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Oscillations-assisted resonant spin-flavor precession and time variations of the solar-neutrino flux
Volume 257, number 1,2
PHYSICS LETTERS B
21 March 1991
Oscillations-assisted resonant spin-flavor precession and time variations of the solar-neutrino flux E.Kh. Akhmedov Kurchatov Institute of Atomic Energy, SU-123 182 Moscow, USSR
Received 7 September 1990
We consider the combined effect of the resonant spin-flavor precession (RSFP) and resonant oscillations of neutrinos in the convective zone of the sun. It is shown that although the oscillations cannot give rise to the apparent time variations of the solar neutrino flux in the 37C1experiment, they can efficiently assist the RSFP to do so. As a result, smaller values of the neutrino transition magnetic moments are required, which are compatible with the recently derived astrophysical upper bound. We also remark on new experimental results on solar neutrinos.
1. There is some evidence that the counting rate o f solar neutrinos in the 37C1 experiment o f Davis a n d his collaborators varies with time in anticorrelation with solar activity [ 1,2 ]. Although this p h e n o m e n o n needs further confirmation, indications in its favor are fairly strong. A m o n g various m e c h a n i s m s which might account for the a p p a r e n t t i m e variations o f the solar-neutrino flux, the most plausible ones are those connected with the magnetic m o m e n t s o f the neutrinos. I f such moments are large enough, in the periods o f high activity the toroidal magnetic field o f the convective zone o f the sun will rotate a significant fraction o f lefth a n d e d VeLinto the sterile right-handed neutrinos VcR (V~R, V,R) or into the right-handed antineutrinos 9,R (9~a), which cannot be detected in the 37C1 experim e n t [3 ]. O f particular interest is the s p i n - f l a v o r precession VeL~; ~?~R(V~R)dUe to the flavor-off-diagonal ( t r a n s i t i o n ) magnetic m o m e n t s ltij ( i # j ) , since m a t t e r can resonantly amplify it [ 4 - 7 ]. Such a resonant s p i n - f l a v o r precession ( R S F P ) is analogous to the resonant neutrino oscillations ( R N O ) in m a t t e r [ 8,9 ]. F o r the neutrino spin (or s p i n - f l a v o r ) precession inside the sun to be efficient, the magnetic m o m e n t s o f neutrinos must be > 10-'~/ta p r o v i d e d the magnetic field in the convective zone is >~4 0 - 6 0 kG. In the case o f the flavor-conserving neutrino spin precession this is necessary to overcome mattersuppression effects a n d to gain a sizeable precession
phase [3,10], while in the case o f the R S F P a sufficiently large transition m o m e n t is n e e d e d to ensure the adiabaticity o f the neutrino conversion [ 4 - 7 ] . The most stringent constraint on the neutrino magnetic m o m e n t s was d e r i v e d from the energy loss o f white dwarfs and helium s t a r s , / ~ < 1 × 10-11//a [1 1-3] ~ , which applies to both the diagonal a n d transition moments. Recently, the d a t a on helium stars have been reexamined yielding a m o r e severe b o u n d [ 18 ] /A~< 3 X 10-121tB •
( 1)
Thus the question arises as to whether the neutrino spin or s p i n - f l a v o r precession with the magnetic moments being that small can still account for the apparent time variations o f the solar-neutrino flux. It is this question that we address in the present paper. O u r answer is yes, at least for the RSFP. In general, a three times smaller magnetic m o m e n t requires a three times stronger magnetic field to produce the same precession effect. Although magnetic fields as strong as 100 k G have been considered possible in the convective zone o f the sun [ 19 ], it would at There are also more restrictive upper bounds on/zv from the observation of the neutrino signal from the supernova SN 1987A, but it has been argued that these bounds are avoidable [ 14,15 ]. Moreover, they are not applicable to the transition magnetic moments of Majorana and Zeldovich-Konopinski-Mahmoud neutrinos [ 16,17 ].
