Model-independent study for the tau-neutrino electromagnetic dipole moments in $e^+e^- \to \nu_\tau \bar \nu_\tau \gamma$ at the CLIC

We conduce a study to probe the sensitivity of the process $e^+e^-\rightarrow (\gamma, Z) \to \nu_\tau \bar \nu_\tau \gamma$ to the total cross section, the magnetic moment and the electric dipole moment of the tau-neutrino in a model-independent way. For this study, the beam polarization facility at the Compact Linear Collider (CLIC) along with the typical center-of-mass energies $\sqrt{s}=380-3000\hspace{0.8mm}GeV$ and integrated luminosities ${\cal L}=10-3000\hspace{0.8mm}fb^{-1}$ is considered. We estimate the sensitivity at the $95\%\hspace{1mm}$ Confidence Level (C.L.) and systematic uncertainties $\delta_{sys}=0, 5, 10\hspace{1mm}\%$ on the dipole moments of the tau-neutrino. It is shown that the process under consideration $e^+e^-\rightarrow (\gamma, Z) \to \nu_\tau \bar \nu_\tau \gamma$ is a good prospect for study the dipole moments of the tau-neutrino at the CLIC. Also, our study illustrates the complementarity between CLIC and other $e^+e^-$ and $pp$ colliders in probing extensions of the Standard Model, and shows that the CLIC at high energy and high luminosity provides a powerful means to sensitivity estimates for the electromagnetic dipole moments of the tau-neutrino.


I. INTRODUCTION
Investigations of the theory and phenomenology of neutrino electromagnetic properties continue to be a very active field of interests to both theoretical and experimental physicists.
In particular, the study of the Magnetic Moment (MM) and the Electric Dipole Moment (EDM) of the neutrino has been challenging High Energy Physics community, both Theoretical and Experimental in recent decades. In the original formulation of the Standard Model (SM) [1][2][3] neutrinos are massless particles with zero MM. However, neutrino flavour oscillation experiments from several sources indicate that neutrinos have non-zero mass, which indicates the necessity of extending the SM to accommodate massive neutrinos. In the minimal extension of the SM to incorporate the neutrino mass, the MM of the neutrino is known to be developed in one loop calculation [4,5], and the non-zero mass of the neutrino is essential to get a non-vanishing MM. Furthermore, the SM predicts CP violation, which is necessary for the existence of the EDM of a variety physical systems. The EDM provides a direct experimental probe of CP violation [6][7][8], a feature of the SM and beyond the SM (BSM) physics. The signs of new physics can be analyzed by investigating the electromagnetic dipole moments of the tau-neutrino, such as its MM and EDM.
The present best upper limits on the MM and the EDM of the neutrinos, either set directly by experiments or inferred indirectly from observational evidences combined with theoretical arguments, are several orders of magnitude larger than the predictions of the minimal extension of the SM [4,5,9]. Therefore, if any direct experimental confirmation of non-zero MM is obtained in the laboratory experiments, it will open a window to new physics.
In addition, the dipole moments with the copious amount of neutrinos in the Universe will have significant implications for astrophysics and cosmology, as well as terrestrial neutrino experiments [10,11]. One of the most sensitive experimental observables to the CP violation BSM is the EDM [12][13][14][15]. The search for new sources of CP violation BSM is currently one of the most important fundamental problems of particle physics to be solved. A. Sakharov proposed a solution to this problem [16], the present interaction has to violate a fundamental symmetry of nature: the CP symmetry. The excess of matter over antimatter, or the baryon number asymmetry, was generated in the early Universe by a theory satisfy Sakharov criteria.
Another interesting topics in neutrino physics is to determine its Dirac or Majorana nature. For respond to this, experimentalist are exploring different reactions where the Majorana nature may manifest [17]. About this topic, the study of neutrino magnetic moments is, in principle, a way to distinguish between Dirac and Majorana neutrinos since the Majorana neutrinos can only have flavor changing, transition magnetic moments while the Dirac neutrinos can only have flavor conserving one.
These sensitivity limits exceed by many orders of magnitude the minimally extended SM prediction given by where µ B = e 2me is the Bohr magneton [4,5]. The best world sensitivity bounds for the electric dipole moments d νe,νµ [25] are: For the τ -neutrino, the bounds on their dipole moments are less restrictive, and therefore it is worth investigating in deeper way their electromagnetic properties. The tau-neutrino correspond to the more massive third generation of neutrinos and possibly possesses the largest mass and the largest MM and EDM. As a consequence, this leaves space for the study of new physics BSM.
