Precise predictions for samesign Wboson scattering at the LHC
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Abstract
Vectorboson scattering processes are of great importance for the current runII and future runs of the Large Hadron Collider. The presence of triple and quartic gauge couplings in the process gives access to the gauge sector of the Standard Model (SM) and possible newphysics contributions there. To test any newphysics hypothesis, sound knowledge of the SM contributions is necessary, with a precision which at least matches the experimental uncertainties of existing and forthcoming measurements. In this article we present a detailed study of the vectorboson scattering process with two positivelycharged leptons and missing transverse momentum in the final state. In particular, we first carry out a systematic comparison of the various approximations that are usually performed for this kind of process against the complete calculation, at LO and NLO QCD accuracy. Such a study is performed both in the usual fiducial region used by experimental collaborations and in a more inclusive phase space, where the differences among the various approximations lead to more sizeable effects. Afterwards, we turn to predictions matched to parton showers, at LO and NLO: we show that on the one hand, the inclusion of NLO QCD corrections leads to more stable predictions, but on the other hand the details of the matching and of the partonshower programs cause differences which are considerably larger than those observed at fixed order, even in the experimental fiducial region. We conclude with recommendations for experimental studies of vectorboson scattering processes.
1 Introduction
The VBS process involving two samesign \(\text {W} \) bosons has the largest signaltobackground ratio of all the VBS processes at the Large Hadron Collider (LHC): evidence for it was found at the centreofmass energy of \(8~\,\text {TeV} \) [1, 2, 3], and it has been recently measured at \(13~\,\text {TeV} \) as well [4]. Presently, the measurements of VBS processes are limited by statistics, but the situation will change in the near future. On the theoretical side, it is thus of prime importance to provide predictions with systematic uncertainties at least comparable to the current and envisaged experimental precision [5, 6].
\(\text {W} ^+\text {W} ^+\) scattering is the simplest VBS process to calculate, because the doublecharge structure of the leptonic final state limits the number of partonic processes and total number of Feynman diagrams for each process. Nonetheless, it possesses all features of VBS at the LHC and is thus representative of other VBS signatures. Therefore, it is the ideal candidate for a comparative study of the different simulation tools.
In the last few years, several nexttoleadingorder (NLO) computations have become available for both the VBS process [7, 8, 9, 10, 11, 12, 13] and its QCDinduced irreducible background process [13, 14, 15, 16, 17]. All these VBS computations rely on various approximations, typically neglecting contributions which are expected to be small in realistic experimental setups [12, 18]. Recently, the complete NLO corrections to \(\text {W} ^+\text {W} ^+\) have been evaluated in Ref. [19], making it possible for the first time to study in detail the quality of the VBS approximations at NLO QCD.^{1}
This article starts with the definition of the VBS process in Sect. 2, before describing the approximations of the various computer codes in Sect. 3. In Sect. 4 a leadingorder (LO) study of the different contributions which lead to the production of two samesign \(\text {W} \) bosons and two jets is performed. In the same section predictions for VBS from different tools are compared at the level of the cross section and differential distributions. The comparison is extended to the NLO corrections to VBS in Sect. 5. The effect of the inclusion of matching LO and NLO computations to parton shower (PS) is discussed in Sect. 6. Finally, Sect. 7 contains a summary of the article and concluding remarks.
2 Definition of the process
The second coupling combination of order \(\mathcal {O}{\left( \alpha _\mathrm{s}^2\alpha ^{4}\right) }\) corresponds to diagrams with a gluon connecting the two quark lines, and with the \(\text {W} \) bosons radiated off the quark lines. Because of the different colour structure, this contribution features a different kinematic behaviour than VBS. Nonetheless it shares the same final state, and therefore constitutes an irreducible background to the EW process.
Finally, the third contribution of order \(\mathcal {O}{\left( \alpha _\mathrm{s}\alpha ^{5}\right) }\) is the interference of the two types of amplitudes described above. It is nonzero only for those partonic subprocesses which involve identical quarks or antiquarks. Such a contribution is usually small (\(3\%\)) within typical experimental cuts [19].
In the rest of this article, the notations LO or NLO(QCD) without any specification of coupling powers refer to the contributions at order \(\mathcal {O}{\left( \alpha ^{6}\right) }\) and \(\mathcal {O}{\left( \alpha _\mathrm{s}\alpha ^{6}\right) }\), respectively.
In experimental measurements, special cuts, called VBS cuts, are designed to enhance the EW contribution over the QCD one and to suppress the interference. These cuts are based on the different kinematical behaviour of the contributions. The EW contribution is characterised by two jets with large rapidities as well as a large dijet invariant mass. The two \(\text {W} \) bosons are mostly produced centrally. This is in contrast to the QCD contribution which favours jets in the central region. Therefore, the event selection usually involves rapiditydifference and invariantmass cuts for the jets. Note that, as pointed out in Ref. [19], when considering full amplitudes, the separation between EW and QCD production becomes illdefined. Hence, combined measurements which are theoretically better defined should be preferably performed by the experimental collaborations at the LHC.
3 Details of the calculations
3.1 Theoretical predictions for VBS
We now discuss the various approximations which are implemented in computer programs for the EW contribution at order \(\mathcal {O}{\left( \alpha ^{6}\right) }\). Since we are mostly interested in the scattering of two \(\text {W} \) bosons, which includes the quartic gaugeboson vertex, it may appear justified to approximate the full process by considering just those diagrams which contain the \(2\rightarrow 2\) scattering process as a subpart. However, this set of contributions is not gauge invariant. In order to ensure gauge invariance, an onshell projection of the incoming and outgoing W bosons should be performed. While this can be done in the usual way for the timelike outgoing W bosons, the treatment of the spacelike W bosons emitted from the incoming quarks requires some care. Following Refs. [22, 23] these Wboson lines can be split, the W bosons entering the scattering process can be projected onshell, and the emission of the W bosons from the quarks can be described by vectorboson luminosities. Such an approximation is usually called effective vectorboson approximation (EVBA) [24, 25, 26].
An improvement of such an approximation consists in considering all t and uchannel diagrams and squaring them separately, neglecting interference contributions between the two classes. These interferences are expected to be small in the VBS fiducial region, as they are both phasespace and colour suppressed [12, 18]. The schannel squared diagrams and any interferences between them and the t/uchannels are also discarded. This approximation is often called t/u approximation, VBF, or even VBS approximation. We adopt the latter denomination in the following. This approximation is gaugeinvariant, a fact that can be appreciated by considering the two incoming quarks as belonging to two different copies of the \(\mathrm {SU}(3)\) gauge group.
A further refinement is to add the squared matrix element of the schannel contributions to the VBS approximation.
The approximations performed at LO can be extended when NLO QCD corrections to the order \(\mathcal {O}{\left( \alpha ^{6}\right) }\) are computed. The VBS approximation can be extended at NLO in a straightforward manner for what concerns the virtual contributions. For the realemission contributions special care must be taken for the gluoninitiated processes. The initialstate gluon and initialstate quark must not couple together, otherwise infrared (IR) divergences proportional to schannels appear, which do not match with the ones found in the virtual contributions. The subset of diagrams where all couplings of the initial state gluon to initial state quark are neglected forms a gaugeinvariant subset, with the same argument presented above.
A further refinement is to consider the full real contributions, which include all interferences, and part of the virtual. In particular one can consider only oneloop amplitudes where there is no gluon exchange between the two quark lines and assuming a cancellation of the IR poles.
When considering the full NLO corrections of order \(\mathcal {O}{\left( \alpha _\mathrm{s}\alpha ^{6}\right) }\), besides real and virtual QCD corrections to the EW treelevel contribution of order \(\mathcal {O}{\left( \alpha ^{6}\right) }\) also real and virtual EW corrections to the LO interference of order \(\mathcal {O}{\left( \alpha _\mathrm{s}\alpha ^{5}\right) }\) have to be taken into account. Since some loop diagrams contribute to both types of corrections, QCD and EW corrections cannot be separated any more on the basis of Feynman diagrams, and the cancellation of IR singularities requires the inclusion of all of them [19].
3.2 Description of the programs used

