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%%%%%%%|       ANOMALOUS MAGNETIC MOMENT OF THE MUON       |%%%%%%%%%
%%%%%%%|               IN A COMPOSITE MODEL                |%%%%%%%%%
%%%%%%%|---------------------------------------------------|%%%%%%%%%
%%%%%%%|                  Prasanta Das                     |%%%%%%%%%
%%%%%%%|               Santosh Kumar Rai                   |%%%%%%%%%
%%%%%%%|              Sreerup Raychaudhuri                 |%%%%%%%%%
%%%%%%%|---------------------------------------------------|%%%%%%%%%
%%%%%%%|        Last modified:  February 20, 2001          |%%%%%%%%%
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           {\LARGE\bf Anomalous Magnetic Moment of \\ 
                     the Muon in a Composite Model  
           } \\ \bigskip\bigskip
           {\large\sl Prasanta Das\footnote{E-mail: 
                pdas@iitk.ac.in}
           },
           {\large\sl Santosh Kumar Rai\footnote{E-mail: 
                skrai@iitk.ac.in}
           },
           {\large\sl Sreerup Raychaudhuri\footnote{E-mail: 
                sreerup@iitk.ac.in, sreerup@mail.cern.ch}
           } \\
           {\rm Department of Physics, 
                Indian Institute of Technology, 
                Kanpur 208 016, India.}
    \end{center}
\vspace*{3cm}
    \begin{center}
           {\bf ABSTRACT}
    \end{center}
    \begin{quotation}
           We make a fresh evaluation of contributions to ($g$-2) of 
           the muon in the framework of a preonic model with coloured 
           vectorlike leptons and heavy coloured $Z$ bosons 
           and discuss their implications in the light of the recent 
           Brookhaven measurement. It is shown that the observed 
           deviation can be explained within this framework with masses 
           and couplings consistent with all constraints.
\end{quotation}
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\newpage

Ever since the original QED derivation of Schwinger\cite{Schwinger}, 
the anomalous magnetic moment of the electron --- and the muon --- has
been a testing ground for the correctness of the Standard Model (SM).
The most precise measurement of this parameter has been the recent
measurement\cite{BNLmuon} 
%---------------
\begin{equation}
    a_\mu = \frac{g - 2}{2} = 11 659 202 (14)(6) \times 10^{-10}
\end{equation}
%---------------
by the E821 experiment at the Brookhaven National Laboratory (BNL), 
which is the culmination
of a series of measurements of ever-increasing accuracy over the past
several years. The present measurement is found to deviate from the 
Standard Model (SM) by
\begin{equation}
a_\mu^{\rm exp} - a_\mu^{SM} = (4.3 \pm 1.6) \times 10^{-9}
\end{equation}
which represents a 2.6$\sigma$ excess. Given the fact that the E821
data have been painstakingly collected and analyzed over many years,
this discrepancy is unlikely to be explained away as a mere fluctuation,
as several earlier deviations from the SM have turned out to be.
Accordingly, it appears that the E821 experiment 
has observed the first real
deviation from the SM in experimental data, which makes their result
very exciting. This deviation constitutes an immediate challenge to
theorists to find possible explanations, and several such analyses
have been forthcoming.

It has been pointed out\cite{Marciano} that in order to explain a
discrepancy of the observed magnitude in a low-energy phenomenon like
the muon anomalous magnetic moment, it is necessary to invoke new
physics with particle masses around the electroweak scale as well as
couplings of at least electroweak strength. This makes it immediately
clear that the new physics responsible for the BNL anomaly cannot be
due\cite{Kane} to effects expected at ${\cal O}{\rm (1~TeV)}$, such
as graviton exchanges in theories with extra 
dimensions\cite{Graesser} and possible
non-commutative effects in QED\cite{Jabbari}, which are two of the 
currently
fashionable ideas. On the other hand, supersymmetry, which is the
most popular way of going beyond the SM, can explain the observed
excess, since it predicts new particles with the correct order of
masses and couplings. In fact, several studies have already appeared
in the literature\cite{Kane,SUSY} which map out the portion 
of the supersymmetric parameter space which could give rise to the 
observed effect. 

