Decay of polarized muon at rest as a source of polarized neutrino beam
aa r X i v : . [ h e p - ph ] M a y Decay of polarized muon at rest as a source ofpolarized neutrino beam
S. Ciechanowicz a , W. Sobk´ow a , ∗ , M. Misiaszek b a Institute of Theoretical Physics, University of Wroc law, Pl. M. Born 9,PL-50-204 Wroc law, Poland b M. Smoluchowski Institute of Physics, Jagiellonian University, ul. Reymonta 4,PL-30-059 Krak´ow, Poland
Abstract
In this paper, we indicate the theoretical possibility of using the decay of polarizedmuons 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 meanhere new tests concerning CP violation, Lorentz structure and chirality structure ofthe charged current weak interactions. The main goal is to show how the energy andangular distribution of the electron antineutrinos in the muon rest frame dependson the transverse components of the antineutrino beam polarization. We admit theparticipation of the complex exotic scalar and tensor couplings of the right-chiralityelectron antineutrinos in addition to the standard vector coupling of the left-chiralityones, while the muon neutrinos are always left-chirality. It means that the outgoingelectron antineutrino beam is a mixture of the left- and right-chirality antineutrinosand has the fixed direction of the transverse spin polarization with respect to theproduction plane. Our analysis is model-independent and consistent with the currentupper limits on the non-standard couplings. The results are presented in a limit ofinfinitesimally small mass for all particles produced in the decay.
Key words: polarized muon decay, exotic couplings, transverse antineutrino spinpolarization
PACS: ∗ Corresponding author
Email addresses: [email protected] (S. Ciechanowicz), [email protected] (W. Sobk´ow), [email protected] (M. Misiaszek).
Preprint submitted to Physic Letters B 24 November 2018
Introduction
Decay of polarized muon at rest (DPMaR) is the very appropriate process totest both the time reversal violation (TRV) and the space-time, and chiralitystructure of the purely leptonic charged weak interactions (PLCWI). More-over, we investigate here if the DPMaR can also be used as a strong source ofthe transversely polarized antineutrino (neutrino) beam which would be scat-tered on the polarized electron target (PET). A detailed analysis of new effectsbeyond the Standard Model (SM) of electroweak interactions [1,2,3] is carriedout in [4]. Ciechanowicz et al. show that the scattering of the left-chirality andlongitudinally polarized neutrino beam on the PET would allow to test thepossibility of CP-breaking in the ( ν µ e − ) scattering. The measurement of theazimuthal asymmetry of recoil electrons could detect the CP-violating phasebetween the standard complex vector and axial-vector couplings. The otherproblem considered in [4] concerns the scattering of the transversely polarizedmixture of left-chirality and right-chirality neutrinos on the PET. If such aneutrino beam would be scattered, the dependence on the angle between thetransverse neutrino spin polarization of incoming beam and the transverseelectron polarization of target in the recoil electron energy spectrum could betested. 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 ex-periments. This means that only left-chirality Dirac neutrinos may take partin the charged and neutral current weak interaction. This structure followsamong other from the low energy measurements of the muon decay such as;the spectral shape, angular distribution, and polarization of the outgoing elec-trons (positrons). At present, there is no evidence for the deviations from theSM for the Michel parameters. The neutrino oscillation experiments indicatethe non-zero neutrino mass and provide first evidence for physics beyond theminimal SM. On the other hand, the experimental precision of present testsstill allows the participation of the exotic scalar S, tensor T and pseudoscalarP couplings of the right-chirality Dirac neutrinos beyond the SM [6]. TheKARMEN experiment [7] has measured the energy distribution of electronneutrinos emitted in positive muon decay at rest ( µ + → e + + ν e + ν µ ). Theobtained result is in agreement with the SM prediction on the neutrino Michelparameter ω L = 0. They get for the first time a 90% confidence upper limit of ω L ≤ . | g SRL + 2 g TRL | ≤ .
