Probe of anomalous quartic $WWZ\gamma$ couplings in the photon-photon collisions

In this paper, we examine the potentials of the processes $\gamma \gamma\rightarrow W^{+} W^{-}Z $ and $e^{+}e^{-} \rightarrow e^{+}\gamma^{*} \gamma^{*} e^{-} \rightarrow e^{+} W^{+} W^{-} Z e^{-}$ at the CLIC with $\sqrt{s}=0.5,1.5$ and $3$ TeV to investigate anomalous quartic $WWZ\gamma$ couplings by two different CP-violating and CP-conserving effective Lagrangians. We find $95\%$ confidence level limits on the anomalous coupling parameters at the three CLIC energies and various integrated luminosities. The best limits obtained from the process $\gamma \gamma\rightarrow W^{+} W^{-}Z $ on the anomalous $\frac{k_{0}^{W}}{\Lambda^{2}}$, $\frac{k_{c}^{W}}{\Lambda^{2}}$ and $\frac{k_{2}^{m}}{\Lambda^{2}}$ couplings defined by CP-conserving effective Lagrangians are $[-1.73;\, 1.73]\times 10^{-7}$ GeV$^{-2}$, $[-2.44;\, 2.44]\times 10^{-7}$ and $[-1.89; \, 1.89]\times 10^{-7}$ GeV$^{-2}$, while $\frac{a_{n}}{\Lambda^{2}}$ coupling determined by CP-violating effective Lagrangians is obtained as $[-1.74;\, 1.74]\times 10^{-7}$ GeV$^{-2}$. In addition, the best limits derived on $\frac{k_{0}^{W}}{\Lambda^{2}}$, $\frac{k_{c}^{W}}{\Lambda^{2}}$ and $\frac{k_{2}^{m}}{\Lambda^{2}}$ and $\frac{a_{n}}{\Lambda^{2}}$ from the process $e^{+}e^{-} \rightarrow e^{+}\gamma^{*} \gamma^{*} e^{-} \rightarrow e^{+} W^{+} W^{-} Z e^{-}$ are obtained as $[-1.09;\, 1.09]\times 10^{-6}$ GeV$^{-2}$, $[-1.54;\, 1.54]\times 10^{-6}$ GeV$^{-2}$, $[-1.18;\, 1.18]\times 10^{-6}$ and $[-1.04;\, 1.04]\times 10^{-6}$ GeV$^{-2}$, respectively.

quartic W W Zγ couplings by two different CP-violating and CP-conserving effective Lagrangians.
We find 95% confidence level sensitivities on the anomalous coupling parameters at the three CLIC energies and various integrated luminosities. The best sensitivities obtained from the process γγ → W + W − Z on the anomalous In the literature, the anomalous quartic W W Zγ couplings are usually investigated by two different dimension 6 effective Lagrangians that keep custodial SU(2) c symmetry and local U(1) QED symmetry. The first is CP-violating effective Lagrangian. It is defined by [1] L n = iπα 4Λ 2 a n ǫ ijk W (i) where F µν is the tensor for electromagnetic field strength, α = e 2 4π is the fine structure constant, a n is the dimensionless anomalous quartic coupling constant and Λ is represented the energy scale of new physics. The anomalous W W Zγ vertex function obtained from effective Lagrangian in Eq. 1 is given in Appendix.
Secondly, we apply the formalism of Ref. [2] to examine CP-conserving effective Lagrangian. As can be seen from Eq. 5 in Ref. [2], there are fourteen effective photonic operators related to the anomalous quartic gauge couplings. These operators are identified by fourteen independent couplings k w,b,m 0,c , k w,m 1,2,3 and k b 1,2 . However, the effective interactions in these operators can be expressed in terms of independent Lorentz structures. For example, the W W γγ and ZZγγ interactions can be parameterized in terms of four independent Lorentz structures, Also, among them two are related to ZZZγ operators: The remaining W W Zγ interactions are given as follows with g = e/s W , g z = e/s W c W and V µν = ∂ µ V ν − ∂ ν V µ where s W = sin θ W , c W = cos θ W and V = W ± , Z. The anomalous vertex functions obtained through the CP-conserving anomalous W W Zγ interactions in Eqs. (8)- (12) are given in Appendix.