While it is gratifying to know that supersymmetric models constitute
an acceptable explanation of the muon anomaly, it is also relevant
and interesting to ask what other new physics options can give rise
to such an effect. Suggestions already made in the literature vary 
from a scalar leptoquark exchange\cite{Mahanta,Choudhury} to lepton 
flavour-violation\cite{Huang}, exotic vectorlike fermions\cite{Rakshit}, 
and possible nonperturbative 
effects at the 1 TeV order\cite{Lane,Mahanta2}. Given the 
importance of the BNL result, many more suggestions will surely be 
forthcoming.

In this work, we revive earlier ideas\cite{composite} that the
excess contribution to the muon anomaly could be a signal for
compositeness of the muon and of the weak gauge bosons. Given the
well-known history of finding layers of substructure every time a
significant increase in the resolving power of experiments has
increased, it is surely reasonable to address the question whether
the BNL anomaly could be the first hint of a new layer of
substructure which may be discovered some time in the future.

It is a well-known fact that since the original 
suggestion\cite{PatiSalam}
several -- in fact, one may say, dozens --- of preonic models have
been proposed in the literature. Different models among these have
different virtues and various motivations, and it is not our
purpose to make a comprehensive survey of these models\cite{Lyons}. A
generic feature of all these models, however, is the existence of
{\it excited} leptons and gauge bosons, which are just excited states
of the preonic combinations which make up ordinary leptons and gauge
bosons. These excitations may be simple orbital excitations, which
share the same set of flavour and colour quantum numbers as the SM
particles, or they may have exotic flavour and colour charges. To fix
our ideas, we concentrate on a well-known model, namely the so-called
{\it haplon} model of Fritzsch and Mandelbaum\cite{Fritzsch}, which has 
the prime virtues of simplicity and elegance\footnote{Another popular 
model is the {\it rishon} model of Harari and Seiberg\cite{Harari}. 
This has a simpler preonic spectrum than the haplon model, but by
virtue of combining together several preons at a time it predicts 
many more exotic colour states.}

The haplon model is based on the assumption that the leptons, quarks
and weak gauge bosons of the SM (as also possible neutral and charged 
scalars) are not elementary particles, but are composed of pairs of 
preons called {\it haplons}. The fundamental
symmetry of Nature is $SU(3)_c \times U(1)_{em} \times SU(N)_h$, which, 
of course, makes the photon and the gluons fundamental particles. The
weak interaction is no longer a gauge interaction, but is
interpreted in this scenario as a feeble van der Waals-type effect of 
the preonic
gauge interaction $SU(N)_h$ (A variant with just a $U(1)$ instead of
$SU(N)$ has also been proposed; however, it is hard to see how this 
could lead to preon confinement.) In this scenario, the weak isospin and
hypercharge have a status similar to the isospin and hypercharge 
of the meson and baryon multiplets. The haplon model is. in fact, a
copy of the quark model at a deeper level, with $SU(3)_c \times U(1)_{em}$
replacing $U(1)_{em}$ and $SU(N)_h$ for the preonic interactions replacing 
$SU(3)_c$ for the quark interactions. The basic building blocks of matter,
then, apart from the photon and gluon, are two fermionic preons 
$\alpha({\bf 3}, -\frac{1}{2}, {\bf N})$ and 
$\beta({\bf 3}, \frac{1}{2}, {\bf N})$, and two scalar preons
$x({\bf 3}, -\frac{1}{6}, \bar{\bf N})$ and
$y(\bar{\bf 3}, \frac{1}{2}, \bar{\bf N})$. Using these, we can build 
up the entire SM by taking bound states of pairs of haplons, and 
interpret the three observed fermion families as representing orbital 
excitations of these bound states. Since it is known that sequential or
mirror fermions other that these three families are ruled out by precision
electroweak data\cite{PDG}, any further families must be vectorlike in 
nature. The cause for these being vectorlike, whereas the first three
states are sequential, must be attributed to the dynamics of the bound 
state, which are as yet unknown.

