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     {\arraycolsep 0.14em\begin{eqnarray}}{\end{eqnarray}}
\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\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\phm{\phantom{-}}
\newcommand{\mathbold}[1]{\mbox{\boldmath $\bf#1$}}



\newcommand{\newc}{\newcommand}
\newc{\gsim}{\stackrel{>}{\sim}}
\newc{\lsim}{\stackrel{<}{\sim}}



\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 Higgs boson with mass of arround $\sim 10$ GeV,
could  explain the recent BNL measurement on 
muon $g-2$  anomalous magnetic moment, in the
framework of the minimal Standard Model with two Higgs doublets. 
Such a light Higgs boson should have a mass bounded from below
from the Upsilon mass and from above from the bottonium mass.}




\keywords{Beyond Standard Model}

\preprint{hep-ph/yymmnnn}


\begin{document}

\section{Introduction}

% The g-2 status
The recent reported 2.6$\sigma$~\cite{Prigl:2001cm}  deviation
%%%%%%%%%%%%%%%
\begin{eqnarray}
a_\mu^{exp}-a_\mu^{SM}=426\pm 165 \times 10^{-11}\;,
\label{dev}
\end{eqnarray}
%%%%%%%%%%%%%%%%
of the experimental value
of the muon anomalous
magnetic moment versus the Standard Model (SM) predicted one,
is very exciting. It has also been reported~\cite{Czarnecki:2001pv}
that at 90\% CL a ``New Physics'' contribution of
%%%%%%%%%%%%%%%
\begin{eqnarray}
215 \times 10^{-11} \lsim \delta a_\mu {\rm (New~Physics)} \lsim 637
 \times 10^{-11} \;,
\label{newphys}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%% 
beyond the SM, is needed. This contribution is positive, and of the order  
of the electroweak corrections to the muon anomalous magnetic moment.
%coming from the SM gauge bosons and fermions.
More precisely, the ``New Physics'' needed, has to be 
 of the order of $\frac{G_Fm_\mu^2}{4\sqrt{2}\pi^2}$, where
$G_F=1.16637(1)\times 10^{-5}~{\rm GeV}^{-2}$ is the Fermi constant
and $m_\mu=0.105$ GeV is the mass of the muon. 
Furthermore, one could think that these corrections could directly 
be originated by the perturbative electroweak symmetry breaking mechanism
or in other words  by the Higgs Boson particle. 
However, in the SM, the Higgs boson contribution (in
units of $\frac{G_Fm_\mu^2}{4\sqrt{2}\pi^2}$) to
the muon anomalous magnetic moment is suppressed by a factor of
$m_\mu^2/M_h^2$. Regarding  the smallness of the 
mass of the muon and the largeness of the Higgs mass ($\gsim 115$ GeV)
  the SM Higgs contribution to $\delta a_\mu$
is negligible. 

The above picture is not in general valid 
 in a straightforward extension of the
SM with two Higgs doublets. In this framework, $\delta a_\mu$ could
be enhanced in two ways :  the first
one is that $\delta a_\mu$ is proportional\footnote{Definitions are
 given below.} to
 $\tan^2\beta$  in the case where $\sin(\beta-\alpha)=0$, and
the second, it is in that case where the light Higgs boson can be as
light as  10 GeV without conflict with experiment.
In terms of $\frac{G_Fm_\mu^2}{4\sqrt{2}\pi^2}$,
the overall enhancement is of  order  
\begin{equation}
-\frac{m_\mu^2}{M_h^2}\times \tan^2\beta \times
F(\frac{m_\mu^2}{M_h^2})\simeq 1-10 \;,
\end{equation}
where $F(x)$ is a loop factor which involves logarithms 
of the form $\ln(\frac{m_\mu^2}{M_h^2})$  of order
${\cal O}(-10)$. Such a light Higgs boson with mass $M_h \simeq 10$ GeV
and $30 \lsim \tan\beta \lsim 44$, predicts a muon anomalous magnetic
moment in agreement with the current  BNL experimental value.

