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RESONANT SPIN-FLAVOR PRECESSION OF NEUTRINOS AS A

arXiv:hep-ph/9205244v1 29 May 1992RESONANT SPIN-FLAVOR PRECESSION OF NEUTRINOS AS APOSSIBLE SOLUTION TO THE SOLAR NEUTRINO PROBLEM ∗SISSA Ref. 49/92/EPEugueni Kh.

Akhmedov † ‡Scuola Internazionale Superiore di Studi AvanzatiStrada Costiera 11, I-34014 Trieste, ItalyAbstractRecent developments of the resonant neutrino spin-flavor precession scenario andits applications to the solar neutrino problem are reviewed. We discuss in particularthe possibilities of reconciliation of strong time variations of the solar neutrino fluxobserved in the Homestake 37Cl experiment with little or no time variation seen in theKamiokande II experiment.March 1992∗Talk given at the XII Moriond Workshop ”Massive Neutrinos.

Tests of Fundamental Symmetries”, LesArcs, France, Jan. 25-Feb. 1, 1992†on leave from Kurchatov Institute of Atomic Energy, Moscow 123182, Russia‡E-mail: akhmedov@tsmi19.sissa.it, akhm@jbivn.kiae.su

1IntroductionThere are two issues in the solar neutrino problem:(1) the deficiency of solar neutrinos observed in the Homestake [1], Kamiokande II [2] and,most recently, SAGE [3] experiments;(2) time variation of the solar neutrino flux in anticorrelation with solar activity (11-yrvariations) for which there is a strong indication in the chlorine experiment of Davis andhis collaborators but which is not seen in the Kamiokande data.In this talk I will discuss mainly the second issue with the emphasis on various possi-bilities of conciliation of the strong time variations in the Homestake experiment with no orlittle variation in Kamiokande II.The most natural explanation of the time variation of the solar neutrino flux is relatedto the possible existence of a large magnetic or electric dipole moments of neutrinos, µ ∼10−11µB. As was pointed out by Vysotsky, Voloshin and Okun (VVO) [4, 5], strong toroidalmagnetic field in the convective zone of the sun B⊥could then rotate left-handed electronneutrinos νeL into right-handed νeR which escape the detection.

In the periods of quiet sunthe solar magnetic field is much weaker and the neutrino spin precession is less efficientwhich explains the 11-yr variation of the neutrino flux.Subsequently, it was noted [5, 6] that the matter effects can suppress the neutrino spinprecession.The reason for this is that νeL and νeR are not degenerate in matter sinceνeL interact with medium whereas νeR are sterile, and their energy splitting reduces theprecession probability.It was also shown [7] that, unlike in the MSW effect case, theadiabaticity may play a bad role for the VVO effect resulting in a reflip of neutrino spin andthus reducing the probability of νeL →νeR transition. In order to break the adiabaticity,the precession length should be large as compared to the characteristic lengths over whichmatter density and magnetic field vary significantly, which gives an upper bound on µB⊥.This parameter should be also bounded from below in order for the precession phase not to1

be too small. Therefore one gets a rather narrow range of allowed values of µB⊥[7].Another interesting possibility is the neutrino spin-flavor precession (SFP) due to theinteraction of flavor-off-diagonal (transition) magnetic or electric dipole moments of neutri-nos µij with transverse magnetic fields [8, 5].

The SFP is the rotation of neutrino spin withits flavor being simultaneously changed. Such a process can occur even for Majorana neu-trinos since the CPT invariance does not preclude the transition magnetic dipole momentsof Majorana particles.

Until recently, the neutrino SFP has not attracted much attentionbecause it was expected to be suppressed by the energy splitting of the neutrinos of differentspecies. If the ”Zeeman energy” µijB⊥is small as compared to the kinetic energy differ-ence ∆m2ij/2E, the SFP probability is heavily suppressed.

However, in 1988 it was notedindependently by the present author [9, 7] and by Lim and Marciano [10] that in matterthe situation can change drastically. Since νeL and right-handed neutrinos or antineutrinosof another flavor interact with matter differently, the difference of their potential energiescan cancel their kinetic energy difference resulting in a resonant amplification of the SFP.Therefore in matter the SFP of neutrinos can be enhanced, unlike the VVO neutrino spinprecession 1.

