\documentclass[12pt]{JHEP}
\usepackage{epsfig}
\usepackage{equations}
\newenvironment{Eqnarray}%
     {\arraycolsep 0.14em\begin{eqnarray}}{\end{eqnarray}}
\def\nicefrac#1#2{\hbox{${#1\over #2}$}}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\beqa{\begin{Eqnarray}}

\def\eeqa{\end{Eqnarray}}
\def\tanb{\tan\beta}
\def\sinb{\sin\beta}
\def\cosb{\cos\beta}
\def\sina{\sin\alpha}
\def\cosa{\cos\alpha}
\def\sinbma{\sin(\beta-\alpha)}
\def\cosbma{\cos(\beta-\alpha)}
\def\sinbmaii{\sin^2(\beta-\alpha)}
\def\cosbmaii{\cos^2(\beta-\alpha)}
\def\hsm{h_{\rm SM}}
\def\mhsm{m_{h_{\rm SM}}}
\def\hl{h}
\def\ha{A}
\def\hh{H}
\def\hpm{H^\pm}
\def\mha{m_{\ha}}
\def\mhl{m_{\hl}}
\def\mhh{m_{\hh}}
\def\mhpm{m_{\hpm}}
\def\mhmax{m_h^{\rm max}}
\def\mz{m_Z}
\def\mw{m_W}
\def\mww{m_W^2}
\def\mzz{m_Z^2}
\def\mt{m_t}
\def\mb{m_b}
\def\amu{\delta a_\mu^{\rm NP}}
\def\ifmath#1{\relax\ifmmode #1\else $#1$\fi}
\def\ls#1{\ifmath{_{\lower1.5pt\hbox{$\scriptstyle #1$}}}}
\def\half{\ifmath{{\textstyle{1 \over 2}}}}
\def\eq#1{eq.~(\ref{#1})}
\def\eqs#1#2{eqs.~(\ref{#1})--(\ref{#2})}
\def\Ref#1{ref.~\cite{#1}}
\def\Refs#1#2{refs.~\cite{#1} and \cite{#2}}
\def\Rref#1{Ref.~\cite{#1}}
\def\Rrefs#1#2{Refs.~\cite{#1} and \cite{#2}}

\def\phm{\phantom{-}}
\def\lsim{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\gsim{\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\newcommand{\mathbold}[1]{\mbox{\boldmath $\bf#1$}}



\title{Can a light Higgs boson explain the recent BNL measurement of the
muon anomalous magnetic moment ?}


\author{Athanasios Dedes \\ 
Physikalisches Institut der Universit\"at Bonn,\\ 
 Nu\ss allee 12, D-53115 Bonn, Germany\\
E-mail: \email{dedes@th.physik.uni-bonn.de}}


\author{Howard E. Haber  \\ 
Santa Cruz Institute for Particle Physics\\ 
University of California, Santa Cruz, CA, 95064 USA\\
E-mail: \email{haber@scipp.ucsc.edu}}




\abstract{A light CP-even Higgs boson with $\mhl\sim 10$~GeV
could explain the recent BNL measurement of the 
muon $g-2$ anomalous magnetic moment, in the
framework of a general two-Higgs-doublet extension of the
Standard Model.  However, the allowed Higgs mass window is
quite small and the corresponding model parameters are very constrained.
The Higgs sector can
contribute significantly to the observed BNL result of $g-2$
only if the $\hl ZZ$ coupling (approximately) vanishes and
$M_{\Upsilon}\lsim\mhl\lsim 2m_B$.}






\keywords{Beyond Standard Model}

\preprint{hep-ph/yymmnnn}


\begin{document}

\section{Introduction}

% The g-2 status
A new experimental value of the muon anomalous
magnetic moment, $a_\mu=\half(g-2)_\mu$, measured at BNL,
was recently reported in \Ref{Prigl:2001cm}.
Comparing the measured value to its predicted value in the
Standard Model,  \Ref{Prigl:2001cm} reported that
%%%%%%%%%%%%%%%
\begin{eqnarray}
a_\mu^{\rm exp}-a_\mu^{\rm SM}=426\pm 165 \times 10^{-11}\;,
\label{dev}
\end{eqnarray}
%%%%%%%%%%%%%%%%
Ref.~\cite{Czarnecki:2001pv} has reviewed the Standard Model
computation of $a_\mu$ and concludes that if deviation
of \eq{dev} can be attributed to new physics [$\amu$],
then at $90\%$ CL, $\amu$ must
lie in the range
%%%%%%%%%%%%%%%
\begin{eqnarray}
215 \times 10^{-11} \lsim \amu \lsim 637
 \times 10^{-11} \;.
\label{newphys}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%% 
This contribution is positive, and is of the order  
of the electroweak corrections to $a_\mu$.
%coming from the SM gauge bosons and fermions.
More precisely, the ``new Physics'' needed, has to be 
of the order of $G_Fm_\mu^2/(4\pi^2\sqrt{2})$, where
$G_F=1.16637(1)\times 10^{-5}~{\rm GeV}^{-2}$ is Fermi's constant.
%and $m_\mu=0.105$ GeV is the mass of the muon. 
In this paper, we consider the possibility that $\amu$
arises entirely from the Higgs sector.  In the SM,
the Higgs boson contribution to $a_\mu$ is further suppressed
(relative to the main electroweak contribution) by a factor of
$m_\mu^2/\mhl^2$.  In light of the recent SM Higgs mass limit,
$\mhl\gsim 113.5$~GeV obtained at the LEP collider \cite{higgslimit},
the SM Higgs contribution to $a_\mu$ is clearly negligible.  