A summary of experimental and theoretical limits on the dipole moments of the tauneutrino are given in Table I [49][50][51]. When is constructed and enter into operation, the γe − and γγ collision modes will be studied. The CLIC will be a multi-TeV collider and will be operate in three energy stages, corresponding to center-ofmass energies √ s = 380, 1500, 3000 GeV , and it is an ideal machine to study new physics BSM.
Motivated by the extensive physical program of the CLIC, we conduce a comprehensive study to probe the sensitivity of the process e + e − → (γ, Z) → ν τντ γ to the total cross section, the MM and the EDM of the tau-neutrino in a model-independent way. For the study, the beam polarization facility at the CLIC along with the typical center-of-mass energies √ s = 380, 1500, 3000 GeV and integrated luminosities L = 10, 50, 100, 300, 500, 1000, 1500, 2000, 3000 f b −1 are considered. In addition, we estimate the sensitivity at the 95% C.L. and systematic uncertainties δ sys = 0, 5, 10 % on the dipole moments of the τ -neutrino. It is shown that the process under consideration e + e − → (γ, Z) → ν τντ γ is a good prospect for study the dipole moments of the tau-neutrino at the CLIC. Furthermore, our study illustrates the complementarity between CLIC and other e + e − and pp colliders in probing extensions of the SM, and shows that the CLIC at high energy and high luminosity provides a powerful means to sensitivity estimates for the electromagnetic dipole moments of the tau-neutrino.
The content of this paper is organized as follows: In Section II, we study the total cross section and the dipole moments of the tau-neutrino through the channel e + e − → (γ, Z) → ν τντ γ. Finally we conclude in Section III. Theoretically the electromagnetic properties of neutrinos best studied and well understood are the MM and the EDM. Despite that the neutrino is a neutral particle, neutrinos can interact with a photon through loop (radiative) diagrams. However, a convenient way of studying its electromagnetic properties on a model-independent way is through the effective neutrino-photon interaction vertex which is described by four independent form factors. The most general expression for the vertex of interaction ν τντ γ is given in Refs. [52][53][54]. For the study of the MM and the EDM of the tau-neutrino we following a focusing as the performed in our previous works [26-34, 36-38, 41, 42] with where e is the electric charge of the electron, m ντ is the mass of the tau-neutrino, q µ is the photon momentum, and F 1,2,3,4 (q 2 ) are the four electromagnetic form factors of the neutrino.
In general the F 1,2,3,4 (q 2 ) are independent form factors, and they are not physical quantities, but in the limit q 2 → 0 they are quantifiable and related to the static quantities corresponding to charge radius, MM, EDM and anapole moment (AM) of the Dirac neutrinos, respectively [44,[55][56][57][58][59][60]. In this paper we study the anomalous MM µ ντ and the EDMt d ντ of the tau-neutrino, which are defined in terms of the F 2 (q 2 = 0) and F 3 (q 2 = 0) independent form factor as follows: as we mentioned above. The form factors corresponding to charge radius and the anapole moment, are not considered in this paper. The corresponding Feynman diagrams for the signal e + e − → (γ, Z) → ν τντ γ are given in Fig. 1. The total cross section of the process e + e − → ν τντ γ with unpolarized electronpositron beam is computed using the CALCHEP 3.6.30 [61] package, which can computate the Feynman diagrams, integrate over multiparticle phase space and event simulation.
Furthermore, in order to select the events we implementing the standard isolation cuts, compatibly with the detector resolution expected at CLIC: we apply these cuts to reduce the background and to optimize the signal sensitivity. In Eq.
(8), p ν T is the transverse momentum of the final state neutrinos, η γ is the pseudorapidity and p γ T is the transverse momentum of the photon. The outgoing particles are required to satisfy these isolation cuts.
Formally, the e + e − → (γ, Z) → ν τντ γ cross section can be split into two parts: where σ BSM is the contribution due to BSM physics, which, in our case it comes from the anomalous vertex ν τντ γ, while σ 0 is the SM prediction. The analytical expression for the squared amplitudes are quite lengthy so we do not present it here. Following the form of Eq. (9), we present numerical fit functions for the total cross section with respect to center-of-mass energy, with unpolarized electron-positron beam and in terms of the independent form factors F 2 (F 3 ).
