Bounds on the non-standard $W^+W^-\gamma$ couplings at the LHeC and the FCC-he

We examine the potential of the $e^-p \to e^-\gamma^*p \to e^-W^-q'X$ ($\gamma^*$ is Weizsacker-Willams photon) reaction to probe the non-standard $W^+W^-\gamma$ couplings at the Large Hadron electron Collider (LHeC) and the Future Circular Collider-hadron electron (FCC-he). We find $95\%$ confidence level bounds on the anomalous coupling $\Delta\kappa_\gamma$ and $\lambda_\gamma$ parameters in the view of effective Lagrangian approach with various values of the integrated luminosity. We assume center-of-mass energies of the electron-proton system $\sqrt{s}= 1.30, 1.98, 7.07, 10\hspace{0.8mm}{\rm TeV}$ and luminosities ${\cal L} = 10-1000 \hspace{0.8mm}{\rm fb^{-1}}$. The best limits obtained from the process $e^-p \to e^-\gamma^*p \to e^-W^-q'X$ on the anomalous $\Delta\kappa_\gamma$ and $\lambda_\gamma$ couplings are $\Delta\kappa_\gamma = |0.00069|$ and $\lambda_\gamma = [-0.0099, 0.0054]$. These bounds show that the process under consideration is a good prospect for the searching of the non-standard $\Delta\kappa_\gamma$ and $\lambda_\gamma$ couplings at the LHeC and the FCC-he. In addition, our results provide complementary information on other results for $\Delta\kappa_\gamma$ and $\lambda_\gamma$ couplings.


I. INTRODUCTION
The Standard Model (SM) of Elementary Particle Physics [1][2][3] is extremely predictive and has been tested in numerous aspects with impressive precision. The SM is a very powerful tool to predict the characteristics, the behavior and the interactions of the elementary particles. Therefore, it is very important to measure particle properties and interactions in the most accurate way possible to better understand the SM, refine it and test its global consistency. In this regard, the anomalous Triple-Gauge-Boson Couplings (aTGC) and anomalous Quartic Gauge Boson Couplings (aQGC) of the W ± boson: W + W − γ, W + W − Z, W Zγ, W γγ, W + W − γγ, W + W − Zγ, W + W − ZZ and W + W − W + W − are a key element in the search for the new physics beyond the SM (BSM), since any discrepancy of the measured value with respect to the predicted could reveal new phenomena other than the SM.
These studies underline the importance to measure the anomalous ∆κ γ and λ γ gauge couplings in several different channels, contexts and colliders. For instance, in hadronhadron, lepton-lepton and hadron-lepton colliders, such as the Large Hadron Collider (LHC), the Compact Linear Collider (CLIC) and the Future Circular Collider (FCC) at CERN. Furthermore, these colliders, operating in the e − γ, γγ, e − γ * , γ * p and γ * γ * modes will be very useful. Some characteristics that distinguish to the future circular colliders from the linear ones are the following: 1) Circular colliders can reach higher-luminosity, especially at low energies. 2) It could start operating as a factory of pairs of Z and W ± bosons.
With these arguments, we determined the production cross-section, as well as modelindependent bounds for the non-standard W + W − γ couplings at the Large Hadron electron Collider (LHeC) and the Future Circular Collider-hadron electron (FCC-he) [33][34][35][36][37][38], through the e − p → e − γ * p → e − W − q ′ X reaction, where γ * is Weizsacker-Willams photon and X represents the proton remnants after deep inelastic scattering. In this study, we use projections for runs at center-of-mass energies of 1.30, 1.98, 7.07 TeV and 10 TeV and total integrated luminosities of 10, 30, 50, 70, 100, 300, 500, 700 and 1000 fb −1 , respectively. In addition, to  [14] characterize possible deviations from the SM predictions on the anomalous ∆κ γ and λ γ couplings, we employ the effective Lagrange technique where the SM is extended by a set of dimension-six operators.
The rest of this paper is organized as follows. In Sect. II, a brief review of the operators in effective Lagrangian is provided. In Sect. III, we compute the total cross-section and derive bounds for the anomalous ∆κ γ and λ γ couplings at the LHeC and the FCC-he. In Sect. IV, we present our conclusions. In this context, in order to analyze the bounds on the non-standard W + W − γ couplings ∆κ γ and λ γ through the process e − p → e − γ * p → e − W − q ′ X, we adopt the effective Lagrangian: where L Therefore, effective Lagrangian relevant to our study on ∆κ γ and λ γ is given by: From Eqs. (3)-(5), D µ is the covariant derivative and Φ is the Higgs doublet field.B µν , and W µν are the U(1) Y and SU(2) L gauge field strength tensors, and the coefficients of these With this methodology, the effective Lagrangian for the non-standard W + W − γ couplings of electroweak gauge bosons take the form [17,39]: where The couplings g γ 1 , κ γ and λ γ are CP-preserving, and in the SM, g γ 1 = κ γ = 1 and λ λ = 0 at the tree level.