Phantom [27] is a dedicated treelevel Monte Carlo for sixparton final states at \(\text {p} \text {p},\, \text {p} \bar{\text {p}}\), and \(\text {e} ^+\text {e} ^\) colliders at orders \(\mathcal O(\alpha ^6)\) and \(\mathcal O(\alpha _\text {s} ^2\alpha ^4)\) including interferences between the two sets of diagrams. It employs complete treelevel matrix elements in the complexmass scheme [28, 29, 30] computed via the modular helicity formalism [31, 32]. The integration uses a multichannel approach [33] and an adaptive strategy [34]. Phantom generates unweighted events at parton level for both the SM and a few instances of beyond the Standard Model (BSM) theories.

WHIZARD [35, 36] is a multipurpose event generator with LO matrixelement generator O’Mega. For QCD amplitudes it uses the colour flow formalism [37]. For NLO QCD calculations [38], where Whizard is in the final validation phase, it provides FKS subtraction terms [39, 40], while virtual matrix elements are provided externally by OpenLoops [41] or Recola [42, 43]. Furthermore, WHIZARD can automatically provide POWHEG matching to parton shower [44]. WHIZARD allows to simulate a huge number of BSM models as well, in particular in terms of higherdimensional operators for VBS processes including means to provide unitarity limits [45, 46].

The program Bonsay [47] consists of a generalpurpose Monte Carlo integrator and matrix elements taken from different sources: Born matrix elements are adapted from the program Lusifer [48], which have been generalised to calculate also real matrix elements. Virtual matrix elements have been calculated using an inhouse matrixelement generator. Oneloop integrals are evaluated using the Collier library [49, 50]. For the results presented here, it uses the VBS approximation at LO and NLO. The virtual corrections are additionally approximated using a doublepole approximation where the final state leptons are assumed to originate from the decay of two resonant Wbosons; see Ref. [51] for a detailed description and Ref. [52] for the onshell projection used. At LO the exact matrix elements can also be used.

The PowhegBox [53, 54, 55] is a framework for matching NLOQCD calculations with parton showers. It relies on the user providing the matrix elements and Born phase space, but automatically constructs FKS [39] subtraction terms and the phase space corresponding to the real emission. For the VBS processes all matrix elements are being provided by a previous version of VBFNLO [17, 56, 57] and hence the approximations used in the PowhegBox are similar to those used in VBFNLO.