This idea cannot take place in the MSSM simply  because the
 Higgs bosons are constraint from certain mass sum rules\footnote{There is 
one exception here : It could be the case that the light Higgs boson
in the MSSM  decays ``invisibly'' into  LSPs (ussually
neutralinos). Although there are strict bounds on
$\sin(\beta-\alpha)-M_h$ plane from LEP searches,
an ``invisible'' Higgs boson with a mass
of few GeV's and $\sin(\beta-\alpha)=0$ is allowed. Such a ``supelight''
higgs
boson contributes to 
the $g-2$ muon anomalous magnetic, in exactly the same way as we describe
here.
However the Higgs boson should decay to very light particles and 
one has propably to consider R-parity violation or to depart from 
the minimal field content by  adding a singlet superfield (NMSSM). This
scenario would be highly fine-tuned.}. 
Although in the exact Supersymmetric limit there are
no contributions~ \cite{Ferrara:1974wb} to the muon $g-2$, this is not
the
case for the softly broken N=1 supersymmetric Standard model where the
neutralino/chargino contributions are proportional to $\frac{m_\mu^2 M_2
\tan\beta}{M_{SUSY}^4} F$ in units of
$\frac{G_Fm_\mu^2}{4\sqrt{2}\pi^2}$.
Thus, for moderate and large $\tan\beta$ values one recovers the
$g-2$ deviation (see the corresponding Refs in~\cite{Prigl:2001cm}). 
Although, the SUSY solution is quite elegant,
we feel that the scenario we propose, is far more
generic and minimal with respect to its field condent. 
 

% Definition of the model

%Model I
% Model II

The two-doublet Higgs sector~\cite{Gunion:1989we} contains eight scalar
degrees of freedom:
one complex $Y=-1$ doublet, {\boldmath $\Phi_d$}$=(\Phi_d^0,\Phi_d^-)$
and one complex $Y=+1$ doublet, {\boldmath
$\Phi_u$}$=(\Phi_u^+,\Phi_u^0)$.  The notation reflects the
form of the MSSM Higgs sector coupling to fermions: $\Phi_d^0$
[$\Phi_u^0$] 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{The
phases of the Higgs fields can be chosen such that the vacuum
expectation values are real and positive.  That is, the tree-level MSSM
Higgs sector conserves CP, which implies that the neutral Higgs mass
eigenstates possess definite CP quantum numbers.}
\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
$v^2\equiv v_d^2+v_u^2={4\mw^2/ g^2}=(246~{\rm GeV})^2$.
Spontaneous electroweak symmetry breaking results in
three Goldstone bosons, which
are absorbed and become the longitudinal components of
the $W^\pm$ and $Z$.  The remaining five physical Higgs particles
consist 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.
The angle $\beta$ is defined as $\tan^{-1}(\frac{v_u}{v_d})$.

The phenomenology of the Higgs sector depends in detail on
the various couplings of the Higgs bosons to gauge bosons, Higgs
bosons and fermions.  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
\beqa\label{hvv}
g\ls{\hl VV}&=& g\ls{V} m\ls{V}\sinbma \nonumber \\[3pt]
           g\ls{\hh VV}&=& g\ls{V} m\ls{V}\cosbma\,,\label{vvcoup}
\eeqa
where
\beq
g\ls V\equiv\begin{cases}
g,& $V=W\,$,\\ g/\cos\theta_W,& $V=Z\,$. \end{cases}
\label{hix}
\eeq
There are no tree-level couplings of $\ha$ or $\hpm$ to $VV$.
The couplings of one gauge boson to two
neutral Higgs bosons:
\beqa g\ls{\hl\ha Z}&=&{g\cosbma\over 2\cos\theta_W}\,,\nonumber \\[3pt]
           g\ls{\hh\ha Z}&=&{-g\sinbma\over 2\cos\theta_W}\,.
           \label{hvcoup}
\eeqa
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
There are two versions of the SM extended with two Higgs doublets
called here Model I and Model II.
The Model I is the case where quarks and leptons do not couple to
the first Higgs doublet (denoted as ${\bf \Phi_d}$ ) but couple
to the second doublet (denoted as ${\bf \Phi_u}$) in a manner
analogous to the minimal Standard Model. The gauge invariant Model I
Yukawa interactions for the muons can be read from 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation} \label{typeone}
-{\cal L}_{\rm Yukawa}= 
%h_t\left[\bar t P_L t \Phi^0_u-\bar t P_L b
%\Phi^+_u\right] +
h_\mu\left[\bar \mu P_L \mu \Phi^0_u-\bar \mu P_L \nu_\mu
\Phi^-_u\right] + {\rm h.c.}\,,
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $P_L\equiv\half(1-\gamma_5)$ is the left-handed projection
operator.  
%[Note that ($\overline\Psi_1 P_L \Psi_2)^\dagger=
%\overline\Psi_2 P_R \Psi_1$, where $P_R\equiv\half(1+\gamma_5)$.] 
In Model II, the Higgs tree-level couplings to fermions obey the
following
property: $\Phi_d^0$ couples exclusively to down-type fermion pairs and
$\Phi_u^0$ couples exclusively to up-type fermion pairs. 
%This pattern
%of Higgs-fermion couplings defines the
%Type-II two-Higgs-doublet model \cite{wise,hhg}.
The gauge-invariant Type-II Yukawa interactions for the muons
 are given by:
\begin{equation} \label{typetwo}
-{\cal L}_{\rm Yukawa}= 
%h_t\left[\bar t P_L t \Phi^0_u-\bar t P_L b
%\Phi^+_u\right] + 
h_\mu\left[\bar \mu P_L \mu \Phi^0_d-\bar \mu P_L \nu_\mu
\Phi^-_d\right] + {\rm h.c.}\,,
\end{equation}

