Search for anomalous quartic $ZZ\gamma\gamma$ couplings in photon-photon collisions

The self-couplings of the electroweak gauge bosons are completely specified by the non-Abelian gauge nature of the Standard Model (SM). The direct study of these couplings provides a significant opportunity to test the validity of the SM and the existence of new physics beyond the SM up to the high energy scale. For this reason, we investigate the potential of the processes $\gamma\gamma\rightarrow ZZ$, $e^{-}\gamma\rightarrow e^{-}\gamma^{*}\gamma \rightarrow e^{-}Z\, Z$ and $e^{+}e^{-} \rightarrow e^{+}\gamma^{*} \gamma^{*} e^{-} \rightarrow e^{+}\, Z\, Z\, e^{-}$ to examine the anomalous quartic couplings of $ZZ\gamma\gamma$ vertex at the Compact Linear Collider (CLIC) with center-of-mass energy $3$ TeV. We calculate $95\%$ confidence level sensitivities on the dimension-8 parameters with various values of the integrated luminosity. We show that the best bounds on the anomalous $\frac{f_{M2}}{\Lambda^{4}}$, $\frac{f_{M3}}{\Lambda^{4}}$, $\frac{f_{T0}}{\Lambda^4}$ and $\frac{f_{T9}}{\Lambda^4}$ couplings arise from $\gamma\gamma\rightarrow ZZ$ process among those three processes at center-of-mass energy of 3 TeV and integrated luminosity of $L_{int}=2000$ fb$^{-1}$ are found to be $[-3.30;3.30]\times 10 ^{-3}$ TeV$^{-4}$, $[-1.20;1.20]\times 10 ^{-2}$ TeV$^{-4}$, $[-3.40;3.40]\times 10 ^{-3}$ TeV$^{-4}$ and $[-1.80;1.80]\times 10 ^{-3}$ TeV$^{-4}$, respectively.


I. INTRODUCTION
The SM of particle physics has been tested with a lot of different experiments for decades and it is proven to be extremely successful. In addition, the discovery of all the particles predicted by the SM has been completed together with the ultimate discovery of the approximately 125 GeV Higgs boson in 2012 at the Large Hadron Collider (LHC) [1,2]. However, we need a new physics beyond the SM to find answers to some fundamental questions, such as the strong CP problem, neutrino oscillations and matter -antimatter asymmetry in the universe. The self-interactions of electroweak gauge bosons are important and more sensitive for new physics beyond the SM. The structure of gauge boson self-interactions is completely determined by the non-Abelian SU(2) L ⊗ U(1) Y gauge symmetry in the SM. Contributions to these interactions, beyond those coming from the SM, will be a supporting evidence of probable new physics. It can be examined in a model independent way via the effective Lagrangian approach. Such an approach is parameterized by high-dimensional operators which induce anomalous quartic gauge couplings that modify the interactions between the electroweak gauge bosons.
In writing effective operators associated to genuinely quartic couplings we employ the formalism of Refs. [3,4]. Imposing global SU(2) L symmetry and local U(1) Y symmetry, dimension-6 effective Lagrangian for the ZZγγ coupling is given by where F µν = ∂ µ A ν − ∂ ν A µ is the tensor for electromagnetic field tensor, and a 0,c are the dimensionless anomalous quartic coupling constants, Λ is a mass-dimension parameter associated with the scale of new physics.
The anomalous quartic gauge couplings come out also from dimension-8 operators. There are three classes of operators containing either covariant derivatives of Higgs doublet (D µ Φ) only, or two field strength tensors and two D µ Φ, or field strength tensors only. The first class operators contain anomalous quartic gauge couplings involving only massive vector boson. We will not examine them since these operators contain only quartic W + W − W + W − , W + W − ZZ and ZZZZ interactions. In the second class, eight anomalous quartic gauge boson couplings are given by [5][6][7] where the field strength tensor of the SU(2) (W µν ) and U(1) (B µν ) are given by Here, τ i (i = 1, 2, 3) are the SU(2) generators, g = e/sinθ W , g ′ = g/cosθ W , e is the unit of electric charge and θ W is the Weinberg angle. The dimension-6 operators can be expressed simply in terms of dimension-8 operators due to their similar Lorentz structures. The following expressions show the relations between the f M i couplings for the ZZγγ vertex and a 0 and the a c couplings, needed to compare with the LEP results; The operators containing four field strength tensors lead to quartic anomalous couplings are as follows where f T 8 and f T 9 are dimensionless parameters which have no dimensions-6 analogue.
The expected design of the future linear collider will include operation also in eγ and γγ modes. In eγ and γγ processes, real photon beams can be generated by converting the incoming e − and e + beams into photon beams through the Compton backscattering mechanism. The maximum collision energy is expected to be 80% for γγ collision and 90% for eγ collision of the original e + e − collision energy. However, the expected luminosities are 15% for γγ collision and 39% for eγ collision of the e + e − luminosities [28]. Also when using directly the lepton beams, quasi-real photons will be radiated at the interaction allowing for processes like eγ * , γγ * and γ * γ * to occur [29][30][31][32][33][34]. Alternatively, a γ * photon emitted from either of the incoming leptons can interact with a laser photon backscattered from the other lepton beam, and the subprocess γγ * → ZZ can take place. Hence, we calculate the process eγ → eγ * γ → eZZ by integrating the cross section for the subprocess γγ * → ZZ over the γ * flux. Furthermore, γ * photons emitted from both lepton beams can collide with each other and the subprocess γ * γ * → ZZ can be produced, and the cross section for the full process ee → eγ * γ * e → eZZe is calculated by integrating the cross section for the subprocess γ * γ * → ZZ over both γ * fluxes. The quasi-real γ * flux in γγ * and γ * γ * collisions is defined by the Weizsacker-Williams approximation (WWA). In the WWA, the electro-production processes includes a small angle of charged particle scattering. The virtuality of γ * photons emitted by the scattering particle is very small. Hence, they are supposed to be almost real.
There is a possibility to reduce the process of electro-production to the photo-production described by the following photon spectrum [32]; where m e is mass of the scattering particle,

