The magnetic moment solution to the solar neutrino problem is investigated in the context of a solar magnetic field with varying phase velocity on the transverse plane. This phase velocity is assumed to successively take values which are approximately proportional to the solar density, giving rise to the appearance of many resonances along the neutrino trajectory. We obtain a magnetic moment necessary for a neutrino reduction compatible with the experimental situation in the range (6-7)×10-13μB, an improvement by a factor of 4-6 relative to the one resonance case under the same field conditions. © 1993.31701/02/15125129Lim, Marciano, (1988) Phys. Rev. D, 37, p. 1368Akhmedov, (1988) Phys. Lett. B, 213, p. 64Balantekin, Hatchell, Loreti, (1990) Phys. Rev. D, 41, p. 3583Pulido, (1992) Phys. Rep., 211, p. 167IFM preprint 19/92, to be publishedBahcall, Ulrich, (1988) Rev. Mod. Phys., 60, p. 297Bahcall, Pinsonneault, (1992) Rev. Mod. Phys., 64, p. 885Davis, (1990) Proc. XXI Cosmic ray Conf., 7, p. 155. , Adelaide, AustraliaAbazov, (1991) Phys. Rev. Lett, 67, p. 3332Hirata, (1991) Phys. Rev. D, 44, p. 2241Anselman, (1991) Phys. Lett. B, 281, p. 376Anselman, (1991) Phys. Lett. B, 281, p. 390Voloshin, Vysotskii, Okun, (1986) Sov. J. Nucl. Phys., 44, p. 440Voloshin, Vysotskii, (1986) Sov. J. Nucl. Phys., 44, p. 544Fujikawa, Shrock, (1980) Phys. Rev. Lett., 45, p. 963Marciano, Parsa, Electroweak Tests of the Standard Model (1986) Annual Review of Nuclear and Particle Science, 36, p. 171Krakauer, (1990) Phys. Lett. B, 252, p. 177. , see alsoWolfenstein, (1978) Phys. Rev. D, 17, p. 2369Wolfenstein, Neutrino oscillations and stellar collapse (1979) Physical Review D, 20, p. 2364Mikheyev, Smirnov, (1985) Sov. J. Nucl. Phys., 42, p. 913Rosen, Gelb, (1986) Phys. Rev. D, 34, p. 969Wudka, Vidal, (1988) UCD-88-40, , University of California preprintAneziris, Schechter, NEUTRINO “SPIN ROTATION” IN A TWISTING MAGNETIC FIELD (1991) International Journal of Modern Physics A, 6, p. 2375Vidal, Wudka, (1990) Phys. Lett. B, 249, p. 473Smirnov, (1991) Phys. Lett. B, 260, p. 161Toshev, (1991) Phys. Lett. B, 271, p. 179Akhmedov, Bychuk, (1989) Sov. Phys. JETP, 68, p. 250IFM preprint 19/92, to be publishedBabu, Mohapatra, Rothstein, (1991) Phys. Rev. D, 44, p. 2265see also E.Kh. Akhmedov, A. Lanza and S.T. Petcov, SISSA preprint 172/92/EP (1992)Mourão, Pulido, Ralston, The neutrino electromagnetic moments and charge radius confront Kamiokande II and Homestake experimental results (1992) Physics Letters B, 285, p. 36

Guzzo M.M.

Pulido J.

English

Repositorio da Producao Cientifica e Intelectual da Unicamp

Physics Letters B 317 (1993) 125-129 
North-Holland PHYSICS LETTERS B 
Twisting magnetic field and multiple resonance mechanism 
in the solar neutrino problem 
M.M. Guzzo a,b,l and J. Pulido c,d 
• Departamento deRaios C6smicos eCronologia, IFGW, Unicamp, Caixa Postal 6165, 13081 Campinas SP, Brazil 
b Instituto de Flsica Te6rica, Unesp, Rua Pamplona, 145, 01405 Silo PauloSP, Brazil 
c IFM, Av. Prof. Gama Pinto, 2, P-1699 Lisboa Codex, Portugal 
d IST, Physics Department, Av. Rovisco Paes, 1)-1096 Lisboa Codex, Portugal 
Received 29 April 1993; revised manuscript received 29 July 1993 
Editor: M. Dine 
The magnetic moment solution to the solar neutrino problem is investigated in the context of a solar magnetic field with varying 
phase velocity on the transverse plane. This phase velocity is assumed to successively take values which are approximately pro- 
portional to the solar density, giving rise to the appearance of many resonances along the neutrino trajectory. We obtain a magnetic 
moment necessary for a neutrino reduction compatible with the experimental situation in the range (6-7) × 10 - t3#e, an improve- 
ment by a factor of 4-6 relative to the one resonance case under the same field conditions. 
