Sensitivity physics expected to the measurement of the quartic WWγγ couplings at the LHeC and the FCC-he

The SU(2)_L × U(1)_Y gauge invariant structure of the standard model (SM) [1–3] specifies the form and strength of the self-interactions of the vector boson fields, particularly the anomalous quartic gauge couplings (aQGC): WWγγ, WWγZ, WWZZ, WWWW, ZZZZ, ZZZγ, ZZγγ, Zγγγ and γγγγ. Studying which processes these aQGC could contribute to may yield further confirmation of the non-abelian gauge structure of the SM or signal the presence of new physics beyond the SM (BSM) in unprobed energy scales. For instance, the following present and future colliders: the Large Hadron Collider (LHC), the High-Luminosity LargeHadron Collider (HL-LHC), the High-Energy Large Hadron Collider (HE-LHC) [4], the Large Hadron electron Collider (LHeC) [5–8], the Future Circular Collider-hadron electron (FCC-he) [9], the International Linear Collider (ILC) [10], the Compact Linear Collider (CLIC) [11], the Circular Electron Positron Collider (CEPC) [12] and the Future Circular Collider e⁺e⁻ (FCC-ee) [13]. All of these colliders contemplate in their physics programs the study of the aQGC.

Over the last few years, the aQGC production processes and single-W and double-W production in hadron–hadron, lepton–hadron and lepton–lepton colliders in different collision modes have attracted attention because future colliders with high energies, high luminosities and cleaner environments may allow experimental studies. Such studies are interesting because they allow further independent testing of the SM, the quartic WWγγ vertex can be probed and the Higgs boson plays an important role in WW channel.

In this paper, we are interested in estimating sensitivity measures in the aQGC f_M,i and f_T,i with i = 0, 1, 2, ..., 7 for the possible energies and luminosities of the LHeC and the FCC-he in its different stages, i.e., $\sqrt{s}$ = 1.30 TeV, 1.98 TeV, $\mathcal{L}=$10 fb⁻¹, 30 fb⁻¹, 50 fb⁻¹, 100 fb⁻¹ and $\sqrt{s}$ = 3.46 TeV, 5.29 TeV, $\mathcal{L}=$100 fb⁻¹, 300 fb⁻¹, 500 fb⁻¹, 1000 fb⁻¹, respectively. As shown in figures 1 and 2, there are 13 Feynman diagrams at the tree level contributing to the process ep → e⁻γ*p → eWγq'X via the subprocess γ*q → Wγq', where $q=u,c,\overline{d},\overline{s}$ and ${q}^{\prime }=d,s,\bar{u},\overline{c}$.

For an experimental and phenomenological review of the measurement evolution of the limits on the aQGC in the context of previous, present and future colliders, such as the LEP at the CERN [14–17], D0 and CDF at the Tevatron [18, 19], ATLAS and CMS at the LHC [20, 21] and in the post-LHC era as the LHeC and the FCC-he [22, 23], the ILC, the CLIC, the CEPC and the FCC-ee, see references [24–57], as well as table 1 of reference [23].

This paper describes searches for the sensitivity physics expected to the measurement of the WWγγ aQGC at the LHeC and the FCC-he using the process ep → e⁻γ*p → eWγq'X. In section 2, we briefly describe the theoretical aspects of the operators in our effective Lagrangian and in section 3, we derive the bounds for the aQGC at the LHeC and the FCC-he. We summarize our conclusions in section 4.

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The effective field theory (EFT) approach is very useful in the absence of a specific model of new physics. An EFT parameterizes the low-energy effects of the new physics to be found at higher energies in a model-independent way.

With this approach, we start from an EFT to probe model-independent sensitivity measures on W⁺W⁻γγ quartic gauge boson vertex. The EFT approach is the natural way to extend the SM such that the gauge symmetries are respected. In addition, the EFT is general enough to capture any BSM physics and provides guidance as to the most likely place to see the effects of new physics.

The measurement of the WWγγ couplings can be made quantitative by introducing a more general WWγγ vertex. For our discussion of phenomenological sensitivities in section 3, we shall use the phenomenological effective Lagrangian which comes from several SU(2)_L × U(1)_Y invariant dimension-8 effective operators [58]:

$$\begin{equation}{\mathcal{L}}_{\text{eff}}=\sum _{i=1}^{2}\frac{{f}_{\mathit{\text{S}},i}}{{{\Lambda}}^{4}}{O}_{\mathit{\text{S}},i}+\sum _{i=0}^{9}\frac{{f}_{\mathit{\text{T}},i}}{{{\Lambda}}^{4}}{O}_{\mathit{\text{T}},i}+\sum _{i=0}^{7}\frac{{f}_{\mathit{\text{M}},i}}{{{\Lambda}}^{4}}{O}_{\mathit{\text{M}},i}.\end{equation} \tag{ 1 }$$

In this equation, the indices S, T and M of the couplings and operators represent three classes of genuine aQGC operators [57]. The f_T,i/Λ⁴ associated operators characterize the effect of new physics on the scattering of transversely polarized vector bosons, and f_M,i/Λ⁴ includes mixed transverse and longitudinal scatterings. A list of these operators is given below.

(i) First class of independent scalar operators:

$$\begin{equation}{O}_{S,0}=\left[{\left({D}_{\mu }{\Phi}\right)}^{{\dagger}}\left({D}_{\nu }{\Phi}\right)\right]{\times}\left[{\left({D}^{\mu }{\Phi}\right)}^{{\dagger}}\left({D}^{\nu }{\Phi}\right)\right],\end{equation} \tag{ 2 }$$

$$\begin{equation}{O}_{S,1}=\left[{\left({D}_{\mu }{\Phi}\right)}^{{\dagger}}\left({D}^{\mu }{\Phi}\right)\right]{\times}\left[{\left({D}_{\nu }{\Phi}\right)}^{{\dagger}}\left({D}^{\nu }{\Phi}\right)\right].\end{equation} \tag{ 3 }$$

(ii) Second class of independent mixed operators:

$$\begin{equation}{O}_{M,0}=\mathrm{Tr}\left[{W}_{\mu \nu }{W}^{\mu \nu }\right]{\times}\left[{\left({D}_{\beta }{\Phi}\right)}^{{\dagger}}\left({D}^{\beta }{\Phi}\right)\right],\end{equation} \tag{ 4 }$$

$$\begin{equation}{O}_{M,1}=\mathrm{Tr}\left[{W}_{\mu \nu }{W}^{\nu \beta }\right]{\times}\left[{\left({D}_{\beta }{\Phi}\right)}^{{\dagger}}\left({D}^{\mu }{\Phi}\right)\right],\end{equation} \tag{ 5 }$$

$$\begin{equation}{O}_{M,2}=\left[{B}_{\mu \nu }{B}^{\mu \nu }\right]{\times}\left[{\left({D}_{\beta }{\Phi}\right)}^{{\dagger}}\left({D}^{\beta }{\Phi}\right)\right],\end{equation} \tag{ 6 }$$

$$\begin{equation}{O}_{M,3}=\left[{B}_{\mu \nu }{B}^{\nu \beta }\right]{\times}\left[{\left({D}_{\beta }{\Phi}\right)}^{{\dagger}}\left({D}^{\mu }{\Phi}\right)\right],\end{equation} \tag{ 7 }$$