0370-2693/91/$ 03.50© 1991 - Elsevier Science Publishers B.V. (North-Holland)
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be desirable to seek some enhancement mechanism which would allow one to achieve the same effect with smaller fields. We shall show that the RNO might provide such a mechanism because they can efficiently amplify the RSFP. 2. For the RSFP to occur, neutrinos must have nonzero flavor-off-diagonal magnetic moments, which implies lepton flavor violation. Therefore, neutrino oscillations must also take place. Thus in general the RSFP should be considered jointly with the RNO. Such a joint consideration was carried out in refs. [20,21 ]. In ref. [20] it was shown that when the resonances of the oscillations and spin-flavor precession overlap, these two effects can either suppress or enhance each other, depending on the degree of the adiabaticity of both the conversions. Since the overlap increases with decreasing neutron number density [20,21 ], one can expect a strong mutual influence of the RNO and RSFP in the convective zone, where the neutron fraction is relatively small. The RSFP can give rise to the time variations provided that the neutrino conversion is nearly adiabatic at the solar maxima and non-adiabatic in the periods of a quiet sun; therefore we are mainly interested in the moderately non-adiabatic regime. In this regime the overlap of the RNO and RSFP resonances improves their adiabaticity, since (i) the widths of both resonances become larger, and (ii) the oscillation and precession lengths become smaller [20]. This implies that the RNO and RSFP enhance each other in the non-adiabatic regime. For the case of the overlapping resonances a simple analytical solution was obtained in ref. [ 20 ] for the moderately non-adiabatic regime. Expressing E / A m 2 through the resonant density Pr (to which it is inversely proportional), one can obtain from eq. (34) ofref. [20] the following expression for the VeLsurvival probability: P ~ cosZ{ 180p~[ (2.3 × lO-3flr/Pr) 2+ tanZ20o ] }.
(2) Here fl,---(ltJlO-tl#a)B±r(kG), BIr being the magnetic field strength at the resonance, 0o is the vacuum mixing angle, and pr is in g / c m 3. To obtain a sizable reduction of the solar ve flux, the argument of the cosine in eq. (2) must be of the order of unity. If 0o were zero, this would require fir> 42 for the resonance occurring at the bottom of 164
21 March 1991
the convective zone (Pr ~ 0.17). For non-vanishing 0o smaller values of fir are needed. For our estimates we shall assume, in accordance with the results of Davis et al. [ 1,2 ], that the maximal counting rate in the 37C1 experiment Qmaxis about 4.2 SNU and Qmin is 2-4 times smaller. This should be compared with the standard solar model (SSM) prediction QssM= 7.9 SNU. One can easily make sure that for pr~ 0.17 and tan 20o,~ 0.5 the maximal value of fir should be about (fir)max'25 for Qmax/amin.~3 and (fir) max ~'~22 for amax/ amin ~ 2. Thus for a factor of 2 (3) variation of the detection rate, a factor of 2 ( 1.7 ) smaller transition magnetic moment is required than in the 0o = 0 case. One can also gain a factor of three in the magnetic moment, but in that case only a ~ 30% variation of the VeLflUXwill result, which is too small. Our estimates apply, strictly speaking, to a definite value of pr, i.e. to a definite neutrino energy. However, moderate changes ofpF do not affect appreciably the results. This is because one of the two terms in the argument of the cosine in eq. (2) is directly proportional to p~, and the other is inversely proportional to it; thus in the case when the two terms are of the same order of magnitude (which is of the major interest to us), the Pr dependence of the whole argument is weakened. From the above estimates it follows that, with a sufficiently large mixing angle, one can gain a factor of two in the transition magnetic moment, so that one still needs a factor of 1.5 increase in magnetic field strength to compensate for a factor of three reduction of the upper bound ( l ) as compared to the previous constraints. However, eq. (2) overestimates the desirable/zejB, value because it completely disregards the oscillations and precession in the radiation zone of the sun. In particular, the magnetic field in the vicinity of the top of the radiation zone can improve the adiabaticity of the RSFP of the neutrinos in the convective zone by enlarging the resonance width [ 16,21,22]. As a result, the required values of I.Ze:Bj_ may be lower than our simple estimates show and, in addition, smaller mixing angles are needed. This can be seen from the results of the numerical calculations. 3. We calculated the VeL evolution inside the sun allowing for both the neutrino oscillations and the spin-flavor precession. For simplicity, we considered the Majorana neutrino case for which/t,-= 0. The
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case of non-resonant (flavor-conserving) neutrino spin precession can also be obtained from our results by formally setting Am 2 = 0. Following refs. [ 16,20 ] we employed the model magnetic field profile inside the sun of the form (see the discussion in ref. [ 16] )
21 March 1991
1.00
0.80
0.60
]2,
/4jB± (r) =B~ [ 7 / ( 7 + x )
0 ~ X ~ 0.65,
=Bo{1-[(X--0.7)/0.312},
0,65
0.40
(3) where x-r/Ro, and 7= 0.1 was taken in all the calculations. The results are shown in figs. 1-3. In our calculations, time variations of the magnetic field in the convective zone were simulated by an order of magnitude decrease of Bo (dashed lines in figs. 1-3). It is seen from fig. l that the mean VeL-flUX suppression factor in the energy range E/Am2,~
0.20
(a) o oq;
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10
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il 10 e
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Fig. 2. Same as fig. 1, but Bo=3× 104 (10-"/taG) (solid line) and Bo=3X 103 ( 1 0 - " / ~ B G ) (dashed line).
