A search for axion-like particles in light-by-light scattering at the CLIC

The virtual production of axion-like particles (ALPs) in the light-by-light scattering at the CLIC collider is studied. Both differential and total cross sections are calculated, assuming interaction of the ALP with photons via CP-odd term in the Lagrangian. The 95\% C.L. exclusion regions for the ALP mass and its coupling constant are given. By comparing our results with existing collider bounds, we see that the ALP search at the CLIC has a great physics potential of searching for the ALPs, especially, in the mass region 1 TeV -- 2.4 TeV, with the collision energy $\sqrt{s} = 3000$ GeV and integrated luminosity $L = 5000$ fb$^{-1}$ for the Compton backscattered initial photons. In particular, our limits are stronger that recently obtained bounds for the ALP production in the light-by-light scattering at the LHC.


I. INTRODUCTION
The notion of the QCD axion is closely related to the strong CP problem, which means the absence of the CP violation in the strong interactions. In its turn, the CP problem arises as a possible solution to the U(1) problem. The QCD Lagrangian in the limit of vanishing masses of u and d quarks has a global symmetry U(2) V × U(2) A = SU(2) I × U(1) Y × SU(2) A × U(1) A . The non-zero quark condensates ūu and d d break down the axial symmetry SU(2) A × U(1) A spontaneously. As a result, four Nambu-Goldstone bosons should appear. But besides light pions, no another light state is present in the hadronic spectrum, since m η ≫ m π . It is called the U(1) problem [1].
The U(1) A symmetry is connected with a transformation of the fermion fields ψ → e iαγ 5 ψ, ψ →ψe iαγ 5 . One possible resolution of the U(1) problem is provided by the Adler-Bell-Jackiw chiral anomaly for the axial current J 5 µ =ψγ µ γ 5 ψ [2]. In the limit of vanishing values of quark masses, it gives where G aµν is a gluon tensor,G aµν = (1/2)ε µνρσ G ρσ a its dual, and N f is a flavor number. However, the problem is not so simple, since the term G aµνG µν a is a total divergency, where A aµ is a gluon field, f abc is a structure constant of the QCD group. A new currentJ 5 µ = J 5 µ −K µ can be introduced which is conserved in the limit m q → 0, but is not gauge-invariant. The chiral anomaly introduce a pure surface integral to the QCD action S QCD where α is a parameter of the chiral transformations. If A aµ = 0 at spatial infinity, then ds µ K µ = 0, ∆S = 0, and U(1) A is an unbroken global symmetry. However, A aµ = 0 can be a pure gage at spatial infinity, [3] A µ = A aµ t a . For such a configuration, ds µ K µ = 0, and consequently U(1) A is not a symmetry of the strong interactions. In the SU(2) QCD ω are classified by the integer n, ω n → e i2πn as r → ∞, and condition (4) is a map of three-dimensional sphere S 3 ∞ on the sphere S 3 1 of the SU(2) group. The winding number n is given by Note that K µ = 4 3g 2 ε µνρσ tr ω∂ ν ω −1 ω∂ ρ ω −1 ω∂ σ ω −1 .
There is an infinite number of the vacuum states characterized by the topological index n.
The condition gA µ r→∞ → −i ∂ µ ωω −1 is a definition of a classical vacuum of the gauges fields. The topological index ν of the instanton solution [4] is equal to the difference of the topological indices of the vacua defined in (5), It means that the instantons realize vacuum-to-vacuum transition. The true or θ-vacuum becomes a superposition of the vacua |n |θ = n e −inθ |n .
As a result, an effective QCD action acquires the θ-term Moreover, an account of the weak interactions adds the following term to the QCD La- and M is the quark mass matrix.θ is invariant under chiral transformation and thus observable. The extra term in (10) breaks P-ant T-invariance but conserves C-invariance, so CP-invariance is violated. Thus, it contributes to the neutron electric dipole moment d n .
The smallness of the angleθ is known as strong CP problem.
