# Precise predictions for \(V+\)jets dark matter backgrounds

## Abstract

High-energy jets recoiling against missing transverse energy (MET) are powerful probes of dark matter at the LHC. Searches based on large MET signatures require a precise control of the \(Z(\nu {\bar{\nu }})+\) jet background in the signal region. This can be achieved by taking accurate data in control regions dominated by \(Z(\ell ^+\ell ^-)+\) jet, \(W(\ell \nu )+\) jet and \(\gamma +\) jet production, and extrapolating to the \(Z(\nu {\bar{\nu }})+\) jet background by means of precise theoretical predictions. In this context, recent advances in perturbative calculations open the door to significant sensitivity improvements in dark matter searches. In this spirit, we present a combination of state-of-the-art calculations for all relevant \(V+\) jets processes, including throughout NNLO QCD corrections and NLO electroweak corrections supplemented by Sudakov logarithms at two loops. Predictions at parton level are provided together with detailed recommendations for their usage in experimental analyses based on the reweighting of Monte Carlo samples. Particular attention is devoted to the estimate of theoretical uncertainties in the framework of dark matter searches, where subtle aspects such as correlations across different \(V+\) jet processes play a key role. The anticipated theoretical uncertainty in the \(Z(\nu {\bar{\nu }})+\) jet background is at the few percent level up to the TeV range.

## 1 Introduction

The signature of missing transverse energy (MET) is one of the most powerful tools in the interpretation of data from hadron colliders. In the Standard Model (SM), MET arises from the neutrinos from the decay of *W* and *Z* bosons, and it can be used in their identification and study, as well as in the identification and study of Higgs bosons, top quarks and other SM particles whose decay products include *W* or *Z* bosons. But MET is also an almost omnipresent feature of theories beyond the SM (BSM), where it can be associated to the decay of new particles to *W* and *Z* bosons, or directly to the production of new stable, neutral and weakly interacting particles. Typical examples are theories with dark matter (DM) candidates, or Kaluza–Klein theories with large extra dimensions. Depending on the details, MET is accompanied by other model-discriminating features, such as the presence of a small or large multiplicity of hard jets, or of specific SM particles. The experimental search for these extensions of the SM relies on a proper modeling of the SM backgrounds to the MET signature. The determination of these backgrounds is ideally done by using data control samples, but theoretical input is often helpful, or even necessary, to extend the experimental information from the control to the signal regions, or to extend the application range of the background predictions and to improve their precision [1, 2, 3].

In this paper we focus on the theoretical modeling of the SM \(V+\) jet backgrounds to inclusive production of large MET recoiling against one or more hadronic jets. These final states address a broad set of BSM models, where the production of an otherwise invisible final state is revealed by the emission of one or more high-\(p_\mathrm {T}\) jets from initial-state radiation, where \(p_\mathrm {T}\) is the momentum in the transverse plane.^{1} Recent publications by ATLAS [5] and CMS [6, 7], relative to LHC data collected at \(\sqrt{s}=13\) TeV, document in detail the current experimental approaches to the background evaluation. The leading background is \(Z(\nu {\bar{\nu }})+\) jet production, followed by \(W(\ell \nu )+\) jet (in particular for \(\ell =\tau \) or when the lepton is outside of the detector).^{2} The experimental constraints on \(Z(\nu {\bar{\nu }})+\) jet production at large MET can be obtained from accurate measurements of \(V+\) jet production processes with visible vector-boson signatures. It is quite obvious, for example, that the measurement of \(Z(\ell ^+\ell ^-)+\) jets with \(\ell =e,\mu \) is the most direct and reliable proxy for \(Z(\nu {\bar{\nu }})+\) jets. This control sample, however, is statistics limited, due to the smaller branching ratio of *Z* bosons to charged leptons relative to neutrinos. To extrapolate the shape of the *Z* spectrum to the largest \(p_\mathrm {T}\) values, therefore, requires a theoretical prediction. The larger statistics of \(W(\ell \nu )+\) jets and \(\gamma +\) jets events makes it possible to directly access the relevant \(p_\mathrm {T}\) range, but the relation between their spectra and the *Z* spectrum needs, once again, theoretical guidance.

The main result of this work is to prove that, thanks to the recent theoretical advances, these goals can be met. This proof requires the analysis of a series of possible effects. On the one hand, the theoretical extrapolation to larger \(p_\mathrm {T}\) of the very precise \(Z(\ell ^+\ell ^-)+\)jets data requires firm control over the shape of the distribution. Several effects, from the choice of parton distribution functions (PDFs) to the choices made for the renormalization and factorization scales used in the calculations, can influence the extrapolation. On the other hand, the level of correlation between the *W*, \(\gamma \) and *Z* spectra must be kept under control. At large \(p_\mathrm {T}\), in particular, large and process-dependent corrections arise due to the growth of the electroweak (EW) corrections, and these may spoil the correlation induced by pure QCD effects. For our analysis we shall use the most up-to-date theoretical predictions available today for the description of vector-boson production at large \(p_\mathrm {T}\). On the QCD side, we rely on the next-to-next-to-leading order (NNLO) calculations, which appeared recently for \(Z+\)jet [8, 9, 10, 11, 12], \(W+\)jet [13, 14] and \(\gamma +\)jet [15, 16] production. On the EW side, we apply full NLO calculations for \(Z+\)jet [17, 18, 19], \(W+\)jet [19, 20] and \(\gamma +\)jet [21] production with off-shell decays of the *Z* and *W* bosons. Given the strong enhancement of EW Sudakov effects in the TeV region, we also include two-loop logarithmic terms at next-to-leading logarithmic (NLL) accuracy for all \(V+\) jet processes [22, 23, 24, 25]. An extensive assessment and discussion of the estimates of missing higher-order terms, and of the relative systematics, is given in the main body of this paper. In particular, in order to address non-trivial issues that arise in the context of dark matter searches, we introduce a global framework for the estimate of theoretical uncertainties in all \(V+\) jet processes, taking into account correlation effects across different processes and \(p_\mathrm {T}\) regions. Also the uncertainties associated with the combination of QCD and EW corrections are discussed in detail.

From the experimental perspective, the determination of the background composition in signal and control regions, and the modeling of other key aspects of experimental analyses (e.g. lepton identification and reconstruction, missing energy, etc.) require a theoretical description of the various \(V+\) jets processes at the particle level. Typically, this is provided by Monte Carlo (MC) samples based on multi-jet merging at LO or NLO QCD, and improvements based on higher-order theoretical calculations can be implemented through reweighting of MC events. For the fit of MC predictions to data, ATLAS and CMS analyses rely on the profile likelihood approach, where experimental and theoretical uncertainties are described in terms of nuisance parameters with Gaussian distributions. In this context, the correlations of theoretical uncertainties across \(p_\mathrm {T}\) bins (shape uncertainties) and across different \(V+\) jets processes play a key role for searches at large MET.

For the implementation of higher-order QCD and EW corrections and for the estimate of theoretical uncertainties in the experimental analysis framework, we propose a procedure based on a one-dimensional reweighting of MC samples. The proposed framework should enable the experiments to carry out their profile likelihood approach, quantifying the impact of the theoretical systematics in their analyses, and validating directly with data the reliability and robustness of the theoretical inputs. In this respect, we would like to stress that, independently of the application to BSM searches, the results in this paper provide a framework for incisive validations of the theoretical calculations. Furthermore, these results might allow for further constraints on PDFs [3, 26].

If the experimental analyses of the MET+jets channel should confirm the usefulness of the approach we propose, the same framework could be adapted to more complex or exclusive final states, in which for example MET is accompanied by a large number of (hard) jets or by specific objects (photons, heavy quarks, Higgs, etc.). These extensions are left for future studies.

The structure of this paper is as follows: In Sect. 2 we introduce the reweighting technique, to incorporate in a MC analysis the effect of higher-order corrections and of their systematic uncertainties including correlations. Section 3 describes details of the setup for our numerical calculations, the employed tools and methods, as well as the detailed definition of physics objects and observables to be used in the context of MC reweighting. In Sect. 4 we discuss higher-order QCD and EW corrections, including the contribution of photon-initiated processes and real vector-boson emission. We present here our approach to the estimate of the various systematics, covering QCD scale, shape and process-dependent uncertainties, as well as uncertainties arising from higher-order EW and mixed QCD–EW corrections. Section 5 contains our summary and conclusions. As detailed in Appendix A, results for all \(V+\)jets processes are available in form of one-dimensional histograms in the vector-boson \(p_\mathrm {T}\) covering central predictions and all mentioned uncertainties. Technical plots on the individual sources of QCD and EW uncertainties are documented in Appendix B.

