Neutrino elastic scattering on polarized electrons as tool for probing neutrino nature

Abstract Possibility of using the polarized electron target (PET) for testing the neutrino nature is considered. One assumes that the incoming electron neutrino (νe) beam is the superposition of left chiral states with right chiral ones. Consequently the non–vanishing transversal components of νe spin polarization may appear, both T-even and T-odd. νes are produced by the low energy monochromatic (un)polarized emitter located at a near distance from the hypothetical detector which is able to measure both the azimuthal angle and polar angle of the recoil electrons, and/or also the energy of the outgoing electrons with a high resolution. A detection process is the elastic scattering of νes (Dirac or Majorana) on the polarized electrons. Left chiral (LC) νes interact mainly by the standard V −A interaction, while right chiral (RC) ones participate only in the non-standard V +A, scalar SR, pseudoscalar PR and tensor TR interactions. We show that a distinction between the Dirac and Majorana νes is possible both for the purely left chiral states and the left-right superposition. In the first case a departure from the standard prediction of the azimuthal asymmetry of recoil electrons is caused by the interferences between the non-standard complex S and T couplings, proportional to the angular correlations (T-even and T-odd) among the polarization of the electron target, the incoming neutrino momentum and the outgoing electron momentum. Such a deviation would indicate the Dirac nature of νes. In the second variant the azimuthal asymmetry, polar distribution and energy spectrum of scattered electrons are sensitive to the interference terms between the standard and exotic interactions, proportional to the various angular correlations (T-even and T-odd) among the transversal νe spin polarization (related to the νe source), the electron target polarization, the incoming νe momentum and the outgoing electron momentum. The basic difference


