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

TL;DR

This paper investigates muon pair production via photon-induced scattering at the CLIC collider within models with extra dimensions, analyzing cross sections and sensitivity bounds for different beam energies and model parameters.

Contribution

It introduces a detailed analysis of muon pair production at CLIC in the context of extra-dimensional models, providing new sensitivity bounds based on collider parameters.

Findings

Cross sections depend on the minimal transverse momenta of muons.

Sensitivity bounds on model parameters are derived for different energies.

Results show potential to probe extra dimensions at CLIC.

Abstract

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.

Equations

$$\frac{dN}{d{E}_{\gamma }}=\frac{{\alpha }_{\mathrm{em}}}{\pi E}\left[\frac{(1-x+x^2/2)}{x}\ln \frac{Q_{\max }^2}{Q_{\min }^2}-\frac{m_e^{2}x}{Q_{\min }^2}\left(1-\frac{Q_{\min }^2}{Q_{\max }^2}\right)\right],$$

$$Q_{\min }^2=\frac{m_e^2{x}^2}{1-x},Q_{\max }^2=2{\mathrm{GeV}}^2.$$

$$Q_{\max }^2=Q_{\min }^2+E^2(1-x){\theta }^2,$$

$$\frac{d{L}^{\gamma \gamma }}{dW}=\underset{y_{\min }}{\overset{y_{\max }}{\int }}dy\frac{W}{2y}{f}_1(\frac{W^2}{4y},Q_1^2)f_2(y,Q_2^2),$$

$$y_{\min }=\frac{W^2}{4(E-m_e)},y_{\max }=E-m_e.$$

$$d\sigma =\underset{W_{\min }}{\overset{W_{\max }}{\int }}dW\frac{d{L}^{\gamma \gamma }}{dW}d{\hat{\sigma }}_{\gamma \gamma \to {\mu }^{+}{\mu }^{-}}(W),$$

$$W_{\min }=2p_{\perp \min },W_{\max }=2(E-m_e),$$

$${\bar{M}}_{\mathrm{Pl}}^2=V_d{\bar{M}}_D^{d+2},$$

$$m_n=\frac{n}{R_c},n=\sqrt{n_1^2+n_2^2+\cdots n_d^2},$$

$$d{s}^2=e^{-2\sigma (y)}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }-d{y}^2,$$

$$S$$

$$+\int d^{4}x\sqrt{|g^{(1)}|}({\mathcal{L}}_1-{\Lambda }_1)+\int d^{4}x\sqrt{|g^{(2)}|}({\mathcal{L}}_2-{\Lambda }_2).$$

$$g_{\mu \nu }^{(1)}(x)=G_{\mu \nu }(x,y=0),g_{\mu \nu }^{(2)}(x)=G_{\mu \nu }(x,y=\pi r_c)$$

$$6{\sigma }^{\prime 2}(y)$$

$$3{\sigma }^{\prime \prime }(y)$$

$$\sigma (y)=\frac{\kappa }{2}(|y|-|y-\pi r_c|)+\frac{\kappa \pi r_c}{2}-C,$$

$$\Lambda$$

$${\Lambda }_1$$

$$\bar{M}_{\mathrm{Pl}}^{2}=\frac{\bar{M}_{5}^{3}}{\kappa}\,e^{2C}\!\left(1-e^{-2\pi\kappa r_{c}}\right)\Big{|}_{\pi\kappa\,r_{c}\gg 1}=\frac{\bar{M}_{5}^{3}}{\kappa}\,e^{2C}\;.$$

$${\mathcal{L}}_{\mathrm{int}}=-\frac{1}{{\bar{M}}_{\mathrm{Pl}}}{h}_{\mu \nu }^{(0)}(x)T_{\alpha \beta }(x){\eta }^{\mu \alpha }{\eta }^{\nu \beta }-\frac{1}{{\Lambda }_{\pi }}\sum _{n=1}^{\infty }{h}_{\mu \nu }^{(n)}(x)T_{\alpha \beta }(x){\eta }^{\mu \alpha }{\eta }^{\nu \beta },$$

