Decay of polarized muon at rest as a source of polarized neutrino beam

In this paper, we indicate the theoretical possibility of using the decay of polarized muons at rest as a source of the transversely polarized electron antineutrino beam. Such a beam can be used to probe new effects beyond standard model. We mean here new tests concerning CP violation, Lorentz structure and chirality structure of the charged current weak interactions. The main goal is to show how the energy and angular distribution of the electron antineutrinos in the muon rest frame depends on the transverse components of the antineutrino beam polarization. Our analysis is model-independent and consistent with the current upper limits on the non-standard couplings. The results are presented in a limit of infinitesimally small mass for all particles produced in the decay.


Introduction
Decay of polarized muon at rest (DPMaR) is the very appropriate process to test both the time reversal violation (TRV) and the space-time, and chirality structure of the purely leptonic charged weak interactions (PLCWI). Moreover, we investigate here if the DPMaR can also be used as a strong source of the transversely polarized antineutrino (neutrino) beam which would be scattered on the polarized electron target (PET). A detailed analysis of new effects beyond the Standard Model (SM) of electroweak interactions [1,2,3] is carried out in [4]. Ciechanowicz et al. show that the scattering of the left-chirality and longitudinally polarized neutrino beam on the PET would allow to test the possibility of CP-breaking in the (ν µ e − ) scattering. The measurement of the azimuthal asymmetry of recoil electrons could detect the CP-violating phase between the standard complex vector and axial-vector couplings. The other problem considered in [4] concerns the scattering of the transversely polarized mixture of left-chirality and right-chirality neutrinos on the PET. If such a neutrino beam would be scattered, the dependence on the angle between the transverse neutrino spin polarization of incoming beam and the transverse electron polarization of target in the recoil electron energy spectrum could be tested. That would be a direct signature of the right-chirality neutrinos. As is well-known, the SM of electro-weak interactions has a vector-axial (V-A) structure [5] which has been put by hand in order to obtain agreement with experiments. This means that only left-chirality Dirac neutrinos may take part in the charged and neutral current weak interaction. This structure follows among other from the low energy measurements of the muon decay such as; the spectral shape, angular distribution, and polarization of the outgoing electrons (positrons). At present, there is no evidence for the deviations from the SM for the Michel parameters. The neutrino oscillation experiments indicate the non-zero neutrino mass and provide first evidence for physics beyond the minimal SM. On the other hand, the experimental precision of present tests still allows the participation of the exotic scalar S, tensor T and pseudoscalar P couplings of the right-chirality Dirac neutrinos beyond the SM [6]. The KARMEN experiment [7] has measured the energy distribution of electron neutrinos emitted in positive muon decay at rest (µ + → e + + ν e + ν µ ). The obtained result is in agreement with the SM prediction on the neutrino Michel parameter ω L = 0. They get for the first time a 90% confidence upper limit of ω L ≤ 0.113, which leads to a limit of |g S RL + 2g T RL | ≤ 0.78 for the interference between the scalar and tensor couplings. The current upper limits on the all non-standard couplings, obtained from the normal and inverse muon decay, are presented in the Table 1 [8]. The coupling constants are denoted as g γ ǫµ , where γ = S, V, T indicates the type of weak interaction, i.e. scalar S, vector V, tensor T; ǫ, µ = L, R indicate the chirality of the electron or muon and the neutrino chiralities are uniquely determined for given γ, ǫ, µ. It means that the neutrino chirality is the same as the associated charged lepton for the V Coupling constants SM Current limits Table 1 Current limits on the non-standard couplings.
interaction, and opposite for the S, T interactions [8]. In the SM, only g V LL is non-zero value.
It is necessary to point out that the existence of the exotic right-chirality neutrinos in the few keV region, that are sterile in SM, can have numerous consequences in astrophysics and cosmology. We mean here the mechanism of neutrino "spin flip" in the Sun's convection zone in order to explain the observed deficit of the solar neutrinos [9]. In addition, the sterile neutrinos could also account for pulsar kicks (high pulsar velocities), could explain all or some fraction of the dark matter in the Universe and would affect emission of supernova neutrinos [10]. Recent analysis carried out by Erwin et al. for the muon decay [11] shows that there exist four-fermion operators that do not contribute to the neutrino mass matrix through radiative corrections. These operators generate the exotic couplings g S,T LR,RL , while all operators generating the vector couplings g V LR,RL contribute to the neutrino mass matrix. So far the CP violation is observed only in the decays of neutral K-and B-mesons [12], and is described by a single phase of the Cabibbo-Kobayashi-Maskawa quark-mixing matrix (CKM) [13]. There is no experimental evidence on the TRV in the PLCWI, e.g. the muon decay and neutrino-electron elastic scattering. However, the baryon asymmetry of the Universe can not be explained by the CKM phase only, and new sources of the CP violation are required [14]. According to the prediction of non-standard models, the effects of new CP-breaking phases could be measured in observables where the SM CP-violation is suppressed, while alternative sources can generate a sizable effect, e.g. the electric dipole moment of the neutron, the transverse lepton polarization in three-body decays of charged kaons K + [15,16], transverse polarization of the electrons emitted in the decay of polarized 8 Li nuclei [17]. The other possibility of measuring the exotic CP-breaking phases is to use the neutrino observables which consist only of the interference terms between the standard coupling of the left-chirality neutrinos and exotic couplings of the right-chirality neutrinos and do not depend on the neutrino mass. We mean here both T-even and T-odd transverse components of the neutrino spin polarization. At present, the direct tests are still impossible. The possible solution can be the scattering of the transversely polarized (anti)neutrino beam, coming from the polarized muon decay at rest, on the PET and the measurement of the maximal asymmetry of the cross section, [4]. Left-right symmetric models (LRSM) and composite models (CM) can be proposed as an example of the non-standard models of purely leptonic weak interactions, in which the exotic couplings of the right-chirality neutrinos (antineutrinos) can appear. Recently TWIST Collaboration [18] has measured the Michel parameter ρ in the normal µ + decay and has set new limit on the W L − W R mixing angle in the LRSM. Their result ρ = 0.75080 ± 0.00044(stat.) ± 0.00093(syst.)±0.00023 is in good agreement with the SM prediction ρ = 3/4, and sets new upper limit on mixing angle |χ| < 0.030 (90% CL). The CM have been proposed to probe the scale for compositeness of quarks and leptons. Lagrangian of the new effective contact interactions (CI) for the muon decay includes among other the contributions from the standard vector coupling of the left-chirality neutrinos and exotic scalar coupling of the right-chirality ones [19]. Our analysis is model-independent and the calculations are made in the limit of infinitesimally small mass for all particles produced in the muon decay. The density operators [20] for the polarized initial muon and for the polarized outgoing electron antineutrino are used, see Appendix A. We use the system of natural units with = c = 1, Dirac-Pauli representation of the γ-matrices and the (+, −, −, −) metric [21].

