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Neutrino oscillations and magnetic moment transitions in a model with a conserved lepton number
Volume 264, number 3,4
PHYSICS LETTERS B
1 August 1991
Neutrino oscillations and magnetic moment transitions in a model with a conserved lepton number Z.G. Berezhiani a, G. Fiorentini a b c d e f
b.c M.
Moretti d,e and S.T. Petcov d,f
Institute of Physics of the Georgian Academy of Sciences, SU-380 077 Tbilisi, USSR Department of Physics, Universit& di Ferrara, 1-44100 Ferrara, Italy lstituto Nazionale di Fisica Nucleare, Sezione di Ferrara, 1-44100 Ferrara, Italy Scuola lnternazionale Superiore di StudiAvanzati, 1-34014 Trieste, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, BG- 1784 Sofia, Bulgaria
Received 7 May 1991
We discuss a three flavour neutrino model with a conserved non-standard lepton number L+ =Le+ (Lu-L.~). In this frame, flavour oscillations and magnetic moment transition can both occur and at the same time this involves only one mass scale. We show that the model can account for the results of both the Chlorine and the Kamioka experiments. In addition the conservation law provides naturally a total suppression of the % flux from the sun.
1. Introduction
The experimental situation concerning solar neutrinos can be summarized as follows: (i) Both in the Chlorine and in the Kamioka experiment the observed signal is significantly smaller than predicted by the standard solar model. For the two experiments, the ratios Zi of the observed signal to the expectation of the standard solar model (SSM), averaged over the data taking period, are respectively [ 1,2 ]
(1.1,2)
(ii) In the Chlorine experiment there are indications that the neutrino signal is (anti) correlated with the solar cycle. For solar cycles 21 and 22, the normalized signals in correspondence of the minima and maxima of solar activity are respectively [ 1 ] (Zc)min=0.53±O.08
,
(Zc)max=O.08+O.03
.
(1.3,4)
(iii) Note that in correspondence of the solar minima, still there is a signal deficit. Mechanisms of neutrinos oscillations, either in vacuum or in matter, can account for (i) and (iii). On the other hand, they hardly can explain the time variation. This feature is most directly connected with the hypothesis of a large neutrino magnetic moment, coupled to the solar magnetic field in the convective zone [ 3 ]. However, this mechanism alone cannot explain the deficit in correspondence of the solar minima. In order to account for all the features simultaneously, it is natural to resort to the so called "hybrid models", i.e. models where both neutrino oscillations and magnetic spin flip transitions are operating. Hybrid models are discussed in the literature in the context of two neutrino flavours, say ve and v~ for definiteness [4 ]. In our opinion, this leads to two unpleasant features: (a) As well known, a sizeable magnetic moment is only possible if the neutrino mass is close to the observational bounds. On the other hand, the magnetic transition is not quenched only if the mass difference between 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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the two neutrino states (which also cannot vanish to allow for flavour oscillation) is very small ~: 6m 2< 10 - 6 (eV) 2. Thus the neutrino mass spectrum involves two vastly different mass scales, which may be difficult to account for theoretically. (b) The second potential difficulty o f hybrid models with two neutrino flavours is related to the absence of any lepton number conservation law. As a consequence, the simultaneous presence of both magnetic and flavour oscillations leads in general to a significant transition among all the four states v~, v,, %, 9,. This may contradict the upper bound on the flux o f % , which can be derived by using the Kamioka data [ 5 ]. A natural way to avoid both these problems consists in considering all the three standard neutrino flavours re, G, V. and assuming a suitable nonstandard lepton number to be conserved:
L± =L~+_ ( L , - L . )
.
(1.5)
This conservation law implies just one mass scale, the mass spectrum consisting of one massive Dirac neutrino and one massless Weyl neutrino [ 6 ]. An unquenched magnetic transition can take place between the two helicity states o f the Dirac neutrino and, on the other hand, the fact that the mass eigenstates are linear superpositions of the different neutrino flavours gives rise to a flavour oscillation. In other words by considering all the three neutrino flavours one can have flavour oscillations even in the presence o f a Dirac neutrino state in the spectrum.
2. The model
For definiteness, we take L+ as a conserved q u a n t u m number. The most general mass term which is consistent with this constraint is Lm = - m OCR( Uee COS a + U,L sin a ) + b.c.
(2.1)
This lagrangian describes one Dirac neutrino with mass m, ~rnL = PeL COS a + P,L sin a ,
~/mR= pCR
(2.2)
and a massless Weyl neutrino, (2.3)
Uo = -- UeLSin o r + U,LCOS a .
For sizeable values of the mixing angle a, experiments at nuclear reactors yield [ 7 ] m ~<0.1 eV. Since significant magnetic moments call for high masses, we will explore the region near this upper bound: m ~ 1 0 - 2 - 1 0 - leV.
(2.4)
In this range of masses the existing data from accelerator experiments on ve~v~ (%--*9,) do not impose any constraint on sin a. The most general magnetic m o m e n t effective lagrangian, consistent with the conservation of L+, has the form
L , = --~ll 1 C V,Ra,~O F~o (PeLCOS ]~"[- V , L s i n
fl) + h . c .
(2.5)
The hamiltonian H which determines the evolution of the ve state in the sun,
i(O/3r) l u(r) ) = H ( r ) l g ( r ) ) ,
(2.6)
where r= ct, is immediately obtained from eqs. (2.1) and (2.5), including the coherent neutrino weak interaction with matter [ 8 ]. We observe that due to the conserved lepton number the hamiltonian decouples into two sub-hamiltonians each coupling only three states. In the mass eigenstates basis (/Vo), [ ~'mL), [ ~//rnR) ) one has ~l This result holds for/t.~ 10 I I ga, B ~ 100 kG. The same results hold also for the resonant spin flip model [ 5 ] if we restrict the magnetic field to the convective zone, which is the only useful assumption to recover the observed time dependence of the solar neutrino flux.
