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\title[Analytic Formulation of 
Neutrino Oscillation Probability in Constant Matter]
{Analytic Formulation 
of Neutrino Oscillation Probability in Constant Matter}

\author{Keiichi Kimura\dag\   
\footnote[3]{Talk presented by Keiichi Kimura}, 
Akira Takamura\dag\ddag \\and Hidekazu Yokomakura\dag}


\address{\dag\ Department of Physics, Nagoya University,
{\small \it Nagoya, 464-8602, Japan}}

\address{\ddag\ Department of Mathematics, 
Toyota National Collage of Technology, 
{\small \it Eisei-cho 2-1, Toyota-shi, 471-8525, Japan}}

\begin{abstract}
In this talk, based on the work \cite{Kimura}, 
we present the simple derivation of exact and analytic 
formula for neutrino oscillation probability. 
We consider three flavor neutrino oscillations 
in matter with constant density. 
\end{abstract}

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The information on the neutrino masses and 
the leptonic mixing angles has been accumulated gradually 
in recent neutrino experiments.
In this situation, Superbeam \cite{JHF} and  
Neutrino factory \cite{nu-factory} experiments 
using artificial neutrino beam 
are planned in near future to determine the unknown parameters. 
In particular, the measurement of the leptonic CP phase 
$\delta$ is the final goal of the neutrino experiments 
and many researches have been done \cite{CP}.
We here focus on the analytic formulation of 
neutrino oscillation probability 
in matter with constant density and the CP dependence of 
the probability. 

We take the channel for $\nu_e \to \nu_{\mu}$ 
transition, which is considered to be golden channel for the 
measurement of CP violation.
In vacuum, 
the oscillation probability is given by 
\begin{eqnarray}
P(\nu_e \to \nu_{\mu})
=-4\sum_{(ij)}^{{\rm cyclic}}{\rm Re}J_{e\mu}^{ij}
\sin^2 \Delta_{ij}^{\prime}
- 2\sum_{(ij)}^{{\rm cyclic}}J
\sin 2\Delta_{ij}^{\prime}
\label{1},
\end{eqnarray}
where 
\begin{eqnarray}
J_{e\mu}^{ij}\equiv U_{ei}U_{\mu i}^*
(U_{ej}U_{\mu j}^*)^*, \quad
J\equiv {\rm Im}\, J_{e\mu}^{12},\quad
\Delta_{ij}^{\prime}\equiv \frac{L}{4E}\Delta_{ij},
\label{2}
\end{eqnarray}
and $U$ is the Maki-Nakagawa-Sakata (MNS) matrix \cite{MNS}.
In matter, the Hamiltonian changes as  
\begin{eqnarray}
\tilde{H}=H+
\frac{1}{2E}
{\rm diag}(a, 0, 0),
\label{3}
\end{eqnarray}
where $a$ is the matter potential.
According to this change, 
the mass eigenvalues and the MNS matrix also change 
to the effective ones.
Their concrete expressions are calculated in 
\cite{Barger1, Zaglauer, Xing}.
However, it is difficult to simplify these expressions 
and understand the CP dependence of the probability exactly.

We here use the method by Naumov \cite{Naumov}, 
Harrison and Scott \cite{Harrison}.
In the method, they notice the invariance of 
the Hamiltonian 
except for $H_{ee}$.
They consider the product of off-diagonal elements 
of Hamiltonian as 
\begin{eqnarray}
{\rm Im}(\tilde{H}_{e\mu}\tilde{H}_{\mu\tau}\tilde{H}_{\tau e})
={\rm Im}(H_{e\mu}H_{\mu\tau}H_{\tau e}), 
\label{4}
\end{eqnarray}
and rewrite the Hamiltonian by using the relation 
\begin{eqnarray}
\tilde{H}=\tilde{U}{\rm diag}(\lambda_1, \lambda_2, 
\lambda_3)\tilde{U}^{\dagger}. 
\label{5}
\end{eqnarray}
As a result, they obtain a simple and elegant identity,
\begin{eqnarray}
\tilde{\Delta}_{12}\tilde{\Delta}_{23}\tilde{\Delta}_{31}
\tilde{J}=\Delta_{12}\Delta_{23}\Delta_{31}J. 
\label{6}
\end{eqnarray}
>From this identity, 
one can see that the matter effect is only included in the effective 
masses and CP phase is included in $J$.
Thus, $\tilde{J}$, which is the product of 
four $\tilde{U}_{\alpha i}$s,
has very simple form.
This suggests that the matter effect and the CP phase 
included in a $\tilde{U}_{\alpha i}$ partially cancel with those in 
other $\tilde{U}_{\beta j}$.
We show that ${\rm Re}\tilde{J}_{e\mu}^{ij}$ is also calculated 
as in the same way.