A natural consequence of having bound pairs of particles belonging to 
either {\bf 3} or $\bar{\bf 3}$ of $SU(3)_c$, is to predict the existence 
of exotic particles such as 
\begin{itemize}
\item Colour-octet $W^\pm$ bosons ($W^+_8 = \bar{\alpha} \beta$ in
a spin-1 state);
\item Colour-octet $Z$ bosons 
($Z_8 = \frac{1}{\sqrt{2}}(\alpha\bar{\alpha} + \beta\bar{\beta})$ in
a spin-1 state);
\item Colour-octet neutral scalars ($H_8 = 
\frac{1}{2}(\alpha\bar{\alpha} + \beta\bar{\beta} + x\bar{x} + y\bar{y}$)
in a spin-0 state);
\item Colour singlet and octet charged scalars ($H^+_8 = 
\bar{\alpha} \beta$ in a spin-0 state);
\item Colour singlet and octet leptoquarks ($\Phi^{+2/3} = \bar{x}y$);
\item Colour sextet quarks ($u_6 = \bar{\alpha}\bar{x}$, 
$d_6 = \bar{\beta}\bar{x}$);
\item Colour-octet leptons ($\ell^-_8 = \bar{\beta}\bar{y}$, 
$\nu_{\ell 8} = \alpha y$). 
\end{itemize}

Since such exotic coloured objects may be produced copiously at hadron 
colliders, it is clear from their non-observation that that they must be 
rather heavy. In fact, direct searches for the coloured fermions leads 
to the following bounds\cite{PDG}: 
$m_{q_6} > 84$ GeV, $m_{\ell_8} > 86$ GeV and
$m_{\nu_8} > 110$ GeV. It is also clear from the non-observation of 
deviations from the SM predictions at hadron colliders that coloured
$W_8$ and $Z_8$ bosons, as well as scalar leptoquarks (coloured or 
otherwise) must be rather heavy, with masses typically of the order of
a few hundred GeV or a few TeV. Direct detection of such objects ---
if they exist --- may, therefore, have to await the commissioning of the
LHC.

We now address the question as to whether the existence of so many 
exotic particles can affect the anomalous magnetic moment of the 
muon\cite{composite}.
It must be noted that the current deviation from the SM is typically
an effect of the order of $10^{-9}$, which makes it comparable to 
relativistic effects in day-to-day life. It is interesting to
speculate that the tiny effect observed at BNL could be the first 
indication of substructure. To this end we have considered the
possible contributions to an excess muon magnetic moment from the
{\it exotic} particle spectrum. Of these, the ones which are of 
greatest interest are of the following three types:
\begin{enumerate}
\item Vertex-type diagrams with a $W^\pm_8$ and/or a $H^\pm$/$H_8^\pm$
together with a neutrino (coloured or ordinary, as the case may be) in 
the loop;
\item Vertex-type diagrams with a $Z^0_8$ and/or a $H^0$/$H_8^0$ 
together with a muon (coloured or ordinary, as the case may be) in the 
loop;
\item Vertex-type diagrams with a scalar leptoquark $\Phi^{\pm 2/3}$ or
$\Phi_8^{\pm 2/3}$ and a quark (ordinary or colour sextet) in the loop.
\end{enumerate}

In this work, we assume that the interactions of scalar bosons, coloured
or otherwise, mimic those of the SM Higgs bosons, {\it i.e.} these scalars 
couple to particles with a strength proportional to the particle mass.
As a result, their coupling to muons is very feeble and hence the 
effects of these scalars need not be considered any further. The situation
is different for scalar leptoquarks, whose couplings have no SM analogy.
While there are strong restrictions on the masses and couplings
of scalar leptoquarks from the Fermilab Tevatron data\cite{PDG}, it has
nevertheless been shown\cite{Mahanta,Choudhury} that scalar leptoquark 
exchanges can produce
the desired excess in $(g - 2)$ of the muon. In this work, however,
we exclude this interesting possibility.

