We analyze the perturbations of the solar magnetic field generated by general solar plasma displacements and investigate their consequences in the left-right conversion probability of neutrinos produced in the Sun. We solve the Hain-Lüst equation and the evolution equations of neutrinos interacting with the solar magnetic field through a nonvanishing neutrino magnetic moment to conclude that the appearance of slow magnetosonic waves (with a period around 100 days) simulate the time behavior of solar neutrino data. © 1995.3574602607Davis, (1990) Proc. of the XXI Int. Cosmic Ray Conference, 7, p. 155. , Adelaide, AustraliaHirata, (1991) Phys. Rev. D, 44, p. 2241Anselmann, (1992) Phys. Lett. B, 285, p. 390Abazov, (1991) Phys. Rev. Lett., 67, p. 3332Voloshin, Vysotsky, Okun, (1986) Yad. Fiz., 44, p. 677Voloshin, Vysotsky, Okun, (1986) Sov. J. Nucl. Phys., 44, p. 440Mikheyev, Smirnov, (1986) Nuovo Cimento C, 9, p. 17Mikheyev, Smirnov, (1985) Yad. Fiz., 42Mikheyev, Smirnov, (1985) Sov. J. Nucl. Phys., 42, p. 913Wolfenstein, (1978) Phys. Rev. D, 17, p. 2369Wolfenstein, (1979) Phys. Rev. D, 20, p. 2634Bahcall, Ulrich, Solar models, neutrino experiments, and helioseismology (1988) Reviews of Modern Physics, 60, p. 298Gloedbloed, Sakanaka, New approach to magnetohydrodynamic stability: I. A practical stability concept (1974) Physics of Fluids, 17, p. 908Bernstein, Frieman, Kruskal, Kulsrud, (1958) Proc. Roy. Soc. A, 224, p. 1Hain, Lüst, (1958) Z. Naturforsch., 13 a, p. 936Akhmedov, Bychuk, (1989) Sov. Phys. JETP, 68, p. 250Smirnov, (1991) Phys. Lett. B, 260, p. 161Future Solar Neutrino Experiments (1995) Nuclear Physics B - Proceedings Supplements, 38, pp. 90-95Tatsuka, Superkamiokande (1990) ICCR-Report 227-90-20Chen, (1985) Phys. Rev. Lett., 55, p. 153

Guzzo M.M.

Reggiani N.

Sakanaka P.H.

Physics Letters B

English

We analyze the perturbations of the solar magnetic field generated by general solar plasma displacements and investigate their consequences in the left-right conversion probability of neutrinos produced in the Sun. We solve the Hain-Lust equation and the evolution equations of neutrinos interacting with the solar magnetic field through a nonvanishing neutrino magnetic moment to conclude that the appearance of slow magnetosonic waves (with a period around 100 days) simulate the time behavior of solar neutrino data.357460260