$$\begin{equation}{O}_{M,4}=\left[{\left({D}_{\mu }{\Phi}\right)}^{{\dagger}}{W}_{\beta \nu }\left({D}^{\mu }{\Phi}\right)\right]{\times}{B}^{\beta \nu },\end{equation} \tag{ 8 }$$

$$\begin{equation}{O}_{M,5}=\left[{\left({D}_{\mu }{\Phi}\right)}^{{\dagger}}{W}_{\beta \nu }\left({D}^{\nu }{\Phi}\right)\right]{\times}{B}^{\beta \mu },\end{equation} \tag{ 9 }$$

$$\begin{equation}{O}_{M,6}=\left[{\left({D}_{\mu }{\Phi}\right)}^{{\dagger}}{W}_{\beta \nu }{W}^{\beta \nu }\left({D}^{\mu }{\Phi}\right)\right],\end{equation} \tag{ 10 }$$

$$\begin{equation}{O}_{M,7}=\left[{\left({D}_{\mu }{\Phi}\right)}^{{\dagger}}{W}_{\beta \nu }{W}^{\beta \mu }\left({D}^{\nu }{\Phi}\right)\right].\end{equation} \tag{ 11 }$$

(iii) Third class of independent transverse operators:

$$\begin{equation}{O}_{T,0}=\mathrm{Tr}\left[{W}_{\mu \nu }{W}^{\mu \nu }\right]{\times}\mathrm{Tr}\left[{W}_{\alpha \beta }{W}^{\alpha \beta }\right],\end{equation} \tag{ 12 }$$

$$\begin{equation}{O}_{T,1}=\mathrm{Tr}\left[{W}_{\alpha \nu }{W}^{\mu \beta }\right]{\times}\mathrm{Tr}\left[{W}_{\mu \beta }{W}^{\alpha \nu }\right],\end{equation} \tag{ 13 }$$

$$\begin{equation}{O}_{T,2}=\mathrm{Tr}\left[{W}_{\alpha \mu }{W}^{\mu \beta }\right]{\times}\mathrm{Tr}\left[{W}_{\beta \nu }{W}^{\nu \alpha }\right],\end{equation} \tag{ 14 }$$

$$\begin{equation}{O}_{T,5}=\mathrm{Tr}\left[{W}_{\mu \nu }{W}^{\mu \nu }\right]{\times}{B}_{\alpha \beta }{B}^{\alpha \beta },\end{equation} \tag{ 15 }$$

$$\begin{equation}{O}_{T,6}=\mathrm{Tr}\left[{W}_{\alpha \nu }{W}^{\mu \beta }\right]{\times}{B}_{\mu \beta }{B}^{\alpha \nu },\end{equation} \tag{ 16 }$$

$$\begin{equation}{O}_{T,7}=\mathrm{Tr}\left[{W}_{\alpha \mu }{W}^{\mu \beta }\right]{\times}{B}_{\beta \nu }{B}^{\nu \alpha },\end{equation} \tag{ 17 }$$

$$\begin{equation}{O}_{T,8}={B}_{\mu \nu }{B}^{\mu \nu }{B}_{\alpha \beta }{B}^{\alpha \beta },\end{equation} \tag{ 18 }$$

$$\begin{equation}{O}_{T,9}={B}_{\alpha \mu }{B}^{\mu \beta }{B}_{\beta \nu }{B}^{\nu \alpha }.\end{equation} \tag{ 19 }$$

In the operators (2)–(19) D_μ is the covariant derivative, Φ denotes the Higgs double field and B^μν, W^μν are the field strength tensors. The O_S,0 and O_S,1 operators given by equations (2) and (3) contain the quartic W⁺W⁻W⁺W⁻, W⁺W⁻ZZ and ZZZZ couplings, which do not concern us here. An exhaustive study on the mechanism to build the dimension-8 operators corresponding to the aQGC is presented in references [32, 40, 41, 56–58].

The phenomenological investigations at ep colliders generally contain usual deep inelastic scattering reactions where the colliding proton dissociates into partons. These reactions have been extensively examined in the literature, but exclusive and semi-elastic processes that are γ*γ* and γ*p have been studied much less. These exclusive and semi-elastic processes have simpler final states with respect to ep processes and thus compensate for the advantages of ep processes such as having a higher center-of-mass energy and luminosity. Here, γ*p processes have effective luminosity and much higher energy compared to γ*γ* process. This may be significant because of the high energy dependencies of the cross-sections containing the new physics parameters and for this reason, γ*p processes are expected to have a high sensitivity to the aQGC.

γ*p processes can be discerned from usual deep inelastic scattering processes by means of two experimental signatures [59]. The first signature is the forward large rapidity gap [60–63]. Quasi-real photons have a low virtuality and scatter with small angles from the beam pipe. Since the transverse momentum carried by a quasi-real photon is small, photon-emitting electrons should also be scattered with small angles and exit the central detector without being detected. This causes a decreased energy deposit in the corresponding forward region. As a result, 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 processes can be rejected by applying a selection cut on this quantity. The second experimental signature is provided by the forward detectors [64–66] which are capable of detecting particles with a large pseudorapidity. When a photon-emitting electron is scattered with a large pseudorapidity, it exceeds the pseudorapidity coverage of the central detectors. In these processes, the electron can be detected by the forward detectors which provides a distinctive signal for γ*p processes. In this context, LHeC Collaboration has a program of forward physics with extra detectors located in a region between a few tens up to several hundreds of meters from the interaction point [66].

In this section, the cross section of the ep → e⁻γ*p → eWγq'X signal is evaluated for the center-of-mass energies and luminosities of the LHeC and the FCC-he with their respective energies and luminosities $\sqrt{s}=1.30,1.98$ TeV, $\mathcal{L}=10-100\enspace {\mathrm{f}\mathrm{b}}^{-1}$ and $\sqrt{s}=3.46,5.29$ TeV, $\mathcal{L}=100-1000\enspace {\mathrm{f}\mathrm{b}}^{-1}$. For ep → e⁻γ*p → eWγq'X signal, we consider leptonic and hadronic decays of the W-boson; W → ν_ll, W → qq' with ν_l = ν_e, ν_μ, l = e⁻, μ and $q=u,c,\overline{d},\overline{s}$, ${q}^{\prime }=d,s,\bar{u},\overline{c}$, respectively.

Formally, the ep → e⁻γ*p → eWγq'X cross section can be split into three parts:

$$\begin{align}\hfill {\sigma }_{\text{tot}}\left(\sqrt{s},\frac{{f}_{M,i}}{{{\Lambda}}^{4}},\frac{{f}_{T,i}}{{{\Lambda}}^{4}}\right)& ={\sigma }_{\text{BSM}}\left(\sqrt{s},\frac{{f}_{M,i}^{2}}{{{\Lambda}}^{8}},\frac{{f}_{T,i}^{2}}{{{\Lambda}}^{8}},\frac{{f}_{M,i}}{{{\Lambda}}^{4}}\frac{{f}_{T,i}}{{{\Lambda}}^{4}}\right)+{\sigma }_{\mathrm{int}}\left(\sqrt{s},\frac{{f}_{M,i}}{{{\Lambda}}^{4}},\frac{{f}_{T,i}}{{{\Lambda}}^{4}}\right)\hfill \\ \hfill & \quad +{\sigma }_{\text{SM}}\left(\sqrt{s}\right), i=0,\enspace \dots ,\enspace 7,\hfill \end{align} \tag{ 20 }$$

where σ_BSM is the contribution due to BSM physics, which in our case comes from the anomalous vertex WWγγ. σ_int is the interference term between SM and the new physics contribution and σ_SM is the SM prediction, respectively.

