and the precision of higher-order QCD calculations has already reached the next-to-next-to-leading order (NNLO) for jet [20, 21]. Also EW corrections can play an important role. Their inclusion is mandatory for any precision measurement. Moreover, EW corrections are especially relevant at the TeV scale, where large logarithms of Sudakov type [22, 23, 24, 25, 26, 27, 28] can lead to NLO EW effects of tens of percent. While NLO predictions for electroweak-boson production in association with a single jet [29, 30, 31, 32, 33, 34, 35, 36, 37] have been available for a while, thanks to the recent progress in NLO automation also multijet calculations at NLO EW became feasible. In particular, various algorithms for the automated generation of one-loop scattering amplitudes have proven to possess the degree of flexibility that is required in order to address NLO EW calculations [38, 39, 40, 41, 42, 43]. Predictions for vector-boson plus multijet production at NLO EW are motivated by the large impact of EW Sudakov effects on BSM signatures with multiple jets  and, more generally, by the abundance of multijet emissions in jets at high energy. First NLO EW predictions for vector-boson production in association with more than one jet have been presented for  and for on-shell -boson production with up to three associated jets at NLO QCD+EW . Independent NLO EW results for jets have been reported in .
In this paper we present new NLO QCD+EW results for jets that involve up to two jets and cover all possible signatures resulting from off-shell vector-boson decays into charged leptons or neutrinos, i.e. we perform full and calculations for jets, jets, jets and jets. For convenience, the above mentioned processes will often be denoted as jet(s) production, while all results in this paper correspond to off-shell jet(s) production.
Our predictions are obtained within the fully automated NLO QCD+EW framework  provided by the OpenLoops [40, 46] generator in combination with the Munich  and Sherpa [48, 49, 50] Monte Carlo programs. Off-shell effects in vector-boson decays are fully taken into account thanks to a general implementation of the complex-mass scheme  at NLO QCD+EW in OpenLoops. This is applicable to any process that involves the production and decay of intermediate electroweak vector bosons, top quarks and Higgs bosons.
Higher-order calculations for jets are obviously relevant for physical observables that involve at least hard jets, but they can play a very important role also for more inclusive observables where less than hard jets are explicitly required. Prominent examples are provided by the inclusive distributions in the transverse momentum () of the leading jet and in the total transverse energy. As is well known, the tails of such distributions receive huge contributions from multijet emissions that tend to saturate the recoil induced by the leading jet, while the vector boson remains relatively soft. As a result, NLO QCD predictions for jet at high jet are plagued by giant -factors , and their accuracy is effectively reduced to leading order due to the dominance of -jet final states with . In this situation it is clear that also NLO EW parton-level results for jet are not applicable as they entirely miss the dominant source of EW higher-order effects, namely Sudakov-type EW corrections to multijet production. At fixed order in perturbation theory, the natural remedy would be given by jet calculations with NNLO QCD and mixed NNLO QCD–EW corrections. Very recently, mixed QCD–EW corrections of to Drell–Yan processes in the resonance region have been presented in [53, 54]. However, a corresponding calculation for jet is clearly out of reach. Thus, as a viable alternative, in this paper we will adopt the multijet merging approach at NLO [55, 56, 57, 58], which allows one to combine NLO simulations of jets matched to parton showers in a way that guarantees parton-shower resummation and NLO accuracy in all phase-space regions with up to resolved jets. While multijet merging methods at NLO QCD—and applications thereof to multijet production [55, 59]—are already well established, in this paper we address the inclusion of NLO EW corrections for the first time. To this end we exploit an approximate treatment of NLO EW corrections, based on exact virtual EW contributions in combination with an appropriate cancellation of infrared singularities. This allows us to implement NLO EW effects in the M
The paper is organised as follows. In Section 2 we provide technical aspects related to the employed tools and the setup of the calculation. Giant -factors for jet production and related issues are recapitulated in Section 3. In Section 4 we present fixed-order NLO QCD+EW predictions for jets including all channels with off-shell or decays to leptons and neutrinos. The merging of NLO QCD+EW predictions for processes with variable jet multiplicity is addressed in Section 5, which starts with an illustration of NLO merging features based on the exclusive-sums approach at parton level. In the following we introduce and validate an approximation of NLO EW corrections which is then used in order to inject NLO EW precision into the M
2 Technical ingredients and setup of the simulations
2.1 Considered processes and perturbative contributions
In this paper we study the production and decay of electroweak bosons () in association with one and two jets at NLO QCD+EW, including off-shell effects and taking into account all decay channels with leptons and neutrinos, i.e. we address off-shell and processes with , , and final states in combination with jets. In the case of charged leptons, only one generation is included, whereas for invisible -boson decays all neutrino species () are taken into account trivially.
