Study on the anomalous quartic $W^+W^-\gamma\gamma$ couplings of electroweak bosons in $e^-p$ collisions at the LHeC and the FCC-he

In this paper, a study is carried out on the $e^-p \to e^-\gamma^* p \to p W^+\gamma \nu_e$ production to probe quartic $W^+W^-\gamma\gamma$ couplings using 10, 100 ${\rm fb^{-1}}$ of $e^-p$ collisions data at $\sqrt{s}$= 1.30, 1.98 GeV at the Large Hadron electron Collider (LHeC) and 100, 1000 ${\rm fb^{-1}}$ with $\sqrt{s}$= 3.46, 5.29 GeV at the Future Circular Collider-hadron electron (FCC-he). Production cross-sections are determined for both at leptonic and hadronic decay channel of the $W$-boson. With the data from future $e^-p$ colliders, it is possible to obtain sensitivity measures at $95\%$ C.L. on the anomalous $f_ {M,i}/\Lambda^4$ and $ f_ {T,i}/\Lambda^4$ couplings which are competitive with the limits obtained by the LHC, as well as with others limits reported in the literature. The production mode $e^-p \to e^-\gamma^* p \to p W^+\gamma \nu_e $ in $e^-p$ collisions offers a window for study the quartic $W^+W^-\gamma\gamma$ electroweak bosons couplings at the LHeC and the FCC-he, which provides a much cleaner collision environment than the LHC.


I. INTRODUCTION
A property of the weak interaction is that its gauge bosons W ± and Z can couple to each other in certain combinations and also to γ. The gauge bosons W ± , Z, and γ through mixing with each other represent some of the Standard Model (SM) [1][2][3] particles most strongly coupled to Electroweak Symmetry Breaking (EWSB). Due to the non-Abelian nature of the SM electroweak theory, gauge bosons interact with each other and the SM predicts the existence of the anomalous Triple Gauge Couplings (aTGC) and the anomalous Quartic Gauge Couplings (aQGC). In particular, the aQGC W W γγ is the main topic in this article.
The LHeC, is one of the proposed colliders in the new energy frontier at the LHC, is to inject an electron beam which will collide head-on with the available proton beam. For a first stage, this ep collision option is for a center-of-mass energy √ s = 1.30 TeV, where e − energy E e = 60 GeV, and the proton beam energy E p = 7 TeV. The second stage can be realised with E e = 140 GeV and E p = 7 TeV. The design for the new collider as well as the details can be found in Refs. [43][44][45][46][47]. Further upgrades to the HE-LHC would provide proton beam energies up to 50 TeV. This is another available option, the FCC-he. This upgrade with E p = 50 TeV, assumes potential reuse of the LHeC with E e = 60 − 140 GeV.
It is s possible that at a later stage the upgrade, with E p = 50 TeV, assumes a maximum of E e = 250 − 500 GeV.
In this paper, we present our results in a model-independent way for the total crosssection of the process e − p → e − γ * p → pW + γν e at the e − γ * mode, as well as limits on W W γγ aQGC assuming L = 10, 100 fb −1 of electron-proton collision data at 1.30 and 1.98 TeV at the LHeC and L = 100, 1000 fb −1 of electron-proton collision data at 3.46 and 5.29 TeV at the FCC-he. For our study, we use an effective Lagrangian approach which provides a generic platform for introducing the effect of new physics Beyond the SM (BSM) by adding additional terms in the Lagrangian of the SM.
The paper is organized as follows: In Section II, we give the general expressions for the effective Lagrangian. In Section III, we evaluate the total cross-section of the reaction e − p → e − γ * p → pW + γν e and derive the 95% C.L. allowed sensitivity measures on the anomalous f M,i /Λ 4 and f T,i /Λ 4 couplings at the LHeC and the FCC-he. In Section IV, we summarize our conclusions.