VBFNLO [17, 56, 57] is a flexible partonlevel Monte Carlo for processes with EW bosons. It allows the calculation of VBS processes at NLO QCD in the VBS approximation, with process IDs between 200 and 290. Samesign \(\text {W} ^+\text {W} ^+jj\) production is provided via the process ID 250. The corresponding schannel contributions are available separately as triboson processes with semileptonic decays, with process IDs between 401 and 492. For the final state studied in this article, only \(\text {W} ^+\text {W} ^+\text {W} ^\) production with a hadronically decaying \(\text {W} ^\) boson, process ID 432, can contribute. These can simply be added on top of the VBS contribution. Interferences between the two are therefore neglected. The usage of leptonic tensors in the calculation, pioneered in Ref. [7], thereby leads to a significant speed improvement over automatically generated code. Besides the SM, also a variety of newphysics models including anomalous couplings of the Higgs and gauge bosons can be simulated.
 MadGraph5_aMC@NLO [58] (henceforth MG5_a MC) is an automatic metacode (a code that generates codes) which makes it possible to simulate any scattering process including NLO QCD corrections both at fixed order and including matching to parton showers, using the MC@NLO method [59]. It makes use of the subtraction method by Frixione, Kunszt and Signer (FKS) [39, 40] (automated in the module MadFKS [60, 61]) for regulating IR singularities. The computations of oneloop amplitudes are carried out by switching dynamically between two integralreduction techniques, OPP [62] or Laurentseries expansion [63], and tensorintegral reduction [64, 65, 66]. These have been automated in the module MadLoop [67], which in turn exploits CutTools [68], Ninja [69, 70], IREGI [71], or Collier [50], together with an inhouse implementation of the OpenLoops optimisation [41]. Finally, scale and PDF uncertainties can be obtained in an exact manner via reweighting at negligible additional CPU cost [72]. The simulation of VBS at NLOQCD accuracy can be performed by issuing the following commands in the program interface:With these commands the complexmass scheme is turned on, then the NLOcapable model is loaded,^{3} finally the process code is generated (note the QCD=0 syntax to select the purelyEW process) and written to disk. No approximation is performed for the Born and realemission matrix elements. Only stronglyinteracting particles circulating in the loops are generated for the virtual matrix element. The version capable of computing both QCD and EW corrections will be released in the future. Such an approximation is equivalent to the assumption that the finite part of those loops which feature EW bosons is zero. In practice, since a part of the contribution to the single pole is also missing, the internal polecancellation check at run time has to be turned off, by setting the value of the IRPoleCheckThreshold and PrecisionVirtualAtRunTime parameters in the Cards/FKS_params.dat file to \(1\).Table 1
Summary of the different properties of the computer programs employed in the comparison
Code
\(\mathcal O(\alpha ^6)\) s, t, u
\(\mathcal O(\alpha ^6)\) interf.
Nonres.
NLO
NF QCD
EW corr. to order \(\mathcal O(\alpha _\text {s} \alpha ^5)\)
Phantom
s, t, u
Yes
Yes
No
–
–
Whizard
s, t, u
Yes
Yes
No
–
–
Bonsay
t, u
No
Yes, virt. no
Yes
No
No
Powheg
t, u
No
Yes
Yes
No
No
VBFNLO
s, t, u
No
Yes
Yes
No
No
MG5_aMC
s, t, u
Yes
Yes
Yes
Virt. no
No
MoCaNLO+Recola
s, t, u
Yes
Yes
Yes
Yes
Yes

The program MoCaNLO+Recola is made of a flexible Monte Carlo program dubbed MoCaNLO and of the matrixelement generator Recola [42, 43]. It can compute arbitrary processes at the LHC with both NLO QCD and EW accuracy in the SM. This is made possible by the fact that Recola can compute arbitrary processes at tree and oneloop level in the SM. To that end, it relies on the Collier library [49, 50] to numerically evaluate the oneloop scalar and tensor integrals. In addition, the subtraction of the IR divergences appearing in the real corrections has been automatised thanks to the Catani–Seymour dipole formalism for both QCD and QED [73, 74]. The code has demonstrated its ability to compute NLO corrections for highmultiplicity processes up to \(2 \rightarrow 7\) [75, 76]. In particular the full NLO corrections to VBS and its irreducible background [19, 77] have been obtained thanks to this tool. One key aspect for these highmultiplicity processes is the fast integration which is ensured by using similar phasespace mappings to those of Refs. [28, 48, 78]. In MoCaNLO+Recola no approximation is performed neither at LO nor at NLO. It implies that, also contributions stemming from EW corrections to the interference are computed.

all s, t, uchannel diagrams are included;

interferences between diagrams of different types are included at LO;

diagrams which do not feature two resonant W bosons are included;

the socalled nonfactorisable (NF) QCD corrections, i.e. the corrections where (real or virtual) gluons are exchanged between different quark lines, are included;

EW corrections to the interference of order \(\mathcal O (\alpha ^5\alpha _\text {s})\) are included. These corrections are of the same order as the NLO QCD corrections to the contribution of order \(\mathcal O (\alpha ^6\)) term.
3.3 Input parameters
The hadronic scattering processes are simulated at the LHC with a centreofmass energy \(\sqrt{s} = 13~\,\text {TeV} \). The NNPDF 3.0 parton distribution functions (PDFs) [79] with five massless flavours,^{4} NLOQCD evolution, and a strong coupling constant \(\alpha _\text {s} \left( M_\text {Z} \right) = 0.118\)^{5} are employed.^{6} Initialstate collinear singularities are factorised according to the \({\overline{\mathrm{MS}}}\) scheme, consistently with what is done in NNPDF.
 The two samesign charged leptons are required to fulfil cuts on transverse momentum, rapidity, and separation in the rapidity–azimuthalangle separation,$$\begin{aligned} p_{\text {T},\ell }> 20\,\text {GeV},\qquad y_{\ell } < 2.5, \qquad \varDelta R_{\ell \ell }> 0.3. \end{aligned}$$(8)
 The total missing transverse momentum, computed from the vectorial sum of the transverse momenta of the two neutrinos, is required to be$$\begin{aligned} p_\mathrm{T, miss} > 40\,\text {GeV}. \end{aligned}$$(9)
 QCD partons (light quarks and gluons) are clustered together using the anti\(k_T\) algorithm [83], possibly using the FastJet implementation [84], with distance parameter \(R=0.4\). We impose cuts on the jets’ transverse momenta, rapidities, and their separation from leptons,VBS cuts are applied on the two jets with largest transverse momentum, unless otherwise stated. In particular, we impose a cut on the invariant mass of the dijet system, as well as on the rapidity separation of the two jets,$$\begin{aligned} p_{\text {T},\text {j}}> 30\,\text {GeV}, \qquad y_\text {j}  < 4.5, \qquad \varDelta R_{\text {j} \ell } > 0.3. \end{aligned}$$(10)if not explicitly stated otherwise.$$\begin{aligned} m_{\text {j} \text {j}}> 500\,\text {GeV},\qquad \varDelta y_{\text {j} \text {j}} > 2.5, \end{aligned}$$(11)

When EW corrections are computed, real photons and charged fermions are clustered together using the anti\(k_T\) algorithm with radius parameter \(R=0.1\). In this case, leptons and quarks are understood as dressed fermions.
4 Leadingorder study
4.1 Contributions
Cross sections at LO accuracy for the three contributions to the process \(\mathrm{p}\mathrm{p}\rightarrow \mu ^+\nu _\mu \mathrm{e}^+\nu _\mathrm{e}\mathrm{j}\mathrm{j}\), obtained with exact matrix elements. These results are for the setup described in Sect. 3.3 but no cuts on \(m_{\text {j} \text {j}}\) and \(\varDelta y_{\text {j} \text {j}}\) are applied. The uncertainties shown refer to the estimated statistical errors of the Monte Carlo integrations
Order  \(\mathcal {O}(\alpha ^6)\)  \(\mathcal {O}(\alpha _\text {s} ^2\alpha ^4)\)  \(\mathcal {O}(\alpha _\text {s} \alpha ^5)\) 