## 2 Reweighting of Monte Carlo samples

*x*,

*x*should be understood as the vector-boson transverse momentum, \(x=p^{(V)}_\mathrm {T}\), while \({\mathbf {y}}\) generically denotes the remaining variables of the fully differential kinematic dependence of the accompanying QCD and QED activity, including both extra jet and photon radiation, as well as leptons and neutrinos from hadron decays. It is implicitly understood that \(\frac{\mathrm{d}}{\mathrm{d}x}\frac{\mathrm{d}}{\mathrm{d}{} \mathbf{y}}\sigma \) depends on

*x*and \({\mathbf {y}}\), while in \(\frac{\mathrm{d}}{\mathrm{d}x}\sigma \) the variables \({\mathbf {y}}\) are integrated out.

Monte Carlo uncertainties, described by \({\varvec{\varepsilon }}_\mathrm {MC}\), must be correlated in the numerator and denominator on the r.h.s. of Eq. (1), while they can be kept uncorrelated across different processes (apart from \(Z(\nu {\bar{\nu }})+\,\mathrm {jet}\) and \(Z(\ell ^+\ell ^-)+\,\mathrm {jet}\)).

We note that, as opposed to an approach based only on ratios of \(p_\mathrm {T}\) distributions, where theory is used for extrapolations across different processes at fixed \(p_\mathrm {T}\), MC reweighting is more powerful as it supports all possible extrapolations across different processes and \(p_\mathrm {T}\) regions. In particular, it makes it possible to exploit \(V+\) jet precision measurements at moderate \(p_\mathrm {T}\) in order to constrain \(Z(\nu {\bar{\nu }})+\mathrm {jet}\) production in the TeV region.

*x*and the binning of its distribution need to be the same in all three terms. Scale choices, QCD and EW input parameters and PDFs should be the same only in the numerator and denominator of

*x*and all other variables \({\mathbf {y}}\),

As concerns the first condition, we note that, depending on the choice of the observable *x*, using state-of-the-art theory calculations that involve higher-order QCD and EW corrections may not guarantee that Eq. (5) is fulfilled. In fact, there are a number of aspects, i.e. resolved multi-jet emissions, the resummation of soft logarithms in the region of small vector-boson \(p_\mathrm {T}\), soft QCD radiation of non-perturbative origin, multiple photon radiation, or neutrinos and charged leptons resulting from hadron decays, for which fixed-order perturbative calculations of \(pp\rightarrow V+\) jet are less accurate than MC simulations.

Thus, the reweighting variable *x* should be defined such as to have minimal sensitivity to the above-mentioned aspects. In this respect, due to its reduced sensitivity to multiple jet emissions, the vector-boson \(p_\mathrm {T}\) is a natural choice. However, in order to fulfil Eq. (5), the region \(p_\mathrm {T}^{(V)}\ll M_V\) should be excluded from the reweighting procedure, unless QCD Sudakov logarithms are resummed to all orders in the theoretical calculations. Moreover, in order to simultaneously fulfill conditions (5) and (6), any aspect of the reconstructed vector-boson \(p_\mathrm {T}\) that is better described at MC level should be excluded from the definition of *x* and included in \({\mathbf {y}}\). This applies, as discussed in Sect. 3, to multiple photon emissions off leptons, and to possible isolation prescriptions for the soft QCD radiation that surrounds leptons or photons. In general, purely non-perturbative aspects of MC simulations, i.e. MPI, UE, hadronization and hadron decays, should be systematically excluded from the definition of the reweighting variable *x*. Thus, impact and uncertainties related to this non-perturbative modeling will remain as in the original MC samples.

It should be stressed that the above considerations are meant for dark matter searches based on the *inclusive* MET distribution, while more exclusive searches that exploit additional information on hard jets may involve additional subtleties. In particular, for analyses that are sensitive to multi-jet emissions, using the inclusive vector-boson \(p_\mathrm {T}\) as the reweighting variable would still fulfil Eq. (5), but the lack of QCD and EW corrections to \(V+2\) jets production in MC simulations could lead to a violation of Eq. (6). In analyses that are sensitive to the tails of inclusive jet-\(p_\mathrm {T}\) and \(H_\mathrm {T}\) distributions this issue is very serious, and QCD+EW corrections should be directly implemented at MC level using multi-jet merging [19].

In general, as a sanity check of the reweighting procedure, we recommend verifying that, for reasonable choices of input parameters and QCD scales, (N)NLO QCD calculations and (N)LO merged MC predictions for vector-boson \(p_\mathrm {T}\) distributions are in reasonably good agreement within the respective uncertainties. Otherwise, in case of significant MC mismodeling of the \(\frac{\mathrm{d}}{\mathrm{d}x}\sigma ^{(V)}\) distribution, one should check the reliability of the MC in extrapolating TH predictions from the reweighting distribution to other relevant observables.

In general, one could check whether the one-dimensional reweighting via the variable *x* in Eq. (1) can in fact reproduce the dependence of the corrections in other kinematic variables that are relevant for the experimental analysis. To this end, distributions of \(\sigma ^{(V)}\) w.r.t. another kinematic variable \(x'\) should be calculated upon integrating Eq. (1). Switching on and off the corrections on the r.h.s. of Eq. (1) in \(\sigma ^{(V)}_{\mathrm {TH}}\) and taking the ratio of the obtained differential cross sections \(\sigma ^{(V)}\), produces the relative correction to the \(x'\) distribution that could be compared to the corresponding result directly calculated from \(\sigma ^{(V)}_{\mathrm {TH}}\).^{3}

Finally, it is crucial to check that state-of-the-art predictions for absolute \(\mathrm {d}\sigma /\mathrm {d}p_{\mathrm {T}}\) distributions agree with data for the various visible final states.

## 3 Setup for numerical predictions

In this section we specify the physics objects (Sect. 3.1), acceptance cuts and observables (Sect. 3.2), input parameters (Sect. 3.3) and tools (Sect. 3.4) used in the theoretical calculations for \(pp\rightarrow W^\pm /Z/\gamma +\) jet.

The definitions of physics objects, cuts and observables–which specify the setup for the reweighting procedure discussed in Sect. 2 – should be adopted both for theoretical calculations and for their Monte Carlo counterpart in the reweighting factor (3). The details of the reweighting setup are designed such as to avoid any possible deficit in the perturbative predictions (e.g. due to lack of resummation at small \(p_\mathrm {T}\)) and any bias due to non-perturbative aspects of Monte Carlo simulations (e.g. leptons and missing energy from hadron decays). Let us also recall that this setup is completely independent of the physics objects, cuts and observables employed in the experimental analyses.

As concerns input parameters and PDFs, the recommendation of Sect. 3.3 should be applied to all QCD and EW higher-order calculations. In particular, it is mandatory to compute (N)NLO QCD and EW corrections in the same EW input scheme, otherwise NLO EW accuracy would be spoiled. Instead, Monte Carlo simulations and the corresponding \(\frac{\mathrm{d}}{\mathrm{d}x}\sigma ^{(V)}_{\mathrm {MC}}\) contributions to the reweighting factor (3) do not need to be based on the same input parameters and PDFs used for theory predictions.

We recommend handling \(W/Z+\mathrm {jet}\) production and decay on the Monte Carlo side as the full processes \(pp\rightarrow \ell \ell /\ell \nu /\nu \nu +\mathrm {jet}\), i.e. with a consistent treatment of off-shell effects, as is done on the theory side.

### 3.1 Definition of physics objects

In the following we define the various physics objects relevant for higher-order perturbative calculations and for the reweighting in the Monte Carlo counterparts in Eq. (3).

#### 3.1.1 Neutrinos

In parton-level calculations of \(pp\rightarrow \ell \ell /\ell \nu /\nu \nu +\mathrm {jet}\), neutrinos originate only from vector-boson decays, while in Monte Carlo samples they can arise also from hadron decays. In order to avoid any bias in the reweighting procedure, only neutrinos arising from *Z* and *W* decays at Monte Carlo truth level should be considered.

#### 3.1.2 Charged leptons

*W*and

*Z*bosons are defined in terms of dressed leptons, and, in accordance with standard experimental practice, both muons and electrons should be dressed. In this way differences between electrons and muons, \(\ell =e,\mu \), become negligible, and the reweighting function needs to be computed only once for a generic lepton flavor \(\ell \).

Similarly as for neutrinos, only charged leptons that arise from *Z* and *W* decays at Monte Carlo truth level should be considered. Concerning QCD radiation in the vicinity of leptons, no lepton-isolation requirement should be imposed in the context of the reweighting procedure. Instead, in the experimental analysis lepton-isolation cuts can be applied in the usual manner.

#### 3.1.3 *Z* and *W* bosons

*W*and

*Z*bosons are defined as

*Z*and

*W*decays are defined as discussed above.

#### 3.1.4 Photons

At higher orders in QCD, photon production involves final-state \(q\rightarrow q\gamma \) splittings that lead to collinear singularities when QCD radiation is emitted in the direction of the photon momentum. Since such singularities are of QED type, they are not canceled by corresponding virtual QCD singularities. Thus, in order to obtain finite predictions in perturbation theory, the definition of the \(pp\rightarrow \gamma +\) jet cross section requires a photon-isolation prescription that vetoes collinear \(q\rightarrow q\gamma \) radiation while preserving the cancellation of QCD infrared singularities.