The detailed Feynman rules for the vector boson interactions in the
haplon model
are given in Ref.~\cite{Gounaris}. An interesting feature of these
is that the interactions of the $W_8^\pm$ are chiral, whereas those
of the $Z_8$ are vectorlike. It is also worth noting that there is
no mixing between the $Z_8$ and the photon, for obvious reasons, and
hence no analogue of the Weinberg angle in $Z_8$ interactions. The
fact that the $Z_8$ interactions are vectorlike allows us to evade
any constraints from the electroweak precision data, provided we
allow the colour octet muon to be degenerate with its coloured
neutrino partner\cite{PDG}.

It is well-known\cite{Marciano} that chiral
interactions lead to contributions to the muon anomalous magnetic 
moment which are strongly suppressed by the ratio of muon mass to the
(large) scale of the new physics, and hence, it in not useful to 
consider the first type of
diagram listed above. On the other hand, it is indeed possible to evaluate 
the new physics contribution to the anomalous magnetic moment of the muon 
at the one-loop level using the vectorlike $Z_8$-$\mu^\pm_8$-$\mu^\mp$
interaction illustrated in Fig.~1. It is worth noting that the coupling
constant $g_8$ --- again arising from preon dynamics --- is unknown but 
is expected to be of electroweak strength, just like the well-known
$Z$-$\mu^\pm$-$\mu^\mp$ coupling.
We have two more unknown quantities to consider in Fig.~1, viz. the mass 
$m_8$ of the octet muon ($\mu_8$), and the mass $M_8$ of the octet 
$Z$-boson ($Z_8$). These are expected to be at least a few hundred GeV
and may well be much larger.

% ------------------------------------------------------------------
\begin{figure}[htb]
\begin{center}
\vspace*{2.4in}
      \relax\noindent\hskip -4.4in\relax{\special{psfile=Fig1.ps}}
\end{center}
\end{figure}
\noindent {\bf Figure 1}.
{\footnotesize\it Illustrating ($i$) the preonic level diagram 
responsible for the $\mu$-$\mu_8$-$Z_8$ vertex 
$\frac{ig_8}{2}\gamma_\lambda$, and ($ii$) the 
corresponding one-loop diagram contributing to $(g - 2)$ of the muon. }
% ------------------------------------------------------------------
\vskip 5pt

The contribution of a heavy fermion and a heavy vector boson to the
anomalous magnetic moment of the muon has been evaluated several times 
before\cite{Leveille}, 
and we skip the details of the derivation. The one-loop-corrected
vertex function of the muon can be written
\begin{equation}
\Gamma_\alpha = F_1(q^2) \gamma_\alpha
            + \frac{i}{2m_\mu} F_2(q^2) \sigma_{\alpha\beta}q^\beta
\end{equation}
from which it follows that
\begin{equation}
a_\mu = \frac{g -2}{2} = \left. F_2(q^2)\right|_{q^2 \to 0} \ .
\end{equation}
Direct evaluation\cite{Leveille} 
of the form factor $F_2(q^2)$ using the Feynman rule
of Fig.~1 and taking account of the colour factor of 8 leads to the 
result (in terms of the unknown parameters $a_8, m_8, M_8$)
\begin{eqnarray}
a_\mu(Z_8) & = &
\left( \frac{\alpha_8}{2\pi} \right) 
\left(\frac{m_\mu m_8}{M_8^2} \right) \frac{1}{(M_8^2 - m_8^2)^4}
\\
&\times & 
\bigg[ 4M_8^8 - 7m_8^2M_8^6 + 3m_8^4M_8^4 - m_8^6M_8^2 + m_8^8 
       + 12 m_8^2 M_8^4 (M_8^2 - m_8^2) \log \frac{m_8}{M_8} 
\nonumber \\
&& + \frac{1}{12}\frac{m_\mu}{m_8} 
\bigg( 8M_8^8 - 38m_8^2M_8^6 + 39m_8^4M_8^4 - 14m_8^6M_8^2 + 5M_8^8 
       - 36m_8^4M_8^4 \log \frac{m_8}{M_8}  \bigg) \bigg] \ ,
\nonumber 
\end{eqnarray}
where $\alpha_8 = g_8^2 /4\pi$. For large values of $m_8$ the terms on
the last line of the above equation may be safely neglected. 

