Measurement of electroweak-induced production of W gamma with two jets in pp collisions at sqrt(s) = 8 TeV and constraints on anomalous quartic gauge couplings

A measurement of electroweak-induced production of W gamma and two jets is performed, where the W boson decays leptonically. The data used in the analysis correspond to an integrated luminosity of 19.7 inverse femtobarns collected by the CMS experiment in sqrt(s) = 8 TeV proton-proton collisions produced at the LHC. Candidate events are selected with exactly one muon or electron, missing transverse momentum, one photon, and two jets with large rapidity separation. An excess over the hypothesis of the standard model without electroweak production of W gamma with two jets is observed with a significance of 2.7 standard deviations. The cross section measured in the fiducial region is 10.8 +/- 4.1 (stat) +/- 3.4 (syst) +/- 0.3 (lumi) fb, which is consistent with the standard model electroweak predictions. The total cross section for W gamma production in association with 2 jets in the same fiducial region is measured to be 23.2 +/- 4.3 (stat) +/- 1.7 (syst) +/- 0.6 (lumi) fb, which is consistent with the standard model prediction from the combination of electroweak- and quantum chromodynamics-induced processes. No deviations are observed from the standard model predictions and experimental limits on anomalous quartic gauge couplings f[M, 0-7] / Lambda^4, f[T, 0-2] / Lambda^4, and f[T, 5-7] / Lambda^4 are set at 95% confidence level.

The VBS processes have some features that can be exploited to better understand the SM in novel phase spaces and to probe new physics or constrain anomalous gauge couplings. For example, phenomenological studies of the EW production of W and Z bosons in association with two jets that exploit the large rapidity gaps between the two jets [23,24]. Also, the VBF process was studied using the Higgs boson production and decay in Ref. [25][26][27][28]. Furthermore, the EW production of Z bosons, Zγ, Zγγ, and same-sign W boson pairs in association with two jets has recently been measured at the LHC [16-18, 20, 21, 29]. Moreover, both the ATLAS and the CMS experiments found evidence for exclusive γγ to W + W − production [15,19], and the ATLAS experiment found evidence for Wγγ triple boson production [30]. All the results are in good agreement with the SM predictions.
In this analysis, we search for EW-induced Wγ production in association with two jets [31] (EW Wγ+2 jets) in the W boson leptonic decay channel (W → ν, = e, µ). This process is expected to have one of the largest cross sections of all the VBS processes and thus is expected to be one of the first VBS processes observable at a hadron collider. As shown in Fig. 1, Wγ production includes several different classes of diagrams: bremsstrahlung of one or two vector bosons and the more interesting VBS EW processes such as in Fig. 1c. The cross sections of EW-induced only and EW+QCD total Wγ processes are measured in a VBS-like fiducial region, where the two jets have a large separation in pseudorapidity. The signal structure of the weak boson scattering events makes VBS processes a good probe of quartic gauge boson couplings. Instead of measuring the SM gauge couplings, which are completely fixed by the SM SU(2) L ⊗ U(1) Y gauge symmetry, we keep the SM gauge symmetry while setting limits on a set of higher dimensional anomalous quartic gauge couplings (aQGCs). More details of the aQGC parameterization can be found in Appendix A.  The production of Wγ+2 jets at the LHC has two major contributions at leading order (LO) in addition to the EW signal process described above: QCD and triple gauge boson WVγ processes, with V = W or Z decaying into a quark-antiquark pair. Because these processes can have the same set of initial and final states, these three contributions interfere. One can suppress this interference by choosing an appropriate phase space for the measurements. The WVγ events reside mainly in the W or Z boson mass window; we require m jj > 200 GeV to eliminate most of this contribution. The EW Wγ+2 jets events favor a larger m jj region than the QCD Wγ+2 jets events do. Calculations using the MADGRAPH program show the interference decreases with increasing m jj and |∆η(j1, j2)|, and can change from constructive to destructive at ∼1 TeV in m jj depending on the choice of renormalization and factorization scales. In the analysis we consider the phase space region with m jj > 700 GeV and |∆η(j1, j2)| > 2.4 to suppress the interference. The interference effect in the fiducial region is estimated to be 4.6% of the total Wγ+2 jets cross section.
