Neutrino elastic scattering on polarized electrons as a tool for probing the neutrino nature
aa r X i v : . [ h e p - ph ] M a y Eur. Phys. J. C manuscript No. (will be inserted by the editor)
Neutrino elastic scattering on polarized electrons as a tool for probingthe neutrino nature
A. Błaut a,1 , W. Sobków b,1 Institute of Theoretical Physics, University of Wrocław, Pl. M. Born 9, PL-50-204 Wrocław, PolandReceived: date / Accepted: date
Abstract
Possibility of using the polarized electron target(PET) for testing the neutrino nature is considered. It is as-sumed that the incoming ν e beam is the superposition of leftchiral (LC) states with right chiral (RC) ones. Consequentlythe non–vanishing transversal components of ν e spin polar-ization may appear, both T-even and T-odd. ν e s are producedby the low energy monochromatic (un)polarized emitter lo-cated at a near distance from the hypothetical detector whichis able to measure both the azimuthal angle and the polar an-gle of the recoil electrons, and/or also the energy of the out-going electrons with a high resolution. A detection processis the elastic scattering of ν e s (Dirac or Majorana) on thepolarized electrons. LC ν e s interact mainly by the standard V − A interaction, while RC ones participate only in the non–standard V + A , scalar S R , pseudoscalar P R and tensor T R interactions. All the interactions are of flavour-conservingtype (FC). We show that a distinction between the Dirac andthe Majorana ν e s is possible both for the longitudinal andthe transversal ν e polarizations.In the first case a departure from the standard prediction ofthe azimuthal asymmetry of recoil electrons is caused bythe interferences between the non-standard complex S and Tcouplings, proportional to the angular correlations (T-evenand T-odd) among the polarization of the electron target,the incoming neutrino momentum and the outgoing electronmomentum. It is shown that such a deviation would indicatethe Dirac ν e nature and the presence of time reversal sym-metry violation (TRSV) interactions. It is remarkable thatthe result is conclusive for all Majorana non–standard cou-plings.In the second case the azimuthal asymmetries, polar distri-bution and energy spectrum of scattered electrons are sensi-tive to the interference terms between the standard and ex- a e-mail: [email protected] b e-mail: [email protected] (corresponding author) otic interactions, proportional to the various angular corre-lations among the transversal ν e spin polarization, the elec-tron target polarization, the incoming ν e momentum and theoutgoing electron momentum. In the particular case of the V − A and S couplings the precise measurement of someobservables, e.g. the spectrum, can distinguish between theDirac and the Majorana ν e s as long as the incoming ν e beamhas non-vanishing transversal polarization. Our model-inde-pendent study is carried out for the flavour ν e eigenstates inthe relativistic ν e limit. One of the basic questions in neutrino physics is whether the ν s are Dirac or Majorana fermions. At present, the neutrino-less double beta decay is viewed as the main tool to investi-gate ν s nature [1–3], however the purely leptonic processes(e.g. the neutrino-electron elastic scattering (NEES)) mayalso shed some light on this problem [4, 5]. Kayser and Lan-gacker have analyzed the ν s nature problem in the contextof non–zero ν s mass and of the standard model (SM) V-Ainteraction [6–10] of only the LC ν s. There is an alternativeopportunity of distinguishing between Majorana and Dirac ν s by admitting the exotic V + A , scalar S , pseudoscalar P and tensor T interactions coupling to the LC and RC ν s inthe leptonic processes within the relativistic ν limit. The ap-propriate tests have been considered by Rosen [11] and Dass[12] (see also [13–24] for other works devoted to the ν na-ture and the non–standard ν properties). The above ideasinvolve the unpolarized detection target. When the target-electrons are polarized by an external magnetic field, onehas a