Probing anomalous quartic $\gamma\gamma\gamma\gamma$ couplings in light-by-light collisions at the CLIC

The anomalous quartic neutral couplings of the $\gamma\gamma\gamma\gamma$ vertex in a polarized light-by-light scattering of the Compton backscattered photons at the CLIC are examined. Both differential and total cross sections are calculated for $e^+e^-$ collision energies 1500 GeV and 3000 GeV. The helicity of the initial electron beams is taken to be $\pm\,0.8$. The unpolarized and SM cross sections for the same values of helicities are also estimated. The 95\% C.L. exclusion limits on two anomalous photon couplings $\zeta_1$ and $\zeta_2$ are calculated. The best bounds on these couplings are found to be $6.85 \times 10^{-16} \mathrm{\ GeV}^{-4}$ and $1.43 \times 10^{-15} \mathrm{\ GeV}^{-4}$, respectively. The results are compared with the exclusion bounds obtained previously for the LHC and HL-LHC. It is shown that the light-by-light scattering at the CLIC, especially the polarized, has a greater potential to search for the anomalous quartic neutral couplings of the $\gamma\gamma\gamma\gamma$ vertex.


Introduction
In the Standard Model (SM), the trilinear gauge couplings (TGCs) [1,2] and quartic gauge couplings (QGCs) [3]- [5] are completely defined by the non-Abelian SU(2) L × U(1) Y gauge symmetry. These couplings have been accurately tested by experiments. A possible deviation from the electroweak predictions can give us important information on probable physics beyond the SM.
Anomalous TGCs and QGCs can be studied in a model independent way in the framework of the effective field theory (EFT) via Lagrangian [6]- [8] L eff = L SM + L (6) + L (8) . (1) The Lagrangian L (6) contains dimension-6 operators. It generates an anomalous contribution to the TGCs and QGCs. Let us underline that the lowest dimension operators that modify the quartic gauge interactions without exhibiting two or three weak gauge boson vertices are dimension-8. The Lagrangian L (8) is a sum of dimension-8 genuine operators, where Λ is a mass-dimension scale associated with new physics, and c i are dimensionless constants. This Lagrangian induces anomalous deviation to the QGCs. It is assumed that the new interaction respects the local SU(2) L × U(1) Y symmetry which is broken spontaneously by the vacuum expectation value of the Higgs field Φ. CP invariance is also imposed. It means that L (8) is invariant under the full gauge symmetry. As a result, the electroweak gauge bosons can appear in the operators O (8) i only from covariant derivatives of the Higgs doublet D µ Φ or from the field strengths B µν , W a µν . There are three classes of dimension-8 operators. The first one contains just D µ Φ. It leads to non-standard quartic couplings of massive vector bosons, W + W − W + W − , W + W − ZZ and ZZZZ. The second class contains two D µ Φ and two field strength tensors. The third class has four field strength tensors only. The dimension-8 operators of the last two classes induce the anomalous quartic neutral couplings of the vertices γγγγ, γγγZ, γγZZ, γZZZ, and ZZZZ. A complete list of dimension-8 operators which lead to anomalous quartic neutral gauge boson couplings is presented in [9]- [11]. In particular, the effective Lagrangian of the operators O (8) i which contributes to the anomalous quartic couplings of the vertex γγγγ looks like see eq. (5) below. The explicit expression for dimension-8 Lagrangian in a broken phase (in which it is expressed in terms of the physical fields W ± , Z and F µν ) can be found, for instance, in [10]. We are interested in an effective Lagrangian for the anomalous γγγγ couplings. It is given by the formula [10] L γγγγ where The QGCs are actively studied for a long time. The anomalous W W ZZ vertex was probed at the LEP [12] (see also [13]) and Tevatron [14] colliders. The L3 Collaboration also searched for the W W ZZ couplings [15]. There have been investigations for the W W γγ couplings at the LHC in [16]- [23]. The possibility of measuring the ZZγγ couplings were studied in [16]- [19], [10] and [24]. Recently, the LHC experimental bounds on QGCs have been presented by the CMS [25] and ATLAS [26] Collaborations. In a number of theoretical papers, search limits for the anomalous vertex W W γγ at future electron-proton colliders have been estimated [27]- [29]. The anomalous QGCs can be also probed at linear e + e − colliders [30], in particular, in the eγ mode [31,32] (W W γγ, ZZγγ and W W Zγ vertices) or γγ mode [33] (W W W W , W W ZZ and ZZZZ vertices), [34] (W W γγ and ZZγγ vertices). Finally, in [35,36] the anomalous quartic couplings of the ZZγγ vertex at the Compact Liner Collider (CLIC) [37,38] have been examined. As one can see, in all these papers the anomalous QGCs with the massive gauge bosons were examined.