$${\Lambda }_{\pi }\simeq {\bar{M}}_{\mathrm{Pl}}{e}^{-\kappa \pi r_c}={\left(\frac{{\bar{M}}_5^3}{\kappa }\right)}^{1/2}{e}^{C-\kappa \pi r_c}.$$

$$J_1(a_{1n})Y_1(a_{2n})-Y_1(a_{1n})J_1(a_{2n})=0,$$

$$a_{1n}=\frac{m_n}{\kappa }{e}^{-C},a_{2n}=\frac{m_n}{\kappa }{e}^{\kappa \pi r_c-C}.$$

$$\bar{M}_{\mathrm{Pl}}^{2}=\frac{\bar{M}_{5}^{3}}{\kappa}\!\left(1-e^{-2\pi\kappa r_{c}}\right)\Big{|}_{\pi\kappa\,r_{c}\gg 1}=\frac{\bar{M}_{5}^{3}}{\kappa}\;.$$

$$m_n=x_n\kappa e^{-\kappa \pi r_c},n=1,2,\dots ,$$

$$\bar{M}_{\mathrm{Pl}}^{2}=\frac{\bar{M}_{5}^{3}}{\kappa}\!\left(e^{2\pi\kappa r_{c}}-1\right)\Big{|}_{\pi\kappa\,r_{c}\gg 1}=\frac{\bar{M}_{5}^{3}}{\kappa}\,e^{2\kappa\pi r_{c}}\;.$$

$$m_n=x_n\kappa ,n=1,2,\dots .$$

$${\bar{M}}_{\mathrm{Pl}}^2={\bar{M}}_5^3{V}_1,$$

$${\mathcal{L}}_{\mathrm{rad}}=\frac{1}{\sqrt{3}{\Lambda }_{\pi }}{T}_{\mu }^{\mu }(x)\varphi (x).$$

$${\Lambda }_{\pi }=100{\left(\frac{{\bar{M}}_5}{\mathrm{TeV}}\right)}^{3/2}{\left(\frac{100\mathrm{MeV}}{\kappa }\right)}^{1/2}\mathrm{TeV}.$$

$$|M{|}^2=|M_{\mathrm{em}}+M_{\mathrm{KK}}{|}^2,$$

$$|M_{\mathrm{em}}{|}^2$$

Full text

Keywords: theories with extra dimensions, Kaluza-Klein gravitons, CLIC collider; dimuon production in $e^{+}e^{-}$ collision, photon-induced scattering.

PACS: 04.50.Cd, 14.80.Rt, 13.66.De.

arXiv:1907.12824

1 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 the 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 much 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. The emission of beamstrahlung and production of background particles at the CLIC is limited to an acceptance level by using flat beams with the small vertical size [6]. It is expected that the CLIC detector will have the high transverse momentum resolution for high-momentum muon tracks (see, for instance, Fig. 7(a) in [6]). One of the main requirements for the CLIC detector is the overall detector coverage (hermeticity), particularly in the forward region [7]. This is needed for lepton identification and missing energy. For particle energy $7.5$ GeV the muon identification efficiency for the CLIC detector was estimated to be about 99% [3].

To work CLIC at maximum efficiency, three energy stages are planned [2]. The first one is at $\sqrt{s}=380$ 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 $\sqrt{s}=1500$ 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, particular 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, the maximum integrated luminosity value is $2500$ fb*-1*. The last stage is that CLIC has reached its maximum center-of-mass energy value $\sqrt{s}=3000$ GeV and integrated luminosity value $L=5000$ fb*-1*. It is possible to do the most precise examinations of the SM. Moreover, it is enabled 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, $e\gamma$ and $\gamma\gamma$ interactions are possible. Such interactions can be formed in two ways: Compton backscattering [8, 9, 10] and photon-induced reactions [11, 12, 13]. In photon-induced reactions, $\gamma\gamma$ and $e\gamma$ interactions can occur spontaneously, 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 interaction 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 $\gamma\gamma\rightarrow X$ approximately with use of the $e^{-}e^{+}\rightarrow e^{-}Xe^{+}$ process [14]. 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 [15, 16, 17, 18, 19, 20, 21, 22].

