Muon pair production via photon-induced scattering at the CLIC in models with extra dimensions

The photon-induced dimuon production $e^+ e^- \to e^+ \gamma \gamma e^- \to e^+ \mu^+ \mu^- e^-$ at the CLIC is studied in the framework of three models with extra dimensions. The electron beam energies 750 GeV and 1500 GeV are considered. The total cross sections are calculated depending on the minimal transverse momenta of the final muons. The sensitivity bounds on the parameters of the models are obtained as functions of the CLIC integrated luminosity.


I. INTRODUCTION
The Standard Model (SM) has been validated by existing experiments including the LHC date and has passed many tests very successfully at the electroweak energy scale.
However, many issues remain open in SM. One of the most fundamental of these problems is the hierarchy problem.This open question involves the large energy gap between the electroweak scale and the gravity scale. One of the fundamental approaches to the solution of the hierarchy problem is the theories that suggest the existence of extra dimensions (EDs). Recently, many articles have been published on these theories, which attracted great attention.
Scientists expect the LHC to elucidate many unanswered physics problems. Nevertheless, this type of collider enables precision measurements due to the nature of the proton-proton collisions. Whereas, interactions of the electrons and positrons with high-luminosity can provide higher precision than the proton-proton interaction with too much background.
Compact Linear Collider (CLIC) is one of the most qualified e + e − colliders. CLIC includes normal conducting accelerating cavities and two-beam acceleration [1]. It is used in a novel two-beam acceleration technique. In this way, accelerating gradients could be obtained as 100 MV/m. To work CLIC at maximum efficiency, three energy stages are planned [2]. First one is at √ s = 380 GeV and can reach the integrated luminosity L = 1000 fb −1 . This era covers Higgs boson, top and gauge sectors. It is possible to search for such SM particles with the high precision [3]. Second operation is at √ s = 1500 GeV. This stage is the highest center-of-mass energy available with a single CLIC drive beam complex. In the second stage, CLIC can give clues to beyond the SM physics. Moreover, detailed Higgs properties such as the Higgs self-coupling and the top-Yukawa coupling and rare Higgs decay channels could be studied [4]. In this stage, maximum integrated luminosity value is 2500 fb −1 . The last stage is that CLIC has reached its maximum center-of-mass energy value √ s = 3000 GeV and integrated luminosity value L = 5000 fb −1 . It is possible to do the most precise examinations of the SM. Moreover, it is enable to discovery beyond the SM heavy particles of mass greater than 1500 GeV [3]. The CLIC potential for new physics is presented in [5].
At the CLIC, as with all linear accelerators, eγ and γγ interactions are possible. Such interactions can be formed in two ways: Compton backscattering [6][7][8] and photon-induced reactions [9][10][11]. In photon-induced reactions, γγ and eγ interactions can occur sponta-neously, unlike the Compton backscattering process. Therefore, photon-induced reactions are much more useful than the Compton backscattering procedure search for new physics beyond the SM. This type of interactions can be studied by the Weizsäcker-Williams approximation (WWA). There are great advantages of using the WWA. Numerical calculations can be easily performed using simple formulas. In addition, this method is useful in experimental searches. Because it allows us to determine events number for the process γγ → X approximately with use of the e − e + → e − Xe + process [12]. Moreover, photon induced reactions have very clean backgrounds since these reactions do not involve interference with weak and strong interactions. There are many phenomenological and experimental studies in the literature on photon-induced process [13][14][15][16][17][18][19][20].
In WWA, the photons have very small virtuality. Therefore, scattered angels of the emitting photons from the electrons path along the actual beam trajectory should be very small. In this approximation, the photon spectrum in incoming electron with the energy E is given by the formula [9] (see also [21] where x = E γ /E is the energy fraction of the photon, m e is the electron mass, α em is the fine structure constant, and In the photon-induced collisions, the luminosity spectrum dL γγ /dW can be found with using WWA as follows The cross section for the process e + e − → e + γγe − → e + µ + µ − e − is obtained by integrating subprocess cross section dσ γγ→µ + µ − (W ) over the photon luminosity spectrum where and p ⊥ min is the minimal transverse momentum of the final muons.
In the present paper, we examine the potential of the photon-induced process e + e − → e + γγe − → e + µ + µ − e − at the CLIC in the framework of three models with EDs. Both flat and warped metrics of the space-time are considered.