However, the Higgs sector contribution to $a_\mu$ could be
considerably enhanced in a two-Higgs-doublet extension of the Standard
Model (2HDM).  This enhancement can arise from two different effects.
First, an enhanced $\hl\mu^+\mu^-$ coupling proportional to the ratio of
Higgs vacuum expectation values, $\tan\beta$, yields a Higgs
contribution to $\amu$ proportional to $\tan\beta^2$.  Second, a
suppressed $\hl ZZ$ coupling, proportional to $\sin(\beta-\alpha$)
[using notation reviewd below], can permit the existence of a CP-even
Higgs boson mass substantially below the present SM Higgs mass
limit.  In units of $G_Fm_\mu^2/(4\pi^2\sqrt{2})$,
the overall enhancement is of order  
\begin{equation}
\frac{m_\mu^2}{\mhl^2}\times \tan^2\beta \times
F(\frac{m_\mu^2}{m_h^2})\simeq 1{\hbox{\rm---}}10 \;,
\end{equation}
where $F(x)$ is a loop factor which involves logarithms 
of the form $\ln(\mhl^2/m_\mu^2) \sim {\cal O}(10)$.
A light CP-even Higgs boson with a mass $\mhl \simeq 10$ GeV
and $30 \lsim \tan\beta \lsim 44$, predicts a muon anomalous magnetic
moment in agreement with the current BNL experimental measurement.

The required 2HDM is not compatable with the Higgs
sector of the minimal supersymmetric extension of the Standard Model
(MSSM).  The reason is that one cannot have a very light $\hl$ with
suppressed $\hl ZZ$ couplings without an observable rate for
$Z\to \hl\ha$, in conflict with LEP 
data \cite{lepsusyhiggs}.\footnote{A possible loophole arises in the
MSSM when $\mha$ is large, $\tan\beta$ is large, and there exists very
strong mixing in the top-squark sector (beyond so-called maximal
mixing).  In this case, radiative corrections can drive $\mhl$ very
light while keeping $|\sin(\beta-\alpha)|\ll 1$.  In such a model, 
the Model II results of this paper apply. However, the required MSSM
parameter regime is rather fine-tuned and probably exhibits
other theoretical problems such as
the existence of charge and color-breaking minima.}
Moreover, the MSSM provides additional mechanisms for generating
significant contributions to $\amu$.  A number of recent 
papers \cite{Everett:2001tq,Feng:2001tr,
Baltz:2001ts,Chattopadhyay:2001vx,Komine:2001fz} have
shown that the recent BNL measurement is compatable with
supersymmetric contributions to $\amu$ involving chargino and
neutralino exchange, over an interesting region of MSSM parameter space.

% Definition of the model

%Model I
% Model II

In this paper, we focus on the possibility that the new physics
contribution to $a_\mu$ arises solely from the Higgs sector.
The two-doublet Higgs sector~\cite{Gunion:1989we} contains eight scalar
degrees of freedom.  It is convenient to distinguish between the
two-doublets by employing 
one complex $Y=-1$ doublet, {\boldmath $\Phi_d$}$=(\Phi_d^0,\Phi_d^-)$
and one complex $Y=+1$ doublet, {\boldmath
$\Phi_u$}$=(\Phi_u^+,\Phi_u^0)$.  To avoid Higgs-mediated flavor
changing neutral currents, we do not allow the most general
Higgs-fermion interaction.  Instead, we impose discrete symmetries
(which may be softly-broken by mass terms), and consider two possible
models.  In Model I, $\Phi_d^0$ couples to both up-type and
down-type quark and lepton pairs, while the coupling of $\Phi_u^0$ to
fermion pairs is absent.\footnote{One can just as well assume that
$\Phi_u^0$ couples to both up-type and
down-type quark and lepton pairs, while the coupling of $\Phi_d^0$ to
fermion pairs is absent.  In this case, the all the results of this
paper would apply simply by replacing $\tan\beta$ with $\cot\beta$.}  
In Model II, $\Phi_d^0$
[$\Phi_u^0$] couples exclusively to down-type [up-type] fermion pairs.
When the Higgs potential is minimized, the neutral components of the
Higgs fields acquire vacuum expectation values:\footnote{In this
paper, we neglect the possibility of significant CP-violation in the
Higgs sector.  In this case, the
phases of the Higgs fields can be chosen such that the vacuum
expectation values are real and positive.}
\beq
\langle {\mathbold{\Phi_d}} \rangle={1\over\sqrt{2}} \left(
\begin{array}{c} v_d\\ 0\end{array}\right), \qquad \langle
{\mathbold{\Phi_u}}\rangle=
{1\over\sqrt{2}}\left(\begin{array}{c}0\\ v_u

