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\begin{document}

\title{Oscillation-enhanced search for new interaction with neutrinos
}

\author{Joe  Sato
\footnote[3]{
 joe@phy.saitama-u.ac.jp}
}

\address{
Department of Physics, Faculty of Science
Saitama University,
Saitama, 338-8570, Japan}




\begin{abstract}
We study how well we can observe the effect of new physics in neutrino
 oscillation experiments.
\end{abstract}

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%\pacs{00.00, 20.00, 42.10}

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\section{Introduction}
Precision measurement on neutrino mass difference and mixing angles for
lepton
will be done in neutrino oscillation experiments of next
generation (see, for example, \cite{NextGeneration}).
According to those study we will know the mass square difference
and the mixing angle relevant to the atomospheric neutrino anomaly
with 3\% and 1\% accuracy, we can measure $U_{e3}$ down to $O(10^{-2})$,
and also we can observe the CP violation effect in near future. 

Till today the main concern about the next generation experiments has
been on these parameters, mixing angles and the mass square
difference. Therefore there arise a question whether we can mesure only
these parameters in those experiments. There are several suggestions
that we may see the effect of flavor-violating new physics\cite{NewPhys}.

In this talk we consider the potential power of the next generation
experiments to observe the effect of new interactions.
For the detail, see ref\cite{OSY}.


\section{Measurement of  Neutrino Oscillation}

All we know in a neutrino factory is that the muons, say, with negative
charge decay at an accumulate ring and wrong sign muons are observed in
a detector located at a length $L$ away just after the time $L/c$, where
$c$ is the light speed.  Therefore if we have sevaral contributions
which give the same signal, We have to
calculate the graph fig.\ref{fig:true} to get the transition rate.
\begin{figure}[h]
\center
\epsfig{file=figtrue.epsi,width=7.5cm}
\caption{Transition rate for ``$\bar\nu_e\rightarrow\bar\nu_\mu$''.}
\label{fig:true}
\end{figure}

Thus there is an interference phenomenon between several amplitudes
in this process.  Through this interference new physics effect gives
a contribution to the transition rate of order of not $O(\epsilon^2)$
but $O(\epsilon)$. We get an enhancement of the effect of
new physics,
that is, we can make oscillation-enhanced search for new
interactions with neutrinos. 


The amplitude for ``neutrino oscillation'' can be
divided into three pieces: 
(1) Amplitude relevant with decay of a 
parent particle denoted as  $A^C_\alpha$,
here $C$ describes the type of interactions. 
For $\mu$ decay, there are two types of interactions, $C=L,R$ 
while for $\pi$ decay we do not need this label.  
$\alpha$ distinguishes the
particle species which easily propagates in the matter
 and make an interaction at a detector. 
(2) Amplitude representing a transition of these propagating
particles, which are usually neutrinos, 
from one species $\alpha$ to another/same $\beta$
, denoted as  $T_{\alpha\beta}$. 
(3) Amplitude responsible for producing a charged lepton $\l$ 
from a propagated particle $\beta$ at a
detector, stood for by $D^I_{\beta l}$. Here $I$ denotes an interaction type.
Using these notations we get the probability to observe a charged lepton
$l^\pm$ at a detector as
\begin{eqnarray}
P_{\mu^{-}\rightarrow l^{+}(l^{-})} &=&
 \left|\sum_{\alpha\beta C I}A^C_\alpha
 T_{\alpha\beta} D_{\beta l^\pm}^I \right|^2.
\end{eqnarray}
Therefore we can consider the effect of new physics separately for
decay, propagation and detection processes.

In each process, we can paramterize new physics effects
as a ratio to the weak interaction, $\epsilon^{s,m,d}_{\alpha\beta}$,
where $s,m,d$ denote ``source''(decay), ``matter''(propagation),
and ``detection'' processes respectively.