78 for the interferencebetween the scalar and tensor couplings. The current upper limits on the allnon-standard couplings, obtained from the normal and inverse muon decay,are presented in the Table 1 [8]. The coupling constants are denoted as g γǫµ ,where γ = S, V, T indicates the type of weak interaction, i.e. scalar S, vectorV, tensor T; ǫ, µ = L, R indicate the chirality of the electron or muon and theneutrino chiralities are uniquely determined for given γ, ǫ, µ . It means thatthe neutrino chirality is the same as the associated charged lepton for the V2 oupling constants SM Current limits | g VLL | > . | g VLR | < . | g VRL | < . | g VRR | < . | g SLL | < . | g SLR | < . | g SRL | < . | g SRR | < . | g TLL | | g TLR | < . | g TRL | < . | g TRR | interaction, and opposite for the S, T interactions [8]. In the SM, only g VLL isnon-zero value.It is necessary to point out that the existence of the exotic right-chiralityneutrinos in the few keV region, that are sterile in SM, can have numerousconsequences in astrophysics and cosmology. We mean here the mechanismof neutrino “spin flip” in the Sun’s convection zone in order to explain theobserved deficit of the solar neutrinos [9]. In addition, the sterile neutrinoscould also account for pulsar kicks (high pulsar velocities), could explain allor some fraction of the dark matter in the Universe and would affect emissionof supernova neutrinos [10].Recent analysis carried out by Erwin et al. for the muon decay [11] showsthat there exist four-fermion operators that do not contribute to the neutrinomass matrix through radiative corrections. These operators generate the exoticcouplings g S,TLR,RL , while all operators generating the vector couplings g VLR,RL contribute to the neutrino mass matrix.So far the CP violation is observed only in the decays of neutral K- andB-mesons [12], and is described by a single phase of the Cabibbo-Kobayashi-Maskawa quark-mixing matrix (CKM)[13]. There is no experimental evidenceon the TRV in the PLCWI, e.g. the muon decay and neutrino-electron elas-tic scattering. However, the baryon asymmetry of the Universe can not beexplained by the CKM phase only, and new sources of the CP violation arerequired [14]. According to the prediction of non-standard models, the effectsof new CP-breaking phases could be measured in observables where the SM3P-violation is suppressed, while alternative sources can generate a sizableeffect, e.g. the electric dipole moment of the neutron, the transverse leptonpolarization in three-body decays of charged kaons K + [15,16], transverse po-larization of the electrons emitted in the decay of polarized Li nuclei [17].The other possibility of measuring the exotic CP-breaking phases is to use theneutrino observables which consist only of the interference terms between thestandard coupling of the left-chirality neutrinos and exotic couplings of theright-chirality neutrinos and do not depend on the neutrino mass. We meanhere both T-even and T-odd transverse components of the neutrino spin polar-ization. At present, the direct tests are still impossible. The possible solutioncan be the scattering of the transversely polarized (anti)neutrino beam, com-ing from the polarized muon decay at rest, on the PET and the measurementof the maximal asymmetry of the cross section, [4].Left-right symmetric models (LRSM) and composite models (CM) can beproposed as an example of the non-standard models of purely leptonic weakinteractions, in which the exotic couplings of the right-chirality neutrinos (an-tineutrinos) can appear. Recently TWIST Collaboration [18] has measured theMichel 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 . ± . stat. ) ± . syst. ) ± . ρ = 3 / | χ | < .