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f GFnex/~ sinaa--GFnn/x/~ H(r)= [
--GFnex/~sincxcost~
\
#Bsin(fl-a)
1 August 1991
-GFnex/~ sin a cos a izB sin(fl-a) m2/2E+GFnex/~cos2o~-Gvnn/x//2 ~Bcos(fl-o~) ~ , /zBcos(fl-a) m2/2E+GFn,/x/2/
(2.7)
where ne and n, are the electron and neutron densities respectively. GF is the Fermi constant and B=B(r) is the transverse component of the solar magnetic field. In the convective zone (r>~RB=0.7 Ro, where R o = 7 X 105 km is the solar radius) we assume n.=~ ne and the density profile to be given by [9] H e ~ F/0
exp ( -
r/LM) ,
(2.8)
with LM ~ 0.1 Ro, and ne (RB) ~ 0.16 NA cm-3, NA being the Avogadro number. Concerning the magnetic field, we will assume that it is confined within the convective zone. In the absence of detailed information on the shape of the field, we use the following parametrization:
B(r) =O(r-Ra)Bo {1 - e x p [ ( --r+RB )/LBI} •
(2.9)
The starting point RB is taken at the beginning of the convective zone. The raising length LB - some fraction of the depth o f the convective zone - and the plateau value Bo are kept as free parameters. Bo varies during the solar cycle, from Bo ~ 0 at the solar minima up to a value Bb~ax in correspondence to the maxima o f solar activity. Note that, according to (2.9), in the very outer region of the sun the field does not fall (as fast) as the coherent weak interaction. In fact we expect the magnetic field to decrease with some power of the distance, whereas the density profile is exponentially decaying. It is useful to introduce a magnetic energy term, e(r) = / t B ( r ) c o s ( f l - a ) ,
(2.10)
and a magnetic energy plateau ~o which is obtained from (2.10) for (/t ~ 10- l OpB, B ~ a× ~ 1 T) one has e ~ax ~ 20/Ro for cos ( f l - a ) = 1.
B=Bo. For the values of interest to us
3. Flavour oscillations and spin flip in the absence of coherent weak interactions As a first stage of the discussion, we neglect in eq. (2.7) the coherent weak interaction. For the values o f interest to us, one has m2/2E>> pB and consequently the ] Uo) state is decoupled from the others. One also has (m2/2E) Ro >> l, so that oscillating terms of argument (rn2/2E)Ro can be replaced by their average value. In these hypotheses, the arrival probabilities P(Vx) of the originally Ve state are P(V e ) = sinaa + cos 4ol COS 2(~)
,
( 3.1 )
P (v,) = ¼sin22a ( 1 + cos2~) ,
( 3.2 )
P(9~) = cos2a sin2~,
(3.3)
where • =/z c o s ( f l - a ) ~
B(r) dr.
(3.4)
The main phenomenological features of the model are immediate from the equations given above: (a) Let us assume that the spectrum of neutrinos produced in the core of the sun, dF/dE, is given correctly by the standard model, apart from an overall factor A accounting for the SSM uncertainties:
dF/dE=A dFSSM/dE, A = 0 . 7 - 1 . 3 .
(3.5) 383
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In this case, the ratio o f the signal in the Chlorine experiment to the expectation of the SSM is simply
Zc = A P ( v e ) .
(3.6)
The signal deficit at solar minima ( qb= 0) calls for a large value of the vacuum mixing angle. Indeed, from eq. (1.3), at the lalevel, one gets 2( 1 - 0 . 6 1 / A ) ~
(3.7)
In the case A = 1 this yields 0.78~
(3.8)
Note that the same results hold for q)= 0 also in the case that coherent weak interactions are included. In other words, the signal deficit at solar minima always implies a large vacuum mixing angle. (b) A significant depletion of the signal in correspondence with the solar maxima can be achieved only for (J~max close to ½n. In fact, smaller values o f q~maxyield a too small time dependence whereas for larger values the signal oscillates with a frequency larger than that o f the solar cycle. As remarked in ref. [ 10 ] this condition requires a fine tuning of the parameters (/~ and B) which are involved. 1 (c) Even for the best case, ~max-~if, a significant depletion of the signal at the solar maxima calls for large values o f the vacuum mixing angle. From ( 1.3 ) and ( 1.4) one has ( Zc" )max/ ( Zc)min =0.15 +-0.07 .
(3.9)
By using eq. ( 3.1 ) with (~)min = 0 and ~bmax = ½n we get, at the 1a level, sin 2 2c~=0.81 + 0 . 1 0 .
(3.10)
(d) In the Kamioka experiment, in addition to re, both v, and 9~ can scatter off electrons. For the Kamioka experiment one has ZK = A [P(ve ) + Wrt(Eth)P(v,) + W~ (Eth)P(v~) ] ,
(3.11 )
where the factors Wx(E~h) depend on the threshold energy and on the neutrino flavour (for Eth=9.3 MeV, I 1 W~=5 and W~=~). For the present model one obviously has ( Z K ) > / ( Z c ) , in agreement with the trend o f the experimental data. Also, the time modulation comes out to be weaker in Kamioka. As an example, by assuming = 0 and q) = ½n at solar m i n i m a and maxima respectively, for a as in (3.10) and a threshold energy E t h = 9.3 we get a modulation (ZK)max/(ZK)min=0.29+_0.7.
