Constraints on anomalous quartic gauge couplings by $\gamma\gamma \to W^+W^-$ scattering

The vector boson scattering (VBS) processes in Large Hadron Collider (LHC) experiments offer a unique opportunity to probe the anomalous quartic gauge couplings (aQGCs). We study the dimension-8 operators contributing to the anomalous $\gamma\gamma WW$ coupling and the corresponding unitarity bounds via the exclusive $\gamma\gamma \to W^+W^-$ production in $pp$ collisions at LHC for a center of mass energy of $\sqrt{s}=13$ TeV. By analysing the kinematical features of the signal, we propose an event selection strategy to highlight the aQGC contributions. Based on the event selection strategy, the statistical significance of the signals are analyzed in detail, and the constraints on the coefficients of the anomalous quartic gauge operators are obtained.


I. INTRODUCTION
After the discovery of the Higgs boson [1], lots of effort has been made in the search of new physics beyond the Standard Model (BSM) at the Large Hadron Collider (LHC).
Among the abundant processes accessible in the LHC, the vector boson scattering (VBS) processes draw a lot of attention because they are sensitive to BSM effects [2,3].In the Standard Model (SM), because of the cancellation among the VBS Feynman diagrams, the cross-sections of VBS do not grow with energy.Such cancellation will be broken when BSM effects are involved [4], therefor the VBS processes at high energies provide great opportunities to search for new physics [5].
In the search of BSM, the Standard Model Effective Field Theory (SMEFT) [6] has emerged as a powerful model-independent approach.In this approach, after integrating out the BSM degree of freedom, the effects of BSM become higher dimensional operators suppressed by energy scale Λ, and the effective Lagrangian is where O 6i and O 8i are dimension-6 and dimension-8 operators.Odd dimension operators are neglected in this letter.The high dimensional operators can contribute to the anomalous trilinear gauge boson couplings (aTGCs) and anomalous quartic gauge boson couplings (aQGCs) which are suitable to be studied via the VBS processes.
In this work, we focus on the aQGCs because the aTGC signals are sensitive else where, e.g. in diboson production processes and vector boson fusion processes [4].Moreover, we consider only the dimension-8 operators contributing to the aQGCs because the dimension-8 operators can introduce decorrelation between aTGCs and aQGCs, i.e. the dimension-6 operators can not contribute to QGCs while not affecting TGCs [2].There are also cases where the dimension-6 operators are absent and the dimension-8 operators are presented.
For example, the Born-Infeld (BI) model [7] with the Lagrangian where i corresponds to one of the 12 generators of the SM SU (3 The evidence of the same sign W W jj channel was found at LHC in 2014 [8], which is the first evidence of the processes involving a QGC.Shortly afterwards, the dimension-8 operators contributing to aQGCs were studied in the VBS processes extensively, for example, in the same sign W W jj channel [9,10], γW jj channel [11], ZZjj channel [12], W Zjj channel [13] and also W + W − jj channel at √ s = 7 and 8 TeV [14].The evidence of exclusive or quasi-exclusive γγ → W + W − process has been observed recently [14].As a supplementary, we study aQGCs in this process at √ s = 13 TeV.The W + W − jj channel receive contributions from γγW W , γZW W , ZZW W and W W W W vertices [15], and one cannot discriminate those vertices by W + W − jj channel alone.Thus, we only consider the vertices contributing to the exclusive γγ → W + W − process in this work. It is well known that the rapid growth of the scattering amplitudes with energy leads to unitarity violation [16].In this work, we calculate the partial wave unitarity bounds of the aQGCs at various proton c.m. energies.We also study the kinematical features of the aQGC signal and the corresponding backgrounds.The signal is found to be sensitive to the M ol cut which is used in the same sign W W jj channel [10].Except for that, the signal has a unique cos(θ ll ) behaviour which provides an efficient cut.Based on the Monte Carlo (MC) simulation, we estimate the constraints and observability of the anomalous γγW W couplings with the current luminosity of LHC.

II. THE DIMENSION-8 ANOMALOUS QUARTIC GAUGE OPERATORS AND γγW W VERTEX
We follow Refs.[15,17] to list all dimension-8 operators contributing to aQGCs, they are with where W ≡ ⃗ σ • ⃗ W /2 with σ the Pauli matrix and ⃗ W = {W 1 , W 2 , W 3 }.Note that, we keep the index of the operators identical to Ref. [17], and therefor the redundant (O M 6 ) or vanishing operators (O T 3,4 ) are not included.The operators contributing to the γγW W interaction can form 5 different vertices , where where The coefficients are V i=0,1,AW are dimension-6 vertices, and V i=2,3,4,AW are dimension-8 vertices which can be introduced by BI model.
One dimension-8 operator contribute to only one vertex, therefor the constraints on α i can be derived by the constraints on dimension-8 operators.The range of α i depends on maximum of each f term.Base on the experimental limits of f M and f T [11], we get the tightest constraints in Table .I.