To optimize the measurement of the electroweak-induced eWγq'X signal and improve the electroweak signal significance, we further consider selections on the following variables to suppress backgrounds. Following is a summary of the baseline selection criteria for the kinematics cuts on the final state particles:

(i) Cuts-0: selected cuts for the p_T:

$$\begin{equation}{\bullet}{p}_{T}^{q}{ >}20\enspace \text{GeV}\quad \left(\text{minimum}\;{p}_{T}\;\text{for}\;\text{the}\;\text{jets}\right),\end{equation} \tag{ 21 }$$

$$\begin{equation}{\bullet}{p}_{T}^{\gamma }{ >}10\enspace \text{GeV}\quad \left(\text{minimum}\;{p}_{T}\;\text{for}\;\text{the}\;\text{photons}\right),\end{equation} \tag{ 22 }$$

$$\begin{equation}{\bullet}{p}_{T}^{l}{ >}10\enspace \text{GeV}\quad \left(\text{minimum}\;{p}_{T}\;\text{for}\;\text{the}\;\text{charged}\;\text{leptons}\right).\end{equation} \tag{ 23 }$$

(ii) Cuts-1: selected cuts for the η:

$$\begin{equation}{\bullet}\vert {\eta }_{q}\vert {< }5\quad \left(\text{maximum}\;\text{rapidity}\;\text{for}\;\text{the}\;\text{jets}\right),\end{equation} \tag{ 24 }$$

$$\begin{equation}{\bullet}\vert {\eta }_{\gamma }\vert {< }2.5\quad \left(\text{maximum}\;\text{rapidity}\;\text{for}\;\text{the}\;\text{photons}\right),\end{equation} \tag{ 25 }$$

$$\begin{equation}{\bullet}\vert {\eta }_{l}\vert {< }2.5\quad \left(\text{maximum}\;\text{rapidity}\;\text{for}\;\text{the}\;\text{charged}\;\text{leptons}\right).\end{equation} \tag{ 26 }$$

(iii) Cuts-2: selected cuts for the ΔR:

$$\begin{equation}{\bullet}{\Delta}{R}_{qq}=0.4\quad \left(\text{minimum}\;\text{distance}\;\text{between}\;\text{jets}\right),\end{equation} \tag{ 27 }$$

$$\begin{equation}{\bullet}{\Delta}{R}_{ll}=0.4\quad \left(\text{minimum}\;\text{distance}\;\text{between}\;\text{leptons}\right),\end{equation} \tag{ 28 }$$

$$\begin{equation}{\bullet}{\Delta}{R}_{\gamma q}=0.4\quad \left(\text{minimum}\;\text{distance}\;\text{between}\;\gamma \;\text{and}\;\text{jet}\right),\end{equation} \tag{ 29 }$$

$$\begin{equation}{\bullet}{\Delta}{R}_{ql}=0.4\quad \left(\text{minimum}\;\text{distance}\;\text{between}\;\text{jet}\;\text{and}\;\text{lepton}\right),\end{equation} \tag{ 30 }$$

$$\begin{equation}{\bullet}{\Delta}{R}_{\gamma l}=0.4\quad \left(\text{minimum}\;\text{distance}\;\text{between}\;\gamma \;\text{and}\;\text{lepton}\right).\end{equation} \tag{ 31 }$$

As we mentioned above, the kinematic cuts given by equations (21)–(31) are applied to reduce the background and to reach higher expected significance for the possible aQGC signal in the process ep → e⁻γ*p → eWγq'X. The sensitivities are investigated using the Monte Carlo simulations with a leading order in MadGraph5_aMC@NLO [67]. The operators described in equations (4)–(19) are implemented into MadGraph5_aMC@NLO through Feynrules package [68] as a Universal FeynRules Output (UFO) module [69].

The future lepton–hadron colliders, such as the LHeC and the FCC-he can be operated as γ*p colliders, in this case the emitted quasi-real photon γ* is scattered with small angles from the beam pipe of e⁻ [70–75]. These processes can be described by the equivalent photon approximation (EPA) [73, 76, 77], using the Weizsacker–Williams approximation. The main idea of EPA is that the electromagnetic interaction of an electron with the complicated field of the proton bunch is replaced by a simpler Compton scattering of this proton with the flux of EPA generated by the electron bunch. For our case, the schematic diagram for the process ep → e⁻γ*p → eWγq'X is given by figure 1 and the Feynman diagrams of the subprocess γ*q → Wγq' are shown in figure 2. In this context, the spectrum of EPA photons is given by [73, 78]:

$$\begin{align}\hfill {f}_{{\gamma }_{1}^{{\ast}}}\left({x}_{1}\right)& =\frac{\alpha }{\pi {E}_{\text{e}}}\left\{\left[\frac{1-{x}_{1}+{x}_{1}^{2}/2}{{x}_{1}}\right]\mathrm{log}\left(\frac{{Q}_{\mathrm{max}}^{2}}{{Q}_{\mathrm{min}}^{2}}\right)-\frac{{m}_{\text{e}}^{2}{x}_{1}}{{Q}_{\mathrm{min}}^{2}}\left(1-\frac{{Q}_{\mathrm{min}}^{2}}{{Q}_{\mathrm{max}}^{2}}\right)\right.\hfill \\ \hfill & \quad \left.-\enspace \frac{1}{{x}_{1}}{\left[1-\frac{{x}_{1}}{2}\right]}^{2}\mathrm{log}\left(\frac{{x}_{1}^{2}{E}_{\text{e}}^{2}+{Q}_{\mathrm{max}}^{2}}{{x}_{1}^{2}{E}_{\text{e}}^{2}+{Q}_{\mathrm{min}}^{2}}\right)\right\}\hfill \end{align} \tag{ 32 }$$

where ${x}_{1}={E}_{{\gamma }_{1}^{{\ast}}}/{E}_{\text{e}}$ and ${Q}_{\mathrm{max}}^{2}$ is the maximum photon virtuality. The minimum value of ${Q}_{\mathrm{min}}^{2}$ is:

$$\begin{equation}{Q}_{\mathrm{min}}^{2}=\frac{{m}_{\text{e}}^{2}{x}_{1}^{2}}{1-{x}_{1}}.\end{equation} \tag{ 33 }$$

Using all of these tools, the total cross sections (see equation (20)) of the ep → e⁻γ*p → eWγq'X signal at the LHeC and the FCC-he are determined by:

$$\begin{equation}\sigma \left(ep\to eW\gamma {q}^{\prime }X\right)=\int {f}_{{\gamma }^{{\ast}}}\left(x\right)\hat{\sigma }\left({\gamma }^{{\ast}}q\to W\gamma {q}^{\prime }\right)\enspace \mathrm{d}x.\end{equation} \tag{ 34 }$$

The total cross section of the process ep → e⁻γ*p → eWγq'X, i.e., $\sigma \left({f}_{M,i}/{{\Lambda}}^{4},{f}_{T,i}/{{\Lambda}}^{4},\sqrt{s}\enspace \right)$ as a function of f_M,i/Λ⁴ and f_T,i/Λ⁴ with i = 0, 1, 2, ..., 7 for the energies of the LHeC with $\sqrt{s}=1.30$ TeV, 1.98 TeV and the FCC-he with $\sqrt{s}=3.46$ TeV, 5.29 TeV are reported in a region defined by the kinematics cuts given in equations (21)–(31).