In general, NLO QCD and EW corrections have to be understood within a mixed coupling expansion in and , where Born and one-loop scattering amplitudes for a given process consist of towers of contributions with a fixed overall order that is distributed among QCD and EW couplings in different possible combinations.
The production and off-shell decay of jet involves a unique LO contribution of and receives NLO QCD corrections of and NLO EW corrections of . Representative Feynman diagrams are illustrated in Figs. 1 and 2. Here it is important to keep in mind a somewhat counter-intuitive feature of NLO EW corrections, namely that real emission at does not only involve photon bremsstrahlung (Fig. (b)b) but also jet final states resulting from the emission of quarks through mixed QCD–EW interference terms (Fig. (c)c).
The LO production and off-shell decay of jets receives contributions from a tower of terms with powers in the strong coupling. The contributions of , and will be denoted as LO, LO mix and LO EW, respectively. The two subleading orders contribute only via partonic channels with four external (anti)quark legs, and the LO EW contribution includes, inter alia, the production of dibosons with semi-leptonic decays. Representative Feynman diagrams for jet production are shown in Figs. 3 and 4. The NLO contributions of and are denoted as NLO QCD and NLO EW, respectively. They are the main subject of this paper, while subleading NLO contributions of or are not considered. Apart from the terminology, let us remind the reader that NLO EW contributions represent at the same time corrections with respect to LO and corrections to LO mix contributions. Therefore, in order to cancel the leading logarithmic dependence on the renormalisation and factorization scales, NLO EW corrections should be combined with LO and LO mix terms.111 LO mix and NLO EW contributions are shown separately in the fixed-order analysis of Section 4, while in the merging framework of Section 5 they are systematically combined.
For what concerns the combination of NLO QCD and NLO EW corrections,
as a default we adopt an additive prescription,
Here, for the case of jet production, is the LO cross section, while and correspond to the and corrections, respectively. Alternatively, in order to identify potentially large effects due to the interplay of EW and QCD corrections beyond NLO, we present results considering the following factorised combination of EW and QCD corrections,
In situations where the factorised approach can be justified by a clear separation of scales—such as where QCD corrections are dominated by soft interactions well below the EW scale—the factorised formula (3) can be regarded as an improved prediction. However, in general, the difference between (2) and (3) should be considered as an estimate of unknown higher-order corrections of QCD–EW mixed type.
Subleading Born and photon-induced contributions of and will also be investigated and partly included in our predictions.
2.2 Methods and tools
Predictions presented in this paper have been obtained with the Monte Carlo frameworks Munich+OpenLoops and Sherpa+OpenLoops, which support in a fully automated way NLO QCD+EW simulations  at parton level and particle level, respectively. Virtual QCD and EW amplitudes are provided by the publicly222The publicly available OpenLoops amplitude library includes all relevant matrix elements to compute NLO QCD corrections, including colour- and helicity-correlations and real radiation as well as loop-squared amplitudes, for more than a hundred LHC processes. Libraries containing NLO EW amplitudes will be provided soon. available OpenLoops program , which is based on a fast numerical recursion for the generation of one-loop scattering amplitudes . Combined with the Collier tensor reduction library , which implements the Denner–Dittmaier reduction techniques [61, 62] and the scalar integrals of , the employed recursion permits to achieve very high CPU performance and a high degree of numerical stability. A sophisticated stability system is in place to rescue potential unstable phase-space points via a re-evaluation at quadrupole precision using CutTools , which implements the OPP method , together with the OneLOop library . As anticipated in the introduction, in order to address the production and decay of unstable particles, the original automation of one-loop EW corrections in OpenLoops  was supplemented by a fully general implementation of the complex-mass scheme .
All remaining tasks, i.e. the bookkeeping of partonic subprocesses, phase-space integration and the subtraction of QCD and QED bremsstrahlung are supported by the two independent and fully automated Monte Carlo generators Munich  and Sherpa [48, 49, 50]. The first one, Munich, is a fully generic and very fast parton-level Monte Carlo integrator, which has been used, mainly in combination with OpenLoops, for various pioneering NLO multi-leg [67, 68, 69, 70] and NNLO applications [71, 72, 73, 74, 75, 76]. Sherpa is a particle-level Monte Carlo generator providing all stages of hadron collider simulations, including parton showering, hadronisation and underlying event simulations. It was used in the pioneering NLO QCD calculations of vector-boson plus multijet production [14, 15, 16, 17, 18, 19], as well as for their matching to the parton shower  and the merging of multijet final states at NLO . For tree amplitudes, with all relevant colour and helicity correlations, Munich relies on OpenLoops, while Sherpa generates them internally with Amegic  and Comix . For the cancellation of infrared singularities both Monte Carlo tools, Munich and Sherpa, employ the dipole subtraction scheme [80, 81]. Both codes were extensively checked against each other, and sub-permille level agreement was found.