\(\sigma \;(\mathrm {fb})\)  \( 2.292 \pm 0.002 \)  \( 1.477 \pm 0.001 \)  \( 0.223 \pm 0.003 \) 
In Fig. 2, these three contributions are shown separately and summed in the differential distributions in the dijet invariant mass \(m_{\text {j} \text {j}}\) and the rapidity difference \(\varDelta y_{\text {j} \text {j}}\). For the dijet invariantmass distribution (left), one can observe that the EW contribution peaks around an invariant mass of about \(80\,\text {GeV} \). This is due to diagrams where the two jets originate from the decay of a W boson (see middle and right diagrams in Fig. 1). Note that these contributions are not present in calculations relying on the VBS approximation as these are only schannel contributions. The EW contribution becomes dominant for dijet invariant mass larger than \(500 \,\text {GeV} \). The same holds true for jetrapidity difference larger than 2.5 (right). This justifies why cuts on these two observables are used in order to enhance the EW contribution over the QCD one. In particular, in order to have a large EW contribution, rather exclusive cuts are required.
4.2 Inclusive comparison
4.3 Comparison in the fiducial region
Cross sections at LO accuracy and order \(\mathcal {O}(\alpha ^6)\). The complete \(2\rightarrow 6\) matrixelement, without any approximation, is employed by Phantom, Whizard, MG5_aMC, and MoCaNLO+Recola. The predictions are obtained in the fiducial region described in Sect. 3.3. The uncertainties shown refer to the estimated statistical errors of the Monte Carlo integrations
Code  \(\sigma \;(\mathrm {fb})\) 

Bonsay  \(1.43636 \pm 0.00002\) 
PowhegBox  \(1.44092 \pm 0.00009\) 
VBFNLO  \(1.43796 \pm 0.00005\) 
Phantom  \(1.4374 \pm 0.0006 \) 
Whizard  \(1.4381 \pm 0.0002 \) 
MG5_aMC  \(1.4304 \pm 0.0007\) 
MoCaNLO+Recola  \(1.43476 \pm 0.00009\) 
In Fig. 5, we show the distributions in the invariant mass (left) and the rapidity difference (right) of the two tagging jets which are key observables for VBS measurements. In both cases we show the absolute distributions in the upper plot, while the lower plot displays the ratio over the predictions of MoCaNLO+Recola, for which we also display the scaleuncertainty band (sevenpoints variation as in Eq. (3.11) of Ref. [19]). For both observables we find a relatively good agreement among the various tools, which confirms the fact that contributions from schannel diagrams as well as interferences are suppressed in the fiducial region. In general, the agreement is at the level of \(1\%\) or below in each bin. We have checked that the same level of agreement holds for other standard differential distributions such as rapidity, invariant mass, or transverse momentum. This means that at LO, in the fiducial volume and for energies relevant to the LHC, the VBS approximation is good to a per cent. This is in agreement with the findings of Sect. 4.2 as the present comparison completely excludes the phasespace region where triboson contributions could have a noticeable impact.
5 Nexttoleading order QCD
5.1 Inclusive comparison
Cross sections at NLO QCD i.e. at order \(\mathcal {O}(\alpha _\text {s} \alpha ^6)\) for the full computation and two approximations. In addition to the cuts of Sect. 3.3, the VBS cuts take the values \(m_{\text {j} \text {j}}>200 \,\text {GeV} \) and \(\varDelta y_{\text {j} \text {j}}>2\). The uncertainties shown refer to the estimated statistical errors of the Monte Carlo programs
Prediction  \(\sigma _{\text {tot}}\,(\text {fb})\)  \(\delta \;(\%)\) 

full  \(1.733\phantom {0} \pm 0.002\phantom {0}\)   
\(t^2 + u^2\)  \(1.6292 \pm 0.0001\)  \(6.0\) 
\(s^2 + t^2 + u^2\)  \(1.7780 \pm 0.0001\)  \(+2.6\) 
We compare three different predictions at NLO QCD: the VBS approximation implemented in Bonsay (dubbed \(t^2+u^2\)), the VBS approximation with the schannel contributions from VBFNLO (dubbed \(s^2+t^2+u^2\)), and the full computation. The full computation employs exact matrix elements meaning that t/u/s interferences, factorisable and nonfactorisable QCD corrections, as well as EW corrections to the order \(\mathcal {O}(\alpha _\text {s} \alpha ^5)\) are included.
The total cross sections within the abovementioned kinematic cuts are shown in Table 4. The \(t^2+u^2\) approximation for NLO QCD predictions is lower by about \(6\%\) than the full calculation. The inclusion of schannel diagrams improves the approximate prediction, leading to an excess at the \(3\%\) level.
In conclusion, both the loose minimum dijet invariantmass cut and the inclusion of QCD radiative corrections render the schannel contributions less suppressed than at LO, making their inclusion mandatory, in order to provide trustworthy predictions at NLO accuracy. In the inclusive region studied here, neglecting schannel contributions, nonfactorisable corrections, and EW corrections can lead to discrepancies of up to \(30\%\) with respect to the full computation. Nevertheless, the VBS approximation at NLO provides a good approximation of full calculations in the kinematic region where \(m_{\text {j} \text {j}} \gtrsim 500 \,\text {GeV} \) and \(\varDelta y_{\text {j} \text {j}} \gtrsim 2.5\)), for both total cross section and differential distributions. This more exclusive region is studied in more detail in the next section.
5.2 Comparison in the fiducial region
Cross sections at NLO accuracy and order \(\mathcal {O}(\alpha _\text {s} \alpha ^6)\). The predictions are obtained in the fiducial region described in Sect. 3.3. The uncertainties shown refer to estimated statistical errors of the Monte Carlo integrations
Code  \(\sigma \;(\mathrm {fb})\) 