4. Inclusion of coherent weak interactions
The neutrino evolution corresponding to the full hamiltonian given in eqs. ( 2 . 7 ) - ( 2 . 9 ) has to be studied numerically. In this section we shall investigate two opposite limiting cases, which can be discussed analytically and yield a qualitative understanding of the complete solution. For the values of the mass we are considering (see eq. ( 2 . 4 ) ) both Gnc and Gn. are (almost) neglibile in comparison with the energy splitting, m 2/2E. The same holds true of the ¢tB terms for the values of interest to us. In these circumstances, the massless neutrino component IVo) is practically decoupled from the others. The problem thus simplifies to the study of a two state system, I ~/JmL) and I q/mR). For this system, the relevant part of the hamiltonian (i.e. apart from terms proportional to the identity) is
h(r)= \e(r) 384
-3(r)]'
(4.1)
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where
A(r) =G(ne COS2Ot-- n n ) / N / ~ •
(4.2)
Note that, since the magnetic m o m e n t couples the left and right components of the same Dirac neutrino, there are no mass terms inhibiting the spin flip transition. We recall that we assume a large mixing angle, c o s 2 a = ½ and in the convective region we use eq. (2.8). We note that, as a consequence of the large vacuum mixing angle, A(r) never vanishes in the convective zone. A non-zero value of A(r) clearly inhibits spin flip transitions, see ref. [ 11 ], with respect to the case without coherent weak interactions. The term ~(R) is defined by means ofeqs. (2.9) and (2.10):
~(r) =O(r--RB)%{1 -- exp[ ( - r + RB) /La]} .
(4.3)
We shall investigate the solution of the problem in terms of the field raising distance LB and of the interaction energy plateau eo which, we recall, varies during the solar cycle from eo ~ 0 at solar m i n i m a up to a m a x i m u m value E~a~ at solar maxima. Qualitatively, for the solution of the problem, one can distinguish three different regions: (a) In the inner part of the sun (say r> ~(r) ] and the solution is immediate. For a state which is [ ~'mL,R) at r = 0 one has
r
(4.4)
(b) In the outer part (say r > R 2 ) the magnetic m o m e n t interaction dominates [A(r)<< E(r)]. Also in this case the analytical solution is immediate. For a state which can be described as [ U/mL,R) at r=R2 one has
r>R2: [~//mL)----~[~//mL) COS~(r)--ilq/mR) s i n ~ ( r ) , [q/mR) ~--i[I//mL) sin ~ ( r ) + [ ~//mR) COS~(r),
(4.5)
where ~(r)= i eB(r')dr'.
(4.6)
R2
(c) In the intermediate region, R l < r
4.1. The adiabatic and the sudden region The transition between the two regions can be taken as adiabatic or sudden depending on the value of the function g(r)=
[EA'-~'dl/2(E2+A2) 3/2 .
(4.7)
For g << 1 (g >> 1 ) the adiabatic (sudden) limit is realized. One has to verify such conditions close to the point r = r* where e (r*) = A (r*). The curves shown in fig. 1 correspond, for cos2a ~ ½, to g ( r * ) = 1, i.e. they denote the transition between the two regimes. 385
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PHYSICS LETTERS B 0.10
0.08
0.06
~
,
-
''I
....
I ....
I ....
I ....
-
--
adia
o.o4--
0.02
I ....
1 August 1991
--
0.00
,,
0
L
....
5
~ , , , I
10
....
16
I,,,,I
20
....
25
30
c o ~, R e
Fig. 1. In the plane (eo,La) we show the borders of the adiabatic and the sudden regimes, defined as g= 1, see eq. (4.7). % is the plateau value of the magnetic interaction energy and LB is its raising length. Ro is the solar radius. The m a i n features o f fig. 1 can be u n d e r s t o o d from the following considerations. Note that g is the sum o f two positive functions, g=g~ +g2, where
g l ( r ) = - ~ A ' / 2 ( ~ 2 + A 2 ) 3/2, g2(r)=~'A/2(~2+A2) 3/2
(4.8)
At r = r*, g~ = [ 4x/~ LM e (r*) ] - ~ and one has gl >/1 for eo ~< (4x/~ LM ) - ~. This explains the region o f small fields where the sudden limit is achieved. F o r i n t e r m e d i a t e values o f the field ( (4xf2 LM ) - ~~/A (rB) ~ 1 0 / R o ) r* is reached while the magnetic interaction energy e is raising and the adiabatic c o n d i t i o n is fulfilled only for sufficiently large values of Lu.
4.2. The adiabatic and the sudden solution If the transition between regions ( a ) and ( b ) is adiabatic, the ( i n s t a n t a n e o u s ) eigenstates o f region ( a ) go into the corresponding ones o f the region ( b ) . After traversing the sun, the probabilities o f the different neutrino flavours are P(ve) = sinac~+ ½ C O S 4OL
,
(4.9)
P(v~) = 3 sin22oz,
(4.10)
P ( 9 ~ ) = ½c o s 2 a ,
(4.11)
where oscillating terms o f argument ( m 2/ 2E) R o >> 1 have been averaged. I f the transition is sudden, when crossing the i n t e r m e d i a t e region (Rl < r < R2) the [ ~mL,R> states do not change. Beyond R2, coherent interactions are irrelevant and the time evolution is the same as discussed in the previous section. 386
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The arrival probabilities are thus given again by eqs. (3.1)-(3.3 ), where the phase • is now given by q~=/2cos(fl-a) f B(R)dR.
(4.12)
R2
This means that the magnetic field contributes to the phase only in the region where it overwhelms the coherent weak interaction.