2.3 Physics objects and selection cuts
For the definition of jets we employ the anti- algorithm  with . More precisely, in order to guarantee infrared safeness in presence of NLO QCD and EW corrections, we adopt a democratic clustering approach [83, 84, 85], where QCD partons and photons are recombined. In order to ensure the cancellation of collinear singularities that arise from collinear photon emissions off charged leptons and quarks, collinear pairs of photons and charged fermions with are recombined via four-momentum addition, and all observables are defined in terms of such dressed fermions. Fermion dressing is applied prior to the jet algorithm, and photons that have been recombined with leptons, as well as (dressed) leptons, are not subject to jet clustering.
After jet clustering QCD jets are separated from photons by imposing an upper bound to the photon energy fraction inside jets. In this case, the cut is applied only to photons that are inside the jet, but outside the technical recombination cone with . The recombination of (anti)quark–photon pairs with represents a technical regularisation prescription to ensure the cancellation of collinear photon–quark singularities. As demonstrated in , this provides an excellent approximation to a more rigorous approach for the cancellation of collinear singularities based on fragmentation functions.
Events will be categorised according to the number of anti- jets with in the transverse-momentum and pseudo-rapidity region
Additionally, for certain observables we present results vetoing a second jet with details explained in the text.
2.4 Input parameters, scale choices and variations
As input parameters to simulate jets at NLO QCD+EW we use the gauge-boson masses and widths 
The latter are obtained from state-of-the art theoretical calculations. For the top quark we use the mass reported in , and we compute the width at NLO QCD,
For the Higgs-boson mass and width  we use
Electroweak contributions to jets involve topologies with -channel top-quark and Higgs propagators that require a finite top and Higgs width. However, at the perturbative order considered in this paper, such topologies arise only in interference terms that do not give rise to Breit–Wigner resonances. The dependence of our results on and is thus completely negligible.
All unstable particles are treated in the complex-mass scheme , where width effects are absorbed into the complex-valued renormalised masses
The electroweak couplings are derived from the gauge-boson masses and the Fermi constant, , using
where the -boson mass and the squared sine of the mixing angle,
are complex-valued. The -scheme guarantees an optimal description of pure SU(2) interactions at the electroweak scale. It is the scheme of choice for jets production, and it provides a very decent description of jets production as well.
The CKM matrix is assumed to be diagonal, while colour effects and related interferences are included throughout, without applying any large- expansion.
For the calculation of hadron-level cross sections we employ the NNPDF2.3 QED parton distributions  which include NLO QCD and LO QED effects, and we use the PDF set corresponding to .333To be precise we use the NNPDF23_nlo_as_0118_qed set interfaced through the Lhapdf library 5.9.1 (Munich) and 6.1.5 (Sherpa) . Matrix elements are evaluated using the running strong coupling supported by the PDFs, and, consistently with the variable flavour-number scheme implemented in the NNPDFs, at the top threshold we switch from five to six active quark flavours in the renormalisation of . All light quarks, including bottom quarks, are treated as massless particles, and top-quark loops are included throughout in the calculation. The NLO PDF set is used for LO as well as for NLO QCD and NLO EW predictions.
In all fixed-order results the renormalisation scale and factorisation scale are set to
where is the scalar sum of the transverse energy of all parton-level final-state objects,
Also QCD partons and photons that are radiated at NLO are included in , and the vector-boson transverse energy, , is computed using the total (off-shell) four-momentum of the corresponding decay products, i.e.
In order to guarantee infrared safeness at NLO EW, the scale (12) must be insensitive to collinear photon emissions off quarks and leptons. To this end, all terms in (12)–(13) are computed in terms of dressed leptons and quarks, while the term in (12) involves only photons that have not been recombined with charged fermions.
Our default scale choice corresponds to , and theoretical fixed-order uncertainties are assessed by applying the scale variations , , , , , , , while theoretical uncertainties of our M
3 Giant -factors and electroweak corrections for jet production
In this section we start our discussion of jets production at NLO QCD+EW by recalling some pathological features of fixed-order calculations for jet. Such observations will provide the main motivation for the multiparticle calculations and the multijet merging approach presented in Sections 4 and 5.