possibility of changing the rate of weak interaction byinverting the direction of magnetic field. This feature is veryimportant in the detection of low energy ν e s because thebackground level would be precisely controlled [25]. PET seems to be a more sensitive laboratory for probing the ν nature and TSRV in the leptonic processes than the unpo-larized target due to the mentioned control of contributionof the interaction to the cross section. It is worth remindingthat the PET has been proposed to test the flavour composi-tion of (anti)neutrino beam [26] and various effects of non–standard physics. We mean the neutrino magnetic moments[27, 28], TRSV in the (semi)leptonic processes [29, 30], ax-ions, spin–spin interaction in gravitation [31–34]. The possi-bility of using polarized targets of nucleons and of electronsfor the fermionic, scalar and vector dark matter detection isalso worth noticing [35–37]. The methods of producing thespin-polarized gasses such as helium, argon and xenon aredescribed in [38, 39].It is also essential to mention the measurements confirm-ing the possibility of realizing the polarized target crystal of Gd SiO (GSO) doped with Cerium (GSO:Ce) [40].Let us recall that there is no difference between Dirac andMajorana ν s in the case of NEES with the standard V-Ainteraction in the relativistic limit, when the target is unpo-larized.The SM does not allow the clarification of the origin of par-ity violation, observed baryon asymmetry of universe [41]through a single CP-violating phase of the Cabibbo-Koba-yashi-Maskawa quark-mixing matrix (CKM) [42] and otherfundamental problems. This has led to the appearance ofmany non-standard models: the left-right symmetric models(LRSM) [43–46], composite models [47–49], models withextra dimensions (MED) [50] and the unparticle models (UP)[51–53]. There is a rich literature devoted to the phenomeno-logical aspects of the non–standard interactions of LC andin particular RC ν s: [54–81]. It is also noteworthy that thecurrent experimental results still leave some space for thescenarios with the exotic interactions.Recently the study of the ν nature with a use of PET in thecase of standard V-A interaction, when the evolution of ν spin polarization in the astrophysical environments is admit-ted, has been carried out in [82, 83].In this paper we consider the elastic scattering of low en-ergy ν e s ( ∼ MeV ) on the polarized electrons of target inthe presence of non-standard complex scalar, pseudoscalar,tensor couplings and V + A interaction as a useful tool fortesting the ν nature. We show how the various types of az-imuthal asymmetry, the polar distribution and the energyspectrum of scattered electrons enable the distinguishing be-tween the Dirac and the Majorana ν e s both for the longitudi-nal and the transversal ν e polarizations, taking into accountTRSV. Both theoretically possible scenarios of physics be-yond SM deal with FC standard and non–standard interac-tions. Our study is model–independent and carried out forthe flavour ν e eigenstates (Dirac and Majorana) in the rela-tivistic limit. One assumes that the monochromatic low en-ergy and (un)polarized ν e emitter with a high activity is placed at a near distance from the detector (or at the detectorcentre). The hypothetical detector is assumed to be able tomeasure both the azimuthal angle φ e and the polar angle θ e of the recoil electrons, and/or also the energy of the outgoingelectrons with a high resolution, Fig.1. We utilize the exper-imental values of standard couplings: c LV = − . , c LA = − .