In WWA, the photons have very small virtuality. Therefore, scattered angels of the emitting photons from the path of the electron along the actual beam trajectory should be tiny. In this approximation, the photon spectrum in incoming electron with the energy $E$ is given by the formula [11] (see also [23])

[TABLE]

where $x=E_{\gamma}/E$ is the energy fraction of the photon, $m_{e}$ is the electron mass, $\alpha_{\mathrm{em}}$ is the fine structure constant, and

[TABLE]

Note that the contribution of the photons with virtualities more than $Q_{\mathrm{max}}^{2}$ is negligible [12]. On the other hand,

[TABLE]

where $E$ is the beam energy, $\theta$ is the electron scattering angle. The photon spectrum (1) is singular at $x\rightarrow 0$. So, we have $\theta\ll 1$, and the validity of the WWA is justified.

In the photon-induced collisions, the luminosity spectrum $dL^{\gamma\gamma}/dW$ can be found with using WWA as follows,

[TABLE]

where $f_{i}=dN/dE_{\gamma i}$, $i=1,2$,

[TABLE]

The cross-section for the process $e^{+}e^{-}\rightarrow e^{+}\gamma\gamma e^{-}\rightarrow e^{+}\mu^{+}\mu^{-}e^{-}$ is obtained by integrating subprocess cross section $d\hat{{\sigma}}_{\gamma\gamma\to\mu^{+}\mu^{-}}(W)$ over the photon luminosity spectrum

[TABLE]

where

[TABLE]

and $p_{\bot\min}$ is the minimal transverse momentum of the final muons.

Precision measurements at the CLIC can be regarded as complementary searches for new physics beyond the SM carried out at the LHC. In the present paper, we examine the potential of the photon-induced dimuon production at the CLIC in the framework of three different models with EDs. Both flat and warped metrics of the space-time are considered. Recently, we have studied the muon pair production in the photon-induced process at the LHC [24]. The LHC discovery limits on 5-dimensional gravity scale for the process $pp\rightarrow p\gamma\gamma p\rightarrow p\mu^{+}\mu^{-}p$ have been calculated. One of the main goals of the present paper is to study the CLIC search limit for the process $e^{-}e^{+}\rightarrow e^{-}\gamma\gamma e^{+}\rightarrow e^{-}\mu^{+}\mu^{-}e^{+}$, and to compare it with the LHC search limit. We will show that the CLIC sensitivity to the effects coming from EDs are noticeably larger than the corresponding LHC bounds.

2 Scenario with extra dimensions and flat metric

One of the 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 [26]. 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 [27]-[29], called ADD, this relation looks like

[TABLE]

where $d$ is the number of EDs, $V_{d}=(2\pi R_{c})^{d}$ is the volume of compact EDs with the radius $R_{c}$, $\bar{M}_{\mathrm{Pl}}=M_{\mathrm{Pl}}/\sqrt{8\pi}$ is the reduced Planck mass, and $\bar{M}_{D}$ is the reduced fundamental gravity scale in $D=4+d$ dimensions, $\bar{M}_{D}=M_{D}/(2\pi)^{d/(d+2)}$. It is assumed that the fundamental gravity scale $M_{D}$ is in the TeV region. The huge gap between the Planck and TeV scales can be justified by the large value of $V_{d}$ (so-called “large EDs”).

The masses of the Kaluza-Klein (KK) gravitons in the ADD model are given by the formula [27]-[29]

[TABLE]

where $n_{i}=0,1,\ldots$ ($i=1,2,\ldots d$). We see that in the scenario with large EDs, the mass splitting $\Delta m_{KK}=1/R_{c}$ is very small. Thus, the mass spectrum of the gravitons can be regarded as continuous.