Introduction
One of the basic questions in the neutrino physics is whether the νs are the Dirac or Majorana fermions. At present the neutrinoless double beta decay is viewed as the main tool to investigate νs nature [1][2][3], however the purely leptonic processes (e.g. the neutrino-electron elastic scattering (NEES)) may also shed some light on this problem [4,5]. Kayser and Langacker have analyzed the νs nature problem in the context of non-zero νs mass and of the standard model (SM) V-A interaction [6][7][8][9][10] of only the LC νs. There is an alternative opportunity of distinguishing between the Majorana and Dirac νs by admitting the exotic V + A, scalar S, pseudoscalar P and tensor T interactions coupling to the LC and RC νs in the leptonic processes within the relativistic ν limit. The appropriate tests have been considered by Rosen [11] and Dass [12]. It is also worthwhile noticing the other interesting papers devoted to the ν nature [13][14][15][16][17][18]. The above ideas involve the unpolarized detection target. When the target-electrons are polarized by an external magnetic field, one has possibility of changing the rate of weak interaction by inverting the direction of magnetic field. This feature is very important in the detection of low energy ν e s because the background level would be precisely controlled [19]. PET seems to be a more sensitive laboratory for probing the ν nature and time reversal symmetry violation in the leptonic processes (TRSV) than the unpolarized target due to the mentioned control of contribution of the interaction to the cross section. It is worth reminding that the PET has been proposed to test the flavor composition of (anti)neutrino beam [20] and various effects of non-standard physics. We mean the neutrino magnetic moments, TRSV in the leptonic processes [21], axions, spin-spin interaction in gravitation [22][23][24][25][26][27][28] The possibility of using polarized targets of nucleons and of electrons for the fermionic, scalar and vector dark matter detection is also worth noticing [29][30][31]. The methods of producing the spin-polarized gasses such as helium, argon and xenon are described in [32,33]. It is also essential to mention the measurements confirming the possibility of realizing the polarized target crystal of Gd 2 SiO 5 (GSO) doped with Cerium (GSO:Ce) [34]. Let us recall that there is no difference between the Dirac and Majorana νs in the case of NEES with the standard V-A interaction in the relativistic limit, when the target is unpolarized. In addition the standard couplings have to be the real numbers as a consequence of the hermiticity condition of interaction lagrangian for the NEES. The SM does not allow to clarify the origin of parity violation, observed barion asymmetry of universe [35] through a single CP-violating phase of the Cabibbo-Kobayashi-Maskawa quark-mixing matrix (CKM) [36] and other fundamental problems. This situation led to the appearance of many non-standard models: the left-right symmetric models (LRSM) [37][38][39][40][41], composite models [42][43][44], models with extra dimensions (MED) [45], the unparticle models (UP) [46][47][48][49][50][51][52][53][54][55][56][57][58] and other schemes outside the SM . It is also noteworthy that the current experimental results still leave some space for the scenarios with the exotic interactions. Recently the study of the ν nature with a use of PET in the case of standard V-A interaction, when the evolution of ν spin polarization in the astrophysical environments is admitted, has been carried out in [83]. In this paper we consider the elastic scattering of low energy ν e s (∼ 1MeV ) on the polarized electrons of target in the presence of non-standard complex scalar, pseudoscalar, tensor couplings and V + A interaction as a useful tool for testing the ν nature. We show how the various types of azimuthal asymmetry, the polar distribution and the energy spectrum of scattered electrons enable to distinguish between the Dirac and Majorana ν e s both for the purely left chiral states (only longitudinal ν e polarization (η ν ) || ) and the left-right superposition (non-zero transversal ν e polarization (η ν ) ⊥ ), taking into account TRSV. Our study is modelindependent and carried out for the flavor-eigenstate (Dirac and Majorana) ν e s in the relativistic limit. One assumes that the monochromatic low energy and (un)polarized ν e emitter with a high activity is placed at a near distance from the detector (or at the detector centre). The hypothetical detector is assumed to be able to measure both the azimuthal angle φ e and polar angle θ e of the recoil electrons, and/or also the en- Fig. 1 Production plane of the ν e beam is spanned by the polarization unit vectorŜ of source and the ν e LAB momentum unit vectorq. Reaction plane is spanned byq and the transverse electron polarization vector of target (η e ) ⊥ for ν e + e − → ν e + e − . θ 1 = π/2 is the angle between the orientation of polarization of the electron targetη e andq, soη e ·q = 0 andη e = (η e ) ⊥ . θ e is the polar angle betweenq and the unit vectorp e of recoil electron momentum. φ e is the angle between (η e ) ⊥ and the transversal component of outgoing electron momentum (p e ) ⊥ .η ν = (sin θ ν cos φ ν , sin θ ν sin φ ν , cos θ ν ).
ergy of the outgoing electrons with a high resolution, Fig.1. We utilize the experimental values of standard couplings: c L V = 1 + (−0.04 ± 0.015), c L A = 1 + (−0.507 ± 0.014) to evaluate the predicted effects [84]. The laboratory differential cross sections (see Appendix 1 for the Majorana ν e s and [21] for the Dirac case) are calculated with the use of the covariant projectors for the incoming ν e s (including both the longitudinal and transversal components of the spin polarization) in the relativistic limit and for the polarized targetelectrons, respectively [85].

Elastic scattering of Dirac electron neutrinos on polarized electrons
We analyze a scenario in which the incoming Dirac ν e beam is assumed to be the superposition of LC states with RC ones. The detection process is the elastic scattering of Dirac ν e s on the polarized target-electrons. LC ν e s interact mainly by the standard V − A interaction and small admixture of non-standard scalar S L , pseudoscalar P L , tensor T L interactions, while RC ones take part only in the exotic V + A and S R , P R , T R interactions. As a result of the superposition of the two chiralities the spin polarization vector have the nonvanishing transversal polarization components, which may give rise to both T-even and T-odd effects. As an example of process in which the transversal ν polarization may be produced, we refer to the ref. [86], where the muon capture by proton has been considered. The amplitude for the ν e e − scattering in low energy region is of the form: where G F = 1.1663788(7) × 10 −5 GeV −2 (0.6 ppm) [87] is the Fermi constant. The coupling constants are denoted as c L,R V , c L,R A , c R,L S , c R,L P , c R,L T respectively to the incoming ν e of left-and right-handed chirality. All the non-standard couplings c R,L S , c R,L P , c R,L T are the complex numbers denoted as c R S = |c R S |e i θ S,R , c L S = |c L S |e i θ S,L , etc. c L,R V , c L,R A coupling constants are the real numbers as a consequence of hermitian interaction lagrangian. Moreover, we take into account the relations between the non-standard complex couplings with left-and right-handed chirality appearing at the level of interaction lagrangian: c L S,T,P = c * R S,T,P .