4.3. Discussion On these grounds we can discuss the arrival probabilities P (vx) during the different phases of the solar cycle. For any LB, at the solar minimum (Eo ~ 0) one is in the sudden limit. The solution is given by eqs. ( 3 . 1 ) (3.3) with ~b=0. As discussed in section 3, a significant depletion of the neutrino signal can be obtained for large mixing angles, as given in eq. (3.8). As the field raises, different regimes (adiabatic, sudden, or intermediate) can be reached depending on the value o f ~ nax and LB. From the phenomenological point of view, the most important features of the adiabatic regime are: (a) The results do not depend on the precise value of/t and B~ ax. Thus a significant modulation of the signal, anticorrelated with the solar cycle, can be obtained without a fine tuning of the magnetic field and magnetic moment, provided that e ~ax and LB are in the adiabatic region. (b) Note that the lowest values of e~ax which satisfy the adiabatic condition (and thus provide a significant time modulation) are of order e~ax ~ 5 / R o , (the inverse of the convective zone dimension). Whichever mechanism one is invoking, one cannot do better, in the sense that for a significant spin flip probability the phase O = coD, where D is the depth of the convective zone, has to exceed unity. (c) On the other hand, the minimum value of P(ve) is ~, which is at the border of the region allowed by the Chlorine data. Also, the signal modulation in the Chlorine experiment (Zc)max/(Zc)rain cannot be smaller than one half, which looks in disagreement with (3.9). In the case that the sudden regime is reached, the same phenomenological considerations as for the "vacuum" case hold. In addition, we note that only the region of high fields (~o > 10/Ro ) is relevant for getting a significant phase q~. In comparison with the adiabatic limit, the sudden case requires EbT M appreciably larger in order to achieve a significant time modulation. On the other hand, the amplitude of the modulation can be much larger, as previously remarked. In summary, the inclusion of matter effects does not alter drastically the picture. One can choose the available parameters so as to reproduce the main features of the experimental results. Furthermore, if adiabaticity holds, one has a natural suppression of the signal at the solar maxima, which does not require a fine tuning of the neutrino magnetic moment and the solar magnetic field. So far we restricted the magnetic field to the convective zone. On the other hand, the existence of a primordial field in the core of the sun, Bin, is not excluded. Of course Bin is decoupled from the eleven year cycle. Before closing this section we note a peculiarity which occurs for Bin :/: 0. For cos2a < ~, A (r) reverses its sign in the core of the sun and this corresponds to a resonance in the core of the sun. If the resonance is crossed adiabatically, all [~mL) transform into I~'mR)" In the convective zone, if the field is high enough, the [ ~'mR) transform back into I~//mL). Thus a positive correlation with the magnetic field would occur.
5. Conclusions
We have developed a phenomenological model with both neutrino flavour oscillations and spin flip transitions, characterized by a lepton number conservation law L+_ =Le +- (L~-L~). This implies a massless Weyl 387
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neutrino and one massive Dirac neutrino. The main consequences of this class of models are: (i) There is a large mixing angle between Ve and % or v~, depending on the choice of the conserved lepton number (L+ or L_ ). (ii) The mass difference Am coincides with the mass of the Dirac neutrino. As such it cannot be too small, in order to allow for a sizeable magnetic moment. A value of order 10-2_ 10-l eV is consistent with bounds from reactor experiments. We note that r e - % oscillation with a large mixing angle and similar Am may be indicated by the recently reported atmospheric neutrino flux anomaly [ 12 ]. (iii) We have seen that the model can account for both the experimental data of the Chlorine and the Kamioka experiment, for a large vacuum mixing angle. The difference between the results of the two experiments can be understood on the basis that in the former only ve are active, whereas in the latter all the neutrino states contribute to the signal. Our discussion was qualitative in that only two limiting cases (the adiabatic and the sudden solution) have been explored. (iv) Obviously the model predicts no % signal from the sun. This is supported by the stringent upper bound on solar 9e flux from the results of the Kamioka experiment [ 6 ]. (v) The suppression of the neutrino signal with respect to the SSM prediction comes out to be independent of the neutrino energy. Thus we expect the Gallium experiment to give the same reduction and modulation as the Chlorine experiment.
Acknowledgement We are deeply indebted to R. Barbieri for several useful suggestions and discussions.
References [ 1 ] R. Davis et al., in: Proc. Intern. Conf. Neutrino '90 (Geneva, June 1990), to appear. [2] K.S. Hirata et al., Phys. Rev. Lett. 65 (1990) 1297. [3] M.B. Voloshin and M.I. Vysotsky, Sov. J. Nucl. Phys. 44 (1986) 544; L.B. Okun, Soy. J. Nucl. Phys. 44 (1986) 546. [4] C.S. Lim and W.J. Marciano, Phys. Rev. D 37 (1988) 1368; E.Kh. Akhmedov, Phys. Lett. B 213 (1988) 64. [5] R. Barbieri, G. Fiorentini, G. Mezzorani and M. Moreni, Phys. Len. B 259 ( 1991 ) 119. [ 6 ] S.T. Petcov, Phys. Lett. B 110 (1982) 245; C.N. Leung and S.T. Petcov, Phys. Lett. B 125 ( 1983 ) 461; M.J. Dugan, G.B. Gelmini, H. Georgi and L.J. Hall, Phys. Rev. Lett. 54 (1985) 2302. [ 7 ] V. Zacek et al., Phys. Rev. D 34 ( 1986 ) 2621; G.S. Vidyakin et al., Sov. Phys. JETP 66 (1987) 243. [8] U Wolfenstein, Phys. Rev. D 17 (1978) 2369. [9] J. Bahcall and R.K. Ulrich, Rev. Mod. Phys. 60 (1988) 298. [ 10] R. Barbieri and G. Fiorentini, in: Proc. second Intern. Workshop on Neutrino telescopes, ed. M. Baldo Ceolin (Venice, 1990). [ 11 ] L.B. Okun, M.B. Voloshin and M.I. Vysotsky, Soy. J. Nucl. Phys. 44 (1986) 1; R. Barbieri and G. Fiorentini, Nucl. Phys. B 304 (1988) 909. [ 12] K.S. Hirata et al., Phys. Lett. B 205 (1988) 416; D. Casper et al., Boston University preprint BU-90-23; see also D. Casper, talk, in: Proc. Third Intern. Workshop on Neutrino telescopes (Venice, 1991 ) to appear.