The resonant spin-flavour conversion [ 1 ] of neu- 
trinos weakly interacting into sterile neutrinos has 
proven to be an effective mechanism [ 2 ] for explain- 
ing the solar neutrino deficit relative to the solar 
standard model prediction [ 3 ] observed by the four 
existing solar neutrino experiments [4 ]. It requires 
the existence of a neutrino magnetic moment [5 ] in 
the range #,~> 10-~2#B which is at least six orders of 
magnitude above the electroweak theory prediction 
[6] and is limited by the firm laboratory bound 
/~, ~< 4X 1 0- l°/~B [ 7 ]. The idea of the mechanism [ 1 ] 
is based on the possible existence of one resonance 
inside the Sun in the neutrino trajectory arising from 
the interplay of the solar density and the quantities 
A2m21/2E (fiavour mass square difference divided by 
the neutrino energy) and/~,B (the product of the 
neutrino magnetic moment by the intensity of the so- 
lar magnetic field). This operates in a way similar to 
the matter oscillation [8 ]. The resonance range is a 
privileged zone for neutrino conversion and in fact 
its existence can reduce by as much as two orders of 
magnitude the necessary magnetic moment for a sig- 
nificant reduction of active neutrinos. One question 
Financial support from CNPq, Brazil. 
that most naturally arises may therefore be whether 
the existence of multiple resonances can allow for a 
further eduction. 
The purpose of this letter is to investigate his pos- 
sibility, by dividing conceptually the neutrino trajec- 
tory in intervals, each containing one resonance only. 
If the conversion probability Pi in each resonance is
admittedly small in connection with a magnetic mo- 
ment smaller than the usually required minimum 
value O ( 10- ~2/~B), the existence of a large number 
of resonances may lead to a sizeable overall probabil- 
ity satisfying P~ O (Y?i). The possibility of a neu- 
trino crossing a large number of resonances along its 
trajectory can only be realized, as will be seen, for a 
solar field B with varying phase velocity. Our analy- 
sis will show that the multiple resonance mechanism 
can provide a large total probability with small val- 
ues of the individual probabilities Pi. Furthermore, 
assuming multiple resonances, the minimum mag- 
netic moment necessary for a suitable solution to the 
solar neutrino problem will be seen to lie almost one 
order of magnitude below the minimum for the con- 
ventional one resonance case. 
As neutrinos travel through the Sun, they will most 
likely experience a magnetic field with varying phase 
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved. 125 
Volume 317, number 1,2 PHYSICS LETTERS B 4 November 1993 
velocity in the transverse plane. Such is the case of 
the twisting magnetic field introduced in the litera- 
ture some time ago [ 9 ]. Although this is the only ex- 
isting candidate for multiple resonances, such a pos- 
sibility was never considered. 
In order to evaluate the transition probability in a 
scenario with multiple resonances, let us take an elec- 
tron neutrino u~ produced in the core of the Sun which 
we will denote by ( 1, 0) on the basis (re, u~) where 
ux refers to a sterile neutrino flavour. After travelling 
through a region containing one resonance, this 
becomes 
(1pP , ) ,  (1, 
where PI is the probability for the v,~ v~ transition 
in this region. Writing this state as the combination 
( l )  (0)  (2, 
(1-P~) 0 +P~ 1 ' 
then, after a second resonance 
(I'~...~(1-P2"~ (~) ( Pu ) (3, 
O] \ P2 / '  - '  1 - /2  ' 
where P2 is the probability for v~-, v~ in the region 
containing the second resonance. Thus the transition 
probability from an active to a sterile neutrino after 
two resonances is 
p~ot __ ( 1 -P~ )/2 +Pt ( 1 -P2) 
--P~ +P2-2P,  Pu. (4) 
Including a third resonance and repeating the argu- 
ment, it is straightforward to derive the total proba- 
bility for three resonances: 
p~ot = PI +/'2 + P3 - 2P1/'2 - 2P1 P3 - 2/'2 P3 
+4P~P2P3 . (5) 
For an arbitrary number of resonances, one therefore 
obtains the general formula for the classical transi- 
tion probability 
Pfn°t = ~ ~ (-2)m-lpilPi2...Pi,.. (6) 
m ~ l i l  ,i2 ,...,im ~ l 
i l  < i2  <. . .< im 
The simplifying assumption of taking approxi- 
mately equal probabilities at each resonance is par- 
ticularly useful to provide an insight on the behav- 
iour of this expression. It may also be not too distant 
from a realistic realization of the multiple resonant 
scenario as will be seen. Instead of eq. (6) we can 
write for equal probabilities P~ =P2 . . . . .  P; . . . . .  