In addition to the main background from QCD Wγ+2 jets production [32], other backgrounds include (1) jets misidentified as photons or electrons, (2) WVγ events with hadronically decaying V bosons (W/Z → jj) and a photon from initial-or final-state radiation, (3) contributions from top quark pairs with a radiated photon, and (4) single top quark events with a radiated photon. The selection criteria are designed to reduce the collective sum of these backgrounds.
In the case of nonzero anomalous couplings, the EW contribution can be greatly enhanced, especially in the high-energy tails of some kinematic distributions; therefore, we require the photon and W boson to have large transverse momenta to obtain better sensitivity.
The paper is organized as follows: Section 2 describes the CMS detector. Section 3 presents the Monte Carlo event simulation and data sample and Section 4 describes the event reconstruction and selection. In Section 5, methods of background modeling are explained. Systematic uncertainties considered in the analysis are discussed subsequently in Section 6. Results of the search for the EW signal and the measured EW and EW+QCD cross sections in the fiducial region are reported in Section 7. Results on anomalous couplings using the W boson transverse momentum distribution are given in Section 8. Finally, Section 9 summarizes the results.

The CMS detector
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter and 13 m length, providing a magnetic field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL). Muons are reconstructed in gas-ionization detectors embedded in the steel flux-return yoke outside the solenoid. Extensive forward calorimetry complements the coverage provided by the barrel and endcap detectors.
The tracking system consists of 1440 silicon pixel and 15 148 silicon strip detector modules and covers the pseudorapidity range |η| < 2.5, providing a transverse momentum p T resolution of about 1.5% at 100 GeV. The electromagnetic calorimeter consists of 75 848 lead tungstate crystals, which provide coverage in |η| < 1.48 in the barrel region (EB) and 1.48 < |η| < 3.00 in the two endcap regions (EE). A preshower detector consisting of two planes of silicon sensors interleaved with three radiation lengths of lead is located in front of the EE. Photons are identified as ECAL energy clusters not linked to the extrapolation of any charged particle trajectory to the ECAL. These energy clusters are merged to form superclusters that are five crystals wide in η, centered around the most energetic crystal, and have a variable width in the azimuthal angle φ. The HCAL consists of a set of sampling calorimeters that utilize alternating layers of brass as absorber and plastic scintillator as active material. It provides coverage for |η| < 3.0. Combined with the forward calorimeter modules, the coverage of hadronic jets is extended to |η| < 5.0. The energy of charged hadrons is determined from a combination of the track momentum and the corresponding ECAL and HCAL energies, corrected for the combined response function of the calorimeters. The energy of neutral hadrons is obtained from the corresponding corrected ECAL and HCAL energies. The muon system includes barrel

Data and simulated samples
The analysis uses a data sample of proton-proton collisions collected at √ s = 8 TeV by the CMS detector in 2012 that corresponds to an integrated luminosity of 19.7 ± 0.5 fb −1 [35].
For all MC samples, a GEANT4-based simulation [50] of the CMS detector is used and the hard-interaction collision is overlaid with a number of simulated minimum-bias collisions. The resulting events are weighted to reproduce the data distribution of the number of inelastic collisions per bunch crossing (pileup). These simulated events are reconstructed and analyzed using the same algorithms as for data. The differences in lepton and photon reconstruction and identification (ID) efficiencies observed between data and simulated events are subsequently corrected with scale factors [51,52].
To improve the precision of the predicted cross section for the signal model, the NLO QCD correction is included with the EW signal process through an NLO/LO cross section K factor of 1.02, determined by using VBFNLO [31,32,[53][54][55]. For QCD Wγ+2 jets production, the K factor is 0.93 and is only applied for the measurement of the EW+QCD cross section, fixing the ratio between EW and QCD components.