III.
ZZ PRODUCTION AT γγ, γ * γ * AND γ * γ COLLISIONS In this section we will display the differential cross sections by considering the contributions of all three types of collisions separately, γγ, γ * γ * and γ * γ, for the ZZ productions through the process γγ → ZZ and the subprocesses γ * γ * → ZZ and γγ * → ZZ. The representative leading order Feynman diagrams of these process are given in Fig. 1 A. γγ collision The total cross section for the process γγ → ZZ has been calculated by using real photon spectrum produced by Compton backscattering of laser beam off the high energy electron beam. We show the total cross section of the process γγ → ZZ depending on the dimension-8 anomalous couplings f M 2 /Λ 4 , f M 3 /Λ 4 , f T 0 /Λ 4 and f T 9 /Λ 4 for √ s= 3 TeV in Fig.   2. In addition to these, the total cross sections as function of anomalous quartic couplings assuming Λ=1 TeV are given in Table I Similarly, sensitivities on f M 2 /Λ 4 and f T 0 /Λ 4 couplings are expected to be more restrictive The γ * γ * → ZZ is generated via the quasi-real photons emitted from both lepton beams collision with each other, and participates as a subprocess in the main process e − e + → e − γ * γ * e + → e − ZZe + . When calculating the total cross sections for this process, we take into account the equivalent photon approximation structure function using the improved Weizsaecker-Williams formula which is embedded in MadGraph. The total cross sections of the process as a function of f M 2 /Λ 4 , f M 3 /Λ 4 , f T 0 /Λ 4 and f T 9 /Λ 4 for √ s= 3 TeV are given in Fig. 3 and tabulated in Table I assuming Λ = 1 TeV.

C. γγ * collision
One of the operating mode of the conventional e + e − machine is the eγ mode. This mode includes γγ * collision of a Weisaczker-Willams photon (γ * ) emitted from the incoming leptons and the laser backscattered photon (γ). Thus, the reaction γγ * → ZZ participates as a subprocess in the main process e − γ → e − γ * γ → e − Z Z. In Fig. 4, we plot the total cross section of the process e − γ → e − γ * γ → e − Z Z as a function of dimension-8 couplings for √ s= 3 TeV. Also, the total cross sections as function of anomalous quartic couplings assuming Λ=1 TeV are given in Table I.