GUZZO, MM

REGGIANI, N

SAKANAKA, PH

Repositorio da Producao Cientifica e Intelectual da Unicamp

SLOW SOLAR MAGNETOSONIC WAVES AND TIME-VARIATION IN THE SOLAR-NEUTRINO DATA

ELSEVIER 
14 September 1995 
Physics Letters B 357 (1995) 602-607 
PHYSICS LETTERS B 
Slow solar magnetosonic waves and time variation in the solar 
neutrino data 
M.M. Guzzo, N. Reggiani, P.H. Sakamka 
Institute de Fisica ‘Gleb Wataghin’, Universidade Estadual de Campinas - UNICAM~ 13083-970 Campinas, &TO Pa&o, Brazil 
Received 19 May 1995 
Editor: M. Dine 
Abstract 
We analyze the perturbations of the solar magnetic field generated by general solar plasma displacements and investigate 
their consequences in the left-right conversion probability of neutrinos produced in the Sun. We solve the Hain-Liist equation 
and the evolution equations of neutrinos interacting with the solar magnetic field through a nonvanishing neutrino magnetic 
moment o conclude that the appearance of slow magnetosonic waves (with a period around 100 days) simulate the time 
behavior of solar neutrino data. 
1. Whether the solar neutrino flux varies with time 
has been an interesting discussion in the context of 
the solar neutrino problem. In fact, time variation in 
the solar neutrino flux has been searched in the solar 
neutrino data [ l-47. Assuming a nonvanishing neu- 
trino magnetic moment the interaction of neutrinos 
with the solar magnetic field [5] could generate an 
anticorrelation of the solar neutrino flux detected at 
the Earth with the solar activity (i.e., long period vari- 
ations of around 11 years). Although Homestake data 
could suggest this anticorrelation, the solar neutrino 
data from other experiments (Kamiokande, Gallex and 
Sage) do not show any compelling evidence for this 
long period variation of the solar neutrino flux. 
Day-night variations of the solar neutrino data (i.e., 
short period fluctuations of these data) have also been 
investigated as a possible indication of the MSW ef- 
fect [ 61 in the Earth. Only real-time experiments, like 
the Kamiokande one, are able to detect such variations 
and no evidence in favor of this possibility was found 
in its data. 
Nevertheless analyzing individually each experi- 
mental point obtained from solar neutrino detectors, 
we see that they are widely dispersed in Homes- 
take [ 11 and 71Ga experimental [3,4] results, pre- 
senting values that vary from approximately 0 to 1 
times the standard solar model theoretical predic- 
tions [ 71. Furthermore observing their experimen- 
tal errors we can conclude that these experimental 
points are not compatible (at lg-level) with each 
other. Kamiokande data [2], on the other hand, do 
not present this dispersion since their experimental 
points are statistically compatible (at la-level) with 
a certain mean value. This apparently different be- 
havior of Homestake and 71Ga experiments and the 
Kamiokande data could suggest that the former have 
observed a certain time variation in solar neutrino 
flux, while the latter show no compelling evidence 
for this variation. 
In this paper we analyze the possibility of such data 
behavior being a consequence of the interaction of 
neutrinos produced in the Sun with a fluctuating so- 
0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved 
SSDIO370-2693(95)00956-6 
M.M. Guuo et al. / Physics Letters B 357 (1995) 602-607 603 
lar magnetic field by means of a nonvanishing neu- 
trino magnetic moment. Using a numerical procedure 
that calculates the fluctuations of the solar magnetic 
field as a consequence of the solar plasma motion [ 81, 
we show that the so-called slow magnetosonic waves 
appear in the solar magnetic field and have a typi- 
cal period of order of one hundred days. Interestingly 
enough we show that these slow magnetosonic waves 
are sufficient to understand the mentioned apparently 
different behavior of the experimental data. 
2. The magnetic waves are calculated assuming that 
they can be generated by small displacements of the 
solar plasma, 5, from an equilibrium configuration. 
The time evolution of these displacements is described 