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Neutrino oscillations and magnetic moment transitions in a model with a conserved lepton number Z.G. Berezhiani a, G. Fiorentini a b c d e f
b.c M.
Moretti d,e and S.T. Petcov d,f
Institute of Physics of the Georgian Academy of Sciences, SU-380 077 Tbilisi, USSR Department of Physics, Universit& di Ferrara, 1-44100 Ferrara, Italy lstituto Nazionale di Fisica Nucleare, Sezione di Ferrara, 1-44100 Ferrara, Italy Scuola lnternazionale Superiore di StudiAvanzati, 1-34014 Trieste, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, BG- 1784 Sofia, Bulgaria
Received 7 May 1991
We discuss a three flavour neutrino model with a conserved non-standard lepton number L+ =Le+ (Lu-L.~). In this frame, flavour oscillations and magnetic moment transition can both occur and at the same time this involves only one mass scale. We show that the model can account for the results of both the Chlorine and the Kamioka experiments. In addition the conservation law provides naturally a total suppression of the % flux from the sun.
1. Introduction
The experimental situation concerning solar neutrinos can be summarized as follows: (i) Both in the Chlorine and in the Kamioka experiment the observed signal is significantly smaller than predicted by the standard solar model. For the two experiments, the ratios Zi of the observed signal to the expectation of the standard solar model (SSM), averaged over the data taking period, are respectively [ 1,2 ]
(1.1,2)
(ii) In the Chlorine experiment there are indications that the neutrino signal is (anti) correlated with the solar cycle. For solar cycles 21 and 22, the normalized signals in correspondence of the minima and maxima of solar activity are respectively [ 1 ] (Zc)min=0.53±O.08
,
(Zc)max=O.08+O.03
.
(1.3,4)
(iii) Note that in correspondence of the solar minima, still there is a signal deficit. Mechanisms of neutrinos oscillations, either in vacuum or in matter, can account for (i) and (iii). On the other hand, they hardly can explain the time variation. This feature is most directly connected with the hypothesis of a large neutrino magnetic moment, coupled to the solar magnetic field in the convective zone [ 3 ]. However, this mechanism alone cannot explain the deficit in correspondence of the solar minima. In order to account for all the features simultaneously, it is natural to resort to the so called "hybrid models", i.e. models where both neutrino oscillations and magnetic spin flip transitions are operating. Hybrid models are discussed in the literature in the context of two neutrino flavours, say ve and v~ for definiteness [4 ]. In our opinion, this leads to two unpleasant features: (a) As well known, a sizeable magnetic moment is only possible if the neutrino mass is close to the observational bounds. On the other hand, the magnetic transition is not quenched only if the mass difference between 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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the two neutrino states (which also cannot vanish to allow for flavour oscillation) is very small ~: 6m 2< 10 - 6 (eV) 2. Thus the neutrino mass spectrum involves two vastly different mass scales, which may be difficult to account for theoretically. (b) The second potential difficulty o f hybrid models with two neutrino flavours is related to the absence of any lepton number conservation law. As a consequence, the simultaneous presence of both magnetic and flavour oscillations leads in general to a significant transition among all the four states v~, v,, %, 9,. This may contradict the upper bound on the flux o f % , which can be derived by using the Kamioka data [ 5 ]. A natural way to avoid both these problems consists in considering all the three standard neutrino flavours re, G, V. and assuming a suitable nonstandard lepton number to be conserved:
L± =L~+_ ( L , - L . )
.
(1.5)
This conservation law implies just one mass scale, the mass spectrum consisting of one massive Dirac neutrino and one massless Weyl neutrino [ 6 ]. An unquenched magnetic transition can take place between the two helicity states o f the Dirac neutrino and, on the other hand, the fact that the mass eigenstates are linear superpositions of the different neutrino flavours gives rise to a flavour oscillation. In other words by considering all the three neutrino flavours one can have flavour oscillations even in the presence o f a Dirac neutrino state in the spectrum.
2. The model
For definiteness, we take L+ as a conserved q u a n t u m number. The most general mass term which is consistent with this constraint is Lm = - m OCR( Uee COS a + U,L sin a ) + b.c.
(2.1)
This lagrangian describes one Dirac neutrino with mass m, ~rnL = PeL COS a + P,L sin a ,
~/mR= pCR
(2.2)
and a massless Weyl neutrino, (2.3)
Uo = -- UeLSin o r + U,LCOS a .
For sizeable values of the mixing angle a, experiments at nuclear reactors yield [ 7 ] m ~<0.1 eV. Since significant magnetic moments call for high masses, we will explore the region near this upper bound: m ~ 1 0 - 2 - 1 0 - leV.
(2.4)
In this range of masses the existing data from accelerator experiments on ve~v~ (%--*9,) do not impose any constraint on sin a. The most general magnetic m o m e n t effective lagrangian, consistent with the conservation of L+, has the form
L , = --~ll 1 C V,Ra,~O F~o (PeLCOS ]~"[- V , L s i n
fl) + h . c .
(2.5)
The hamiltonian H which determines the evolution of the ve state in the sun,
i(O/3r) l u(r) ) = H ( r ) l g ( r ) ) ,
(2.6)
where r= ct, is immediately obtained from eqs. (2.1) and (2.5), including the coherent neutrino weak interaction with matter [ 8 ]. We observe that due to the conserved lepton number the hamiltonian decouples into two sub-hamiltonians each coupling only three states. In the mass eigenstates basis (/Vo), [ ~'mL), [ ~//rnR) ) one has ~l This result holds for/t.~ 10 I I ga, B ~ 100 kG. The same results hold also for the resonant spin flip model [ 5 ] if we restrict the magnetic field to the convective zone, which is the only useful assumption to recover the observed time dependence of the solar neutrino flux.