\end{array}\right)\,,\label{potmin}
\eeq
where the normalization has been chosen such that
$v^2\equiv v_d^2+v_u^2={4\mw^2/ g^2}=(246~{\rm GeV})^2$,
while the ratio $\tan\beta\equiv v_u/v_d$ is a free parameter of the
model.  The physical Higgs spectrum consists of
a charged Higgs pair
%\vspace*{-1pc}
\beq \label{hpmstate}
\hpm=\Phi_d^\pm\sinb+ \Phi_u^\pm\cosb\,,
\eeq
one CP-odd scalar
%\vspace*{-1pc}
\beq \label{hastate}
\ha= \sqrt{2}\left({\rm Im\,}\Phi_d^0\sinb+{\rm Im\,}\Phi_u^0\cosb
\right)\,,
\eeq
and two CP-even scalars:
%\vspace*{-1pc}
\beqa
\hl &=& -(\sqrt{2}\,{\rm Re\,}\Phi_d^0-v_d)\sin\alpha+
(\sqrt{2}\,{\rm Re\,}\Phi_u^0-v_u)\cos\alpha\,,\nonumber\\
\hh &=& (\sqrt{2}\,{\rm Re\,}\Phi_d^0-v_d)\cos\alpha+
(\sqrt{2}\,{\rm Re\,}\Phi_u^0-v_u)\sin\alpha\,,
\label{scalareigenstates}
\eeqa
(with $\mhl\leq \mhh$).
The angle $\alpha$ arises when the CP-even Higgs
squared-mass matrix (in the $\Phi_d^0$---$\Phi_u^0$ basis) is
diagonalized to obtain the physical CP-even Higgs states.


Relevant to our analysis, we note the tree-level couplings of 
lightest CP-even Higgs boson, $\hl$, to gauge
bosons and Higgs bosons:
The couplings of the two CP-even Higgs bosons
to $W$ and $Z$ pairs
are given in terms of the angles $\alpha$ and $\beta$ by
\beq\label{hvv}
g\ls{\hl VV}= g\ls{V} m\ls{V}\sinbma 
\label{vvcoup}
\eeq
where
\beq
g\ls V\equiv\begin{cases}
g,& $V=W\,$,\\ g/\cos\theta_W,& $V=Z\,$, \end{cases}
\label{hix}
\eeq
and
\beq g\ls{\hl\ha Z}={g\cosbma\over 2\cos\theta_W}\,.
           \label{hvcoup}
\eeq


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


The pattern of couplings of the Higgs bosons to fermions depends 
on the choice of model.  However, in this paper we are only concerned
with the coupling of down-type fermions to Higgs bosons.  As a result,
the distinction between Model I and Model II is not required.  
For our analysis, the relevant 
couplings of the neutral Higgs bosons to $b\bar b$ or $\mu^+\mu^-$
relative to the Standard Model
value, $gm_f/2\mw$ [$f$= $b$ or $\mu$], are given by
\begin{eqaligntwo}
 \label{qqcouplings}
\hl b\bar b \;\;\; ({\rm or}~ \hl \mu^+ \mu^-):&~~~ -
{\sin\alpha\over\cos\beta}=\sin(\beta-\alpha)
-\tan\beta\cos(\beta-\alpha)\,,\nonumber\\[3pt]
%
\hh b\bar b \;\;\; ({\rm or}~ \hh \mu^+ \mu^-):&~~~
\phm{\cos\alpha\over\cos\beta}=
\cos(\beta-\alpha)
+\tan\beta\sin(\beta-\alpha)\,,\nonumber\\[3pt]
%
\ha b \bar b \;\;\; ({\rm or}~ \ha \mu^+
\mu^-):&~~~\phm\gamma_5\,{\tan\beta}\,,
\end{eqaligntwo}
(the $\gamma_5$ indicates a pseudoscalar coupling), and the
charged Higgs boson couplings to muon pairs
(with all particles pointing into the vertex) is given by
\beq
\label{hpmqq}
g_{H^- \mu^+ \nu}= {gm_\mu\over{\sqrt{2}\mw}}\
\tan\beta\,P_L\,,
\eeq
where $P_L\equiv\half(1-\gamma_5)$.