Bonsay  \(1.35039 \pm 0.00006\) 
PowhegBox  \(1.3605 \pm 0.0007\) 
VBFNLO  \(1.3916 \pm 0.0001\) 
MG5_aMC  \(1.363 \pm 0.004\) 
MoCaNLO+Recola  \( 1.378 \pm 0.001\) 
The first observation is that the predictions featuring two versions of the VBS approximation (Bonsay and the PowhegBox) are relatively close.^{10} Bonsay uses a doublepole approximation for the virtual matrix element, and it is worth noticing that this approximation seems to be accurate at \(1\%\) level as compared to the PowhegBox. This means that the doublepole approximation on the two W bosons used in Bonsay constitutes a good approximation of the VBSapproximated virtual corrections implemented in the PowhegBox. Both predictions differ by about \(2\%\) with respect to the full computation (MoCaNLO+Recola). The second observation is that the inclusion of schannel contributions seems to have a significant impact. Indeed, their inclusion (as done in VBFNLO) approximates the full computation by a per cent. The main contribution due to the schannel diagrams thereby consists of realemission contributions, where one of the two leading jets is formed by one quark, or possibly also both quarks, originating from the Wboson decay, and the second one by the extra radiation emitted from the initial state. In such configurations, the hadronicallydecaying W boson can become onshell and hence yield larger contributions than at LO, where the invariant mass cut on the two jets forces the boson into the far offshell region. However, the agreement between MoCaNLO+Recola and VBFNLO is mostly accidental, as the inclusion of interference effects and some nonfactorisable corrections (in the real corrections) in MG5_aMC brings the prediction down and closer to the VBS approximation. Not unexpectedly none of the approximations used here agrees perfectly with the full calculation of MoCaNLO+Recola at NLO. Nevertheless, the disagreement seems never to exceed \(2\%\) at the fiducial crosssection level.
In Figs. 10, 11 and 12, several differential distributions are shown. All these predictions are performed at NLO accuracy at the order \(\mathcal {O}(\alpha _\text {s} \alpha ^6)\). In the upper panel, the absolute predictions are shown while in the lower panel, the ratio with respect to the full predictions are displayed. The band corresponds to a sevenpoints variation of the factorisation and renormalisation scales (as defined in Eq. (3.11) of Ref. [19]).
In the end, the quality of the VBS approximations is good up to \(10\%\) in the fiducial region. These differences are larger than those at LO.
The contributions from the schannel amplitude can be sizeable especially at low invariant mass for the two tagging jets (comparing the predictions of VBFNLO against the ones of Bonsay and Powheg). This can be explained by the fact that schannel contributions are less suppressed at NLO. As real radiation, an extra gluonjet can be radiated from any of the stronglyinteracting particles while the two quarks originating from the Wboson decay can be recombined in a single jet. Therefore, the jet requirements (\( m_{\text {j} \text {j}} > 500\,\text {GeV} \) and \(\varDelta y_{\text {j} \text {j}} > 2.5\)) that were suppressing schannel contributions at LO are partially lifted with the inclusion of a third jet at NLO. Such an effect has also been observed for top–antitop production in the lepton+jet channel at NLO QCD [85].
In phasespace regions where the schannel contributions are sizeable their interference with the t/uchannel can be of similar size. This can be observed by comparing the predictions of VBFNLO against the ones of MG5_aMC.
Finally, the effect of EW corrections and nonfactorisable contributions in the virtual corrections are usually small. But they can be relatively large (about \(10\%\)) for large transverse momentum of the hardest jet. These highenergy region of the phase space are where EW Sudakov logarithms become large. Nonetheless these regions are rather suppressed and thus these effects are hardly visible at the level of the cross section.
6 Matching to parton shower
We now discuss how different predictions compare when the matching to parton shower is included, both at LO (i.e. at order \(\mathcal O (\alpha ^6)\)) and at NLOQCD (i.e. at order \(\mathcal O (\alpha ^6\alpha _\text {s})\)) accuracy. For such a comparison we expect larger discrepancies than what we found at fixed order, as a consequence of the different matching schemes, parton showers employed, and of other details of the matching (such as the choice of the parton shower initial scale). Among the codes capable of providing fixedorder results, presented before, MG5_aMC, the PowhegBox, and VBFNLO can also provide results at (N)LO+PS accuracy. For VBFNLO matched to Herwig and the PowhegBox, we restrict ourselves to show results only in the VBS approximation, i.e. the schannel contributions are neglected here. Besides, also Phantom and Whizard are used for LO+PS results.
MG5_aMC, which employs the MC@NLO [59] matching procedure, is used together with Pythia8 [86] (version 8.223) and Herwig7 [87, 88] (version 7.1.2). For the latter, the default angularordered shower is employed. The same parton showers are employed for the LO results of Phantom. Pythia8 is also employed for the LO results of Whizard. For the PowhegBox, the namesake matching procedure is employed [53, 54], together with Pythia8 (version 8.230). VBFNLO serves as a matrixelement and phasespace provider for the Matchbox module [89] of Herwig7 [87, 88], using an extended version of the Binoth Les Houches Accord interface [90, 91, 92]. The Matchbox module makes it possible to choose between MC@NLOlike and Powheglike matching. As parton shower, both the default angularordered shower as well as the dipole shower can be employed. We use here the subtractive (MC@NLOtype) matching to these partonshower algorithms. Whenever Pythia8 is used, the Monash tune [93] is selected. Multipleparton interactions are disabled.
Results are presented within the cuts described in Sect. 3.3, applied after shower and hadronisation (this implies that jets are obtained by clustering stable hadrons, and not QCD partons). It follows that at the eventgeneration level, looser cuts (or even no cuts at all) must be employed in order not to bias the results. This also implies that the tagging jets, whose momenta enter in the renormalisation and factorisation scales, Eq. (7), are now defined without imposing the \(\varDelta R_{\text {j} \ell }\) cut. The effect of this change is below one per cent at the level of the fiducial cross sections at NLO.
Cross sections at LO+PS accuracy. The MG5_aMC results with \(\varGamma _\mathrm{resc}\) are rescaled to account for the effect related to the Wboson width computed by MadSpin (see the text for details). The uncertainties shown refer to estimated statistical errors of the Monte Carlo integrations
Code  \(\sigma \;(\mathrm {fb})\) 