(a) E/Am 2, MeV/eV2
1.00
10 '°
E/Am 2. MeV/eV2
I
....
~0
'0
E/Am 2, MeV/eV2
Fig. 1. VeL survival probabilities at the surface o f the sun versus
E/Am 2. B, = 1X 106 ( 10-"lt~ G); B o = 2 X 104 ( 1 0 - " # G ) (solid line) and B o = 2 × 10a ( 1 0 - " / t a G ) (dashed line). (a) 0o=0; (b) tan 200 = 0.4.
(5X 107-1.5X 10 s) MeV/eV 2 varies with time between 0.55 and 0.3 for tan20o=0.4, and only between 1 and 0.85 for 0o=0. Thus the desirable time variations can-be achieved for gej being compatible with the bound ofeq. ( 1 ) and maximal magnetic field in the convective zone about 60 kG. If the magnetic field is a factor of 1.5 higher, one can get the mean suppression factor in the same energy domain varying with time between 0.55 and 0.18 (fig. 2b). A more precise statement can be made after the direct calculation of the chlorine detection rate. For the neutrino survival probabilities of fig. 1 and Am2= 10 - 7 e V 2 such a calculation yields the detection-rate suppression factor varying between 0.46 and 0.26 for tan 200=0.4 (fig. lb) and only between 0.99 and 0.61 for vanishing mixing angle (fig. 1a). For the survival probabilities of fig. 2 the detection-rate suppression factors vary between 0.46 and 0. l 1 for tan 200= 0.4 165
Volume 257, n u m b e r 1,2
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PHYSICS LETTERS B
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,
. . . . . . . .
,
. . . . . . . .
,
the low activity periods. Even for 0o=0, the RSFP requires smaller magnetic moments: in the case of the non-resonant precession one needs r > 500p to overcome the matter-suppression effects, whereas the adiabaticity condition of the RSFP requires fir > 100v/~. This gives a factor of two gain in fl for the RSFP occurring at the bottom of the convective zone.
. . . . . . .
0.80
0.60
0.40
0.20
o.oq
. . . .
,,
10 5
10 6
10 7
10 8
10 9 2 E/#,m
10 ~° 2
,
MeV/eV
1.00
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t
0.40
.,'~
0.20
o oq;,
I0 5
I0 6
I0 7
10
a E/Am
I0' I0'° 2 , MeV/eV 2
Fig. 3. Same as fig. 2, but B~ = 5 × 106 ( 10- H #B G). For comparison, the pure R N O (B~ = B0 = 0) case is also shown (doned line).