II. DIMENSION-8 OPERATORS SET RELEVANT FOR e − p → e − γ * p → pW + γν e A suitable and relatively modern approach to observe the effects of new BSM physics in a model-independent formalism is to use an effective Lagrangian description of the SM.
Starting from our present theoretical, phenomenological and experimental understanding, treating the SM in a effective Lagrangian approach is an well-motivated starting point since we have no present evidence of BSM physics. In practice, this means defining a scale, Λ, of new physics higher than the energy scale being probed in the experiment and using the fields of the SM to write higher dimension operators in addition to dimension-4 operators of the SM. Following the context of Refs. [54][55][56], the effective Lagrangian as well as the classes of genuine aQGC operators [57] of dimension-8 for W W γγ vertex are the following [58]: T-type operators Here, 18 operators of dimension-8 which are classified in independent scalar operators, independent mixed operators and independent transverse operators are given in Table I. Table   I contains the Wilson coefficient Λ 4 which is relationship with a W 0,c /Λ 2 couplings as follows [5,57,59]:

III. CROSS-SECTION MEASUREMENTS AT THE LHEC AND THE FCC-HE
To investigate the effect of dimension-8 operators we focus on the process W -boson production in association with a neutrino plus a photon at the LHeC and the FCC-he.
Dimension-8 operators given in Table I directly enter into the process e − p → e − γ * p → pW + γν e by modifying the W W γγ vertex. The part of σ(e − p → e − γ * p → pW + γν e ) that grows with the energy is controlled by dimension-8 operators alone. To quantitatively evaluate the effects we can expect at the electron-proton level, we turn to numerics. For which we consider the following cinematic cuts transverse momentum for the photon and charged leptons p γ T and p l T , the rapidity for the photons and charged leptons η γ and η l , distance 6 between leptons and distance between γ and lepton ∆R ll and ∆R γl with the purpose of reducing the background and improve the sensitivity of the signal. To reconstruct the signal, we require at least one electron or muon with p l T > 10 GeV, least one photon with p γ T > 10 GeV, rapidity for the photons and leptons with η γ,l < 2.5 and distance between leptons and distance between γ and leptons with ∆R γl,ll = 0.4. These selection cuts are summarized in Table II. A. Photoproduction at the LHeC and the FCC-he Photon interactions have been extensively studied at HERA [65], LEP [66], Tevatron [67] and LHC [68], in processes involving exchange of quasi-real photons collinear to the incoming lepton. In a similar manner, a significant fraction of a lepton-hadron collisions at the LHeC and the FCC-he will involve quasi-real photon interactions. The LHeC and the FCC-he can to some extend be considered as a high energy eγ * , γ * p and γ * γ * collisions. On this topic, the futures lepton-hadron colliders offer excellent new opportunities for the study of high energy particle collisions, thus significantly extending the physics capabilities of an lepton-hadron collider. With this options, a large number of new and exciting measurements become accessible with a eγ * , γ * p and γ * γ * collisions. Because the photons couple directly to all fundamental fields carrying the electromagnetic current leptons, quarks, W ′ s, etc.. High energy eγ * , γ * p and γ * γ * collisions will provide a comprehensive laboratory for exploring virtually every aspect of the SM and BSM physics. A review of the studies made on eγ * , γ * p and γ * γ * collisions physics on future colliders it is made in Refs. [54][55][56][69][70][71][72][73][74][75][76][77][78][79][80][81][82].
It is appropriate to mention that the studies of photon interactions at the LHC are possible due to experimental signatures of events involving photon exchanges such as the presence of very forward scattered protons and of large rapidity gaps in forward directions. However, to tag efficiently photon induced processes and to keep backgrounds under control, some processes require very forward proton detectors [83]. The photon induced processes have been measured in pp collisions at Tevatron-Fermilab using the large rapidity gap signature.
The exclusive two-photon production of lepton pairs and the diffractive photoproduction of J/ψ mesons were studied in Refs. [84][85][86], respectively. In both cases clear signals were obtained with low backgrounds.