The resonant spin-flavor precession (RSFP) of neutrinos has also some moreadvantages as compared to the VVO mechanism:• the adiabaticity plays a good role for the RSFP increasing the conversion probability,and therefore the µijB⊥should be bounded only from below; the required magnitude of thisparameter is a factor of 2 −3 smaller than that for the VVO effect;• some energy dependence of the neutrino conversion seems to be necessary to reconcilethe Homestake and Kamiokande II data (see below). The RSFP probability has the desiredenergy dependence whereas the VVO neutrino spin precession is energy independent.Although the above arguments disfavor the VVO effect as a solution of the solar neutrinoproblem, they do not rule it out, given the uncertainties of the experimental data.1The VVO neutrino spin rotation can also be resonantly enhanced provided the magnetic field twistsalong the neutrino trajectory, see [11, 12] and below.2

2General features of RSFP of neutrinosThe RSFP of neutrinos is analogous to the resonant neutrino oscillations [13, 14], but differsfrom the latter in a number of important respects. The main features of this effect havebeen discussed in detail in my talk at the last Moriond meeting [15], and so I will just brieflymention them here.The magnetic-field induced mixing of νeL and νµR(¯νµR) can be described by the mixingangle θ,tan 2θ =2µeµB⊥√2GF(Ne −αNn) −∆m2eµ2Ecos 2θ0(1)Here Ne and Nn are the electron and neutron number densities, α = 1/2 for Dirac neutrinosand 1 for Majorana neutrinos, GF is the Fermi constant, and θ0 is the ordinary neutrinomixing angle in vacuum.

The resonant density is defined as a density at which the mixingangle θ becomes π/4:√2GF(Ne −αNn)|r = ∆m2eµ2Ecos 2θ0(2)The efficiency of the νeL→νµR(¯νµR) transition is defined by the degree of the adiabaticitywhich depends on both the neutrino energy and magnetic field strength at the resonance:λ ≡π∆rlr= 8E∆m2eµ(µeµB⊥r)2Lρ(3)Here∆r = 8EµeµB⊥r∆m2eµLρ(4)is the resonance width, lr = π/µeµB⊥is the precession length at the resonance and Lρ isthe characteristic length over which matter density varies significantly in the sun. For theRSFP to be efficient, λ should be > 1.

In non-uniform magnetic field the field strength atresonance B⊥r depends on the resonance coordinate and so, through eq. (2), on neutrinoenergy.

Therefore the energy dependence of the adiabaticity parameter λ in eq. (3) is, ingeneral, more complicated than just λ ∼E, and is defined by the magnetic field profile3

inside the sun 2. The main difficulty in the analyses of the RSFP as a possible solution ofthe solar neutrino problem is that this profile is essentially unknown, so that one is forcedto use various more or less plausible magnetic field configurations.In the adiabatic regime (λ >> 1), the νeL survival probability isP(νeL →νeL) = 12 + 12 cos 2θi cos 2θf + 12 sin 2θi sin 2θf cosZ tfti∆E(t) dt(5)where∆E =vuut"√2GF(Ne −αNn) −∆m2eµ2Ecos 2θ0#2+ (2µeµB⊥)2(6)Here θi and θf are the mixing angles (1) at the neutrino production point and on the surfaceof the sun respectively.

If the νeL are produced at a density which is much higher than theresonant one, θi ≈0 and the survival probability (4) becomesP(νeL →νeL) ≈cos2 θf(7)Since the magnetic field becomes very weak at the sun’s surface, the mixing angle θf ≈π/2,and so the νeL survival probability is very small in the adiabatic regime. The adiabaticityparameter λ in eq.

(3) depends drastically on the magnetic field strength at resonance, whichgives a natural explanation of time variations of the solar magnetic flux in anticorrelationwith solar activity.The RSFP requires non-vanishing flavor-off-diagonal magnetic dipole moments of neu-trinos and so is only possible if the neutrino flavor is not conserved. Therefore neutrinooscillations must also take place, and in general one should consider the SFP and oscilla-tions of neutrinos jointly.

This have been done in a number of papers both analytically[16, 17] and numerically [10, 16, 18, 19, 17]. It was shown that a subtle interplay betweenthe RSFP and the MSW resonant neutrino oscillations can occur.

In particular, althoughthe resonant neutrino oscillations cannot give rise to the time variations of the solar neutrinoflux, they can assist the RSFP to do so by improving the adiabaticity of the latter [17].2Note that for the MSW effect the adiabaticity parameter is inversely proportional to E [13].4

3Neutrino spin precession in twisting magnetic fieldsIf the magnetic field changes its direction along the neutrino trajectory, this can resultin new interesting phenomena. In particular, new kinds of resonant neutrino conversionsbecome possible, the energy dependence of the conversion probability can be significantlydistorted and the lower limit on the value of µB⊥required to account for the solar neutrinoproblem can be slightly relaxed [11, 12].