It is well known that NLO QCD predictions for jet production [7, 8, 9, 34] suffer from a very poor convergence of the perturbative expansion, which manifests itself in the form of giant -factors  at large jet transverse momenta. In this kinematic regime the NLO QCD cross section is dominated by dijet configurations where the hardest jet recoils against a similarly hard second jet, while the vector boson remains relatively soft. Such bremsstrahlung configurations are effectively described at LO, with correspondingly large scale uncertainties. Moreover, in this situation NLO EW calculations for jet are meaningless, as they completely miss EW correction effects for the dominating dijet configurations.
The above mentioned anomalies are clearly manifest in Fig. 5, where NLO QCD and EW effects in jet444 A similar behaviour is encountered also in the various other channels with jet final states. A more detailed discussion of the interplay between QCD and EW corrections in the presence of giant -factors, for the case of jets production, can be found in Section 6.1 of . are plotted versus the transverse momenta of the reconstructed vector boson, defined in terms of their decay products, i.e. for , and of the leading jet. While overall QCD corrections to the -boson distribution are moderate (at the level of 40-50%) they strongly increase in the tail of the distribution reaching 100% at 3 TeV. In the case of the jet- the QCD corrections show a clear pathological behaviour growing larger than several 100% in the multi-TeV region. In the distribution, NLO EW corrections present a consistent Sudakov shape, with corrections growing negative like and reaching a few tens of percent in the tail. However, as reflected in the sizeable disparity between additive QCD+EW and multiplicative QCDEW combinations, the large size of NLO QCD and NLO EW effects suggests the presence of important uncontrolled mixed NNLO QCD–EW corrections. In the case of the jet- distribution these problems become dramatic. Besides the explosion of NLO QCD corrections, in the multi-TeV range we observe a pathological NLO EW behaviour, with large positive corrections instead of negative Sudakov effects. On one side, similarly as for the giant QCD -factor, this feature can be attributed to hard dijet configurations that enter the NLO EW bremsstrahlung through mixed QCD–EW terms of (see Fig. (c)c). On the other side, EW Sudakov effects are completely suppressed due to the absence of one-loop corrections for jet configurations.
In principle, the pathological behaviour of NLO predictions can be avoided by imposing a cut that renders the jet cross section sufficiently exclusive with respect to extra jet radiation. For instance, as shown in the right plot of Fig. 5, suppressing bremsstrahlung effects with a veto against dijet configurations with angular separation leads to well-behaved QCD predictions and a standard NLO EW Sudakov behaviour, with up to 40% corrections at .
Thus, giant -factors and related issues can be circumvented through a jet veto. However, in order to obtain a precise theoretical description of inclusive jets production at high , it is clear that fixed-order NLO QCD+EW calculations for one-jet final states have to be supplemented by corresponding predictions for multijet final states. This task, as well as the consistent merging of NLO QCD+EW cross sections with different jet multiplicity, will be the subject of the rest of this paper.
4 Fixed-order predictions for jet production
In this section we present numerical results for jet production, including NLO QCD and EW corrections, as well as subleading Born and photon-induced contributions.
4.1 NLO QCD+EW predictions
In the following, we discuss a series of fixed-order NLO QCD+EW results for jets including leptonic decays, i.e. we investigate the processes jets, jets, jets and jets at 13 TeV. We will focus on the effect of EW corrections on the spectra of reconstructed vector bosons, charged leptons and jets. Such observables are of direct relevance as a background for many searches for new physics including dark matter at the LHC. Instead of presenting the four processes and their higher-order corrections independently, we will mostly show them together for the different observables in order to highlight important similarities and investigate possible differences. Additionally, for jets we show distributions in the transverse mass and missing energy, while for jets we show the distribution in the invariant mass of the leptonic decay products. Predictions for further kinematic observables are presented in Appendix A.555Our NLO EW predictions for jets have been compared in detail against the results of . Good agreement was found within the small uncertainties due to the different treatment of photons and b-quark induced processes.
Figure 6 displays results for the transverse-momentum spectra of the reconstructed (off-shell) vector bosons. For all processes NLO QCD corrections are remarkably small, and even in the tails scale uncertainties hardly exceed 10%. In contrast, NLO EW corrections feature a standard Sudakov behaviour and become very large at high . They exceed QCD scale uncertainties already at a few hundred GeV and reach about at 2 TeV. Due to the small size of QCD corrections, for all processes we observe a good consistency between NLO QCD+EW and NLO QCDEW results. As expected, QCD and EW corrections for jets turn out to be very similar to the ones observed in the corresponding calculation of  where the boson was kept on-shell.