P,,=P 
~ (2  " - l (n )P  " ,  ptot= - ) (7) 
m~l m 
which, for instance, taking n=100, P=0.01 gives 
Pt~to =0.43. The first term in eq. (7) is the sum of all 
probabilities n.P while the following ones provide al- 
ternatively negative and positive contributions which 
converge to zero. The net result is thus a reduction to 
a number within the same order of magnitude of the 
quantity n.P. 
The implementation f the multiple resonance sce- 
nario is most naturally done, as referred to above, in 
the case of the so called twisting magnetic field or, 
more generally, a field with varying phase velocity on 
the transverse plane. The neutrino evolution equa- 
tion is in such cases [10-11 ]
d : Bo (8) '~ux/]=~16,B,e i" --Bo "" J]\vx/" 
Here ~ is the field phase, B(t) = (B~cos ~0, Blsin ~, 0), 
B~ its modulus, Bo = GF (Are -- N.) / x//-2 and, along the 
neutrino trajectory, the neutron density N. = ~N. with 
the electron density Are decreasing exponentially [ 3 ]. 
Applying the unitary transformation 
0 
vx] \ 0 e-i*/2J\vx] ' (9) 
one can eliminate the imaginary part of the evolution 
Hamiltonian, so eq. (8) becomes 
. d (v , ) '=(Bo-½~ I~,,B1 ~(v,~' (10) 
1~\Vx/  \ #,.B~ -Bo+½~/kvx] " 
It is therefore seen that a non-vanishing phase veloc- 
ity ~ opens the possibility for one or more resonances 
to occur through the condition of equal diagonal 
elements: 
~=2Bo. (ll) 
For a field whose direction changes permanently 
and irregularly on a plane perpendicular to the direc- 
tion of motion, as is possibly the case, this condition 
126 
Volume 317, number 1,2 PHYSICS LETTERS B 4 November 1993 
may be verified a large number of times along the 
neutrino trajectory. We note that the effect of the 
phase velocity would be a direct increase in the tran- 
sition probability if instead it appeared in the off-di- 
agonal elements of eq. (10). It therefore plays a sim- 
ilar role to the mass square difference in the MSW 
mechanism [ 8 ] or the spin flip with only one reso- 
nance [ 1 ]. Thus it may only indirectly provide an 
increase in the transition probability through the 
multiple use of the resonance condition ( 11 ). 
