Search for the anomalous electromagnetic moments of the tau lepton through electron-photon scattering at the CLIC

We have examined the anomalous electromagnetic moments of the tau lepton in the processes $e^{-}\gamma \to \nu_e\tau\bar{\nu}_\tau$ ($\gamma$ is the Compton backscattering photon) and $e^{-}e^{+} \to e^{-}\gamma^* e^{+} \to \nu_{e}\tau \bar{\nu}_\tau e^+$ ($\gamma^*$ is the Weizsacker-Williams photon) with unpolarized and polarized electron beams at the CLIC. We have obtained 95$\%$ confidence level bounds on the anomalous magnetic and electric dipole moments for various values of the integrated luminosity and center-of-mass energy. Improved constraints of the anomalous magnetic and electric dipole moments have been obtained compared to the LEP sensitivity.


I. INTRODUCTION
The spin and magnetic moment of the electron arise from the aberration of atoms embedded in a magnetic field, in which fine structure is observed by using spectroscopic methods [1,2]. The magnetic moment of the electron with spin s, electric charge e and mass m can be written in the general form as µ = g (eℏ/2mc) s, where the coefficient g, being α fine structure constant, is stated as g ∼ = 2 1 + (α/2π) − 3.28 (α/π) 2 ∼ = 2.0023192. The g is the Landé g-factor for the electron's spin. The obtainment of g = 2 is possible in consideration of the Dirac equation. The deviation from the Dirac value a e = (g − 2) /2 measured by spin precession methods is known as the anomalous magnetic moment. The first detailed investigation on anomalous magnetic moment of the electron was perfomed within the framework of Quantum Electrodynamics (QED) using radiative corrections by Schwinger in 1948, as α e = α/2π [3]. Ever since, many experimental and theoretical works have been devoted to improve for a-values of particles [4]. It should be noted that the studies on anomalous magnetic moment of particles have provided a precision test of the Standard Model (SM) and gives point of view "new physics" effects [5]. However the anomalous magnetic moment a µ of the muon is important to examine the SM and in establishing of alternative theories to the SM. While a e and a µ can be determined by spin precession experiments the anomalous magnetic moment of the tau cannot be measured by same method due to short lifetime 2.906 × 10 −13 s of the tau lepton [6]. So, no direct measurement of the tau lepton's anomalous magnetic moment exists up to now. Producing tau lepton is performed by collision experiments with accuracy. The theoretical value of the a τ from QED by considering higher loop corrections is given as a SM τ = 0.001177 [7,8]. The experimental bounds on the the a τ are provided by, L3: −0.052 < a τ < 0.058, OPAL: −0.068 < a τ < 0.065, DELPHI: −0.052 < a τ < 0.013 collaborations at the LEP at 95% C.L. [9][10][11].
The CP violation effects have important results on the coupling of the tau lepton. K o L mesons enable a perfect experimental system to test CP invariance. A pure sample of the long-lived species can be formed by using a long enough beam. In such a system, if a 2π decay is observed, it is clear that CP has been violated [12]. Here this phenomenon can be adapted to the SM by including an experimental phase factor to Cabibbo-Kabayashi-Maskawa matrix in the quark sector [13]. However, while there is no question CP violation in the lepton sector in the SM, CP violation in the quark sector leads to form a very small contribution to CP violation of the leptonic interactions via multi-loop effects. At least three-loop are needed to enable corresponding contribution of the SM to the electric dipol moment of the tau lepton, in which estimated value range is obtained as |d τ | ≤ 10 −34 e cm [14]. Neutrino mixing with different masses are another source of CP violating in the lepton sector as similar to the CP violation in the quark sector [15].
Additional sources beyond the SM for the CP violation in the lepton sector are leptoquark [16,17], SUSY [18], left-right symmetric [19,20] and Higgs-Multiplets [21,22]. In this study we have examined the processes eγ → ν e τν τ (γ is the Compton backscattering photon) and e − e + → e − γ * e + → ν e τν τ e + (γ * is the Weizsacker-Williams photon). There are three Feynman diagrams of these processes to be used as mediators W + , τ, e. τ τ γ vertex belongs to only a diagram in these Feynman diagrams. The main motivation of the present study is on how to investigate τ τ γ vertex contribution with anomalous electromagnetic form factors to the SM. In this manner, the electromagnetic vertex factor in characterizing the interaction of tau lepton with photon can be parametrized by with σ νµ = i 2 (γ ν γ µ − γ µ γ ν ), where q and m τ are the photon momentum and the mass of tau lepton, respectively. F 1,2,3 (q 2 ) are the electric charge, the anomalous magnetic dipole and the electric dipole form factors of the tau lepton. As well known, when F 1 = 1, F 2 = F 3 = 0, the electromagnetic vertex factor is reduced to vertex factor in the SM. However, due to τ τ γ vertex, in other words, contributions from loop effects or arising from the new physics, F 2 and F 3 could not be taken as zero [23][24][25]. When considering the limiting case of q 2 → 0, the form factors become in which new case relates to the static properties of the fermions.
On the other hand, it should be noted that the Compact Linear Collider (CLIC) is a collider with the range 0.5 − 3 TeV in order to perform e + e − collision that is planned to be constructed at a future date [26]. In this study, as previously mentioned briefly, we have probed the anomalous magnetic and electric dipole moments of the tau lepton by considering at a future linear collider with high-energy and high-luminosity, such as the CLIC [27]. Beam polarization could play essential role in the next linear colliders as well as RHIC and HERA.
It is expected that 80 % polarization of lepton beam can be achievable at the future linear colliders [28]. In this work, we take into account one beam can be 80% polarization (-80% means that eighty of percent are left polarized).
In this paper, we have examined the anomalous magnetic and electric dipole moments of the tau lepton through the processes e − γ → ν e τν τ and e − e + → e − γ * e + → ν e τν τ e + . Linear colliders can be constructed γγ and eγ collider modes with real photons. This real photon beam is obtained by the Compton backscattering of laser photons off linear electron beam.
Moreover, most of these photon beams can be high energy region [29,30]. The linear colliders make it possible to eγ * and γ * γ * interactions to examine new physics beyond the SM. The photons which are emitted from the incoming electrons scatter at very small angels from the beam pipe. Therefore, these photons have very low virtuality and we say that these photons "almost-real". These processes take place as follows: Almost-real photons (γ * ) are emitted from the incoming electrons (positrons) and interact other photon or electrons (positrons).
Then, the γ * γ * or eγ * interacts with each other to produce X through the subprocess γ * γ * → X or e − γ * → X. These photon distributions can be examined phenemenologically via the WWA approximation in the literature [36][37][38][39][40][41][42][44][45][46]. The WWA has simple formulas then it provides to get simple numerical estimations [50]. Morever, this method gives a facility in the experiments because it permits to give cross section for the process e − γ * → X approximately through the study of the main e − e + → e − γ * e + → Xe + process. Here, X represents particles obtained in the final state. Also, these interactions have a very clean experimental conditions, since they contain only electromagnetic interactions. With these motivations, we have obtained the sensitivity bounds on new physics parameters through the e − γ → ν e τν τ and e − e + → e − γ * e + → ν e τν τ e + in next subsections.
In the next section, we briefly outline of our numerical calculation details and results.
Final section is devoted to our summary and conclusions.