In Fig. 7 we plot, where applicable, the spectra of the hardest lepton. The behaviour of the QCD and EW corrections is very similar to the one observed for the of the reconstructed vector bosons. Clearly, the observed large Sudakov corrections are a result of the TeV scale dynamics that enter the production of a high- vector boson, while they are hardly affected by vector-boson decay processes, which occur at much smaller energy scales.
Figures 8 and 9 present distributions in the transverse momenta of the hardest and second-hardest jet, respectively. Again, the perturbative QCD expansion turns out to be very stable, with scale uncertainties that hardly exceed 10%. In these jet- distributions we observe smaller NLO EW corrections as compared to the case of the vector-boson spectrum. This is due to the fact that and bosons carry larger SU(2) charges as compared to gluons and quarks inside jets. Thus, the largest EW Sudakov corrections arise when the vector-boson is highest, while very hard jets in combination with less hard vector bosons yield less pronounced EW Sudakov logarithms. We also find that, at a given , the second jet always receives larger EW corrections than the first jet. Quantitatively, the EW corrections to the different jet and jet processes are rather similar. Thus, corresponding ratios are expected to be only mildly sensitive to EW (or QCD) corrections.
In Figure 10 we show distributions in the transverse mass, , and in the missing transverse energy (i.e. the spectrum of the neutrino) for jet production. Both observables are of paramount importance in many BSM searches, especially in the high-energy regime. Again, QCD effects and uncertainties turn out to be rather mild. As far as EW corrections are concerned, at large transverse masses we observe only a minor impact, which does not exceed and remains at the level of QCD scale uncertainties. In contrast, and as expected, the missing-energy distributions follow the behaviour of the lepton- distribution shown in Fig. 7, and NLO EW corrections reach about at 2 TeV.
Finally, in Fig. 11 we turn to the differential distribution in the invariant mass, , of the lepton pair produced in jets. The plotted range corresponds to the event selection specified in Table 1 and does not extend up to the high-energy region, where EW Sudakov effects would show up. However, the NLO EW corrections are sensitive to QED radiation off the charged leptons and shift parts of the cross section from above the Breit–Wigner peak to below the peak. The observed shape of the EW corrections is qualitatively very similar to the well-known NLO EW corrections to neutral-current Drell–Yan production [91, 92]. In this kinematic regime, QCD corrections are very small and always below 10%, while scale uncertainties are as small as a few percent.
In summary, NLO QCD+EW effects for jets turn out to be completely free from the perturbative instabilities that plague NLO predictions for jet production: the perturbative QCD expansion is very well behaved, and NLO EW corrections feature, as expected, Sudakov effects that become very large at the TeV scale, especially for jet configurations where the highest transverse momentum is carried by the electroweak vector boson.
4.2 Subleading Born and photon-induced contributions
In this section we quantify the numerical impact of subleading Born and photon-induced ( and initial states) contributions to jet production with leptonic decays, i.e. tree-level contributions of and .666The subleading Born contributions of are dominated by diboson production with semi-leptonic decays. In order to avoid a double counting between diboson and jets processes we do not include those contributions in any of our predictions in the following sections.
Figures 12 and 13 illustrate the subleading contributions for the distributions in the transverse momenta of the reconstructed vector-boson and of the hardest jet, respectively. Although mostly suppressed by several orders of magnitude at small energies, at large energies -initiated production can have a sizable impact on the spectrum of the vector boson, whereas the LO mix contribution grows up to several tens of percent in the multi-TeV region of the jet- spectrum.These effects can both be understood as induced by PDFs: in current PDF fits including QED corrections  the photon density at high Bjoerken strongly increases, while at the same time a relative increase of quark PDFs over the gluon PDF induces an enhancement of the four-quark channel (which involves LO mix terms) over the two-quark channel. Although strongly suppressed in the full range, it is interesting to note that the LO mix contributions to the spectrum of the reconstructed vector bosons feature a different behaviour in the case of vs. and vs. production (see Fig. 12). In all cases we observe a sign flip that results from the interference of resonant EW diagrams with non-resonant QCD amplitudes (see the discussion of “pseudo resonances” in ). However, the location of the sign flip and the subsequent onset of a sizable negative contribution is significantly displaced in the different related processes. This can be attributed to the fact that the position of the sign flip is very sensitive to phase-space boundaries and the relative yields of the various contributing partonic channels, which in turn is sensitive to differences in the PDF luminosities that enter the various processes.