507 to evaluate the predicted effects [84], where theindexes V , A and L denote the vector, the axial interactionsand the left-handed chirality of ν e s, respectively. We assumethe values of exotic couplings which are compatible with theconstraints on non–standard interactions obtained in our pre-vious paper [85] and by various authors: [86, 87]. The labo-ratory differential cross sections (see Appendix 1 for Majo-rana ν e s and [30] for the Dirac case) are calculated with theuse of the covariant projectors for the incoming ν e s (includ-ing both the longitudinal and the transversal components ofthe spin polarization) in the relativistic limit and for the po-larized target-electrons, respectively [88]. We analyze a scenario in which the incoming Dirac ν e beamis assumed to be the superposition of LC states with RCones. The detection process is the elastic scattering of Dirac ν e s on the polarized target-electrons. The relative orienta-tion of the incoming ν e , the polarized electron target and theoutgoing electron is depicted in Fig. 1, where the notationfor the relevant quantities is explained.LC ν e s interact mainly by the standard V − A interactionand a small admixture of the non-standard scalar S L , pseu-doscalar P L , tensor T L interactions, while RC ones take partonly in the exotic V + A and S R , P R , T R interactions. As a re-sult of the superposition of the two chiralities the spin polar-ization vector has the non-vanishing transversal polarizationcomponents, which may give rise to both T-even and T-oddeffects. As an example of the process in which the transver-sal ν polarization may be produced, we refer to the ref. [89],where the muon capture by proton has been considered. Theamplitude for the ν e e − scattering in low energy region is inthe form: M D ν e e − = G F √ { ( u e ′ γ α ( c LV − c LA γ ) u e )( u ν e ′ γ α ( − γ ) u ν e )+ ( u e ′ γ α ( c RV + c RA γ ) u e )( u ν e ′ γ α ( + γ ) u ν e ) (1) + c RS ( u e ′ u e )( u ν e ′ ( + γ ) u ν e )+ c RP ( u e ′ γ u e )( u ν e ′ γ ( + γ ) u ν e )+ c RT ( u e ′ σ αβ u e )( u ν e ′ σ αβ ( + γ ) u ν e )+ c LS ( u e ′ u e )( u ν e ′ ( − γ ) u ν e )+ c LP ( u e ′ γ u e )( u ν e ′ γ ( − γ ) u ν e ) ( (cid:1) ) (cid:2)(cid:0) (cid:3) ( (cid:4) ) (cid:5) q (cid:6)(cid:7) (cid:8)(cid:9) z ( (cid:10) ) e (cid:11) e (cid:12) e p e p e ( (cid:13) ) e (cid:14) Fig. 1
The reaction plane is spanned by the ν e LAB momentum unitvector ˆ q and the electron polarization vector of the target ˆ η e . θ isthe angle between ˆ η e and ˆ q (on the plot θ = π / η e = ( ˆ η e ) ⊥ ). θ e is the polar angle between ˆq and the unit vector ˆ p e ofthe recoil electron momentum. φ e is the angle between ( ˆ η e ) ⊥ and thetransversal component of outgoing electron momentum ( ˆ p e ) ⊥ . φ ν and θ ν are the azimuthal and the polar angles of the unit polarization vectorof the incoming neutrino, ˆ η ν = ( sin θ ν cos φ ν , sin θ ν sin φ ν , cos θ ν ) . + c LT ( u e ′ σ αβ u e )( u ν e ′ σ αβ ( − γ ) u ν e ) } , where G F = . ( ) × − GeV − ( . ppm ) [90] isthe Fermi constant. The coupling constants are denoted as c L , RV , c L , RA , c R , LS , c R , LP , c R , LT respectively to the incoming ν e ofleft- and right-handed chirality. All the non-standard cou-plings c R , LS , c R , LP , c R , LT are the complex numbers denoted as c RS = | c RS | e i θ S , R , c LS = | c LS | e i θ S , L , etc. Reality of c L , RV , c L , RA cou-pling constants follows from the hermiticity of the interac-tion lagrangian; for the same reason we take into account therelations between the non-standard complex couplings withleft- and right-handed chirality, c LS , T , P = c ∗ RS , T , P . All resultsare stated in terms of R couplings. The fundamental difference between Majorana and Dirac ν e s arises from a fact that Majorana ν e s do not participatein the vector V and tensor T interactions. This is a directconsequence of the ( u , υ ) –mode decomposition of the Ma-jorana field. The amplitude for NEES on PET for Majoranalow energy ν e s is as follows: M M ν e e − = G F √ {− ( u e ′ γ α ( c V − c A γ ) u e )( u ν e ′ γ α γ u ν e ) (2) + ( u e ′ u e ) (cid:2) c LS ( u ν e ′ ( − γ ) u ν e ) + c RS ( u ν e ′ ( + γ ) u ν e ) (cid:3) + ( u e ′ γ u e ) (cid:2) − c LP ( u ν e ′ ( − γ ) u ν e ) + c RP ( u ν e ′ ( + γ ) u ν e ) (cid:3) } . We see that the ν e contributions from A , S , P are multipliedby the factor of 2 as a result of the Majorana condition. Theindexes L , ( R ) for the standard interactions are omitted. Itmeans that both LC