Moreover, if the neutrino oscillations are also takeninto account, the transitions νe →¯νe can become resonant, and the order of the RSFP andMSW resonances can be interchanged [20].Since the main features of the resonant neutrino spin-flip transitions in twisting magneticfields are discussed in some detail in the contributions of Krastev and Toshev in this volume,I will confine myself to a new development which was not covered in their talks.A few years ago, Vidal and Wudka [21] claimed that the field rotation effects can greatlyenhance the neutrino spin-flip probability and reduce the needed value of µB⊥by a feworders of magnitudes. In [11, 12] it was shown that this result is incorrect and typically therequired value of µB⊥can only be reduced by a factor 2–3 (see also [22, 23] in which theprocess without matter effects was considered).

However, in these papers it was not provedthat there cannot exist a rotating field configuration giving stronger enhancement of thespin-flip probability and larger gain in the µB⊥parameter. Recently, Moretti [24] has founda severe constraint on the transition probability which eliminates even this possibility.

Theeffective Hamiltonian describing the evolution of the system of left handed νeL and righthanded neutrino of the same or another flavor νR in a twisting magnetic field isH =V (t)/2µB⊥eiφ(t)µB⊥e−iφ(t)−V (t)/2(8)where V(t) is just the denominator of the r.h.s. of eq.

(1), and the angle φ(t) defines thedirection of the magnetic field in the plane orthogonal to the neutrino momentum. The5

transition probability P(νeL →νR) turns out to have the following upper bound [24]:P(νeL →νR; t) ≤µZ t0 B⊥(t′) dt′(9)The analogous result can also be obtained for the neutrino oscillations in matter as well asfor the evolution of any other two-level system.4RSFP and antineutrinos from the sunIf both the SFP and oscillations of neutrinos can occur, this will result in the conversion ofa fraction of solar νe into ¯νe [10, 16, 25, 26]. For Majorana neutrinos, the direct νe →¯νeconversions are forbidden since the CPT invariance precludes the diagonal magnetic momentµee.

However, this conversion can proceed as a two-step process in either of two ways:νeLoscill.−→νµLSFP−→¯νeR(10)νeLSFP−→¯νµRoscill.−→¯νeR(11)One can then consider two possibilities:(1) both oscillations and SFP take place inside sun [10, 16, 25]. The amplitudes of theprocesses (10) and (11) have opposite signs since the matrix of the magnetic moments ofMajorana neutrinos is antisymmetric.

Therefore there is a large cancellation between thesetwo amplitudes (the cancellation is exact in the limit of vanishing neutron density Nn), andthe probability of the νe →¯νe conversion inside the sun turns out to be about 3–5% evenfor large mixing angles θ0 [16, 25]. (2) Only the RSFP transition νeL →¯νµR occurs in the sun with an appreciable proba-bility whereas the oscillations of neutrinos proceed mainly in vacuum on their way betweenthe sun and the earth [eq.

(11)]. For not too small neutrino mixing angles the probabilityof the νe →¯νe conversion can then be quite sizable [26].In [27] the background events in the Kamiokande II experiment were analysed and astringent bound on the flux of ¯νe from the sun was obtained: Φ(¯νe) ≤(0.05 −0.07)Φ(νe).6

This poses a limit on the models in which both the RSFP and neutrino oscillations occur: themixing angle θ0 should be less than 6−8◦. This rules out the models with the large magneticmoments of pseudo Dirac neutrinos including those with only one neutrino generation forwhich θ0 is the mixing between νeL and sterile ¯νeL [28, 29].

However, the models with aconserved lepton charges Le ± (Lµ −Lτ) are not excluded even though the mixing angle isπ/4, since the νe →¯νe conversion probability vanishes identically in this case [30].The ¯νe production due to the combined effect of the RSFP and oscillations of neutrinoscan be easily distinguished from the other mechanisms of ¯νe generation (like ν →¯ν+Majorondecay) since (i) the neutrino flux should vary in time in direct correlation with solar activity,and (ii) the neutrino energy is not degraded in this case [16, 25]. The ¯νe flux from the sun ofthe order of a few per cent of the expected νe flux should be detectable in the forthcomingsolar neutrino experiments like BOREXINO, SNO and Super-Kamiokande [25, 26, 31].5Reconciling the Homestake and Kamiokande II dataIt has been mentioned above that while there is a strong indication in favor of time variationof the neutrino detection rate in the Homestake data, the Kamiokande experiment does notsee such a time variation.