and between 0.99 and 0.35 for 0o=0. The role of the inner magnetic field can be seen by comparing figs. 2 and 3. From our results one can also find the VeLsuppression factors for the non-resonant precession (VVO mechanism [3]). These are obtained in the E/Am2--,oo limit (in fact, they correspond to E/Am2> 10 9 MeV/eV2). It can be seen from figs. 13 that for the ranges of the parameters we consider, the VeL suppression factor varies typically between 1 and 0.8 (or between 0.9 and 0.7 for the stronger fields, fig. 3 ), irrespective of the v mixing angle. For Bo > 60 (in units 10- ~#a kG) one can get a suppression factor about 0.5, but again in the quiet sun it will be nearly 1. This means not only that the RSFP has the advantage as compared to the non-resonant conversion that it can be amplified by the RNO, but it also has the welcome feature of lower neutrino fluxes at 166
21 March 1991
4. Finally we would like to remark on some recent experimental results on solar neutrinos. It was claimed by the Kamiokande II Collaboration that the ratio of the average detection rate of the solar neutrinos over the three years of observation to the SSM prediction is 0.46 _ 0.05 (stat.) _+0.06 (syst.), and the time variations, if any, do not exceed 30% [23 ]. This seems to be at variance with the results of the 37C1 experiment of Davis et al., in which much stronger time variations are observed. However, one can readily make sure that these two results can be reconciled in the RSFP scenario provided that the lowenergy v contributions to the 37C1 experiment are strongly suppressed. This is because in that case weaker suppression of the 8B v's is needed, i.e. their flux may be higher. Since the V,'s and V~'s do contribute to the ve reaction which is employed in the Kamiokande II experiment (though with 6-7 times smaller cross sections), a 30% time variation in the Kamiokande II detection rate may be compatible with a factor 2-3 variation in the a7Cl experiment [22]. For example, for A m 2 = 10 - 7 e V 2 and the neutrino survival probabilities of fig. lb the calculation yields the detection rate suppression factors for the neutrino-electron scattering which vary between 0.61 and 0.39. This is in accordance with the Kamiokande II data [23]. One can expect strong suppression of the 71Ga counting rate in this case, which is consistent with the first results of the SAGE Collaboration [ 24 ]. A drastic depletion of the low-energy VeLflUX we need in this scenario may be either due to the strong inner magnetic field (fig. 3a), or due to the RNO (figs. lb, 2b and 2b). In conclusion, we showed that the resonant v oscillations can efficiently enhance the resonant spin-flavor precession allowing the sizable time variations of the solar-neutrino flux with the neutrino transition magnetic moments being compatible with the bound of eq. ( 1 ). Our results are in accordance with all the
Volume 257, number 1,2
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a v a i l a b l e e x p e r i m e n t a l d a t a o n solar n e u t r i n o s , inc l u d i n g t h o s e o f K a m i o k a n d e II. T h e a u t h o r is grateful to Z . G . B e r e z h i a n i , L.B. O k u n , A.Yu. S m i r n o v a n d M.I. V y s o t s k y for useful discussions.
References [ 1 ] R. Davis Jr., talk XXIst Intern. Conf. on Cosmic-ray physics (Adelaide, Australia, 1990 ). [2 ] K. Lande, talk Neutrino 90, 14th Intern. Conf. on Neutrino physics and astrophysics (CERN, Geneva, June 1990 ). [3] M.B. Voloshin, M.I. Vysotsky and L.B. Okun, Sov. Phys. JETP 64 (1986) 446. [4] E.Kh. Akhmedov, preprint IAE-4568/1 (January 1988). [5] E.Kh. Akhmedov, Yad. Fiz. 48 (1988) 599. [ 6 ] C.-S. Lim and W.J. Marciano, Phys. Rev. D 37 ( 1988 ) 1368. [7] E.Kh. Akhmedov, Phys. Lett. B 213 (1988) 64. [8] L. Wolfenstein, Phys. Rev. D 17 (1978) 2369. [9] S.P. Mikheyev and A.Yu. Smirnov, Sov. J. Nucl. Phys. 42 (1985) 913.
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[10] R. Barbieri and G. Fiorentini, Nucl. Phys. B 304 (1988) 909. [ 11 ] S.I. Blinnikov, preprint ITEP-I 9 (1988). [ 12] M. Fukugita and S. Yazaki, Phys. Rev. D 36 (1987) 3817. [ 13 ] G.G. Raffelt and D.S.P. Dearborn, Phys. Rev. D 37 ( 1988 ) 549. [ 14] M.B. Voloshin, Phys. Lett. B 209 (1988 ) 360. [ 15 ] R. Barbieri, R.N. Mohapatra and T. Yanagida, Phys. Lett. B213 (1988) 69. [16] E.Kh. Akhmedov and O.V. Bychuk, Sov. Phys. JETP 68 (1989) 250. [ 17 ] M. Leurer and J. Liu, Phys. Lett. B 219 (1989) 304. [ 18 ] G.G. Raffelt, Phys. Rev. Lett. 64 (1990) 2856. [19] V.N. Krivodubsky, Pis'ma Astron. Zh. 13 (1987) 803. [20] E.Kh. Akhmedov, Soy. Phys. JETP 68 (1989) 690. [21 ] H. Minakata and H. Nunokawa, Phys. Rev. Left. 63 (1989) 121. [22] E.Kh. Akhmedov, preprint 5017/1 (1990); Yad. Fiz., in press. [23 ] Y. Totsuka, talk Neutrino 90, 14th Intern. Conf. on Neutrino physics and astrophysics (CERN, Geneva, June 1990). [24] V.N. Gavrin, talk Neutrino 90, 14th Intern. Conf. on Neutrino physics (CERN, Geneva, June 1990).