030 (90% CL ). The CM havebeen proposed to probe the scale for compositeness of quarks and leptons. La-grangian of the new effective contact interactions (CI) for the muon decayincludes among other the contributions from the standard vector coupling ofthe 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 limitof infinitesimally small mass for all particles produced in the muon decay. Thedensity operators [20] for the polarized initial muon and for the polarized out-going electron antineutrino are used, see Appendix A. We use the system ofnatural units with ~ = c = 1, Dirac-Pauli representation of the γ -matricesand the (+ , − , − , − ) metric [21]. To show how the energy and angular distribution of the electron antineutrinosmay depend on the angle between the transverse antineutrino spin polarizationand 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 noticethat 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 bythe direction of the initial muon polarization ˆ η µ and of the outgoing electron4 - targetsource ^ ^ ^ ^ h m q h ´ m j n e q h ^n - Fig. 1. Figure shows the production plane of the ν e -antineutrinos for the process of( µ − → e − + ν e + ν µ ), η ⊥ ν - the transverse polarization of the outgoing antineutrino.The production plane is spanned by the direction of the initial muon polarization ˆ η µ and of the outgoing antineutrino momentum ˆq .SM Nonstandard Interference of Interference ofcouplings (NC) SM with NC SM with NCa b g VLL g SLL g VLL g S ∗ LL = 0 g VLL g S ∗ LL = 0 g SLR g VLL g S ∗ LR = 0 g VLL g S ∗ LR = 0 g SRL g VLL g S ∗ RL = 0 g VLL g S ∗ RL = 0 g SRR g VLL g S ∗ RR = 0 g VLL g S ∗ RR = 0 g TLL g VLL g T ∗ LL = 0 g VLL g T ∗ LL = 0 g TLR g VLL g T ∗ LR = 0 g VLL g T ∗ LR = 0 g TRL g VLL g T ∗ RL = 0 g VLL g T ∗ RL = 0 g TRR g VLL g T ∗ RR = 0 g VLL g T ∗ RR = 0Table 2Table shows non-vanishing interferences between the g VLL and g S,Tǫµ ( ǫµ = LL, LR, RL, RR ) couplings after a computation of traces for two cases: a. Polarizedmuon and polarized electron antineutrino. b. Polarized muon and polarized muonneutrino. antineutrino momentum ˆq , Fig. 1. We admit a presence of the exotic scalar g SLR and tensor g TLR couplings in addition to the standard vector g VLL coupling.It means that the outgoing electron antineutrino flux is a mixture of the left-chirality antineutrinos produced in the g VLL weak interaction and the right-chirality ones produced in the g SLR and the g TLR weak interactions. Becauseour analysis is carried out in the limit of vanishing antineutrino mass, theleft-chirality antineutrino has positive helicity, while the right-chirality one5as negative helicity, see [22]. The muon neutrino is always left-chirality bothfor the g VLL and the g SLR , g TLR couplings (muon neutrino has negative helicity,when m ν µ → g VLL is non-zero value. The table 2 displaysexplicitly that there are only two non-zero interferences between the standardcoupling g VLL and exotic couplings, i. e. g SLR and g TLR . Because we allow for thenon-conservation of the combined symmetry CP, all the coupling constants g VLL , g
SLR , g
TLR are complex. The amplitude for the polarized muon decay is ofthe form: M µ − = G F √ { g VLL ( u e γ α (1 − γ ) v ν e )( u ν µ γ α (1 − γ ) u µ )+ g SLR ( u e (1 + γ ) v ν e )( u ν µ (1 + γ ) u µ ) (1)+ g TLR u e σ αβ (1 + γ ) v ν e )( u ν µ σ αβ (1 + γ ) u µ ) } , where v ν e and u e ( u µ and u ν µ ) are the Dirac bispinors of the outgoingelectron antineutrino and electron (initial muon and final muon neutrino),respectively. G F = 1 . × − GeV − [8] is the Fermi constant. Thecoupling constants are denoted as g VLL and g SLR , g
TLR respectively to the chiralityof the final electron and initial muon. The formula for the the energy andangular distribution of the electron antineutrinos in the muon rest frame,including interference terms between the standard g VLL and exotic g SLR , g
