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\begin{document}
    
    \title[]{Lepton flavour violating $\mu$-e conversion rate for various nuclei}
    
    \author{%
    Ryuichiro Kitano\dag\ddag\footnote[3]{%
    Present address:
    Institute for Advanced Study,
    Princeton,
    NJ 08540,
    USA
    }, 
    Masafumi Koike\dag 
    and
    Yasuhiro Okada\dag\ddag
    }
    
    \address{\dag\ %
    Theory Group,
    KEK,
    1-1 Oho, Tsukuba, Ibaraki 305-0801,
    Japan}
    \address{\ddag\ %
    Department of Particle and Nuclear Physics,
    The Graduate University for Advanced Studies,
    1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
    }
    
%     \eads{mkoike@post.kek.jp}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
    \begin{abstract}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	The coherent $\mu$-e conversion rate for general LFV
	interactions for various nuclei is calculated.
	%
	The branching ratio increases for the light nuclei such as $Z
	\lesssim 30$, are largest for $Z = 30$ -- $60$, and decreases
	for heavy nuclei with $Z \gtrsim 60$.  Although this tendency
	of $Z$ dependence is the same for different types of coupling
	constants, there are significant differences in $Z$ dependence
	of branching ratios.
	%
	The atomic number dependence of the conversion ratio
	calculated here is useful to distinguish theoretical models
	with lepton flavor violation.
	
    \end{abstract}
    
    %Uncomment for PACS numbers title message
    %\pacs{11.30.Fs, 14.60.Ef}
    
    % Uncomment for Submitted to journal title message
    %\submitto{\jpg}
    
    % Comment out if separate title page not required
    %\maketitle
    
    \section{Introduction}
    %
    The search for the lepton flavour violation (LFV) is one of the
    most interesting subjects in particle physics.
    %
    The $\mu$-e conversion in muonic atoms, or neutrinoless muon
    capture, is an expected signal of the LFV to be observed.
    %
    The $\mu$-e conversion rate has been calculated by several authors
    \cite{theory-coherent,theory-incoherent}, and its experimental
    upper bound is given to be $6.1 \times 10^{-13}$ \cite{SINDRUMII}.
    %
    Further experiments are ongoing and planned to improve the bound
    down to $10^{-17} - 10^{-18}$ \cite{MECO,PRISM}.
    %
    The PRISM project \cite{PRISM} in the JAERI-KEK joint project is
    of particular interest among them since it is planned to make use
    of a muon storage ring, the muon beam from which is free from
    contamination of electrons and pions.
    %
    It will then be possible to carry out the experiments with various
    target atom, and thus to optimize the target atom for the search.
    %
    We thereby evaluate the $\mu$-e conversion rate for nuclei of a
    wide range of atomic numbers.
    %
    The atomic number dependence of the conversion ratio calculated
    here turns out to be useful to distinguish theoretical models with
    LFV.
    %
    More details are found in the reference \cite{Kitano:2002mt}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
    \section{Formula of $\mu$-e conversion rate}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    It is necessary to specify the interaction Lagrangian and the
    initial and final states in order to calculate the $\mu$-e
    conversion amplitudes and rates.
    %
    The LFV interaction Lagrangian is taken to be most general form
    which contributes to the $\mu$-e conversion in nuclei
    \cite{Kuno:1999jp}.
    %
    It consists of photonic interaction terms and four-Fermi
    interaction terms, and the magnitude of coupling constant in each
    term depends on the interaction of the new physics in which the
    lepton flavour conservation is violated.
    %
    The initial state in the $\mu$-e conversion process is the 1s
    state of the muonic atom, and the final electron state is the
    eigenstate with an energy of $m_\mu - \varepsilon_b$, where
    $\varepsilon_b$ is the binding energy of the 1s state of muonic
    atom.
    %
    The final state of the nucleus is taken to be same as the initial
    state, which is referred to as the coherent process; the fraction
    of this process is generally larger than incoherent one
    approximately by a factor of the mass number of the target nuclei.
    