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21 March 1991
Oscillations-assisted resonant spin-flavor precession and time variations of the solar-neutrino flux E.Kh. Akhmedov Kurchatov Institute of Atomic Energy, SU-123 182 Moscow, USSR
Received 7 September 1990
We consider the combined effect of the resonant spin-flavor precession (RSFP) and resonant oscillations of neutrinos in the convective zone of the sun. It is shown that although the oscillations cannot give rise to the apparent time variations of the solar neutrino flux in the 37C1experiment, they can efficiently assist the RSFP to do so. As a result, smaller values of the neutrino transition magnetic moments are required, which are compatible with the recently derived astrophysical upper bound. We also remark on new experimental results on solar neutrinos.
1. There is some evidence that the counting rate o f solar neutrinos in the 37C1 experiment o f Davis a n d his collaborators varies with time in anticorrelation with solar activity [ 1,2 ]. Although this p h e n o m e n o n needs further confirmation, indications in its favor are fairly strong. A m o n g various m e c h a n i s m s which might account for the a p p a r e n t t i m e variations o f the solar-neutrino flux, the most plausible ones are those connected with the magnetic m o m e n t s o f the neutrinos. I f such moments are large enough, in the periods o f high activity the toroidal magnetic field o f the convective zone o f the sun will rotate a significant fraction o f lefth a n d e d VeLinto the sterile right-handed neutrinos VcR (V~R, V,R) or into the right-handed antineutrinos 9,R (9~a), which cannot be detected in the 37C1 experim e n t [3 ]. O f particular interest is the s p i n - f l a v o r precession VeL~; ~?~R(V~R)dUe to the flavor-off-diagonal ( t r a n s i t i o n ) magnetic m o m e n t s ltij ( i # j ) , since m a t t e r can resonantly amplify it [ 4 - 7 ]. Such a resonant s p i n - f l a v o r precession ( R S F P ) is analogous to the resonant neutrino oscillations ( R N O ) in m a t t e r [ 8,9 ]. F o r the neutrino spin (or s p i n - f l a v o r ) precession inside the sun to be efficient, the magnetic m o m e n t s o f neutrinos must be > 10-'~/ta p r o v i d e d the magnetic field in the convective zone is >~4 0 - 6 0 kG. In the case o f the flavor-conserving neutrino spin precession this is necessary to overcome mattersuppression effects a n d to gain a sizeable precession
phase [3,10], while in the case o f the R S F P a sufficiently large transition m o m e n t is n e e d e d to ensure the adiabaticity o f the neutrino conversion [ 4 - 7 ] . The most stringent constraint on the neutrino magnetic m o m e n t s was d e r i v e d from the energy loss o f white dwarfs and helium s t a r s , / ~ < 1 × 10-11//a [1 1-3] ~ , which applies to both the diagonal a n d transition moments. Recently, the d a t a on helium stars have been reexamined yielding a m o r e severe b o u n d [ 18 ] /A~< 3 X 10-121tB •
( 1)
Thus the question arises as to whether the neutrino spin or s p i n - f l a v o r precession with the magnetic moments being that small can still account for the apparent time variations o f the solar-neutrino flux. It is this question that we address in the present paper. O u r answer is yes, at least for the RSFP. In general, a three times smaller magnetic m o m e n t requires a three times stronger magnetic field to produce the same precession effect. Although magnetic fields as strong as 100 k G have been considered possible in the convective zone o f the sun [ 19 ], it would at There are also more restrictive upper bounds on/zv from the observation of the neutrino signal from the supernova SN 1987A, but it has been argued that these bounds are avoidable [ 14,15 ]. Moreover, they are not applicable to the transition magnetic moments of Majorana and Zeldovich-Konopinski-Mahmoud neutrinos [ 16,17 ].