*n*are free parameters that allow one to control the amount of allowed QCD radiation in the vicinity of the photon.

The photon-isolation prescription is applicable to QCD as well as to EW higher-order corrections. At NLO EW, \(\gamma +\) jet production involves bremsstrahlung contributions with two final-state photons. In this case, at least one isolated photon is required. The other photon might become soft, guaranteeing the cancellation of related soft and collinear singularities in the virtual EW corrections. In case of two isolated photons in the final state, the hardest photon is considered. In particular, an explicit photon-isolation prescription is mandatory at NLO EW in order to prevent uncanceled singularities from \(q\rightarrow q\gamma \) splittings in the \(\mathcal {O}(\alpha ^2\alpha _{\mathrm {S}})\) mixed EW–QCD contributions from \(qq\rightarrow qq\gamma \) and crossing-related channels.

As a consequence of \(q\rightarrow q\gamma \) collinear singularities and the need to apply a photon-isolation prescription, QCD corrections to \(pp\rightarrow \gamma +\) jet behave differently as compared to \(Z/W+\) jet production. Such differences can be important even at the TeV scale, where one might naively expect that massive and massless vector bosons behave in a universal way from the viewpoint of QCD dynamics. Instead, the presence of collinear \(q\rightarrow q V\) singularities at (N)NLO QCD implies a logarithmic sensitivity to the vector-boson masses, which results, respectively, in \(\ln (R_0)\) and \(\ln (p_{\mathrm {T},V}/M_V)\) terms for the case of massless and massive vector bosons at \(p_{\mathrm {T},V}\gg M_{W,Z}\).

*Z*-boson mass, i.e.

*R*approximation. In this way, using a smooth isolation with \(R_0=R_{\mathrm {dyn}}(p_{\mathrm {T},\gamma },\varepsilon _0)\) mimics the role of the

*Z*- and

*W*-boson masses as regulators of collinear singularities in \(Z/W+\)jet production at high \(p_\mathrm {T}\), while using a fixed cone radius \(R_0\) would correspond to an effective \(M_{\gamma j}\) cut well beyond \(M_{Z,W}\), resulting in a more pronounced suppression of QCD radiation in \(\gamma +\) jet production as compared to \(Z/W+\) jet.

^{4}in this study we use the dynamic cone isolation defined through Eqs. (10) and (11), with parameters

In Fig. 2 we present a comparison of the NLO QCD *K*-factors for \(W/Z+\) jet and \(\gamma +\) jet production with dynamic and fixed-cone isolation. For \(p_{\mathrm {T},\gamma }<290\) GeV, where both isolation prescriptions correspond to a fixed cone radius, the QCD corrections to \(pp\rightarrow \gamma +\) jet grow rapidly with decreasing \(p_{\mathrm {T}} \). At low \(p_{\mathrm {T}} \), due to the smaller cone size, fixed isolation (\(R_0=0.4\)) leads to more pronounced corrections as compared to dynamic isolation (\(R_0=1.0\)), but the slopes of the corresponding \(\gamma +\) jet *K*-factors are quite similar to each other and very different as compared to the ones for \(pp\rightarrow W/Z+\) jet. In the case of fixed isolation, this difference persists also in the high-\(p_{\mathrm {T}} \) regime (apart form the accidental agreement of *K*-factors at \(p_{\mathrm {T},V}\approx 800\) GeV). Instead, in the case of dynamic photon isolation, at large \(p_{\mathrm {T}} \) the QCD corrections to \(\gamma +\) jet and \(W/Z+\) jet production turn out to be remarkably similar, both in shape and size. As expected, the onset of this universal behavior is located close to \(p_{\mathrm {T},\gamma }=290\) GeV, where the isolation radius \(R_{0,\mathrm {dyn}}\) starts varying with \(p_{\mathrm {T}} \) in a way that rejects QCD radiation with \(M_{\gamma j}\lesssim M_{W,Z}\). The differences between \(\gamma +\) jet and \(W/Z+\) jet *K*-factors remain as small as a few percent up to the TeV scale.

#### 3.1.5 QCD partons and photons inside jets

In order to avoid any bias due to the different modeling of jets in MC simulations and perturbative calculations, theory calculations and reweighting should be performed at the level of inclusive vector-boson \(p_\mathrm {T}\) distributions, without imposing any requirement on the recoiling jet(s). Predictions presented in this study are thus independent of specific jet definitions or jet cuts.

Concerning the composition of the recoil, we observe that, at NLO EW, \(q\rightarrow q\gamma \) splittings can transfer an arbitrary fraction of the recoiling momentum from QCD partons to photons. In particular, in \(pp\rightarrow V\gamma j\) contributions of \(\mathcal {O}(\alpha ^2\alpha _{\mathrm {S}})\), the photon can carry up to 100% of the recoil momentum. Such contributions involve soft QCD singularities that are canceled by including also virtual QCD corrections to \(pp\rightarrow V\gamma \). In order to minimize double counting with diboson production,^{5} \(V\gamma \) production at LO is not included in the EW corrections to \(pp\rightarrow Vj\). In practice, as demonstrated in Fig. 3, the relative weight of \(pp\rightarrow V\gamma \) at \(\mathcal {O}(\alpha ^2)\) versus \(pp\rightarrow Vj\) at \(\mathcal {O}(\alpha \alpha _{\mathrm {S}})\) is well below the percent level. Thus the impact of \(\mathcal {O}(\alpha ^2\alpha _{\mathrm {S}})\) contributions from hard \(V\gamma \) production, which are included in this study, should be completely negligible.

### 3.2 Cuts and observables

^{6}

Process | Extra cuts | Observable | Comments |
---|---|---|---|

\(pp\rightarrow \ell ^+\nu _\ell +\) jet | None | \(p_{\mathrm {T},\ell ^+\nu _\ell }\) | \(\ell = e\) or \(\mu \) |

\(pp\rightarrow \ell ^-{\bar{\nu }}_\ell +\) jet | None | \(p_{\mathrm {T},\ell ^-{\bar{\nu }}_\ell }\) | \(\ell = e\) or \(\mu \) |

\(pp\rightarrow \nu _\ell {\bar{\nu }}_\ell +\) jet | None | \(p_{\mathrm {T},\nu _\ell {\bar{\nu }}_\ell }\) | \(\ell =e+\mu +\tau \) |

\(pp\rightarrow \ell ^+\ell ^-+\) jet | \(m_{\ell \ell } > 30\,\text {GeV} \) | \(p_{\mathrm {T},\ell ^+\ell ^-}\) | \(\ell = e\) or \(\mu \) |

\(pp\rightarrow \gamma +\) jet | \(p_{\mathrm {T},\gamma }\) |

For leptons and MET we do not apply any \(p_\mathrm {T}\) or rapidity cuts. Moreover, we do not impose any restrictions on QCD radiation in the vicinity of leptons and MET. Also QCD radiation is handled in a fully inclusive way, i.e. the presence of a recoiling jet is not explicitly required, and, as discussed in Sect. 3.1, at NLO EW the recoil can be entirely carried by a photon. Here we want to stress again that of course the particle-level analysis of the reweighted Monte Carlo samples can (and will) involve a more exclusive event selection than used for the reweighting itself.

*Z*and

*W*decays only a single lepton generation is considered. For \(pp\rightarrow \ell ^+\ell ^-+\) jet an extra invariant-mass cut is applied in order to avoid far off-shell contributions, especially from \(\gamma ^*\rightarrow \ell ^+\ell ^-\) at low invariant mass. The relatively low value of the lower cut, \(m_{\ell \ell }> 30\,\text {GeV} \), is intended to minimize cross section loss due to photon radiation that shifts events from the

*Z*-peak region down to lower invariant mass (see Fig. 4). This choice guarantees a reduced sensitivity with respect to the modeling of QED radiation.

### 3.3 Input parameters, PDFs and QCD scales

Input parameters and PDFs employed for theoretical predictions in this study are specified in the following. Let us recall that, as discussed in Sect. 2, Monte Carlo samples used in the experimental analyses do not need to be generated with the same input parameters and PDFs used for higher-order theoretical predictions.

