Probing anomalous $\gamma\gamma\gamma Z$ couplings through $\gamma Z$ production in $\gamma\gamma$ collisions at the CLIC

We have estimated the sensitivity to the anomalous couplings of the $\gamma\gamma\gamma Z$ vertex in the $\gamma\gamma\rightarrow\gamma Z$ scattering of the Compton backscattered photons at the CLIC. Both polarized and unpolarized collisions at the $e^+e^-$ energies 1500 GeV and 3000 GeV are addressed, and anomalous contributions to helicity amplitudes are derived. The differential and total cross sections are calculated. We have obtained 95\% C.L. exclusion limits on the anomalous quartic gauge couplings (QGCs). They are compared with corresponding bounds derived for the $\gamma\gamma\gamma Z$ couplings via $\gamma Z$ production at the LHC. The constraints on the anomalous QGCs are one to two orders of magnitude more stringent that at the HL-LHC. The partial-wave unitarity constraints on the anomalous couplings are examined. It is shown that the unitarity is not violated in the region of the anomalous QGCs studied in the paper.


Introduction
In our previous paper [1] we probed the anomalous quartic gauge couplings (QGCs) in the γγ → γγ process at the Compact Linear Collider (CLIC) [2,3].Both the unpolarized and polarized light-by-light scatterings were considered, and the bounds on QGCs were obtained.The neutral anomalous quartic couplings are of particular interest.The anomaly interactions γZZZ, γγZZ, and γγγZ at the LHC were analyzed in [4]- [9].The LHC experimental bounds on QGCs were presented by the CMS [10] and ATLAS [11] Collaborations (see also [14]).The bounds on the anomalous γγγZ vertex can be also derived from the constraints on the B(Z → γγγ) branching ratio obtained at the LEP [12] and LHC [13].As for e + e − colliders, they may operate in eγ and γγ modes [15].The bounds on QGCs in e + e − , eγ and γγ collisions were given in [16]- [20].In particular, the limits on the quartic couplings for the vertex γγγZ were derived in [21] using LEP 2 data for the reactions e + e − → γγγ, γγZ.A similar analysis for the exclusive γZ production with intact protons at the LHC was done in [22].The search for virtual SUSY effects in the process γγ → γZ at high energies was presented in [23].
As one can see, the anomalous γγγZ vertex urgently needs to be examined in high energy e + e − collisions.That is why, in the present paper we study the process (see Fig. 1) where p 1 , p 2 , p 3 , p 4 are boson momenta, µ, ν, ρ, α are boson Lorentz indices, and ingoing particles are real polarized photons generated at the CLIC by the laser Compton backscattering [24]- [26].Our main goal is to derive bounds on γ Z γ γ anomaly couplings for the vertex γγγZ which can be reached at the CLIC using both polarized and unpolarized photon beams.The great potential of the CLIC in probing new physics is well-known [27]- [29].Let us underline that a physical potential of a linear high energy e + e − collider may be significantly enhanced, provided the polarized beams are used [30,31].
Let λ e be the helicity of the initial electron beam, while λ 0 be the helicity of the ingoing laser photon beam.In our calculations, we will consider two sets of these helicities, with opposite sign of λ e , (λ (1)  e , λ 0 ; λ (2)  e , λ 0 ) = (0.8, 1; 0.8, 1) , (λ (1)  e , λ 0 ; λ (2)  e , λ where the superscripts 1 and 2 enumerate the beams.We will work in the effective field theory framework.Previously effective Lagrangians were used in [32]- [35] for examining the γγγZ interaction in the Z → γγγ decay, as well as in [36], [21], and [22].Anomalous quartic gauge couplings (QGCs) are induced at the dimension-six level already.However, they are not independent of anomalous trilinear gauge couplings.That is why, in our paper, we study anomalous QGCs which enter the effective Lagrangian at dimension-eight without contributing to anomalous trilinear gauge interactions.The paper is organized as follows.In the next section, the effective Lagrangian is described, and Feynman rules for the anomalous γγγZ vertex are presented.The helicity amplitudes are studied in Sec. 3. In Sec. 4, both differential and total cross sections for the process (1) are calculated, and bounds on the QGCs are given.In Sec. 5, unitarity constraints on anomalous quartic couplings are obtained.In Appendix A, polarization tensors for the vertex γγγZ are listed.The explicit expressions for the anomalous contributions to the helicity amplitudes are given in Appendix B. Some formulas for Wigner's d-function are collected in Appendix C. Finally, in Sec.6, we summarize our results and give conclusions.