and RC ν e s may take part in the aboveinteractions. All the other assumptions are the same as forthe Dirac case. In this section we analyze the possibility of distinguishingthe Dirac from the Majorana ν e s through probing the az-imuthal asymmetries, A , A y , A θ e , of recoil electrons. Theasymmetry functions are defined by the following formulas: A ( Φ ) : = Φ + π R Φ d σ d φ e d φ e − Φ + π R Φ + π d σ d φ e d φ e Φ + π R Φ d σ d φ e d φ e + Φ + π R Φ + π d σ d φ e d φ e , (3) A y ( Φ ) : = Φ + π R Φ d σ d φ e dy d φ e − Φ + π R Φ + π d σ d φ e dy d φ e Φ + π R Φ d σ d φ e dy d φ e + Φ + π R Φ + π d σ d φ e dy d φ e , (4) A θ e ( Φ ) : = Φ + π R Φ d σ d φ e d θ e d φ e − Φ + π R Φ + π d σ d φ e d θ e d φ e Φ + π R Φ d σ d φ e d θ e d φ e + Φ + π R Φ + π d σ d φ e d θ e d φ e . (5)These observables are functions of the asymmetry angle Φ (the interpretation of Φ follows from the definitions of eqs.(3-5) and Fig. 1: Φ is measured with respect to the trans-verse electron polarization vector of target ( ˆ η e ) ⊥ ); by Φ max we denote the location of their maxima. In what follows weshall use the normalized, dimensionless kinetic energy of therecoil electrons y , defined by y ≡ T e E ν = m e E ν ( θ e )( + m e E ν ) − cos ( θ e ) , (6) ° ° ° ° ° ° A y ( Φ m a x ) , A θ e ( Φ m a x ) θ e y ° ° ° ° ° ° A y ( Φ m a x ) , A θ e ( Φ m a x ) θ e y ° ° ° ° ° ° A y ( Φ m a x ) , A θ e ( Φ m a x ) θ e y Fig. 2
Dirac (or Majorana) ν e with V − A interaction: plot of A y ( Φ max ) as a function y (dotted line) and A θ e ( Φ max ) as a function of θ e (solidline) for ˆ η ν · ˆ q = − E ν = MeV , Φ max = π /
2: upper plot for θ = . θ = π /
2; lower plot for θ = π − . where T e is the kinetic energy of the recoil electron, E ν is theincoming ν e energy, m e is the electron mass. Fig. 2 showsthe asymmetries A y ( Φ max ) , A θ e ( Φ max ) for the standard V-A interaction with ˆ η ν · ˆ q = −
1. Although the orientation ofthe asymmetry axis is fixed at Φ max = π / y and θ e . We see that themaximum values of A y ( Φ max ) and A θ e ( Φ max ) also dependon the angle θ between ˆ η e and ˆ q : they grow from 0 . θ = . .
54 for θ = π − . ν e s therefore the asymmetries Φ m a x Δθ ST,R ( D ) , Δθ SP,R ( M ) A ( Φ m a x ) Fig. 3
Upper plot (dashed line) is the dependence of Φ max on ∆θ ST , R ( D ) for Dirac ν e in case of V − A , S R and T R interactions when | c RS | = | c RT | = .
2. Upper plot (dotted line) is the dependence of Φ max on ∆θ SP , R ( M ) for Majorana ν e in case of V − A , S R and P R cou-plings when | c RS | = | c RP | = .
2. Both scenarios assume ˆ η ν · ˆ q = − θ = π / , E ν = MeV . Lower plot is the dependence of A ( Φ max ) on ∆θ ST , R ( D ) for Dirac ν e (dashed line) and on ∆θ SP , R ( M ) for Majorana ν e (dotted line), respectively, with same assumptions as for Φ max . can not discriminate between the two ν types even if thetarget-electrons are polarized.The presence of non–standard S , T , P complex couplingsof Dirac ν e s with ˆ η ν · ˆ q = − q , ˆ p e , ( ˆ η e ) ⊥ vectors.These terms not only play a role in distinguishing betweenthe Dirac and the Majorana ν e s but in the Dirac case theyallow to search for the effects of TRSV in NEES. Fig. 3shows how the asymmetry axis location Φ max (upper plot)and the magnitude of A ( Φ max ) (lower plot) depend on thephase differences ∆θ ST , R ( D ) = θ S , R − θ T , R for the Dirac ν e s(dashed lines) and ∆θ SP , R = θ S , R − θ P , R ( M ) for the Majorana ν e s (dotted lines) when θ = π /
2. To explain the origin ofthe difference we give the formulas for A ( Φ ) with assumedvalues of θ = π / θ ν = π , the experimental standard cou-plings and E ν = MeV . For the Dirac scenario with V − A , S R and T R interactions, it reads: A ( S , R )( T , R ) D ( Φ ) = − (cid:26)(cid:20) . ( − sin ( Φ )( | c RS || c RT | cos ( ∆θ ST , R )+ | c RT | + . ) − | c RS || c RT | sin ( ∆θ ST , R ) cos ( Φ ) (7) A ( Φ m a x ) Φ m a x ϕ ν ϕ ν Fig. 4
Superposition of LC ν e s with RC ones in presence of non–standard couplings with ˆ η ν · ˆ q = − .