With respect to the large impact of the -initiated production at large vector-boson , one should, however, keep in mind that the photon PDF is still very poorly constrained in this regime . Therefore, we do not include these contributions in any of the predictions for jets production in the rest of the paper.
Having a merging of different jet multiplicities in mind, we want to note that the LO mix contributions to jet production discussed here are in fact identical with the QCD–EW mixed bremsstrahlung contributions to jet production. The multijet merging approach introduced in the next section guarantees a consistent inclusion of such effects without double counting.
5 Multijet merged predictions for jets at NLO QCD+EW
In order to address the need of NLO QCD+EW accuracy for observables that receive sizable contributions from multijet radiation, in this section we introduce an approach that allows one to readily implement NLO QCD+EW effects in the context of multijet merging. The benefits of multijet merging are first illustrated through a naïve combination of fixed-order calculations for jet and jet production based on exclusive sums. Subsequently, we introduce an approximate treatment of EW corrections, based on infrared-subtracted virtual contributions, which allows us to include EW corrections in the M
5.1 Combining jets with exclusive sums
From the discussion of giant -factors in Section 3 it should be clear that a theoretically well behaved and phenomenologically sensible prediction of inclusive jets cross sections can only be achieved combining NLO QCD+EW cross sections for jet and multijet production. In this section, using a naïve merging approach based on exclusive sums  we illustrate how the combination of one- and two-jet NLO samples can stabilise the perturbative QCD convergence of one-jet inclusive observables and guarantee a consistent behaviour of EW corrections. Exclusive sums consist of combinations of fixed-order NLO calculations with variable jet multiplicity, where double counting is avoided by imposing appropriate cuts on the jet transverse momenta . To combine jet and jet samples, we use the dimensionless variable
where and are the transverse momenta of the first two jets in the acceptance region (4), and if there is only one jet within the acceptance. The exclusive sum is built by imposing a cut that separates the phase space into complementary regions, and . In order to avoid a double counting of topologies with two hard jets, the jet sample is restricted to the region , which corresponds to one-jet topologies, whereas the jet sample is used to populate the region, characterised by the presence of two hard jets.
In Figures 14 and 15 we present leading-jet and vector-boson distributions for inclusive jets and jets production, where the one- and two-jet contributions are combined using a separation cut . In the distribution of the hardest jet we observe, as expected, that above a few hundred GeV the impact of two-jet topologies is overwhelming. In contrast, for GeV their contribution tends to be suppressed by the the acceptance cut on the second jet, GeV, which effectively corresponds to . Thanks to the fact that the huge contributions from two-jet topologies are included starting from Born level and supplemented by NLO QCD+EW corrections, the exclusive-sums approach leads to a drastic improvement of the perturbative convergence as compared to fixed-order predictions for inclusive jet production in Fig. 5 (left). In fact, in the full range considered we observe moderate NLO QCD corrections and scale uncertainties. Moreover, NLO EW effects in Fig. 14 feature a consistent Sudakov behaviour, with corrections around 2 TeV. Including also QCD–EW mixed Born terms of (LO mix) in the two-jet sample, we observe that at the TeV scale their contribution becomes sizable and can even overcompensate the negative effects of EW Sudakov type. Apart from these quantitative considerations, it is important to realise that mixed Born contributions in the two-jet region () represent the natural continuation of NLO mixed bremsstrahlung in the one-jet region (). Their inclusion is thus crucial for a consistent combination of different jet multiplicities.
In the vector-boson distribution (Fig. 15) we observe that, similarly as in Fig. 14, the relative weight of jet topologies grows with up to about 300 GeV as a result of the acceptance cut on the second jet. However, in contrast to the case of the jet , in the region of high vector-boson , where the separation cut comes into play, we see that one-jet contributions become increasingly important again. This indicates that the higher a boost of the boson is required by the observable, the less likely it is to have two jets of comparable , leading to a hierarchical pattern of QCD radiation. In this situation NLO calculations for jet prodution are expected to be reliable, and in fact we find that inclusive jet predictions and exclusive sums provide similarly well behaved results. In both cases the quality of the perturbative QCD expansion turns out to be good, and in the multi-TeV regime we observe the usual negative NLO EW effects, which can become as large as . We also note that, as compared to fixed-order jet inclusive results in Fig. 5 (left), exclusive sums lead to a smaller difference between the QCD+EW and QCDEW prescriptions. Finally, at high vector-boson we find that, consistent with the subleading role of two-jet topologies, mixed Born contributions to jets are irrelevant.