MG5_aMC+Pythia8  1.352 ± 0.003 
MG5_aMC+Herwig7  1.342 ± 0.003 
MG5_aMC+Pythia8, \(\varGamma _\mathrm{resc}\)  1.275 ± 0.003 
MG5_aMC+Herwig7, \(\varGamma _\mathrm{resc}\)  1.266 ± 0.003 
Phantom+Pythia8  1.235 ± 0.001 
Phantom+Herwig7  1.258 ± 0.001 
VBFNLO+Herwig7Dipole  1.3001 ± 0.0002 
Whizard+Pythia8  1.229 ± 0.001 
Cross sections at NLO+PS accuracy. The MG5_aMC results with \(\varGamma _\mathrm{resc}\) are rescaled to account for the effect related to the Wboson width computed by MadSpin (see the text for details). For VBFNLO+Herwig7Dipole, the threepoint scale uncertainties are shown, while for MG5_aMC+Pythia8 the two displayed uncertainties are respectively the ninepoint scale uncertainty and the PDF one. The uncertainties shown refer to estimated statistical errors of the Monte Carlo integrations
Code  \(\sigma \;(\mathrm {fb})\) 

MG5_aMC+Pythia8  \(1.491 ^{+1\%}_{2\%} {}^{+2\%}_{2\%} \pm 0.004\) 
MG5_aMC+Herwig7  \(1.427 \pm 0.003\) 
MG5_aMC+Pythia8, \(\varGamma _\mathrm{resc}\)  \(1.407 \pm 0.003\) 
MG5_aMC+Herwig7, \(\varGamma _\mathrm{resc}\)  \(1.346 \pm 0.002\) 
PowhegBox+Pythia8  \(1.3642 \pm 0.0004\) 
VBFNLO+Herwig7Dipole  \(1.3389 ^{+0\%}_{1\%} \pm 0.0006\) 
VBFNLO+Herwig7  \(1.3067 \pm 0.0006\) 
We now present the results of predictions matched to parton showers. The total rates within VBS cuts are displayed in Tables 6 and 7, at LO and NLO accuracy respectively. For MG5_aMC, the numbers with \(\varGamma _\mathrm{resc}\) are rescaled to take into account the width effects described in the above paragraph. At NLO accuracy, for MG5_aMC + Pythia8 and VBFNLO+Herwig7Dipole, we also quote theoretical uncertainties. For the former, we show both PDF and scale uncertainties,^{11} obtained via exact reweighting [72] by varying independently the renormalisation and factorisation scales by a factor of two around the central value, Eq. (7) (ninepoint variations). For the latter, we show the threepoint scale uncertainties, obtained by considering correlated variations of the renormalisation, factorisation, and shower starting scale. Theory uncertainties should have very little dependence on the tool employed. We observe that, once the width effect is taken into account, total rates from different tools agree within some per cents (\(\le 7\%\)), both at LO and NLO. Larger discrepancies, however, appear for differential observables, which we discuss in the following. Theory uncertainties on the total rates are very small, regardless of whether scale variations are estimated with independent or correlated variations of the renormalisation and factorisation scales. Concerning differential distributions, for each observable we display results in two plots, shown sidebyside. In the plot on the left (right), (N)LO+PS predictions are shown with different colours in the main frame. In the inset, these predictions are compared in both cases with a fixedorder prediction at NLO accuracy (obtained with VBFNLO, i.e. the VBS approximation with schannel). For the differential observables, the MG5_aMC predictions are not rescaled to compensate for the width effect mentioned above. As for the table, we show theoretical uncertainties for the NLO+PS samples obtained with VBFNLO and MG5_aMC: again, for the first the band corresponds to threepoint variations, while for the second the darker (lighter) band corresponds to ninepoint scale variations (plus PDF uncertainties, linearly added).
The next observable that we study is the invariant mass of the two tagging jets, shown in Fig. 14. For this observable, both at LO+PS and NLO+PS, the spread of predictions matched with parton shower is rather small (\(\lesssim 10\%\), if one compensates for the \(6\%\) width effect for MG5_aMC). The LO+PS predictions tend to be significantly softer than the fixed NLO one, with an effect of about \(\,30\%\) at the end of the displayed range. At NLO+PS, this effect is mitigated, owing to the better description of the first QCD emission which is now driven by the realemission matrix element. For this observable (and all the others which are NLO accurate) the effect of reducing the shower scale is negligible, hence it is not shown.
The rapidity difference between the two tagging jets, shown in Fig. 15, has some similarities with the invariantmass distribution. At LO+PS all predictions show the tendency to deplete the largeseparation region with respect to the fixedorder prediction, in a quantitatively similar way, except for VBFNLO+Herwig7 where the effect is mitigated. At NLO+PS, when the extra radiation is described by the real matrix element, such an effect is greatly reduced. A notable exception is the PowhegBox prediction, which still shows a suppression at large separations. Since such a suppression is already there for the PowhegLHE sample, it is very likely that it is driven by the way the first emission is generated. A minor effect in the same direction is visible in the last two bins of the MG5_aMC+Herwig7 prediction (although with rather large statistical uncertainties).
If we consider the rapidity of the second jet, Fig. 18, we observe again rather small differences among tools, with the tendency towards a general stabilisation at NLO+PS. However, some (small) differences in the shape remain at NLO+PS, which are worth to be briefly discussed: predictions obtained with MG5_aMC are very close to the fixedorder prediction; the PowhegBox displays an enhancement of the central region, and a consequent suppression in the peripheral region, while VBFNLO shows an opposite behaviour. However, the effect is rather small, with the largest departure from the fixedorder prediction being at most \(10\%\).^{14}
Finally, focusing on the third jet, we conclude the list of differential observables by showing the Zeppenfeld variable defined in Eq. (12), Fig. 19. This variable is closely related to the third jet rapidity, and small (large) values of z correspond to central (peripheral) rapidities. In general, for observables which involve the third jet, one can clearly see a degradation of the agreement among the various tools, because of the poorer perturbative description of these observables. The Zeppenfeld variable is a striking example: both at LO and NLO, the tendency of Pythia8 to generate more hard and central radiation, corresponding to low values of z, is clearly visible. Such an effect, which is related to the way Pythia8 deals with the recoil of the radiation in VBF(VBS)type processes, can be mitigated by setting SpaceShower:dipoleRecoil = on in the Pythia8 input file.^{15} It is interesting to notice that the effect survives beyond the first emission, as it can be observed by comparing PowhegLHE with Powheg+ Pythia8, and that it is only marginally attenuated when the shower scale is reduced. A similar behaviour of Pythia8 has also been observed in the study of EW production of a \(\text {Z} \) boson in association with two jets (see the recent CMS measurement, Ref. [97] Figure 12), where the experimental data seem to prefer the description by Herwig++ [98, 99]. The central enhancement is a bit mitigated if NLO+PS tools are used (compare LO+PS and NLO+PS from MG5_aMC+Pythia8 with the fixedNLO prediction), however even at NLO+PS the central region (\(z_{j_3}<0.5\)) is cursed by huge differences between tools. Large differences, reaching a factor 2, persist also away from the central region. These findings are consistent with behaviour displayed in Refs. [11, 100, 101, 102, 103] where the behaviour of NLO matching in VBS processes has been reported.
In conclusion, the comparison of tools including matching to parton shower clearly shows the benefits of the inclusion of NLO corrections: for most observables described effectively at NLO accuracy differences between tools are at (or below) the \(10\%\) level. Some exceptions exist, e.g. the rapidity separation of the two tagging jets, which on the one hand clearly suggest not to rely on a single tool/parton shower, and on the other make it worth investigating more in detail the way QCD radiation is generated, e.g. when fullydifferential computations at NNLO will become available (for VBF Higgs production, see Refs. [104, 105]). It is a remarkable fact that, even for those observables that display small discrepancies, the theoretical uncertainty obtained via scale variations (renormalisation, factorisation, and shower scale) systematically underestimates the spread of predictions. We note that in the only VBF process where NNLO corrections are known, i.e. VBF Higgs production [104, 105], the NLO scaleuncertainty band does not include the NNLO prediction. This suggests that the NLO scale variation underestimates the size of the perturbative uncertainty. Again, this stresses the need to employ at least two different tools in order to obtain a more realistic estimate of theoretical uncertainties. Finally, the size of discrepancies for observables that are described at a lower perturbative accuracy, notably those related to the third jet, suggests that experimental analyses should rely as little as possible on those observables and, in any case, use conservative estimates of the theory uncertainties. On the one hand, in order to improve the description of these observables, a simulation of VBS+j at NLO accuracy, currently unavailable but within the reach of modern automated tools, is certainly desirable. On the other hand, measurements of processes with similar colour flow (EW production of a single vector boson plus jets, VBF, \(\ldots \)) can certainly help in order to discriminate which tools perform better in the comparison with data [97, 106].
7 Conclusions and recommendations
In the present article, a detailed study of the process \(\text {p} \text {p} \rightarrow \mu ^+\nu _\mu \mathrm{e}^+\nu _\mathrm{e}\,\text {j} \text {j} +\mathrm {X}\) at the LHC has been presented, mainly focused on the EW production mechanism which involves the scattering of massive vector bosons. Until very recently, when the complete calculation became available for the NLO QCD corrections (order \(\mathcal O (\alpha _\text {s} \alpha ^6)\)), the socalled VBS approximation was the standard for this kind of simulations. For this reason, various theoretical predictions have been compared to the full computation, both in a typical VBS fiducial region and also in more inclusive phase space. We have precisely quantified the differences that arise for several physical observables, in particular for the dijet invariant mass and the rapidity separation of the leading two jets. This is the first time that such an indepth study is performed. Besides the study of fixedorder predictions, we have also investigated the impact of parton showering. To that end, several LO and NLO event generators which are able to perform matching to parton showers have been employed, and various observables have been thoroughly compared. While in general observables which are described at NLO accuracy show reasonable agreement among the tools, larger differences can appear for those observables described at a lower accuracy, such as those that involve the third jet. In particular such differences are quite prominent in the centralrapidity region, and are the largest for those simulations which employ Pythia8. The effect has been understood, and it can be partially mitigated by changing the recoil scheme of Pythia8 to distribute momenta within initial–final colour connections. These findings make it worth to further investigate these issues not only in the theoretical community, but also by experimental collaborations, for example by measuring related observables for similar processes.