The great potential of the CLIC in probing new physics is well-known [39]- [41]. At the CLIC, it is possible to investigate not only e + e − scattering but also eγ and γγ collisions with real photons. In the present paper, we will examine the possibility of searching for anomalous γγγγ couplings in the light-by-light (LBL) scattering with ingoing Compton backscattered (CB) photons at the CLIC. Both unpolarized and polarized initial photons will be considered. The first evidence of the process γγ → γγ was observed by the ATLAS and CMS Collaborations in high-energy ultra-peripheral PbPb collisions [42,43]. The LBL collisions at the LHC have been studied in [44,45]. Recently, the LBL scattering at the CLIC induced by axion-like particles has been examined [46,47].

Light-by-light scattering in effective field theory
The e + e − colliders may operate in eγ and γγ modes [48]. Hard real photon beams at the CLIC can be generated by the laser Compton backscattering. When soft laser photons collide with electron beams, a large flux of photons, with a great amount of the parent electron energy, is produced. Let E 0 and λ 0 be the energy and helicity of the initial laser photon beam, while E e and λ e be the energy and helicity of the electron beam before CB. In our calculations, two sets of these helicities, with opposite sign of λ e , will be considered, namely 0 ) = (0.8, 1; 0.8, 1) , where the superscripts 1 and 2 enumerate the beams. The helicity of the photon with energy E γ obtained by the Compton backscattering of the laser photons with helicity λ 0 off the electron beam is given by the formula , m e being the electron mass.
The spectrum of the CB photons is defined by the helicities λ 0 , λ e and dimensionless variables x, r, ζ as follows where The maximum possible value of x is equal to The laser beam energy is chosen to maximize the backscattered photon energy E γ . This can be achieved if one puts ζ ≃ 4.8, then x max ≃ 0.83. γ γ γ γ Figure 1: The diphoton production in the collision of the backscattered photons at the CLIC via anomalous quartic coupling.
The LBL scattering of the CB photons happens as shown in Fig. 1. Its differential cross section is expressed in terms of the CB photon spectra, their helicities, and helicity amplitudes [49] where γ /E e are the energy fractions of the CB photon beams, , p ⊥ is the transverse momentum of the outgoing photons.
√ s is the center of mass energy of the e + e − collider, while √ sx 1 x 2 is the center of mass energy of the backscattered photons. We will apply the cut on the rapidity of the final state photons |η γγ | < 2.5.
The physical potential of linear e + e − colliders may be enhanced if the polarized beams are used [50,51]. As will be seen below, it is exactly so in our case. For comparison, similar results for unpolarized electron beams (λ (1,2) e = 0) will be also presented. Our calculations have shown that the total cross sections are almost indistinguishable from the SM ones for √ s = 380 GeV (the first energy stage of the CLIC). That is why, we will focus on the energies √ s = 1500 GeV (the second energy stage of the CLIC) and √ s = 3000 GeV (the third energy stage of the CLIC). The expected integrated luminosities for these baseline CLIC energy stages [51] are presented in Tab. 1. 8   2  1500  2500  2000  500  3  3000  5000  4000  1000   Table 1: The CLIC energy stages and integrated luminosities for the unpolarized and polarized initial electron beams.
We have calculated the differential cross sections dσ/dm γγ , where m γγ is the invariant mass of the outgoing photons. Each of the amplitudes is a sum of the anomaly and SM terms, As the SM background, we have taken into account both W -loop and fermionloop contributions The explicit analytical expressions for the SM helicity amplitudes in the right-hand side of eq. (13), both for the fermion and W -boson terms, are too long. That is why we do not present them here. They can be found in [46]. The differential cross sections as functions of the photon invariant mass m γγ are shown in Figs. 2 and 3. We imposed the cut on the rapidity of the outgoing photons, |η γγ | < 2.5. The left, middle and rights panels of these figures correspond to the electron beam helicities λ e = 0.8, λ e = −0.8, and λ e = 0, respectively. Note that the anomaly amplitude is pure real, while the SM one is mainly imaginary. As a result, the interference contribution to the differential cross section is relatively very small for any values of m γγ in the region m γγ > 200 GeV. If, for instance, √ s = 1500 GeV, λ e = 0.8, ζ 1 = 10 −13 GeV −4 , ζ 2 = 0, and m γγ = 500 GeV, the anomaly, SM, and interference terms of the cross section are equal to 3.45×10 −4 fb/GeV, 5.90×10 −3 fb/GeV, and 9.05×10 −5 fb/GeV, respectively. For √ s = 3000 GeV, the same values of λ e , ζ 1,2 , and m γγ = 1000 GeV we find, correspondingly, 1.05 × 10 −2 fb/GeV, 1.64 × 10 −3 fb/GeV, and 1.80 × 10 −4 fb/GeV. For both √ s, and any value of λ e , the anomaly differential cross sections become to dominate the SM background at about m γγ > 750 GeV for ζ 1 = 10 −13 GeV −4 , ζ 2 = 0. For the couplings ζ 1 = 0, ζ 2 = 10 −13 GeV −4 it takes place in the region m γγ > 960 GeV. For the same √ s and ζ 1,2 , the differential cross section with λ e = 0.8 becomes larger than the differential cross section with the opposite beam helicity λ e = −0.8 and unpolarized one, as m γγ grows. A possible background with fake photons from decays of π 0 , η, and η ′ is negligible in the signal region. The leading part of the anomalous cross section is proportional to s 2 . However, it does not mean that the unitarity is violated for the region of the anomalous QGCs considered in our paper. As it is shown in [52], the anomalous quartic couplings of the order of 10 −13 GeV −4 do not lead to unitarity violation for the collision energy below 3 TeV.