95: dependence of A ( Φ max ) on φ ν (solid line) and Φ max on φ ν (dashed line) for E ν = MeV , θ = π / V − A and S R when | c RS | = . , θ S , R =
0; middle left plot for Majorana case of V − A with S R when | c RS | = . , θ S , R =
0; lower left plot for Dirac case of V − A with T R when | c RT | = . , θ T , R =
0. TRSV: upper right plot for Dirac scenariowith V − A and S R when | c RS | = . , θ S , R = π /
2; middle right plot forMajorana case of V − A with S R when | c RS | = . , θ S , R = π /
2; lowerright plot for Dirac case of V − A and T R when | c RT | = . , θ T , R = π / ° ° ° ° ° ° A θ e ( Φ m a x ) ° ° ° ° ° ° A θ e ( Φ m a x ) ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° θ e ° ° ° ° ° ° θ e Fig. 5
Superposition of LC ν e s with RC ones in presence of non–standard couplings with ˆ η ν · ˆ q = − .
95: plot of A θ e ( Φ max ) as a functionof θ e for the case of V − A with S R when E ν = MeV , φ ν = ν e , right column for Majorana ν e ; upper plot for θ = .
1; middle plot for θ = π /
2; lower plot for θ = π − .
1; solidline for | c RS | = . θ S , R =
0; dotted line for | c RS | = . θ S , R = π / | c RS | = . θ S , R = π / ° ° ° ° ° ° A θ e ( Φ m a x ) ° ° ° ° ° ° A θ e ( Φ m a x ) ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° θ e ° ° ° ° ° ° θ e Fig. 6
Superposition of LC ν e s with RC ones in presence of non–standard couplings with ˆ η ν · ˆ q = − .
95: plot of A θ e ( Φ max ) as a functionof θ e for the case of V − A with S R when E ν = MeV , φ ν = π / ν e , right column for Majorana ν e ; upper plot for θ = .
1; middle plot for θ = π /
2; lower plot for θ = π − .
1; solid linefor | c RS | = . θ S , R =
0; dotted line for | c RS | = . θ S , R = π / | c RS | = . θ S , R = π / − .
369 sin ( Φ )) (cid:21) / (cid:20) ( | c RS | + | c RT | + . ) + . ( | c RS | + | c RT | + . ) + ( | c RS | + | c RT | + . ) − | c RS || c RT | cos ( ∆θ ST , R ) + . (cid:21)(cid:27) . The corresponding formula for A ( Φ ) in the Majorana casewith V − A , S R , P R and V + A couplings is of the form: A ( All ) M ( Φ ) = − (cid:26) .
354 sin ( Φ ) (cid:20) | c RP || c RS | cos ( ∆θ SP , R ) − . (cid:21) / (cid:20) ( | c RP | + . | c RS | + . ) (cid:21)(cid:27) . It can be noticed that these expressions explicitly reveal dif-ferent dependence on the azimuthal angle Φ in the two cases.Thus, it is not possible to reproduce the dashed curve of Fig.3, which describes the Dirac ν beam, by using the all pos-sible non-standard (flavour-conserving) Majorana ν interac-tions.Another possibilities are related to the ˆ η ν · ˆ q = − ν e beam is the super-position of LC ν s with RC ones and there is an experimen-tal control of the angle φ ν connected with ( ˆ η ν ) ⊥ , we havenew opportunities of testing the ν e nature and TRSV. Fig. 4illustrates the asymmetry A in this case. It probes the depen-dence of A ( Φ max ) (solid line) and the asymmetry axes loca-tion Φ max (dashed line) on φ ν . The plot compares behaviorof the Dirac ( S R , T R ) and the Majorana ( S R ) ν s additionally taking into account TRSC (left column) and TRSV (rightcolumn) options. The detection of a non–trivial dependenceof the asymmetry on the φ ν shown in Fig. 4 would indi-cate the existence of exotic scalar or tensor couplings of RC ν s. The precise measurement of the magnitude of A ( Φ max ) would help to detect TRSV, however it can be difficult todistinguish between the Dirac and the Majorana ν e s.The Figs. (5-6) display the impact of θ and φ ν on the pos-sible values of the asymmetry A θ e ( Φ max ) for the scalar in-teractions. We see that this measurement is sensitive to thepresence of exotic couplings, offers the possibility of thedistinction between the Dirac and the Majorana ν e s, and al-lows the detection of TRSV effects. For the sake of the il-lustration of the variety of possible outcomes we point onthe most noticable characteristics of the diagrams presentedin Fig. 5. The maximum values of A θ e ( Φ max ) for θ = . .