3 Scenario with one warped extra dimension

The Randall-Sundrum (RS) scenario with one ED and warped metric is based on the following background metric [30]

[TABLE]

where $\eta_{\mu\nu}$ is the Minkowski tensor with the signature $(+,-,-,-)$, and $y$ is a compactified extra coordinate. The periodicity condition $y=y+2\pi r_{c}$ is imposed, and the points $(x_{\mu},y)$ and $(x_{\mu},-y)$ are identified. Thus, we obtain a model of gravity in a slice of the AdS5 space-time compactified to the orbifold $S^{1}\!/Z_{2}$ with the size $\pi r_{c}$. Since this orbifold has two fixed points, $y=0$ and $y=\pi r_{c}$, 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 [30]

[TABLE]

Here $G_{MN}(x,y)$ is the 5-dimensional metric, $M,N=0,1,2,3,4$. The quantities

[TABLE]

are induced metrics on the branes, $\mu=0,1,2,3$. $\Lambda$ is a five-dimensional cosmological constant, while $\Lambda_{1}$ and $\Lambda_{2}$ are tensions on the branes. $\mathcal{L}_{1}$ $\mathcal{L}_{2}$ are brane Lagrangians, and $G=\det(G_{MN})$, $g^{(i)}=\det(g^{(i)}_{\mu\nu})$. From the RS action (3) one gets 5-dimensional Einstein-Hilbert’s equations

[TABLE]

In what follows, the reduced 5-dimensional gravity scales will be used, $\bar{M}_{5}=M_{5}/(2\pi)^{1/3}$. Let us underline that equations (13), (14) contain only derivatives of the function $\sigma(y)$ and that equation (14) is symmetric with respect to the branes.

As it was shown in details in [31] (see also [32]), a general solution of equations (13), (14) is given by

[TABLE]

where the parameter $\kappa$ with a dimension of mass defines a five-dimensional scalar curvature $\mathcal{R}^{(5)}$, and $C$ is $y$-independent quantity.111The term $\kappa\pi r_{c}/2$ is introduced in (15) for convenience only. In addition, the following fine tuning

[TABLE]

must be realized [31]. From now on, it will be assumed that $\kappa>0$, $\pi\kappa\,r_{c}\gg 1$. Then the hierarchy relation is of the form

[TABLE]

The interactions of the gravitons $h_{\mu\nu}^{(n)}$ with the SM fields on the physical (TeV) brane are given by the effective Lagrangian

[TABLE]

were $T^{\mu\nu}(x)$ is the energy-momentum tensor of the SM fields, and the coupling constant of the massive modes is

[TABLE]

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, [32])

[TABLE]

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

[TABLE]

By taking different values of $C$ in eq. (15), 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 [30] with the hierarchy relation

[TABLE]

In order (23) to be satisfied, one has to put $\bar{M}_{5}\sim\kappa\sim\bar{M}_{\mathrm{Pl}}$ [30]. The graviton masses, as one can see from (21)-(22), are given by the formula

[TABLE]

where $x_{n}$ are zeros of $J_{1}(x)$. The coupling constant (20) will be of the order of one TeV, if we put $\kappa r_{c}\simeq 11.3$. It is in agreement with our assumption $\pi\kappa\,r_{c}\gg 1$. Then the lightest graviton resonance has a mass of order one–few TeV.

Taking $C=\pi\kappa\,r_{c}$, we come to the RS-like model with a small curvature (RSSC model). For the first time, it was studied in [33], see also [31]-[32], [34]-[35]. In such a model, the hierarchy relations take the form

[TABLE]

It is thanks to the exponential factor in (25) that the mass hierarchy can be satisfied even for moderate values of $\bar{M}_{5}$ and $\kappa$. For instance, this relation holds, if one puts $\bar{M}_{5}\sim 1$ TeV, $\kappa\sim 1$ GeV, and $\kappa\,r_{c}=10.2$. On the contrary, the RS1 hierarchy relation (23) does not admit the parameters $\kappa$, $\bar{M}_{5}$ to lie in these region. The mass spectrum of the KK gravitons in the RSSC model, as it follows from (21)-(22), is defined as

[TABLE]

Note that in the limit $\kappa\rightarrow 0$, the hierarchy relation for the flat metric with one ED (8) is reproduced from (25)

[TABLE]

where $V_{1}=2\pi r_{c}$ is the volume of ED. At the same time, $\Lambda_{\pi}\rightarrow\bar{M}_{\mathrm{Pl}}$, and $m_{n}\rightarrow n/r_{c}$ [32].