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f GFnex/~ sinaa--GFnn/x/~ H(r)= [
--GFnex/~sincxcost~
\
#Bsin(fl-a)
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-GFnex/~ sin a cos a izB sin(fl-a) m2/2E+GFnex/~cos2o~-Gvnn/x//2 ~Bcos(fl-o~) ~ , /zBcos(fl-a) m2/2E+GFn,/x/2/
(2.7)
where ne and n, are the electron and neutron densities respectively. GF is the Fermi constant and B=B(r) is the transverse component of the solar magnetic field. In the convective zone (r>~RB=0.7 Ro, where R o = 7 X 105 km is the solar radius) we assume n.=~ ne and the density profile to be given by [9] H e ~ F/0
exp ( -
r/LM) ,
(2.8)
with LM ~ 0.1 Ro, and ne (RB) ~ 0.16 NA cm-3, NA being the Avogadro number. Concerning the magnetic field, we will assume that it is confined within the convective zone. In the absence of detailed information on the shape of the field, we use the following parametrization:
B(r) =O(r-Ra)Bo {1 - e x p [ ( --r+RB )/LBI} •
(2.9)
The starting point RB is taken at the beginning of the convective zone. The raising length LB - some fraction of the depth o f the convective zone - and the plateau value Bo are kept as free parameters. Bo varies during the solar cycle, from Bo ~ 0 at the solar minima up to a value Bb~ax in correspondence to the maxima o f solar activity. Note that, according to (2.9), in the very outer region of the sun the field does not fall (as fast) as the coherent weak interaction. In fact we expect the magnetic field to decrease with some power of the distance, whereas the density profile is exponentially decaying. It is useful to introduce a magnetic energy term, e(r) = / t B ( r ) c o s ( f l - a ) ,
(2.10)
and a magnetic energy plateau ~o which is obtained from (2.10) for (/t ~ 10- l OpB, B ~ a× ~ 1 T) one has e ~ax ~ 20/Ro for cos ( f l - a ) = 1.
B=Bo. For the values of interest to us
3. Flavour oscillations and spin flip in the absence of coherent weak interactions As a first stage of the discussion, we neglect in eq. (2.7) the coherent weak interaction. For the values o f interest to us, one has m2/2E>> pB and consequently the ] Uo) state is decoupled from the others. One also has (m2/2E) Ro >> l, so that oscillating terms of argument (rn2/2E)Ro can be replaced by their average value. In these hypotheses, the arrival probabilities P(Vx) of the originally Ve state are P(V e ) = sinaa + cos 4ol COS 2(~)
,
( 3.1 )
P (v,) = ¼sin22a ( 1 + cos2~) ,
( 3.2 )
P(9~) = cos2a sin2~,
(3.3)
where • =/z c o s ( f l - a ) ~
B(r) dr.
(3.4)
The main phenomenological features of the model are immediate from the equations given above: (a) Let us assume that the spectrum of neutrinos produced in the core of the sun, dF/dE, is given correctly by the standard model, apart from an overall factor A accounting for the SSM uncertainties:
dF/dE=A dFSSM/dE, A = 0 . 7 - 1 . 3 .
(3.5) 383
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In this case, the ratio o f the signal in the Chlorine experiment to the expectation of the SSM is simply
Zc = A P ( v e ) .
(3.6)
The signal deficit at solar minima ( qb= 0) calls for a large value of the vacuum mixing angle. Indeed, from eq. (1.3), at the lalevel, one gets 2( 1 - 0 . 6 1 / A ) ~
(3.7)
In the case A = 1 this yields 0.78~
(3.8)
Note that the same results hold for q)= 0 also in the case that coherent weak interactions are included. In other words, the signal deficit at solar minima always implies a large vacuum mixing angle. (b) A significant depletion of the signal in correspondence with the solar maxima can be achieved only for (J~max close to ½n. In fact, smaller values o f q~maxyield a too small time dependence whereas for larger values the signal oscillates with a frequency larger than that o f the solar cycle. As remarked in ref. [ 10 ] this condition requires a fine tuning of the parameters (/~ and B) which are involved. 1 (c) Even for the best case, ~max-~if, a significant depletion of the signal at the solar maxima calls for large values o f the vacuum mixing angle. From ( 1.3 ) and ( 1.4) one has ( Zc" )max/ ( Zc)min =0.15 +-0.07 .
(3.9)
By using eq. ( 3.1 ) with (~)min = 0 and ~bmax = ½n we get, at the 1a level, sin 2 2c~=0.81 + 0 . 1 0 .
(3.10)
(d) In the Kamioka experiment, in addition to re, both v, and 9~ can scatter off electrons. For the Kamioka experiment one has ZK = A [P(ve ) + Wrt(Eth)P(v,) + W~ (Eth)P(v~) ] ,
(3.11 )
where the factors Wx(E~h) depend on the threshold energy and on the neutrino flavour (for Eth=9.3 MeV, I 1 W~=5 and W~=~). For the present model one obviously has ( Z K ) > / ( Z c ) , in agreement with the trend o f the experimental data. Also, the time modulation comes out to be weaker in Kamioka. As an example, by assuming = 0 and q) = ½n at solar m i n i m a and maxima respectively, for a as in (3.10) and a threshold energy E t h = 9.3 we get a modulation (ZK)max/(ZK)min=0.29+_0.7.