We have noted above that only light Higgs bosons with enhanced
couplings to down-type fermions can contribute appreciably to $\amu$.
To avoid the LEP Higgs mass limit, such a light Higgs boson should be
almost decoupled from $ZZ$.  This implies that either $\hl$ is light,
with $|\sin(\beta-\alpha)|\ll 1$ [see \eq{vvcoup}] or $\ha$ is light
(since $\ha$ has no tree-level coupling to vector boson pairs).  In
the next section, we will show that a light $\ha$ makes a {\it
negative} contribution to $\amu$ and thus is not compatible with the
recent BNL measurement.  Hence, we focus on a 2HDM model in which only
$\hl$ is light and $\sin(\beta-\alpha)\simeq 0$.  From
\eq{qqcouplings}, we see that if $\sin(\beta-\alpha)\simeq 0$, then
the coupling of $\hl$ to down-type fermions is proportioanl
$\tan\beta$.  Thus, in the region of large $\tan\beta$ and small
$\sin(\beta-\alpha)$, the contribution of a light CP-even Higgs boson
of the 2HDM may yield a significant correction to $\amu$ without
being in conflict with the LEP Higgs search.

Finally, we note that in the parameter region cited above, the heavier
Higgs bosons, $\hh$, $\ha$, $\hpm$, cannot be arbitrarily heavy.  If
one attempts to take such a limit, one finds that there must be some
Higgs quartic self-couplings that become significantly larger than 1.
That is, this model does not possess a decoupling limit.  However,
the model stays weakly coupled as long as the heavier Higgs states are
not too much larger than $v=246$~GeV.   In contrast, in the limit of
$\cos(\beta-\alpha)=0$, the coupling of $\hl$ reduce to those of the
Standard Model Higgs boson.  This decoupling limit can be formally
reached by taking the masses of $\hh$, $\ha$, $\hpm$ to be arbitrarily
large.  The resulting low-energy effective theory is just the
SM with one Higgs doublet.  Of course, as we have noted above, in this
limit, the contribution of $\hl$ to $\amu$ is negligible.\footnote{We  
demonstrate in the next section that in the decoupling limit
[where $\cos(\beta-\alpha)\simeq 0$], if the masses of
$\hh$, $\ha$ and $\hpm$ are large, but not too large, 
and $\tanb\gg 1$, then the heavy Higgs
states dominate the Higgs sector contribution to $\amu$.
Nevertheless, the overall Higgs contribution is significantly smaller
than the electroweak contribution and thus could not be a source of
the new physics contribution to $a_\mu$.}


\section{Higgs boson corrections to the muon anomalous magnetic
  moment}

The first calculation of the electroweak corrections to the muon
anomalous magnetic moment was done by Weinberg and
Jackiw~\cite{Jackiw:1972jz} and 
by Fujikawa, Lee and Sanda~\cite{Fujikawa:1972fe}. A very nice
overview of these corrections in a general field theory framework can
be found in Ref.\cite{Leveille:1978rc}. In a two-Higgs-doublet model,
these corrections can be obtained from 
\Rrefs{haber}{Krawczyk:1997sm}\footnote{We correct
a small error in the expression for the charged Higgs contribution
given in \Ref{haber}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\delta a_\mu^h &=& \frac{G_Fm_\mu^2}{4\pi^2 \sqrt{2}}~
\biggl (\frac{\sin{\alpha}}{\cos{\beta}}\biggr )^2 ~R_h~ F_h(R_h) \; \\
\delta a_\mu^H &=& \frac{G_Fm_\mu^2}{4\pi^2 \sqrt{2}}~
\biggl (\frac{\cos{\alpha}}{\cos{\beta}}\biggr )^2 ~R_H~ F_H(R_H) \; \\
\delta a_\mu^A &=& \frac{G_Fm_\mu^2}{4 \pi^2 \sqrt{2}}~
\tan^2\beta ~R_A~ F_A(R_A) \; \\
\delta a_\mu^{H^\pm} &=& \frac{G_Fm_\mu^2}{4 \pi^2 \sqrt{2}}~
\tan^2\beta ~R_{H^\pm}~ F_{H^\pm}(R_{H^\pm},R_\nu) \;
\label{amuch}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%
where $R_{h,H,A,H^\pm}\equiv m_\mu^2/m_{h,H,A,H^\pm}^2$,
$R_\nu\equiv m_\nu^2/m_{H^\pm}^2$ and 
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
F_{h,H}(R_{h,H}) &=& \int_0^1 dx\, \frac{2 x^2-x^3}{R_{h,H} x^2-x+1}
 \;, \\
F_{A}(R_{A}) &=& \int_0^1 dx\, \frac{-x^3}{R_{A} x^2-x+1}
 \;, \\
F_{H^\pm}(R_{H^\pm},R_\nu) &=& \int_0^1 dx\, \frac{x^2-x^3}{R_{H^\pm}
 x^2+(1-R_{H^\pm}-R_\nu) x+ R_\nu}
 \;, 
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%
The neutrino mass is negligible, so henceforth we set $R_\nu=0$.
Since $R_{h,H,A,H^\pm}\ll 1$, one can easily expand the above
integrals in the corresponding small parameter.  In the next two
subsections, we write out the leading terms in this expansion, which
are very accurate in the Higgs mass range of interest.\footnote{The
plots shown in this paper are based on the exact values of the above
integrals.}   