^{7}

For the calculation of hadron-level cross sections at (N)NLO QCD + (n)NLO EW we employ the LUXqed_plus_PDF4LHC15_nnlo_100 PDF set, which is based on PDF4LHC NNLO PDFs [33, 34, 35, 36, 37, 38] supplemented with QED effects [39]. The same PDF set, and the related \(\alpha _{\mathrm {S}}\) value, is used throughout, i.e. also in the relevant LO and NLO ingredients used in the estimate of theoretical uncertainties. At the level of precision discussed in this study also the uncertainty on the value of \(\alpha _{\mathrm {S}}\) becomes relevant. Given 1% uncertainty on the measured value of \(\alpha _{\mathrm {S}}\) this results in an overall 1–2% normalization uncertainty on the differential \(p_{\mathrm {T}} \) distributions. However, one should keep in mind that in the process ratios this uncertainty cancels completely and thus it is irrelevant for background estimates in DM searches at high-MET. Consistently with the five-flavor number scheme employed in the PDFs, *b*-quarks are treated as massless partons, and channels with initial-state *b*-quarks are taken into account. All light quarks, including bottom quarks, are treated as massless particles, and top-quark loops are included up to NLO throughout. Matrix elements at (N)NLO are evaluated using the five-flavor running of the strong coupling supported by the PDFs and, for consistency, top-quark loops are renormalized in the decoupling scheme. For the NNLO QCD coefficient no top-quark loops are considered.

For the assessment of PDF uncertainties the PDF4LHC prescription [33] is adopted. In addition to standard PDF variations, also additional LUXqed variations for the photon PDF are applied. For more details see more details in Sects. 4.3–4.4.

### 3.4 Computational frameworks

The theoretical predictions presented in Sect. 4 include corrections up to NNLO QCD and NLO EW, as well as Sudakov EW effects at \(\mathcal {O}(\alpha ^2)\). They have been obtained by means of a variety of methods and tools, as detailed in the following.

The NLO QCD and NLO EW calculations for all \(pp\rightarrow V+\) jet processes have been performed with Munich+OpenLoops and/or Sherpa+OpenLoops. In these automated frameworks [19, 40, 41] virtual amplitudes are provided by the OpenLoops program [42, 43], combined with the Collier tensor reduction library [44] or with CutTools [45]. The remaining tasks are supported by the two independent and fully automated Monte Carlo generators Munich [46] and Sherpa [47, 48, 49, 50]. Additionally, we carefully validated the NLO EW predictions against the results of Refs. [17, 18, 20]. The NLO EW calculations for \(pp\rightarrow V+2\) jets performed to test the factorization of QCD and EW corrections have been checked against the one of Ref. [51] for \(pp\rightarrow Z+2\) jets in Ref. [21]. The NLO EW amplitudes for all \(V+\)jet processes in OpenLoops have been supplemented with the one- and two-loop analytical Sudakov logarithms of Refs. [22, 23, 24, 25, 52].

The NNLO QCD predictions for \(Z+\)jet production have been obtained with the parton-level event generator NNLOjet, which provides the necessary infrastructure to perform fully differential calculations at NNLO using the antenna subtraction formalism [53, 54, 55, 56, 57, 58, 59, 60, 61]. The computation of \(pp\rightarrow W+\)jet through NNLO is based on the *N*-jettiness subtraction scheme for NNLO calculations [13]. The above-cut contribution within the *N*-jettiness subtraction was obtained using Munich+OpenLoops. The NNLO QCD prediction for the \(pp\rightarrow \gamma +\)jet process is based on the calculations of Refs. [15, 16] and has been obtained using MCFM [62]. In order to ensure the correctness of the numerical implementation of cuts and other parameters in the NNLO codes, a detailed comparison has been performed at the level of the NLO QCD results as described above.

## 4 Higher-order QCD and EW predictions

To illustrate the effect of higher-order corrections and uncertainties we present a series of numerical results for \(pp\rightarrow V+\) jet at a center-of-mass energy of 13 TeV in the setup specified in Section 3. In particular, \(pp\rightarrow \gamma +\) jet predictions are based on the dynamic photon isolation (13). As anticipated in Sect. 3.1, this prescription provides a very convenient basis for the systematic modeling of the correlation of QCD uncertainties between the various \(V+\) jet production processes (see Sect. 4.1).

Vector-boson \(p_\mathrm {T}\) spectra are plotted starting at 80 GeV, but for the sake of a complete documentation data sets are provided above 30 GeV (see Appendix A). However, we note that in the region of \(p_\mathrm {T}\lesssim 100\) GeV there are potential sources of systematics that we are not controlling or even discussing, as they would require a separate study. These arise from the resummation of QCD Sudakov logarithms or from non-perturbative effects (e.g. an order \(\Lambda _{\mathrm {QCD}}\) average shift of the vector-boson \(p_\mathrm {T}\) associated with the asymmetry of color flow in the final state). Furthermore, as shown later, a reliable correlation between the *Z* / *W* spectra and the photon spectrum requires \(p_\mathrm {T}\) to be large enough so that fragmentation contributions in \(\gamma +\)jet production become small. We also expect that in the \(p_\mathrm {T}\) regions up to a few hundred GeV the statistics are sufficient to guarantee that experimental analyses of missing-\(E_\mathrm {T}\) backgrounds can entirely rely on the direct measurement of the *Z* spectrum measured via \(Z\rightarrow \ell ^+\ell ^-\). As a result, we believe that our conclusions on the systematic uncertainties are most reliable and useful for experimental applications in the region of \(p_\mathrm {T}\) larger than 100–200 GeV.

### 4.1 Higher-order QCD predictions

^{8}\(\mathcal {O}(\alpha \alpha _{\mathrm {S}}^3)\). However, as ingredients for the assessment of some theory uncertainties, also LO and NLO QCD contributions will be used.

*K*-factors, while LO predictions on the r.h.s. of Eq. (27) are taken at the central scale, \({\varvec{\mu }}_0=(\mu _{R,0},\mu _{F,0})\). For the central scale we adopt the commonly used choice

#### 4.1.1 Pure QCD uncertainties

*K*-factors this corresponds to

*K*-factors, also the LO

*K*-factor differs from 1.

*K*-factors \(K^{(V)}_{\mathrm {N}^k\mathrm {LO}}(x)\) and their uncertainties \(\delta ^{(i)}K^{(V)}_{\mathrm {N}^k\mathrm {LO}}(x)\) depend only very weakly

^{9}on

*V*at high \(p_\mathrm {T}\), and in this situation the small process-dependent part of QCD

*K*-factors can be used as an estimator of the degree of correlation across processes. To this end we consider the highest available term in the perturbative expansion,

*K*-factors with respect to \(Z+\) jet production,

*K*-factor differences between processes. The choice of \(Z+\) jet production as reference process in Eq. (38) is arbitrary, but changing the reference process has very little impact on process correlations since the resulting overall shift in \(\delta ^{(3)} K^{(V)}_{\mathrm {N}^k\mathrm {LO}}(x)\) cancels to a large extent in ratios of \(V+\) jet cross sections.

The above prescription should be regarded as conservative, since parts of the available *K*-factors are downgraded from the status of known higher-order corrections to uncertainties. However, thanks to the fact that the \(V+\) jet *K*-factors of the same order *k* are strongly correlated, \(\delta ^{(3)} K^{(V)}_{\mathrm {N}^k\mathrm {LO}}(x) \ll \varDelta K^{(V)}_{\mathrm {N}^k\mathrm {LO}}\), the resulting losses of accuracy in the nominal \(\mathrm {N}^k\mathrm {LO}\) predictions for individual processes are rather small.

For the application to experimental analyses, it is important to keep in mind that the above modeling of process correlations assumes a close similarity of QCD effects between all \(pp\rightarrow V+\) jet processes, which is achieved, in the present study, by means of the dynamic photon isolation (13). Thus, as discussed in Sect. 3.1, experimental analyses that employ a different photon-isolation approach require an additional \(\gamma +\) jet specific uncertainty.

#### 4.1.2 Numerical results

Predictions for \(V+\)jet distributions and their ratios at LO, NLO and NNLO QCD are presented in Figs. 6, 7 and 8 as well as in Figs. 18 and 19 (see Appendix B). In Figs. 7, 18 and 19, scale uncertainties (33), shape uncertainties (35), and process-correlation uncertainties (38) are shown separately, while in Figs. 6 and 8 the three QCD uncertainties are combined in quadrature. Here and in the following *W* denotes \(W^+\) and \(W^-\) combined.

At high transverse momentum, we find that QCD corrections and uncertainties for the various \(V+\,\)jet production processes behave in a very similar way. At NLO the corrections amount to 40–60% with residual uncertainties around 10–20%, while NNLO corrections increase the cross section by 5–10% and reduce the combined uncertainty to 3–10%. Scale variations \(\delta ^{(1)}K_{\mathrm {N}^k\mathrm {LO}}\) and shape variations \(\delta ^{(2)}K_{\mathrm {N}^k\mathrm {LO}}\) are the dominant sources of uncertainty in \(p_{\mathrm {T}} \)-distributions. Their contributions are very similar across \(V+\,\)jet processes. Thus in the ratios scale and shape variations largely cancel, and the process-correlation uncertainty \(\delta ^{(3)}K_{\mathrm {N}^k\mathrm {LO}}\) tends to dominate.

