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\begin{document}

\title{An Overview on  Muon Physics }

\classification{14.60.Ef }
\keywords      {Lepton Flavor Violation; Muon}

\author{Giancarlo Piredda}{
  address={INFN Sezione di Roma, Dipartimento di Fisica, Universit\'a Sapienza}
}



\begin{abstract}
 {A  summary of the present status of the charged lepton flavor violation search with muons is presented.  The relevant measurements  $(g-2)_{\mu}$,  the $\mu^+\rightarrow e^+ \gamma$ decay and the direct muon-to-electron conversion projects are briefly reviewed.}    

\end{abstract}

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\section{Introduction}
It would be very difficult to condense a review on Muon Physics
in a restricted time. Therefore I decided to review the role of the muon lepton in the quest for 
New Physics (NP).
The NP can be pursued either by studying carefully processes well predicted 
by the Standard Model (SM) or by searching for processes completely forbidden in the SM.
The $(g-2)_{\mu}$ precision measurement
belongs to the former category whereas the Flavor Violation in the charged lepton sector (cLFV)  belongs to the latter.
 In fact, even if the flavor mixing in the neutral sector has been established since one decade with the observation
 of  neutrino oscillations,
the corresponding branching ratio for the $\mu^+\rightarrow e^+ \gamma$ is in the order of $10^{-54}$ in the SM, i.e.
practically zero. 
%\begin{figure}[htbp]
\begin{figure} [! b]
\resizebox{1.0\columnwidth} {!}
%\begin{center}
{\includegraphics{Mu2egvsg-2.pdf}}
%\includegraphics[width=0.5\textwidth]{Mu2egvsg-2.pdf}
%\includegraphics{muegvsg-2}
\caption{BR($\mu^+\rightarrow e^+ \gamma$)  vs $ \Delta a_{\mu}$. The green lines indicate the one sigma range 
of the E821 experiment result. The red area represents the constraints coming from B decays.}
\label{fig:g-2}
 %\end{center}
\end{figure}
On the contrary, many models on the SUSY-GUT or SO(10) domain predict branching ratios in the order of $10^{-12}-10^{-13}$
and therefore improving on the upper limit (or even better detecting a signal) could be an unambiguous epiphany of NP. 
Moreover, there is a clear and very important correlation between the two processes, as shown in fig \ref {fig:g-2} taken 
 from the paper by  Isidori {\it et al.} \cite{Isidori}.
 The  interesting physics case is well established once one realizes that the
expected  $\mu^+\rightarrow e^+ \gamma$ branching ratio in correspondence of the best determination so far 
of $\Delta a_\mu$, the difference between the measured and the expected $(g-2)_{\mu}$ obtained by the  E821 experiment,  
is >  $\sim 0.2\cdot10^{-11}$, namely  almost  within reach! 
Indeed it looks like  great discoveries could be  around the corner.
\section{The $(g-2)_{\mu}$ projects}
What is the experimental situation? The best result, so far,  has been obtained by the E821 Collaboration \cite{E821} at Brookhaven 
at the beginning of the decade. In terms of $ \Delta a_{\mu}$, it reads 
$ \Delta a_{\mu}=(297\pm 79)10^{-11}$, corresponding to an experimental accuracy on $a_{\mu}$ of $0.54 ppm$. 
For the future, there is a proposal by almost the same (but enlarged) Collaboration  to move the E821 apparatus to Fermilab, 
in order to  exploit the more intense muon beam to reduce the error by a factor of 4 i.e. down to about
$0.1 ppm$.  Another Collaboration aims at a similar sensitivity with a low energy muon beam to be available in the J-Parc
complex.
\section{The search for the $\mu^+\rightarrow e^+ \gamma$ decay}
\par
Where do we stand with the search for the charged lepton flavor violation?
Great attention nowadays in the scientific community is focused at the MEG experiment \cite{Meg}.
MEG is currently taking data on the world's most intense dc muon beam at Paul Scherrer Institut (Switzerland).
  \begin{figure}[! b]
\resizebox{0.9\columnwidth}{!}
{\includegraphics{MEG_EV.pdf}}
\caption{Pictorial view of a 'signal-like' event acquired by the MEG detector. Top: a positron tracked in the
spectrometer hits two bars in the TC. Bottom: schematic of the energy released by the photon in the Liquid Xenon
Calorimeter}
\label{fig:picture}
\end{figure}
 Together with the world's largest liquid Xenon photon detector of $\sim$ 800 liter and a gradient-field superconducting positron spectrometer, MEG is able to distinguish the decay of a muon at rest into a photon and positron from the normal Michel decay and the prompt background due to the radiative muon decay. The experimental task is, in principle, straightforward as one has to detect a simultaneous
 back-to-back photon and positron pair of equal energy corresponding to half of the muon mass, as shown pictorially
