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\begin{document}

\title[Matter profile effect in a neutrino factory]{
Matter profile effect in a neutrino factory:\\
Yet another correlation in the analysis on CP violation
at a neutrino oscillation experiment}

\author{Toshihiko Ota\footnote[1]{e-mail:toshi@higgs.phys.kyushu-u.ac.jp}}

\address{Department of Physics, Kyushu University,\\ 
         Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan}


\begin{abstract}
 We investigate the effect induced by the variation of the density
profile of the Earth's interior in a long baseline neutrino oscillation
experiment.
 
 We show that there is a strong correlation between the average
density value and the first Fourier coefficient of the matter profile
function in the analyses on the oscillation probability.
 We also point out two facts,
 the most essential part of the matter profile is the
first Fourier mode of the matter profile function, 
 the Earth models based on the study of the seismology include large
uncertainty for the first Fourier mode.
 This means that the matter profile effect induces extra uncertainty for
average matter parameter. 

 Taking into account this extra uncertainty, we make numerical
calculations for the sensitivity to CP violation search.  
 The results show that the CP sensitivity is impaired in large $U_{e3}$ 
region.
\end{abstract}
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%13.15.+g,   % neutrino interactions
%14.60.Lm,   % ordinary neutrino (e, mu, tau)
%14.60.Pq,   % neutrino mass and mixing
%91.35.$-$x  % Earth's interior structure and properties
%}
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\section{Correlation between the average matter and the matter profile}
To pick out the most essential part of the effect induced by the 
matter profile, 
we separate the matter effect into two parts, the average matter effect,
$a_{0}$, and the remaining part, $\delta a (x)$, which depends on the 
position $x$. After that, we expand the matter profile part into the
Fourier series:
\begin{eqnarray}
\delta a (x) = 
 \sum_{\begin{minipage}{1.2cm} 
         {\footnotesize $ n = - \infty \vspace{-0.1cm}\\ 
         \hspace*{0.2cm} 
           n \ne 0 $}
       \end{minipage}}^{\infty} 
       a_{n} e^{-i p_{n} x},
\qquad
p_{n} \equiv \frac{2 \pi}{L} n .
\end{eqnarray}  
We establish the existence of the correlation between the parameter for
the average matter density, $a_{0}$, and the first Fourier coefficient of the
matter profile function, $a_{1}$.
In a high energy experiment, only a first few Fourier modes are
essential.
This fact can be understood by the resonance conditions \cite{OS}, 
and this is consistent with the research using different method \cite{Winter}.


Here, we present one example to understand intuitively this correlation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Fig: oscillation probability                   %
%            ``PREM vs constant''                %
%                                in L = 7,332 km %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[th]
\unitlength=1cm
\hspace*{1cm}
\begin{picture}(10,7)
\includegraphics[width=10cm]{PREM-vs-a0-and-a1-dm31-25e-4.epsi}
\put(0,0.5){\LARGE $E$[GeV]}
\put(-11,2.5){\rotatebox{90}{\LARGE $P_{\nu_{e} \rightarrow \nu_{\mu}}$}}
\put(-5.8,5){constant}
\put(-5.5,4.6){profile}
\put(-7,4.9){\begin{minipage}{1cm}
            \begin{eqnarray}
            \left\{ 
              \begin{array}{c}
                     \\
                     \\
                     \\
                     \\
              \end{array} 
            \right. \nonumber
            \end{eqnarray}
           \end{minipage}}
\put(-4.2,5.5){$\rho_{0} = 4.2{\rm[g/cm^{3}]}$}
\put(-4.2,4.9){$\rho_{0} = 4.4{\rm[g/cm^{3}]}$}
\put(-4.2,4.3){$\rho_{0} = 4.6{\rm[g/cm^{3}]}$}
\put(-4.2,3.6){up-to $a_{1}$ profile}
\put(-4.6,3){full-PREM profile}
\end{picture}
\caption{% 
Oscillation probability for $\nu_{e} \rightarrow \nu_{\mu}$ in $L=7,332$
km.
The dotted, dash, dash-dot lines are calculated using the constant
 matter profile whose values of the matter parameter are 4.2, 4.4, 4.6 
$\rm{g/cm^3}$, respectively.  
The dash-dot-dot line is adopted the up-to-first mode profile whose
values are $\rho_0=4.2$ $\rm{g/cm^3}$ and $\rho_1=-0.32$ $\rm{g/cm^3}$.
The solid line is used the full-PREM matter profile.
We set the oscillation parameters as $\sin \omega = 0.5$, $\sin \psi =1/\sqrt{2}$,
 $\sin \phi = 0.1$, $\Delta m_{31}^{2} = 2.5 \times 10^{-3}{\rm eV}^{2}$, 
$\Delta m_{21}^{2} = 5.0 \times 10^{-5} {\rm eV}^{2}$,
$\delta = \pi / 2$.}
\label{Fig:oscProb-PREM-vs-a0}
\end{figure}
The oscillation probabilities for $\nu_{e} \rightarrow \nu_{\mu}$ with
the different methods to deal with the matter effect are shown 
in Fig.\ref{Fig:oscProb-PREM-vs-a0}, where the baseline length is 7,332 km. 
 This figure shows that the constant profile with the average density
following PREM (dotted line) cannot reproduce the behavior of the
oscillation probability with the full-PREM profile (solid line).  
It reveal that if the shift of the constant value is allowed, 
then a good fitting with the full-PREM calculation can be realized at 
$\rho_{0} = 4.4$ $\rm{g/cm^{3}}$.  
This fact means that the shift of the constant matter parameter can 
pretend the matter profile effect that is almost equal to the effect 
induced by the first Fourier modes in this baseline length,
which can be understood in the comparison of the dash-dot-dot line with 
solid line.  