The results of our calculations of the total cross sections σ(m γγ > m γγ,min ), where m γγ,min is the minimal invariant mass of the outgoing photons, are  Anomaly; 0, GeV SM s=3000 GeV, Unpolarized Figure 3: The same as in Fig. 2, but for the e + e − collider energy √ s = 3000 GeV.
is several times large than the total cross section with the opposite beam helicity. Note, however, that for λ e = 0.8 the CLIC expected integrated luminosities are four times smaller than those for λ e = −0.8, for both values of e + e − collision energy, see Tab. 1.
To calculate the exclusion region, we use the following formula for the exclusion significance [53] Here s and b represent the total number of signal and background events, respectively, and δ is the percentage systematic error. In the limit δ → 0 expression (16) is simplified to be We define the regions S excl 1.645 as the regions that can be excluded at the 95% C.L.
For the polarized LBL scattering at the CLIC, the exclusion bounds on the anomalous photon couplings are presented in Tabs. 2, 3 using the cut GeV.
m γγ > 1000 GeV. Note that the values of the expected integrated luminosities depend on the energy √ s. As one can see from these tables, for both energies the exclusion bounds on couplings ζ 1 and ζ 2 weakly depend on the helicity of the initial electron beams. Previously, the discovery potential for the LBL scattering at the 14 TeV LHC has been estimated in [54]- [56]. As was shown in [55], the 14 TeV LHC 95% C.L. exclusion limits on ζ 1 and ζ 2 couplings are 1.5 × 10 −14 GeV −4 and 3.0 × 10 −14 GeV −4 , respectively, for L = 300 fb −1 integrated luminosity. For L = 3000 fb −1 (HL-LHC), the values are twice smaller, 7.0 × 10 −15 GeV −4 and 1.5 × 10 −14 GeV −4 . The sensitivity in the (ζ 1 , ζ 2 ) plane is shown in Fig. 8 taken from [56]. As one can see from Tab. 2, our CLIC bounds on the couplings ζ 1 , ζ 2 for the LBL scattering with √ s = 1500 GeV are comparable with the HL-LHC bounds [56]. However, for √ s = 3000 GeV our lower bounds on ζ 1 , ζ 2 are approximately one order of magnitude smaller than the HL-LHC ones, see Tab. 3.

Conclusions
In the present paper, we have examined the anomalous quartic neutral couplings of the γγγγ vertex in the polarized light-by-light collisions of the Compton backscattered photons at the CLIC. Both the second and third stages of the CLIC are considered with the collision energies √ s = 1500 GeV and √ s = 3000 GeV, respectively. The helicity of the initial electron beam  Figure 7: The same as in Fig. 6, but for √ s = 3000 GeV and L = 5000 fb −1 .
was taken to be λ e = ± 0.8. The unpolarized case (λ e = 0) has been also considered. We used the SU(2) L × U(1) Y effective Lagrangian describing the contribution to the anomalous quartic neutral gauge boson couplings. Its part, relevant to the anomalous γγγγ vertex (4), expressed in terms of the physical fields, contains two couplings ζ 1 , ζ 2 of dimension −4.
We have calculated both the differential and total cross sections of the light-by-light scattering γγ → γγ, with the cut imposed on the rapidity of the final photons, |η γγ | < 2.5. The plots for two values of the collision energy √ s and three values of the electron beam helicity λ e (including the unpolarized case) are presented. The anomaly and SM contributions to the cross sections are presented separately. The CLIC exclusion sensitivity bounds on the anomaly coupling constants ζ 1 and ζ 2 , coming from the process γγ → γγ, are calculated for three values of the systematic error, δ = 0%, δ = 5%, and δ = 10%. To reduce the SM background, we imposed the cut on the invariant mass of the outgoing photons, m γγ > 1000 GeV.
For the unpolarized LBL scattering at the CLIC, the 95% C.L. exclusion regions are shown in Figs. 6   0.8, and δ = 10%, our bounds on ζ 1 , ζ 2 have appeared to be approximately one order of magnitude stronger than the corresponding HL-LHC bounds obtained for √ s = 14 TeV and integrated luminosity L = 3000 fb −1 in [56].
All said above allows us to conclude that the LBL scattering at the CLIC, especially the polarized, has a great physical potential in searching for the anomalous quartic neutral couplings of the γγγγ vertex. Figure 8: The LHC sensitivity in the (ζ 1 , ζ 2 ) plane. In particular, the red region can be probed at the 95% C.L. using proton tagging at the LHC. The white region is inaccessible. The figure is taken from Ref. [56].