As we mentioned above, scenarios like the LHeC and the FCC-he [69][70][71][72][73][74] offer an unique opportunity to build ep collider, which can also be operated in γp collisions. These conversions are made by converting the incoming electrons or protons into an intense beam of high-energy photons. In addition, the ep colliders also provide the opportunity to examine γ * γ * , γ * e and γ * p modes with quasi-real photons through the Equivalent Photon Approximation (EPA) [83,87,88], using the Weizsacker-Williams Approximation (WWA).
On the other hand, the phenomenological investigations at lepton-hadron colliders generally contain usual deep inelastic scattering reactions where the colliding hadron dissociates into partons. These reactions have been extensively studied in the literature, while the processes elastic and semi-elastic, such as γ * γ * and γ * p have been much less studied. These processes have simpler final states with respect to lepton-hadron processes. In this case, these processes compensate for the advantages of lepton-hadron processes such as having high center-of-mass energy and high luminosity. In addition, eγ * have effective luminosity and much higher energy compared to the process γ * γ * collisinos. This may be significant because of the high energy dependencies of the cross-section containing the new physics parameters. For all the aforementioned, it is expected that the γ * p collisions to have a high sensitivity to the W W γγ aQGC.
Regarding eγ * collisions these can be discerned from usual deep inelastic scattering collisions by means of two experimental signatures. First signature is the forward large rapidity gap. Quasi-real photons have a low virtuality and scattered with small angles from the beam pipe. As the transverse momentum carried by a quasi-real photon is small, photon-emitting proton should also be scattered with small angles and exit the central detector without being detected. This causes a decrease in the energy deposit in the corresponding forward region.
As a result of this, one of the forward regions of the central detector has a significant lack of energy. This defines the forward large-rapidity gap and usual ep deep inelastic collisions can be rejected by applying a selection cut on this quantity. Second experimental signature is provided by the forward detectors. Forward detectors are capable to detect particles with a large pseudorapidity. When a photon emitting from proton is scattered with a large pseudorapidity, it exceeds the pseudorapidity coverage of the central detectors. The detection of this proton by the forward detectors provides a distinctive signal for eγ * collisions. In this regard, the LHeC Collaboration has a program of forward physics with extra detectors located in a region between a few tens up to several hundreds of metres from the interaction point [89]. 8 B. The total cross-section for one exchanged quasi-real photon γ * photons emitted from proton beams collide with the incoming electron, and eγ * collisions are generated. The process e − γ * → W + γν e participates as a subprocess in the process e − p → e − γ * p → pW + γν e . In addition, the diagram of the process e − p → e − γ * p → pW + γν e is given in Fig. 1. The Feynman diagrams for the subprocess e − γ * → W + γν e are shown in Fig. 2. Therefore, we find the total cross section of the main process e − p → e − γ * p → pW + γν e by integrating the cross section for the subprocess e − γ * → W + γν e . The total cross section of this process can be written as Here, the spectrum of EPA photons f γ * (x) is defined as follows [87,90]: GeV 2 is the maximum virtuality of the photon and Q 2 min is: In addition, the explicit form of function ϕ contained in Eq. (7) is: where explicitly a, b, c and y are: We calculate the total cross-section of the reaction e − p → e − γ * p → pW + γν e through the expression given by Eq. (6) and in the presence of dimension-8 operators using the MadGraph5 aMC@NLO [91] package. This requires events that pass the selection cuts to have p T transverse moment, η rapidity and ∆R distance between particles as specified in Table II Table   IV as well as by Fig. 10.
To close this subsection, it is worth mentioning that our results shows that a nonzero aQGC enhances the production cross-section at large energies of the ep system with respect to the SM prediction, as can be seen in Figs. 3-10. Furthermore, for the aQGC search, a restricted region of p T , η and ∆R is used, that is say, the fiducial region is defined in Table II. This is chosen to reduce the contribution of the background and to improve the sensitivity of the signal.
