Sensitivity on the Dipole Moments of the τ-Neutrino at e + e − Colliders : ILC and CLIC

We study the sensitivity on the anomalous magnetic and electric dipole moments of the τ-neutrino at a high-energy and highluminosity linear electron positron collider, such as the ILC or CLIC, through the reaction e+e− → ]]γ. We obtain limits on the dipole moments at the future linear colliders energies. For integrated luminosities of 500fb and center of mass energies between 0.5 and 3 TeV, the future e+e− colliders may improve the existing limits by two or three orders of magnitude.


Introduction
In the standard model (SM) [1][2][3] extended to contain righthanded neutrinos, the neutrino magnetic moment induced by radiative corrections is unobservably small,  ] ≃ 3.1 × 10 −19 ( ] /1 eV)  [4].Current limits on these magnetic moments are several orders of magnitude larger, so that a magnetic moment close to these limits would indicate a window for probing effects induced by new physics beyond the SM [5].Similarly, a neutrino electric dipole moment will also point to new physics and will be of relevance in astrophysics and cosmology, as well as terrestrial neutrino experiments [6].Some bounds on the neutrino magnetic moment are shown in Table 1.
Gould and Rothstein [7] reported in 1994 a bound on  ]  obtained through the analysis of the process  +  − → ]], near the -resonance, with a massive neutrino and the SM  +  − and ]] couplings.
At higher , near the  pole  ≈  2   , the dominant contribution involves the exchange of a  boson.The dependence on the magnetic moment ( ]  ) and the electric dipole moment ( ]  ) comes from the radiation of the photon observed by the neutrino or antineutrino in the final state.However, in order to improve the limits on the magnetic moment and the electric dipole moment of the tau-neutrino, in our calculation of the process  +  − → ]] we consider the contribution that involves the exchange of a virtual photon.In this case, the dependence on the magnetic moment comes from a direct coupling to the virtual photon, and the observed photon is a result of initial-state Bremsstrahlung.The Feynman diagrams which give the most important contribution to the cross section are shown in Figure 1.
Our aim in the present paper is to analyze the reaction  +  − → ]] in the framework of the standard model minimally extended to include massive Dirac neutrinos and we attribute an anomalous magnetic moment (MM) and an electric dipole moment (EDM) to a massive tau-neutrino.This process sets limits on the tau-neutrino MM and EDM.In this paper, we take advantage of this fact to set limits on  ]  and  ]  for integrated luminosities of 500 fb −1 and center of mass energies between 0.5 and 3 TeV, that is to say, in the next generation of linear colliders, namely, the International Linear Collider (ILC) [8] and the Compact Linear Collider (CLIC) [9].
angular range (white histogram).From the same figure in this angular interval, that is, −0.7 < cos   < 0.7, we see that only 6 events were found; this is the real background, not 14 events.In this case a simple method [11][12][13] is that at 1 level (68% C.L.) for a null signal the number of observed events should not exceed the fluctuation of the estimated background events:  =   +√  .Of course, this method is good only when   is sufficiently large (i.e., when the Poisson distribution can be approximated with a gaussian [11][12][13]) but for   > 10 it is a good approximation.This means that at 1, 2, and 3 level (68%, 90%, and 95% C.L.) the limits on the nonstandard parameters are found replacing the equation for the total number of events expected  =   + √  in the expression  = ( ]  ,  ]  )L.The distributions of the photon energy and the cosine of its polar angle are consistent with SM predictions.This paper is organized as follows.In Section 2 we present the calculation of the process  +  − → ]] in the context of the standard model minimally extended to include massive Dirac neutrinos.Finally, we present our results and conclusions in Section 3.

The Total Cross Section
In this section we calculate the total cross section for the reaction  +  − → ]].The respective transition amplitudes are thus given by where is the neutrino electromagnetic vertex,  is the charge of the electron,   is the photon momentum, and  1,2,3 ( 2 ) are the electromagnetic form factors of the neutrino, corresponding to charge radius, MM, and EDM, respectively, at  2 = 0 [19,20], while    is the polarization vector of the photon.() and   (  ) stand for the momentum of the virtual neutrino (electron) and antineutrino (positron), respectively.
The MM and EDM give a contribution to the total cross section for the process  +  − → ]] of the following form: Advances in High Energy Physics 3 where   ≡ sin 2   and   , cos   are the energy and the opening angle of the emitted photon.
The expression given for the first term of (3) corresponds to the cross section previously reported by Gould and Rothstein [7], while the second term comes from the contribution that involves the exchange of a virtual photon and the observed photon is radiated to the initial electron or positron.