012 in the Dirac case (dashed line in left up-per plot), and to 0 .
035 in the Majorana case (dashed line inright upper plot) in comparison to the standard expectationof 0 . θ = π / A θ e ( Φ max ) may decrease to around 0 .
02 for the Majo-rana ν e s (solid line in middle right plot), while the standardprediction gives 0 .
08 at θ e = π /
6. The maximum value of A θ e ( Φ max ) for θ = π − . . − .
17 for theDirac ν e s (lower left plot) and to 0 . − .
27 in the Majoranacase (lower right plot), compared to the standard expectationof 0 .
54. In addition one can observe an offset of the maxi-mum of A θ e ( Φ max ) to higher values of θ e particularly for theMajorana ν e s. Similar features can be observed in Fig. 6 for φ ν = π / ν e s by the measurement of observables dependenton ( ˆ η ν ) ⊥ would be extremely difficult. In order to measure A θ e ( Φ max ) one should determine the location of Φ max bycounting the events from Φ to Φ + π and from Φ + π to Φ + π (for various Φ ) at fixed θ e (and a particular config-uration of φ ν ). In this way Φ max and A θ e ( Φ max ) are foundaccording to their definitions. These measurements have tobe repeated for different θ e s. The experimental curve drawnwith respect to θ e should fit to one of the curves in Figs. 2, 5,6. The measurement of A ( Φ max ) proceeds in a similar way,but now θ e is not fixed - the azimuthal orientation of ( ˆ η ν ) ⊥ described by φ ν is fixed instead. Events within the azimuthalangle from Φ to Φ + π and from Φ + π to Φ + π for all θ e are counted. The repetitions of the measurements for differ-ent φ ν s give a curve which should fit to one of the curves inFig. 4. ° ° ° ° ⨯ d σ / d θ e θ e Fig. 7
Dirac (or Majorana) ν e with V − A interaction, ˆ η ν · ˆ q = − E ν = MeV : plot of d σ / d θ e as a function of θ e for different valuesof θ ; dotted line for θ =
0; solid line for θ = π /
2; dashed line for θ = π . y ⨯ d σ / d y Fig. 8
Dirac (or Majorana) ν e with V − A interaction ˆ η ν · ˆ q = − E ν = MeV : plot of d σ / dy as a function of y for different values of θ : solidline for θ =
0; dashed line for θ = π /
2; dotted line for θ = π . In this section we explore the ν e nature problem by using theelectron energy spectrum and the polar angle distribution ofscattered electrons. To begin with, it is worth recalling thatthe above observables do not allow one to differentiate be-tween the Dirac and the Majorana ν e s in the case of the stan-dard V − A interaction in the relativistic limit; see Figs. (7-8)plotted for θ = , π / , π .If one assumes that the ν e source produces the superposi-tion of LC with RC ν s, the cross sections d σ / d θ e , d σ / dy for the detection of Dirac and Majorana ν e s contain the in-terferences between LC and RC ν s proportional to the var-ious angular correlations (T-even and T-odd) among ( ˆ η ν ) ⊥ ,ˆ q , ˆ p e , ( ˆ η e ) ⊥ vectors. Consequently the linear contributions ⨯ d σ / d y Fig. 9 d σ / dy as a function of y for V − A and S R scenario with c RS = . θ = π /
2; ˆ η ν · ˆ q = − η ν · ˆ q = − .