One possible solution to this problem is a spontaneously broken CP. However, we know that experimental data are in excellent agreement with the CKM-model in which the CP is explicitly broken. The elegant solution of the CP mystery of the SM is provided by the Peccei-Quinn (PQ) mechanism with a new, spontaneously broken approximate global U(1) PQ symmetry [6]. As it is shown in [7,8] it leads to a light neutral pseudoscalar particle, the axion a, which is the Nambu-Goldstone boson of the broken U(1) PQ symmetry. The idea is to replace the CP-violating termθ by the CP-conserving axion. Namely, the axion field can de redefined to absorb the parameterθ. In fact, the axion replaces the QCD theta parameter by a dynamical quantity, thereby explaining of non-observation of the strong CP violation. Thus, the PQ mechanism is a compelling solution to the strong CP problem.
In the PQWW scheme [6]- [8] an extra Higgs doublet is used, and the axion mass is related to the electroweak symmetry breaking scale. There are two models in which the PQ symmetry is decoupled from the electroweak (EW) scale and is spontaneously broken. It results in axions with extremely weak couplings ("invisible" axion). One of the models is the KSVZ model [9]- [10] with one Higgs doublet in which the axion is introduced as the phase of an EW singlet scalar field. This scalar is coupled to an additional heavy quark, and its coupling is induced by the interaction of the heavy quarks with other fields. In the DFSZ model [11]- [12] two Higgs doublets are used, as well as an additional EW singlet scalar. The latter is coupled to the SM fields through its interaction with the the Higgs doublets.
The axion also appears in the context of the string theory [13]- [15]. In the string theory spin-zero particles must couple to a photon field, since all couplings are defined by the expectation value of scalar fields. This implies the existence of the P-odd term in the Lagrangian proportional to where F µν is the electromagnetic tensor,F µν = (1/2)ε µνρσ F ρσ its dual, and a is the QCD axion or axion-like particle (ALP) [16]. APLs can also appear in theories with spontaneously broken symmetries [17]- [18] or in GUT [19]. Lately, a number of new theoretical schemes with the axion as a basic quantity was developed [20]- [27]. For a review on the axions and APLs, see [28]- [31] and references therein.
Both theory and phenomenology of the axions were also studied in large [32]- [35] and warped [36]- [38] extra dimensions (EDs). In an ED framework, the mass of the axion becomes independent of the scale associated with the breaking of the PQ symmetry. It means that the axion mass can be treated independently of its couplings to the SM fields.
The very low mass and small coupling axion and/or ALP are a leading dark matter (DM) candidate, since their properties, allow them to be stable and difficult-to-detect. Both axions and ALPs can be produced in the early Universe and therefore constitute the most of the cold DM in the Universe [39]- [40] (see also recent papers [41]- [46]). The relevance of the QCD axion and, more generally, of ALPs in astrophysics and cosmology is of particular interest [47]- [52]. Many axion DM experiments are in progress [53]- [60] (see also [61]).
The axion phenomenology involves such phenomena as stellar evolution, axion mediated forces, dark matter detection, axion decays, axion-photon conversion, so-called "light shining trough the wall", etc.
There is a broad experimental program aiming to search for the QCD axion via its coupling to the SM. On the other hand, many ALP searches assume their strong couplings to the electromagnetic term F µνF µν as in eq. (12). In terrestrial experiments, bounds on very low mass axions and small mass axions were obtained [62]- [67]. The coupling of the ALPs to other gauge bosons are also studied (see for instance, [68]). Note that the ALPs are not directly relevant for the QCD axion. Therefore, heavy APLs can be detected at colliders, in particular, in a light-by-light scattering [69]- [72]. As it was shown in [73], searches at the LHC with the use of the proton tagging technique can constrain the ALP masses in the region 0.5 TeV-2 TeV.