The ratio plots (Fig. 8) allow one to appreciate small differences in the QCD dynamics of the various \(V+\) jet processes. As reflected in the *Z* / *W* ratio, the NLO and NNLO corrections for the corresponding processes are almost identical, with differences below 1–2% up to one TeV. Only at very large \(p_{\mathrm {T}} \) the NLO and also NNLO corrections to \(W+\)jet grow faster than in the case of \(Z+\)jet. This results in an increase of the process-correlation uncertainty \(\delta ^{(3)}K_{\text {NLO}}\) up to about \(5\%\) beyond \(p_{\mathrm {T}} =2~\text {TeV} \).

As can be seen in the \(Z/\gamma \) and \(W/\gamma \) ratios, the higher-order QCD corrections to \(\gamma +\)jet production behave very similarly as for \(Z+\) jet and \(W+\) jet production at large \(p_{\mathrm {T}} \). This is the result of the dynamic photon isolation (13), which guarantees that the differences in the NLO and NNLO corrections remain below 3–4% for \(p_{\mathrm {T}} >200\,\) GeV. Instead, at lower \(p_{\mathrm {T}} \) the behavior of \(\gamma +\) jet production changes drastically due to mass effects, which results in sizable process-correlation uncertainties.^{10} Note that for \(p_{\mathrm {T}} \approx 300~\text {GeV} \) the NLO process-correlation uncertainty in \(pp\rightarrow \gamma +\)jet is accidentally very small (see Fig. 18) yielding a pinch in the total QCD uncertainty for the \(Z/\gamma \) and the \(W/\gamma \) ratios (see also Fig. 19). However, one should keep in mind that an additional analysis-dependent photon-isolation uncertainty (see Sect. 3.1) has to be considered for these ratios.

### 4.2 Electroweak corrections

*W*and

*Z*decays, they are applied only at the level of \(pp\rightarrow V+\) jet production, including off-shell decays at LO.

- (a.1)
virtual EW corrections to \(q{\bar{q}}\rightarrow Vg\);

- (a.2)
\(q{\bar{q}}\rightarrow Vg\gamma \) photon bremsstrahlung;

- (a.3)
virtual QCD corrections to \(q{\bar{q}}\rightarrow V\gamma \), which are needed to cancel soft-gluon singularities from (a.2) if the final-state QCD partons are allowed to become unresolved;

- (a.4)
\(q {\bar{q}}\rightarrow Vq'{\bar{q}}'\) bremsstrahlung, which contributes at \(\mathcal {O}(\alpha ^2\alpha _{\mathrm {S}})\) through the interference of \(\mathcal {O}(eg_S^2)\) and \(\mathcal {O}(e^3)\) tree amplitudes in the same-flavor case, \(q=q'\);

- (a.5)
\(\gamma q\rightarrow V q g\) photon-induced quark bremsstrahlung,

^{11}at \(\mathcal {O}(\alpha ^2\alpha _{\mathrm {S}})\), which plays the dual role of NLO EW correction to the \(q{\bar{q}}\rightarrow Vg\) channel and NLO QCD correction to the \(\gamma q\rightarrow Vq\) channel. As discussed in Sect. 4.3, given the relatively small impact of \(\gamma q\rightarrow V q\) processes at \(\mathcal {O}(\alpha ^2)\), photon-induced contributions of \(\mathcal {O}(\alpha _{\mathrm {S}}\alpha ^2)\) will not be included in the present study; - (a.6)
real-boson emission, i.e. \(pp\rightarrow VV'j\), contributes at \(\mathcal {O}(\alpha ^2\alpha _{\mathrm {S}})\). As discussed in Sect. 4.5, in order to avoid double counting with diboson production, such contributions should be treated as separate background samples and not as part of the EW corrections to \(pp\rightarrow Vj\).

^{12}[63, 64]

In this work we will employ the explicit NLL Sudakov results of Refs. [22, 23, 24, 25, 52], which have been implemented, in addition to exact NLO QCD+NLO EW amplitudes, in the OpenLoops matrix-element generator [19, 40]. Let us recall that the results of Refs. [22, 23, 24, 25, 52] are based on the high-energy limit of virtual one- and two-loop corrections regularized with a fictitious photon mass of order \(M_W\). This generates logarithms of the form \(\alpha ^n\ln ^k({\hat{s}}/M^2_W)\), which correspond to the combination of virtual one- and two-loop EW corrections plus corresponding photon radiation contributions up to an effective cut-off scale of order \(M_W\). In the case of \(V+\) jet production, for physical observables that are inclusive with respect to photon radiation, this approximation is accurate at the one-percent level [21, 22, 25].

#### 4.2.1 Pure EW uncertainties

#### 4.2.2 Numerical results

Contrary to the case of QCD corrections, higher-order EW effects have a significant impact on the shapes of \(p_{\mathrm {T}} \) distributions as well as a pronounced dependence on the scattering process. This behavior is mainly due to the \(p_{\mathrm {T}} \) dependence of EW Sudakov logarithms and their dependence on the SU(2) charges of the produced vector bosons.

As can be seen in Fig. 9, the vector-boson \(p_{\mathrm {T}} \) spectra receive negative EW corrections that grow with \(p_{\mathrm {T}} \) and become very sizable in the tails. At the TeV scale, NLO EW effects reach 20–50% for \(Z+\)jet and \(W+\)jet production, and 10–15% for \(\gamma +\)jet production. As expected from exponentiation, NNLO Sudakov logarithms have positive sign. Thus they compensate in part for the impact of NLO EW corrections.

In Fig. 9 exact NLO EW results are also compared to the NLL Sudakov approximation at the same order, denoted \(\mathrm {nLO}\,{\mathrm {EW}}\). The observed agreement indicates that the Sudakov approximation at NLO works very well, thereby supporting the usage of EW Sudakov logarithms at \(\mathrm {NNLO}\). Moreover, the fact that \(\mathrm {nNLO}\,{\mathrm {EW}}\) results are well consistent with NLO predictions supplemented by the corresponding uncertainties (57) provides an important confirmation of the goodness of the proposed approach for the estimate of EW uncertainties.

As shown in Fig. 11, the various ratios of \(p_{\mathrm {T}} \) distributions and their shape receive significant EW corrections, with the largest effects observed in the \(Z(\ell ^+\ell ^-)/\gamma \) and \(W/\gamma \) ratios. In these ratios the remaining combined EW uncertainties are at the level of few percent in the TeV range, reaching about \(5\%\) for \(p_{\mathrm{T}, V}\simeq 2\) TeV. Interestingly, also the \(Z(\ell ^+\ell ^-)/Z(\nu {\bar{\nu }})\) and \(W^-/W^+\) ratios receive non-negligible EW corrections. In the case of the \(W^-/W^+\) ratio this is due to the behavior of mixed QCD–EW interference contributions at high \(p_{\mathrm {T}} \), which yield relevant (negative) contributions in \(W^{+}+\)jet production but less in \(W^{-}+\)jet production. As for the \(Z(\ell ^+\ell ^-)/Z(\nu {\bar{\nu }})\) ratio, the observed EW effects can be attributed to \(p_{\mathrm {T}} \)-migration effects induced by QED radiation off leptons. At moderate \(p_{\mathrm {T},Z}\), the invariant mass of photon-lepton pairs that lie inside the recombination cone \(\varDelta R_{\ell \gamma }<0.1\) is well below \(M_Z\). Thus a significant fraction of the \(Z\rightarrow \ell ^+\ell ^-\gamma \) phase space does not undergo photon-lepton recombination, and photon radiation results in a negative mass and momentum shift for the \(\ell ^+\ell ^-\) system. The *Z*-mass shift is typically not sufficient to push \(Z\rightarrow \ell ^+\ell ^-\gamma \) events outside the inclusive \(m_{\ell \ell }\) window defined in Sect. 3.2. However, the reduction of the reconstructed \(p_{\mathrm {T},\ell \ell }\) results in a negative correction to the \(Z(\ell ^+\ell ^-)/Z(\nu {\bar{\nu }})\) ratio. Vice versa, for \(p_{\mathrm {T},Z}\gtrsim 1\) TeV the recombination cone \(\varDelta R_{\ell \gamma }<0.1\) covers photon-lepton invariant masses up to \(p_{\mathrm {T},Z}\varDelta _{\ell \gamma }> M_Z\), i.e. beyond the \(Z\rightarrow \ell ^+\ell ^-\gamma \) phase space. As a result, \(p_{\mathrm {T},\ell \ell }\) starts capturing a non-negligible amount of ISR QED radiation, which results in a positive shift of \(p_{\mathrm {T},\ell \ell }\) and thus in a positive correction to the \(Z(\ell ^+\ell ^-)/Z(\nu {\bar{\nu }})\) ratio. Note that the quantitative impact of such corrections depends on the choice of the \(m_{\ell \ell }\) mass window. Thus, for a consistent implementation of the predictions presented in this study it is crucial to reweight MC samples using the \(m_{\ell \ell }\) window defined in Sect. 3.2. Moreover, in order to guarantee a consistent extrapolation of QED radiative effects to the \(m_{\ell \ell }\) window employed in experimental analyses, it is mandatory to employ MC samples that account for QED radiation off leptons.