    
    Then the formula of the $\mu$-e conversion rate can be written in
    terms of the five overlap integrals $D, S^{\mathrm{(p, n)}}$ and
    $V^{\mathrm{(p, n)}}$ as
    %
    \begin{eqnarray}
	\omega_{\mathrm{conv}}
	& = &
	2 G_{\mathrm{F}}^2
	\left|
	A_{\mathrm{R}}^{\ast} D
	+ \tilde{g}_{\mathrm{LS}}^{\mathrm{(p)}} S^{\mathrm{(p)}}
	+ \tilde{g}_{\mathrm{LS}}^{\mathrm{(n)}} S^{\mathrm{(n)}}
	+ \tilde{g}_{\mathrm{LV}}^{\mathrm{(p)}} V^{\mathrm{(p)}}
	+ \tilde{g}_{\mathrm{LV}}^{\mathrm{(n)}} V^{\mathrm{(n)}}
	\right|^2
	\nonumber \\
	& + &
	2 G_{\mathrm{F}}^2
	\left|
	A_{\mathrm{L}}^{\ast}  D
	+ \tilde{g}_{\mathrm{RS}}^{\mathrm{(p)}} S^{\mathrm{(p)}}
	+ \tilde{g}_{\mathrm{RS}}^{\mathrm{(n)}} S^{\mathrm{(n)}}
	+ \tilde{g}_{\mathrm{RV}}^{\mathrm{(p)}} V^{\mathrm{(p)}}
	+ \tilde{g}_{\mathrm{RV}}^{\mathrm{(n)}} V^{\mathrm{(n)}}
	\right|^2.
	\label{eq:omega-final}
    \end{eqnarray}
    %
    Here $G_{\mathrm{F}}$ is the Fermi constant, and $A_{ \mathrm{R,
    L} }^{\ast}$ and $\tilde{g}_{\mathrm{LS, LV, RS, RV}}^{\mathrm{(p,
    n)}}$ are the coupling constants relevant to the LFV interaction.
    %
    The overlap integrals are defined by
    %
    \begin{eqnarray}
	D
	= 
	\frac{4}{\sqrt{2}} m_\mu 
	\int_{0}^{\infty} \mathrm{d} r r^{2}
	[ - E(r) ]
	( g_{ \mathrm{e} }^{-} f_{\mu}^{-} + f_{ \mathrm{e} }^{-} g_{\mu}^{-} ),
	\label{eq:D} \\
	%
	S^{\mathrm{(p)}}, V^{ \mathrm{(p)} } =
	\frac{1}{2 \sqrt{2}} 
	\int_{0}^{\infty} \mathrm{d} r r^{2}
	Z \rho^{ \mathrm{(p)} }
	( g_{ \mathrm{e} }^{-} g_{\mu}^{-} \mp f_{ \mathrm{e} }^{-} f_{\mu}^{-} ),
	\label{eq:SVp}
	\\
	S^{ \mathrm{(n)} }, V^{ \mathrm{(n)} } =
	\frac{1}{2 \sqrt{2}} 
	\int_{0}^{\infty} \mathrm{d} r r^{2}
	(A - Z) \rho^{\mathrm{(n)}}
	( g_{ \mathrm{e} }^{-} g_{\mu}^{-} \mp f_{ \mathrm{e} }^{-} f_{\mu}^{-} ),
	\label{eq:SVn}
    \end{eqnarray}
    %
    where %
    $m_{\mu}$ is the muon mass, %
    $E(r)$ is the electric field in nuclei, %
    $\rho^{ \mathrm{(p, n)} }(r)$ is the normalized density
    distribution of the proton and neutron in the nuclei, and %
    $g_{ \mu, \mathrm{e} }^{-}(r)$ and $f_{ \mu, \mathrm{e} }^{-}(r)$
    are the radial part of the wave function of the muon and the
    electron.
    %
    The minus and plus sign in (\ref{eq:SVp}) and (\ref{eq:SVn})
    corresponds to $S^{ \mathrm{(p, n)} }$ and $V^{ \mathrm{(p, n)}
    }$, respectively.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
    \section{Numerical results}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    
    Figure \ref{fig}(\textit{a}) shows the five overlap integrals as
    functions of the atomic number $Z$.
    %
    \begin{figure}[t]
	\begin{center}
	    \epsfxsize=120mm
	    \epsfbox{koike-figure.eps} 
	\end{center}
	\caption{ \label{fig}%
	(\textit{a}): The overlap integrals are plotted as functions
	of the atomic number $Z$.  Neutron density distributions are
	assumed to be equal to charge density distributions.
	%
	(\textit{b}): The $\mu$-$e$ conversion ratios for typical
	theoretical models are plotted as functions of the atomic
	number $Z$.  The solid, the long dashed, and the dashed lines
	represent the cases where the photonic dipole, scalar, and
	vector operator dominates, respectively.  The proton and
	neutron distribution are taken to be the same, and the
	conversion ratios are normalized by the conversion ratio in
	aluminum nuclei ($Z = 13$).%
	}
    \end{figure}
    %
    Here the proton density is taken from the reference \cite{table},
    and the neutron density is assumed to be equal to the proton
    density.
    %
    This is not necessarily a good approximation to calculate $S^{
    \mathrm{(n)} }$ and $V^{ \mathrm{(n)} }$ for heavy nuclei, but the
    result turns out to be consistent within 10\% to the result
    obtained from the experimental neutron distribution data
    \cite{Kitano:2002mt}.
    %
    We see that the overlap integrals increase as functions of $Z$ for
    light nuclei up to $Z \sim 30$, and saturate or decrease for heavy
    nuclei.
    %
    Although the tendency is the same, each overlap integral has
    different $Z$ dependence, especially for heavy nuclei.
    