Fermion masses are generated when the neutral Higgs components acquire
vacuum expectation values.  Inserting \eq{potmin} into \eq{typetwo}
yields a relation between the muon mass and its Yukawa coupling:
\beqa
h_\mu &=&  {\sqrt{2}\,m_\mu\over v_u}={\sqrt{2}\, m_\mu\over
  v\sin\beta}\,,
\;\;\;\;\; {\rm Model~ I} \;, \\
h_\mu &=& {\sqrt{2}\,m_\mu\over v_d}={\sqrt{2}\, m_\mu\over
v\cos\beta}\,,\;\;\;\;\; {\rm Model~ II}\;.
\eeqa
The couplings
of the physical Higgs bosons to the muon is
obtained from \eqs{typeone}{typetwo} 
by using \eqs{hpmstate}{scalareigenstates}.
In particular, the couplings of the neutral Higgs bosons to $\mu\bar \mu$
relative to the Standard Model
value, $gm_f/2\mw$, are given by the following Table :
%%%%%%%%%%%%%%%%%TABLE%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beq
\begin{tabular}{|c|c|c|}\hline
     & {\rm Model~ I} & {\rm Model~ II} \\ \hline
$ g_{h\mu^+\mu^-}$  & $-\frac{\cos\alpha}{\sin\beta}$ &
$\frac{\sin\alpha}{\cos\beta}$ \\ \hline
$ g_{H\mu^+\mu^-} $ &$ -\frac{\sin\alpha}{\sin\beta}$ &
$-\frac{\cos\alpha}{\cos\beta}$ \\ \hline 
$g_{A\mu^+\mu^-}$ 
& $-\gamma_5 \cot\beta$ & $\gamma_5 \tan\beta$ \\ \hline
\end{tabular}
\label{table1}
\nonumber
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(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
\begin{tabular}{|c|c|c|}\hline
 & {\rm Model~ I} & {\rm Model~ II} \\ \hline
$g_{H^-\mu^+\nu_\mu}$ & $-\frac{g \cot\beta}{\sqrt{2}{M_W}}m_\mu P_L$ &
$\frac{g\tan\beta}{\sqrt{2} M_W} m_\mu P_L$\\ \hline
\end{tabular}
\label{table2}
\nonumber
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Some very useful trigonometric relations are in order :
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
 \label{qqcouplings}
-{\sin\alpha\over\cos\beta}&=&\sin(\beta-\alpha)
-\tan\beta\cos(\beta-\alpha)\,,\nonumber\\[3pt]
%
{\cos\alpha\over\sin\beta}&=&\sin(\beta-\alpha)
+\cot\beta\cos(\beta-\alpha)\,,\nonumber\\[3pt]
%
{\cos\alpha\over\cos\beta}&=&
\cos(\beta-\alpha)
+\tan\beta\sin(\beta-\alpha)\,,\nonumber\\[3pt]
%
{\sin\alpha\over\sin\beta}&=&\cos(\beta-\alpha)
-\cot\beta\sin(\beta-\alpha)\,.
%
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%
We remind the reader, that we need the Higgs couplings to muons to be 
enhanced by $\tan\beta$ (or $\cot\beta$ for $\tanb < 1$) and 
a very light Higgs boson which does not couple to the Z-Gauge boson 
and thus almost undetectable at LEP. 
From the above tables and the trigonometric 
relations it is obvious that this is satisfied {\it if and only
  if},
\begin{equation}
\sin(\beta-\alpha)=0 \;.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%
In that case from eq.(\ref{hvv}) we obtain, $g_{hZZ}=0$ and
%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
& &g_{h\mu^+\mu^-}\;\;,\;\; g_{A\mu^+\mu^-}\;\;,\;\;g_{H^-\mu^+\nu_\mu}
\propto \tan\beta \;\;:\;\;\;  {\rm Model~I} \;\\
& & g_{h\mu^+\mu^-}\;\;,\;\; g_{A\mu^+\mu^-}\;\;,\;\;g_{H^-\mu^+\nu_\mu}
\propto \cot\beta \;\;:\;\;\; {\rm Model~II} \;
\label{hmumuI}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
and $|g_{H\mu^+\mu^-}|=1$ in both models. 