5.2 Virtual approximation of NLO EW corrections
As discussed in the following, virtual EW corrections with an appropriate infrared subtraction can provide a fairly accurate approximation of exact NLO EW effects. The fact that such an approximation does not require the explicit integration of subtracted real-emission matrix elements represents an important technical simplification. In particular, since Born and infrared-subtracted EW virtual contributions live on the same -parton phase space, the combination of contributions with variable jet multiplicity can be realised with a multijet merging approach of LO complexity. The main physical motivation for a virtual EW approximation is given by the fact that Sudakov EW logarithms—the main source of large NLO EW effects at high energy—arise only from virtual corrections. Moreover, in various cases, such as for vector-boson production in association with one  or two jets , it turns out that a virtual EW approximation can provide percent-level accuracy for a wide range of observables and energy scales, also well beyond the kinematic regions where Sudakov EW logarithms become large.
Motivated by these observations, we adopt the following virtual approximation for the NLO EW corrections to jet production,
Here, stands for the Born contribution of , and denotes the exact one-loop EW corrections of . The cancellation of virtual infrared singularities is implemented through the term, which represents the NLO EW generalisation of the Catani–Seymour operator [95, 96, 41]. This latter term does not contain the EW and operators. It results from the endpoint term of the analytic integration over all dipole subtraction terms of , which arise from the insertion of QED and QCD dipole kernels in squared Born matrix elements and QCD–EW mixed Born terms, respectively.
In the following the shorthand will be used to denote the virtual EW approximation of (15). The accuracy of this approximation is illustrated in Figures 16–19 by comparing it to exact NLO EW results for various (physical and unphysical) differential observables in jet production.777Process-dependent correction factors are introduced in Figures 16–19 such that the integrated NLO QCD+EW predictions match the complete NLO QCD+EW results. These factors are for +jets, for +jets and for +jets. Exact and approximate results are compared both for the case of a conventional NLO calculation for jet () and combining NLO predictions for jets with exclusive sums (). Exclusive sums provide a quantitative indication of the accuracy of the approximation in a framework that mimics, although in a rough way, the multijet merging approach that will be adopted in Sections 5.3–5.5.
For the various processes and distributions in Figures 16–19 the approximation turns out to be in generally good agreement with exact NLO predictions. The most striking exception is given by the and invariant-mass distributions in the off-shell region below the Breit–Wigner peak. In this case, real QED radiation off the charged leptons leads to corrections of a few tens of percent, which can not be reproduced by the approximation as exclusive real photon emission is not included. In contrast, for distributions in the transverse momentum of the vector bosons or of the charged leptons that arise from their decays, we observe very good agreement, typically at the 1–2% level, from low up to the multi-TeV region.
The leading-jet distribution represents a special case. Here, the approximation performs quite well up to about 500 GeV, but at the TeV scale it is plagued by sizable inaccuracies. We have checked that this is largely due to the contribution of mixed bremsstrahlung, i.e. to the QCD–EW interference between matrix elements that describe the real emission of QCD partons at . Such contributions are not covered by the approximation, while in a standard NLO EW calculation for jet () they can reach 30–50% in the multi-TeV region. In contrast, in the exclusive-sums approach mixed bremsstrahlung is suppressed by the separation cut between 1-jet and 2-jet regions (), and the discrepancy between exact EW corrections and approximation is reduced to less than 10% at 3 TeV. On the one hand, this level of agreement can be further improved by lowering the value of the separation cut. Thus in our implementation of multijet merging we will adopt a merging cut that corresponds to in the multi-TeV region. On the other hand, for a realistic description of EW effects, it is clear that the sizable contribution from mixed bremsstrahlung should be included also above the merging cut. In the M
In summary, in absence of kinematic constraints that confine vector bosons in the off-shell regime below the resonance region, combining the approximation of (15) with mixed bremsstrahlung contributions can reproduce NLO EW predictions with an accuracy of 1–2% up to transverse momenta of the order of 1 TeV or more.
As a basis to combine NLO EW corrections with multijet merging, in this section we recapitulate the essential features of the M
In the leading-order formulation of the M
where summarises the relevant squared Born matrix elements convoluted with PDFs and summed/averaged over all partonic degrees of freedom. The theta function ensures that all partons in the matrix elements correspond to resolved jets, while denotes a truncated vetoed parton shower that is restricted to the unresolved regions, , as explained in more detail below. For the highest matrix-element multiplicity, , the region is inclusive with respect to higher-order radiation. Thus, the -veto is relaxed to a -veto, and the truncated parton shower can fill the whole phase space below.