Cross sections of the process ep → e⁻γ*p → eWγq'X as a function of aQGC f_M,i/Λ⁴ (f_T,i/Λ⁴) are given in figures 3–10. For evaluation of the total cross sections, the leptonic and hadronic decays of the W-boson in the final state are considered. The total cross sections for each coupling are evaluated while fixing the other couplings to zero.

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The corresponding expected cross sections after acceptance cuts for the process ep → e⁻γ*p → eWγq'X give the value $\sigma \left({f}_{T,6}/{{\Lambda}}^{4},\sqrt{s}\enspace \right)\simeq 1{0}^{5}$ pb for |f_T,6/Λ⁴| = 1 × 10⁻⁸  GeV⁻⁴ with the hadronic decay channel of the W-boson. The cross section is $\sigma \left({f}_{T,6}/{{\Lambda}}^{4},\sqrt{s}\enspace \right)\simeq 1{0}^{4}$ pb for the leptonic decay channel of the W-boson in the final state.

In tables 1 and 2, we illustrate the total cross sections in the fiducial region given by equations (21)–(31) for the process ep → e⁻γ*p → eWγq'X with the different f_M,i/Λ⁴ and f_T,i/Λ⁴ couplings, and for the future energies of the LHeC and the FCC-he.

Table 1. Summary of the total cross-sections of the process ep → e⁻γ*p → eWγq'X for $\sqrt{s}=1.30,1.98$ TeV at the LHeC and $\sqrt{s}=3.46,5.29$ TeV at the FCC-he depending on thirteen anomalous couplings obtained by dimension-8 operators. The total cross-sections for each coupling are calculated with the values of 1 × 10⁻⁸  GeV⁻⁴ and 5 × 10⁻⁹  GeV⁻⁴ at the LHeC and the FCC-he, respectively.

σ(ep → e−γ*p → eWγq'X) (pb)
	LHeC		FCC-he
	Leptonic channel		Leptonic channel
SM	7.27 × 10−3	2.18 × 10−2	2.27 × 10−2	6.38 × 10−2
Couplings	$\sqrt{s}=1.30$ TeV	$\sqrt{s}=1.98$ TeV	$\sqrt{s}=3.46$ TeV	$\sqrt{s}=5.29$ TeV
fM0/Λ4	2.29 × 10−2	3.98 × 10−1	7.40 × 10−2	2.80
fM1/Λ4	1.66 × 10−2	2.29 × 10−1	6.30 × 10−2	2.07
fM2/Λ4	6.79 × 10−1	1.62 × 101	2.29	1.17 × 102
fM3/Λ4	4.47 × 10−1	9.14	1.83	8.61 × 101
fM4/Λ4	5.84 × 10−2	1.25	1.94 × 10−1	9.00
fM5/Λ4	4.35 × 10−2	7.27 × 10−1	1.70 × 10−1	6.70
fM7/Λ4	1.05 × 10−2	7.69 × 10−2	3.29 × 10−2	5.71 × 10−1
fT0/Λ4	9.55 × 10−1	3.59 × 101	5.39	5.85 × 102
fT1/Λ4	2.61	8.10 × 101	1.96 × 101	1.97 × 103
fT2/Λ4	3.25 × 10−1	1.02 × 101	2.39	2.36 × 102
fT5/Λ4	1.01 × 101	3.88 × 102	5.79 × 101	6.31 × 103
fT6/Λ4	2.79 × 101	8.71 × 102	2.12 × 102	2.10 × 104
fT7/Λ4	3.45	1.10 × 102	26.13	2.51 × 103

Table 2. Summary of the total cross-sections of the process ep → e⁻γ*p → eWγq'X for $\sqrt{s}=1.30,1.98$ TeV at the LHeC and $\sqrt{s}=1.30,1.98$ TeV at the FCC-he depending on thirteen anomalous couplings obtained by dimension-8 operators. The total cross-sections for each coupling are calculated with the values of 1 × 10⁻⁸  GeV⁻⁴ and 5 × 10⁻⁹  GeV⁻⁴ at the LHeC and the FCC-he, respectively.

σ(ep → e−γ*p → eWγq'X) (pb)
	LHeC		FCC-he
	Hadronic channel		Hadronic channel
SM	1.54 × 10−2	4.51 × 10−2	4.94 × 10−2	1.34 × 10−1
Couplings	$\sqrt{s}=1.30$ TeV	$\sqrt{s}=1.98$ TeV	$\sqrt{s}=3.46$ TeV	$\sqrt{s}=5.29$ TeV
fM0/Λ4	5.79 × 10−2	9.38 × 10−1	1.79 × 10−1	2.50
fM1/Λ4	6.05 × 10−2	6.89 × 10−1	7.34 × 10−1	5.88
fM2/Λ4	1.84	3.84 × 101	5.70	1.02 × 102
fM3/Λ4	2.08	2.82 × 101	2.98 × 101	2.49 × 102
fM4/Λ4	1.54 × 10−1	2.98	4.79 × 10−1	7.88
fM5/Λ4	1.80 × 10−1	2.22	2.32	1.92 × 101
fM7/Λ4	2.92 × 10−2	2.15 × 10−1	2.26 × 10−1	1.59
fT0/Λ4	2.27	7.24 × 101	1.51 × 101	5.07 × 102
fT1/Λ4	1.14 × 101	2.15 × 102	5.81 × 102	7.07 × 103
fT2/Λ4	1.21	2.50 × 101	5.11 × 101	6.54 × 102
fT5/Λ4	2.42 × 101	7.76 × 102	1.62 × 102	5.45 × 103
fT6/Λ4	1.22 × 102	2.31 × 103	6.28 × 103	7.65 × 104
fT7/Λ4	1.30 × 101	2.68 × 102	5.50 × 102	7.02 × 103

From figures 3–10 and tables 1 and 2, it is clear that the cross section projects a greater dependence with respect to the f_T,6/Λ⁴ and f_T,5/Λ⁴ couplings than the f_M,7/Λ⁴, f_M,0/Λ⁴, f_M,1/Λ⁴, etc. There is also a difference in the measured cross section of up to an order of magnitude between the leptonic and hadronic cases. The cross sections are evaluated in a region defined by the kinematic cuts given by equations (21)–(31).

Table 3 shows the effects of cuts on the cross-section values of SM and some aQGC. As mentioned above, the kinematic cuts given by equations (21)–(31) are applied to reduce the background and to reach higher expected significance for the possible aQGC signal in the process ep → e⁻γ*p → eWγq'X. To compare the SM cross section and the total cross sections, we have taken the values of the couplings as 5 × 10⁻¹⁰  GeV⁻⁴ for center-of-mass energy of 1.98 TeV and 5 × 10⁻¹¹  GeV⁻⁴ for center-of-mass energy of 5.29 TeV. For example, regarding the effect of cuts at 5.29 TeV for hadronic decay process, after cut-0 set is applied, the cut efficiency is about 4% for the SM background which has the same final state with signal and after applying cut-1 and cut-2 sets, the efficiency is reduced by 75%. After the cuts are selected, the SM cross section is more suppressed with respect to the cross sections including the aQGC. Consequently, sensitivities are better when cuts are applied.