We would like  to comment here on the other extreme limit where
$\cos(\beta-\alpha)=0$. In this case it is
%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
& &g_{H\mu^+\mu^-}\;\;,\;\; g_{A\mu^+\mu^-}\;\;,\;\;g_{H^-\mu^+\nu_\mu}
\propto \tan\beta \;\;:\;\;\;  {\rm Model~I} \;\\
& & g_{H\mu^+\mu^-}\;\;,\;\; g_{A\mu^+\mu^-}\;\;,\;\;g_{H^-\mu^+\nu_\mu}
\propto \cot\beta \;\;:\;\;\; {\rm Model~II} \;
\label{hmumuII}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
and $|g_{h\mu^+\mu^-}|=1$.
An interesting aspect 
of both models is that as far as the higgs coupling to the muons
is concerned, they are  dual to one each other
%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
{\rm Model~I}~~~\tan\beta \leftrightarrow \cot\beta~~~{\rm Model~II}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%
Thus, investigating one of the two models is enough. 
We follow the notation of Model II.










\section{Higgs boson corrections to the muon $g-2$ anomalous magnetic
  moment}

\input{mdm}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The first calculation of the electroweak corrections to the $g-2$ 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 the Type II 2HDM framework
these corrections can be read from~\cite{haber}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\delta a_\mu^h &=& \frac{G_Fm_\mu^2}{4 \sqrt{2} \pi^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 \sqrt{2} \pi^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 \sqrt{2} \pi^2}~
\tan^2\beta ~R_A~ F_A(R_A) \; \\
\delta a_\mu^{H^\pm} &=& \frac{G_Fm_\mu^2}{4 \sqrt{2} \pi^2}~
\tan^2\beta ~R_{H^\pm}~ F_{H^\pm}(R_{H^\pm},R_\nu) \;
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%
where $R_{h,H,A,H^\pm}=\frac{m_\mu^2}{M_{h,H,A,H^\pm}^2}$,
$R_\nu=\frac{m_\nu^2}{M_{H^\pm}^2}$ 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}
%%%%%%%%%%%%%%%%%%%%%%%
Although we find approximate 
analytical solutions for the above integrals, we
solve them numerically in the figures presented. 
The small neutrino mass does not affect our results 
and we set it zero for simplicity.