### 4.3 Photon-induced production and QED effects on PDFs

Higher-order QCD and EW calculations for \(pp\rightarrow V+\) jet require PDFs at a corresponding accuracy level, i.e. including also QED corrections. The effect of QED interactions on parton densities is twofold. Firstly they introduce a photon parton distribution and so open up partonic channels such as \(\gamma q \rightarrow V q'\). Secondly they modify the quark (and even gluon) PDFs both through QED effects in the initial conditions and especially in the DGLAP evolution.

*W*boson in the

*t*-channel. Such contributions are suppressed by a relative factor \(\alpha /\alpha _S\) and can be treated at LO, which corresponds to \(\gamma q\rightarrow Vq\) at \(\mathcal {O}(\alpha ^2)\) or, if necessary, at NLO QCD, i.e. up to order \(\mathcal {O}(\alpha ^2\alpha _{\mathrm {S}})\). This order comprises:

- (b.1)
virtual QCD corrections to \(\gamma q\rightarrow Vq\);

- (b.2)
\(\gamma g\rightarrow Vq{\bar{q}}\) quark bremsstrahlung;

- (b.3)
\(\gamma q\rightarrow Vqg\) gluon bremsstrahlung.

Figure 12 illustrates the impact of photon-induced \(V+\) jet production at LO according to three recent PDF sets that implement QED corrections. Effects of the order of 5–10% for \(W+\)jet can be observed in the TeV region if CT14qed_inc [65] or LUXqed PDFs [39] are used. Much larger effects are found with NNPDF30qed [66, 67]. The impact of photon-induced production to \(Z+\)jet (and also \(\gamma +\)jet) processes on the other hand is negligible [17, 18].

For the description of PDFs and their uncertainties we will use the LUXqed PDFs and their intrinsic uncertainties, given that this set of parton distributions implements a model-independent, data-driven determination of the photon distribution. From Fig. 12 one sees that the LUXqed uncertainties for \(\gamma p\rightarrow V+\) jet are small. Using the CT14qed_inc PDFs, based on a non-perturbative model with limited data-based constraints for the inelastic contribution, would result in fairly similar photon-induced cross sections but somewhat larger uncertainties (not shown) as compared to LUXqed PDFs. The NNPDF30qed parton distributions are model independent and data driven, but they are based on a different approach from LUXqed for deducing the photon distribution from data, which results in large uncertainties in the photon-induced component, of the order of \(100\%\) for \(pp\rightarrow \ell ^+\nu _\ell +\) jet at \(p_{\mathrm {T},\ell }=1\,\)TeV [20].

We have verified that the NLO QCD corrections to photon-induced production have an impact at the percent level relative to \(\mathcal {O}(\alpha ^2)\) and can safely be omitted. This implies that \(\gamma p\rightarrow V+\) jet can be regarded as independent processes. Thus photon-induced \(V+\) jet production can be either included through the parton-level predictions provided in this study or handled as separate background processes through dedicated MC simulations.

Concerning the size of the QED effects on the QCD partons, Fig. 13 examines the two main parton luminosities that contribute to the \(Z+\)jet process, i.e. \(g\Sigma =2 \sum _i (\mathcal{L}_{gq_i} + \mathcal{L}_{g{\bar{q}}_i})\) (which dominates) and \(q{\bar{q}}=2\sum _i \mathcal{L}_{q_i{\bar{q}}_i}\) (which accounts for the remaining \(15\%{-}30\%\)). It shows the ratio of these luminosities in LUXqed_plus_PDF4LHC15_nnlo relative to the PDF4LHC15_nnlo set on which it is based. The ratio is given as a function of half the partonic invariant mass, *M* / 2, which is commensurate with the \(p_{\mathrm {T}}\) of the *Z*.

### 4.4 PDF uncertainties

*Z*/

*W*ratio the PDF uncertainties cancel almost completely and remain below \(0.5(2)\%\) up to \(p_{\mathrm {T}} \approx 800(1500)\) GeV. In the \(Z/\gamma \) and \(W/\gamma \) ratios the PDF uncertainties are at the level of 1–\(2\%\) up to \(p_{\mathrm {T}} \approx 1300\) GeV, while the \(W^-/W^+\) ratio is subject to PDF uncertainties beyond \(5\%\) already for \(p_{\mathrm {T}} \gtrsim 1\) TeV, driven by uncertainties on the

*u*/

*d*ratio at large Bjorken-

*x*[3].

### 4.5 Real-boson emission

Inclusive diboson production (in particular \(pp\rightarrow VV'+\)jets) can be understood as the real-emission counterpart to NLO EW corrections to \(pp\rightarrow V+\) jet. Both contributions are separately finite and well defined if \(V'=W,Z\). Although they are expected to cancel against each other to a certain (typically small) extent, in practice one should only make sure that both types of processes, \(pp\rightarrow V+\) jet and \(pp\rightarrow VV'\)(+jets) with leptonic and hadronic decays of the \(V'\), are included in the analysis, and, in order to avoid double counting, contributions of type \(VV'\)(+jets) should be included in separate diboson MC samples and not as EW correction effects in \(V+\) jets samples. Unless a very strong cancellation is observed (which is typically not the case), there is no reason to worry about the possible correlation of uncertainties in \(V+\) jets and \(VV'\)(+jets) production, i.e. one can treat the respective uncertainties as uncorrelated.

As concerns the accuracy of MC simulations of \(pp\rightarrow VV'\)(+jets), it is important to notice that a large diboson background to inclusive vector-boson production at high \(p_\mathrm {T}\) is expected to arise from \(pp\rightarrow VV'j\) topologies with a hard back-to-back *Vj* system accompanied by a relatively soft extra vector boson. This calls for a reliable description of \(VV'+\) jet including QCD (and possibly EW) corrections. Thus we recommend the use of merged diboson samples that include at least one extra jet at matrix-element level. At the TeV scale, the EW corrections to \(pp\rightarrow VV'+\) jet can become quite large [69, 70] and should ultimately be included, together with the corresponding QCD corrections [71, 72, 73, 74, 75, 76, 77, 78].

### 4.6 Combination of QCD and electroweak corrections

*K*-factor in Eq. (63) is due to the fact that QCD and EW correction factors are normalized to \(\sigma _{\text {LO} \,{\mathrm {QCD}}}^{(V)}({\varvec{\mu }}_0)\) and \(\sigma _{\text {LO} \,{\mathrm {QCD}}}^{(V)}({\varvec{\mu }})\), respectively.

^{13}

*K*-factor, i.e.

The prescription (64) is motivated by the factorization of QCD corrections from the large Sudakov-enhanced EW corrections at high energies [64] and by the observation that in cases where the multiplicative and additive approach are far apart from each other, such as in the presence of giant *K*-factors [19, 81], the former turns out to be much more reliable. In general, when QCD and EW corrections are simultaneously enhanced, the \(\mathcal {O}(\alpha \alpha _{\mathrm {S}})\) mixed terms that are controlled by the multiplicative prescription can become quite significant. We also note that, thanks to the fact that the relative EW correction factors \(\kappa _{{\mathrm {EW}}}^{(V)}(x)\) are essentially insensitive to QCD scale variations, the scale dependence of the multiplicative combination (64) is similar to pure \(\mathrm {N}^k\mathrm {LO}\) QCD predictions. In contrast, the additive approach (63) can suffer from sizable scale uncertainties when EW corrections become large.

^{14}

*K*-factors for \(V+2\) jet and \(V+1\) jet production. To this end, we consider the identity

*N*-jettiness [82] resolution parameter \({\tau }_{\mathrm {cut}}\), as described in more detail below, and the above equation should be understood as the definition of \(\delta \kappa ^{(V)}_{\mathrm {NNLO}\,{\mathrm {mix}}}(x,{\tau }_{\mathrm {cut}})\), which will be used as estimator of \(\delta \kappa ^{(V)}_{\mathrm {NNLO}\,{\mathrm {mix}}}(x)\) in Eq. (70). In Eq. (71) we use the notation \(\kappa _{\text {NLO} \, {\mathrm {EW}}}^{V+1\,\mathrm {jet}}(x)=\kappa _{\text {NLO} \, {\mathrm {EW}}}^{(V)}(x)\), and we keep the \(\mu \)-dependence as implicitly understood, since the term \(\delta \kappa ^{(V)}_{\mathrm {NNLO}\,{\mathrm {mix}}}(x,{\tau }_{\mathrm {cut}})\) is expected to be quite stable with respect to scale variations. Instead, the \({\tau }_{\mathrm {cut}}\) parameter plays an important role since it acts as a cut-off of infrared QCD singularities in the regions where the second jet becomes soft or collinear. Based on the universal behavior of IR QCD effects, such singularities are expected to factorize into identical singular factors on the left- and the right-hand side of Eq. (71). Thus, while the \(\delta \kappa ^{(V)}_{\mathrm {NNLO}\,{\mathrm {mix}}}(x,{\tau }_{\mathrm {cut}})\) term on the right-hand side depends on \({\tau }_{\mathrm {cut}}\), this dependence is expected to be free from large \({\tau }_{\mathrm {cut}}\)-logarithms and thus reasonably mild.