Table 3. Effects of selected cuts on the cross-sections of the process σ(ep → e⁻γ*p → eWγq'X) for SM and BSM for various anomalous couplings at 1.98 TeV and 5.29 TeV. The cross-sections are calculated with the values of 5 × 10⁻¹⁰  GeV⁻⁴ and 5 × 10⁻¹¹  GeV⁻⁴ at the LHeC and the FCC-he, respectively. The hadronic decay of W-boson is considered.

σ(ep → e−γ*p → eWγq'X) (pb)
	LHeC				FCC-he
	$\sqrt{s}=1.98$ TeV				$\sqrt{s}=5.29$ TeV
Couplings	No cuts	Cuts-0	Cuts-1	Cuts-2	No cuts	Cuts-0	Cuts-1	Cuts-2
SM	6.18	0.11	0.08	0.046	15.91	0.58	0.23	0.13
fM0/Λ4	6.42	0.12	0.08	0.05	18.16	0.57	0.25	0.14
fM2/Λ4	7.10	0.29	0.19	0.14	19.51	1.37	0.25	0.14
fM5/Λ4	7.65	0.12	0.09	0.05	16.53	0.59	0.23	0.13
fT0/Λ4	6.91	0.53	0.30	0.22	16.68	3.65	0.44	0.17
fT6/Λ4	12.31	8.25	7.69	5.72	80.92	65.2	33.71	7.47
fT7/Λ4	6.22	1.19	0.94	0.69	21.42	8.80	3.31	0.81

In summary, the ep → e⁻γ*p → eWγq'X cross section in the presence of anomalous couplings increases rapidly with the electron–proton increasing center-of-mass energy.

We perform χ² analysis to obtain the sensitivity measures on the anomalous f_M,i/Λ⁴ and f_T,i/Λ⁴ couplings. χ² is defined as follows:

$$\begin{equation}{\chi }^{2}\left({f}_{M,i}/{{\Lambda}}^{4},{f}_{T,i}/{{\Lambda}}^{4}\right)={\left(\frac{{\sigma }_{\text{SM}}\left(\sqrt{s}\right)-{\sigma }_{\text{BSM}}\left(\sqrt{s},{f}_{M,i}/{{\Lambda}}^{4},{f}_{T,i}/{{\Lambda}}^{4}\right)}{{\sigma }_{\text{SM}}{\delta }_{\text{st}}}\right)}^{2},\end{equation} \tag{ 35 }$$

where ${\sigma }_{\text{SM}}\left(\sqrt{s}\right)$ is the cross section of the SM and ${\sigma }_{\text{BSM}}\left(\sqrt{s},{f}_{M,i}/{{\Lambda}}^{4},{f}_{T,i}/{{\Lambda}}^{4}\right)$ is the BSM cross section. ${\delta }_{\text{st}}=\frac{1}{\sqrt{{N}_{\text{SM}}}}$ is the statistical error and N_SM is the number of events:

$$\begin{equation}{N}_{\text{SM}}={\mathcal{L}}_{\mathrm{int}}{\times}{\sigma }_{\text{SM}}.\end{equation} \tag{ 36 }$$

Here, we assume the integrated luminosities ${\mathcal{L}}_{\mathrm{int}}=10-100 {\mathrm{f}\mathrm{b}}^{-\mathrm{1}}$ for the LHeC and ${\mathcal{L}}_{\mathrm{int}}=100-1000 {\mathrm{f}\mathrm{b}}^{-\mathrm{1}}$ for the FCC-he.

Tables 4 and 5 summarize all sensitivity measures on the dimension-8 aQGC obtained from ep → e⁻γ*p → eWγq'X data with the leptonic decay of the W-boson in the final state at center-of-mass energies of $\sqrt{s}$ = 1.30, 1.98 TeV at the LHeC and $\sqrt{s}$ = 3.46, 5.29 TeV at the FCC-he. For these sensitivity measures, all parameters except the one shown are fixed to zero. The results for leptonic final state at $\sqrt{s}$ = 3.46 TeV and $\sqrt{s}$ = 5.29 TeV given in table 5 are better values compared to those obtained for the $\sqrt{s}$ = 1.30 TeV and $\sqrt{s}$ = 1.98 TeV presented in table 4. A similar behavior can be seen for the hadronic decay of W-boson given in tables 6 and 7. The sensitivity measures of f_T6/Λ⁴ = [−1.10; 1.60]  TeV⁻⁴ with $\sqrt{s}$ = 3.46 TeV and f_T6/Λ⁴ = [−4.37; 5.51] × 10⁻¹  TeV⁻⁴ with $\sqrt{s}$ = 5.29 TeV in table 7 are the most stringent. These sensitivity measures are also approximately an order of magnitude more stringent than those obtained at the LHeC and the FCC-he through the main ep → e⁻γ*p → eWγq'X reaction. [22, 23].

Table 4. Sensitivity measures on aQGC at the 95% C. L. via ep → e⁻γ*p → eWγq'X for $\sqrt{s}=1.30,1.98$ TeV at the LHeC.