II. SCENARIO WITH EXTRA DIMENSIONS AND FLAT METRIC
One of promising possibilities to go beyond the SM is to consider a theory in a space-time with extra spatial EDs. Such an approach is motivated by the (super)string theory [24]. One of the main goals of the theories with EDs is to explain the hierarchy relation between the electromagnetic and Planck scales. In the model proposed by Arkani-Hamed, Dimopolous, Dvali and Antoniadis [25]- [27], called ADD, this relation looks likē where d is the number of EDs, V d = (2πR c ) d is the volume of compact EDs with the radius where n i = 0, 1, . . . (i = 1, 2, . . . d). We see that in the scenario with large EDs the mass splitting ∆m KK = 1/R c is very small. Thus, the mass spectrum of the gravitons can be regarded as continuous.

III. SCENARIO WITH ONE WARPED EXTRA DIMENSION
The Randall-Sundrum (RS) scenario with one ED and warped metric is based on the following background metric [28] where η µν is the Minkowski tensor with the signature (+, −, −, −), and y is a compactified extra coordinate. The periodicity condition y = y + 2πr c is imposed, and the points (x µ , y) and (x µ , −y) are identified. Thus, we obtain a model of gravity in a slice of the AdS 5 spacetime compactified to the orbifold S 1 /Z 2 with the size πr c . Since this orbifold has two fixed points, y = 0 and y = πr c , two branes can be put at these points. They are called Planck and TeV brane, respectively. All SM fields are assumed to live on the TeV brane.
The classical action of the RS scenario looks like [28] Here G M N (x, y) is the 5-dimensional metric, M, N = 0, 1, 2, 3, 4. The quantities are induced metrics on the branes, µ = 0, 1, 2, 3. Λ is a five-dimensional cosmological constant, while Λ 1 and Λ 2 are tensions on the branes. L 1 L 2 are brane Lagrangians, and µν ). From the RS action (10) one gets 5-dimensional Einstein-Hilbert's equations In what follows, the reduced 5-dimensional gravity scales will be used, Let us underline that equations (12), (13) contain only derivatives of the function σ(y) and that equation (13) is symmetric with respect to the branes.
As it was shown in details in [29] (see also [30]), a general solution of equations (12), (13) is given by where the parameter κ with a dimension of mass defines a five-dimensional scalar curvature R (5) , and C is y-independent quantity. 2 In addition, the following fine tuning must be realized [29]. From now on, it will be assumed that κ > 0, πκ r c ≫ 1. Then the hierarchy relation is of the form The interactions of the gravitons h (n) µν with the SM fields on the physical (TeV) brane are given by the effective Lagrangian were T µν (x) is the energy-momentum tensor of the SM fields, and the coupling constant of the massive modes is The graviton masses m n are defined from the boundary conditions imposed on wave functions of the KK excitations. They result in the equation (see, for instance, [30]) where J 1 (x) and Y 1 (x) are the Bessel functions of the first and second kind, respectively, and the following notations are introduced By taking different values of C in eq. (14), we come to quite different physical models within the framework with the warped metric. In particular, for C = 0, we come to the original RS1 model [28] with the hierarchy relation In order (22) to be satisfied, one has to putM 5 ∼ κ ∼M Pl [28]. The graviton masses, as one can see from (20)- (21), are given by the formula where x n are zeros of J 1 (x). The coupling constant (19) will be of the order of one TeV, if we put κr c ≃ 11.3. It is in agreement with our assumption πκ r c ≫ 1. Then the lightest graviton resonance has a mass of order one-few TeV.
Taking C = πκ r c , we come to the RS-like model with a small curvature (RSSC model).
For the first time, it was studied in [31], see also [29]- [30], [32]- [33]. In such a model, the hierarchy relations takes the form It is thanks to the exponential factor in (24) Note that in the limit κ → 0, the hierarchy relation for the flat metric with one ED (7) is where V 1 = 2πr c is the volume of ED. At the same time, Λ π →M Pl , and m n → n/r c [30].
Thus, from the point of view of a 4-dimensional observer, the models with C = 0 and C = κπr c are quite different physical models. The experimental signature of the RS1 model is a production of heavy resonances, while the signature of the RSSC model is a deviation of cross sections from SM predictions.