Effective Lagrangian
It is appropriate to describe the anomalous γγγZ interaction by means of an effective Lagrangian.Given parity is conserved and gauge invariance is valid, there are only two independent operators with dimension 8. Following [32,33], we take the Lagrangian with the operators where with the operators where with the operators where Fµν = (1/2) ε µνρσ F ̺σ , and Zµν = (1/2) ε µνρσ Z ̺σ .Using integration by parts and equations of motion, one can easily obtain the following relations between two bases for the effective Lagrangian [35], We also have the relations [22] The above listed equations enable us to relate anomalous coupling in eqs.(3), (5), and (7).In particular, we find The Feynman rules for the effective anomalous vertex, resulting from the Lagrangian (3), are given by [33] where P denotes possible permutations (p 1 , µ) ↔ (p 2 , ν) ↔ (p 3 , ρ), and all momenta in the γγγZ vertex are assumed to be incoming ones.Correspondingly, the polarization tensor is equal to Electromagnetic gauge invariance results in equations p µ 1 P µνρα = p ν 2 P µνρα = p ρ 3 P µνρα = 0. Note that terms proportional to p µ 1 , p ν 2 , p ρ 3 are omitted in (12), since they do not contribute to the matrix element, see eq. ( 18) below.Explicit expressions for the tensors P (1,i) µνρα and P (2,i) µνρα are presented in Appendix A. To calculate helicity amplitudes for the process (1), one has to make the replacement p 3 → −p 3 in the Feynman rules for the γγγZ vertex given by eqs.(12), (13), and (A.1)-(A.11).

Helicity amplitudes
We work in the c.m.s. of the colliding real photons, p 1 + p 2 = 0, where the momenta are given by Here E = k 2 + m 2 Z , with m Z being the mass of the Z boson.The Mandelstam variables of the process (1) are where θ is a scattering angle in the c.m.s.Note that s + t + u = m 2 Z .In the chosen system the polarization vectors are equal to They obey the orthogonality condition ε λ µ (k)k µ = 0. Correspondingly, we get the helicity vectors of the final photon and Z boson, The matrix element of the process (1) with the definite helicities of the incoming and outgoing bosons can be written as where the polarization tensor P µνρα is given by eq. ( 13).We have calculated the anomalous helicity amplitudes, and present their explicit expressions in Appendix B. Using these expressions, we obtain the unpolarized amplitude squared where With a help of relations (11), we get from (19) the differential cross section where β = 1 − m 2 Z /s, in a full agreement with eq. ( 2.3) in [22].To estimate a SM background, we take analytical expressions for the SM helicity amplitudes from Appendix A in [23].Both W boson loops [41,42] and charged fermion loops [41,43] contribute to these amplitudes.As shown in [23], for s > (250 GeV) 2 the dominant SM amplitudes A λ 1 λ 2 λ 3 λ 4 are the Wloop non-flip amplitudes A W ++++ (s, t, u) and A W +−+− (s, t, u) = A W +−−+ (s, u, t).Almost negligible are A W +++0 (s, t, u) and A W +−+0 (s, t, u) = A W +−−0 (s, u, t).The rest are even smaller.The fermion-loop amplitudes are comparable only to very small W -loop amplitudes [23].Similar properties of the SM helicity amplitudes are also valid for the process γγ → γγ [44].
Another possible background comes from the SM process γγ → γl + l − where the invariant mass of the lepton pair, m l + l − , is close to the Z boson mass m Z .We have obtained the cross section of the process to be of order 10 −3 fb for |m l + l − −m Z | < 10 GeV.So, this background can be safely ignored.