0370-2693/91/$ 03.50© 1991 - Elsevier Science Publishers B.V. (North-Holland)
163
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be desirable to seek some enhancement mechanism which would allow one to achieve the same effect with smaller fields. We shall show that the RNO might provide such a mechanism because they can efficiently amplify the RSFP. 2. For the RSFP to occur, neutrinos must have nonzero flavor-off-diagonal magnetic moments, which implies lepton flavor violation. Therefore, neutrino oscillations must also take place. Thus in general the RSFP should be considered jointly with the RNO. Such a joint consideration was carried out in refs. [20,21 ]. In ref. [20] it was shown that when the resonances of the oscillations and spin-flavor precession overlap, these two effects can either suppress or enhance each other, depending on the degree of the adiabaticity of both the conversions. Since the overlap increases with decreasing neutron number density [20,21 ], one can expect a strong mutual influence of the RNO and RSFP in the convective zone, where the neutron fraction is relatively small. The RSFP can give rise to the time variations provided that the neutrino conversion is nearly adiabatic at the solar maxima and non-adiabatic in the periods of a quiet sun; therefore we are mainly interested in the moderately non-adiabatic regime. In this regime the overlap of the RNO and RSFP resonances improves their adiabaticity, since (i) the widths of both resonances become larger, and (ii) the oscillation and precession lengths become smaller [20]. This implies that the RNO and RSFP enhance each other in the non-adiabatic regime. For the case of the overlapping resonances a simple analytical solution was obtained in ref. [ 20 ] for the moderately non-adiabatic regime. Expressing E / A m 2 through the resonant density Pr (to which it is inversely proportional), one can obtain from eq. (34) ofref. [20] the following expression for the VeLsurvival probability: P ~ cosZ{ 180p~[ (2.3 × lO-3flr/Pr) 2+ tanZ20o ] }.
(2) Here fl,---(ltJlO-tl#a)B±r(kG), BIr being the magnetic field strength at the resonance, 0o is the vacuum mixing angle, and pr is in g / c m 3. To obtain a sizable reduction of the solar ve flux, the argument of the cosine in eq. (2) must be of the order of unity. If 0o were zero, this would require fir> 42 for the resonance occurring at the bottom of 164
21 March 1991
the convective zone (Pr ~ 0.17). For non-vanishing 0o smaller values of fir are needed. For our estimates we shall assume, in accordance with the results of Davis et al. [ 1,2 ], that the maximal counting rate in the 37C1 experiment Qmaxis about 4.2 SNU and Qmin is 2-4 times smaller. This should be compared with the standard solar model (SSM) prediction QssM= 7.9 SNU. One can easily make sure that for pr~ 0.17 and tan 20o,~ 0.5 the maximal value of fir should be about (fir)max'25 for Qmax/amin.~3 and (fir) max ~'~22 for amax/ amin ~ 2. Thus for a factor of 2 (3) variation of the detection rate, a factor of 2 ( 1.7 ) smaller transition magnetic moment is required than in the 0o = 0 case. One can also gain a factor of three in the magnetic moment, but in that case only a ~ 30% variation of the VeLflUXwill result, which is too small. Our estimates apply, strictly speaking, to a definite value of pr, i.e. to a definite neutrino energy. However, moderate changes ofpF do not affect appreciably the results. This is because one of the two terms in the argument of the cosine in eq. (2) is directly proportional to p~, and the other is inversely proportional to it; thus in the case when the two terms are of the same order of magnitude (which is of the major interest to us), the Pr dependence of the whole argument is weakened. From the above estimates it follows that, with a sufficiently large mixing angle, one can gain a factor of two in the transition magnetic moment, so that one still needs a factor of 1.5 increase in magnetic field strength to compensate for a factor of three reduction of the upper bound ( l ) as compared to the previous constraints. However, eq. (2) overestimates the desirable/zejB, value because it completely disregards the oscillations and precession in the radiation zone of the sun. In particular, the magnetic field in the vicinity of the top of the radiation zone can improve the adiabaticity of the RSFP of the neutrinos in the convective zone by enlarging the resonance width [ 16,21,22]. As a result, the required values of I.Ze:Bj_ may be lower than our simple estimates show and, in addition, smaller mixing angles are needed. This can be seen from the results of the numerical calculations. 3. We calculated the VeL evolution inside the sun allowing for both the neutrino oscillations and the spin-flavor precession. For simplicity, we considered the Majorana neutrino case for which/t,-= 0. The
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case of non-resonant (flavor-conserving) neutrino spin precession can also be obtained from our results by formally setting Am 2 = 0. Following refs. [ 16,20 ] we employed the model magnetic field profile inside the sun of the form (see the discussion in ref. [ 16] )
21 March 1991
1.00
0.80
0.60
]2,
/4jB± (r) =B~ [ 7 / ( 7 + x )
0 ~ X ~ 0.65,
=Bo{1-[(X--0.7)/0.312},
0,65
0.40
(3) where x-r/Ro, and 7= 0.1 was taken in all the calculations. The results are shown in figs. 1-3. In our calculations, time variations of the magnetic field in the convective zone were simulated by an order of magnitude decrease of Bo (dashed lines in figs. 1-3). It is seen from fig. l that the mean VeL-flUX suppression factor in the energy range E/Am2,~
0.20
(a) o oq;
10
10
~
10
7 '
I '
10 8
10 ~
::"'""
1.00
0.80
,", 0.60
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0.40
0.80
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0.40
b) 10 5
il 10 e
10 ~
10 B E/Am
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"
........ i 10
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i
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i
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....