TLR couplings with ˆ η µ · ˆq = 0 is of the form: d Γ dyd Ω ν = d Γ dyd Ω ν ! ( V ) + d Γ dyd Ω ν ! ( S + T ) + d Γ dyd Ω ν ! ( V S + V T ) , (2) d Γ dyd Ω ν ! ( V ) = G F m µ π ( | g VLL | y (1 − y )(1 + ˆ η ν · ˆ q )(1 + ˆ η µ · ˆ q ) ) , (3) d Γ dyd Ω ν ! ( S + T ) = G F m µ π (1 − ˆ η ν · ˆ q ) | g SLR | y ((cid:20) (3 − y ) − (1 − y ) ˆ η µ · ˆ q (cid:21) + 4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g TLR g SLR (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) (15 − y ) − (13 − y ) ˆ η µ · ˆ q (cid:21)) , (4) d Γ dyd Ω ν ! ( V S + V T ) = G F m µ π y (1 − y ) (5) · ( Re ( g VLL g S ∗ LR )( η ⊥ ν · ˆ η µ ) + Im ( g VLL g S ∗ LR ) η ⊥ ν · ( ˆq × ˆ η µ ) − (cid:20) Re ( g VLL g T ∗ LR )( η ⊥ ν · ˆ η µ ) + Im ( g VLL g T ∗ LR ) η ⊥ ν · ( ˆq × ˆ η µ ) (cid:21)) , m µ is the muon mass, y = E ν m µ is the reduced antineutrino energy, itvaries from 0 to 1, ˆ η ν · ˆ q = +1 is the longitudinal polarization of the left-chirality electron antineutrino for the standard g VLL , while ˆ η ν · ˆ q = − g S,TLR couplings.It is necessary to point out that the above formula is presented after theintegration over all the momentum directions of the outgoing electron andmuon neutrino. If the ˆ η µ · ˆq = 0 the interference part can be rewritten in thefollowing way: d Γ dyd Ω ν ! ( V S + V T ) = G F m µ π | η ⊥ ν || η ⊥ µ || g VLL || g SLR | (6) · (cid:26) cos ( φ − α V S ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g TLR g SLR (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos ( φ − α V T ) (cid:27) y (1 − y ) , where φ is the angle between the direction of η ⊥ ν and the direction of η ⊥ µ ; α V S ≡ α LLV − α LRS , α V T ≡ α LLV − α LRT are the relative phases between the g VLL and g SLR , g
TLR couplings.We see that in the case of the transversely polarized antineutrino beam comingfrom the polarized muon decay, the interference terms between the standardcoupling g VLL and exotic g S,TLR couplings do not vanish in the limit of vanishingelectron-antineutrino and muon-neutrino masses. This independence on themass makes the measurement of the relative phases α V S , α
V T between thesecouplings possible. The interference part, Eq. (6), includes only the contri-butions 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 the ˆ q . It canbe noticed that the relative phases α V S , α
V T different from 0 , π would indi-cate the CP violation in the CC weak interaction. Using the current data [8],we calculate the upper limit on the magnitude of the transverse antineutrinopolarization and lower bound for the longitudinal antineutrino polarization,see [22]: | η ⊥ ν | = 2 q Q νL (1 − Q νL ) ≤ . , ˆ η ν · ˆ q = 2 Q νL − ≥ . , (7) Q νL = 1 − | g SLR | − | g TLR | ≥ . , (8)where Q νL is the probability of obtaining the left-chirality (anti)neutrino.The Fig. 2 shows the plot of the d Γ d Ω ν as a function of the azimuthal angle φ for ˆ η µ · ˆ q = 0 , ˆ η ν · ˆ q = 0 . , | η ⊥ ν | = 0 . , | η ⊥ µ | = 1 , y = 2 / , | g SLR | =0 . , | g TLR | = 0 . , | g VLL | = 0 . α V S = π/ , α V T = π/ ig. 2. Plot of the d Γ d Ω ν as a function of φ : a) solid line is for the SM; b) CP violation, α V S = π/ , α V T = π/ α V S = 0 , α
V T = 0(long-dashed line) .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 direc-tions (with | g VLL | = 1 , ˆ η ν · ˆ q = +1) is the same as the Eq. (7) in [22] (with Q νL = 1 , ω L = 0 , η L = 0, neglecting the masses of the neutrinos and of theelectron as well as radiative corrections). We see that for ˆ η µ · ˆ q = − S + T ) survives: d Γ dyd Ω ν ! ( S + T ) = G F m µ π (1 − ˆ η ν · ˆ q ) | g SLR | y (1 − y ) ( (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g TLR g SLR (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ) . (9)It means that the electron antineutrino beam emitted in the direction antipar-allel to the muon polarization direction includes only the exotic right-chiralityantineutrinos with ˆ η ν · ˆ q = −
1. If the exotic interactions g SLR , g