The events are selected by using single-lepton triggers with p T thresholds of 24 GeV for muons and 27 GeV for electrons. The overall trigger efficiency is 90% (94%) for the electron (muon) data, with a small dependence on p T and η. Charged-particle tracks are required to originate from the event primary vertex, defined as the reconstructed vertex within 24 cm (2 cm) of the center of the detector in the direction along (perpendicular to) the beam axis that has the highest value of p 2 T summed over the associated charged-particle tracks. The events are also required to have either one muon or one electron; events with additional charged leptons are excluded. The muon candidates are reconstructed with information from both the silicon tracker and from the muon detector by means of a global fit [33]. They are required to satisfy a requirement on the PF-based relative isolation, which is defined as the ratio of the p T sum of all other PF candidates reconstructed in a cone ∆R = √ (∆η) 2 + (∆φ) 2 = 0.3 (0.4) around the candidate electron (muon) to the p T of the candidate, and is corrected for contributions from pileup [51]. The selection efficiency is approximately 96%. Muons with p T > 25 GeV and |η| < 2.1 are included in the analysis. The electron candidates are reconstructed by associating a charged particle track originating from the event primary vertex with superclusters of energy depositions in ECAL [51]. They must also satisfy the PF-based relative isolation be smaller than 0.15. The ID and isolation selection efficiency is approximately 80%. The electron candidates are further required to satisfy p T > 30 GeV and |η| < 2.5, excluding the transition region between the ECAL barrel and endcaps, 1.44 < |η| < 1.57, because the reconstruction of electrons in this region has lower efficiency. To suppress the Z → e + e − background in the electron channel, where one electron is misidentified as a photon, a Z boson mass veto of |m eγ − M Z | > 10 GeV is applied.
A well-identified and isolated photon is also required for the event selection [52]. Photons are reconstructed from superclusters and are required to satisfy a number of criteria aimed at rejecting misidentified jets. They have to have a small ratio of hadronic energy in the HCAL that is matched in (η, φ) to the electromagnetic energy in the ECAL; small shower shape variable σ ηη , which quantifies the lateral extension of the shower along the η direction [51]; small PF-based charged and neutral photon isolations including pileup corrections [56]; and an electron-track veto to reduce electron misidentification. With these requirements the photon ID and isolation efficiency is about 70%. The resulting photon candidates are further required to satisfy p γ T > 22 GeV and must be in the barrel region with |η sc | < 1.44, where η sc refers to the supercluster η, corresponding to a fiducial region in the ECAL barrel excluding the outer barrel ECAL rings of crystals.
Jets are reconstructed from PF particles [56,57] using the anti-k T clustering algorithm [58] with a distance parameter of 0.5. Only charged particles with tracks originating from the primary vertex are considered for clustering. Jets from pileup are identified and removed with a pileup jet identification algorithm [59], based on both vertex information and jet shape information. Jets are required to satisfy a set of loose ID criteria designed to eliminate jets originating from Single-lepton (e, µ) trigger |M eγ − M Z | > 10 GeV (electron channel) Lepton, photon ID and isolation p j1 noisy channels in the calorimeter [60]. Pileup collisions and the underlying event can contribute to the energy of the reconstructed jets. A correction based on the projected area of a jet on the front face of the calorimeter is used to subtract the extra energy deposited in the jet coming from pileup [61,62]. Furthermore, the energy response in η and p T is corrected, and the energy resolution is smeared for simulated samples to give the same response as observed [63]. An event is selected if it has at least two jets, with the leading jet p T > 40 GeV, second-leading jet p T > 30 GeV, and each jet within |η| < 4.7. These two jets are denoted as "tag jets". To suppress the WVγ background, m jj is required to be at least 200 GeV.
In addition, the event should have | p miss T | > 35 GeV. The reconstructed transverse mass of the leptonically decaying W boson, defined as is the azimuthal angle between the lepton momentum and the p miss T , is required to exceed 30 GeV [64]. We reconstruct the leptonic W boson decay by solving for the longitudinal component of the neutrino momentum and using the mass of the W boson as a constraint. In the case of complex solutions in this reconstruction, we choose the real part of the solution, and if there are two real solutions, we choose the solution that gives a neutrino momentum vector that is closer to the longitudinal component of the corresponding charged lepton momentum.
Mismeasurement of jet energies can generate | p miss T |. To eliminate events in which this mismeasurement may generate an apparent large | p miss T |, the azimuthal separation between each of the tag jets and the p miss T is required to be larger than 0.4 rad. Additionally, to suppress the top quark backgrounds, we require that the tag jets fail a b tagging requirement of the combined secondary vertex algorithm [65] with a misidentification rate of 1%.