\subsection{Non-decoupling limit: $\sin(\beta-\alpha)=0$}

In section 1, we argued that the most significant Higgs contribution
to $\amu$ (consistent with the LEP Higgs search data) arises in the
parameter regime in which $\sinbma\simeq 0$ and $\tanb\gg 1$.  
Setting  $\sinbma=0$ and keeping only the leading terms in $R$, 
the total Higgs sector contribution to $a_\mu$
is given by:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{eqnarray}
&& \delta a_\mu^{\rm Higgs}=  \delta a_\mu^h+\delta a_\mu^H
+\delta a_\mu^A+
\delta a_\mu^{H^\pm}  \nonumber \\[5pt]
&& \qquad\simeq\frac{G_Fm_\mu^2}{4\pi^2 \sqrt{2}}
\tan^2\beta \bigg \{
\frac{m_\mu^2}{\mhl^2} \biggl[\ln\left(\frac{\mhl^2}{m_\mu^2}\right)
-\frac{7}{6} \biggr] -\frac{m_\mu^2}{\mha^2} \biggl[
\ln\left(\frac{\mha^2}{m_\mu^2}\right) -\frac{11}{6}\biggr]-
\frac{m_\mu^2}{6m_{H^{\pm}}^2} \biggr \} \;. \nonumber \\
\label{hfull}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Note that the 
logarithms appearing in \eq{hfull} always dominate the corresponding
constant terms when the Higgs masses are larger than 1 GeV.
It is then clear that $\ha$ and $\hpm$ exchange contribute a
negative value to $\amu$.  Since our goal is to explain the BNL $g-2$
measurement which suggests a positive value for $\amu$, we should take
$\mha$ and $\mhpm$ large (masses above 100~GeV are sufficient)
in order that the corresponding
$\ha$ and $\hpm$ negative contributions are 
neglibly small.\footnote{Grifols 
and Pascual~\cite{Grifols:1980yk}
found that for a very light charged Higgs boson, 
the two-loop contribution to $a_\mu$ is positive and
can be larger in magnitude than the one-loop
result given in \eq{amuch}:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\delta
 a_\mu^{H^\pm}=a_\mu^{H^\pm}\hbox{\rm {(1--loop)}}+
\frac{1}{180}\left(\frac{\alpha}{\pi}\right)^2
\left(\frac{m_\mu}{m_{H^\pm}}\right)^2+{\cal
 O}\biggl [\left(\frac{m_\mu}{m_{H^\pm}}\right)^4 
\ln\left(\frac {m_\mu}{m_{H^\pm}}\right)
 \biggr ]  \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%5
However, the LEP bound
on the charged Higgs mass \Ref{lepsusyhiggs},
$\mhpm>80.5$~GeV, implies that both the one and two-loop charged Higgs
contribution to $\amu$ are negligible.}
If $\amu$ is to be a consequence of the Higgs sector, it must be entirely
due to the contribution of the light CP-even Higgs boson.
Note that the heavier
CP-even Higgs, $\hh$, does not give a contribution proportional to
$\tan\beta$ (as shown in section 1); thus its contribution
to $\amu$ can be neglected in \eq{hfull}.
Thus, to a good approximation,
   