*x*-spectrum and within an appropriately chosen \({\tau }_{\mathrm {cut}}\) range. Thanks to the cancellation of IR QCD singularities in Eq. (72), the resulting \(\xi ^{(V)}\) coefficients should be reasonably stable with respect to the choice of the resolution parameter. Thus, \({\tau }_{\mathrm {cut}}\) can be varied in a rather wide range. In principle one could even consider the \({\tau }_{\mathrm {cut}}\rightarrow 0\) limit of Eq. (73). However, given that two-loop mixed EW–QCD contributions are not taken into account, this limit does not converge towards the full NNLO result corresponding to \({\tau }_{\mathrm {cut}}=0\). Moreover, for very small values of \({\tau }_{\mathrm {cut}}\) the numerator and denominator of \(\kappa ^{V+2\,\mathrm {jets}}_{\text {NLO} \, {\mathrm {EW}}}(x,{\tau }_{\mathrm {cut}})\) are dominated by universal \({\tau }_{\mathrm {cut}}\)-logarithms that should cancel against virtual two-loop terms, and since such logarithms factorize, their dominance can result in an underestimation of non-factorizing effects. Vice versa, excessively large values of \({\tau }_{\mathrm {cut}}\) can lead to an overestimation of non-factorizing effects. This is due to the fact that increasing \({\tau }_{\mathrm {cut}}\) enhances the difference between EW \(\kappa \)-factors in Eq. (73) but also suppresses the cross section of the \(V+2\)-jet subprocess, rendering it a less and less significant estimator of the behavior of mixed corrections for inclusive \(V+\) jet production. Thus, excessively small or large values of \({\tau }_{\mathrm {cut}}\) should be avoided.

*N*-jettiness cut parameter [82]. More precisely, we use the dimensionless one-jettiness parameter

^{15}The \(q_k\) denote the four-momenta of any such final-state parton, and \(\sqrt{\hat{s}}\) is the partonic center-of-mass energy. All quantities are defined in the hadronic center-of-mass system.

The rather small values of the \(\xi ^{(V)}\) coefficients confirm that the bulk of the EW and QCD corrections factorize. However, in the case of \(W+\) jet and \(\gamma +\) jet production, the relative size of non-factorizing corrections appears to be rather significant. This is due to the behavior of the EW \(\kappa \)-factors in the multi-TeV region, where the difference between the EW \(\kappa \)-factors for \(pp\rightarrow V+1\) jet and \(pp\rightarrow V+2\) jet is enhanced by the presence of mixed EW–QCD interference contributions in channels of type \(qq\rightarrow qq V\) (see the contributions of type a.5 in Sect. 4.2). More precisely, EW–QCD interference effects of \(\mathcal {O}(\alpha _{\mathrm {S}}\alpha ^2)\) enhance the EW corrections to \(pp\rightarrow V+1\) jet as a result of the opening of the *qq* channel at NLO EW, while in \(pp\rightarrow V+2\) jet the EW *K*-factor is not enhanced since the *qq* channel is already open at LO. Based on this observation, and also on the fact that the main effect of the opening of the *qq* channel is already reflected in the NLO QCD *K*-factor for \(V+1\,\)jet production, the above-mentioned EW–QCD interference effects could be excluded from the factorization prescription (64) and treated as a separate contribution. As illustrated by the dashed curves in Fig. 15, this approach would lead to a drastic reduction of non-factorizing effects, especially for \(\gamma +\) jet production. Nevertheless, given that the effects observed in Fig. 15 are subdominant with respect to current PDF and statistical uncertainties, in the present study we refrain from implementing such a splitting.

### 4.7 Combination of QCD and EW corrections with related uncertainties

In Fig. 16 we compare the additive and multiplicative combinations of QCD and EW corrections showing also the corresponding uncertainty estimate (77) for various \(V+\)jet processes.

## 5 Summary and conclusions

The precise control of SM backgrounds, and notably of \(pp \rightarrow Z(\nu {\bar{\nu }})+\) jets, is crucial in order to maximize the potential of MET+jets searches at the LHC. Such backgrounds can be predicted directly using QCD and EW calculations. Alternatively, QCD and EW calculations can be used to relate them to experimental data for similar \(V+\) jet production processes, i.e. \(pp\rightarrow \gamma +\) jets, \(pp \rightarrow W(\ell \nu )+\) jets and \(pp \rightarrow Z(\ell ^+\ell ^-)+\) jets.

In this article we have presented predictions for inclusive vector-boson \(p_\mathrm {T}\) distributions based on the most advanced calculations available today, bringing together results from a number of groups so as to have perturbative QCD to NNLO accuracy, EW corrections to NLO accuracy and additionally the inclusion of two-loop EW Sudakov logarithms.

A substantial part of our study concerned uncertainty estimates. In particular we proposed and applied various new approaches for uncertainty estimates and correlations across processes and \(p_\mathrm {T}\) regions.

We defined the uncertainties due to normal QCD scale variations in a way that gives a strong correlation across different \(p_\mathrm {T}\) regions, Eq. (33). We then supplemented it with a shape uncertainty that is anti-correlated across \(p_\mathrm {T}\), Eqs. (35)–(36). To address the long-standing problem of evaluating the correlations between uncertainties for different processes, we separated the uncertainty into process-independent and process-dependent components. The universal component was taken to be composed of the overall scale and shape uncertainties for the reference \(Z+\mathrm {jet}\) process. The process-dependent component, which is generally small, was determined by considering the difference between suitably normalized *K*-factors for the different processes, Eq. (38). This amounts to a conservative choice of taking the uncertainty on ratios as the difference between the best available prediction and the one at one order lower.

Special attention was devoted to the correlation of \(Z/W+\) jet and \(\gamma +\) jet production. In that case a substantial non-universal contribution is associated with the masslessness of the photon and the need to control collinear divergent \(q\rightarrow q\gamma \) radiation through a photon-isolation prescription. We introduced a novel photon-isolation prescription with a dynamically chosen isolation radius, Eq. (11), designed to suppress \(q\rightarrow \gamma q\) radiative effects in a way that is similar to the effect of the masses of the *Z* and *W* bosons in the case of \(q\rightarrow V q\) splittings at large \(p_\mathrm {T}\). Such a dynamic isolation allows one to split \(\gamma +\)jet production into a quasi-universal part, which can be treated on the same footing as \(Z+\mathrm {jet}\) and \(W+\mathrm {jet}\) production, and a non-universal part which is kept uncorrelated. The non-universal part is given by the difference between the cross sections with conventional and dynamic photon-isolation prescriptions.

For pure EW corrections we considered three uncertainty sources for unknown higher-order contributions. These address unknown Sudakov logarithms beyond NNLO and/or NLL accuracy, as well as unknown hard (non-Sudakov) EW corrections beyond NLO and process-correlation effects.

One potentially large source of uncertainty arises from mixed QCD and EW corrections, given that both \(\mathcal {O}(\alpha _{\mathrm {S}})\) and \(\mathcal {O}(\alpha )\) NLO corrections can be large and that the \(\mathcal {O}(\alpha \alpha _{\mathrm {S}})\) NNLO corrections are not currently known. We chose a multiplicative scheme for combining EW and QCD corrections. To obtain an estimate of unknown \(\mathcal {O}(\alpha \alpha _{\mathrm {S}})\) corrections not captured by this factorized Ansatz, we studied the NLO EW corrections to \(V+2\) jet production, which represent the real–virtual part of a full \(\mathcal {O}(\alpha \alpha _{\mathrm {S}})\) calculation for \(V+\) jet production. Based on this analysis, we concluded that it is reasonable to assume that the multiplicative combination of QCD and EW corrections describes the full \(\mathcal {O}(\alpha \alpha _{\mathrm {S}})\) correction with a relative uncertainty that varies between 10 and 20% for \(pp\rightarrow W/Z+\) jet and 40% for \(pp\rightarrow \gamma +\) jet.

Overall, QCD corrections are substantial, a few tens of percent at NLO, and up to \(10\%\) at NNLO. The NNLO results are consistent with the NLO predictions within our prescription for the uncertainty bands of the latter. This is true not just for absolute cross sections and their shapes, but also for ratios of cross sections. These ratios are remarkably stable across LO, NLO and NNLO QCD corrections; see Fig. 8. Using dynamic photon isolation, this statement holds true also for the \(\gamma +\mathrm {jet}\) process at \(p_\mathrm {T}\gtrsim 300~\text {GeV} \).