Separation between pairs of objects in the event is required: ∆R jj , ∆R jγ , ∆R j , and ∆R γ > 0.5. All the requirements described above ensure the quality of the identified final states and comprise the baseline selections for the analysis. Table 1 summarizes these criteria.
To optimize the measurement of the EW-induced Wγ+2 jets signal and improve the EW signal significance, we further consider selections on the following variables to suppress backgrounds: the Zeppenfeld variable [23], |y Wγ − (y j1 + y j2 )/2|, calculated using the rapidities (y) of the Wγ system and the two jets; the azimuthal separation between the Wγ system, which combines the four momenta of the W boson and the photon, and the dijet system |∆φ Wγ,jj |; the dijet invariant mass m jj ; and the pseudorapidity separation between the tag jets |∆η(j1, j2)|. These additional requirements are chosen as follows:

5 Background estimation 5 Background estimation
The dominant background comes from QCD Wγ+jets production. It is estimated using simulation and is normalized to the number of events in data in the region 200 < m jj < 400 GeV. The data/simulation normalization factors 0.77 ± 0.05 (muon channel) and 0.77 ± 0.06 (electron channel) are consistent with the K factor of 0.93 ± 0.27 obtained with VBFNLO. For the combined measurement of the EW+QCD cross section, the contribution of QCD Wγ+jets is taken directly from simulation (scaled by the K factor) since this contribution is then no longer a background.
The background from misidentified photons arises mainly from W+jets events where one jet satisfies the photon ID criteria. The estimation is based on events similar to the ones selected with the baseline selection described in Section 4, except that the photon must fail the tight photon ID and satisfy a looser ID requirement. This selection ensures that the photon arises from a jet, but still has kinematic properties similar to a genuine photon originating from the primary vertex. The selected events are then normalized to the number of events satisfying the tight photon ID and weighted with the probability of a jet to be misidentified as a photon. The misidentification probability is calculated as a function of photon p T in a manner similar to that described in Ref. [66]. The method uses the shapes of the σ ηη and PF charged isolation distributions, which differ for genuine and misidentified photons. The fraction of the total background in the signal region contributed by this source decreases with p γ T , from a maximum of 33% (p T ≈ 22 GeV) to 6% (p T > 135 GeV).
The γ+jets events contribute to the background when the jet is misidentified as a muon or electron. The contribution is found to be negligible in the muon channel, but can be significant in the electron channel, especially in the low-m jj region. A control data sample is selected, in a similar way to that discussed in the previous paragraph, from the PF relative isolation sideband with a very loose electron ID requirement. Events in this control sample are then normalized to the events with signal selection and weighted with the misidentification probability for a jet to satisfy the electron selections. This probability is determined from a three-component fit to the | p miss T | distribution considering the γ+jets misidentified events, QCD Wγ+jets events, and misidentified photon events, as explained in more detail in Ref. [64]. The γ+jets background contribution in electron channel is estimated to be 7% of the total yield for the baseline selections and negligible in the EW signal region.
Other background contributions are small and are estimated from simulation. The contributions from top quark pair and single top quark production, each in association with a photon, are suppressed with the b quark veto and represents only 3.4% of the total event yield in the EW signal region. The Z(→ )γ(+jets) events can contribute if one of the decayed leptons is undetected, resulting in | p miss T |. The predicted cross sections of the Zγ and WV processes decrease with increasing m jj and contribute about 2% of the total SM prediction in the EW signal region. Figure 2 shows three m jj distributions in orthogonal, but signal-like, regions obtained by inverting each of three signal selection criteria: |∆η(j1, j2)| < 2.4; |y Wγ − (y j1 + y j2 )/2| > 0.6; and |∆φ Wγ,jj | < 2.6 rad. Each of these regions is enriched in QCD production of Wγ+jets events and, to a lesser degree, background having a jet misidentified as a photon. They confirm our modeling of those backgrounds.   Figure 2: The m jj distributions in orthogonal, but signal-like, regions obtained by inverting the signal selection criteria: |∆η(j1, j2)| < 2.4; |y Wγ − (y j1 + y j2 )/2| > 0.6; and |∆φ Wγ,jj | < 2.6 rad. Events from electron and muon channels are combined. Backgrounds from jets misidentified as photons (Jets → γ) and jets misidentified as electrons (Jets → e) are estimated from data as described in the text. The diboson contribution includes WV(+γ) and Zγ(+jets) processes. The top quark contribution includes both the ttγ and single top quark processes. The signal contribution is shown on top of the backgrounds. The last bin includes the overflow events. The shaded area represents the total uncertainty in the simulation, including statistical and systematic effects.