As found in Ref. [19], the NLO EW corrections of order \(\mathcal {O}{\left( \alpha ^{7}\right) }\) are the dominant NLO contribution to the process \(\text {p} \text {p} \rightarrow \mu ^+\nu _\mu \mathrm{e}^+\nu _\mathrm{e}\,\text {j} \text {j} +\mathrm {X}\). It is thus highly desirable to combine them with NLOQCD predictions matched with parton shower, or at least to include them into experimental analyses. Since, as shown in Ref. [77], these large EW corrections originate from the Sudakov logarithms which factorise, we recommend to combine them with QCD corrections in a multiplicative way. The estimate of missing higherorder EW corrections can be obtained, in a first approximation, by considering \(\pm \delta ^2_\mathrm{NLO EW}\),^{16} while the missing higherorder mixed QCDEW corrections can be estimated by taking the difference between the multiplicative and additive prescriptions. For more detailed studies of the combination of QCD and EW higherorder corrections, see e.g. Ref. [107] in the context of toppair production, or Ref. [108] for SM backgrounds for dark matter searches at the LHC.

For the typical fiducial region used by experimental collaborations for their measurements, the agreement between the approximations and the full calculation is satisfactory given the current experimental precision, as well as the one foreseen for the near future [5, 6, 109]. Nonetheless, care has to be taken when using such approximations, in particular if more inclusive phasespace cuts are used.