In Ref.\cite{YetAnother}, 
we show the existence of this correlation analytically 
and conclude that there is a strong correlation between $a_{0}$ and
$a_{1}$ in the case that the baseline length is less than 7,500 km. 
We also show that 
the reasonable error assumed by the seismological Earth models can
induce huge, a few hundred percent, uncertainty of $a_{1}$. 
This correlation suggests that the uncertainty 
in $a_1$ gives an extra uncertainty to $a_{0}$,
so the uncertainty of $a_{1}$ affects the CP sensitivity.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Source of an extra uncertainty for the matter effect}
Figure \ref{Fig:CPsensitivityPlot} indicates the lower boundary of 
$\Delta m_{21}^{2}$ and $|U_{e3}|$ $(=\sin \phi)$ to reject the
hypotheses that $\delta$ are $0$ and $\pi$ at 99\% level of significance 
when nature takes $\delta = \pi/2$ in the case that the baseline length
is 3,000 km and muon energy is 30 GeV.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Fig: region which the correlation exists       %
%                 for neutrino and anti-neutrino %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[th]
\unitlength=1cm
\hspace*{1cm}
\begin{picture}(10,6.8)
 \put(-0.2,5.9){\Large$\Delta m_{21}^{2}{\rm [eV^{2}]}$}
 \put(10,1.2){\Large $\sin \phi$}
 \put(9.8,0.7){\Large $(=|U_{e3}|)$}
 \includegraphics[width=10cm]{Phi-dm21-3000km.epsi}
\end{picture}
\caption{%
Sensitivity reach for CP violation with $\delta = \pi/2$,
the baseline length 3,000km, and the muon energy 30 GeV, 
where we assume
 $10^{21}$ muon decays and 40 kt detector.
The dotted curve is calculated using the up-to-fist-mode profile 
where the parameter uncertainty, $\Delta (a_{0}) = 3\%$, 
$\Delta (a_{1}) = 200\%$, $\sin \phi = 0 \sim 0.16$,
$\Delta(\sin \psi) = 1\%$, $\Delta (\Delta m_{31}^{2}) = 3 \%$, $\Delta (\sin \omega)= 5\%$, and $\Delta (\Delta m_{21}^{2})= 5\%$ are assumed.
The reference values for the oscillation parameters except $\Delta m_{21}^{2}$ and $\sin \phi$ are the same as Fig.\ref{Fig:oscProb-PREM-vs-a0}. 
The solid, dash-dot and dash curves are calculated using the
constant matter profile whose uncertainties are $\Delta a_{0} = 0\%, 3\%,$
and $5\%$, respectively and uncertainty for the other parameters is the same
as the dotted line.}
 \label{Fig:CPsensitivityPlot}
\end{figure}
Including the matter profile effect and its uncertainty induces an extra
uncertainty to the parameter of the average matter.
This fact can be explained by the correlation between $a_0$ and $a_1$.
This extra uncertainty gives rise to an extra
absorption of the signal of the CP violating effect.  
We know that the effect of the fake CP signal induced by the matter
effect becomes more serious as $|U_{e3}|$ is larger, so if $|U_{e3}|$ 
will be measured just under the current boundary, we will have to take 
the effect induced by the density profile of the Earth more seriously.
If we can lower the detection threshold the shorter baseline and the lower
energy option can avoid this disadvantage.
On the contrary, if small $|U_{e3}|$ will be established, 
the long baseline and high energy option is more advantageous because of
large statistics.


We would like to stress that the uncertainty of the matter effect is not
so small when we introduce the matter profile effect. 
Therefore, we should regard it more seriously whatever the statistics 
we use, especially in the case that the main part of the path of the
neutrino beam is the upper mantle and transition zone (up to 670 km in
depth) which include large uncertainty for the density profile.
We should deal with the matter effect much more cautiously.\\


%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{9}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{OS}
   T. Ota and J. Sato
      Phys. Rev. {\bf D63} (2001) 093004 [arXiv:hep-ph/0011234].

\bibitem{Winter}
   W. Winter, in these proceedings.

\bibitem{YetAnother}
   T. Ota and J. Sato, in preparation.
 
\end{thebibliography}

\end{document} 