Using the system of equations (10), it is straight- 
forward to get the second order differential equation 
for the evolution of Ax, the amplitude for re--* vx. It is 
well known that the quantity Bo introduced in eq. (8) 
decreases xponentially along the neutrino trajectory 
[ 3]. Since however we divide this trajectory in rela- 
tively short intervals we will assume, within limits, a 
constant Bo in each interval as well as a constant phase 
velocity ~ and field magnitude BI. In this way we ob- 
tain a simple Laplace equation for Ax~ 
d2Axi +[(B0 i -  " 2 2 2 ½¢i) + Iz,Bli]Axi =0 (12) 
dt 2 
This is the equation of motion describing the propa- 
gation of neutrinos within the ith region, each being 
characterized bya fixed value of Boi, Bu and ~. It is 
also assumed, as we have seen, that the resonance 
condition ( 11 ) is satisfied once and only once in each 
of these sections of the trajectory. Hence, on the basis 
of these assumptions, the transition probability for 
veo v~ in one interval calculated from the solution of 
eq. (12) is the more accurate, the closer one stays to 
the resonance condition within the interval. In other 
words the expression 
ei( u~ Ux) = [axe[ 2 
2 2 
-½¢ i )  +~B.  (Bo; • 2 2 2 
xsin2([(Bo~-½~,)2+lt2B~d~/2At,) (13) 
(where Att is the length of the interval ) gives a better 
approximation ofthe actual probability, the stronger 
the inequality 
(Bo ; -  "2  2 2 ½¢,) << #,B~; (14) 
is fulfilled throughout the whole interval. In this way 
a "quasi-resonance" condition remains valid within 
this length. For short intervals this condition is of 
course likelier to be ensured than for comparatively 
larger ones. Let us therefore assume that eq. (14) 
holds in each section of the trajectory and thus the 
probability Pi is accurately given by its maximum 
value 
Pi=sin2(lt~BliAti) . (15) 
In order to get a rough but safe estimate of the 
maximum allowed value of At~ for the approximation 
to be accurate, let us consider a linearly decreasing 
density extending to both sides within the vicinity of 
the resonance. Thus 
Bo = Bo~exp ( - f i t )  ~Bo~( 1- f i t ) .  (16) 
Here fl= 1/0.09Rs with Rs being the solar radius 
(Rs=696000 km) and t=0 at the ith resonance 
point. Applying the resonance condition (11 ), we 
obtain from the inequality (14) 
max At~ ~ 1.28X 101SeV - l  ltvBu ~, (17) 
The phase velocity ~ entering eq. (17) is therefore 
fixed by the density at resonance: 
q), = 2Bo, = x/~GF" ~Ne, (18) 
which, using [1] Ne~=2.4×lO26exp(-x~/O.09) 
cm -a, becomes 
, ,=2.5× 10- ' lexp(~)eV,  (19) 
where x; is the fraction of the solar radius at the res- 
onance point, x~=r~/Rs. In other words, we are as- 
suming the phase velocity of the field successively to
reach along the neutrino trajectory a value given by 
eq. (19), so that at each such point the ith resonance 
occurs. 
We now investigate under what conditions the to- 
tal transition probability can be maximized for a given 
field and magnetic moment. A maximum probability 
must obviously result from a balance between the 
maximization ofthe intervals ize and the number of 
intervals: the large one is, the smaller the other. If the 
regions are too long we approach the well known one 
resonance situation [2 ], whereas if they become too 
small, the probabilities P,., eq. (15) become negligi- 
ble. For the solar field we take one of the Akhmedov 
model distributions [ 12 ] 
127 
Volume 317, number 1,2 PHYSICS LETTERS B 4 November 1993 
BI(x)=IO'×( 0.2 ~2 G 
\x+0.2]  
for 0~<x~<0.65, 
2 
= 105×[1- (~)  ] G 
for 0.65 <x~< 1 . (20) 
Using eqs. ( 15 ) and ( 17 ) -  ( 20 ) we first calculated 
P~ (max Ate) at each given interval. The minimum of 
these quantities was chosen as a common value for 
the probabilities o that the simplified formula, eq. 
(7), for the total probability could be applied. In this 
way the corresponding interval length At~ is often 
much smaller than max At`.. In table 1 we list the val- 
ues of max At,, eq. ( 17 ), the value At`. obtained in this 
way at several points of  the trajectory, the corre- 
sponding values ~(x~), eq. (19) and the magnetic 
field, eq. (20). In this way we are in the most favour- 
able case for applying eq. (7). For #~=7× 10-13/za, 
we have PI~P2,,~...~Pn~P~2×IO -3, while for 
/z~=6× 10-~3/~B, P~ 10 -3. The results ofeq. (7) for 
these two values of/z~ are listed in table 2, together 
with the approximate number of  resonances encoun- 
tered by the neutrinos, for these being produced at 
0.05, 0.06 and 0.1 of  the solar radius. This approxi- 
mate number of  resonances refers of  course to the 
whole neutrino trajectory, from their production 
point up to the surface of  the Sun. 
A comparison of the results for the total suppres- 
sion probability in the multiple resonance case with 
the one resonance case [2 ] shows that, under the same 
field conditions, an analogous probability is obtained 
for a neutrino magnetic moment 4-6 times smaller. 