Here, remember the expression for the oscillation probability 
in (\ref{1})-(\ref{2}).
One can see that $\tilde{U}_{ei}$ always appears with 
$\tilde{U}_{\mu i}^*$.
Namely, we only have to calculate 
$\tilde{U}_{ei}\tilde{U}_{\mu i}^*$ not $\tilde{U}_{\alpha i}$ 
to obtain the oscillation probability.
We construct three identities on 
$\tilde{U}_{ei}\tilde{U}_{\mu i}^*$ by using the invariance 
of the Hamiltonian.
First one is obtained from the unitarity as 
\begin{eqnarray}
\tilde{U}_{e1}\tilde{U}_{\mu 1}^*
+\tilde{U}_{e2}\tilde{U}_{\mu 2}^*
+\tilde{U}_{e3}\tilde{U}_{\mu 3}^*=0. 
\label{7}
\end{eqnarray}
Second one is obtained from 
$\tilde{H}_{e\mu}=H_{e\mu}$. 
The Hamiltonian is rewritten by the MNS matrix from (\ref{5}) as 
in the previous case and we obtain 
\begin{eqnarray}
\lambda_1 \tilde{U}_{e1}\tilde{U}_{\mu 1}^*
+\lambda_2 \tilde{U}_{e2}\tilde{U}_{\mu 2}^*
+\lambda_3 \tilde{U}_{e3}\tilde{U}_{\mu 3}^*
=p,
\label{8}
\end{eqnarray}
where $p\equiv H_{e\mu}/(2E)$.
Third one is obtained from 
$\tilde{H}_{\tau \mu}\tilde{H}_{e\tau}
-\tilde{H}_{e\mu}\tilde{H}_{\tau \tau}
=H_{\tau \mu}H_{e\tau}-H_{e\mu}H_{\tau \tau}$ as
\begin{eqnarray}
\lambda_2 \lambda_3 \tilde{U}_{e1}\tilde{U}_{\mu 1}^*
+\lambda_3 \lambda_1 \tilde{U}_{e2}\tilde{U}_{\mu 2}^*
+\lambda_1 \lambda_2 \tilde{U}_{e3}\tilde{U}_{\mu 3}^*
=q,
\label{9}
\end{eqnarray}
where $q\equiv (H_{\tau \mu}H_{e\tau}-H_{e\mu}H_{\tau \tau})/(2E)^2 $.
It is noted that these are independent of the matter effects.
Regarding these three identities as the equations on 
$\tilde{U}_{ei}\tilde{U}_{\mu i}^*$, we simply obtain 
the expression \footnote{Similar expression is also seen 
in Ref. \cite{Ohlsson, Harrison2}} as 
\begin{eqnarray}
\tilde{U}_{ei}\tilde{U}_{\mu i}^*
=\frac{p\lambda_i+q}
{\tilde{\Delta}_{ji}\tilde{\Delta}_{ki}}.
\label{10}
\end{eqnarray}
Note that the matter effect is included only 
in the effective masses as in the case of $\tilde{J}$.
How about the CP phase ?
In the standard parametrization 
\begin{eqnarray}
\hspace{-0.5cm}
U=\left(
\begin{array}{ccc}
c_{13}c_{12} & c_{13}s_{12} & s_{13}e^{-i\delta} \\
-c_{23}s_{12}-s_{23}s_{13}c_{12}e^{i\delta}
& c_{23}c_{12}-s_{23}s_{13}s_{12}e^{i\delta}
& s_{23}c_{13} \\
s_{23}s_{12}-c_{23}s_{13}c_{12}e^{i\delta}
& -s_{23}c_{12}-c_{23}s_{13}s_{12}e^{i\delta}
& c_{23}c_{13}
\end{array}
\right),
\label{11}  
\end{eqnarray}
the CP phase is not included in the expressions for $\lambda_i$ 
at all \cite{Barger1}.
On the other hand, 
$p$ and $q$ are given by  
\begin{eqnarray}
p=\Delta_{21} U_{e2}U_{\mu 2}^*
+\Delta_{31} U_{e3}U_{\mu 3}^*, \quad 
q=\Delta_{21} \Delta_{31} U_{e1}U_{\mu 1}^*.
\label{12}
\end{eqnarray}
Substituting (\ref{12}) into (\ref{10}), we find that  
the oscillation probability has the form  
\begin{eqnarray}
P(\nu_e \to \nu_{\mu})
=A\cos \delta
+B\sin \delta+C 
\label{13}
\end{eqnarray}
with respect to $\delta$ \cite{Kimura}.
Note that the probability has 
only linear terms of $\cos\delta$ and $\sin\delta$ 
exactly\footnote{It is shown in Ref. \cite{Krastev} 
that the CP-odd part is linear in $\sin\delta$.}.
This is one of our main results.
We can further calculate  
$A$, $B$ and $C$ as 
\begin{eqnarray}
\hspace{-0.5cm}&&A=\sum_{(ijk)}^{{\rm cyclic}}\frac{-8J_r \Delta_{21}
[\Delta_{31}\lambda_k(\lambda_k-\Delta_{31})
+A_k]}
{\tilde{\Delta}_{jk}^2\tilde{\Delta}_{ki}^2}
\cos \tilde{\Delta}_{ij}^{\prime}
\sin \tilde{\Delta}_{jk}^{\prime}
\sin \tilde{\Delta}_{ki}^{\prime}, 
\label{14-1} \\
\hspace{-0.5cm}&&B=\frac{8\Delta_{12} \Delta_{23} \Delta_{31}}
{\tilde{\Delta}_{12} \tilde{\Delta}_{23}
\tilde{\Delta}_{31}} J_r
\sin\tilde{\Delta}_{12}^{\prime}
\sin\tilde{\Delta}_{23}^{\prime}
\sin\tilde{\Delta}_{31}^{\prime}, 
\label{14-2} \\
\hspace{-0.5cm}&&C=\sum_{(ij)}^{{\rm cyclic}}
\frac{-4[s_{13}^2 s_{23}^2 c_{13}^2
\Delta_{31}^2 \lambda_i \lambda_j
+C_{ij}]}
{\tilde{\Delta}_{ij} \tilde{\Delta}_{12}
\tilde{\Delta}_{23}\tilde{\Delta}_{31}}
\sin^2 \tilde{\Delta}_{ij}^{\prime}, 
\label{14-3}
\end{eqnarray}
where $A_k$ and $C_{ij}$ include the higher terms of 
$\Delta_{21}$ and can be neglected in the limit 
$\Delta_{21}\ll \Delta_{31}$. 
See \cite{Kimura} for their concrete expressions.