4. Inclusion of coherent weak interactions
The neutrino evolution corresponding to the full hamiltonian given in eqs. ( 2 . 7 ) - ( 2 . 9 ) has to be studied numerically. In this section we shall investigate two opposite limiting cases, which can be discussed analytically and yield a qualitative understanding of the complete solution. For the values of the mass we are considering (see eq. ( 2 . 4 ) ) both Gnc and Gn. are (almost) neglibile in comparison with the energy splitting, m 2/2E. The same holds true of the ¢tB terms for the values of interest to us. In these circumstances, the massless neutrino component IVo) is practically decoupled from the others. The problem thus simplifies to the study of a two state system, I ~/JmL) and I q/mR). For this system, the relevant part of the hamiltonian (i.e. apart from terms proportional to the identity) is
h(r)= \e(r) 384
-3(r)]'
(4.1)
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where
A(r) =G(ne COS2Ot-- n n ) / N / ~ •
(4.2)
Note that, since the magnetic m o m e n t couples the left and right components of the same Dirac neutrino, there are no mass terms inhibiting the spin flip transition. We recall that we assume a large mixing angle, c o s 2 a = ½ and in the convective region we use eq. (2.8). We note that, as a consequence of the large vacuum mixing angle, A(r) never vanishes in the convective zone. A non-zero value of A(r) clearly inhibits spin flip transitions, see ref. [ 11 ], with respect to the case without coherent weak interactions. The term ~(R) is defined by means ofeqs. (2.9) and (2.10):
~(r) =O(r--RB)%{1 -- exp[ ( - r + RB) /La]} .
(4.3)
We shall investigate the solution of the problem in terms of the field raising distance LB and of the interaction energy plateau eo which, we recall, varies during the solar cycle from eo ~ 0 at solar m i n i m a up to a m a x i m u m value E~a~ at solar maxima. Qualitatively, for the solution of the problem, one can distinguish three different regions: (a) In the inner part of the sun (say r
r
(4.4)
(b) In the outer part (say r > R 2 ) the magnetic m o m e n t interaction dominates [A(r)<< E(r)]. Also in this case the analytical solution is immediate. For a state which can be described as [ U/mL,R) at r=R2 one has
r>R2: [~//mL)----~[~//mL) COS~(r)--ilq/mR) s i n ~ ( r ) , [q/mR) ~--i[I//mL) sin ~ ( r ) + [ ~//mR) COS~(r),
(4.5)
where ~(r)= i eB(r')dr'.
(4.6)
R2
(c) In the intermediate region, R l < r
4.1. The adiabatic and the sudden region The transition between the two regions can be taken as adiabatic or sudden depending on the value of the function g(r)=
[EA'-~'dl/2(E2+A2) 3/2 .
(4.7)
For g << 1 (g >> 1 ) the adiabatic (sudden) limit is realized. One has to verify such conditions close to the point r = r* where e (r*) = A (r*). The curves shown in fig. 1 correspond, for cos2a ~ ½, to g ( r * ) = 1, i.e. they denote the transition between the two regimes. 385
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0.08
0.06
~
,
-
''I
....
I ....
I ....
I ....
-
--
adia
o.o4--
0.02
I ....
1 August 1991
--
0.00
,,
0
L
....
5
~ , , , I
10
....
16
I,,,,I
20
....
25
30
c o ~, R e
Fig. 1. In the plane (eo,La) we show the borders of the adiabatic and the sudden regimes, defined as g= 1, see eq. (4.7). % is the plateau value of the magnetic interaction energy and LB is its raising length. Ro is the solar radius. The m a i n features o f fig. 1 can be u n d e r s t o o d from the following considerations. Note that g is the sum o f two positive functions, g=g~ +g2, where
g l ( r ) = - ~ A ' / 2 ( ~ 2 + A 2 ) 3/2, g2(r)=~'A/2(~2+A2) 3/2
(4.8)
At r = r*, g~ = [ 4x/~ LM e (r*) ] - ~ and one has gl >/1 for eo ~< (4x/~ LM ) - ~. This explains the region o f small fields where the sudden limit is achieved. F o r i n t e r m e d i a t e values o f the field ( (4xf2 LM ) - ~~
4.2. The adiabatic and the sudden solution If the transition between regions ( a ) and ( b ) is adiabatic, the ( i n s t a n t a n e o u s ) eigenstates o f region ( a ) go into the corresponding ones o f the region ( b ) . After traversing the sun, the probabilities o f the different neutrino flavours are P(ve) = sinac~+ ½ C O S 4OL
,
(4.9)
P(v~) = 3 sin22oz,
(4.10)
P ( 9 ~ ) = ½c o s 2 a ,
(4.11)
where oscillating terms o f argument ( m 2/ 2E) R o >> 1 have been averaged. I f the transition is sudden, when crossing the i n t e r m e d i a t e region (Rl < r < R2) the [ ~mL,R> states do not change. Beyond R2, coherent interactions are irrelevant and the time evolution is the same as discussed in the previous section. 386
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The arrival probabilities are thus given again by eqs. (3.1)-(3.3 ), where the phase • is now given by q~=/2cos(fl-a) f B(R)dR.
(4.12)
R2
This means that the magnetic field contributes to the phase only in the region where it overwhelms the coherent weak interaction.