Therefore, the fourteen effective photonic operators related to the anomalous quartic gauge couplings can be appropriately rewritten in terms of the above independent Lorentz where the coefficients that parametrise the strength of the anomalous quartic gauge couplings are expressed as where . For this study, we take care of the five coefficients k W i (i = 0, c, 1, 2, 3) defined in Eqs. (18)- (20) corresponding to the W W Zγ vertex. However, these parameters are correlated with those coupling constants that describe W W γγ, ZZγγ and ZZZγ couplings [2]. Thus, the anomalous W W Zγ coupling should be dissociated from the other anomalous quartic couplings to obtain the only non-vanishing W W Zγ vertex. For the non-vanishing of the only W W Zγ vertex, we can apply additional restrictions on k j i parameters. One of the possible restrictions, proposed in [3], to verify this is to set k m 2 = −k m 3 and other parameters(k w,b,m 0,c , k w 1,2,3 , k m 1 and k b 1,2 ) to zero. As a result of this choice, Eq. (13) reduces to only non-vanishing W W Zγ couplings as follows The current experimental sensitivities on a n /Λ 2 parameter derived from CP-violating effective Lagrangian through the process e + e − → W + W − γ at the LEP are obtained by L3, OPAL and DELPHI collaborations. These are L3 : −0.14 GeV −2 < a n Λ 2 < 0.13 GeV −2 , OPAL : −0.16 GeV −2 < a n Λ 2 < 0.15 GeV −2 , DELPHI : −0.18 GeV −2 < a n Λ 2 < 0.14 GeV −2 (24) at 95% confidence level [4][5][6].
Besides, the CERN LHC provides current experimental sensitivities on only k W 0 Λ 2 and k W c Λ 2 couplings given in Eqs. (18)- (19) which are related to the anomalous quartic W W Zγ couplings within CP-conserving effective Lagrangians [7]. The results obtained for these couplings at 95% C. L. through the process qq ′ → W (→ ℓν)Z(→ jj)γ at √ s = 8 TeV with an integrated luminosity of 19.3 fb −1 are given as follows and The LHC is anticipated to answer some of the unsolved questions of particle physics.
However, it may not provide high precision measurements due to the remnants remaining after the collision of the proton beams. A linear e + e − collider with high luminosity and energy is the best option to complement and to extend the LHC physics program. The CLIC is one of the most popular linear colliders, planned to carry out e + e − collisions at energies from 0.5 TeV to 3 TeV [23]. To have its high luminosity and energy is quite important with regards to new physics research beyond the SM. Since the anomalous quartic W W Zγ couplings described through CP-violating and CP-conserving effective Lagrangians have dimension-6, they have very strong energy dependences. Thus, the anomalous cross section containing the W W Zγ vertex has a higher energy than the SM cross section. In addition, the future linear collider will possibly generate a final state with three or more massive gauge bosons. Hence, it will have a great potential to examine anomalous quartic gauge boson couplings.
Another possibility expected for the linear colliders is to operate this machine as γγ and γe colliders. This can be performed by converting the incoming leptons into intense beams of high-energy photons [24,25]. On the other hand, γ * γ * and γ * e processes at the linear colliders arise from quasi-real photon emitted from the incoming e + or e − beams. Hence, γ * γ * and γ * e processes are more realistic than γγ and γe processes. The photons in these processes are defined by the Equivalent Photon Approximation (EPA) [26][27][28][29][30]. In the EPA, the quasi-real photons are scattered at very small angles from the beam pipe, so they have low virtuality. For this reason, they are supposed to be almost real. Moreover, the EPA has a lot of advantages: First, it provides the skill to reach crude numerical predictions via simple formulae. In addition, it may principally ease the experimental analysis because it enables one to achieve directly a rough cross section for γ * γ * → X process via the examination of the main process e + e − → e + Xe − . Here, X represents objects produced in the final state. The production of high mass objects is specially interesting at the linear colliders. Furthermore, the production rate of massive objects is limited by the photon luminosity at high invariant mass.
In conclusion, these processes have a very clean experimental environment, since they have no interference with weak and strong interactions. Up to now, the photon-induced processes for the new physics searches were investigated through the EPA at the LEP, Tevatron, LHC and CLIC in literature .