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


Expanding the integrals in powers of R,
the leading contribution of the Higgs sector for  
$\sin(\beta-\alpha)=0$  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 \\[2mm]
& &-\frac{G_Fm_\mu^2}{4 \sqrt{2} \pi^2}~\tan^2\beta \bigg \{
\frac{m_\mu^2}{M_h^2} \biggl [\frac{7}{6}+\ln(\frac{m_\mu^2}{M_h^2}) 
\biggr ] -\frac{m_\mu^2}{M_A^2} \biggl [
\frac{11}{6}+\ln(\frac{m_\mu^2}{M_A^2}) \biggr ]+ 
\frac{1}{6}\frac{m_\mu^2}{M_{H^{\pm}}^2} \biggr \} \; \nonumber \\
\label{hfull}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Note the minus sign out of Eq.(\ref{hfull}). Also note that the 
logarithms encountered are negative and always larger than 
the constant term when the Higgs masses are larger than 1 GeV.
Having that in our mind, it is obvious that a possible light CP-Odd
Higgs does not help since it gives negative $\delta a_\mu$.
The cituation
of both $A$ and $h$ being light does not help either, since 
the logarithms associated  with them cancel out. Furthermore, 
the charged Higgs boson enters\footnote{Actually Grifols 
and Pascua~\cite{Grifols:1980yk}
have found that the  two-loop corrections,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\delta
 a_\mu^{H^\pm}=\frac{1}{180}(\frac{\alpha}{\pi})^2
(\frac{m_\mu}{m_{H^\pm}})^2+{\cal
 O}\biggl [(\frac{m_\mu}{m_{H^\pm}})^4 \ln(\frac {m_\mu}{m_{H^\pm}})
 \biggr ]  \;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%5
are bigger than the 1-loop ones. But again a very light charged
higgs of the order of few GeV's is required and this case is
experimentally
excluded.} in Eq.(\ref{hfull}) without a logarithm
and a 
light charged
Higgs boson less than 45 GeV is excluded by LEP I searches.
The other CP-even Higgs does not give contributions proportional to
$\tan\beta$ (see previous section) and thus its contribution
to $\delta a_\mu$ is negligible. 

   

%%%%%%%%%%%%%%%%%% FIG 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\FIGURE[t]{\centerline{\epsfig{figure=non_decoup.ps,height=4in,angle=90}}
\caption{Contour plot of the  predicted 1-loop Higgs contributions
to the muon $g-2$ anomalous magnetic moment, 
$\delta \alpha_\mu$(higgs) (in units of $10^{-11}$), 
 on 
$M_h-\tan\beta$ plane, in the 2HDM(II) scenario. We present results
with  $\sin(\beta-\alpha)=0$ as this is motivated in the text.
The other Higgs masses have been set to
$M_{H^\pm}=M_A=M_H$=200 GeV but their variation for above 100 GeV do
not affect the results (see Eq.(\ref{hfull})). 
The dashed line represents the central
difference between the experimental value and the SM predicted one,
 $\delta_{\alpha_\mu}=426\pm 165\times 10^{-11}$, as reported
from BNL~\cite{Prigl:2001cm}. Contours of 90\% CL $215\times 10^{-11}\le
{\rm (New ~Physics) } \le 637 \times 10^{-11}$~\cite{Czarnecki:2001pv} 
need to recover the
current experimental value, are  shown.
The region on the left side and behind the blue line is excluded
by CUSB Collaboration~\cite{CUSB}.
The region above the solid red line 
is experimentally excluded by DELPHI LEP I searches~\cite{Delphi}.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The only possibility left is a very light Higgs boson
which we advocate here.
In that case, its contribution to the muon $g-2$ is given in a very
good approximation by 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{eqnarray}
\delta a_{\mu}({\rm higgs})\simeq 
\delta a_\mu^{{h}} &=& -\frac{G_Fm_\mu^2}{4 \sqrt{2} \pi^2}
(\frac{m_\mu^2}{M_h^2}) \tan^2\beta
\bigg [
\frac{7}{6}+\ln(\frac{m_\mu^2}{M_h^2}) \biggr ] \;.
\label{app1}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%
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 actually the new 
physics we need in Eq.(\ref{newphys}). 
The full numerical 1-loop light Higgs boson  
contribution to muon $g-2$, $\delta
a_\mu^h{\rm (higgs)}$ (in units of $10^{-11}$), is presented 
in contour lines on the plane $M_h-\tan\beta$ in Fig.1.
The other Higgs masses have been set to
$M_{H^\pm}=M_A=M_H$=200 GeV but their variation for above 100 GeV do
not affect the results. The experimental bounds are also presented
in the figure (see discussion below). As it is clear, a light
Higgs boson of 10 GeV mass and $30\lsim \tan\beta \lsim 44$
is capable to give the neccesary contributions to the currently
measured value of the muon anomalous magnetic moment. 
We will turn back into this point in the last section where the mass 
region where  such a light Higgs might exist, will be set.