$\sqrt{\mathrm{s}}$ = 1.30 TeV, leptonic channel
Couplings (TeV−4)	10 fb−1	30 fb−1	50 fb−1	100 fb−1
fM0/Λ4	[−3.27;3.28] × 103	[−2.48;2.49] × 103	[−2.18;2.19] × 103	[−1.84;1.85] × 103
fM1/Λ4	[−3.51;4.57] × 103	[−2.56;3.62] × 103	[−2.20;3.26] × 103	[−1.78;2.85] × 103
fM2/Λ4	[−5.02;5.01] × 102	[−3.81;3.80] × 102	[−3.36;3.35] × 102	[−2.82;2.81] × 102
fM3/Λ4	[−5.39;6.99] × 102	[−3.93;5.54] × 102	[−3.38;4.99] × 102	[−2.74;4.35] × 102
fM4/Λ4	[−1.81;1.82] × 103	[−1.37;1.38] × 103	[−1.21;1.22] × 103	[−1.01;1.02] × 103
fM5/Λ4	[−2.56;1.92] × 103	[−2.03;1.39] × 103	[−1.84;1.19] × 103	[−1.61;0.97] × 103
fM7/Λ4	[−0.91;0.70] × 104	[−0.72;0.51] × 104	[−0.65;0.44] × 104	[−0.57;0.35] × 104
fT0/Λ4	[−4.18;4.26] × 102	[−3.17;3.24] × 102	[−2.78;2.86] × 102	[−2.33;2.41] × 102
fT1/Λ4	[−2.43;2.66] × 102	[−1.82;2.05] × 102	[−1.59;1.82] × 102	[−1.32;1.55] × 102
fT2/Λ4	[−0.66;0.79] × 103	[−0.49;0.62] × 103	[−0.42;0.55] × 103	[−0.35;0.48] × 103
fT5/Λ4	[−1.27;1.30] × 102	[−9.64;9.88] × 101	[−8.47;8.71] × 101	[−7.10;7.35] × 101
fT6/Λ4	[−7.57;7.94] × 101	[−5.71;6.08] × 101	[−5.00;5.37] × 101	[−4.18;4.55] × 101
fT7/Λ4	[−2.08;2.34] × 102	[−1.55;1.81] × 102	[−1.35;1.61] × 102	[−1.12;1.38] × 102
$\sqrt{\mathrm{s}}$ = 1.98 TeV
fM0/Λ4	[−8.86;8.89] × 102	[−6.73;6.76] × 102	[−5.92;5.95] × 102	[−4.97;5.00] × 102
fM1/Λ4	[−1.07;1.28] × 103	[−0.79;1.00] × 103	[−0.68;0.90] × 103	[−0.56;0.78] × 103
fM2/Λ4	[−1.35;1.34] × 102	[−1.03;1.02] × 102	[−9.00;8.99] × 101	[−7.57;7.56] × 101
fM3/Λ4	[−1.66;1.92] × 102	[−1.24;1.49] × 102	[−1.07;1.33] × 102	[−0.88;1.14] × 102
fM4/Λ4	[−4.86;4.88] × 102	[−3.69;3.70] × 102	[−3.25;3.26] × 102	[−2.73;2.75] × 102
fM5/Λ4	[−0.71;0.59] × 103	[−0.55;0.44] × 103	[−0.50;0.38] × 103	[−0.43;0.31] × 103
fM7/Λ4	[−2.57;2.14] × 103	[−2.01;1.58] × 103	[−1.80;1.37] × 103	[−1.55;1.12] × 103
fT0/Λ4	[−9.00;9.03] × 101	[−6.83;6.86] × 101	[−6.01;6.04] × 101	[−5.05;5.08] × 101
fT1/Λ4	[−5.91;6.09] × 101	[−4.47;4.65] × 101	[−3.92;4.10] × 101	[−3.28;3.47] × 101
fT2/Λ4	[−1.60;1.78] × 102	[−1.20;1.38] × 102	[−1.04;1.23] × 102	[−0.86;1.05] × 102
fT5/Λ4	[−2.69;2.80] × 101	[−2.03;2.14] × 101	[−1.78;1.89] × 101	[−1.49;1.60] × 101
fT6/Λ4	[−1.77;1.89] × 101	[−1.33;1.45] × 101	[−1.17;1.28] × 101	[−0.97;1.09] × 101
fT7/Λ4	[−5.05;5.29] × 101	[−3.81;4.05] × 101	[−3.34;3.58] × 101	[−2.79;3.03] × 101

Table 5. Sensitivity measures on aQGC at the 95% C. L. via ep → e⁻γ*p → eWγq'X for $\sqrt{s}=3.46,5.29$ TeV at the FCC-he.

$\sqrt{\mathrm{s}}$ = 3.46 TeV, leptonic channel
Couplings (TeV−4)	100 fb−1	300 fb−1	500 fb−1	1000 fb−1
fM0/Λ4	[−6.70;6.81] × 102	[−5.08;5.19] × 102	[−4.47;4.57] × 102	[−3.75;3.85] × 102
fM1/Λ4	[−0.62;0.88] × 103	[−0.44;0.71] × 103	[−0.38;0.64] × 103	[−0.30;0.57] × 103
fM2/Λ4	[−1.03;1.00] × 102	[−7.87;7.63] × 101	[−6.95;6.71] × 101	[−5.86;5.62] × 101
fM3/Λ4	[−1.11;1.13] × 102	[−8.42;8.65] × 101	[−7.40;7.63] × 101	[−6.21;6.43] × 101
fM4/Λ4	[−3.75;3.77] × 102	[−2.85;2.87] × 102	[−2.51;2.52] × 102	[−2.11;2.12] × 102
fM5/Λ4	[−4.17;4.08] × 102	[−3.18;3.09] × 102	[−2.80;2.71] × 102	[−2.36;2.27] × 102
fM7/Λ4	[−1.78;1.39] × 103	[−1.40;1.01] × 103	[−1.26;0.88] × 103	[−1.10;0.71] × 103
fT0/Λ4	[−6.71;6.93] × 101	[−5.08;5.30] × 101	[−4.45;4.67] × 101	[−3.73;3.95] × 101
fT1/Λ4	[−3.49;3.52] × 101	[−2.64;2.68] × 101	[−2.33;2.36] × 101	[−1.95;1.98] × 101
fT2/Λ4	[−0.89;1.14] × 102	[−0.65;0.90] × 102	[−0.56;0.81] × 102	[−0.45;0.71] × 102
fT5/Λ4	[−2.02;2.03] × 101	[−1.53;1.55] × 101	[−1.35;1.36] × 101	[−1.13;1.15] × 101
fT6/Λ4	[−0.94;1.19] × 101	[−0.69;0.94] × 101	[−0.60;0.84] × 101	[−0.48;0.73] × 101
fT7/Λ4	[−0.28;0.33] × 102	[−0.20;0.26] × 102	[−0.18;0.23] × 102	[−0.15;0.20] × 102
$\sqrt{\mathrm{s}}$ = 5.29 TeV
fM0/Λ4	[−1.19;1.24] × 102	[−9.00;9.45] × 101	[−7.90;8.34] × 101	[−6.61;7.05] × 101
fM1/Λ4	[−1.28;1.53] × 102	[−0.95;1.20] × 102	[−0.82;1.07] × 102	[−0.67;0.92] × 102
fM2/Λ4	[−1.84;1.82] × 101	[−1.40;1.38] × 101	[−1.23;1.22] × 101	[−1.04;1.02] × 101
fM3/Λ4	[−2.06;2.22] × 101	[−1.54;1.70] × 101	[−1.35;1.51] × 101	[−1.12;1.28] × 101
fM4/Λ4	[−6.62;6.68] × 101	[−5.02;5.09] × 101	[−4.42;4.48] × 101	[−3.71;3.77] × 101
fM5/Λ4	[−0.83;0.72] × 102	[−0.65;0.54] × 102	[−0.58;0.47] × 102	[−0.49;0.39] × 102
fM7/Λ4	[−2.98;2.76] × 102	[−2.29;2.07] × 102	[−2.03;1.81] × 102	[−1.73;1.50] × 102
fT0/Λ4	[−8.18;8.23]	[−6.21;6.26]	[−5.46;5.51]	[−4.59;4.64]
fT1/Λ4	[−4.37;4.60]	[−3.29;3.52]	[−2.88;3.11]	[−2.41;2.64]
fT2/Λ4	[−1.23;1.36] × 101	[−0.92;1.05] × 101	[−0.81;0.93] × 101	[−0.67;0.79] × 101
fT5/Λ4	[−2.41;2.60]	[−1.81;1.99]	[−1.58;1.77]	[−1.31;1.50]
fT6/Λ4	[−1.22;1.53]	[−0.89;1.21]	[−0.77;1.08]	[−0.63;0.94]
fT7/Λ4	[−3.83;4.07]	[−2.88;3.13]	[−2.52;2.77]	[−2.10;2.35]

Table 6. Sensitivity measures on aQGC at the 95% C. L. via ep → e⁻γ*p → eWγq'X for $\sqrt{s}=1.30,1.98$ TeV at the LHeC.