Systematic uncertainties
The background rate of QCD Wγ+jets production is measured in the low-m jj control region and extrapolated to the signal region. The rate uncertainty includes the statistical uncertainty as well as the uncertainties due to the misidentification probability of jets as photons or leptons. This uncertainty is 6.2% (7.1%) for the muon (electron) channel. In addition, when extrapolating from the control region to the signal region, the shape dependence on theoretical parameters affects the normalization of the QCD Wγ+jets distribution at high m jj . This extrapolation uncertainty is calculated by using different MC samples with matching and renormalization/factorization scales varied up and down by a factor of two. Contributions of all the shapes are normalized in the control region and the largest absolute difference from the nominal one in the signal region is considered as the uncertainty, this is about 20% for m jj ≈ 1 TeV.

EW Wγ+2 jets signal and cross section measurements
The uncertainty on the misidentification probability of jets as electrons is estimated by considering both the | p miss T | fit uncertainty and shape uncertainty and is estimated to be 40%. There are three contributions to the uncertainties in the misidentified photon background: the statistical uncertainty, the variation in the choice of the charged isolation sideband, and the σ ηη shape in the sample of events with objects misidentified as photons. The combined uncertainty, calculated in p γ T bins, increases from 13% at p γ T ≈ 25 GeV to 54% for p γ T ≈ 135 GeV. The uncertainty in the measured value of the integrated luminosity is 2.6% [35]. Jet energy scale and resolution uncertainties contribute via selection thresholds for the jet p T and m jj . We consider the uncertainties in different intervals of m jj , giving a combined uncertainty varying from 12 to 31% with increasing m jj in the signal region. A small difference in | p miss T | resolution [67] between data and simulation affects the signal selection efficiency by less than 1%. The uncertainties due to the lepton trigger efficiency and reconstruction and the selection efficiencies are estimated to be 1% and 2%, respectively. Photon reconstruction efficiency and energy scale uncertainties contribute to the signal selection efficiency at the 1% level. The uncertainty from the b jet veto procedure is 2% in the data/simulation efficiency correction factor [65]. This uncertainty has an effect of 8% on the ttγ background, 23% on the single top quark background, and a negligible effect on the signal. The theoretical uncertainty in the ttγ and Zγ+jets production cross section is 20% [14].
The theoretical uncertainty is evaluated with VBFNLO by varying the renormalization and factorization scales, each by factors of 1/2 and 2 with the requirement that the two scales remain equal. The envelope of all the variations is taken as the uncertainty. The uncertainty related to the PDF is calculated using the CTEQ6.1 [68] PDF uncertainty sets, following the prescription of Ref. [68]. For EW Wγ+2 jets and possible aQGC signal yield, this uncertainty is found to be 20% with scale variations and 2.8% with PDF sets. For QCD Wγ+2 jets, this is 29% with scale variations and 4.2% with PDF sets. The theoretical uncertainties due to scale and PDF choices affect the expected m jj shape and introduce an uncertainty in the cross section measured by fitting the m jj distribution. In addition, they affect the signal and the selection acceptance and efficiency. Extrapolation from the selected region to the fiducial cross section region, defined in Section 7, introduces an uncertainty of 1% in the measured fiducial cross section.

EW Wγ+2 jets signal and cross section measurements
A search for the SM EW Wγ+2 jets signal is performed based on the binned m jj distribution, as shown in Fig. 3, for both the muon and electron channels, using only the two rightmost bins corresponding to m jj > 700 GeV. The EW-and QCD-induced Wγ+2 jets production is modeled at LO, neglecting interference, with NLO QCD corrections to the cross section applied through their K factors.