95 (middle plot),ˆ η ν · ˆ q = ν e s. from the non–standard interactions allow us to distinguishbetween the Dirac and the Majorana ν e s, and search forTRSV, see Figs. (9-10).Fig. 9 shows that it is possible to distinguish the neutrino na-ture in the case of V − A and S interactions. Scalar couplingsfor both, the Dirac and the Majorana ν s, are constrained togive the same value of the spectrum at some specified valueof y (in Fig. 9 y = . | c RS | = . ν e (which, under the above con-straint, determines a value of | c RS | for the Dirac ν e ). Thetransversal polarization of the incoming ν e plays here a cru-cial role, because only for ( ˆ η ν ) ⊥ = θ affects the energy spec-trum of recoil electrons in the presence of interferences re-lated to transversal component of the neutrino spin polariza- ⨯ d σ / d y ⨯ d σ / d y y ⨯ d σ / d y Fig. 10
Superposition of LC ν e s with RC ones in presence of non–standard couplings with ˆ η ν · ˆ q = − .
95: dependence of d σ / d y on y for different values of θ when E ν = MeV , φ ν =
0. Upper plot for θ =
0; middle plot for θ = π /
2; lower plot for θ = π ; dashed linefor Dirac ν e with V − A and T R , | c RT | = . , θ T , R =
0; dotted line forMajorana ν e with V − A with S R , | c RS | = . , θ S , R =
0; solid line forDirac ν e with V − A and S R when | c RS | = . , θ S , R = tion, both for the Dirac and the Majorana ν e s; the plot shouldbe compared with Fig.8. Small deviations for the low energyrecoil electrons in the case of Dirac scenario with V − A and T interactions when θ = , π / θ = π is too small to beclearly visible (lower plot). It is worth noting that Figs. (9-10) have been made at fixed azimuthal angle φ ν =
0; changesof φ ν would affect the spectrum and polar distributions. We have studied the two theoretically possible scenarios ofthe physics beyond the standard model in which flavour-conserving standard and non–standard interactions of bothleft chiral and right chiral electron neutrinos were introduced.We have shown that the various types of the azimuthal asym-metries of the recoil electrons, the energy spectrum and thepolar angle distribution of the scattered electrons can in prin-ciple discern between the Dirac and the Majorana ν e s inter-acting with PET both for the longitudinal and the transversal ν e polarizations. The high-precision measurements of thesequantities may shed some light on the fundamental problemsof the ν nature and TRSV in the leptonic processes. In theparticular case of the V − A and S couplings the spectrumcan distinguish the two types of the neutrinos as long as theincoming ν e beam has non–vanishing transversal polariza-tion. But even for the longitudinally polarized ν e beam theasymmetry observable A ( Φ ) can identify the Dirac scenariowith V − A , S and T interactions independently of any as-sumed Majorana non–standard couplings.The proposed new tests require intensive monochromaticlow-energy ν e sources, large PET, and detectors enabling ameasurement of the azimuthal angle and the polar angle ofthe recoil electrons with the high angular resolution. Propo-sitions of the relevant detectors have been discussed in theliterature [91–95]. In turn high-resolution measurements ofthe spectrum of low energy outgoing electrons demand de-tectors with the ultra low detection threshold and backgroundnoise. Some interesting concepts of various (monochromatic) ν e sources