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Fig. 2. Same as fig. 1, but Bo=3× 104 (10-"/taG) (solid line) and Bo=3X 103 ( 1 0 - " / ~ B G ) (dashed line).
(a) E/Am 2, MeV/eV2
1.00
10 '°
E/Am 2. MeV/eV2
I
....
~0
'0
E/Am 2, MeV/eV2
Fig. 1. VeL survival probabilities at the surface o f the sun versus
E/Am 2. B, = 1X 106 ( 10-"lt~ G); B o = 2 X 104 ( 1 0 - " # G ) (solid line) and B o = 2 × 10a ( 1 0 - " / t a G ) (dashed line). (a) 0o=0; (b) tan 200 = 0.4.
(5X 107-1.5X 10 s) MeV/eV 2 varies with time between 0.55 and 0.3 for tan20o=0.4, and only between 1 and 0.85 for 0o=0. Thus the desirable time variations can-be achieved for gej being compatible with the bound ofeq. ( 1 ) and maximal magnetic field in the convective zone about 60 kG. If the magnetic field is a factor of 1.5 higher, one can get the mean suppression factor in the same energy domain varying with time between 0.55 and 0.18 (fig. 2b). A more precise statement can be made after the direct calculation of the chlorine detection rate. For the neutrino survival probabilities of fig. 1 and Am2= 10 - 7 e V 2 such a calculation yields the detection-rate suppression factor varying between 0.46 and 0.26 for tan 200=0.4 (fig. lb) and only between 0.99 and 0.61 for vanishing mixing angle (fig. 1a). For the survival probabilities of fig. 2 the detection-rate suppression factors vary between 0.46 and 0. l 1 for tan 200= 0.4 165
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the low activity periods. Even for 0o=0, the RSFP requires smaller magnetic moments: in the case of the non-resonant precession one needs r > 500p to overcome the matter-suppression effects, whereas the adiabaticity condition of the RSFP requires fir > 100v/~. This gives a factor of two gain in fl for the RSFP occurring at the bottom of the convective zone.
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Fig. 3. Same as fig. 2, but B~ = 5 × 106 ( 10- H #B G). For comparison, the pure R N O (B~ = B0 = 0) case is also shown (doned line).
and between 0.99 and 0.35 for 0o=0. The role of the inner magnetic field can be seen by comparing figs. 2 and 3. From our results one can also find the VeLsuppression factors for the non-resonant precession (VVO mechanism [3]). These are obtained in the E/Am2--,oo limit (in fact, they correspond to E/Am2> 10 9 MeV/eV2). It can be seen from figs. 13 that for the ranges of the parameters we consider, the VeL suppression factor varies typically between 1 and 0.8 (or between 0.9 and 0.7 for the stronger fields, fig. 3 ), irrespective of the v mixing angle. For Bo > 60 (in units 10- ~#a kG) one can get a suppression factor about 0.5, but again in the quiet sun it will be nearly 1. This means not only that the RSFP has the advantage as compared to the non-resonant conversion that it can be amplified by the RNO, but it also has the welcome feature of lower neutrino fluxes at 166
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4. Finally we would like to remark on some recent experimental results on solar neutrinos. It was claimed by the Kamiokande II Collaboration that the ratio of the average detection rate of the solar neutrinos over the three years of observation to the SSM prediction is 0.46 _ 0.05 (stat.) _+0.06 (syst.), and the time variations, if any, do not exceed 30% [23 ]. This seems to be at variance with the results of the 37C1 experiment of Davis et al., in which much stronger time variations are observed. However, one can readily make sure that these two results can be reconciled in