We search for an enhancement in the rate of Wγ+2 jets production due to EW-induced production, treating non-Wγ and QCD-induced Wγ+2 jets production as background. The expected signal and background yields after the selections are shown in Table 2.
The measured yield of data events is well described by the theoretical predictions, which include the EW contribution. A CL s based method [69][70][71] is used to estimate the upper limit on the EW signal strength µ sig , which is defined as the ratio of the measured to the expected signal yield. Combining four m jj bins from the two decay channels gives an upper limit of 4.3 times the SM EW prediction at a 95% confidence level (CL), compared to an expected limit of 2.0 from the background-only hypothesis.  Figure 3: The m jj distribution in the muon (left) and electron (right) channels, in which the signal region lies above 700 GeV, indicated by the horizontal thick arrows. Backgrounds from jets misidentified as photons (Jets → γ) and jets misidentified as electrons (Jets → e) are estimated from data as described in the text. The diboson contribution includes WV(+γ) and Zγ(+jets) processes. The top quark contribution includes both the ttγ and single top quark processes. The signal contribution is shown on top of the backgrounds. The last bin includes the overflow events. The shaded area represents the total uncertainty in the simulation, including statistical and systematic effects. The measured signal strength can be translated into the fiducial cross section σ fid using the generated cross sections of the simulated samples σ gen and an acceptance acc for the total cross section from the fiducial region to the signal region: σ fid = σ gen µ sig acc . The fiducial cross section is reported in a region defined as follows: • p j1 T > 30 GeV, |η j1 | < 4.7; • p j2 T > 30 GeV, |η j2 | < 4.7; • m jj > 700 GeV, |∆η(j, j)| > 2.4; • p T > 20 GeV, |η | < 2.4; • p γ T > 20 GeV, |η γ | < 1.4442; This phase space corresponds to the acceptance of the CMS detector, with a minimal number of additional selections on m jj and |∆η(j, j)| to ensure that the VBS contribution is large. It does not include requirements on the Zeppenfeld variable and the |∆φ Wγ,jj | variable, which are applied at the reconstruction level. The acceptance corrections for these selections are 0.289 ± 0.001 for the EW cross section and 0.174 ± 0.002 for the QCD one, where we include both PDF and scale uncertainties.

Limits on anomalous quartic gauge couplings
Following Ref. [72], we parameterize the aQGCs in a formalism that maintains SU(2) L ⊗ U(1) Y gauge symmetry and leads to 14 possible dimension-eight operators that contribute to the signal. The L M,5 operator is found to be non-Hermitian and needs to be replaced by a summation of the original and its Hermitian conjugate (see Appendix A for the definition). The presence of aQGCs should lead to an enhancement of the EW Wγ+2 jets cross section, which should become more pronounced at the high-energy tails of some distributions. As shown in Fig. 4, the p W T distribution is sensitive to the aQGCs and therefore is used to set limits. We choose a p W T distribution binned over the range 50-250 GeV, with the overflow contribution included in the last bin. The shape of the distribution at high p W T is used to extract aQGC limits. These limits are not sensitive to small variations in the number of bins or range used for the p W T distribution. The events are selected with the baseline selections from Section 4, with the following additional requirements: |y Wγ − (y j1 + y j2 )/2| < 1.2, |∆η(j1, j2)| > 2.4, and p γ T > 200 GeV. A tight p γ T selection is applied to reach higher expected significance for the possible aQGC signal in the EW Wγ+2 jets process.
The stringent selections above lead to increased statistical uncertainties in the estimations of the backgrounds. The second largest uncertainty comes from the scale variations in the predicted aQGC signal. Other uncertainties include the signal PDF choice, integrated luminosity, trigger efficiency, and lepton and photon efficiencies.