TLR are presentin the DPMaR, the right-chirality antineutrinos (with negative helicity for m ν e →
0) are no longer ”sterile”.After the integration of the Eqs. (3, 4, 6), the muon lifetime is as follows:8 = 192 π m µ G F | g VLL | + | g SLR | + 3 | g TLR | ! . (10)Because the muon lifetime is measured observable, so the admittance of theexotic g S,TLR couplings means that the standard coupling g VLL should be de-creased in order to the sum ( | g VLL | + | g SLR | + 3 | g TLR | ) was a constant value.If the antineutrino beam comes from the unpolarized muon decay, the energyand angular distribution of the electron antineutrinos consists only of twoparts; standard ( V ) and exotic ( S + T ), i. e. Eqs. (3, 4) for ˆ η µ · ˆq = 0. If oneputs E ν = m µ / y = 1) in both parts, the standard ( V ) part vanishes,while the exotic ( S + T ) one survives. In this paper, we have shown that the energy and angular distribution of theelectron-antineutrinos in the muon rest frame can be sensitive to the interfer-ence terms between the standard left- and exotic right-chirality antineutrinos,proportional to the transverse components of the antineutrino beam polariza-tion. The magnitude of the azimuthal asymmetry caused by the interferencesis 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 interactionsin addition to the standard vector interaction in the DPMaR indicates thepossibility of producing the mixture of the left- and right-chirality electronantineutrinos with the assigned direction of the transverse antineutrino spinpolarization with respect to the production plane. Such polarized beam couldbe scattered on the PET in order to measure the CP-violating effects causedby the exotic couplings of the right-chirality antineutrinos g S,TLR (or the right-chirality neutrinos) in the purely leptonic processes.We have noticed that for ˆ η µ · ˆ q = −
1, the energy and angular distribution ofthe electron antineutrinos consists only of the exotic part ( S + T ). It meansthat if the SM prediction is correct, no signal should be detected for theelectron antineutrino beam emitted in the direction antiparallel to the muonpolarization direction. We see that the magnitude of this contribution is verysmall 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 lon-gitudinally polarized (anti)neutrino beam and to measure the dependence ofthe antineutrino energy spectrum on the ˆ η µ · ˆ q . So far no such tests have beencarried out.The observation of the right-handed current interaction is important for in-terpreting results on the neutrinoless double beta decay [23].9e plan to search for the other polarized (anti)neutrino beams, which couldbe interesting from the point of observable effects caused by the exotic right-chirality states. We expect some interest in the neutrino laboratories workingwith 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 teefor Scientific Research LNGS/103/2006 and 1 P03D 005 28.
The formula for the the spin polarization 4-vector of massive antineutrino S ′ moving with the momentum q is as follows: S ′ = ( S ′ , S ′ ) , (11) S ′ = | q | m ν ( ˆ η ν · ˆq ) , (12) S ′ = − (cid:18) E ν m ν ( ˆ η ν · ˆq ) ˆq + ˆ η ν − ( ˆ η ν · ˆq ) ˆq (cid:19) , (13)where ˆ η 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 limitof vanishing antineutrino mass m ν is given by:lim m ν → Λ ( s ) ν = lim m ν → (cid:26) [( q µ γ µ ) − m ν ] [1 + γ ( S ′ µ γ µ )] (cid:27) (14)= 12 (cid:26) ( q µ γ µ ) [1 − γ ( ˆ η ν · ˆq ) + γ ( ˆ η ν − ( ˆ η ν · ˆq ) ˆq ) · γ ] (cid:27) (15)= 12 (cid:26) ( q µ γ µ ) h − γ ( ˆ η ν · ˆq ) − γ S ′⊥ · γ i (cid:27) , (16)where S ′⊥ = (cid:16) , η ⊥ ν = ˆ η ν − ( ˆ η ν · ˆq ) ˆq (cid:17) . We see that in spite of the singularities m − ν in the polarization four-vector S ′ , the density operator Λ ( s ) ν remains finiteincluding the transverse component of the antineutrino spin polarization [20]. References [1] S. L. Glashow, Nucl. Phys. , 579 (1961).
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