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\newcommand{\CL} {C.L.}
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\newcommand{\eVq}{\text{eV}^2}
\newcommand{\Sol}  {\mathsc{sol}}
\newcommand{\Atm}  {\mathsc{atm}}
\newcommand{\Chooz}{\mathsc{chooz}}
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\newcommand{\Dma}{\Delta m^2_{\Atm}}
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\newcommand{\AHEP}{Instituto de F\'{\i}sica Corpuscular --
  C.S.I.C./Universitat de Val{\`e}ncia \\
  Edificio Institutos de Paterna, Apt 22085,
  E--46071 Val{\`e}ncia, Spain\\}
\newcommand{\IZMIRAN}{The Institute of Terrestrial Magnetism, 
Ionosphere and \\  
Radio Wave Propagation of the Russian Academy of Sciences,\\
  IZMIRAN, Troitsk, Moscow region, 142190, Russia\\}
\begin{document}
\preprint{IFIC/02-37}
%
\title{Large mixing angle oscillations as a probe of the deep solar interior}
%
\author{C. Burgess}
\email{cliff@hep.physics.mcgill.ca}
\affiliation{Physics Department, McGill University,\\
               3600 University Street;
               Montr\'eal, Qu\'ebec, Canada, H3A 2T8\\}

 \author{N. S. Dzhalilov}
\affiliation{\IZMIRAN} 
% 
\author{M. Maltoni}
\affiliation{\AHEP}
%
\author{T. I. Rashba}
\affiliation{\AHEP} \affiliation{\IZMIRAN}
%
\author{V. B. Semikoz}
\email{semikoz@ific.uv.es}
\affiliation{\AHEP} \affiliation{\IZMIRAN}
%
\author{M. Tortola}
\affiliation{\AHEP}
%
\author{ J.~W.~F. Valle}
\email{http://alpha.ific.uv.es/~ahep/}
\affiliation{\AHEP}

%\date{DDMM2002}
\date{\today}

\begin{abstract}

  We re-examine the sensitivity of solar neutrino oscillations to
  noise in the solar interior using the best current estimates
  of neutrino properties. Our results show that the measurement of
  neutrino properties at KamLAND can provide new information
  about fluctuations in the solar environment on scales to which
  standard helioseismic constraints are largely insensitive. We
  argue that a resonance between helioseismic and Alfv\'en waves
  might provide a physical mechanism for generating these
  fluctuations and, if so, neutrino-oscillation measurements could
  be used to constrain the size of magnetic fields deep within
  the solar radiative zone.

\end{abstract}
\maketitle

%\section{Introduction}

Current solar~\cite{sol02,Ahmad:2002jz,Fukuda:2002pe} and
atmospheric~\cite{atm02,Fukuda:1998mi} neutrino data give compelling
evidence that neutrino conversions take place. For the simplest case
of oscillations, the relevant parameters are two mass-squared
differences $\Delta m^2_{\mathrm{sol}}$ and $\Delta
m^2_{\mathrm{atm}}$, three angles $\theta_{12}$, $\theta_{23}$,
$\theta_{13}$ plus a number of CP violating
phases~\cite{Schechter:1980gr}.
%
One knows fairly well now that $\theta_{23}$ is nearly maximal (from
atmospheric data) and that the preferred solar solution for
$\theta_{12}$ is the so-called large mixing angle (LMA)
solution~\cite{Gonzalez-Garcia:2000aj}, while the third angle
$\theta_{13}$ is strongly constrained by the result of reactor
experiments~\cite{Apollonio:1999ae}. The CP phases are completely
unknown at present.
%
A recent analysis of solar and atmospheric data in terms of
neutrino oscillations is given in ref.~\cite{Maltoni:2002ni}, and
finds the currently-favored LMA solution of the solar neutrino
problem has
%
\begin{equation}
  \label{eq:lmabfp}
   \tan^2\theta_{\Sol} = 0.44\,, \quad
    \Dms = 6.6\times 10^{-5}~\eVq
\end{equation}
   %
and corresponds to oscillations into active neutrinos.