• For It is worth mentioning that in equations for the total cross section (10)- (12), the coefficients of F 2 (F 3 ) given the anomalous contribution, while the independent terms of F 2 (F 3 ) correspond to the cross section at F 2 = F 3 = 0 and represents the SM total cross section magnitude.
C. Sensitivty estimates on the µ ντ and d ντ with unpolarized electron-positron beam Based on the formulas given by Eqs. (10)-(12), we make model-independent sensitivity estimates for the total cross section of the signal as well as for the anomalous MM µ ντ and EDM d ντ of the τ -neutrino at the CLIC. To carry out this task, we consider the acceptance cuts given in Eq. (8) and we take into account the systematic uncertainties δ sys = 0, 5, 10 % for the collider. In addition, to sensitivity estimates on the parameters of the process e + e − → ν τντ γ, we use the χ 2 function [26,27,48,[62][63][64][65][66][67] where σ BSM ( √ s, µ ντ , d ντ ) is the total cross section including contributions from the SM and new physics, δ st = 1 √ N SM is the statistical error and δ sys is the systematic error. The number of events is given by As stated in the Introduction, to carry out our study we considered the typi- tends to zero, recovering the value of the SM as it is shown in Eq. (12).
Sensitivity contours at the 95% C.L. in the F 3 − F 2 plane for the signal e + e − → ν τντ γ with center-of-mass energies √ s = 380, 1500, 3000 GeV and luminosities L = 10, 100, 500, 1500, 3000 f b −1 are given in Figs. 4-6. As highlighted in Fig. 6, the three most sensitive contours for F 2 and F 3 they are the corresponding ones for high energy and high luminosity of √ s = 3000 GeV and L = 3000 f b −1 .

We show our results in
where P e − (P e + ) is the polarization degree of the electron (positron) beam, while σ −+ stands for the cross section for completely left-handed polarized e − beam P e − = −1 and completely the process e + e − → ν τντ γ for √ s = 380 GeV and P e − = P e + = 0%.  For our sensitivity study, we assuming for definiteness an electron-positron beam polarization (P e − , P e + ) = (−80%, 60%) in the estimated range of the expected CLIC operation setup. Besides the polarized beams we consider the isolation cuts given for Eq. (8).
The numerical fit functions for the total cross sections of the process e + e − → ν τντ γ, following the form of Eq.(9) with polarized electron-positron beam, and in terms of the independent form factors F 2 (F 3 ) are given by: the process e + e − → ν τντ γ for √ s = 1500 GeV and P e − = P e + = 0%.

III. CONCLUSIONS
In this paper, we have sensitivity estimates on the total cross section and on the dipole moments µ ντ and d ντ through the process e + e − → ν τντ γ at the future CLIC. Furthermore, the process is analyzed for two scenarios motivated by the strong advantage in searching for new physics BSM: a) unpolarized electron-positron beam (P e − , P e + ) = (0, 0) and b) polarized electron-positron beam (P e − , P e + ) = (−80%, 60%). In the first scenario, the unpolarized the process e + e − → ν τντ γ for √ s = 380 GeV , P e − = −80% and P e − = 60%.
90% C.L. √ s = 380 GeV δ sys = 0% δ sys = 5% δ sys = 10%  Comparing each scenario shows that the cross section is enhanced for 100 pb for the case of polarized electron-positron beam. The option of upgrading the incoming electron and the positron beam to be polarized has the power to enhance the potential of the machine.
In addition to these, the results for the sensitivity contours in the F 2 − F 3 plane for the unpolarized and polarized case are presented (see Figs. 4-6 and 9-11). the process e + e − → ν τντ γ for √ s = 1500 GeV , P e − = −80% and P e + = 60%.
In conclusion, the process itself is very useful to sensitivity probing on the dipole moments of the tau-neutrino and illustrates the complementarity between CLIC and other e + e − and pp colliders for probing extensions of the SM. Furthermore, we hope that this work will motivate further studies of the e + e − → ν τντ γ process, using in particular polarized electropositron beams.   Fig. 6, but for P e − = −80% and P e + = 60%.