by the linearized equation of motion [ 91 
pa = F(P) 
at2 ’ (1) 
where 
F(5) = V(ypV.S) - b x (V x Bo) 
-Box (Vxb), (2) 
p is the density, p is the pressure, y = 1.0, BO is 
the equilibrium magnetic field and b is the magnetic 
perturbation generated by the displacement 5 given by 
b=V x (5x Ba). (3) 
Considering an exponential time dependence for the 
displacement 5, g( r, 8, z, t) = e-“‘“‘t( r, 19, z), the fre- 
quency of the wave w and the displacement 5 can be 
found by solving the eigenvalue equations 
-pw2.$ = F(t). (4) 
We are interested in the behavior of the magnetic 
field inside the Sun along the trajectory of the neu- 
trinos that reach the Earth. This trajectory is local- 
ized around the plane of the solar equator. We can use 
therefore cylindrical coordinates to solve Eq. (4) to 
obtain the magnetic perturbations in this region. We 
consider the plane of the equator coincident with one 
of the planes of the cylinder perpendicular to the z 
axis. Employing cylindrical coordinates (Y, 8, z ) with 
periodicity in the coordinate z the displacement 5 can 
be written as 
g(r,e,z,t) =Cp,k(r)exp[i(mB-kz --cut)]. 
mk 
(5) 
In cylindrical coordinates the linearized equations 
of motion present a very useful feature: for each 
Fourier component (m, k) they separate out into a 
second-order differential equation for lr and two rela- 
tions expressing 5~ and cZ in terms of tr and dl,/dr 
[ 81. This property permits us to obtain all the com- 
ponents of the displacement 5, and, consequently, all 
the components of the fluctuation of the magnetic 
field solving just the differential equation for t,.. This 
differential equation was first obtained by Hain and 
Lust [lo] and is given by 
f [f(r) $(&)I + g(r) (rSr) = 0, 
where 
(6) 
yp+B,2(W2-w$)(W2-w;) 
f(r) = ~ (7) r (w2 - wT>(w2 - w;> ’ 
rg( r) = (pw2 - F2) - ($) 2 (n~~~~H~2 
+rd 
[ 
2kBosGa 
dr r2(p2w4 - Ha) 
] -r-f-(+)“, (8) 
F2 2 YP F2 w2 = - A 
P’ 
w =-- 
’ w+B; P’ 
(9) 
2 
w1,2 = 
H(w + B;) 
2P 
x If l-4 
{ [ 
YPF~ 
(yp + B;j2H I) ’ (10) 
a = pw2(yp + B,2) - ypF2, 
m 
F = ;Boe +kBoz, (11) 
H=$+k’, 
m 
G = yBoz - kBoe. (12) 
This equation is solved imposing appropriate bound- 
ary conditions. In this case we can use for the bound- 
ary condition at r = 1 the experimental data of the 
magnetic field in the solar surface indicating a magni- 
tude of order lo-100 G (and can reach up to 0( 103) 
in sunspots). At r = 0 the boundary condition is given 
by the expected behavior of <r which is t,.( r -+ 0) = 
rlmlel. The Ham-Lust equation has singularities when 
604 M.M. Guzzo et al. /Physics Letters B 357 (19951602-607 
f(r) = 0, that is, when w2 = wi or w2 = w$. In the 
interval 0 i r 5 1 the functions wi and wi take con- 
tinuous values that define ranges of values of w that 
correspond to improper eigenvalues. So, the eigenval- 
ues of the Ham-Lust equation must be searched in the 
regions where w2 = wi and w2 = wg, which are called 
AlfvCn and slow continua, respectively. 
We have to choose an equilibrium profile for the 
magnetic field. Nevertheless it is very difficult to ex- 
perimentally investigate the inner solar magnetic field 
because the solar material is opaque. Therefore large 
uncertainties are associated with models for the mag- 
netic field in the sun. We take the model proposed by 
Akhmedov and Bychuk in Ref. [ II] : 
Be(r) = 
I 
0.2 2 
a1 ~ 
( > r + 0.2 
for 0 < P 5 0.7, 
u2,1- (r ,.&I 2 ] (13) G 
for 0.7<ril, 
where at M 105-IO7 and a2 M 104-lo5 in such a 
way that the continuity of the magnetic field at the 
point r = 0.7 is satisfied (r is the radial distance from 
the center of the Sun normalized by the solar radius 
6.96 x lo5 km). We are interested in the behavior of 
the magnetic field in a transverse plane to the neutrino 
trajectory once that this is the field component felt 
by the neutrinos. Using cylindrical coordinates, the 
transverse plane to the neutrino trajectory is given by 
the z and 6’ components of the magnetic field, B, and 
Bo. The equilibrium magnetic field BO is chosen in the 
z direction. We assume the matter density distribution, 