The upcoming KamLAND experiment \cite{kamland} is likely to bring
neutrino physics to a new level, by probing neutrino oscillations
using terrestrial neutrino sources. This will be revolutionary for two
reasons. First, it will decisively distinguish among the various
proposed solutions of the solar neutrino problem, such as the
possibility of neutrino
spin-flavor-precession~\cite{Schechter:1981hw}, or non-standard
neutrino matter interactions~\cite{Guzzo:2001mi}, which may arise in
models of neutrino mass~\cite{NSImodels}. Second, it will bring to
fruition one of the initial motivations for studying solar neutrinos
in the first place \cite{Bahcall}: the use of solar neutrinos to infer
the equilibrium properties of the solar core.

In this article we make the following two points:

$\bullet$
%
We show that if, as seems likely, the KamLAND results largely
support LMA neutrino oscillations, then the comparison between
KamLAND and solar neutrino experiments will provide new
information about fluctuations in the solar core on much shorter
scales than those which existing constraints (like
helioseismology) can presently probe.

$\bullet$
%
We propose a physical process which could arise in the core, that
may produce fluctuations on the scales to which solar neutrinos
are sensitive.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{3mm}
%
\noindent{\em Sensitivity to Fluctuations:}
%
The evolution of solar neutrinos in the presence of a fluctuating
solar matter density has been considered previously in
ref.~\cite{Balantekin:1996pp,Bamert:1997jj,Nunokawa:1996qu,Burgess:1996}.
In a nutshell, these studies show that neutrino oscillations in the
Sun can be influenced by density fluctuations provided that two
conditions are satisfied {\em at the position of the
  MSW~\cite{Wolfenstein:1977ue} oscillation:} ($i$) The fluctuation's
correlation length, $L_0$, is comparable to the local neutrino
oscillation length, $L_{\rm osc} \sim 100$ km and, ($ii$) the
fluctuation's amplitude, $\xi$, is at least roughly 1\%.  These
conclusions are summarized in Fig.~\ref{fig:noisyLMA}, where we show
how the electron-neutrino survival probability depends on the
fluctuation amplitude, $\xi$, given optimal choices for $L_0 = 100$ km
and neutrino oscillation parameters fixed at the best fit in
eq.~(\ref{eq:lmabfp}).
%
 \begin{figure}[htb!]
  \begin{center}
 \includegraphics[width=0.49\textwidth,height=3.8cm]{penergy.eps}
    \caption{Effect of random matter density perturbations on the
      LMA solution. The left panel is noiseless case, middle and right
      panels correspond to $\xi=4 \%$ and $\xi=8 \%$, respectively.}
\vglue-.5cm
    \label{fig:noisyLMA}
  \end{center}
\end{figure}

These early estimates can now be sharpened in view of our better
understanding of neutrino-oscillation parameters. To illustrate this
we have performed a global analysis of the solar data, including
radiochemical experiments (Chlorine, Gallex-GNO and SAGE) as well as
the latest SNO data in the form of 17 (day) + 17 (night) recoil energy
bins (which include CC, ES and NC contributions,
see~\cite{Maltoni:2002ni})~\cite{Ahmad:2002jz} and the
Super-Kamiokande spectra in the form of 44 bins~\cite{Fukuda:2002pe}.
%
The sensitivity of the neutrino signal to the level of solar density
fluctuations is shown in Fig.~\ref{fig:chi2fit}. Fixing the
oscillation parameters at the current best-fit LMA point given in
eq.~(\ref{eq:lmabfp}), leads to the bound shown in the left panel.  The
three curves denote the bounds on $\xi$ as a function of the
correlation length $L_0$ at 90\% C.L. (solid line), 95\% C.L. (dashed)
and 99\% C.L.  (dash-dotted).  In contrast, taking into account only
that the current neutrino data favor LMA oscillations (i.~e. leaving
the neutrino oscillation parameters vary freely inside this region)
leads to much weaker bounds on the noise, indicated in the right panel
of Fig.~\ref{fig:chi2fit}.
%
 \begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=.49\textwidth,height=5cm]{chi2bf.eps}
    \caption{Sensitivity of present  solar neutrino data to the
      ``solar noise level'' $\xi$ as a function of the correlation
      length $L_0$, assuming neutrino oscillation parameters fixed at
      the present best fit (left panel) and varied inside the current
      LMA region (right panel).}
    \label{fig:chi2fit}
  \end{center}
\end{figure}
%
Comparing the two panels one sees the importance of a precise
determination of the neutrino oscillation parameters in obtaining a
constraint on the magnitude of fluctuations.
%
This highlights the importance of a better determination of the solar
neutrino oscillation parameters at KamLAND (the collaboration aims at
a precision of 10 \% in $\Dms$ and 30 \% in
$\tan^2\theta_\Sol$~\cite{kamland}).  This will directly translate to
an enhanced sensitivity to solar density fluctuations with respect to
the current one (right panel in Fig.~\ref{fig:chi2fit}).