%%%%%%%%%%%%%%%%%% FIG 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\FIGURE[t]{\centerline{\epsfig{figure=non_decoup.ps,height=4in,angle=90}}
\caption{Contour plot of the  predicted one-loop Higgs sector contribution
to the muon anomalous magnetic moment, 
$\delta \alpha_\mu^{\rm Higgs}$ (in units of $10^{-11}$).
in the 2HDM with $\sin(\beta-\alpha)=0$, and
$\mhh=\mha=\mhpm=200$~GeV.
(There is little sensitivity to the heavier Higgs masses.)
The dashed line contour corresponds to the central value of
$\delta a_\mu\equiv a_\mu^{\rm exp}-a_\mu^{\rm SM}$, as reported 
in \Ref{Prigl:2001cm}.
The contours lines marked 215 and 637 correspond to 90\% CL limits
for the contrubution of new physics to $a_\mu$~\cite{Czarnecki:2001pv}.
The dark-shaded (red) region is excluded
by CUSB Collaboration~\cite{CUSB}.
The light-shaded (yellow) region 
is experimentally excluded by DELPHI LEP I search
for $e^+e^-\to b\bar b\hl$~\cite{Delphi}.
In the small hatched region (green) nestled between the two excluded
shaded regions centered around $\mhl\simeq 10$~GeV, the Higgs sector
contributes a value for $\amu$ within the $90\%$ CL allowed range
[\eq{newphys}].}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\beq
\delta a_{\mu}^{\rm Higgs}\simeq 
\delta a_\mu^{{h}} \simeq \frac{G_Fm_\mu^2}{4\pi^2 \sqrt{2}}
\left(\frac{m_\mu^2}{\mhl^2}\right) \tan^2\beta
\bigg[
\ln\left(\frac{\mhl^2}{m_\mu^2}\right)-\frac{7}{6} \biggr ] \;.
\label{app1}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%

One can easily check that a light Higgs boson with a mass of
arround 10 GeV and with $\tan\beta=35$ gives $\delta a_{\mu}^{\rm
 Higgs}\simeq 280 \times 10^{-11}$ which is precisely the new 
physics contribution we need in Eq.(\ref{newphys}). 
Contour lines corresponding to a full numerical 
evaluation of the Higgs sector one-loop 
contribution to $\delta
a_\mu^{\rm Higgs}$ [in units of $10^{-11}$], is presented
in Fig.~1, for $\sinbma=0$ and $\mhh=\mha=\mhpm=200$~GeV. 
The results are insensitive to the value of the heavy Higgs masses
above 100~GeV.
The relevant experimental bounds are also displayed
in the figure (see discussion below). Clearly, a light
Higgs boson of 10 GeV mass and $30\lsim \tan\beta \lsim 45$
will yield a value for $\amu$ within the $90\%$ CL allowed range
[\eq{newphys}].  In section 3, we review the experimental limits for
light CP-even masses and show that if $\sinbma\simeq 0$, and $\tanb$
in the range required to explain the BNL measurement of $a_\mu$,
then $\mhl$ must lie in a very narrow range centered around 10~GeV.




\subsection{Decoupling Limit: $\cos(\beta-\alpha)=0$}

In the decoupling limit, where $\cosbma\simeq 0$ and $\mha\gg\mz$,
the couplings of the light Higgs boson,
$\hl$, are (nearly) identical to those of the one-Higgs-doublet Standard Model.
As a result, the LEP Higgs mass bound of $\mhl\gsim 113.5$~GeV applies.
In this limit, the $\hh$ couplings to fermions are enhanced by $\tanb$
[see \eq{qqcouplings}].
Thus, the Higgs sector contribution to $\amu$ is given by
\eq{hfull}, with $\mhl$ replaced by $\mhh$.  In the decoupling limit,
$\mhh\simeq\mha\simeq \mhpm$ (the mass diferences are of ${\cal
O}(\mz^2/\mha)$).  Setting $\cosbma=0$ and $\mhh=\mha=\mhpm$, we find
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beq
\delta a_\mu^{\rm Higgs} \simeq  \frac{G_Fm_\mu^2}{4 \pi^2\sqrt{2}}
\left(\frac{m_\mu^2}{\mha^2}\right)\tan^2\beta \biggl[
\frac{1}{2}-\left(\frac{2m_\mu^2}{\mha^2}\right) 
\ln\left(\frac{\mha^2}{m_\mu^2}\right)
 \biggr ]\;.
\label{app2}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%
The contribution of $\hl$ is not $\tanb$--enhanced and is thus
negligible.  In is interesting to note that for values of
$\mha\lsim\mhl\tanb$, the heavier (``decoupled'')
Higgs bosons actually dominate in the Higgs sector contribution
to $\amu$.\footnote{If we formally take $\mha\to\infty$, we recover
the Standard Model Higgs contribution to $a_\mu$.}
However, for 100~GeV $<\mha< 1000$~GeV, and $30<\tanb<100$,
the Higgs sector contribution to $a_\mu$ ranges from about $5\times
10^{-12}$ to $5\times 10^{-14}$, which is
three to five orders of magnitude below what
is needed to explain the BNL measured value of $a_\mu$.  