The search is performed for each aQGC parameter separately, while setting all other parameters to their SM values. Each signal sample, representing a different aQGC prediction, is generated at LO using the reweight method in MADGRAPH [37]. For each aQGC case, we compute the aQGC/SM event yield ratios for all p W T bins from this sample and use these ratios to rescale the SM signal shape to the enhanced aQGC shape. Then we consider the following test statistic: where the likelihood function is constructed in two lepton channels and then combined for the calculation. The α term represents the aQGC point being tested, and θ the nuisance parameters. Theθ nuisance parameters correspond to the maximum of the likelihood at the point α, whilê α andθ correspond to the global maximum of the likelihood. This test statistic is assumed to follow a χ 2 distribution [73,74]. One can therefore extract the limits directly by using the delta log-likelihood function ∆NLL = t α /2 [75]. Table 4 lists 95% CL exclusion limits for all parameters.
Because of the nonrenormalizable nature of higher-dimensional operators, any nonzero aQGC parameter violates unitarity at high energies. An effective theory is therefore only valid at low energies, and we need to check that the energy scale we probe is less than a new physics scale and does not violate unitarity. Sometimes a form factor is introduced to unitarize the high-energy contribution within that energy range; however, the form factor complicates the limit-setting procedure and makes it difficult to compare results among experiments. We use VBFNLO without any form factors to calculate the unitarity bound corresponding to the maximum aQGC enhancements, which would conserve unitarity within the range of energies probed at the 8 TeV LHC [53,76]. We find that unitarity is violated in many cases. We compare our results, in a consistent way, with existing limits on aQGC parameters in Fig. 5, where the aQGC convention used in VBFNLO has been transformed to the one that is used in our analysis. Existing competitive limits include the results from WVγ production [14], same-sign WW production [17], exclusive γγ → WW production at the ATLAS and the CMS experiments [15, 19, 77], and Wγγ production at the ATLAS experiment [30]. The limits on the a W 0 /Λ 2 and a W C /Λ 2 couplings in these references are transformed to ours by using Eq.

Summary
A search for EW-induced Wγ+2 jets production and aQGCs has been presented based on events containing a W boson that decays to a lepton and a neutrino, a hard photon, and two jets with large pseudorapidity separation. The data analyzed correspond to an integrated luminosity of 19.7 fb −1 collected in proton-proton collisions at √ s = 8 TeV with the CMS detector at the LHC. An excess is observed above the expectation from QCD-induced Wγ+2 jets and other backgrounds, with an observed (expected) significance of 2.7 (1.5) standard deviations. The corresponding cross section within the VBS-like fiducial region is measured to be 10.8 ± 4.1 (stat) ± 3.4 (syst) ± 0.3 (lumi) fb, which is consistent with the SM prediction of EWinduced signal. In the same fiducial region, the total cross section for Wγ+2 jets is measured to be 23.2 ± 4.3 (stat) ± 1.7 (syst) ± 0.6 (lumi) fb, which is consistent with the SM EW+QCD prediction. Exclusion limits for aQGC parameters f M,0−7 /Λ 4 , f T,0−2 /Λ 4 , and f T,5−7 /Λ 4 are set at 95% CL. Competitive limits are obtained for several parameters and first limits are set on the f M,4 /Λ 4 and f T,5−7 /Λ 4 parameters.

A Anomalous quartic gauge coupling parameterization
Gauge boson self-interactions are fixed by the gauge symmetries of the SM. To investigate possible deviations from the SM, we parameterize the aQGCs in a formalism that maintains the SU(2) L ⊗ U(1) Y gauge symmetry. As a natural extension to the SM, the lowest order pure anomalous quartic couplings arise from dimension-eight operators. This analysis adopts the following effective Lagrangian containing such aQGCs [72]: where Φ represents the Higgs doublet, B µν and W i µν are the associated field strength tensors of the U(1) Y and SU(2) L gauge symmetries, and W µν ≡ ∑ j W j µν σ j /2. The f T /Λ 4 associated operators characterize the effect of new physics on the scattering of transversely polarized vector bosons, and f M /Λ 4 includes mixed transverse and longitudinal scatterings; however, pure longitudinal scattering effects do not occur in the Wγ final state due to the presence of the photon. The listed operators include all contributions to the WWγγ and WWZγ vertices. In this paper, we set c = 1 to describe energy, momentum, and mass in units of GeV.
Any nonzero value in aQGCs will lead to tree-level unitarity violation at sufficiently high energy and could be unitarized with a suitable form factor; however the unitarization depends on the detailed structure of new physics, which is not known a priori. Following Ref.