    The conversion rate $\omega_{ \mathrm{conv} }$ and the branching
    ratio $B_{ \mathrm{\mu N \to e N} } \equiv \omega_{ \mathrm{conv}
    } / \omega_ {\mathrm{capt} }$, where $\omega_{\mathrm{capt} }$ is
    the rate of the ordinary muon capture, can be straightforwardly
    calculated through equation (\ref{eq:omega-final}) once a
    theoretical model with LFV is specified.
    %
    We consider the following three cases: %
    (i) Only the photonic dipole LFV operator is relevant; %
    (ii) Only the scalar four-Fermi LFV operator is relevant; %
    (iii) Only the vector four-Fermi LFV coupling constants %
    $\tilde g_{ \mathrm{LV} }^{ \mathrm{(p)} }$ and/or %
    $\tilde g_{ \mathrm{RV} }^{ \mathrm{(p)} }$ are relevant.%
    %
    Figure \ref{fig}(\textit{b}) presents the $\mu$-e conversion
    branching ratio normalized by the value for aluminum ($Z = 13$).
    %
    We can see that, for all three types, the branching ratio
    increases with $Z$ for $Z \lesssim 30$, is largest for $30
    \lesssim Z \lesssim 60$, and decreases for $Z \gtrsim 60$.
    %
    It is also seen that the conversion ratios are very different in
    heavy nuclei depending on the type of interaction.
    %
    From this property we may be able to distinguish models beyond the
    Standard Model through several experiments with different targets.
    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
    \ack
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    The authors are grateful to J.~Hisano for useful discussions.
    %
    This work is supported by the JSPS Research Fellowships for Young
    Scientists (R.~K. and M.~K.).
    %
    The work of Y.~O. is supported in part by a Grant-in-Aid of the
    Ministry of Education, Culture, Sports, Science and Technology,
    Government of Japan (No.~13640309), priority area ``Supersymmetry
    and Unified Theory of Elementary Particles'' (No.~707).
    