\vspace{3mm}

Two objections were believed to limit this kind of analysis.  First,
on the observational side, the success of helioseismology seemed to
preclude the existence of fluctuation amplitudes larger than 1\% in
size. Second, on the theoretical side, no known physics of the solar
core could generate fluctuations large enough to be detected. Of these
two, the first is the more serious, since our inability to guess a
source of fluctuations in such a complicated environment is much less
worrying than is a potential conflict with helioseismic data.
Nonetheless, in the remainder of this letter we argue why neither of
these objections can rule out the possibility of having large enough
density fluctuations without undergoing more careful scrutiny.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{3mm}
%
\noindent{\em Helioseismic Bounds:}
%
Helioseismology~\cite{Castellani:1997pk,Christensen-Dalsgaard:2002ur}
is rightfully celebrated as a precision tool for studying the
inner properties of the Sun. Careful measurements have provided
precise frequencies for numerous oscillation modes, and these may
be compared with calculations of these frequencies given assumed
density and temperature profiles for the solar interior.
Constraints on solar properties arise because careful comparison
between theory and measurements gives agreement only if the
assumed profiles are within roughly 1\% of the predictions of the
best solar models.

For the present purposes, the weak link in this train of argument lies
in the details of the inversion process which obtains the solar
density profile given an observed spectrum of helioseismic
frequencies. This inversion is only possible if certain smoothness
assumptions are made about solar properties, due to the inevitable
uncertainties which arise in the observed solar helioseismic
oscillation pattern.  As a result helioseismology severely constrains
the existence of density fluctuations, but only those which vary over
very long scales $\gg$ 1000 km
\cite{Castellani:1997pk,Christensen-Dalsgaard:2002ur}.  In particular,
the measured spectrum of helioseismic waves is largely insensitive to
the existence of density variations whose wavelength is short enough
-- on scales close to $L_{\rm osc} \sim 100$ km, deep within the
solar core -- to be of interest for neutrino oscillations. In
particular, we claim that such density variations with amplitudes as
large as 10\% cannot yet be ruled out.

\vspace{3mm}
%
\noindent{\em  Fluctuation Mechanism:}
%
A mechanism which might produce density variations of the required
amplitude and correlation length arises once helioseismology is
reconsidered in the presence of magnetic fields, which are normally
neglected in helioseismic analyses \cite{Couvidat:2002bs}.  Generally,
the neglect of magnetic fields in helioseismology is very reasonable
since the expected magnetic field energy densities, ${\bf B}^2/8\pi$,
are much smaller than are gas pressures and other relevant energies.

In \cite{newhelio} we study helioseismic waves in the Sun, and find,
contrary to naive expectations, that reasonable magnetic fields in the
radiative zone \cite{Parker} can appreciably affect the profiles of
helioseismic $g$-modes as a function of solar depth. In particular,
density profiles due to these waves tend to form comparatively narrow
spikes at specific radii within the Sun, corresponding to radii where
the frequencies of magnetic Alfv\'en modes cross those of
buoyancy-driven ($g$-type) gravity modes. Due to this resonance,
energy initially in $g$-modes is directly pumped into the Alfv\'en
waves, causing an amplification of the density profiles in the
vicinity of the resonant radius. This amplification continues until it
is balanced by dissipation, resulting in an unexpectedly large density
variation at the resonant radii. Furthermore, these level crossings
only occur with $g$-modes, and so typically occur deep within the
solar radiative zone. They also do not affect substantially the
observed $p$-modes, which makes it unlikely that these resonances
alter standard analyses of helioseismic data (which ignore solar
magnetic fields), in any significant way.
%
 \begin{figure}[ht]
  \begin{center}
    \includegraphics[height=5cm,width=.48\textwidth]{rsR.eps}
    \caption{The positions of Alfv\'en/$g$-mode resonances as 
      a function of mode number, for various magnetic field
      strengths.}
    \label{fig:zepsR}
  \end{center}
\end{figure}
%