%%%%%%%%%%%%%%%%%% FIG 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\FIGURE[t]{\centerline{\epsfig{figure=decoup.ps,height=4in,angle=90}}
%\caption{Contour plot of the  predicted one-loop Higgs sector contribution
%to the muon anomalous magnetic moment, 
%$\delta \alpha_\mu^{\rm Higgs}$ (in units of $10^{-14}$).
%
%Contour plot of the   Higgs 1-loop contributions
%to the muon $g-2$ anomalous magnetic moment $\delta_{\alpha_\mu}$(higgs)
%(in units of $10^{-14}$), on a plane $M_A-\tan\beta$, 
%in the decoupling limit, {\it i.e.,} $\cos(\beta-\alpha)=0$..
%The light Higgs boson mass has been fixed to its current experimental
%bound, $M_h=115$ GeV. The heavy CP-even and the charged Higgses are
%taken to be equal to $M_A$. Note the residual enhancement with the 
%$\tan\beta$ even in the region of large $M_A$.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%






%%%%%%%%%%%%%%%%%% FIG 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\FIGURE[t]{\centerline{\epsfig{figure=decoup_app.ps,height=5in,angle=90}}
%\caption{Contour plot of the  1-loop Higgs contributions
%to the muon g-2 anomalous magnetic moment in the decoupling limit.
% We use the approximation of
%eq.(\ref{app2}).}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{CUSB and LEP I constraints on the light Higgs boson}

Let us consider a 2HDM model in which $\sinbma=0$ and $\tanb$ is
large.  The tree-level $\hl ZZ$ coupling is absent, which implies that
the LEP Higgs search based on $e^+e^-\to Z\to Z\hl$ does not impose
any significant constraints on the Higgs mass.\footnote{Presumably,
radiative corrections would lead to a small effective value for
$\sinbma$.  The LEP Higgs search yields an excluded region in the 
$\sinbma$--$\mhl$ plane, and suggest that for $\mhl\sim 10$~GeV, 
$\sinbma\lsim 0.14$ \cite{Sopczak:1995zy}.}  Moreover, $\tanb$ must be
large if the Higgs sector is to be the source for $\amu$ observed by
BNL. However, there are a number of constraints on light Higgs masses 
that do not rely on the $\hl VV$ coupling.  For Higgs bosons with
$\mhl\lsim 5$~GeV, the Standard Model Higgs boson was ruled out by a
variety of arguments that were summarized in \Ref{Gunion:1989we}.  For
5~GeV $\lsim\mhl\lsim 10$~GeV, the relevant
Higgs boson constraint can be derived 
from the absence of Higgs production in $\Upsilon\to\hl\gamma$.

An experimental search for this process by the
CUSB Collaboration~\cite{CUSB} found no candidates.  The Higgs mass
limit obtained from this results depends on the theoretical
prediction.  In addition to the tree-leve prediction of \Ref{wil},
there are three classes of corrections that have been explored in the
literature: hard ${\cal O}(\alpha_s)$ QCD
corrections~\cite{vys,nason}, relativistic corrections to the
non-relativistic treatment of the $b\bar b$ bound state
\cite{biswas,azn}, and bound state threshold corrections \cite{wu}.
The theoretical picture that emerges is confused.  The hard QCD
corrections are large and suggest that ${\cal O}(\alpha_s^2)$
corrections could be significant.  Second, relativistic effects
enter at the same order as the ${\cal O}(\alpha_s)$ corrections [both
are of ${\cal O}(v^2/c^2)$, and thus the two must be treated
consistently.  Finally,  Faldt, Osland and Wu~\cite{wu} argue that
strong cancellations can occur among various contributions in the
threshold region, leading to an additional suppression in rate of 
about 14 for $\mhl=8.5$~GeV (and even a larger suppression
as $\mhl\to M_{\Upsilon}$).  The application of the theoretical
analysis of $\Gamma(\Upsilon\to\hl\gamma)$ to the CUSB data suggests
that values of $\mhl\lsim 5$--7~GeV can be ruled out, although a
precise upper limit cannot be obtained due to 
the theoretical uncertainties outlined above.

The above discussion was relevant for limits on the mass of the
Standard Model Higgs boson.  In the 2HDM model of interest here,
$\tanb\gg 1$, and the prediction for $\Gamma(\Upsilon\to\hl\gamma)$ is
enhanced by a factor of $\tan^2\beta$.  This factor dwarfs the theoretical
uncertainties discussed above except for values of $\mhl$ very close
to $M_\Upsilon$.  For values of $\tanb\geq 10$, the CUSB data can
reliably rule out Higgs masses up to about 8~GeV; the excluded region
is indicated by the dark (red) shaded region in Fig.~1.  But, for
Higgs masses above 8~GeV, we see that $\tanb>30$ if the Higgs sector
contribution to $\amu$ lies in the $90\%$ CL range specified in
\eq{newphys}.  For such large values of $\tanb$,
the predicted rate for
$\Upsilon\to\hl\gamma$ is increased by at least three orders of magnitude.
This factor dwarfs the theoretical uncertainties discussed above
except for values of $\mhl$ very close to $M_\Upsilon$.  Thus, we
obtain an upper bound of $\mhl\gsim M_\Upsilon$.