The interaction Lagrangian of the radion field $\phi(x)$ on the visible brane looks like

[TABLE]

In the RSSC model, the mass scale $\Lambda_{\pi}$ is given by [36]

[TABLE]

As it follows from Eqs. (28), (29), there is no problem with the radion field $\phi$, since for $\kappa\sim 0.1\div 1$ GeV, $\bar{M}_{5}\sim 1$ TeV, the quantity $1/(\sqrt{3}\Lambda_{\pi})$ is very small, and, consequently, the coupling of the radion to the SM fields is firmly suppressed. Let us underline that the magnitude of all “graviton” cross-sections (with either real or virtual production of the KK gravitons) is defined by the fundamental gravity scale $\bar{M}_{5}$, which is a few TeV, not by the coupling $\Lambda_{\pi}$.

To conclude, from the point of view of a 4-dimensional observer, the models with $C=0$ and $C=\kappa\pi r_{c}$ 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.

4 Photon-induced dimuon production

Let us consider the subprocess $\gamma\gamma\rightarrow\mu^{+}\mu^{-}$ 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 [37]

[TABLE]

where

[TABLE]

The quantity $S(s)$ contains summation over $s$-channel massive KK excitations which can be calculated without specifying process, $\hat{s}$, $\hat{t}$ are Mandelstam variables of the subprocess $\gamma\gamma\rightarrow\mu^{+}\mu^{-}$, and $e^{2}=4\pi\alpha_{\mathrm{em}}$.

In the ADD model, this sum is given by222We use the definition of the graviton field of [38]: $g_{AB}=\eta_{AB}+\sqrt{2}\bar{M}_{D}^{-1-d/2}h_{AB}$.

[TABLE]

where the masses $m_{n}$ are defined by Eq. (9). Since the sum is infinite for $d\geqslant 2$, an ultraviolet procedure is needed [39]-[40]. In the Han-Lykken-Zhang (HLZ) convention [38], the sum of virtual KK exchanges is replaced by the integral in variable $m_{n}$ with the ultraviolet cutoff $M_{S}$, that results in

[TABLE]

where $x=M_{S}/\sqrt{s}$, and

[TABLE]

In what follows, we put $M_{S}=M_{D}$.

In the Hewett convention, sum (34) is replaced by [39]

[TABLE]

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 $\lambda=\pm 1$ is taken in analogy with the standard parametrization for contact interactions.

Note that in the Giudice-Rattazzi-Wells convention [40]

[TABLE]

where $\Lambda_{T}$ is a cutoff scale. We will not use this approximation for $\mathcal{S}_{\mathrm{ADD}}(s)$ in our numerical analysis.

In the RS scenario, the contribution of $s$-channel gravitons is given by the sum

[TABLE]

Here $\Gamma_{n}$ denotes the total width of the graviton with the KK number $n$ and mass $m_{n}$ [35]

[TABLE]

where $\rho=0.09$.

In the RS1 model, taking into account that the KK resonances are very heavy, we put

[TABLE]

The contribution from other resonances to the sum (39) is negligible.

In the RSSC model, graviton sum (39) can be calculated analytically [35]

[TABLE]

where

[TABLE]

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.

5 Numerical analysis and results

The main goal of this section is to calculate the deviations of the cross-sections from the SM predictions in several models with EDs and to estimate the CLIC 95% C.L. search limit for the photon-induced process $e^{+}e^{-}\rightarrow e^{+}\gamma\gamma e^{-}\rightarrow e^{+}\mu^{+}\mu^{-}e^{-}$. We assume 100% efficiency in the reconstruction and identification of the final muons. The expected collision energy $\sqrt{s}$ of the CLIC is 380 GeV (1st stage), 1500 GeV (2nd stage) or 3000 GeV (3rd stage), with the integrated luminosities for unpolarized 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 $\sqrt{s}=380$ GeV. That is why we will present our result for $\sqrt{s}=1500$ GeV and $\sqrt{s}=3000$ GeV only.