Fig.~\ref{fig:zepsR} plots the positions of the Alfv\'en/$g$-mode
resonant layers as a function of an integer mode label $n$, for
various values of a hypothetical magnetic field. What this figure
shows is that there are very many such level crossings, whose
position varies most quickly with radius within the Sun near the
solar centre. The superposition of several different modes results
in a series of relatively sharp spikes in the radial density
profile at the radii where these resonances take place. From the
point of view of exiting neutrinos, passage through these
successive helioseismic resonances mimics the passage through a
noisy environment whose correlation length is the spacing between
the density spikes.

In \cite{newhelio} we make several estimates of the size of these
waves in the Sun. Fig.~\ref{fig:corr} shows how the spacing between
the spikes varies as a function of their position within the Sun.
Because the spikes are quite narrow, we find the energy cost of
producing them with amplitudes as large as 10\% can be much less than
the prohibitively large values which were required to obtain similar
amplitudes for g-waves alone~\cite{Bamert:1997jj}.  It is remarkable
that the position at which this spacing is close to 100 km lies near
$r = 0.12 \; R_\odot$ for a very wide range of magnetic fields. On the
other hand, we have observed~\cite{newhelio} that, for magnetic fields
which are of order 10 kG, the spacing of resonances is near 100 km for
a very wide range of radii -- including the neutrino resonance region,
$r \sim 0.3 \; R_\odot$.

Are such large radiative-zone magnetic fields possible? Very little is
directly known about magnetic field strengths within the radiative
zone. The only generally-applicable bound there is due to
Chandrasekar, and states that the magnetic field energy must be less
than the gravitational binding energy: $B^2/8\pi <
GM^2_\odot/R_\odot^4$, or $B < 10^8$ G. A stronger bound is also
possible if one assumes the solar magnetic field to be a relic of the
primordial field of the collapsing gas cloud from which the Sun
formed. In this case it has been argued that central fields cannot
exceed around 30 G \cite{Parker}. (Still stronger limits, $B <
10^{-3}$ G, are possible \cite{MestelWeiss} if the solar core should
be rapidly rotating, as is sometimes proposed.) Since the initial
origin of the central magnetic field is unclear, we believe any
magnetic field up to the Chandrasekar bound should be entertained.

%

 \begin{figure}[ht]
  \begin{center}
    \includegraphics[width=0.48\textwidth]{rs-rs-small.eps}
    \caption{Solar density correlation length, versus
      distance from solar center, for different magnetic fields.}
    \label{fig:corr}
  \end{center}
\end{figure}
%

The above mechanism has many of the features required to produce
density fluctuations in the Sun which may be relevant to neutrino
oscillations. Although much more study is required to establish
whether these resonances really occur and affect neutrinos, we believe
their potential existence substantially reinforces the general
motivation for using neutrino oscillations to directly probe
short-wavelength density fluctuations deep within the solar core.

\vspace{3mm}

This work was supported by Spanish grants BFM2002-00345, by the
European Commission RTN network HPRN-CT-2000-00148, by the European
Science Foundation network grant N.~86, by Iberdrola Foundation (VBS)
and by INTAS grant YSF 2001/2-148 and CSIC-RAS agreement (TIR). C.B.'s
research is supported by grants from NSERC (Canada) and FCAR (Quebec).
M.M.\ is supported by contract HPMF-CT-2000-01008.  VBS, NSD and TIR
were partially supported by the RFBR grant 00-02-16271.

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\end{document}
