The EW corrections to \(V+\)jet cross sections amount to a few tens of percent in the TeV region; see Fig. 9. In the ratios they cancel only in part, due to the sensitivity of EW effects to the SU(2) charges of the produced vector bosons. At the TeV scale, the NNLO Sudakov logarithms can reach the several percent level and their systematic inclusion is an important ingredient in order to achieve percent precision at very high \(p_\mathrm {T}\).

*Z*/

*W*ratio remaining uncertainties are at the level of only 1–2% up to 1 TeV and below 5% up to 2 TeV. Similarly, the \(Z/\gamma \) ratio is constrained at the 5% level up to 2 TeV. Noteworthy, including the NNLO QCD corrections the process ratios remain very stable and in particular within the uncertainty estimates based on NLO QCD. This reflects the fact that QCD uncertainties are very well under control: taking at face value the NNLO QCD systematics we are at the level of a few percent all the way up to the multi-TeV scale (see Fig. 8), and at large \(p_{\mathrm {T}} \) we are dominated by EW and PDF uncertainties. The latter are below the perturbative uncertainties in all nominal distributions and all but the \(W^-/W^+\) ratio, where a precise measurement at high \(p_{\mathrm {T}} \) could help to improve PDF fits. In this respect, we note that the theoretical uncertainty for the \(W^-/W^+\) ratio is entirely dominated by mixed QCD–EW effects and is most likely overestimated due to our conservative assumption of keeping such uncertainties uncorrelated across processes (see Sect. 4.6).

We also discussed photon-induced contributions and QED corrections to PDFs. In this context, for a precise prediction of the \(\gamma \)-PDF we have advocated the use of the LUXqed_plus_PDF4LHC15_nnlo PDFs, which implement a data-driven determination of the \(\gamma \)-PDF. For a consistency treatment of \(\mathcal {O}(\alpha )\) effects in the PDFs, the LUXqed_plus_PDF4LHC15_nnlo distributions should be used in all photon-, quark-, and gluon-induced channels.^{16} Photon-induced effects are negligible in \(Z+\) jet and \(\gamma +\) jet production, but their impact on \(pp\rightarrow W+\) jet, and thus on the *W* / *Z* and \(W/\gamma \) ratios, can reach the 5% level at the TeV scale^{17} (see Fig. 12).

Our predictions are provided in the form of tables for the central predictions and for the different uncertainty sources. Each uncertainty source is to be treated as a 1-standard deviation uncertainty and pragmatically associated with a Gaussian-distributed nuisance parameter.

The predictions are given at parton level as distributions of the vector boson \(p_\mathrm {T}\), with loose cuts and inclusively over other radiation. They are intended to be propagated to an experimental analysis using Monte Carlo parton shower samples whose inclusive vector-boson \(p_\mathrm {T}\) distribution has been reweighted to agree with our parton-level predictions. The impact of additional cuts, non-perturbative effects on lepton isolation, etc., can then be deduced from the Monte Carlo samples. The additional uncertainties associated with the Monte Carlo simulation are expected to be relatively small, insofar as the vector-boson \(p_\mathrm {T}\) distribution that we calculate is closely connected to the main experimental observables used in MET\(+\)jets searches.

Some caution is needed in implementing the results of this paper: for example the uncertainty prescriptions are tied to the use of the central values that we provide. If an experiment relies on central values that differ, e.g. through the use of MC samples that are not reweighted to our nominal predictions, then the uncertainty scheme that we provide may no longer be directly applicable. Furthermore, for searches that rely on features of the event other than missing transverse momentum, one should be aware that our approach might need to be extended. This would be the case notably for any observable that relies directly on jet observables, whether related to the recoiling jet or vetoes on additional jets.

Overall, it is possible to obtain precise theoretical control both for vector-boson \(p_\mathrm {T}\) distributions, and for their ratios, at the level of a few percent. We expect this precision, across a wide range of \(p_\mathrm {T}\), to be of significant benefit in MET+jets searches, notably enabling reliable identification or exclusion of substantially smaller BSM signals than was possible so far. In fact, since the release of the first version of this paper, the background estimates we propose here have been adopted in analyses by ATLAS [83] and CMS [84].

## Footnotes

- 1.
For a recent comprehensive review of DM models leading to this class of signatures; see e.g. [4].

- 2.
Other backgrounds (such as QCD multijets, \(t\bar{t}\) or pairs of gauge bosons) are suppressed, and their contribution to the overall uncertainty is well below the percent level.

- 3.
This procedure should be restricted to variables \(x'\) that can be described with good accuracy both in perturbative calculations and in the MC simulations.

- 4.
The same isolation prescription used for theory predictions should be applied also to their MC counterparts \(\mathrm {d}\sigma _\mathrm{MC}/\)d

*x*in the context of the reweighting procedure. - 5.
Diboson backgrounds, including \(pp\rightarrow V\gamma \), can be included through separate Monte Carlo samples in the experimental analyses.

- 6.
See e.g. the comparison of NNLOPS against fixed-order predictions in Fig. 3 of Ref. [29].

- 7.
Besides loop diagrams with top quarks and Higgs bosons, the NLO EW corrections to \(pp\rightarrow W^\pm +\) jet receive \(\mathcal {O}(\alpha ^2\alpha _{\mathrm {S}})\) bremsstrahlung contributions from \(qb\rightarrow q' W^\pm b\) channels that involve

*s*-channel top-quark propagators and thus require a finite top-quark width, for which we use the NLO QCD value \(\Gamma _t=1.339~\text {GeV} \). However, at the perturbative order considered in this study, such topologies arise only in QCD–EW interference terms that do not give rise to Breit–Wigner resonances. The dependence of our results on \(\Gamma _t\) is thus completely negligible. - 8.
Here and in the following we adopt a power counting that does not include the extra factor \(\alpha \) associated with vector-boson decays.

- 9.
For what concerns process correlations, it is crucial that (apart from the \(M_V\) dependence) all \(V+\) jet processes are evaluated using equivalent dynamical scales.

- 10.
In this regime, which is not the main focus of the present study, the process-correlation uncertainty (38) ceases to be a meaningful uncertainty estimate.

- 11.
Note that, in spite of the fact that we present them as separate terms in Eq. (25), \(\gamma \)-induced contributions and NLO EW corrections to \(pp\rightarrow V+\) jet are interconnected at \(\mathcal {O}(\alpha ^2\alpha _{\mathrm {S}})\).

- 12.
Here, in order to discuss qualitative features of Sudakov logarithms, we adopt a generic and rather schematic representation of the asymptotic high-energy limit. In particular, we do not consider some aspects, such as the helicity dependence of the corrections or SU(2) soft-correlation effects. However, in the numerical analysis all relevant aspects are consistently included.

- 13.
- 14.
As discussed below, the goodness of this naive Ansatz will be justified by fitting it to a realistic estimator of \(\delta \kappa ^{(V)}_{{\mathrm {mix}}}(x)\).

- 15.
In order to guarantee a proper cancellation of QCD and EW singularities, the jet algorithm is applied to all QCD partons and photons, excluding photons that are recombined with leptons, as well as the leading identified photon in case of the \(\gamma +\)jets process.

- 16.
This is automatically achieved by reweighting MC samples generated with arbitrary PDFs with our complete \(\mathrm {N}^k\mathrm {LO}\,{\mathrm {QCD}}\times \mathrm {nNLO}\,{\mathrm {EW}}\) predictions based on LUXqed_plus_PDF4LHC15_nnlo PDFs. Vice versa, restricting the reweighting to pure EW corrections and using MC samples based on different PDFs can lead to inconsistencies at \(\mathcal {O}(\alpha )\).

- 17.
Note that photon-induced contributions are not included in the summary plots of Fig. 17.

## Notes

### Acknowledgements

We wish to thank Frank Krauss, Keith Ellis, Christian Gütschow, Sarah Malik, Fabio Maltoni, Holger Schulz and Graeme Watt for valuable discussions. This research was supported in part by the UK Science and Technology Facilities Council, the Swiss National Science Foundation (SNF) under Contracts 200020-162487, CRSII2-160814, and BSCGI0-157722, and by the Research Executive Agency (REA) of the European Union under the Grant Agreements PITN-GA-2012-316704 (“HiggsTools”), PITN–GA–2012–315877 (“MCnet”), and the ERC Advanced Grants MC@NNLO (340983) and LHCtheory (291377). R.B. is supported by the DOE Contract DE-AC02-06CH11357. F.P. is supported by the DOE Grants DE-FG02- 91ER40684 and DE-AC02-06CH11357. C.W. is supported by the National Science Foundation through Award number PHY-1619877. The research of J.M.C. is supported by the US DOE under Contract DE-AC02-07CH11359. The work of S.D. is supported by the German Federal Ministry for Education and Research (BMBF). This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. We also acknowledge support provided by the Center for Computational Research at the University at Buffalo and the Wilson HPC Computing Facility at Fermilab.

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