\vskip -15mm \begin{flushright} \vskip -15mm {%Fermilab-Pub-0?/??-T\\ BUHEP-01-3\\ hep-ph/0102131\\} \vskip 5mm \end{flushright} {\Large{\hskip 0.38truein New Model--Independent Limit \\ on Muon Substructure}}\\
\centerline{{590 Commonwealth Avenue, Boston, MA 02215}}
If the discrepancy between the theoretical and newly measured values of the muon's anomalous magnetic moment is ascribed to muon substructure, there results an improved model--independent limit on its energy scale, $1.2\,\tev < \Lambda_\mu < 3.2\,\tev$ at 95\%~C.L.
hep-ph0102131.tex
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\begin{document}
\title{
\vskip -15mm
\begin{flushright}
\vskip -15mm
{%\small Fermilab-Pub-0?/??-T\\
\small BUHEP-01-3\\
\small hep-ph/0102131\\}
\vskip 5mm
\end{flushright}
{\Large{\bf \hskip 0.38truein
New Model--Independent Limit \\ on Muon Substructure}}\\
}
\author{
\centerline{{Kenneth Lane\thanks{lane@physics.bu.edu}}}\\ \\
\centerline{{Department of Physics, Boston University,}}\\
\centerline{{590 Commonwealth Avenue, Boston, MA 02215}}\\
}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace*{.3in}
\begin{abstract}
If the discrepancy between the theoretical and newly measured values of the
muon's anomalous magnetic moment is ascribed to muon substructure, there
results an improved model--independent limit on its energy scale, $1.2\,\tev
< \Lambda_\mu < 3.2\,\tev$ at 95\%~C.L.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

The Muon ($g-2$) Collaboration has announced a new measurement of the
anomalous magnetic moment of the positive muon~\cite{gtwo},
%
\be\label{eq:meas} 
a_{\mu^+} = \half(g-2)_{\mu^+} = 11\,\,659\,\,202(14)(6) \times 10^{-10}
\ts\ts (1.3\ts {\rm ppm}) \ts.
\ee
%
The value currently expected in the standard model is~\cite{smvalue}
%
\be\label{eq:SM}
a_{\mu}({\rm {SM}}) = 11\,\,659\,\,159.6(6.7)\times 10^{-10}
\ts\ts (0.57\ts {\rm ppm})\ts.
\ee
%
Using the world--average experimental value of the muon's anomalous moment,
there is now a discrepancy with theory of 2.6 standard deviations:
%
\be\label{eq:diff}
\delta a_\mu \equiv a_\mu(\rm exp) - a_{\mu}({\rm {SM}}) = 43(16) \times
10^{-10}\ts.
\ee
%
This error is smaller than that of previous measurements by about a factor of
three~\cite{previous}.

If the muon is a composite fermion at the scale $\Lambda_\mu \gg m_\mu$,
there is a contribution to its magnetic moment of~\cite{composite}
%
\be\label{eq:delta}
\delta a_\mu(\Lambda_\mu) \simeq {m_\mu^2 \over{\Lambda_\mu^2}} \ts.
\ee
%
If $\delta a_\mu$ is ascribed to muon substructure, the scale is $\Lambda_\mu
\simeq 1.6\,\tev$. It is more appropriate, however, to take advantage of the
new smaller error by interpreting $\delta a_\mu$ to limit $\Lambda_\mu$. For
example, its 95\%~C.L. range is
%
\be\label{eq:limit}
1.2\,\tev < \Lambda_\mu < 3.2\,\tev \ts.
\ee
%
If, as expected, analysis of the 2000 data for $g-2$ decreases its
statistical error by half, but the central value and other errors do not
change, this range will become $1.3\,\tev < \Lambda_\mu < 2.3\,\tev$

The importance of this bound is its model--independence. It requires no
assumption on the compositeness of other quarks or leptons. Limits that do
assume $\Lambda_e \simeq \Lambda_\mu$ and $\Lambda_{u,d} \simeq \Lambda_\mu$
come from $e^+ e^- \ra \mu^+\mu^-$ and $\ol qq \ra
\mu^+\mu^-$~\cite{elp}. They are $\Lambda_e \simeq \Lambda_\mu \simge
4$--$5\,\tev$ and $\Lambda_{u,d} \simeq \Lambda_\mu \simge
3$--$4\,\tev$~\cite{pdg}. These lower bounds are more stringent, but they are
also less incisive. There is no reason {\it a priori} for the equality of the
first and second--generation lepton substructure scales.

