Model independent study for the anomalous quartic $WW\gamma\gamma$ couplings at Future Electron-Proton Colliders

The Large Hadron Electron Collider and the Future Circular Collider-hadron electron with high center-of-mass energy and luminosity allow to better understand the Standard Model and to examine new physics beyond the Standard Model in the electroweak sector. Multi-boson processes permit for a measurement of the gauge boson self-interactions of the Standard Model that can be used to determine the anomalous gauge boson couplings. For this purpose, we present a study of the process $ep \rightarrow \nu_{e} \gamma \gamma j$ at the Large Hadron Electron Collider with center-of-mass energies of 1.30, 1.98 TeV and at the Future Circular Collider-hadron electron with center-of-mass energies of 7.07, 10 TeV to interpret the anomalous quartic $WW\gamma\gamma$ gauge couplings using a model independent way in the framework of effective field theory. We obtain the sensitivity limits at $95\%$ Confidence Level on 13 different anomalous couplings arising from dimension-8 operators.


I. INTRODUCTION
New physics beyond the Standard Model (SM) refers to the theoretical developments needed to clarify the deficiencies of the SM, such as neutrino oscillations, matter-antimatter asymmetry, the origin of mass, the strong CP problem and the nature of dark energy and dark matter. For this reason, new physics models beyond the SM are investigated in various processes at colliders. One of the ways to research new physics models is to study the where F µν = ∂ µ A ν − ∂ ν A µ represents the electromagnetic field tensor, a 0 and a c are the anomalous coupling parameters.
Dimension-8 operators are described by using a linear representation of the spontaneously broken gauge symmetry of the SM. In this case, the anomalous quartic gauge boson couplings are built by extending the SM Lagrangian with terms including dimension-8 operators as this is the lowest dimension that defines the quartic gauge boson couplings without exhibiting triple gauge-boson couplings [2]. Therefore, the linear effective Lagrangians can be given as follows There are 17 different operators that define the anomalous quartic gauge boson couplings.
The indices S, T and M of the couplings represent three class operators.
The first class of these operators, two independent operators including covariant derivative of the Higgs doublet are generated by The O S0 and O S1 operators involve quartic W W W W , W W ZZ and ZZZZ couplings.
Seven operators in second class derive the anomalous quartic gauge boson couplings that are obtained by thinking two electroweak field strength tensors and two covariant derivatives of the Higgs doublet Here, the field strength tensors of W µν and B µν gauge fields are expressed as where τ i (i = 1, 2, 3) shows the SU(2) generators, g = e/sinθ W , g ′ = g/cosθ W , e and θ W are the unit of electric charge and the Weinberg angle, respectively.
The final class have 8 operators that consist of four field strength tensors. These operators generate the following quartic anomalous couplings: All quartic gauge boson couplings altered with dimension-8 operators are presented in Table I.
The present experimental sensitivities on the anomalous f M 0 Λ 4 , f M 1 Λ 4 , f M 2 Λ 4 and f M 3 Λ 4 couplings arising from dimension-8 operators through the process pp → pγ * γ * p → pW W p [36] at center-of-mass energy of √ s = 8 TeV using data corresponding to an integrated luminosity of 19.7 fb −1 at the LHC are reported by CMS Collaboration. These are at 95% Confidence Level.
However, Ref. [37] supplies the most restrictive limits on the anomalous couplings which are related to the anomalous W W γγ quartic couplings derived with operators given by Eqs. 11-14 and 16-21. The results obtained for these couplings at 95% Confidence Level through the process pp → W γjj at √ s = 8 TeV with an integrated luminosity of 19.7 fb −1 are listed as Recently, a lot of work that are experimental or theoretical were done using dimension-6 operators for the anomalous quartic W W γγ couplings. Dimension-6 operators can be determined in terms of dimension-8 operators with the simple relations. The relations between f M i and a 0,c couplings are given as follows [38]  The context of this study is planned as follows: In Section II, we perform numerical analysis of the process ep → ν e γγj at the LHeC and FCC-he to obtain limits on the anomalous quartic couplings. Finally, we discuss conclusions in Section III.