Compact Linear Collider (CLIC) is the linear collider that is planned to accelerate and collide electrons and positrons at maximally 3 TeV center-of-mass energy [74]. In CLIC, it is possible to obtain accelerating gradients of 100 MV/m. Three energy states are considered to operate CLIC at maximum efficiency [75]. The √ s = 380 GeV is the first one and it is possible to reach the integrated luminosity L = 1000 fb −1 . This energy stage cover Higgs boson, top quark, and gauge sectors. It is planned to examine such SM particles with high precision [76]. The second one has √ s = 1500 GeV center-of-mass energy and 2500 fb −1 integrated luminosity. At this stage, it is enable to investigate beyond the SM physics. Also, a detailed analysis of the Higgs boson can be made such as the Higgs self-coupling and the top-Yukawa coupling and rare Higgs decay channels. [77]. The third stage of the CLIC has a maximum center-of-mass energy value √ s = 3000 GeV and integrated luminosity value L = 5000 fb −1 . At this stage, the most precise examinations of the SM is possible. Moreover, it is enabled to discovery beyond the SM heavy particles of mass greater than 1500 GeV [76]. The new physics search potential of the CLIC is presented in [78].
At the CLIC it is also possible to study photon-induced processes in γγ and eγ collisions.
In this type of processes, the photons are emitted from the incoming electron beams. The photons scatter at tiny angels from the beam pipe. Hence, they have very low virtuality, that is why these photons are called "almost-real".
The first evidence of the subprocess γγ → γγ was observed by the ATLAS collaboration in high-energy ultra-peripheral PbPb collisions [79]. The same process was also reported by the CMS Collaboration [80]. Recently, the ATLAS collaboration have published the evidence of the light-by-light scattering with the certainty of 8,2 sigma [81]. The analysis of the exclusive and diffractive γγ production in PbPb collisions was done in [82]. We have examined a possibility to constrain the parameters of the model with a warp ED in the photon-induced process pp → pγγp → p ′ γγp ′ at the LHC [83]. Previously, the photoninduced processes in EDs were studied in [84]- [85].
In the present paper, we propose to search for the ALP a in the exclusive light-by-light scattering at the lepton collider CLIC.
In the next section differential and total cross sections are calculated as functions of the ALP mass m a and its coupling f . It enables us to estimate the CLIC exclusion regions for both types on the initial photons.

II. LIGHT-BY-LIGHT VIRTUAL PRODUCTION OF ALP
The ALP couples to the SM photons via were f Note that, in contrast to the true QCD axion, the mass and couplings of the ALP are independent parameters. In what follows, we assume that only the CP-odd interaction term is realized in (13) with f (−) a = f . As for possible contribution from the CP-even term in (13), it is discussed in the section Conclusions.
The cross section of the diphoton production at the CLIC can be found as the integration where f γ/e (y) is the photon spectrum, and dσ(γγ → γγ) is the unpolarized differential cross section of the subprocess γγ → γγ. The explicit expressions for the photon spectrum are given below. The differential cross section is the following sum of helicity amplitudes squared Here and below s, t and u are the Mandelstam variables of the diphoton system. Each of the helicity amplitudes is a sum of the ALP and SM terms, The explicit expressions of the pure ALP amplitudes can be found in [71]. In particular, where Γ a is the total width of the ALP, and is its decay width into two photons. Correspondingly, we have [71] ReM (a) ReM (a) ReM (a) An account of the ALP width Γ a is mainly important in a vicinity of the point s ∼ m 2 a . That is why, it is omitted in the deniminators of the last two terms in the first row of eq. (22).
The SM (electroweak) amplitude is a sum of the fermion and W boson one-loop ampli- They amplitudes M f ++++ (s, t, u) and M W ++++ (s, t, u) are calculated in [86]- [87] (see also [85]) where e f is the fermion electric charge in units of the proton charge.

A. Compton backscattered photons
In addition to e + e − collisions, eγ and γγ interactions with real photons can be examined at the CLIC. For this process, real photons could be constructed by the Compton backscattering of laser photons off linear electron beam. Most of these real photons have high energy. The Compton backscattered (CB) photons give a spectrum which is defined as follows [88]- [89] f γ/e (x) = 1 g(ζ) 1 where and Note that x max = ζ/(1 + ζ). Here E γ is the energy of the backscattered photon, E 0 and E e are energies of the incoming laser photon and electron, respectively. x max reaches 0.83 when ζ = 4.8.