$\sqrt{\mathrm{s}}$ = 1.30 TeV, hadronic channel
Couplings (TeV−4)	10 fb−1	30 fb−1	50 fb−1	100 fb−1
fM0/Λ4	[−2.37;2.41] × 103	[−1.80;1.84] × 103	[−1.59;1.62] × 103	[−1.32;1.36] × 103
fM1/Λ4	[−1.95;2.59] × 103	[−1.41;2.06] × 103	[−1.21;1.86] × 103	[−0.98;1.62] × 103
fM2/Λ4	[−3.68;3.61] × 102	[−2.81;2.73] × 102	[−2.47;2.40] × 102	[−2.09;2.01] × 102
fM3/Λ4	[−2.93;3.98] × 102	[−2.12;3.17] × 102	[−1.82;2.87] × 102	[−1.47;2.51] × 102
fM4/Λ4	[−1.31;1.33] × 103	[−9.92;10.13] × 102	[−8.72;8.93] × 102	[−7.31;7.53] × 102
fM5/Λ4	[−1.44;1.07] × 103	[−1.15;0.77] × 103	[−1.04;0.66] × 103	[−0.91;0.54] × 103
fM7/Λ4	[−5.18;3.85] × 103	[−4.12;2.79] × 103	[−3.73;2.39] × 103	[−3.26;1.93] × 103
fT0/Λ4	[−3.28;3.29] × 102	[−2.49;2.50] × 102	[−2.19;2.20] × 102	[−1.84;1.85] × 102
fT1/Λ4	[−1.31;1.63] × 102	[−0.96;1.28] × 102	[−0.83;1.15] × 102	[−0.68;1.00] × 102
fT2/Λ4	[−4.06;4.98] × 102	[−2.98;3.91] × 102	[−2.58;3.50] × 102	[−2.11;3.03] × 102
fT5/Λ4	[−9.54;10.51] × 101	[−7.14;8.11] × 101	[−6.23;7.20] × 101	[−5.17;6.13] × 101
fT6/Λ4	[−3.85;5.15] × 101	[−2.79;4.09] × 101	[−2.40;3.70] × 101	[−1.94;3.24] × 101
fT7/Λ4	[−1.23;1.52] × 102	[−0.90;1.20] × 102	[−0.78;1.08] × 102	[−0.64;0.93] × 102
$\sqrt{\mathrm{s}}$ = 1.98 TeV
fM0/Λ4	[−6.76;6.89] × 102	[−5.13;5.25] × 102	[−4.50;4.63] × 102	[−3.78;3.90] × 102
fM1/Λ4	[−7.10;8.91] × 102	[−5.21;7.02] × 102	[−4.49;6.30] × 102	[−3.66;5.47] × 102
fM2/Λ4	[−1.05;1.04] × 102	[−7.96;7.86] × 101	[−7.02;6.91] × 101	[−5.91;5.80] × 101
fM3/Λ4	[−1.07;1.38] × 102	[−0.78;1.09] × 102	[−0.67;0.98] × 102	[−0.54;0.86] × 102
fM4/Λ4	[−3.76;3.79] × 102	[−2.85;2.89] × 102	[−2.51;2.54] × 102	[−2.10;2.14] × 102
fM5/Λ4	[−4.90;3.93] × 102	[−3.86;2.89] × 102	[−3.46;2.49] × 102	[−3.00;2.03] × 102
fM7/Λ4	[−1.79;1.42] × 103	[−1.41;1.04] × 103	[−1.27;0.90] × 103	[−1.10;0.73] × 103
fT0/Λ4	[−7.54;7.62] × 101	[−5.72;5.80] × 101	[−5.03;5.11] × 101	[−4.23;4.30] × 101
fT1/Λ4	[−3.99;4.85] × 101	[−2.94;3.80] × 101	[−2.54;3.40] × 101	[−2.08;2.94] × 101
fT2/Λ4	[−1.17;1.43] × 102	[−0.86;1.12] × 102	[−0.74;1.01] × 102	[−0.61;0.87] × 102
fT5/Λ4	[−2.31;2.32] × 101	[−1.75;1.76] × 101	[−1.54;1.55] × 101	[−1.29;1.31] × 101
fT6/Λ4	[−1.30;1.39] × 101	[−0.98;1.07] × 101	[−0.86;0.94] × 101	[−0.71;0.80] × 101
fT7/Λ4	[−3.50;4.42] × 101	[−2.57;3.49] × 101	[−2.21;3.13] × 101	[−1.80;2.72] × 101

Table 7. Sensitivity measures on aQGC at the 95% C. L. via ep → e⁻γ*p → eWγq'X for $\sqrt{s}=3.46,5.29$ TeV at the FCC-he.

$\sqrt{\mathrm{s}}$ = 3.46 TeV, hadronic channel
Couplings (TeV−4)	100 fb−1	300 fb−1	500 fb−1	1000 fb−1
fM0/Λ4	[−5.12;5.13] × 102	[−3.89;3.90] × 102	[−3.42;3.44] × 102	[−2.88;2.89] × 102
fM1/Λ4	[−1.93;2.58] × 102	[−1.40;2.05] × 102	[−1.20;1.85] × 102	[−0.97;1.62] × 102
fM2/Λ4	[−0.87;0.70] × 102	[−0.68;0.52] × 102	[−0.61;0.45] × 102	[−0.53;0.37] × 102
fM3/Λ4	[−2.94;3.93] × 101	[−2.13;3.12] × 101	[−1.83;2.82] × 101	[−1.48;2.47] × 101
fM4/Λ4	[−2.78;2.89] × 102	[−2.10;2.21] × 102	[−1.84;1.95] × 102	[−1.54;1.65] × 102
fM5/Λ4	[−1.53;1.02] × 102	[−1.24;0.73] × 102	[−1.13;0.62] × 102	[−1.00;0.49] × 102
fM7/Λ4	[−5.29;3.76] × 102	[−4.24;2.71] × 102	[−3.85;2.32] × 102	[−3.39;1.86] × 102
fT0/Λ4	[−4.62;5.00] × 101	[−3.47;3.85] × 101	[−3.03;3.41] × 101	[−2.52;2.90] × 101
fT1/Λ4	[−0.74;0.81] × 101	[−0.56;0.62] × 101	[−0.49;0.55] × 101	[−0.40;0.47] × 101
fT2/Λ4	[−2.26;2.99] × 101	[−1.64;2.37] × 101	[−1.41;2.14] × 101	[−1.14;1.87] × 101
fT5/Λ4	[−1.42;1.51] × 101	[−1.07;1.16] × 101	[−0.94;1.02] × 101	[−0.78;0.87] × 101
fT6/Λ4	[−2.12;2.62]	[−1.56;2.05]	[−1.34;1.84]	[−1.10;1.60]
fT7/Λ4	[−7.18;8.88]	[−5.27;6.98]	[−4.55;6.26]	[−3.72;5.42]
$\sqrt{\mathrm{s}}$ = 5.29 TeV
fM0/Λ4	[−1.53;1.58] × 102	[−1.15;1.21] × 102	[−1.01;1.07] × 102	[−0.85;0.90] × 101
fM1/Λ4	[−0.90;1.12] × 102	[−0.66;0.88] × 102	[−0.57;0.79] × 102	[−0.46;0.68] × 102
fM2/Λ4	[−2.41;2.33] × 101	[−1.84;1.77] × 101	[−1.62;1.55] × 101	[−1.37;1.30] × 101
fM3/Λ4	[−1.30;1.84] × 101	[−0.94;1.47] × 101	[−0.80;1.34] × 101	[−0.64;1.18] × 101
fM4/Λ4	[−8.44;8.67] × 101	[−6.39;6.62] × 101	[−5.61;5.84] × 101	[−4.70;4.93] × 101
fM5/Λ4	[−6.23;4.81] × 101	[−4.93;3.51] × 101	[−4.44;3.02] × 101	[−3.87;2.45] × 101
fM7/Λ4	[−2.19;1.83] × 102	[−1.71;1.35] × 102	[−1.53;1.17] × 102	[−1.32;0.96] × 102
fT0/Λ4	[−1.04;1.10] × 101	[−0.78;0.84] × 101	[−0.68;0.74] × 101	[−0.57;0.63] × 101
fT1/Λ4	[−2.53;3.21]	[−1.85;2.53]	[−1.60;2.27]	[−1.30;1.97]
fT2/Λ4	[−0.82;1.10] × 101	[−0.59;0.88] × 101	[−0.51;0.79] × 101	[−0.41;0.69] × 101
fT5/Λ4	[−3.17;3.30]	[−2.39;2.52]	[−2.10;2.23]	[−1.75;1.89]
fT6/Λ4	[−8.17;9.32] × 10−1	[−6.08;7.23] × 10−1	[−5.29;6.44] × 10−1	[−4.37;5.51] × 10−1
fT7/Λ4	[−2.35;3.48]	[−1.68;2.81]	[−1.43;2.56]	[−1.14;2.27]