A second bound on $\mhl$ can be derived from the non-observation of 
Higgs bosons at LEP in the process $e^+e^-\to b\bar b\hl$.
The cross-section for this process depends on the $\hl b\bar b$ Yukawa
coupling.  In the 2HDM model with $\sin(\beta-\alpha)=0$, the $\hl
b\bar b$ coupling is enhanced (relative to its SM value) by 
$\tanb$.\footnote{We assume that $\mha$ is heavy enough so that
$e^+e^-\to \hl\ha$ is not observed at LEP.}  A preliminary analysis 
of DELPHI based on the search for $e^+e^-\to b\bar b\hl$, $\hl\to
\tau^+\tau^-$, $b\bar b$ finds no evidence for Higgs production and
excludes the light-shaded (yellow) region of Fig.~1.  Note that the
lower limit on $\tanb$ is weaker for $\mhl<2m_B$, where $B$ is the
lightest $B$-meson [$m_B=5.279$~GeV].  In this mass region, the
dominant decay of $\hl$ is into $\tau^+\tau^-$.  For values of
$\mhl>2m_B$, the Higgs decays primarily into $b\bar b$, and the DELPHI
limits completely exclude the region of parameter space in which the
Higgs sector contribution to $\amu$ lies in the $90\%$ CL range 
specified in \eq{newphys}.


Finally, regarding other constraints,
it has been shown in \cite{Chankowski:1999ta}
that even in the presence of
such a light higgs boson, the two Higgs doublets model,
can describe the electroweak data with the same precision to that 
given by the SM. It has also been shown~\cite{Krawczyk:1999kk}
that the Wilczek process
$Z\rightarrow h\gamma$ constraints 
$\tan\beta \lsim 75$ for $M_h=10$ GeV. This bound is weaker than
the most recent LEP I constraints \cite{Delphi} depicted in Fig.1.



\section{Final Results and Conclusions}

If we combine the experimental searches for a light Higgs boson,
 we conclude that a light Higgs boson
can be responsible for the observed
the $2.6\sigma$ deviation of the BNL measurement of the muon 
anomalous magnetic moment at the 90\% CL in the framework of a
two-Higgs-doublet model only if the model parameters satisfy the
following requirements:
%%%%%%%%%%%%%%%%%%
\beqa
&& m_{\Upsilon}\lsim M_h \lsim 2 m_B \;, \nonumber \\
&& \sinbma\simeq 0 \;, \nonumber \\
&& 30\lsim\tanb\lsim 45 \;.
\label{result}
\eeqa
%%%%%%%%%%%%%%%%%%

One noteworthy consequence of a Higgs boson in this mass regime was 
discussed in \Rrefs{haber}{egns}.  Namely, there would be mixing
between the $\hl$ and the $0^{++}$ $b\bar b$ bound states
$\chi\ls{b0}(1P)$ and $\chi\ls{b0}(2P)$.  As a result, the decay
$\chi\ls{b0}\to\tau^+\tau^-$ should be prominent.  The predicted
rate is roughly
\beq
{\Gamma(\chi\ls{b0}\to\tau^+\tau^-)\over\Gamma(\chi\ls{b0}\to~{\rm hadrons})}
={2.5\times 10^{-7}~{\rm GeV}^2\over
(m_\chi-\mhl)^2}\tan^4\beta\:,
\eeq
which is valid for $\mhl$ near $m_\chi$ but separated by a few Higgs
widths.\footnote{%  
If the two masses are within a Higgs width, then the mixing
of the two states will be close to maximal \cite{egns}, 
and the corresponding $\tau^+\tau^-$ branching ratio of both 
eigenstates would be close to
100\% due to the large $\tanb$ enhancement.}
Due to the large $\tanb$ enhancement, the predicted branching ratio for 
$\chi\ls{b0}\to\tau^+\tau^-$ can be substantial.  Remarkably, the
Particle Data book \cite{pdg} provides no data on possible 
decay modes of the $\chi\ls{b0}$ other than radiative decays, 
$\chi\ls{b0}\to\Upsilon\gamma$, $\Upsilon'\gamma$.
 
Apart from a careful study of $\chi\ls{b0}$ decays, the 2HDM model
specified by \eq{result} could be confirmed or ruled out
by experiments at future colliders
in the future colliders~\cite{Krawczyk:2000kf}, or by a further
improvement of the BNL measurement of the muon anomalous magnetic moment. 

\vspace*{0.5cm}

{\it A.D. would like to acknoweledge financial support from the
Network RTN European Program HPRN-CT-2000-0014
``Physics Across the Present Energy Frontier: Probing the Origin of
Mass.''  H.E.H. would like to thank H.-P. Nilles and H.K. Dreiner for
their hospitality during his visit to the Physikalisches Institut der
Universit\"at Bonn.  H.E.H. is supported in part by a grant from the
U.S. Department of Energy.
}


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\end{thebibliography}

\end{document}