p, usually accepted to fit the predictions of the standard 
solar model [ 71, i.e., an approximately monotonically 
decreasing exponential function in the radial direction 
from the center to the surface of the Sun [7]. We 
consider the pressure to be proportional to the density. 
The results of our calculations indicate that a known 
kind of stable fluctuations are found in the region of 
squared frequencies of order w2 M lo-t5 se2, corre- 
sponding to a period of around 100 days. These fluctu- 
ations, localized out of the Alfven and slow continua 
and characterized y V . ,$ # 0, are usually referred 
to as slow magnetosonic waves. A set of these results 
are shown in Fig. 1, where the displacement are nor- 
malized such that 151 5 1. Using g in Eq. (3) one 
builds the perturbations to be summed to the equilib- 
rium magnetic field, Eq. ( 13). 
Since we are assuming a nonvanishing neutrino 
magnetic moment, the interaction of neutrinos with a 
magnetic field will be given by the evolution equa- 
tions [ 121 
X (14) 
where VL (VR) is the left (right) handed component 
of the neutrino field, Am = Irni - rnij is their squared 
mass difference, E is the neutrino energy, GF is the 
Fermi constant, N,(r) is the electron number density 
distribution and I BJ. ( r) 1 is the transverse component 
of magnetic field given by @_ = (Bo + b, ) 2 + bi. 
Solving Eqs. (14) we can calculate the survival 
probability of left-handed neutrinos P ( v~ + VL) pro- 
duced in the Sun to arrive at the Earth after interact- 
ing with the solar magnetic field 1 Bl (r) I perturbated 
by the fluctuations calculated in this paper. We can 
therefore compare this survival probability with exper- 
imental data. We will analyze data coming from the 
two oldest solar neutrino experiments, namely, Home- 
stake and Kamiokande, from 1987 up to now, when 
both collected data contemporary. Our analysis relies 
on the fact that while the Homestake experiment takes 
one experimental point each 70 days approximately, 
Kamiokande puts the results of its observations over 
a period of around 260 days together. 
In Fig. 2a we put the experimental panorama of 
both experiments which is to be compared with our 
calculations (shown in Fig. 2b). In this figure, solid 
circles correspond to Homestake measurements while 
solid squares are related to Kamiokande. Note that 
no experimental errors are shown in Fig. 2a, but they 
are around 0.2 (in units of theoretical solar neutrino 
flux) for Homestake and around 0.1 for Kamiokande. 
To solve Eq. (14) we use the values 10P7 eV2 and 
3 x 10-12,u~ for the mass squared difference Am and 
the neutrino magnetic moment ,u,,. Note however that 
M.M. Guzzo et al./Physics Letters B 357 (1995) 602-607 605 
-0.5 - 
-’ 01 1 0 0.2 0.4 0.6 0.8 1 
r r 
Fin. 1. Displacements .f, characterized bv the wave numbers m = 2, k = 10m7 and frequencies shown in the figure, obtained by solving 
the Ha&L&t Eq. (6)- 
the relevant quantity to build Fig. 2b is the product 
pLL, 1BI (I-) 1 which appears in the evolution equations 
( 14). The mentioned value of the neutrino magnetic 
moment ,u,, requires at = 5 x lo6 and a2 = 2.5 x lo5 
in Eq. ( 13). We can diminish these values for at and 
a2 if we increase the neutrino magnetic moment p, 
by the same factor. In our calculations the perturba- 
tion of the magnetic field is obtained through Eq. (3)) 
using the displacement g, characterized by m = 2, k = 
10e7, and w2 = 9.41 x 10-15, presented in Fig. 1. The 
results present in Fig. 2b are the survival probability 
integrated in time over periods of 70 (solid circles) 
and 260 (solid squares) days. 
Comparing Figs. 2a and 2b we see that some general 
features of the referred experimental data can be found 
also in the calculated survival probability. In partic- 
ular we observe that this survival probability varies 
from 0.15 to 0.65 for Homestake while its amplitude 
is much smaller for Kamiokande. 
3. Neutrinos have been indicated as a possible 
source of experimental information on the inner part 
of stars and supernovae, which is in general very dif- 
ficult to obtain through other observational methods. 