    \section*{References}
    \begin{thebibliography}{9}
	%%%%%
	\bibitem{theory-coherent}
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	1959
	\textit{Phys. Rev. Lett} \textbf{3} 111;
	%%%%%
	Marciano W J and Sanda A I
	1977
	%``The Reaction Mu- Nucleus $\to$ E- Nucleus In Gauge Theories,''
	\textit{Phys. Rev. Lett.} \textbf{38} 1512;
	%%%%%
	Shanker O
	1979
	%``Z Dependence Of Coherent Mu E Conversion Rate In Anomalous 
	%Neutrinoless Muon Capture,''
	\textit{Phys. Rev.} D \textbf{20} 1608;
	%%%%%
	Czarnecki A \textit{et al.}
	%``Coherent muon electron conversion in muonic atoms,''
	\textit{Preprint} hep-ph/9801218;
	%%%%%
	Czarnecki A \textit{et al.}
	1998 
	\textit{Proc. of Workshop on Physics at the 1st Muon Collider
	and at the Front End of the Muon Collider, AIP Conf. Proc. No. 435}
	edited by S H Geer and R Raja %
	(New York: American Institute of Physics)
	p~409
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\bibitem{theory-incoherent}
	Chiang H C \textit{et al.}
	1993
	\textit{Nucl. Phys.} A \textbf{559} 526;
	%%%%%
	Kosmas T S \textit{et al.}
	1994
	\textit{Nucl. Phys.} A \textbf{570} 637;
	%%%%%
	Kosmas T S \textit{et al.}
	1996
	%``(mu-, e-) conversion: A Symbiosis of
	% particle and nuclear physics,''
	\textit{Phys. Rept.}  \textbf{264} 251;
	%%%%%
	Kosmas T S \textit{et al.}
	1998
	\textit{Phys. Atom. Nucl.}  \textbf{61} 1161;
	%%%%%
	Kosmas T S \textit{et al.}
	2000
	%``The inclusive mu - e conversion process with
	%renormalized  quasiparticle random-phase approximation,''
	\textit{Nucl. Phys.} A {\bf 665} 183;
	%%%%%
	Siiskonen T \textit{et al.}
	2000
	%``Realistic nuclear matrix elements for the lepton-flavor
	% violating mu- $\to$ e- conversion in AL-27 and Ti-48,''
	\textit{Phys. Rev.} C {\bf 62} 035502;
	%%%%%
	Kosmas T S
	2001
	%``On the nuclear dependence of the mu- e- conversion branching ratio,''
	\textit{Preprint} nucl-th/0108045
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\bibitem{SINDRUMII}
	Wintz P
	1998
	\textit{Proc. 1st Int. Symp. on Lepton and Baryon Number Violation}
	edited by H V Klapdor-Kleingrothaus and I V Krivosheina
	(Bristol/Philadelphia: Institute of Physics)
	p~534
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\bibitem{MECO}
	Bachman M {\it et al.} (MECO Collaboration)
	1997
	\textit{Experimental proposal E940 to Brookhaven National Laboratory
	Alternating Gradient Synchrotron}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\bibitem{PRISM}
	Kuno Y
	2000
	\textit{Proc. of The 2nd Int. Workshop on Neutrino Oscillations and their Origin}
	edited by Y Suzuki {\it et al.}
	(Singapore: World Scientific)
	p~253
	%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\cite{Kitano:2002mt}
	\bibitem{Kitano:2002mt}
	%	Kitano R, Koike M and Okada Y
	Kitano R \textit{et al.}
	2002
	%``Detailed calculation of lepton flavor violating muon electron
	%conversion rate for various nuclei,''
	\textit{Phys. Rev.} D \textbf{66} 096022
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\bibitem{Kuno:1999jp}
	For review, see
	Kuno Y and Okada Y
	2001
	%``Muon decay and physics beyond the standard model,''
	\textit{Rev. Mod. Phys.}  \textbf{73} 151
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\bibitem{table}
	De Jager C W \textit{et al.}
	1987
	%``Nuclear Charge And Magnetization Density Distribution Parameters
	%  From Elastic Electron Scattering,''
	\textit{Atom. Data Nucl. Data Tabl.} \textbf{36} 495;
	%%%%%
	Fricke G {\it et al.}
	1995
	\textit{Atom. Data Nucl. Data Tabl.} \textbf{60} 177
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \end{thebibliography}
    
\end{document}