II. NUMERICAL ANALYSIS
In our calculations, we analyze signals and backgrounds of the process ep → ν e γγj by using MadGraph5 − aMC@NLO [41] in which the anomalous quartic couplings are implemented through FeynRules package [42] through dimension-8 effective Lagrangians related to the anomalous quartic W W γγ couplings. The CTEQ6L1 set is used to define the proton structure functions [43]. In order to obtain limits on the 13 different anomalous couplings arising from dimension-8 operators, we investigate the process ep → ν e γγj. Symbolic diagram of this process is presented in Fig. 1.
A set of cuts used for the analysis of signal and background events in the process ep → ν e γγj including the anomalous quartic W W γγ interactions is impose as follows where p T is the transverse momentum of the final state particles, η is the pseudorapidity and ∆R is the separation of the final state particles.
To have a comprehensive investigation on the cross section behavior, we present the analytical form of cross sections including the anomalous couplings, where σ SM shows the SM cross section, σ int and σ N P are the interference term between SM and the new physics contribution, and the pure new physics contribution, respectively. In this analysis, we suppose that only one of the anomalous parameters deviate from the SM at any given time. For 13 different anomalous couplings, we estimate the cross sections of the SM and signals after applied kinematic cuts used for the process ep → ν e γγj at the LHeC and FCC-he are given in Table II-III. As seen from Tables II-III In order to investigate the limits on the anomalous quartic W W γγ couplings, we consider χ 2 test with one-parameter sensitivity analysis. For this purpose, the χ 2 test is described as follows where δ = 1 √ In Tables IV-VII, we present the sensitivities on the anomalous quartic W W γγ couplings for different center-of-mass energies and integrated luminosities. From Tables it is clear that increasing the integrated luminosity as well as center-of-mass energy provides more restricted limits on all the anomalous quartic couplings. Comparing the results in Table IV with the corresponding data in Table VII, there is an improvement in our limits up to several orders of magnitude with increasing integrated luminosity and center-of-mass energy. The best limits on these couplings are given for FCC-he with 10 TeV in Table VII. As shown from Tables IV-V, since the LHeC has less center-of-mass energy and less luminosity than the LHC, sensitivity limits on 13 different anomalous quartic W W γγ parameters obtained from our work are worse than the experimental limits. In Table VI, we present the sensitivity limits the process ep → ν e γγj at 7.07 TeV FCC-he. As can be seen from this Table, our best sensitivities on these couplings are up to one order of magnitude better than the sensitivities derived in Ref. [36]. The most important results on couplings given in Table VII are  ×10 −2 TeV −4 which is up to a factor of 10 2 better than the experimental limit. It can be seen from these results that the anomalous quartic W W γγ couplings can be examined with very good sensitivity at FCC-he. One consideration when examining the anomalous quartic boson couplings is to isolate only one of these quartic couplings. For example, an important advantage of the process pp → pγ * γ * p → pW W p [36] through the subprocess γ * γ * → W W at the LHC is that it isolates W W γγ coupling from the other quartic couplings. In addition, we can easily see that the process ep → ν e γγj we examine isolates the W W γγ coupling.
Photons in the final state of the process ep → ν e γγj at the LHeC and FCC-he have the advantage of being identifiable with high purity and efficiency. The diphoton channels are especially sensitive for new physics beyond the SM in terms of modest backgrounds, excellent mass resolution and the clean experimental signature.
As far as we can see from the literature, this study is the first report on the anomalous quartic couplings determined by effective Lagrangians at ep colliders. Moreover, we consider that this paper will motivate further works to investigate the another anomalous quartic couplings at ep colliders.
Consequently, the process ep → ν e γγj is very beneficial to sensitivity studying on the anomalous quartic W W γγ couplings and illustrates the complementarity between LHC and future ep colliders for probing extensions of the SM.