The truncated vetoed shower supplements multijet matrix elements with Sudakov suppression factors that render resolved jet emissions equally exclusive as shower emissions. In combination with the CKKW scale choice [99, 100], this guarantees a smooth transition from matrix-element to parton-shower predictions across and ensures the restauration of the parton shower’s resummation properties in the matrix-element region. As a result, the dependence of physical observables is kept at a formally subleading level with respect to the logarithmic accuracy of the parton shower. The implementation of the above aspects of the merging procedure requires, for each multijet event, the determination of a would-be shower history consisting of a core process, characterised by a certain core scale , and a series of subsequent branchings at scales . In the M
The truncated parton shower in (17) starts at the resummation scale and is stopped and restarted at each reconstructed branching scale . At each stage a kernel corresponding to the actual partially clustered configuration is used. Finally, the shower terminates at the infrared cutoff, . The Sudakov form factor that guarantees the exclusiveness of -jet contributions is generated by vetoing the entire event in case of any resolved emission () of the truncated shower for . Since the role of the Sudakov suppression is to avoid double counting between contributions with different numbers of resolved jets, unresolved emissions () are not vetoed.101010Note that, for -jet configurations, in spite of , also truncated shower emissions with can give rise to unresolved jets with due to the different nature of the shower evolution variable and the -measure .
The factorisation scale is set equal to the core scale, , while the strong coupling in multijet Born matrix elements is evaluated at the renormalisation scale , defined through
where and are the overall factors for the LO cross section of the actual multijet process and for the related core process, respectively.
In the case of jets, the shower history is determined by stepwise clustering of multijet events based on the relative probability of all possible QCD and EW splitting processes, using matrix-element information to select allowed states only.111111For example, in a configuration identifying a splitting would be allowed by the parton shower and preferred in many regions of phase space over the alternatives. However, this would lead to a configuration and, thus, identifying such a splitting needs to be prevented. In particular, also the creation of vector bosons and their (off-shell) decays are treated as possible splitting processes. Thus the clustering of multijet events terminates with three possible core processes: , and . The corresponding default core scales in Sherpa read121212The core scale is driven by the smallest Mandelstam invariant, i.e. by the scale associated with the dominant topology in the core process. In practice is fairly close to the jet transverse momentum after clustering.
Note that excluding EW splittings from the clustering procedure would always lead to a Drell–Yan core process and a core scale , which is clearly inappropriate at high transverse momenta. Including all QCD and EW splittings in the clustering algorithm is thus crucial for the consistent determination of the hard core processes and the related scale. In particular, it allows for shower histories where multijet production proceeds via hard dijet production and subsequent soft vector-boson emission, which corresponds to the dominant mechanism of jets production at high jet .
As discussed in more detail below, the term corresponds to so-called soft events in M
Similarly as in the LO case, soft and hard events in (20) are used as seeds of truncated vetoed parton showers with starting scale and a veto against emissions with . The veto is relaxed when the maximum jet multiplicity is reached. In M
Here, denotes exact Catani–Seymour subtraction terms. They are used to generate emissions with hardness , which arise from -parton configurations, and they match the full-colour infrared singularity structure of real-emission matrix elements. The remaining terms in (21) describe intermediate emissions with hardness that arise from partially clustered configurations with partons and corresponding Catani–Seymour kernels in the usual leading-colour approximation of the parton shower. The matching of the truncated vetoed parton shower to the first NLO emission results in the following expression for hard events,
where stands for real-emission matrix elements. The soft term in (20) reads
It comprises a Born contribution, , a term consisting of virtual QCD corrections and initial-state collinear counterterms,141414Such contributions correspond to the dependent part of the integrated operator in the Catani–Seymour approach. and the integrated subtraction terms (21) associated with the truncated parton shower. Similarly as for LO merging, we set , and the renormalisation scale is chosen according to (18).
5.4 Extension of M
Let us now turn to the extension of the M
Besides all relevant tree plus virtual amplitudes and Catani–Seymour counterterms—which are already available in Sherpa+OpenLoops in the framework of fixed-order NLO QCD+EW automation—a complete implementation of M
Here is the usual NLO QCD soft term (23), and denotes QCD–EW mixed Born contributions of . The terms and represent the renormalised virtual corrections of and the NLO EW generalisation of the Catani–Seymour operator, respectively, as discussed in Section 5.2.
The term cancels all infrared divergences in the virtual EW corrections. This corresponds to an approximate and fully inclusive description of the emission of photons and QCD partons at