%%%%%%%%%%%%%%%%%% FIG 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\FIGURE[t]{\centerline{\epsfig{figure=non_decoup_app.ps,height=5in,angle=90}}
%\caption{Contour plot of the  1-loop Higgs contributions
%to the muon g-2 anomalous magnetic moment. We use the approximation of
%eq.(\ref{app1}).}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





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

The couling of the Z-gauge boson to the light Higgs boson, $g_{ZZh}$
is maximal and the lower LEP II bound\footnote{For large
$\tan\beta$ values this lower bound becomes  weaker ($\gsim 91$ GeV) 
but it does not make any difference in  the
results since it is already too heavy and  its contribution to $\delta
a_\mu$ is not
enhanced by $\tan^2\beta$.} on the light Higgs mass,
$M_h\gsim 115$ GeV, is applicable. 
The same approximate formula as in Eq.(\ref{hfull}) is
applied, with the light Higgs Boson $h$ being replaced by the heavy,
$H$, one. In the limit
where all the other Higgses are equal, $M_A=M_H=M_{H^\pm}$, we 
obtain 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\delta a_\mu{{\rm (higgs)}} &\simeq & \frac{G_Fm_\mu^2}{4 \sqrt{2} \pi^2}
(\frac{m_\mu^2}{M_A^2})\tan^2\beta \bigg [
\frac{1}{2}+2 (\frac{m_\mu^2}{M_A^2}) \ln(\frac{m_\mu^2}{M_A^2})
 \biggr ]\;.
\label{app2}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
What is interesting here is that even though the Higgs masses are
quite large, $M_A\rightarrow \infty$, there is still an enhancement
of $\delta a_\mu$ with $\tan^2\beta$ (note that the logarithmic
contribution
is suppressed). 
The situation is even clearer if we look at 
Fig.2 where contour lines of the muon $g-2$ (in units of $10^{-14}$)
are plotted on a plane of $M_A-\tan\beta$. 
The conributions to $\delta a_\mu$ are three orders of magnitude
smaller than those in the non-decoupling case and thus negligible 
(for reasonable values of $\tan\beta$).
For fixed $\tan\beta$,  $\delta a_\mu{{\rm (higgs)}}$
 dies out with a rate
$\sim \frac{m_\mu^2}{M_A^2}$ and the decoupling is manifest, {\it
  i.e.,} we recover the SM-case.


%%%%%%%%%%%%%%%%%% FIG 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\FIGURE[t]{\centerline{\epsfig{figure=decoup.ps,height=4in,angle=90}}
\caption{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}

As we have already discussed, the light higgs does not
couple to the Z-boson in the case of $\sin(\beta-\alpha)=0$ 
and thus the SM LEP bound of 115 GeV does not apply.
In principle, such a Higgs boson can be very light, or
in other words : it has been  escaped detection from the past Higgs
searches. However, there are two major constraints on a
possible Higgs mass, of arround $~10$ GeV, we quickly review
here.