In addition to the standard interpretation of EW signal versus QCD background, combined measurements should also be presented as they are better defined theoretically. In fact, while at LO the interference term can be included in the background component, at NLO the separation of EW and QCD components becomes more blurred, as, e.g. at the order \(\mathcal {O}{\left( \alpha _\text {s} \alpha ^{6}\right) }\) both types of amplitudes contribute. Therefore, a combined measurement including the EW, QCD, and interference contributions is desirable. Note that with such a measurement a comparison to the SM would be straightforward and still be sensitive to the EW component. In addition, the QCD component could be subtracted based on a welldefined Monte Carlo prediction.

Since the inclusion of NLO QCD corrections gives a better control of extra QCD radiation and reduces the ambiguities related to the matching details and/or the parton shower employed, we encourage the use of NLOaccurate event generators in experimental analyses. In doing so, special care should be employed in order to estimate the theoretical uncertainties, as the standard prescription based on renormalisation and factorisationscale variation is clearly inadequate. Rather, different combinations of generators and parton showers should be employed.

The present study has focused on the orders \(\mathcal {O}{\left( \alpha ^{6}\right) }\) at LO and \(\mathcal {O}{\left( \alpha _\text {s} \alpha ^{6}\right) }\) at NLO. NLO computations and publiclyavailable tools also exist for the QCDinduced process [13, 14, 15, 16, 17, 19, 58].

For practical reasons, we have focused on the \(\mathrm{W}^+\mathrm{W}^+\) signature. Nonetheless, the observed features (e.g. validity of the VBS approximation or comparison of theoretical predictions matched to parton shower) should be qualitatively similar for other VBS signatures with massive gauge bosons. For these other signatures, similar quantitative studies should be performed.
Footnotes
 1.
 2.
The name VBS is used even though not all Feynman diagrams involve the scattering of vector bosons.
 3.
Despite the loop_qcd_qed_sm_Gmu model also includes NLO counterterms for computing EW corrections, it is not yet possible to compute such corrections with the current public version of the code.
 4.
For the process considered, no bottom (anti)quarks appear in the initial or final state at LO and NLO, as they would lead to top quarks, and not light jets, in the final state.
 5.
Note that the PowhegBox uses its own implementation of the two loop running for \(\alpha _\mathrm{s}\).
 6.
The corresponding identifier lhaid in the program LHAPDF6 [80] is 260000.
 7.
By default, the renormalisation and factorisation scales employed in the PowhegBox slightly differ from the ones defined in Eq. (7), as the momenta of two finalstate quarks in the underlying Born event are employed instead of those of the two hardest jets.
 8.
In Fig. 4, the level of the accuracy of the predictions in each bin is around a per mille.
 9.
The bin in the topleft corner of the righthandside plot of Fig. 7 suffers from large uncertainty (\(30\%\)) while the other errors are at the percent level.
 10.
The VBFNLO prediction omitting schannel contributions amounts to \(1.3703 \pm 0.0001\) fb. This differs from the PowhegBox prediction mainly due to the different choice of scales used in the PowhegBox (cf. footnote 7).
 11.
A preliminary study on PDF uncertainties in VBS has appeared in Ref. [47].
 12.
At LO, the choice of such a scale is arbitrary and usually driven by common sense, as it is the case for the factorisation and renormalisation scales. At NLO, one has the freedom to change the shower scale without losing formal NLO accuracy within the MC@NLO matching, provided the Monte Carlo counterterms are also consistently updated. In the Powheg matching, the shower scale of the socalled \(\tilde{B}\) events is fixed to the transverse momentum of the radiation generated according the Powheg Sudakov factor, while it can be changed in the remnant events.
 13.
The reduction of the shower scale for MG5_aMC+Herwig7 gives no visible effect on any of the observables discussed in this work.
 14.
If the setting SpaceShower:dipoleRecoil = on (discussed in the following) is used when Pythia8 is employed together with the PowhegBox, the enhancement at central rapidities and the depletion at small value of transverse momentum are partially compensated.
 15.
This requires version \(\ge 8.230\). Note that such a setting is not compatible with the NLO matching in MG5_aMC (but it is compatible with the Powheg matching). Also, this setting has other effects, though smaller, on the rapidity spectra of the two hardest jets.
 16.
The quantity \(\delta _\mathrm{NLO EW}\) is defined through the relation \(\sigma _\mathrm{NLO EW} = \sigma _\mathrm{LO} \left( 1+ \delta _\mathrm{NLO EW}\right) \).
Notes
Acknowledgements
The authors are grateful to the members of the VBSCan collaboration for several discussions and comments on this work. The author also thank the Pythia8 developers, in particular Stefan Prestel, Torbjörn Sjöstrand, and Peter Skands for discussions and clarifications about the thirdjet rapidity spectrum. MZ would like to thank Andreas Papaefstathiou, Christian Reuschle and David Grellscheid for their help with Herwig7. RF, HSS, and MZ thank the MadGraph5_aMC@NLO authors for discussions, and they are particularly grateful to Valentin Hirschi for comments on this manuscript. The authors would like to acknowledge the contribution of the COST Action CA16108 which initiated this work. Moreover, this work was supported by several STSM Grants from the COST Action CA16108. Many authors acknowledge hospitality from Nikhef, where part of this work has been performed. BB, AD, and MP acknowledge financial support by the German Federal Ministry for Education and Research (BMBF) under Contract no. 05H15WWCA1 and the German Research Foundation (DFG) under reference number DE 623/61. SD and CS acknowledge support by the state of BadenWürttemberg through bwHPC and the DFG through Grant no. INST 39/9631 FUGG and Grant DI 784/3. The work of BJ was supported in part by the Institutional Strategy of the University of Tübingen (DFG, ZUK 63) and in part by the BMBF under Contract number 05H2015. AK acknowledges financial support by the Swiss National Science Foundation (SNF) under Contract 200020175595. MR acknowledges funding from the European Union’s Horizon 2020 research and innovation programme as part of the Marie SkłodowskaCurie Innovative Training Network MCnetITN3 (Grant agreement no. 722104). HSS is supported by the ILP Labex (ANR11IDEX000402, ANR10LABX63). GZ is supported by ERC Consolidator Grant HICCUP (no. 614577). MZ is supported by the Netherlands National Organisation for Scientific Research (NWO).
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