Numerical results
The differential cross section of the process γγ → γZ depends on spectra of the Compton backscattered (CB) photons f γ/e (x i ), their helicities ξ(E , and helicity amplitudes [1,19], 0 0 where γ /E e and x 2 = E γ /E e are the energy fractions of the CB photon beams, , and p ⊥ is the transverse momentum of the outgoing particles.Note that is the invariant energy of the backscattered photons.The explicit expressions for f γ/e (x i ) and ξ(E γ , λ 0 ) can be found in [1].The differential cross sections are shown in Figs. 2, 3 as functions of the invariant mass of the γZ system.We have imposed the cut on the rapidity of the final bosons, |η| < 2.5, and considered the region m γZ > 250 GeV.As one can see, the anomalous cross sections dominate the SM one for m γZ > 600 GeV.The effect is more pronounced for the collision energy √ s = 3000 GeV, especially as m γZ grows.Note that for √ s = 3000 GeV the differential cross sections depend weakly on electron beam helicity λ e .In Figs. 4, 5 the total cross sections are presented depending on m γZ,min , minimal invariant mass of two outgoing bosons.The anomalous contribution dominates both the interference one and SM cross section.The ratio of the total cross section to the SM one grows with an increase of m γZ , being more than one order of magnitude at large m γZ .The knowledge of the total cross sections and planned CLIC integrated luminosities [31] enables us to calculate the exclusion regions for the QGCs.In our study we consider leptonic (electrons and muons) decays of the Z boson.Let s(b) be the total number of signal (background) events, and δ the percentage systematic error.The number of events is defined as σ × L ×   GeV.The left, middle and right panels correspond to the electron beam helicities λ e = 0.8, −0.8, and 0, respectively.On each plot the curves denote (from the top downwards) the differential cross sections for the couplings g 1 = 10 −12 GeV −4 , g 2 = 0, and g 1 = 0, g 2 = 10 −12 GeV −4 , the anomalous contributions for the same values of couplings, the SM cross section.
B(Z → e, µ).The exclusion significance is given by [45] S We define the regions S excl 1.645 as a regions that can be excluded at the 95% C.L. in the process γγ → γZ at the CLIC.To reduce the SM background, we impose the cut m γZ > 1000 GeV, in addition to the bound |η| < 2.5.The expected integrated luminosity at the CLIC can be found, for   instance, in [31].
It is worth considering the unpolarized case first.One can obtain from eq. ( 19) that the anomalous contribution to the unpolarized total cross section is proportional to the coupling combination 3g 2  1 − 4g 1 g 2 + 4g 2 2 , provided terms proportional to m 2 Z /s ≪ 1 are neglected in it.In such a case, the exclusion regions are ellipses in the plane (g 1 − g 2 ) rotated clockwise through the angle 0.5 arctan 8 ≃ 41.4 • around the origin.It is clear that our process is slightly more sensitive to the coupling g 2 rather than to g 1 .Our 95% C.L. exclusion regions for anomalous QGCs for the unpolarized process γγ → γZ at the CLIC are shown in Figs. 6, 7. The results are presented for δ = 0, δ = 5%, and δ = 10%.
In Tabs. 1, 2 we show the exclusion bounds on the couplings g 1 and g 2 for three values of the electron beam helicity λ e and corresponding integrated luminosity L. Let us underline that this time we did not neglect the terms proportional to m 2 Z , both for unpolarized and for polarized reactions.As   Recently, the bounds on the anomalous quartic couplings for the vertex γγγZ were obtained via γZ production with intact protons in the forward region at the LHC [22].To examine this process, the effective Lagrangian (7) was used with the anomalous couplings ζ, ζ.Both for integrated luminosity 300 fb −1 and high luminosity 3000 fb −1 sensitivities were found to be similar, ζ, ζ ∼ 1 × 10 −13 at the 95% C.L. Taking into account the relations between couplings ζ, ζ and our couplings g 1 , g 2 , (11), we expect that the sensitivities of g 1 , g 2 ∼ 8 × 10 −13 can be reached at the LHC (HL-LHC).These values   should be compared with the CLIC bounds in Tabs. 1 and 2. Note that the expected sensitivity from the Z → γγγ decay search at the LHC [13] is approximately three orders of magnitude smaller than that obtained in [22].
Partial-wave unitarity in the limit s ≫ (m 1 + m 2 ) 2 requires that Using orthogonality of the d-functions (C.2), we find the partial-wave amplitude Here and in what follows, z = cos θ.Note that