The first lower bound  comes  
from CUSB Collaboration~\cite{CUSB}, 
where an extensive search 
for a monochromatic photon in radiative decays
$\Upsilon(1S)\rightarrow h\gamma$ where 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{\Gamma(\Upsilon\rightarrow h\gamma)}{\Gamma(\Upsilon\rightarrow
  \mu^+\mu^-)} &=&\frac{G_F m_b^2 \tan^2\beta
  (M_{\Upsilon}^2-M_h^2)}{4\sqrt{2}\pi \alpha}\biggl |\frac{T(\Upsilon\rightarrow h\gamma)}{T(\Gamma(\Upsilon\rightarrow
  \mu^+\mu^-)} \biggr |^2 \nonumber \\ & &
\times \biggl (1-\frac{4\alpha_s}{3\pi}
  a_H(z)+{\cal O}(a_s^2) \biggr )
\label{ups}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
had led to a lower bound,
%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
600~{\rm MeV} \lsim M_h \lsim 5~{\rm GeV}\;\;\;\;, \;\;\; {\rm
  for~}~\tan\beta=1\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%
The first term~\cite{wil,wu,biswas,azn} in Eq.(\ref{ups}) 
contains  the matrix elements $T$ (six diagrams in time order
perturbation theory~\cite{wu}) and the second~\cite{vys,nason}
the QCD corrections with $\alpha_H(z)$ being a loop factor 
with $z=1-\frac{M_h^2}{4m_b^2}$. The QCD corrections reduce
the rate by 84\% for $M_h<<M_{\Upsilon(1S)}$. 
Some comments are in order here. In the first calculation, 
done by Wilczek~\cite{wil},  one of the two dominant diagrams was considered.
In fact, as Faldt, Osland and Wu~\cite{wu} had
pointed out,  strong cancellations can occur between the two diagrams
leading to a suppresion factor to the rate of about 14 for $M_h=8.5$ GeV
(the suppression factor has a tendency of being increased while
$M_h\rightarrow M_{\Upsilon}$). Their calculation is valid only 
when the Higgs mass is arround the Upsilon mass 
($7.5~{\rm GeV}~ \lsim M_h \lsim M_{\Upsilon}$).
These supressions and the large QCD corrections and even problems
with renormalization scheme dependencies had led to a cloudy situation,
and  no reliable SM bound on the Higgs mass had arised from CUSB. 
However, as the reader might have already noticed, in the 2HDM we have
an extra huge enhancement proportional to $\tan^2\beta$. We feel like,
whatever large theoretical uncertainties\footnote{Unfortunately the
calculations towards that direction had been 
stopped since collider searches took
over.}  involved in the calculations
of the radiative Upsilon decays, the ratio (\ref{ups})
cannot be suppressed by a factor 
of $\tan^2\beta \sim 10^2-30^2$ (!), which is the case we are dealing with.  
We conclude that the lower bound on the 
Higgs boson mass  is just at the
Upsilon threshold
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
M_h\gsim M_{\Upsilon(1S)}=9.46~{\rm GeV}~~~{\rm for}~~~\tan\beta \gsim 10 \;.
\label{exp1}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%% 
This bound is denoted with the left area behind the 
yellow line in Fig.(1).  

The second lower bound on the Higgs mass  arises
from (DELPHI) LEP I searches~\cite{Delphi}. The only process which
is capable to constraint our very light Higgs boson is the Yukawa
process~\footnote{The pair production process is not kinematically
  allowed here
since as we have already explained,
 all the other Higgses should be above the Z-threshold in order 
to obtain the correct $\delta a_\mu$.}. Preliminary limits on
the Yukawa production ($Z\rightarrow f\bar{f}h$) of light neutral
Higgs
above the $2m_\tau$ threshold were set : $2m_\tau<M_h<2m_B$ from
$4\tau$ (see left corner bounded from the red line in our fig.(1))
topologies and $M_h>2m_B$ from 4b topologies (see right corner above
the red line in fig.(1)). It turns out that LEP I might have already
produced the higgs boson without seeing it. Unfortunately, LEP searches
cannot ruled out or confirm  the existence of such a light
Higgs boson and only a bound, 
%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
M_h \gsim 2 m_B=10.6 {\rm ~GeV}&~~~&{\rm for~ \tan\beta \gsim 28}\;
 \nonumber  \\  
M_h \gsim m_{\Upsilon}&~~~&{\rm for~ \tan\beta \gsim 44} \;,
\label{exp2}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
can be set.

 













\section{Combined Results and Conclusions}

If we combine the experimental searches for a light Higgs boson,
(\ref{exp1}) and (\ref{exp2}) we conclude that the only explanation for
the $2.6\sigma$ deviation  of the BNL measurement in the minimal two Higgs
doublet SM model (Model I or II) comes from a light Higgs boson in the range 
%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
m_{\Upsilon}\le M_h \le 2 m_B \;.
\label{result}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%
with $\tan\beta$ being in the region $30\lsim \tan\beta \lsim 44$.
Such a Higgs could be confirmed or ruled out 
in the future colliders~\cite{Krawczyk:2000kf}, or by the future
improvement
of the BNL~\cite{Prigl:2001cm} $g-2$ 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''.}


\begin{thebibliography}{99}

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%%%%%%%%%%%%%%%%%%chapter 2 %%%%%%%%%%%%%%%%%%%%
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%\cite{Fujikawa:1972fe}
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%%%%%%%%chapter 3%%%%%%%%%%%%%%%%%%%%%
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%%CITATION = PHLTA,B175,223;%%



\bibitem{Krawczyk:2000kf}
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%%CITATION = HEP-PH 0009201;%%



\end{thebibliography}

\end{document}