5.1 ADD model

In the ADD model with the Han-Lykken-Zhang (HLZ) convention [38] the invariant part of the subprocess $\gamma\gamma\rightarrow\mu^{+}\mu^{-}$ is given by equations (35), (36). The parameters are the number of EDs $d$ and the cutoff scale $M_{S}$. The latter is believed to be of the order of the $(4+d)$-dimensional gravity scale $M_{D}$. The results of our calculations of the total cross-sections are shown in Figs. 1-4 as functions of the minimal transverse momenta of the final muons $p_{\mathrm{t,min}}$. As one can see from these figures, the deviations from the SM take place only for $p_{\mathrm{t,min}}\gtrsim 200$ GeV for both energies. The curves correspond to different values of $d$ and $M_{S}$. As one can see, for $M_{S}=2$ TeV and $M_{S}=4.5$ TeV, the cross-sections have nontrivial dependence on $d$. The interference term of the cross sections could be relatively large in absolute value and negative for some mediate $p_{t,\min}$ (150-450 GeV). As a result, in this $p_{t,\min}$ region cross-section slightly rises with $d$ for some values of $M_{S}$, as one can see in Figs. 1, 4.

We have calculated the statistical significance $SS$ using formula [42]

[TABLE]

where $S(B)$ is a number of the signal (background) events. Note that $SS\simeq S/\sqrt{B}$ for $S\ll B$. We have assumed that the uncertainty of the background is negligible.

The main contribution to the SM background comes from the SM process $e^{-}e^{+}\rightarrow e^{-}\gamma\gamma e^{+}\rightarrow e^{-}\mu^{+}\mu^{-}e^{+}$, going via subprocess $\gamma\gamma\rightarrow\mu^{+}\mu^{-}$. Note that the other SM process $e^{-}e^{+}\rightarrow e^{-}ZZ(WW)e^{+}\rightarrow e^{-}\mu^{+}\mu^{-}e^{+}$ may not be taken into account as a background, since as given in [43], the $Z$-$Z$ luminosity function is 100 times smaller than the photon-photon luminosity function.

It can be considered that the dimuon pair-production $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ also contributes to the background when the final-state electrons in the photon-induced processes are almost along the beamlines, and they can not be detected. As it is shown in Appendix A, the total cross section of this process $\sigma(p_{t}>30\mathrm{\ GeV})$ is very small with respect to our main SM background. Moreover, the final state $\mu^{+}\mu^{-}$ in the s-channel process $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ has a quite different $p_{t}$ distribution. Hence, it can be easily discerned from our process.

The 95% C.L. search limits for two CLIC energies are presented in Figs. 5, 6 as functions of the number of EDs and CLIC integrated luminosity for $\sqrt{s}=1500$ GeV and $\sqrt{s}=3000$ GeV. The bounds on $M_{S}$ are as large as 3629(3593) GeV for $d=2(6)$ can be achieved for the CLIC energy $\sqrt{s}=3000$ GeV and integrated luminosity $L=5000\mathrm{\ fb}^{-1}$.

The cross-sections in the Hewett convention of the ADD model [39] depend on the ultraviolet cutoff $M_{H}$ and sign of the parameter $\lambda$ in (37). They are shown in Figs. 7, 8 as functions of $p_{\mathrm{t,min}}$, both for positive and negative sign of the parameter $\lambda$ in (37). As one can see from these figures, for $p_{\mathrm{t,min}}>300$ GeV the SM and KK contributions can be defined well.

The 95% C.L. bounds on the cutoff scale $M_{H}$ as functions of the CLIC integrated luminosity are given in Figs. 9, 10. They demonstrate us that the case $\lambda=-1$ is clearly preferable to the case $\lambda=1$. The bound $M_{H}=2204$ GeV can be achieved for $\sqrt{s}=3000$ GeV, $L=5000\mathrm{\ fb}^{-1}$. Note that in the Hewett scheme there is no dependence on the number of EDs. The bounds obtained should be compared with the LHC bounds on the parameters in the ADD model (see, for instance, [44]).