Results and Conclusions
In order to evaluate the integral of the total cross section as a function of the parameters of the model, that is to say,  ]  and  ]  , we require cuts on the photon angle and energy to avoid divergences when the integral is evaluated at the important intervals of each experiment.We integrate over   from 44.5 ∘ to 135.5 ∘ and   from 15 GeV to 100 GeV.Using the following numerical values: sin 2   = 0.2314,   = 91.18GeV, and Γ  = 2.49 GeV, we obtain the cross section  = ( ]  ,  ]  ).
As was discussed in [7,10,21,22],  ≈ ( ]  ,  ]  )L, where  =   + √  is the total number of  +  − → ]] events expected at 1, 2, and 3 level as is mentioned in the introduction and L = 500 fb −1 according to the data reported by the ILC and CLIC [8,9].Taking this into consideration, we can obtain a limit for the tau-neutrino magnetic moment with  ]  = 0.
The previous analysis and comments can readily be translated to the EDM of the -neutrino with  ]  = 0.The resulting limits for the EDM as a function of the center of mass energy are shown in Table 2.
In order to see the sensitivity of the photon energy and the magnetic moment to new physics, in Figure 2 we plot the differential cross section versus (  ,  ]  ) for the limits of the magnetic moment given in Table 2.We see that the differential cross section strongly depends on both the photon energy   and the magnetic moment  ]  .
Figure 3 shows the total cross section for  +  − → ]] as a function of the center of mass energy √ and different values representative of the magnetic moment, which are reported in the literature, that is to say,  ]  = 3.3 × 10 −6   (L3), 5.4 × 10 −7   (BEBC (CERN)), 1.68 × 10 −8   (Table 2).Starting from a center of mass energy just greater than the  mass, a minimum around √ ≃ 200 GeV occurs due to the SM boson resonance tail on the high energies.For different values of the parameter  ]  the shape of the curves does not change and there is only a shift of these depending on the value of the magnetic moment.Finally, in Figure 4 we present the dependence of the sensitivity limits of the magnetic moment  ]  with respect to the collider luminosity L for three different values of the center of mass energy, √ = 0.5, 1, 1.5 TeV.
In conclusion, we have found that the process  +  − → ]] in the context of the standard model minimally extended to include massive Dirac neutrino at the high energies and luminosities expected at the ILC/CLIC colliders can be used to probe for bounds on the magnetic moment  ]  and electric dipole moment  ]  .In particular, we can appreciate that the 95% C.L. sensitivity limits expected for the magnetic moment at 0.5-1.5 TeV center of mass energies already can provide  proof of these bounds of order 10 −8 -10 −9 , that is to say, 2-3 orders of magnitude better than those reported in the literature:  ]  < 3.3 × 10 −6   (90% C.L.) from a sample of  +  − annihilation events collected with the L3 detector at the  resonance corresponding to an integrated luminosity of 137 pb −1 [10];  ]  ≤ 2.7 × 10 −6   (95% C.L.) at  2 =  2  from measurements of the  invisible width at LEP [19];  ]  ≤ 2.62 × 10 −6 in the effective Lagrangian approach at the  pole [36];  ]  < 1.83 × 10 −6   (90% C.L.) from the analysis of  +  − → ]] at the -pole, in a class of  6 inspired models with a light additional neutral vector boson [37]; from the order of  ]  <  (1.1 × 10 −6   ) Akama et al. derive and apply model-independent limits on the anomalous magnetic moments and the electric dipole moments of leptons and quarks due to new physics [38].However, the limits obtained in [38] are for the tau-neutrino with an upper bound of   < 18.2 MeV which is the current experimental limit.It was pointed out in [38], however, that the upper limit on the mass of the electron neutrino and data from various neutrino oscillation experiments together imply that none of the active neutrino mass eigenstates is heavier than approximately 3 eV.In this case, the limits given in [38] are improved by seven orders of magnitude.The limit  ]  < 5.4 × 10 −7   (90% C.L.) is obtained at  2 = 0 from a beam-dump experiment with assumptions on the   production cross section and its branching ratio into ]  [39], thus severely restricting the cosmological annihilation scenario [40].Our results in Table 2 compare favorably with the limits obtained by the L3 Collaboration [10], and with other limits reported in the literature [7,10,19,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].
Finally, it seems that in order to improve these limits it might be necessary to study direct CP-violating effects [41].In addition, the analytical and numerical results for the cross section could be of relevance for the scientific community.

Figure 2 :Figure 3 :
Figure 2: The differential cross section for  +  − → ]] as a function of   and  ]  .

Figure 4 :
Figure 4: Dependence of the sensitivity limits at 95% C.L. for the anomalous magnetic moment for three different values of the center of mass energy in the process  +  − → ]].

Table 1 :
Bounds on the neutrino magnetic moment.