Elastic scattering of Majorana electron neutrinos on polarized electrons
The fundamental difference between the Majorana and Dirac ν e s arises from a fact that the Majorana ν e s do not participate in the vector V and tensor T interactions. This is a direct consequence of the (u, υ)-mode decomposition of the Majorana field. The amplitude for the NEES on the PET for the Majorana low energy ν e s is as follows: We see that the ν e contributions from A, S, P are multiplied by the factor of 2 as a result of the Majorana condition. The as a function y (dotted line) and A θe (Φ max ) as a function of θ e (solid line) forη ν ·q = −1, E ν = 1 MeV , Φ max = π/2: upper plot for θ 1 = π/18; middle plot for θ 1 = π/2; lower plot for θ 1 = 17π/18.
indexes L, (R) for the standard V − A and non-standard V + A interactions are omitted. It means that both LC and RC ν e s may take part in the above interactions. All the other assumptions are the same as for the Dirac case.

Distinguishing between Dirac and Majorana neutrinos through azimuthal asymmetries of recoil electrons
In this section we analyze the possibility of distinguishing the Dirac from Majorana ν e s through probing the azimuthal ; θ e = π/12 (dotted line); θ e = π/6 (solid line); θ e = π/3 (dashed line).
asymmetries of recoil electrons (defined in the Appendix 2). Let us remind that the Dirac and Majorana ν e s can not be discriminated in the case of V-A interaction withη ν ·q = −1 in the relativistic ν limit, even if the target-electrons are polarized. The illustration of this regularity are the Fig. 2 and Fig.3. The Fig. 2 shows how the asymmetries A y (Φ max ), A θ e (Φ max ) depend on the angle θ 1 betweenη e andq. For simplicity, Fig. 1 is made for θ 1 = π/2. We see that the maximum values of A y (Φ max ) and A θ e (Φ max ) grow from 0.008 at θ e = π/6 for θ 1 = π/18 (upper plot) to 0.42 at θ e = π/12 for θ 1 = 17π/18 (lower plot). Although the magnitude of the asymmetries may change, orientation of the asymmetry axis is fixed at Φ max = π/2. This is also illustrated on the plot of d 2 σ /dφ e d θ e in Fig.3. When one departs from the pure V − A interaction and, still assuming fully longitudinal polarization of incoming ν e s (η ν · q = −1), one introduces the non-standard couplings in the detection process, the asymmetries A(Φ max ), A y (Φ max ) and A θ e (Φ max ) can distinguish between the Dirac and Majorana ν e s in the vanishing ν e mass limit. The Fig. 4 displays that the azimuthal distribution of recoil electrons for the Dirac ν e s has the maximum at φ e = π for θ 1 = π/4, π/2, 3π/4 (upper plot), while for the Majorana ones the maximum is shifted to φ e = 0 = 2π (lower plot). The presence of nonstandard S, T, P complex couplings of Dirac ν e s produces, among other terms, the non-vanishing triple angular correlations composed ofq,p e , (η e ) ⊥ vectors. It allows to search for the effects of TRSV in the NEES. In the Majorana case the interference between V − A and V + A interactions proportional to T-even correlations only survives. The Fig. 5 shows how the asymmetry axis location Φ max (upper plot) and the magnitude of A(Φ max ) (lower plot) depend on the phase differences ∆ θ ST,R = θ S,R − θ T,R (dashed lines) and ∆ θ PT,R = θ P,R − θ T,R (dotted lines) for θ 1 = π/2. For illustrative purposes, we present the formula on A(Φ) with θ 1 dependence for the Dirac scenario with V − A, S R and T R interactions when θ ν = π, assuming the experimental values of standard couplings, We see that for θ 1 = 0 = π the asymmetry vanishes. The case of V − A with P R and T R interactions when θ ν = π has been added to show the differences between both scenarios. The observation of departure of the asymmetry axis location from Φ = π/2 would indicate the Dirac ν e s and signalize the possibility of TRSV. The other asymmetry A y (Φ max ) shown in Fig. 6  It is necessary to point out that from an experimental point of view a searching for the differences between the Dirac and Majorana ν e s by the measurement of observables dependent on (η ν ) ⊥ related to the production process would be extremely difficult. In order to measure A θ e (Φ max ) one should determine the location of Φ max by counting the events along the azimuthal angle (at fixed θ e for any configuration of φ ν ) from Φ to Φ + π and from Φ + π to Φ + 2π (for various Φ); in this way Φ max and A θ e (Φ max ) would be found according to its definition. These measurements have to be repeated for different θ e s. The drawn curve with respect to θ e should fit to a one of the curves on the Fig. 6. The measurement of A(Φ max ) would proceed in the similar way as above, but now θ e is not fixed (azimuthal orientation of (η ν ) ⊥ of the incoming ν e s described by φ ν is fixed instead). One counts events along azimuthal angle from Φ to Φ + π and from Φ + π to Φ + 2π for all θ e . The repetition of the measurements for different φ ν would give the curve with respect to φ ν which should fit to a one of the curves on the Fig. 7.