IV. PHOTON-INDUCED DIMUON PRODUCTION
Let us consider the subprocess γγ → µ + µ − of the photon-induced dimuon production in e + e − collision. It's matrix element squared is the sum of electromagnetic, KK graviton and interference terms [34] where The quantity S(s) contains summation over s-channel massive KK excitations which can be calculated without specifying process,ŝ,t are Mandelstam variables of the subprocess γγ → µ + µ − , and e 2 = 4πα em .
In the ADD model this sum is given by 3 where the masses m n are defined by Eq. (8). Since the sum is infinite for d 2, an ultraviolet procedure is needed [36]- [37]. In the Han-Lykken-Zhang (HLZ) convention [35], the sum of virtual KK exchanges is replaced by the integral in variable m n with the ultraviolet cutoff M S , that results in where x = M S / √ s, and In what follows, we put M S = M D .
In the Hewett convention, sum (31) is replaced by [36] 3 We use the definition of the graviton field of [35]: where M H is the unknown mass scale, presumably of order M D The exact relationship between scales M H and M D is not calculable without knowledge of the full theory. The parameter λ = ±1 is taken in analogy with the standard parametrization for contact interactions.
Note that in the Giudice-Rattazzi-Wells convention [37] where Λ T is a cutoff scale. We will not use this approximation for S ADD (s) in our numerical analysis.
In the RS scenario the contribution of s-channel gravitons is given by the sum Here Γ n denotes the total width of the graviton with the KK number n and mass m n [33] where ρ = 0.09.
In the RS1 model, taking into account that the KK resonances are very heavy, we put The contribution form other resonances to the sum (36) is negligible.
In the RSSC model, graviton sum (36) can be calculated analytically [33] S where As was already mentioned above, in the RSSC model the KK graviton exchanges should lead to the deviations of the cross sections from the SM predictions.

V. NUMERICAL ANALYSIS AND RESULTS
The main goal of this section is to calculate the deviations of the cross sections from the SM predictions in a number of models with EDs and to estimate the CLIC 95% C.L. search limit for the photon-induced process e + e − → e + γγe − → e + µ + µ − e − . The expected collision energy √ s of the CLIC is 380 GeV (1st stage), 1500 GeV (2nd stage) or 3000 GeV (3rd stage), with the integrated luminosities for unporalized beams to be equal to 1000 fb −1 , 2500 fb −1 , and 5000 fb −1 , respectively, as mentioned above. Our numerical results have shown that for the same values of the parameters of the models, the deviations from the SM are much smaller for √ s = 380 GeV. That is why, we will present our result for √ s = 1500 GeV and √ s = 3000 GeV only. We have calculated the statistical significance SS using formula [39]

B. RS model
We have also calculated the cross sections in the Randall-Sundrum model [28] using formula (38).

VI. CONCLUSIONS
In the present paper we have studied the photon-induced dimuon production e + e − → e + γγe − → e + µ + µ − e − at the CLIC in a number of models with EDs. Among these mod- for the photon-induced process pp → pγγp → pµ + µ − p have been calculated in our recent paper [42]. The LHC bounds obtained there are noticeably lower that our CLIC bounds.
Let us underline, the great advantage of the CLIC collider is that it has very clean backgrounds. Moreover, the CLIC detectors don't need additional equipment for Weizsäcker-Williams photon-induced collisions analyzed in the present paper. That is why, we think that studying such reactions at the CLIC could be one of most important physical tasks.
It would be interesting to compare our results on the total cross section and CLIC search limits with the corresponding predictions for the processes e + γ → e + γ and e + e − → e + γγe − → e + µ + µ − e − , where γ is the Compton backscattering photon [6]. It will be a subject of our separate publication.