Unitarity bounds on coupling g 1 (g 2 = 0)
To obtain a unitarity bound on the coupling g 1 , we put g 2 = 0. Let us note that due to eq. ( 26), it is sufficient to examine the helicity amplitudes with ++++ , M +−−+ , and M (1) ++−− , since the rest are suppressed by small factor m Z / √ s or zero.
1. λ 1 = λ 2 = λ 3 = λ 4 = 1, then λ = µ = 0.The helicity amplitude is given by the first of equations (B.4) As a result, we get , and we find from (B.4) We follow the derivation of eq. ( 35) and come to the inequality Table 3: Unitarity constraints on the anomalous couplings when just one coupling is non-zero (second and third columns), and when both couplings are non-vanishing (fourth and fifth columns for the couplings of the same sign, sixth and seventh columns for the couplings of opposite signs).The numerical values of the bounds are given for the collision energy √ s = 1500(3000) GeV.

Conclusions
In the present paper, the CLIC discovery potential for exclusive γZ production in the scattering of the Compton backscattered photons at the e + e − collision energies 1500 GeV and 3000 GeV is studied.We have shown that such a process provides an opportunity of searching for the anomalous quartic neutral gauge couplings for the γγγZ vertex at the CLIC.Both unpolarized and polarized initial electron beams are examined.To describe the anomalous quartic gauge couplings we used the effective Lagrangian which conserves gauge invariance.Although quartic gauge couplings are already induced at the dimension-six level, we considered the effective Lagrangian with CP conserving dimension-eight operators without contributing to anomalous trilinear gauge interactions.
We have derived the explicit expressions for the anomalous contributions to the helicity amplitudes of the process γγ → γZ.After that the differential and total cross sections are calculated depending on m Zγ , the invariant mass of the γZ system.It is shown that the anomalous contribution dominates both the interference and SM cross sections.Moreover, the ratio of the total cross section to the SM one grows with the increase of m Zγ , being more approximately one order of magnitude at large m γZ .
It enabled us to obtain the exclusion regions for the anomalous couplings with the systematic errors of 0%, 5%, and 10%.We have considered the Z boson decay into leptons (electron and muons).For both couplings, g 1,2 , the best bounds are equal to approximately 4.4 × 10 −14 GeV −4 and 5.1 × 10 −15 GeV −4 , for the e + e − energies 1500 GeV and 3000 GeV, respectively.They are achieved when electron beam helicity is equal to 0.8.We have checked that the unitarity is not violated in the region of the couplings considered in the paper.Our best bound on the anomalous couplings for the collision energy 3000 GeV is roughly two orders of magnitude stronger than the limits which can be reached at the LHC and HL-LHC.This points to a great potential of the CLIC and other future leptonic colliders to probe the anomalous γγγZ couplings.

Figure 2 :
Figure 2: The differential cross sections for the process γγ → γZ as functions of the invariant mass of the outgoing bosons for the CLIC energy √ s = 1500

Figure 4 :
Figure4: The total cross sections for the process γγ → γZ as functions of the minimal invariant mass of the outgoing bosons for the e + e − collider energy √ s = 1500 GeV.The left, middle and right panels correspond to the electron beam helicities λ e = 0.8, −0.8, and 0, respectively.On each plot the curves denote (from the top downwards) the total cross sections for the couplings g 1 = 10 −12 GeV −4 , g 2 = 0, and g 1 = 0, g 2 = 10 −12 GeV −4 , the anomalous contributions for the same values of couplings, the SM cross section.

Table 1 :
The 95% C.L. exclusion limits on the anomalous quartic couplings g 1 and g 2 for the collision energy √ s = 1500 GeV, and the cut m γZ > 1000

Table 2 :
The same as in Tab. 1, but for the energy √ s = 3000 GeV and different values of the integrated luminosities.