5.2 RS model

We have also calculated the cross-sections in the Randall-Sundrum model [30] using formula (41). The corresponding curves are shown in Figs. 11, 12 as functions of the mass $m_{1}$ of the lightest graviton for three values (0.01, 0.05, 0.1) of the ratio

[TABLE]

The oscillations of the curves in these figures correspond to the resonance character of the invariant part of the $s$-channel amplitude in the RS model (41). Let us indicate on the strong $\beta$-dependence of the cross section for $m_{1}\lesssim 1.5(2.0)$ TeV at $\sqrt{s}=1500(3000)$ GeV. For $\beta=0.01$ the deviations from the SM are negligible for all $m_{1}$.

The estimation for the $95\%$ C.L. parameter exclusion region is shown in Figs. 13, 14 for three expected values of the CLIC integrated luminosity. For both energies, the value 500 GeV was taken as the minimal transverse momenta of the final muons. Due to the oscillation behavior of the total cross sections, their values are closed to the SM cross sections for all $0.01\leqslant\beta\leqslant 0.1$ in the mass interval $m_{1}=750-1070$ GeV for $\sqrt{s}=1500$ GeV, see Fig. 11. As a result, the exclusion region has the “hole” in these mass region, as one can see in Figs. 13. As for invariant energy $\sqrt{s}=3000$ GeV, similar oscillations of the cross section presented in Fig. 12 result in the “bumps” on the exclusion curves in the $m_{1}=865-1170$ GeV region, see Fig. 14.

The best lower bound which can be achieved, is equal to $m_{1}=2629$ GeV for $\beta=0.1$. The present experimental bounds on $m_{1}$ are stronger [45]. Thus, we don’t expect that, instead of the very high integrated luminosity of the CLIC 3rd stage, the existing experimental bounds on $m_{1}$ could be improved in the photon-induced dimuon production.

5.3 RSSC model

Finally, we have estimated the total cross-sections for the RS-like model with the small curvature (RSSC model) using formulas (42), (43). As one can see in Figs. 15, 16, the total cross sections weakly depend on the curvature parameter $\kappa$ for all values of the 5-dimensional Planck scale $\bar{M}_{5}$. It is a well-known feature of the RSSC model, provided the condition $\kappa\ll\bar{M}_{5}$ is satisfied [31]-[35].

The CLIC search bounds for the scale $\bar{M}_{5}$ in the RSSC model are shown in Figs. 17, 18. The value $\bar{M}_{5}=2534$ TeV can be achieved for $\sqrt{s}=3000$ GeV, $L=5000\mathrm{\ fb}^{-1}$. Let us stress that the parameter $\bar{M}_{5}$ has quite different magnitudes in the RS and RSSC models. In the RS model, the bounds on the parameter set ($\beta,m_{1}$) are searched for. On the contrary, in the RSSC model one can directly obtain bounds on the 5-dimensional Planck scale $\bar{M}_{5}$, while dependence on the curvature parameter $\kappa$ is rather weak. Note that the LHC bound on $D$-dimensional scale $M_{D}$ (see, for instance, [44]) cannot be applied to our lower limits on the scale $\bar{M}_{5}$, since the RSSC model cannot be regarded as a small distortion of the ADD model with one ED [35].

6 Conclusions

It is expected that LHC gives many answers to some fundamental problems of particle physics. However, high precision experiments could not be done at the LHC due to the strong interactions which spoil the proton structure through the proton-proton interactions. Therefore, the lepton collider with high luminosity and energy is a good candidate to complement and develop the LHC physics studies. Because the CLIC has high energy and luminosity, it is extremely important to investigate energy-dependent new physics beyond the SM, such as extra-dimensional models, axion-like particle production, anomalous gauge couplings, etc.