It still cannot rule out a small (≤30%) time variation. Thereforea question naturally arises as to whether it is possible to reconcile large time variations inthe Homestake 37Cl experiment with small time variation in the water ˇCerenkov experiment.There are two major differences between these two experiments which could in principle giverise to different time variations of their detection rates:(1) Homestake experiment utilizes the νe −37Cl charged current reaction, while in theKamiokande detector ν−e scattering is used which is mediated by both charged and neutralcurrents;(2) the energy threshold in the Homestake experiment is 0.814 MeV so that it is sensitiveto high energy 8B, intermediate energy 7Be and partly to low energy pep neutrinos; at the7

same time the energy threshold in the Kamiokande II experiment is 7.5 MeV and so it isonly sensitive to the high-energy fraction of the 8B neutrinos.In [32, 17] it was noted that if the lower-energy neutrino contributions to the chlorinedetection rate are suppressed stronger than that of high-energy neutrinos, the latter can varyin time with smaller amplitude and still fit the Homestake data. In that case one can expectweaker time variations in the Kamiokande II experiment.

The desired suppression of thelow-energy neutrino flux can be easily explained in the framework of the RSFP scenario as aconsequence of flavor-changing spin-flip conversion due to a strong inner magnetic field, theexistence of which seems quite plausible [33]. The alternative possibility is the suppression oflow energy neutrinos by the MSW effect when RSFP and the resonant neutrino oscillationsoperate jointly.

Another important point is that due to the RSFP solar νe are convertedinto ¯νµR or ¯ντR which are sterile for the chlorine detector but can be detected (though witha smaller cross section) by water ˇCerenkov detectors. This also reduces the amplitude of thetime variation in the Kamiokande II detector.

If both these factors are taken into account,it becomes possible to reconcile the Homestake and Kamiokande data; one can expect alow signal in the gallium experiments in this case since they are primarily sensitive to lowenergy neutrinos whose flux is supposed to be heavily suppressed [32, 17].A similar possibility has been recently considered by Babu, Mohapatra and Rothstein[34] and by Ono and Suematsu [35]. They pointed out that due to the energy dependenceof the RSFP neutrino conversion probability, lower-energy neutrinos can exhibit strongertime variations (i.e.

stronger magnetic field dependence) than the higher-energy ones. Infact, this is very natural in the RSFP scenario: with increasing neutrino energy the width ofthe resonance increases [see eq.

(4)] and at sufficiently high energies it can be a significantfraction of the solar radius. The neutrino production point can then happen to be insidethe resonant region, which reduces the conversion efficiency.

The different magnetic fielddependence of the Homestake and Kamiokande II detection rates is illustrated by the figures8

which we borrowed from ref. [34].Fig.

1. (a) Expected event rate in chlorine as a function of the convective zone magnetic field.

Here∆m2 = 7.8×10−9 eV2, the maximal value of the magnetic field in the core B1 = 107 G and µ = 2×10−11µB. (b) The same as (a) but for the Kamiokande event rate.It should be noted that the ordinary VVO neutrino spin precession lacks energy de-pendence which is required to get smaller time variation in the Kamiokande II experiment.Moreover, it converts νeL into sterile νeR (unless the neutrinos are Zeldovich-Konopinski-Mahmoud particles) which do not contribute to the ν −e cross section.

However, for theVVO scenario yet another possibility of reconciliation of the Homestake and Kamiokandedata exists. In order to get sizable magnetic moments of neutrinos, µ ≈10−11µB, one hasto go beyond the Standard Model.

Most of the models producing large neutrino magneticmoments are based on various extensions of the Standard Model containing new chargedscalars. In these models right-handed sterile neutrinos can interact with electrons via scalarexchange and therefore can contribute to the ν −e reaction which increases the signal in theKamiokande II detector and reduces the amplitude of its time variation [36].

Note that themodels giving large transition neutrino magnetic moments usually also contain new scalarsand therefore the same mechanism can be operative in case of the RSFP as well.9

6ConclusionWe conclude that the resonant neutrino spin-flavor precession mechanism provides a viableexplanation of the solar-neutrino problem which complies with all the existing experimentaldata and yields a number of interesting predictions for the forthcoming experiments.AcknowledgementThe author is deeply indebted to Scuola Internazionale Superiore di Studi Avanzati wherethis report was written for kind hospitality and financial support.References[1] R. Davis, Jr., In Neutrino ’88, Proc. XIXth Int.

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