III. UNITARITY BOUND
The aQGC contributions grow significantly at high energies.On one hand, this feature indicates that at higher energies, the VBS process is ideal to search for aQGCs, on the other hand.The cross-section of VBS with aQGCs will violate unitarity at certain energy, which indicates that as the energy scale grows the new physics particles degrees of freedom will emerge, and SMEFT is not valid.To avoid the violation of unitarity, the coefficients of the operators will be constrained, which is the unitarity bound.
For the γγ → W + W − process, 36 different helicity amplitudes can be obtained by partial wave expansion.The number of amplitudes can be reduced by using . For simplicity we denote ŝ = (p γ1 + p γ2 ) 2 , note that ŝ is not the c.m. energy of protons.It is only necessary to keep the terms at the leading order O(ŝ 2 ), which grow fastest with ŝ.The leading terms are list in Table .II.The tightest bounds are √ ŝ is related to √ s by photon distribution functions [21].To extract √ ŝ, we analysis the γγ → l + l − ν ν process with photons from protons based on Monte Carlo (MC) simulation using the MadGraph5_aMC@NLO toolkit [22][23][24][25].We run MC simulations for √ s = 13 (14) TeV at LHC [27], 27 TeV at HE-LHC [28], 50 TeV at FCC-hh [29] and 100 TeV at SppC [30].
We find that the energies of photons grow very slowly with √ s.
TABLE II: The helicity ampltidues at the order of O(s 2 ).Experimentally, VBS is characterized by the presence of a pair of vector bosons and two forward jets.The SM process pp → jjℓ + ℓ − ν ν is treated as the irreducible background, the typical Feynman diagrams at tree level are shown in Fig. 1, which can be categorized into The standard VBF/VBS cuts are used to discriminate the VBS from non-VBS process, but we need to also to discriminate aQGC VBS from SM VBS.It has been studied in the same sign W -boson process that M o1 is sensitive to anomalous quartic gauge operators except for O S i [10].This variable is defined as which provides a very efficient discrimination between signal and backgrounds as shown in Fig. 4. (a).We select events with M o1 > 500 GeV.
The W ± bosons in VBS process should be dominantly back-to-back.For energetic W ± bosons, the flight directions of leptons are close to ± bosons.Therefor the leptons should also be dominantly which leads to a small cos(θ ℓℓ ) where θ ℓℓ is the angle between the leptons.The differential cross-sections as functions of cos(θ ℓℓ ) are shown in  .IV.The basic cuts are from the MadGraph5_aMC@NLO toolkit.

B. Significance of the signal
In the significance analysis, non-VBS aQGC diagrams (Fig. 2. (b)) and all possible interference effects are included.In this case, the total cross-section with one vertex at a time denoted as σ i , is approximately a bilinear function of α i .After scanning over the parameter σ(fb) α 0 = 0.12 α 1 = 0.2 α 2 = 2.9 α 3 = 2.3 α 4 = 5.9 SM  space of α i in Table I, we can obtain the σ i by fitting.
This indicates that the cross-section is reduced by the contributions of the interference and s-channel diagrams (Fig. 2. (b)).
We calculate statistical significance (SS) using SS ≡ N S / √ N S + N B where N S and N B are the numbers of signal and background events, respectively.With SS, we calculate the expected constraints on the vertices and display the results of the constraints in Fig. 6 and  V. SUMMARY Among the processes measured at LHC, the VBS processes provide excellent opportunities to study the structure of QGCs and possible effects of BSM.In this letter, we investigate  the dimension-8 operators contributing to anomalous γγW W coupling via the VBS process γγ → W + W − at the 13 TeV.The corresponding γγW W vertices are investigated, the unitarity bounds of those vertices are analyzed.Our analysis shows that the range of coefficients we picked satisfy the unitarity at 13 TeV.To study the observability of the operators, we analyse the signals of aQGCs and backgrounds based on the CMS detector simulation for jjl + l − ν ν final state.Compared with the SM backgrounds, the γγW W aQGC has unique kinematical features.We found that the M o1 and cos(θ ℓℓ ) are sensitive variables which cut the SM backgrounds efficiently.For the significance analysis, we take into account the schannel aQGC effects and the interference between the signal and SM backgrounds.The contribution of s-channel diagrams induced by the aQGC and the interference effects will decrease the cross section.Such correction is found to be sizable, and should be considered to ensure an accurate measurement.Based on the SS calculated, the expected constraints on the γγW W vertices are obtained in Fig. 6 and Table .V. The γγ → W + W − process is found to be sensitive to the V 0,AW and V 1,AW vertices corresponding to the O M i operators, and the constraints can be tighten by more than one order of magnitude at 13 TeV LHC with current luminosity.

FIG. 1 :FIG. 2 :FIG. 3 :
FIG. 1: The backgrounds are the processes contributing to jjl + l − ν ν final states in the SM.The typical EW-VBS diagrams are shown in (a), EW-non-VBS diagrams are shown in (b), and the typical QCD diagrams are shown in (c).

FIG. 4 :
FIG. 4: The differential cross-sections of the SM background and signals as a function of M o1 (left panel) and cos(θ ℓℓ ) (right panel) after the standard VBS/VBF cuts.

Fig. 4 .
Fig. 4. (b), and we choose the cut as cos(θ ℓℓ ) < −0.75.The efficiency of these cuts are listed in Table.IV.The basic cuts are from the MadGraph5_aMC@NLO toolkit.

ACKNOWLEDGMENT
This work was supported in part by the National Natural Science Foundation of China under Grants No.11905093, No.11847019 and No.11947066, the Natural Science Foundation of the

TABLE I :
The constraints on vertices and the corresponding limits on the dimension-8 operators.

TABLE III :
The unitarity bounds on the vertices at different c.m. energies.
Since√ ŝ is a distribution, one can set the unitarity bounds in a statistical way.We set the bounds by requiring 95% events are at the valid region in the sense of unitarity.The corresponding √ ŝ and the bounds on the coefficients are listed in Table.III.It can be found that the ranges of the coefficients in Table.I indeed satisfy the unitarity bounds.

TABLE IV :
Signal and background cross sections (in fb) with consecutive cuts for theℓ + ℓ − jj + /E final states at √ s= 13 TeV.

TABLE V :
The constraints on the vertices at √ s = 13 TeV and at luminosity 137 fb −1 .