 in fig. \ref{fig:picture}.
 The dominant background, however,   consists of accidental events made  of a positron coming from a Michel decay and a photon from a radiative decay, Bremsstrahlung or positron annihilation in flight. As a consequence, a detector is required with excellent spatial, time and energy resolution.
 The MEG apparatus, put in operation after a decade of intense $R\&D$ work, represents today the 
 best achievable technology to 
improve by two orders of magnitude the previous MEGA \cite{Mega} upper limit, published in the year 2000,
 Br ($\mu^+\rightarrow e^+ \gamma)<1.2 \cdot10^{-11}$. 
 The present preliminary MEG upper limit, at $90\%$ confidence level, is $1.5 \cdot 10^{-11}$, \cite{Nishiguchi} based on the 2009 data sample, which represents about $10\%$ of the total foreseen statistics. 
 It is worth noticing that the experiment sensitivity (estimated  in signal-free data samples) is $0.6\cdot10^{-11}$, raising the very interesting
 question about the true nature of the discrepancy.
 Anyway, thanks to a  long 2010  data taking, which  is supposed to at least double the present statistics ,
  we will  know the answer rather soon.
The MEG schedule is to continue data taking in 2011-2012 and  to improve the experimental resolutions in order to have the best achievable control on the accidental background in the attempt to reach a sensitivity of the order of $10^{-13}$.
\section{Direct muon-to-electron conversion}
  The neutrino-less muon capture by a nucleus with consequent emission of an electron 
  (otherwise known as {\it direct muon-to-electron conversion})  
  could represents a step forward in the long quest for the cLFV search.
   The conversion is not only complimentary to $\mu^+\rightarrow e^+ \gamma$  but, because it could proceed through alternative diagrams , 
   can  also give independent information about the cLFV nature.
  The technique, pioneered by the Sindrum-II Collaboration  \cite{sindrum}, basically consists in letting  an intense pure muon beam 
impinge on an appropriate thin target and see whether any isolated electron with an energy almost equal to the muon mass could be
detected. The challenge is the exploitation of an intense pulsed uncontaminated beam followed by a suitable apparatus able to efficiently detect the emerging monochromatic electron and separate it from the high energy tail of the abundant decay-in-orbit background.
Two major projects are at the moment under approval \cite{Kuno}, both aiming at a sensitivity of the order of $10^{-16}-10^{-17}$ in the
rate $\mu^-N\rightarrow e^- N$ with respect to the capture rate,
which corresponds to $10^{-14}-10^{-15}$ of the Br ($\mu^+\rightarrow e^+ \gamma$) scale.
\par
The two proposals are quite similar, although they differ concerning the beam transport system. In fact, the Fermilab Mu2e collaboration proposes the use of a straight solenoid with a gradient field, whereas the J-Parc based COMET collaboration intends to utilize a U-shape solenoid which allows a reduced trigger rate.
Both Collaborations plan  to take data  about five years from now. 
\section{Conclusions}
The cLFV field looks indeed very interesting. MEG will provide new results in the incoming
months, and  in the mid-term new g-2 determinations are expected. New 
 muon-to-electron conversion projects are in preparation which will confirm or disproof the previous
 results. Muons played a fundamental role in the construction of the SM. We believe that it
 could be the same in going beyond it.
\begin{theacknowledgments}
I wish to thank the many colleagues who helped in various ways in the preparation of this review.
Particular credits are due to G. Cavoto, Y.Kuno, S.Mihara, D.Nicol\'o, P.Paradisi, F.Renga, and N. Saito.
I wish to  thank very much the Organizers of this Workshop that was held in a 
 beautiful place and managed with great efficiency.
\end{theacknowledgments}

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\begin{thebibliography}{999}
\bibitem{Isidori}
G.Isidori {\it et al.} Phys. Rev. D75 (2007) 115019
\bibitem{sindrum}
 W.Bertl  {\it et al.} (Sindrum-II Collaboration) EPJ C 47 (2006) 337-346
 \bibitem{Meg}
 A.Baldini {\it et al.} (MEG Collaboration), Research Proposal to PSI R-99-05, 1999
 \bibitem{Mega}
 M.L.Brooks {\it et al.} [Mega Collaboration], Phys. Rev. Lett. 83, (1999) 1521.
\bibitem{E821}
G.W.Bennett {\it et al.} (E821 Collaboration)  Phys.Rev.Lett. 92(2004) 1618102 
\bibitem{masiero}
A.Masiero {\it et al.} NJP 6 (2004) 202 
\bibitem{Nishiguchi}
H.Nishiguchi, WG4,This Workshop
\bibitem{Kuno}
Y.Kuno, WG4, This Workshop


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