IV. BOUNDS ON ANOMALOUS QUARTIC COUPLINGS
The SM cross section of the processes γγ → ZZ, e − γ → e − γ * γ → e − Z Z and e + e − → e + γ * γ * e − → e + Z Z e − is quite small, because the process γγ → ZZ and the subprocesses γ * γ → ZZ and γ * γ * → ZZ are not allowed at the tree level. They are only allowed at loop level and can be neglected. On the other hand, as stated in Ref. [66], Here we consider that only one of the anomalous couplings changes at any time.
As can be seen from best sensitivities on f T 0 /Λ 4 and f T 9 /Λ 4 couplings in Fig. 5 are far beyond the sensitivities of the LHC. As can be seen from Table II, when the luminosity reduction factor is  TeV −2 , respectively. However, the best sensitivities on a 0 Λ 2 and ac Λ 2 couplings for L int = 590 fb −1 at √ s = 3 TeV at the CLIC are at the order of 10 −2 TeV −2 [14]. Also, Refs. [68,69] have  Table II, we show the best sensitivity bounds at 95% C. L. of  Table II, our best sensitivities on a 0,c Λ 2 couplings by examining the process γγ → ZZ are about 10 5 times better than the sensitivities calculated in Refs. [20,67]. Our bounds can set more stringent sensitivity by three orders of magnitude with respect to the best sensitivity derived from the CLIC with √ s = 3 TeV. Finally, we can understand from Table   IV that the best bounds obtained through the process γγ → ZZ with integrated luminosity 2000 fb −1 at √ s = 3 TeV improve the sensitivities of f M 2 Λ 4 and f M 3 Λ 4 couplings by up to a factor of 10 4 compared to Refs. [68,69]. However, we compare our results with the sensitivities of Ref. [68] which investigates phenomenologically f T 9 /Λ 4 coupling via pp → ZZ +2j → 4l+2j process at √ s= 14 (33) TeV with 300 (3000) fb −1 luminosity. The bound on f T 9 /Λ 4 coupling at √ s= 33 TeV with L int = 3000 fb −1 is [−2.50; 2.50] TeV −4 which is up to a factor of 10 3 worse than our best bound. However, it can be seen from Fig. 6 that bounds on f T 9 Λ 4 coupling obtained from the process e − γ → e − γ * γ → e − Z Z are more restrictive than the bounds on Λ 4 and f T 0 Λ 4 couplings. The best sensitivities obtained for four different couplings from the process γγ → ZZ in Fig. 5 are approximately an order of magnitude more restrictive with respect to the main process e + e − → e + γ * γ * e − → e + Z Z e − in Fig. 7 which is obtained by integrating the cross section for the subprocess γ * γ * → ZZ over the effective photon luminosity. Although the luminosity reduction factor is taken into account in γγ and eγ collision modes, the results show that γγ collisions give the best bounds to test anomalous quartic gauge couplings with respect to γ * γ * and γγ * collisions. Principally, the sensitivity of the processes to anomalous couplings rapidly increases with the center-of-mass energy and the luminosity.

V. CONCLUSIONS
CLIC is envisaged as a high energy e + e − collider having with very clean experimental conditions and being free from strong interactions with respect to the LHC. In addition, the number of SM events vanishes for γγ → ZZ, e − γ → e − γ * γ → e − Z Z and e + e − → e + γ * γ * e − → e + Z Z e − processes. Therefore, the observation of a few events at the final state of such processes would be an important sign for anomalous quartic couplings beyond the SM. For these reasons, we have estimated the improvement of sensitivity to anomalous quartic ZZγγ couplings with dimension-8 as function of collider energies and luminosities through the processes γγ → ZZ, e − γ → e − γ * γ → e − Z Z and e + e − → e + γ * γ * e − → e + Z Z e − . As a result, the CLIC as photon-photon collider provides an ideal platform to examine anomalous quartic ZZγγ gauge couplings at high energies and luminosities. 103.5 f 2