% ------------------------------------------------------------------
\begin{figure}[htb]
\begin{center}
\vspace*{2.8in}
      \relax\noindent\hskip -4.4in\relax{\special{psfile=Fig2.ps}}
\end{center}
\end{figure}
\vspace*{0.5in}
\noindent {\bf Figure 2}.
{\footnotesize\it Illustrating the regions in the $M_8 - \alpha_8$ plane
which can explain the BNL data. Values of $\alpha_8$ are marked inside 
the relevant shaded region. Dark (light) shading denotes the allowed 
region at 1(2)$\sigma$. There is no upper bound on $M_8$ at the
3$\sigma$ level because the deviation from the SM is at 2.6$\sigma$. }
% ------------------------------------------------------------------
\vskip 5pt

We are now in a position to make a phenomenological analysis of the
haplon model vis-\'a-vis the E821 data. Our numerical results are
set out in Fig.~2, where we have plotted, for three different choices
of $\alpha_8$, the region in the $M_8$--$m_8$ plane which can explain
the BNL excess. The darker-shaded regions correspond to the data with
errors at the 1$\sigma$ level, while the lighter-shaded regions correspond
to the 2$\sigma$ level. It is of great interest to note that large values
of the exotic particle masses $m_8$ and $M_8$ can actually yield the
correct order of magnitude of the anomalous magnetic
moment\footnote{The apparent
non-decoupling behaviour of the excess contribution is corrected when we
go to larger values of the masses than are shown in Fig.~2.}.


We see, therefore, that it is possible to interpret the observed
deviation of the muon anomalous moment from the SM prediction as a
signal for a preonic structure of the weak gauge bosons provided the
effective coupling $\alpha_8$ lies in the range 0.001 to 0.1. The
former represents a superweak-type interaction, while the latter 
represents a strong interaction. However, it is gratifying to note
that the most favourable region in the parameter space is indeed when
$\alpha_8$ is close to the weak coupling strength, {\it i.e.} 
around 0.01, this being an 
eminently desirable feature, given the philosophy adopted in the 
haplon model. In fact, several interesting phenomenological features 
of the model emerge. For example, the exotic particle masses which
can explain the BNL result are rather large and can easily evade 
any constraints coming from Fermilab Tevatron data. In fact, to 
see any manifestation of these exotic particles and their interactions 
at low energies will generically require a measurement which is precise 
to the same level as the E821 experiment, which would explain why the
observed excess in the muon anomalous magnetic moment could be a
single isolated hint of a composite structure of leptons and weak
bosons.

To summarize, then, we have made a phenomenological analysis of the recent
BNL measurement and shown how it can be interpreted as a signal for
heavy exotic coloured lepton and $Z$-boson states arising in a preonic 
model. Though, admittedly, this is not a unique way in
which the observed deviation from the SM can be interpreted, nor is the
haplon model the only preonic model, but our result is
nevertheless exciting, since it means that the old idea of substructure
remains one of the new physics options which remain viable in the 
post-BNL scenario. This does not depend very crucially on our choice
of model (which was essentially for simplicity). Whether this 
explanation is the correct one, or whether the postulated substructure
is anywhere near the correct theory is
something only time and new data from other processes can indicate.

\bigskip\bigskip

\noindent {\bf Acknowledgments}: The authors collectively thank
Pankaj Jain and Gautam Sengupta for discussions and SR would like to 
thank Sourendu Gupta and Rohini M. Godbole for prompt information 
concerning the new BNL measurement.

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