In fact a magnetic moment of  order (6-7)  × 10-13/z a 
can give the necessary suppression of  active neutri- 
nos compatible with the experimental situation, to be 
compared w i th /z~ (2 -4 )× 10-12//B in the one res- 
onance case. A different feature of  the multiple reso- 
nance framework with respect o both MSW and one 
resonance mechanisms i  that all neutrinos are equally 
suppressed, regardless of  their energy, and the loca- 
tion of the resonances is also energy independent. This 
is a consequence of the fact that the present scenario 
works independently of mass differences between 
Table 1 
The values of the magnetic field, eq. (20), at a fraction xof the solar adius, the phase velocity (19 ), the maximum interval length (17 ) 
and the value of Ati used in the calculation of the probability P~ in the corresponding i terval. 
x= r/Rs B1 (x) ( X 106G) ~(x) (eV) max Att(eV-l ) Atl(eV- 1 ) 
0.04 6.9 1.61 X 10 -ll 2.22X 1012 1.70X 1012 
0.05 6.4 1.44X 10 -11 2.29X 1012 1.84X 1012 
0.06 5.9 1.28X 10 -11 2.37X 1012 1.99X 1012 
0.1 4.4 8.26X 10 -12 2.75X 1012 2.75X 1012 
0.2 2.5 2.72X 10 -12 4.74X 1012 4.74X 1012 
0.3 1.6 8.95X 10 -13 9.21 X 1012 7.37X 1012 
0.4 1.1 2.93X 10 -13 1.97X 1013 1.05X 1013 
0.5 0.82 9.69X 10 -14 4.34X 1013 1.44X 1013 
0.6 0.62 3.19X 10 -14 1.01X l014 1.90X 1013 
Table 2 
The number of resonances ncountered byneutrinos produced at 0.05, 0.06 and 0.1 of the solar adius and their total expected transition 
probability into sterile ones for two values of the magnetic moment. The solar magnetic field is taken from eq. (20). 
x= r /Rs  Approximate number ptot 
of resonances 
/~v= 7 X IO-13#B #,--6X I0-13~js 
0.05 400 0.41 0.28 
0.06 380 0.39 0.27 
0. I 320 0.36 0.24 
128 
Volume 317, number 1,2 PHYSICS LETTERS B 4 November 1993 
neutrino flavours and does not require neutrino mass. 
From eq. (15) it is clear that there is the possibil ity 
of  an anti-correlation of the neutrino flux with the 
sunspot activity in the convective zone of  the Sun. 
This t ime dependence has been claimed by the 
Homestake xperiment [4 ] but the Kamiokande data 
do not seem to confirm it. In the one resonance case, 
such a strong effect in one experiment and its absence 
in the other can be explained by the assumption of a 
resonance located in the convective zone for the ber- 
i l l ium neutrino sector only. This would lead us to ex- 
pect an even stronger t ime dependence in gallium ex- 
periments [4,13], as the Be fraction is more 
important there. In the multiple resonance ease, 
however, where the location of  the resonances i de- 
termined by the field phase velocity and not by the 
mass difference, a possible reliable explanation lies 
in the comparison of  the neutrino scattering effects 
in Homestake and Kamiokande. While Homestake 
observes a purely weak process, the neutr ino-elec- 
tron scattering observed by Kamiokande has an elec- 
tromagnetic ontribution from the magnetic mo- 
ment which does not flutuate in time. So the 
Kamiokande ffect is expected to be smoother. How- 
ever, this smoothening is only apparent if 
#~ i> 10- t°/tB [ 14 ]. Although this range possibly leads 
in the present scenario to probabil it ies that are far 
too large, the large experimental uncertanties in- 
volved allow us to invoke this explanation. 
The phase of solar magnetic field probably rotates 
in space and in t ime along the neutrino trajectory but 
in an unknown fashion. It is therefore possible that it 
may successively reach values on the way of  the neu- 
trinos which are proport ional  to the local solar den- 
sity. Such a coincidence is not unlikely to be realized 
in practice but it requires ome fine tuning to ensure 
that the phase velocity, whose range of  variation is 
unknown, goes through the required values at the ap- 
propriate locations. Thus this mechanism, although 
possible in practice, is not necessarily impl ied by the 
existence of  a variable magnetic field phase. This sce- 
nario thus offers a viable alternative for a solution to 
the solar neutrino problem within the assumption of 
a non-zero neutrino magnetic moment.  
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One of  us (M.M.G.)  would like to thank the Insti- 
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129 


Twisting Magnetic Field And Multiple Resonance Mechanism In The Solar Neutrino Problem

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Twisting Magnetic Field And Multiple Resonance Mechanism In The Solar Neutrino Problem

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