As one of the merit of our formula, one can simply reproduce the 
well-known approximate formulas given in \cite{Sato, Cervera} 
from (\ref{13})-(\ref{14-3}).
Another merit is that we can evaluate $\delta$ 
directly in terms of the probabilities.
The probability of anti-neutrino has also the same form as 
that of neutrino 
\begin{eqnarray}
P(\bar{\nu}_e \to \bar{\nu}_{\mu})
=\bar{A}\cos \delta
+\bar{B}\sin \delta+\bar{C}.
\label{15}
\end{eqnarray}
>From (\ref{13}) and (\ref{15}), 
we obtain the analytic expressions for $\delta$ as 
\begin{eqnarray}
\sin \delta&=&\frac{\{\bar{A} P(\nu_e \to \nu_{\mu})-A 
P(\bar{\nu}_e \to \bar{\nu}_{\mu})\}
-(\bar{A} C-A \bar{C})}{\bar{A} B-A \bar{B}}, 
\label{16} \\
\cos \delta&=&\frac{\{\bar{B} P(\nu_e \to \nu_{\mu})-
B P(\bar{\nu}_e \to \bar{\nu}_{\mu})\}
-(\bar{B} C-B \bar{C})}{\bar{B} A-B \bar{A}}. 
\label{17}
\end{eqnarray}
If the probabilities of neutrino and anti-neutrino are 
precisely determined by future experiments, 
we can calculate the value of $\delta$ analytically
by using (\ref{16})-(\ref{17})\footnote{
It is pointed out that there are ambiguities for 
the measurement of CP phase $\delta$ due to the eight-fold 
degeneracy of other parameters \cite{Burguet, Minakata, 
Fogli, Barger2}.
However, we do not discuss the problem here.}. 
Thus, we expect that our formula is useful to understand the 
features of neutrino physics and to obtain 
the information on the leptonic CP phase.

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\end{thebibliography}





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