4.3. Discussion On these grounds we can discuss the arrival probabilities P (vx) during the different phases of the solar cycle. For any LB, at the solar minimum (Eo ~ 0) one is in the sudden limit. The solution is given by eqs. ( 3 . 1 ) (3.3) with ~b=0. As discussed in section 3, a significant depletion of the neutrino signal can be obtained for large mixing angles, as given in eq. (3.8). As the field raises, different regimes (adiabatic, sudden, or intermediate) can be reached depending on the value o f ~ nax and LB. From the phenomenological point of view, the most important features of the adiabatic regime are: (a) The results do not depend on the precise value of/t and B~ ax. Thus a significant modulation of the signal, anticorrelated with the solar cycle, can be obtained without a fine tuning of the magnetic field and magnetic moment, provided that e ~ax and LB are in the adiabatic region. (b) Note that the lowest values of e~ax which satisfy the adiabatic condition (and thus provide a significant time modulation) are of order e~ax ~ 5 / R o , (the inverse of the convective zone dimension). Whichever mechanism one is invoking, one cannot do better, in the sense that for a significant spin flip probability the phase O = coD, where D is the depth of the convective zone, has to exceed unity. (c) On the other hand, the minimum value of P(ve) is ~, which is at the border of the region allowed by the Chlorine data. Also, the signal modulation in the Chlorine experiment (Zc)max/(Zc)rain cannot be smaller than one half, which looks in disagreement with (3.9). In the case that the sudden regime is reached, the same phenomenological considerations as for the "vacuum" case hold. In addition, we note that only the region of high fields (~o > 10/Ro ) is relevant for getting a significant phase q~. In comparison with the adiabatic limit, the sudden case requires EbT M appreciably larger in order to achieve a significant time modulation. On the other hand, the amplitude of the modulation can be much larger, as previously remarked. In summary, the inclusion of matter effects does not alter drastically the picture. One can choose the available parameters so as to reproduce the main features of the experimental results. Furthermore, if adiabaticity holds, one has a natural suppression of the signal at the solar maxima, which does not require a fine tuning of the neutrino magnetic moment and the solar magnetic field. So far we restricted the magnetic field to the convective zone. On the other hand, the existence of a primordial field in the core of the sun, Bin, is not excluded. Of course Bin is decoupled from the eleven year cycle. Before closing this section we note a peculiarity which occurs for Bin :/: 0. For cos2a < ~, A (r) reverses its sign in the core of the sun and this corresponds to a resonance in the core of the sun. If the resonance is crossed adiabatically, all [~mL) transform into I~'mR)" In the convective zone, if the field is high enough, the [ ~'mR) transform back into I~//mL). Thus a positive correlation with the magnetic field would occur.
5. Conclusions
We have developed a phenomenological model with both neutrino flavour oscillations and spin flip transitions, characterized by a lepton number conservation law L+_ =Le +- (L~-L~). This implies a massless Weyl 387
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neutrino and one massive Dirac neutrino. The main consequences of this class of models are: (i) There is a large mixing angle between Ve and % or v~, depending on the choice of the conserved lepton number (L+ or L_ ). (ii) The mass difference Am coincides with the mass of the Dirac neutrino. As such it cannot be too small, in order to allow for a sizeable magnetic moment. A value of order 10-2_ 10-l eV is consistent with bounds from reactor experiments. We note that r e - % oscillation with a large mixing angle and similar Am may be indicated by the recently reported atmospheric neutrino flux anomaly [ 12 ]. (iii) We have seen that the model can account for both the experimental data of the Chlorine and the Kamioka experiment, for a large vacuum mixing angle. The difference between the results of the two experiments can be understood on the basis that in the former only ve are active, whereas in the latter all the neutrino states contribute to the signal. Our discussion was qualitative in that only two limiting cases (the adiabatic and the sudden solution) have been explored. (iv) Obviously the model predicts no % signal from the sun. This is supported by the stringent upper bound on solar 9e flux from the results of the Kamioka experiment [ 6 ]. (v) The suppression of the neutrino signal with respect to the SSM prediction comes out to be independent of the neutrino energy. Thus we expect the Gallium experiment to give the same reduction and modulation as the Chlorine experiment.
Acknowledgement We are deeply indebted to R. Barbieri for several useful suggestions and discussions.
References [ 1 ] R. Davis et al., in: Proc. Intern. Conf. Neutrino '90 (Geneva, June 1990), to appear. [2] K.S. Hirata et al., Phys. Rev. Lett. 65 (1990) 1297. [3] M.B. Voloshin and M.I. Vysotsky, Sov. J. Nucl. Phys. 44 (1986) 544; L.B. Okun, Soy. J. Nucl. Phys. 44 (1986) 546. [4] C.S. Lim and W.J. Marciano, Phys. Rev. D 37 (1988) 1368; E.Kh. Akhmedov, Phys. Lett. B 213 (1988) 64. [5] R. Barbieri, G. Fiorentini, G. Mezzorani and M. Moreni, Phys. Len. B 259 ( 1991 ) 119. [ 6 ] S.T. Petcov, Phys. Lett. B 110 (1982) 245; C.N. Leung and S.T. Petcov, Phys. Lett. B 125 ( 1983 ) 461; M.J. Dugan, G.B. Gelmini, H. Georgi and L.J. Hall, Phys. Rev. Lett. 54 (1985) 2302. [ 7 ] V. Zacek et al., Phys. Rev. D 34 ( 1986 ) 2621; G.S. Vidyakin et al., Sov. Phys. JETP 66 (1987) 243. [8] U Wolfenstein, Phys. Rev. D 17 (1978) 2369. [9] J. Bahcall and R.K. Ulrich, Rev. Mod. Phys. 60 (1988) 298. [ 10] R. Barbieri and G. Fiorentini, in: Proc. second Intern. Workshop on Neutrino telescopes, ed. M. Baldo Ceolin (Venice, 1990). [ 11 ] L.B. Okun, M.B. Voloshin and M.I. Vysotsky, Soy. J. Nucl. Phys. 44 (1986) 1; R. Barbieri and G. Fiorentini, Nucl. Phys. B 304 (1988) 909. [ 12] K.S. Hirata et al., Phys. Lett. B 205 (1988) 416; D. Casper et al., Boston University preprint BU-90-23; see also D. Casper, talk, in: Proc. Third Intern. Workshop on Neutrino telescopes (Venice, 1991 ) to appear.
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