Transversely Polarized Electron-Antineutrino Beam
To show how the energy and angular distribution of the electron antineutrinos may depend on the angle between the transverse antineutrino spin polarization and the muon polarization vectors, we assume that the DPMaR (µ − → e − + ν e + ν µ ) is a source of the electron antineutrino beam. It is worth to notice that if one takes into account the positive muon decay (µ + → e + + ν µ + ν e ), the electron neutrino beam is produced. The production plane is spanned by the direction of the initial muon polarizationη µ and of the outgoing electron  Table 2  Table shows  antineutrino momentumq, Fig. 1. We admit a presence of the exotic scalar g S LR and tensor g T LR couplings in addition to the standard vector g V LL coupling. It means that the outgoing electron antineutrino flux is a mixture of the leftchirality antineutrinos produced in the g V LL weak interaction and the rightchirality ones produced in the g S LR and the g T LR weak interactions. Because our analysis is carried out in the limit of vanishing antineutrino mass, the left-chirality antineutrino has positive helicity, while the right-chirality one has negative helicity, see [22]. The muon neutrino is always left-chirality both for the g V LL and the g S LR , g T LR couplings (muon neutrino has negative helicity, when m νµ → 0). In the SM, only g V LL is non-zero value. The table 2 displays explicitly that there are only two non-zero interferences between the standard coupling g V LL and exotic couplings, i. e. g S LR and g T LR . Because we allow for the non-conservation of the combined symmetry CP, all the coupling constants g V LL , g S LR , g T LR are complex. The amplitude for the polarized muon decay is of the form: where v νe and u e (u µ and u νµ ) are the Dirac bispinors of the outgoing electron antineutrino and electron (initial muon and final muon neutrino), respectively. G F = 1.16639(1) × 10 −5 GeV −2 [8] is the Fermi constant. The coupling constants are denoted as g V LL and g S LR , g T LR respectively to the chirality of the final electron and initial muon. The formula for the the energy and angular distribution of the electron antineutrinos in the muon rest frame, including interference terms between the standard g V LL and exotic g S LR , g T LR couplings withη µ ·q = 0 is of the form: where m µ is the muon mass, y = 2Eν mµ is the reduced antineutrino energy, it varies from 0 to 1,η ν ·q = +1 is the longitudinal polarization of the leftchirality electron antineutrino for the standard g V LL , whileη ν ·q = −1 is the longitudinal polarization of the right-chirality electron antineutrino for the exotic g S,T LR couplings. It is necessary to point out that the above formula is presented after the integration over all the momentum directions of the outgoing electron and muon neutrino. If theη µ ·q = 0 the interference part can be rewritten in the following way: where φ is the angle between the direction of η ⊥ ν and the direction of η ⊥ µ ; T are the relative phases between the g V LL and g S LR , g T LR couplings.
We see that in the case of the transversely polarized antineutrino beam coming from the polarized muon decay, the interference terms between the standard coupling g V LL and exotic g S,T LR couplings do not vanish in the limit of vanishing electron-antineutrino and muon-neutrino masses. This independence on the mass makes the measurement of the relative phases α V S , α V T between these couplings possible. The interference part, Eq. (6), includes only the contributions from the transverse component of the initial muon polarization η ⊥ µ and the transverse component of the outgoing antineutrino polarization η ⊥ ν . Both transverse components are perpendicular with respect to theq. It can be noticed that the relative phases α V S , α V T different from 0, π would indicate the CP violation in the CC weak interaction. Using the current data [8], we calculate the upper limit on the magnitude of the transverse antineutrino polarization and lower bound for the longitudinal antineutrino polarization, see [22]: where Q ν L is the probability of obtaining the left-chirality (anti)neutrino. The Fig. 2 shows the plot of the dΓ dΩν as a function of the azimuthal angle φ forη µ ·q = 0,η ν ·q = 0.992, |η ⊥ ν | = 0.126, |η ⊥ µ | = 1, y = 2/3, |g S LR | = 0.088, |g T LR | = 0.025, |g V LL | = 0.998. The short-dashed line illustrates the possible effect of the CP violation for the relatives phases α V S = π/2, α V T = π/2, . while the long-dashed line represents the case of the CP conservation for the α V S = 0, α V T = 0. We note that the Eq. (3) after integration over all the antineutrino directions (with |g V LL | = 1,η ν ·q = +1) is the same as the Eq. (7) in [22] (with Q ν L = 1, ω L = 0, η L = 0, neglecting the masses of the neutrinos and of the electron as well as radiative corrections). We see that forη µ ·q = −1 only the exotic part (S + T ) survives: It means that the electron antineutrino beam emitted in the direction antiparallel to the muon polarization direction includes only the exotic right-chirality antineutrinos withη ν ·q = −1. If the exotic interactions g S LR , g T LR are present in the DPMaR, the right-chirality antineutrinos (with negative helicity for m νe → 0) are no longer "sterile".
After the integration of the Eqs. (3,4,6), the muon lifetime is as follows: Because the muon lifetime is measured observable, so the admittance of the exotic g S,T LR couplings means that the standard coupling g V LL should be decreased in order to the sum (|g V LL | 2 + 1 4 |g S LR | 2 + 3|g T LR | 2 ) was a constant value. If the antineutrino beam comes from the unpolarized muon decay, the energy and angular distribution of the electron antineutrinos consists only of two parts; standard (V ) and exotic (S + T ), i. e. Eqs. (3,4) forη µ ·q = 0. If one puts E ν = m µ /2 (i. e. y = 1) in both parts, the standard (V ) part vanishes, while the exotic (S + T ) one survives.