II. CROSS SECTIONS AND NUMERICAL ANALYSIS
All numerical calculations in this study were evaluated using the computer package couplings can be given by In addition, the total cross sections containing k m 2 couplings are obtained as follows Finally, the total cross sections including a n couplings can be written by where σ SM is the SM cross section, σ int is the interference terms between SM and the anomalous contribution, and σ ano is the pure anomalous contribution. The interference terms in total cross sections given in Eqs. (27)-(28) related to CP-conserving effective Lagrangians are negligibly small compared to pure anomalous terms. Nevertheless, we took into account the effect of all interference term in the numerical calculations. However, the total cross section depends only on the quadratic function of a n anomalous coupling defined by CP-violating effective Lagrangians, since anomalous coupling a n does not interfere with the SM amplitude.
The quasi-real photons emitted from both lepton beams collide with each other, and representing the main process is given in Fig. 1. When calculating the total cross sections for this process, we used the equivalent photon spectrum described by the EPA which is embedded in CalcHEP. The total cross sections of the process e + e − → e + γ * γ * e − → e + W + W − Ze − as functions of anomalous The total cross section for the γγ → W + W − Z process has been calculated by using real photon spectrum produced by Compton backscattering of laser beam off the high energy electron beam. In Figs. 6-8, we plot the total cross section of the process γγ → W + W − Z as a function of anomalous couplings for √ s = 0.5, 1.5 and 3 TeV energies. The total cross section depending on the anomalous an Λ 2 of the process γγ → W + W − Z for the three center of mass energies are plotted in Fig. 9.
The kinematical distributions of final state particles can give further information about how we can separate among the different anomalous interactions. In this context, some distributions of the final state W and Z bosons are plotted for illustrative purposes using close to sensitivity of the anomalous couplings Figs. 10-21. We show the transverse momentum distributions of Z boson in the final states using anomalous couplings in Fig. 10 and using k m 2 Λ 2 and an Λ 2 anomalous couplings in Fig. 11 for We plot the rapidity distributions of the W + boson for two processes using the anomalous Especially, as can be seen from Fig. 15, the anomalous interactions for an Λ 2 coupling cause the production of more W + bosons in the central region.
In order to distinguish the different anomalous couplings with the SM, we illustrate the cosθ W distributions of W + for two processes where θ W is polar angle of W + with respect to the beam pipe. We show the cosθ W distributions with the anomalous couplings In order to probe the sensitivity to the anomalous quartic W W Zγ couplings, we use one and two-dimensional χ 2 analysis: where σ AN is the cross section including new physics effects, δ stat = 1 √ N and N is the number of SM events. The number of events for the processes γγ → W + W − Z and e + e − → where L int is the integrated luminosity. In addition, we assume that the the leptonic decay channel of Z boson with branching ratio is BR(Z → ℓl) = 0.067 and the hadronic decay channel of W boson with the branching ratio is BR(W → qq ′ ) = 0.676. In our calculations, one of the anomalous quartic couplings is assumed to deviate from their SM values (the others fixed to zero) at the one-dimensional χ 2 analysis, while two anomalous quartic couplings ( k W 0,c Λ 2 ) are assumed to deviate from their SM values at the two-dimensional χ 2 analysis. In this case, we take into account χ 2 value corresponding to the number of observable.
In Tables I-IV,  As can be seen in Table I, the process γγ → W + W − Z improves the sensitivities of k W 0 Λ 2 and k W c Λ 2 by up to a factor of 10 2 compared to the LHC [7]. The expected best sensitivities on an Λ 2 in Table II are far beyond the sensitivities of the existing LEP. However, we compare our results with the sensitivities of Ref. [13], in which the best sensitivities on and an Λ 2 couplings by examining the two processes e + e − → W − W + γ and e + e − → e + γ * e − → e + W − Zν e at the 3 TeV CLIC are obtained. We observed that the sensitivities obtained on k W 0 Λ 2 and k W c Λ 2 are at the same order with those reported in the Ref. [13] while sensitivities on k m 2 Λ 2 and an Λ 2 are 2 and 5 times better than the sensitivities calculated in Ref. [13], respectively. Our sensitivities on k m 2 Λ 2 can set more stringent sensitive by two orders of magnitude with respect to the best sensitivity derived from W Zγ production at the LHC with √ s = 14 TeV and the integrated luminosity of L = 200 fb −1 [17].