In the present paper we have studied the photon-induced dimuon production $e^{+}e^{-}\rightarrow e^{+}\gamma\gamma e^{-}\rightarrow e^{+}\mu^{+}\mu^{-}e^{-}$ at the CLIC in a number of models with EDs. Among these models are: (i) ADD model with the Han-Lykken-Zhang convention [38]; (ii) the ADD model with the Hewett convention [39], (iii) the original Randall-Sundrum model [30], (iv) the Randall-Sundrum-like model with the small curvature [33]. The total cross sections have been calculated for energies $\sqrt{s}=1500$ GeV and $\sqrt{s}=3000$ GeV. It enabled us to obtain $95\%$ C.L. search limits on the parameters of the models as functions of the CLIC integrated luminosity $L$.

In order to suppress the SM background, we imposed the cut on the transverse momenta of the final muons, $p_{t}>500$ GeV. Although the KK terms dominate the SM term for $p_{t}>800$ GeV, our calculations show that the CLIC sensitivity bounds for the cuts $p_{t}>800$ and $p_{t}>500$ GeV are almost the same for small $L$, and differ by less than 10% for very high $L$. Note that the number of events becomes very small for $p_{t}>800$. That is why, for the calculations of the exclusion regions, we used the cut $p_{t}>500$ GeV.

The best bounds have been derived for the $e^{+}e^{-}$ collision energy $\sqrt{s}=3000$ GeV and integrated luminosity $L=5000\mathrm{\ fb}^{-1}$. For the ADD model with the HLZ convention, we have obtained $M_{S}\geqslant 3629$ GeV (Fig. 6), while for the ADD model with the Hewett convention, we have got that $M_{H}\geqslant 2204$ GeV (Fig. 10). The best limits for the parameters ($\beta,m_{1}$) of the RS model are presented in Fig. 14. The bounds on the fundamental gravity scale $\bar{M}_{5}$ in the RSSC model is of considerable interest, since so far there are no experimental limits on the parameters of this RS-like model. The best bound on this model parameter shown in Fig. 18. Note that the LHC discovery limits on $\bar{M}_{5}$ for the photon-induced process $pp\rightarrow p\gamma\gamma p\rightarrow p\mu^{+}\mu^{-}p$ have been calculated in our recent paper [24]. The LHC bounds obtained there are noticeably lower than our CLIC bounds.

The gravity and collider constraints on parameters of the models with EDs lie in the TeV range. For our calculations, we have used parameters that are of the order of the current experimental bounds. Three models we analyzed have different metrics and/or different physical contents. From this perspective, we have demonstrated the potentials of CLIC photon-induced reactions for three different models with EDs.

The searches for effects of new physics at the LHC and CLIC can be regarded as complementary searches. 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 $e^{+}\gamma\rightarrow e^{+}\gamma$ and $e^{+}e^{-}\rightarrow e^{+}\gamma\gamma e^{-}\rightarrow e^{+}\mu^{+}\mu^{-}e^{-}$, where $\gamma$ is the Compton backscattering photon [8]. It will be a subject of our separate publication.

Acknowledgments

This work is partially supported by the Scientific Research Project Fund of Sivas Cumhuriyet University under project number “F-596”.

Appendix A

One of possible SM backgrounds is the process without electron pair in the final states. In Fig. 19 the differential cross section for the process $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ is shown in comparison with that for the SM process $e^{+}e^{-}\rightarrow e^{+}\mu^{+}\mu^{-}e^{-}$. As one can see, the former is relatively small up to $p_{t}=500(1000)$ GeV for $\sqrt{s}=1500(3000)$ GeV. As a result, with the cut $p_{t}>30$ GeV we obtain $\sigma(e^{+}e^{-}\rightarrow\mu^{+}\mu^{-})$ = 6.14 fb and 1.52 fb, for $\sqrt{s}=1500$ GeV and $\sqrt{s}=3000$ GeV, respectively, while $\sigma(e^{+}e^{-}\rightarrow e^{+}\mu^{+}\mu^{-}e^{-})$ = 509 fb and 677 fb for the same energies.