\section*{Acknowledgements}

I thank Robert Carey for a discussion of the new $g-2$ result of
Ref.~\cite{gtwo} and Estia Eichten, Bill Marciano, Jim Miller, and Lee
Roberts for valuable comments. This research was supported in part by the
Department of Energy under Grant~No.~DE--FG02--91ER40676.

\begin{thebibliography}{99}
%
\bibitem{gtwo} H.~N.~Brown, {\it et al.}, ($g-2$) Collaboration,
  submitted to Physical Review Letters; hep-ex/0102017, v2.
%
%
\bibitem{smvalue} A.~Czarnecki and W.~J.~Marciano, Nucl.~Phys. (Proc.~Supp.)
  {\bf B76}, 245 (1999), and hep-ph/0010194 v2, and references therein.
%
%
\bibitem{previous} H.~N.~Brown {\it et al.}, ($g-2$) Collaboration,
  Phys.~Rev.~{\bf D62}, 091101 (2000); R.~M.~Carey, {\it et al.}, ($g-2$) Collaboration, Phys.~Rev.~Lett.~{\bf 82},
  1632 (1999); J.~Bailey {\it et al.}, Nucl.~Phys.~{\bf B150}, 1 (1979).
%
%
\bibitem{composite} E.~Eichten, K.~Lane and J.~Preskill,
  Phys.~Rev.~Lett.~{\bf 45}, 225 (1980); \\ K.~Lane, Physica Scripta {\bf
  23}, 1005 (1981); further references are cited in
  Ref.~\cite{smvalue}. Equation~(\ref{eq:delta}) represents a normalization
  of $\Lambda_\mu$ just as the conventional $4\pi/\Lambda^2$ does in contact
  interactions induced by substructure.
%
%
\bibitem{elp} E.~J.~Eichten, K.~D.~Lane, and M.~E.~Peskin,
Phys.~Rev.~Lett.~{\bf 50}, 811 (1983).
%
%
\bibitem{pdg} D.~E.~Groom {\it et al.}, Review of Particle Physics,
  Eur.~Phys.~J.~{\bf C15}, 1 (2000).
\end{thebibliography}

\vfil\eject



\end{document}

hep-ph0102131.extracted_bib
\begin{thebibliography}{99}
%
\bibitem{gtwo} H.~N.~Brown, {\it et al.}, ($g-2$) Collaboration,
  submitted to Physical Review Letters; hep-ex/0102017, v2.
%
%
\bibitem{smvalue} A.~Czarnecki and W.~J.~Marciano, Nucl.~Phys. (Proc.~Supp.)
  {\bf B76}, 245 (1999), and hep-ph/0010194 v2, and references therein.
%
%
\bibitem{previous} H.~N.~Brown {\it et al.}, ($g-2$) Collaboration,
  Phys.~Rev.~{\bf D62}, 091101 (2000); R.~M.~Carey, {\it et al.}, ($g-2$) Collaboration, Phys.~Rev.~Lett.~{\bf 82},
  1632 (1999); J.~Bailey {\it et al.}, Nucl.~Phys.~{\bf B150}, 1 (1979).
%
%
\bibitem{composite} E.~Eichten, K.~Lane and J.~Preskill,
  Phys.~Rev.~Lett.~{\bf 45}, 225 (1980); \\ K.~Lane, Physica Scripta {\bf
  23}, 1005 (1981); further references are cited in
  Ref.~\cite{smvalue}. Equation~(\ref{eq:delta}) represents a normalization
  of $\Lambda_\mu$ just as the conventional $4\pi/\Lambda^2$ does in contact
  interactions induced by substructure.
%
%
\bibitem{elp} E.~J.~Eichten, K.~D.~Lane, and M.~E.~Peskin,
Phys.~Rev.~Lett.~{\bf 50}, 811 (1983).
%
%
\bibitem{pdg} D.~E.~Groom {\it et al.}, Review of Particle Physics,
  Eur.~Phys.~J.~{\bf C15}, 1 (2000).
\end{thebibliography}