Conclusions
In this paper, we have shown that the energy and angular distribution of the electron-antineutrinos in the muon rest frame can be sensitive to the interference terms between the standard left-and exotic right-chirality antineutrinos, proportional to the transverse components of the antineutrino beam polarization. The magnitude of the azimuthal asymmetry caused by the interferences is illustrated in the Fig. 2. The observation of the dependence on the angle φ would be a direct signature of the right-chirality antineutrinos in the DP-MaR. The admittance of the exotic scalar and tensor charged weak interactions in addition to the standard vector interaction in the DPMaR indicates the possibility of producing the mixture of the left-and right-chirality electron antineutrinos with the assigned direction of the transverse antineutrino spin polarization with respect to the production plane. Such polarized beam could be scattered on the PET in order to measure the CP-violating effects caused by the exotic couplings of the right-chirality antineutrinos g S,T LR (or the rightchirality neutrinos) in the purely leptonic processes. We have noticed that forη µ ·q = −1, the energy and angular distribution of the electron antineutrinos consists only of the exotic part (S + T ). It means that if the SM prediction is correct, no signal should be detected for the electron antineutrino beam emitted in the direction antiparallel to the muon polarization direction. We see that the magnitude of this contribution is very small compared with the dominant one from the standard vector interaction, basing on the current limits obtained for the non-standard couplings. The DPMaR may also be used to produce the strong left-chirality and longitudinally polarized (anti)neutrino beam and to measure the dependence of the antineutrino energy spectrum on theη µ ·q. So far no such tests have been carried out. The observation of the right-handed current interaction is important for interpreting results on the neutrinoless double beta decay [23].
We plan to search for the other polarized (anti)neutrino beams, which could be interesting from the point of observable effects caused by the exotic rightchirality states. We expect some interest in the neutrino laboratories working with polarized muon decay and neutrino beams, e.g. KARMEN, PSI, TRI-UMF.

Acknowledgments
This work was supported in part by the grants of the Polish Commit tee for Scientific Research LNGS/103/2006 and 1 P03D 005 28.

4
Four-vector antineutrino polarization and density operator The formula for the the spin polarization 4-vector of massive antineutrino S ′ moving with the momentum q is as follows: whereη ν -the unit 3-vector of the antineutrino polarization in its rest frame. The formula for the density operator of the polarized antineutrino in the limit of vanishing antineutrino mass m ν is given by: where S ′⊥ = 0, η ⊥ ν =η ν − (η ν ·q)q . We see that in spite of the singularities m −1 ν in the polarization four-vector S ′ , the density operator Λ (s) ν remains finite including the transverse component of the antineutrino spin polarization [20].