The presence of new physics characterized by the parameters f M,i /Λ 4 and f T,i /Λ 4 may be quantified by a simple χ 2 method that varies the parameters (f M,i /Λ 4 , f T,i /Λ 4 ) and is based on: In Eq. (14), σ SM ( √ s) is the cross-section of the SM and σ BSM ( √ s, f M,i /Λ 4 , f T,i /Λ 4 ) is the BSM cross-section, while δ st = 1 √ N SM is the statistical error and N is the number of events: To get an idea of the LHeC and FCC-he constraining power, in Tables V-VI, Table 4 of Ref. [92] for the CMS Collaboration at the LHC.
Another paper presents a study of vector boson scattered in W W , W Z, and ZZ channels using pp collisions at √ s = 13 TeV and integrated luminosity of 35.9 ± 0.9 fb −1 collected with the CMS detector at the LHC [93].

V. SUMMARY AND CONCLUSIONS
In this work, in the effective Lagrange approach, we study the e − p → e − γ * p → pW + γν e channel at the LHeC and the FCC-he as a way to perform sensitivity measures on the total cross-section and on the anomalous f M,i /Λ 4 and f T,i /Λ 4 couplings. We focus on new physics as equal to 1 × 10 −8 and 5 × 10 −9 GeV −4 , respectively. The total cross-sections for each coupling are calculated while fixing the other couplings to zero with the selections cuts defined in Table II. effects that grow with energy, parameterized by dimension-8 effective operators within the effective Lagrange framework. In particular, we identify 13 operators that induce a growth with energy.
To get a quantitative idea of the sensitivity of our results, we give a summary of the projected for the total cross-section of the process e − p → e − γ * p → pW + γν e in Tables III-IV and Figs. 3-10, as well as 95% C.L. sensitivity measure on 13 operators listed in Tables  TeV at the FCC-he for each coupling f M,i /Λ 4 or f T,i /Λ 4 fixing one at a time. In our computation, we apply the selections cuts defined in Table II which is efficient in reducing the backgrounds while preserving most of the signal. An interesting feature of our results is the impact of the dimension-8 operators.
Since the aQGC W W γγ described through effective Lagrangian have dimension-8, they have very strong energy dependence. Therefore, the anomalous cross section containing the W W γγ vertex has a higher energy than the SM cross section. In addition, the future ep collider will possibly generate a final state with two or more massive gauge bosons. Hence, it will have a great potential to investigate aQGC. High-energy accelerated e − and p beams at these colliders radiate quasi-real photons, and thus eγ * , γ * p and γ * γ * collisions are produced from the e − p process itself. Therefore, ep colliders will provide an important opportunity to probe eγ * , γ * p and γ * γ * collisions at high energies. These collisions for the new physics TeV [93]. Also, via the observation of electroweak production of same-sign W -boson pairs in the two jet and two same-sign lepton final state in proton-proton collisions at 13 TeV [98].
In Ref. [96,99], is discuss the feature of the signals of aQGC and sensitivities to the aQGC in the pp → W γjj channel at the LHC √ s = 13 TeV. As well as with other limits reports with a projection at the LHeC and the FCC-he through the ep → ν e γγjj reaction [54], the process ep → e − γ * γ * p → e − W + W − p [55] and of the ep → e − γ * p → eW γq ′ X → eν l lq ′ X [56] signal. In addition, of other limits reports in the literature [39, 40, 54-56, 94, 95].
We conclude by mentioning that our projections at the LHeC and the FCC-he are interpreted in the approach of dimension-8 effective field theory operators through the e − p → e − γ * p → pW + γν e channel. Confidence intervals are derived for all 13 parameters of aQGC this analysis is sensitive to. In this sense, our results indicate that the e − p → e − γ * p → pW + γν e production is convincing for searching for the dimension-8 opera-