The three operators of dimension-six, given by Eq.
(2) are related to the anomalous triple gauge boson couplings via [18,40,41]: with From the effective Lagrangian given in Eq. (6), the Feynman rule for the anomalous W + W − γ vertex function most general CP-conserving and that is consistent with gauge and Lorentz invariance of the SM is given by [17]: where p 1 represents the momentum of the photon and p 2 and p 3 symbolize the momenta of W ± bosons. The first three terms in Eq. (10) corresponds to the SM couplings, while the terms with ∆κ γ and λ γ give rise to the anomalous triple gauge boson couplings.
Several searches on these anomalous ∆κ γ and λ γ couplings of W + W − γ vertex were performed by the LEP, Tevatron and LHC experiments, as shown in Table I. At the LHeC and the FCC-he, the aTGC can be directly probed via single-W ± production. Therefore, in this work we focus on the process e − p → e − γ * p → e − W − q ′ X, as well as with its corresponding sub-processes (see Figs. 1 and 2): where l + = e + , µ + ; ν l = ν e , ν µ and γ * is the Weizsacker-Williams photon. It is worth noting that in Eqs. (11)-(19), we take into account interactions between different flavors of the quarks due to off-diagonal elements of the Cabibbo-Kobayashi-Maskawa matrix.
In this type of γ * p interactions the emitted quasi-real photon γ * is scattered with small angles from the beam pipe of e − [42][43][44][45][46][47]. As these photons have low virtuality, they are almost on the mass shell. These processes can be described by the Equivalent Photon Approximation (EPA) [45,48,49], using the Weizsacker-Williams Approximation (WWA).
The EPA has a lot of advantages such as providing the skill to reach crude numerical predictions via simple formulae. Furthermore, it may principally ease the experimental analysis because it enables one to directly achieve a rough cross-section for γ * p → X process via the examination of the main process e − p → e − Xp where X represents objects produced in the final state. The production of high mass objects is particularly interesting at the e − p colliders and the production rate of massive objects is limited by the photon luminosity at high invariant mass while the γ * p process at the e − p colliders arises from quasi-real photon emitted from the incoming beams. However, using the EPA, new physics searches BSM have been theoretically and experimentally examined at the LEP, the Tevatron and the LHC .
In the EPA, the energy spectrum of the photons emitted from electron beam is given by the following analytical formula [45,71]: where x 1 = E γ * e /E e is the energy fraction transferred from the electron to the photon and Q 2 max = 2 GeV 2 is maximum virtuality of the photon. The minimum value of the Q 2 min is given by: Therefore, the total cross-section of the e − p → e − γ * p → e − W − q ′ X reaction is determined through: For the numerical evaluation of the total cross-section, we have used CTEQ6L1 [72] for the parton distribution functions and the W + W − γ vertex is embedded in CalcHEP package [71] together with the energy spectrum of the photon. In all calculations in this paper, we assume that the center-of-mass energies of the electron-proton system are 1.30, 1.98, 7.07 and 10 TeV, respectively.
In Figs. 3 and 4, we plot the total cross sections of the e − p → e − γ * p → e − W − q ′ X process.
We apply the following basic cuts to reduce the background and optimize the signal: T (e + ) > 20 GeV, T (ν e ) > 15 GeV, T (d) > 20 GeV, T (ū) > 20 GeV, These graphs describe the behavior of the total cross-section σ(∆κ γ , λ γ , √ s) in the context of effective Lagrangians, and as a function of the anomalous parameters ∆κ γ /λ γ , maintaining fixed the center-of-mass energy of the collider, which is assumed for values of √ s = 1.30, 1.98, 7.07 and 10 TeV. It is observed that the total cross-section σ(∆κ γ , λ γ , √ s) increases to a value of the order of ∼ 90 pb (70 pb), on a scale of center-of-mass energies of 1.30 up to 10 TeV. From this analysis it is concluded that the total cross-section to production single of vector bosons W − depends significantly on the center-of-mass energy of the collider, as well as of the anomalous parameters ∆κ γ and λ γ , which means that the effective area of the collision increases for very high energy ranges. In summary, the aTGC and the parameters of the colliders both tend to increase the cross-section of the single W − gauge boson production.
The bounds obtained from our analysis on the ∆κ γ and λ γ parameters are comparable to the previous most stringent limits from other single-boson and diboson analyses. Furthermore, our results indicate (see Figs. 3-4, and Table II) that the e − p → e − γ * p → e − W − q ′ X process is potentially suitable for probe the non-standard W + W γ couplings at the LHeC and the FCC-he, and with cleaner environments, compared to those for the case of hadron colliders.
The topic is interesting and under-explored, so it is of great importance to have theoretical and phenomenological interest to motivate experimental collaborations to measure this very interesting sector of QED. A prominent advantage of the e − p → e − γ * p → e − W − q ′ X process is that it isolates anomalous W + W − γ couplings. It allows to study W + W − γ couplings independent from W + W − Z as well as from W + W − γγ.