are under debate [96–102]. A preliminary studyof the feasibility of the electron polarized scintillating GSOtarget has been carried out by [40]. In order to make the de-tection of ( ˆ η ν ) ⊥ -dependent effects possible, further studieson the appropriate choice of ν e source which would take intoaccount the exotic couplings of RC ν e s are needed. This isnecessary to explain the basic role of production processesin generating ν e beam with non–zero transversal polariza-tion and to control the azimuthal angle φ ν . The controlledproduction of ν e beam with the fixed direction of ( ˆ η ν ) ⊥ with respect to the production plane is impossible to date,thus the alternative option with the (un)polarized ν e sourcegenerating only the longitudinally polarized ν e s seems to bepresently more available. ν e s on PET The laboratory differential cross section for Majorana ν e s,when ˆ η e ⊥ ˆq ( θ = π / d σ dyd φ e = (cid:18) d σ dyd φ e (cid:19) V − A + (cid:18) d σ dyd φ e (cid:19) ( S , P ) R (8) + (cid:18) d σ dyd φ e (cid:19) S R V − A + (cid:18) d σ dyd φ e (cid:19) P R V − A (cid:18) d σ dyd φ e (cid:19) V − A = B ( c A (cid:20) ( ˆ η e ) ⊥ · ( ˆp e ) ⊥ ˆ η ν · ˆ q ( y − ) (9) · s y (cid:18) m e E ν + y (cid:19) + y − y + + m e E ν y (cid:21) + c V (cid:20) ( ˆ η e ) ⊥ · ( ˆp e ) ⊥ ˆ η ν · ˆ q y s y (cid:18) m e E ν + y (cid:19) + y − y + − m e E ν y (cid:21) + c V c A (cid:20) ( ˆ η e ) ⊥ · ( ˆp e ) ⊥ s y (cid:18) m e E ν + y (cid:19) ( y − )+ y ( y − ) ˆ η ν · ˆ q (cid:21)) , (cid:18) d σ dyd φ e (cid:19) ( S , P ) R = B (cid:26) y (cid:18) y + m e E ν (cid:19) | c RS | + y | c RP | (10) − y s y (cid:18) m e E ν + y (cid:19) ( ˆ η e ) ⊥ · ( ˆp e ) ⊥ ˆ η ν · ˆ q · (cid:2) Re ( c RS ) Re ( c RP ) + Im ( c RS ) Im ( c RP ) (cid:3) (cid:27) , (cid:18) d σ dyd φ e (cid:19) S R V − A = B ( c V s y (cid:18) m e E ν + y (cid:19) (11) · (cid:18) ( ˆ η ν ) ⊥ · ( ˆq × ( ˆp e ) ⊥ ) Im ( c RS ) + ( ˆ η ν ) ⊥ · ( ˆp e ) ⊥ Re ( c RS ) (cid:19) − c A (cid:18) E ν m e y + (cid:19)" ( ˆp e ) ⊥ · (( ˆ η e ) ⊥ × ( ˆ η ν ) ⊥ ) Im ( c RS ) · s y (cid:18) m e E ν + y (cid:19) + y ( ˆ η e ) ⊥ · ( ˆp e ) ⊥ (cid:18) ( ˆ η ν ) ⊥ · ( ˆq × ( ˆp e ) ⊥ ) · Im ( c RS ) + ( ˆ η ν ) ⊥ · ( ˆp e ) ⊥ Re ( c RS ) (cid:19) − Im ( c RS ) · ˆq · (( ˆ η e ) ⊥ × ( ˆ η ν ) ⊥ ) + m e E ν ( ˆ η e ) ⊥ · ( ˆ η ν ) ⊥ Re ( c RS ) ! , (cid:18) d σ dyd φ e (cid:19) P R V − A = B c A y ( E ν m e y ( ˆ η ν ) ⊥ · ( ˆp e ) ⊥ (cid:18) Im ( c RP ) (12) · ( ˆ η e ) ⊥ · ( ˆq × ( ˆp e ) ⊥ ) + ( ˆ η e ) ⊥ · ( ˆp e ) ⊥ Re ( c RP ) (cid:19) + ( ˆ η ν ) ⊥ · ( ˆp e ) ⊥ (cid:18) ( ˆ η e ) ⊥ · ( ˆq × ( ˆp e ) ⊥ ) Im ( c RP ) + Re ( c RP ) · ( ˆ η e ) ⊥ · ( ˆp e ) ⊥ (cid:19) + ( y − )( ˆ η e ) ⊥ · ( ˆ η ν ) ⊥ Re ( c RP )) − s y (cid:18) m e E ν + y (cid:19) ( ˆp e ) ⊥ · (( ˆ η e ) ⊥ × ( ˆ η ν ) ⊥ ) Im ( c RP ) ) , where B ≡ (cid:0) E ν m e / π (cid:1) (cid:0) G F / (cid:1) . ˆ η ν is the unit 3-vector of ν e spin polarization in its rest frame, Fig. 1. ( ˆ η ν · ˆ q ) ˆq is thelongitudinal component of ν e spin polarization. | ˆ η ν · ˆ q | = | − Q ν L | , where Q ν L is the probability of producing the LC ν e . We see that the interference terms between standard V − A and exotic S R , P R couplings depend on the transversal ν e spin polarization ( ˆ η ν ) ⊥ related to the production process(similar regularity as in the Dirac case [30]). References
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