A. Analysis with Compton Backscattered Photons
In this subsection, we show the numerical results for the e − γ → ν e τν τ . All numerical analysis, we have used the CalcHEP package [47,48]. This program allows automatic calculations of the distributions and cross sections in the SM as well as their extensions at the tree level. We have considered √ s = 1.4 TeV and 3 TeV CLIC center-of-mass energies in our calculations. We carry out the following cuts on the final state leptons in all of our calculations |η| < 2.5. In sensitivity analysis, we take into account χ 2 method, where σ(F 1 , F 2 ) is the total cross section which includes SM and new physics, δ stat = 1/ √ N is the statistical error and N = σ SM × BR × L int . We assume that tau lepton decays into hadrons hence we take BR = 0.65 in all calculations.
In Table I where σ ab represents the obtained cross section with fixed helicities a for positron and b for the electron. P e − and P e + are the polarization degree of the electron and positron, respectively. In our numerical calculations we take into account only left-handed electrons since the examined process includes only a weak interaction. Hence, we have applied the electron polarization as P e − = −80%. To this purpose, we give 95% C.L. sensitivity bounds on the anomalous a τ and d τ couplings in Table II here  (P e − = −80%) extends to wider regions than the cases of the unpolarized electron beams.

B. Analysis with Weizsacker-Williams Photons
We have analyzed the anomalous dipole moments of the tau lepton via the main process e − e + → e − γ * e + → ν e τν τ e + in this subsection. In Table III, we present the 95% C.L. sensitivity bounds on the anomalous a τ and d τ parameters for the unpolarized electron beams. We can obtain from the table that the sensitivity bounds of the anomalous couplings enhance with the increasing center-of-mass energy and luminosity. The obtained bounds for the a τ are also better than the current experimental limits. On the other hand, bounds with Compton backscattered photon (Tabel I) sensitive than the bounds for Weizsacker-Williams approximation (Tabel III). Main reason of this situation is the Compton backscattered photon spectrum gives higher effective than the Weizsacker-Williams photon spectrum in high energy regions [29,33]. However, the application of the WWA gives a lot of benefits in experimental and phenomenological studies as mentioned in Section I. We present the 95% C.L. sensitivity bounds on the anomalous a τ and d τ parameters for the P e − = −80% polarized electron beams in Table IV Fig.7 (Fig.8) same as the Fig. 5 (Fig.6) but for the polarized electron beams. Limits for polarized case strong compared with unpolarized case, however, that bounds weak compared with the compton backscattered case.

III. CONCLUSION
We have analyzed tau lepton anomalous dipole moments through the processes e − γ → ν e τν τ and e − e + → e − γ * e + → ν e τν τ e + . These processes have very clean environment.
Therefore, deviation of the anomalous couplings from the expected values of the SM would evidence the existence of the new physics. We have found that e − γ → ν e τν τ process give better bounds than the other. However, the processes which are include γγ and e − γ interactions require new apparatus. On the other hand, e − γ * and γ * γ * are occurred spontaneously without any new apparatus.
Additionally, we have used polarized and unpolarized electron beam in our study. We have seen that the polarization enhances the sensitivity bounds. Our results for the a τ better than the current experimental limits. Based on the obtaining of this paper, it is concluded that, CLIC provides new opportunities for examination of the tau physics beyond the SM with using eγ and eγ * mode.

IV. ACKNOWLEDGEMENTS
This work has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK) in the framework of Project No. 115F136. [50] C. Carimalo, P. Kessler and J. Parisi, Phys.Rev. D 20 1057 (1979).