We start from the case when the initial photons in the subprocess γγ → γγ are the CB photons, whose spectrum is given by formulas (34)  The differential cross sections for the process γγ → γγ for the CB initial photons is shown in Fig. 2 as functions of the transverse momenta of the final photons p t . The ALP mass m a and its coupling f are chosen to be equal to 1200 GeV and 10 TeV, respectively. In order to reduce the SM background, we have imposed the cut W = m γγ > 200 GeV. The curves are presented for two values of the ALP branching into two photons Br = Br(a → γγ).
For this differential cross sections, the virtual production of the ALP dominates the SM light-by-light subprocess for p t > 100 GeV region. The total cross sections σ(p t > p t,min ) as functions of the minimal transverse momenta of the final photons p t,min are shown in Fig. 3.
It can be seen from this figure that the deviation from the SM gets higher as the p t -cut increases. Moreover, while the SM value decreases until the value of p t,min = 500 GeV, the total cross section value is almost unchanged.   6. In these figures we have applied the cut p t > 500 GeV in order to suppress SM cross sections relative to total cross sections as we analyzed from the Fig. 3. In this analysis, we have used the following statistical significance (SS) formula [90], Here S and B are the numbers of the signal and background events, respectively. It can be It is assumed that the uncertainty of the background is negligible. Our obtained exclusion regions should be compared with the current exclusion regions on the ALP coupling and ALP mass presented in Fig. 7, especially with that obtained for the process pp → p(γγ → γγ)p at the LHC [71]. This comparison demonstrates the great potential of the light-by-light scattering at the CLIC. The estimation for the 95% C.L.
This figure shows the exclusion f −1 < 5.5 × 10 −2 TeV −1 for the ALP mass interval 10 GeV-800 GeV, while the light-by-light scattering at the LHC gives the bound f −1 < 4 × 10 −1 TeV −1 for the same mass interval. Moreover, we have obtained the very strong upper bound on f −1 which is of order of 10 −4 TeV −1 for the mass range m a = 1000 − 1200 GeV. The best limit for the pp → p(γγ → γγ)p is of the order of 10 −2 TeV −1 for the mass range m a = 600 − 800 GeV, as seen from Fig. 7. Fig. 6 is the same as Fig. 5, but for √ s = 3000 GeV and L = 5000 fb −1 . It demonstrates the wider exclusion regions. In particular, it shows the exclusion f −1 < 3 × 10 −2 TeV −1 for the ALP mass interval 10 GeV-800 GeV.
The stronger bounds on f −1 have been obtained which are of the order of 10 −4 TeV −1 for the mass range m a = 1000 − 2400 GeV and Br(a → γγ) = 1.
In the WWA, the photons have the following spectrum Here m e shows the electron mass, Q 2 = −q 2 is the photon virtuality, x = E γ /E e is the ratio of the photon energy to the energy of the incoming electron, and α is the fine structure constant.
In our case, X = γγ (see Fig. 8), and the WWA spectrum of the photons is given by formula (38). In addition to the backgrounds mentioned in subsection A, possible backgrounds also came from γγ → e + e − γγ and ZZ-induced processes. The first one was estimated in [81] to be below 1%. The second background may not be taken into account since the ZZ luminosity is approximately 100 times smaller than the γγ luminosity [104].
The results of our calculations of the differential and total cross sections are presented in Figs GeV-3000 GeV for both values of the collision energy √ s.  Fig. 13 shows the 95% C.L. exclusion region in the (m a − f −1 ) plane for √ s = 3000 GeV and L = 5000 fb −1 . In the mass region 10 GeV-1000 GeV the bounds on f −1 are of the order of 10 −1 TeV −1 . In the mass range 1000 GeV-1500 GeV, these bounds reach the value 1 × 10 −3 TeV −1 . For both √ s, the bounds are much weaker than those for the CB initial photons. one in (13), with the same coupling f −1 .
By comparing our exclusion regions with other collider exclusion regions, we may conclude that the ALP search at the CLIC has the great physics potential of searching for the ALPs, especially, in the mass region 1 TeV -2.4 TeV, for the collision energy √ s = 3000 GeV and integrated luminosity L = 5000 fb −1 . In particular, our bounds are much stronger that recently obtained bounds for the ALP virtual production in the process p(γγ → γγ)p at the LHC [71].