The γ * γ * collision of CLIC with √ s = 3 TeV and L int = 590 fb −1 investigates the CP-conserving and CP-violating anomalous W W Zγ coupling with a far better than the experiments sensitivities. One can see from Table III  couplings calculated with the help of the process e + e − → e + γ * γ * e − → e + W + W − Ze − are less sensitive than the results of Ref. [13]. On the other hand, our k m 2 Λ 2 and an Λ 2 couplings obtained from this process have similar sensitivities as Ref. [13].
In Figs. 22-24, we present 95% C.L. contours for anomalous k W 0 Λ 2 and k W c Λ 2 couplings for the process e + e − → e + γ * γ * e − → e + W + W − Ze − at the CLIC for various integrated luminosities and center-of-mass energies. As we can see from We can compare the obtained sensitivities on anomalous couplings by using statistical The obtained sensitivities using signal significance at 5 σ are approximately 1.5 times better than the best sensitivities obtained from χ 2 analysis at 95% C.L..

III. CONCLUSIONS
The linear e − e + colliders will provide an important opportunity to probe eγ and γγ collisions at high energies. In eγ and γγ collisions, high energy real photons can be obtained by converting the incoming lepton beams into photon beams via the Compton backscattering mechanism. In addition, high-energy accelerated e − and e + beams at the linear colliders radiate quasi-real photons, and thus eγ * and γ * γ * collisions are produced from the e − e + process itself. Therefore, eγ * and γ * γ * collisions at these colliders can occur spontaneously apart from eγ and γγ collisions. In the literature, Refs. [14,17] only examined the sensitivities on an Λ 2 couplings through the process γγ → W + W − Z at future linear colliders. As stated in Ref. [15], the γγ collisions can examine the sensitivities on an Λ 2 with a higher precision with respect to the eγ and e − e + collisions. For this reason, we compare our sensitivities with the results of Ref. [13]. For an Λ 2 couplings, γγ collisions at the 3 TeV CLIC with an integrated luminosity of 590 fb −1 enable us to improve the sensitivities by almost a factor of five with respect to sensitivities coming from e − e + collisions. Also, our sensitivities show that γγ collisions provide anomalous k m 2 Λ 2 couplings with a better than the e − e + collisions. On the other hand, we can see that the sensitivities on k W 0 Λ 2 and k W c Λ 2 expected to be obtained for the future γγ colliders with √ s = 3 TeV are roughly 2 times worse than the sensitivities in Ref. [13]. We find that the sensitivities obtained for four different and an Λ 2 couplings from the process γγ → W + W − Z are approximately an order of magnitude more restrictive with respect to the main process e + e − → e + γ * γ * e − → e + W + W − Ze − which is obtained by integrating the cross section for the subprocess γ * γ * → W + W − Z over the effective photon luminosity. The process γγ(γ * γ * ) → W + W − Z includes only interactions between the gauge bosons, causing more apparent possible deviations from the expected value of SM [15]. Therefore, in this paper, we analyze the CP-conserving parameters Λ 2 and CP-violating parameter an Λ 2 on the anomalous quartic W W Zγ gauge couplings through the processes γγ → W + W − Z obtained by laser-backscattering distributions and e + e − → e + γ * γ * e − → e + W + W − Ze − derived by EPA distributions at the CLIC. The γγ collisions seem to be the best place to test k m 2 Λ 2 and an Λ 2 which are the anomalous quartic couplings involving photons. Therefore, the CLIC as photon-photon collider provides an ideal platform to examine anomalous quartic W W Zγ gauge couplings at high energies.

Appendix: The anomalous vertex functions derived from CP-violating and CPconserving effective Lagrangians
The anomalous W + (p α 1 )W − (p β 2 )Z(k ν 2 )γ(k µ 1 ) vertex function obtained from CP-violating effective L n Lagrangian is given below (A.1) can be written as follows, respectively          Λ 2 and an Λ 2 couplings through the process e + e − → e + γ * γ * e − → e + W + W − Ze − at the CLIC with √ s = 1.5 and 3 TeV for various integrated lumi- an