the RSFP scenario provided that the lowenergy v contributions to the 37C1 experiment are strongly suppressed. This is because in that case weaker suppression of the 8B v's is needed, i.e. their flux may be higher. Since the V,'s and V~'s do contribute to the ve reaction which is employed in the Kamiokande II experiment (though with 6-7 times smaller cross sections), a 30% time variation in the Kamiokande II detection rate may be compatible with a factor 2-3 variation in the a7Cl experiment [22]. For example, for A m 2 = 10 - 7 e V 2 and the neutrino survival probabilities of fig. lb the calculation yields the detection rate suppression factors for the neutrino-electron scattering which vary between 0.61 and 0.39. This is in accordance with the Kamiokande II data [23]. One can expect strong suppression of the 71Ga counting rate in this case, which is consistent with the first results of the SAGE Collaboration [ 24 ]. A drastic depletion of the low-energy VeLflUX we need in this scenario may be either due to the strong inner magnetic field (fig. 3a), or due to the RNO (figs. lb, 2b and 2b). In conclusion, we showed that the resonant v oscillations can efficiently enhance the resonant spin-flavor precession allowing the sizable time variations of the solar-neutrino flux with the neutrino transition magnetic moments being compatible with the bound of eq. ( 1 ). Our results are in accordance with all the
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a v a i l a b l e e x p e r i m e n t a l d a t a o n solar n e u t r i n o s , inc l u d i n g t h o s e o f K a m i o k a n d e II. T h e a u t h o r is grateful to Z . G . B e r e z h i a n i , L.B. O k u n , A.Yu. S m i r n o v a n d M.I. V y s o t s k y for useful discussions.
References [ 1 ] R. Davis Jr., talk XXIst Intern. Conf. on Cosmic-ray physics (Adelaide, Australia, 1990 ). [2 ] K. Lande, talk Neutrino 90, 14th Intern. Conf. on Neutrino physics and astrophysics (CERN, Geneva, June 1990 ). [3] M.B. Voloshin, M.I. Vysotsky and L.B. Okun, Sov. Phys. JETP 64 (1986) 446. [4] E.Kh. Akhmedov, preprint IAE-4568/1 (January 1988). [5] E.Kh. Akhmedov, Yad. Fiz. 48 (1988) 599. [ 6 ] C.-S. Lim and W.J. Marciano, Phys. Rev. D 37 ( 1988 ) 1368. [7] E.Kh. Akhmedov, Phys. Lett. B 213 (1988) 64. [8] L. Wolfenstein, Phys. Rev. D 17 (1978) 2369. [9] S.P. Mikheyev and A.Yu. Smirnov, Sov. J. Nucl. Phys. 42 (1985) 913.
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[10] R. Barbieri and G. Fiorentini, Nucl. Phys. B 304 (1988) 909. [ 11 ] S.I. Blinnikov, preprint ITEP-I 9 (1988). [ 12] M. Fukugita and S. Yazaki, Phys. Rev. D 36 (1987) 3817. [ 13 ] G.G. Raffelt and D.S.P. Dearborn, Phys. Rev. D 37 ( 1988 ) 549. [ 14] M.B. Voloshin, Phys. Lett. B 209 (1988 ) 360. [ 15 ] R. Barbieri, R.N. Mohapatra and T. Yanagida, Phys. Lett. B213 (1988) 69. [16] E.Kh. Akhmedov and O.V. Bychuk, Sov. Phys. JETP 68 (1989) 250. [ 17 ] M. Leurer and J. Liu, Phys. Lett. B 219 (1989) 304. [ 18 ] G.G. Raffelt, Phys. Rev. Lett. 64 (1990) 2856. [19] V.N. Krivodubsky, Pis'ma Astron. Zh. 13 (1987) 803. [20] E.Kh. Akhmedov, Soy. Phys. JETP 68 (1989) 690. [21 ] H. Minakata and H. Nunokawa, Phys. Rev. Left. 63 (1989) 121. [22] E.Kh. Akhmedov, preprint 5017/1 (1990); Yad. Fiz., in press. [23 ] Y. Totsuka, talk Neutrino 90, 14th Intern. Conf. on Neutrino physics and astrophysics (CERN, Geneva, June 1990). [24] V.N. Gavrin, talk Neutrino 90, 14th Intern. Conf. on Neutrino physics (CERN, Geneva, June 1990).
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