Distinguishing between Dirac and Majorana neutrinos via spectrum and polar angle distribution of scattered electrons
In this section we explore the ν e nature problem by using the electron energy spectrum and polar angle distribution of scattered electrons. To begin with, it is worth recalling that the above observables do not allow to differentiate between the Dirac and Majorana ν e s in the case of standard V − A interaction in the relativistic limit; see Figs. (10)(11) which are made for θ 1 = 0, π/2, π. If one assumes that the ν e source produces the superposition of LC with RC νs and one has the fixed location of (η ν ) ⊥ with respect to the production plane, the cross sections dσ /dθ e , dσ /dy for the detection of Dirac and Ma- 8.

Conclusions
We have shown that the various types of the azimuthal asymmetries of recoil electrons, the energy spectrum and the polar angle distribution of scattered electrons are sensitive to the differences between the effects caused by the Dirac and Majorana ν e s interacting with PET in the presence of exotic interactions, both for θ ν = π and θ ν = π with (η ν ) ⊥ = 0. The high-precision measurements of these quantities can shed some light on the fundamental problems of ν nature and TRSV in the leptonic processes. It is necessary to stress that new tests require the strong low-energy (monochromatic) ν e sources, the large PET, and the detectors sensitive to the measurement of the azimuthal angle and polar angle of recoil electrons with the high angular resolution. The proposals of this type of detectors have been discussed in the literature [88][89][90][91][92][93]. The high-resolution measurements of the spectrum of low energy outgoing electrons need the detectors with the ultra low threshold and background. The interesting concepts of various (monochromatic) ν e sources are also worth noticing [94][95][96][97][98][99]. A preliminary study on the feasibility of electron polarized scintillating GSO target has been carried out by [34]. In order to make the detection of (η ν ) ⊥ -dependent effects feasible, further studies on the appropriate choice of ν e source, in which the exotic couplings of RC ν e s in addition to the LC ones take part, are needed to explain the basic role of production process in generating ν e beam with non-zero (η ν ) ⊥ and in controlling the angle φ ν , Fig. 1. Today the controlled production of ν e beam with the fixed direction of (η ν ) ⊥ with respect to the production plane is still impossible, so the variant with the use of unpolarized ν e source generating only the longitudinally polarized ν e s seems to be more available.