Table 8 illustrates sensitivity measures on aQGC at the 95% C. L. via ep → e⁻γ*p → eWγq'X with the EPA for various Q_max values with $\sqrt{s}=5.29$ TeV at the FCC-he. The EPA factorize the dependence on virtuality of the photon from the cross-section of the photon-induced process (γ*γ* and γ*p collisions) to the equivalent photon flux. However, Q_max dependence of new physics parameters has also been studied in the literature. Ref. [79] has examined Q_max dependence of the cross sections with the EPA for the process pp → pγ*γ*p → pτ⁺τ⁻p without anomalous couplings of tau lepton. They found that the cross sections for Q_max = (1–2)  GeV do not appreciably change. In addition, the cross sections of the process ee → eγ*γ*e → eeττ at the CLIC for values of Q_max = (1.41–8)  GeV are obtained in reference [80]. The potential of the process ep → eγ*γ*p → eτ⁻τ⁺p at the LHeC and the FCC-he to examine non-standard τ⁻τ⁺γ coupling in a model independent way by means of the effective Lagrangian approach is investigated in reference [81]. In that study, Q_max = 100  GeV is assumed and in our case, we consider the maximum photon virtuality Q_max = 100  GeV, where this value is the default value in MadGraph5_aMC@NLO. We calculate the Q_max dependency on the aQGC for the highest center-of-mass energy. In table 8, sensitivity measures on aQGC at the 95% C.L. via the process ep → eγ*p → eWγq'X for the hadronic decay of W-boson for $\mathcal{L}=1000\enspace {\mathrm{f}\mathrm{b}}^{-1}$, $\sqrt{s}=5.29$ TeV and Q_max = 1.41, 8  GeV are obtained. We conclude that Q_max dependence on the aQGC does not significantly change.

We now compare our findings with the other results in the literature which used different cuts and different channels. In reference [22], a detailed study of the LHeC and the FCC-he sensitivity to the anomalous f_M,i/Λ⁴ and f_T,i/Λ⁴ couplings in ν_eγγq production was carried out. Using a χ² analysis and kinematic cuts for the final state particles in ν_eγγq production, they obtained limits on the thirteen different anomalous couplings arising from dimension-8 operators. In reference [23], limits were obtained from diboson production at both the LHeC and the FCC-he and on the anomalous f_M,i/Λ⁴ and f_T,i/Λ⁴ couplings considering the process e⁻p → e⁻γ*γ*p → e⁻W⁺W⁻p with the subprocess γ*γ* → W⁺W⁻. These limits are weaker by about a factor of 3 or 5 and up to an order of magnitude than our results. CMS Collaboration [82, 83] at the LHC with $\sqrt{s}=8$ TeV and to an integrated luminosity of 19.7 fb⁻¹ searches for exclusive or quasi-exclusive WW production via the signal topology pp → p*W⁺W⁻p* where the p* indicates that the final state protons either remain intact (exclusive or elastic production), or dissociate into an undetected system (quasi-exclusive or proton dissociation production). Their research is translated into upper limits on the aQGC operators f_M,0,1,2,3/Λ⁴ (dimension-8). From its investigations, CMS Collaboration measures the electroweak-induced production of W and two jets, where the W boson decays leptonically, and experimental limits on aQGC f_M,0−7/Λ⁴, f_{T,0−2,5−7}/Λ⁴ are set at 95% C.L. [82, 83]. On the other hand, ATLAS Collaboration at the LHC studied the production of WVγ events in eνμνγ, eνqqγ and μνqqγ final states with ${\mathcal{L}}_{\mathrm{int}}=20.2\enspace {\mathrm{f}\mathrm{b}}^{-\mathrm{1}}$ of proton–proton collisions with $\sqrt{s}=8$ TeV [21].

The calculations on the production cross section in this paper are derived for the eWγq'X final states at the LHeC with center-of-mass energies $\sqrt{s}=1.30,1.98$ TeV and the FCC-he with $\sqrt{s}=3.46,5.29$ TeV in the fiducial regions given by equations (21)–(31). Our results are summarized through figures 3–10 and in tables 1–3. Furthermore, we show individual upper sensitivity measures obtained for the aQGC f_M,0−5,7/Λ⁴ and f_{T,0−2,5−7}/Λ⁴ at 95% C.L. both at leptonic and hadronic decay channel of the W-boson in tables 4–7. As can be seen in the results, the process gives strong constraints on aQGC sensitivity measures at high energy region and high luminosities.

In conclusion, we explore the phenomenological aspects of the anomalous WWγγ couplings via the process ep → e⁻γ*p → eWγq'X at the LHeC and the FCC-he. These couplings are defined through a phenomenological effective Lagrangian. The major goal of these measurements will be the confirmation of the new physics BSM. If the energy scale of the new physics responsible for the non-standard gauge boson couplings f_M,i/Λ⁴ and f_T,i/Λ⁴ is the center-of-mass energy of 5.29 TeV and the integrated luminosity of 1000 fb⁻¹, these couplings are expected to be no larger than $\mathcal{O}\left(1{0}^{-1}\right)$. Our results, as well as our expectations, indicate that with cleaner environments, appropriate fiducial regions, high energies and high luminosities for future colliders will be possible to obtain stronger upper sensitivity measures on the anomalous WWγγ couplings.

AGR and MAHR thank SNI and PROFEXCE (México). The numerical calculations reported in this paper were partially performed at TUBITAK ULAKBIM, High Performance and Grid Computing Center (TRUBA resources).