The solar magnetic field behavior in the very interior 
of the Sun has no direct experimental evidence other 
than the solar neutrino flux observations (if we as- 
sume a nonvanishing neutrino magnetic moment). In 
this paper we analyze the possibility of interpreting 
these observations as evidence of the existence of 
slow magnetosonic waves in the Sun. We calculate the 
consequences of general solar plasma displacement 
606 M.M. Guzzo et aL/Physics Letters B 357 (1995) 602407 
x 0.9 s 
- 1 B 0.6 
jy 0.7 c -I 
P 0.6 - 
a . 
3 O5 ..o . . 0.4 - 
z 0.3 - 
$ 0.2 - 
r r 1 
k- O.it 1986 1966 1990 1992 1994 
Years 
Fig. 2. (a) Neutrino flux at the Earth, normalized by the solar 
standard model predictions, measured in Homestake (circles) and 
in Kamiokande (squares) experiments. (b) Theoretical survival 
probability obtained solving Eq. ( 11)) calculated for Homestake 
(circles) and Kamiokande (squares) experimental conditions. 
on the solar magnetic field and how they influence 
the detectable solar neutrino flux. Note that the crn- 
cial point of our results is the fact that such slow 
magnetosonic waves have an approximate 100 days 
period, which is the right period to leads us to the 
time behavior of solar neutrino data presented in Fig. 
2a. Interesting enough, this period is not an imposi- 
tion in our calculation but derives naturally from the 
solution of the Hain-Lust equation, Eq. (6). 
Two remarks are in order. Although statistical fluc- 
tuations cannot, in the light of the present data, be 
discarded to explain the behavior of the solar neutrino 
data, it is not possible also to discard the time behavior 
of these experimental observations as a consequence 
of a physical effect in the Sun. 
Finally we say that an analysis of the short pe- 
riod (around lo-100 days) time variations in the 
Kamiokande data could be of interest to obtain new 
information on the possible existence of solar slow 
magnetic waves. Nevertheless a conclusive analysis 
of the consequences of slow magnetic waves will 
have to wait for more precise experimental data to 
be obtained in future solar neutrino experiments (see 
[ 131 and references therein). Experiments like Su- 
perkamiokande [ 141 and SNO [ 151 will measure the 
spectrum of high energy neutrinos in a quite accurate 
way. Time variations of the neutrino flux having a pe- 
riod of approximately 100 days could be interpreted 
as the effect of slow solar magnetosonic waves. This 
effect can be distinguished from flux modulations 
due to seasonal variation in Sun-Earth distance and 
vacuum oscillations. This is the reason why it will 
be possible to investigate flux variations of neutri- 
nos with a fixed energy. Such time variations will be 
different if neutrinos have evolution equations like 
( 14) or evolve as in MSW [ 61 or vacuum oscillation 
phenomenon. Note, for instance, that the two free pa- 
rameters ,z, and Am appear in different elements of 
the evolution matrix ( 14). Therefore, differently from 
what happens in the pure oscillation phenomenon, 
where the corresponding free parameters, vacuum 
mixing angle and squared mass difference appear, in 
the same elements in the evolution matrix, fitting fu- 
ture experimental data using the expected theoretical 
probability equations will allow precise determination 
of Am and the value of the average of the product 
,u~ 1 BI I. Together with other possible model indepen- 
dent analyses we will be able to use solar neutrino 
data as a source of information about the physics of 
neutrinos as well as the inner part of the Sun. 
The authors would like to thank Funda@o de Am- 
paro a Pesquisa do Estado de Sao Paul0 (FAPESP) 
and Conselho National de Desenvolvimento Cientifi- 
co e Tecnologico (CNPq) for several financial sup- 
ports. 
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Slow Solar Magnetosonic Waves And Time Variation In The Solar Neutrino Data

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Slow Solar Magnetosonic Waves And Time Variation In The Solar Neutrino Data

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