\section{Analysis}

As an example of the sensitivity of nuetrino factory,
Fig. \ref{fig:nuenumu_EMu_U10} is shown.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Fig. nu_{e} to nu_{mu}, epsilon_{e mu}         %
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\begin{figure*}[th]
\unitlength=1cm
\begin{picture}(12,4)
\put(1.75,-0.4){$E_{\mu}$[GeV]}
\put(5.8,-0.4){$E_{\mu}$[GeV]}
\put(9.8,-0.4){$E_{\mu}$[GeV]}
\put(13.85,-0.4){$E_{\mu}$[GeV]}
\put(-0.5,1.3){\rotatebox{90}{$L$[km]}}
\includegraphics[width=16cm]{EtoMu-EpsilonEMu-U10.epsi}
\put(-13.8,3.1){\rotatebox{30}{1}}
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\put(-13,2){\rotatebox{25}{0.05}}
\put(-12.8,1.4){\rotatebox{25}{0.01}}
\color[rgb]{0,0,0}
\end{picture}
%\vspace{0.3cm}
\caption{ Contour plots of the required date size in unit
of $10^{21}$ decay muon $\times$ 100kt detector to observe 
the new physics effects concerning $\epsilon_{e \mu}^{s,m}$ 
at 90\% C.L.
in $\nu_{e} \rightarrow \nu_{\mu}$ channel 
when there is no uncertainty for theoretical parameters.
From left to right: 
$(\epsilon_{e \mu}^{s}, \epsilon_{e \mu}^{m}) = 
 (3.0 \times 10^{-3},   0)                   $, 
$(3.0 \times 10^{-3} i, 0)                   $, 
$(0,                    3.0 \times 10^{-3})  $, 
$(0,                    3.0 \times 10^{-3} i)$. 
Here each oscillation parameter is assumed to have
 10\% uncertainty.}
\label{fig:nuenumu_EMu_U10}
\end{figure*}


\section{Summary}

For a neutrino factory:
\begin{itemize}
 \item In $\nu_{\alpha} \rightarrow \nu_{\beta}$, $(\alpha=e, \mu,$
       $\beta=e, \mu, \tau)$ appearance channel, 
       the observable effects of new $(V-A)(V-A)$
       interactions come only  from $\epsilon_{\alpha \beta}^{s,m}$.
       The others are too small or too vulnerable against the 
       adjustment of the theoretical parameters.
       Note that $\delta$ and $\epsilon$'s phase are correlated.
       Namely the measured values are a certain combination of $\delta$
       and $\epsilon$.

 \item In $\nu_{\mu} \rightarrow \nu_{\mu}$ disappearance channel,
       we can measure $\epsilon_{\mu \tau}^{s,m}$ depending on their phase.
       In other words, the signal includes information of the phase.
       Furthermore, there is no correlation between $\delta$ and
       $\epsilon$, so the measurement tells us directly
       the phase concerning the lepton-flavor violating process.
       In $\nu_{e} \rightarrow \nu_{e}$ disappearance channel, we can
       not get anything for new interactions in the oscillation
       enhanced way.

 \item The $\chi^{2}$ is proportional to $|\epsilon|^{2}$.
       The expected sensitivity is to $|\epsilon| \ge \mathcal{O}
       (10^{-4})$ by using this methodology. 
 
 \item When the situations that new interactions exist not only
       in the source but also in the matter effect are considered, 
       we can easily understand the sensitivity by simply 
       adding each effect.

  \item Oscillation-enhanced effects for the $(V-A)(V+A)$ type interactions are
       strongly suppressed by $m_{e}/m_{\mu}$, 
       so we can not get an advantage over a direct measurement.
\end{itemize}

For an upgraded conventional beam:
\begin{itemize}
 \item We do not have to care the types of new interactions in the source.
       The analyses for the feasibility are similar to that 
       of $(V-A)(V-A)$ type for a neutrino factory.
       In the assumed energy and baseline region, there is no
       sensitivity to the new effect in matter.   

 \item The $\epsilon$'s for a conventional beam have  different dependence
       from those for a neutrino factory on new interactions. Therefore,
       the comparison between two methods makes clear the species of 
       new physics.

\end{itemize}


\begin{thebibliography}{9}
\bibitem{NextGeneration} JHF-Kamioka project, Y. Itow {\it et al.},
hep-ex/0106019.  Neutrino Factory and Muon Collider Collab., D. Ayres
{\it et al.}, physics/9911009;    C. Albright {\it et al.},
     hep-ex/0008064;

\bibitem{NewPhys} M.C. Gonzalez-Garcia, Y. Grossman, A. Gusso, and
Y. Nir, Phys.  Rev.  {\bf D64} (2001) 096006 [hep-ph/0105159];
A. M. Gago, M. M. Guzzo, H. Nunokawa, W. J. C. Teves, and R. Zukanovich
Funchal Phys.  Rev. {\bf D64} (2001) 073003 [hep-ph/0105196];
J. W. Valle, and P. Huber, hep-ph/0108193.

\bibitem{OSY} T. Ota, J. Sato, and N. Yamashita,
Phys. Rev. {\bf D65} (2002) 093015
 []hep-ph/0112329].


\end{thebibliography}


\end{document}

