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\author{\sf Athanasios Dedes}
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%%%%%%%%%%local defs
\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\amu{\delta a_\mu^{\rm NP}}
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\begin{document}
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\lhead{}
\rhead{\sf  \thepage}
\headrulewidth 2pt
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\chead{\sf{\usebox{\Title}}}
\date{30/6/2000}
\def\diad#1{\def\diadvalue{#1}}\def\diadvalue{}
\diad{\sf University of Bonn}
\lfoot{\diadvalue}
%\cfoot{\mbox{\epsfxsize=2.3cm\epsfysize=2.3cm
%\epsffile{../ral/ppd.ps}}}
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%   \mbox{\epsfxsize=1.3cm\epsfysize=1.3cm\epsffile{clrc.ps}} &
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\Trtitle{\sf  Can the Higgs sector contribute 
significantly to the muon anomalous magnetic moment?}
%
\Large
\sf
%
\begin{titlepage}


\setlength{\fboxrule}{2pt}

\vskip1cm

\begin{center}
{\LARGE \Red{ From the Planck Scale to the Electroweak
Scale, \\ La Londe-les-Maures, France, 11 - 16 May 2001}}\\[0.4cm]
%                           Collider Physics          }} \\
%{\LARGE \Red{ 19 - 24 September 1999,
%St John's College, Durham }}\\[1cm]
\Green{Athanasios Dedes}\\
\end{center}
\vskip 0.2 cm
\begin{center}
{\large   \sl   University of Bonn, Germany  } \\
\end{center}

\vskip2.0cm

\begin{center}
\fcolorbox{blau}{groguenc}{\textcolor{blau}{\fontsize{10mm}{16}
\begin{tabular}{c}
\sf       \phantom{  ..... ............... .....}     \\
\sf        \Red{{\bf Can the Higgs sector contribute }} \\
\sf \Red{{ \bf  significantly  to the   }}\\
\sf     \Red{{\bf  muon anomalous magnetic moment?  }} \\
% \\ \sf \Red{{\bf at hadron colliders in the MSSM}}    \\
\sf       \phantom{  ..... ............... .....}     \\
\end{tabular}}}
\end{center}

\vskip0.5cm
\begin{center}
\Green{In collaboration with:}\\[2mm]
\Red{H.~ E.~ Haber, {\tt hep-ph/0102297}, JHEP {\sl to appear}} \\
%\Red{A. Dedes and S. Moretti, Nuclear Phys. B to appear} \\[2mm]
%\Red{A. Dedes and M. Pospelov, Phys. Rev. D61:116010,2000}\\[2mm]
%\Red{S. Abel, A. Dedes and H. Dreiner, JHEP0005:013,2000}\\
\end{center}


\vskip0.9cm

\LARGE{ \Blue{We show :}}
\Large{
\begin{enumerate}
\item \Blue{A light CP-even Higgs boson with \Red{$\mhl\sim{\cal
        O}(10)$~GeV}  could
explain the recent BNL measurement of the muon anomalous magnetic
moment }\\
\item \Blue{Framework : General CP-conserving
two-Higgs-doublet extension of
the Standard Model with no tree-level flavor-changing neutral Higgs 
couplings  }\\
\item \Blue{ Significant contribution  to muon $g-2$
 \sl{\Red {without violating known experimental constraints}} {\rm  ~only
if the \Red{$\hl ZZ$} coupling (approximately) \Red{vanishes} and
\Red{$M_{\Upsilon}\lsim\mhl\lsim 2m_B$}.} }\\
%\item \Blue{ Suggestions }
\end{enumerate}
}


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%\epsffile{../ral/ppd.ps}}  \\
% \end{tabular}
% --
%\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{titlepage}


%
% ----------------------------------------
%
% ----------------------------------------
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Theory , Channels ...
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\include{phases}
%\include{strong_CP}
%\include{edms_rparity}
%\include{bonn_conclusions}
%
\newpage
%%%%%%%%%%%%%%%% PAGE 1 %%%%%%%%%%%%%%%%%

\Green{$\bullet$} \Blue{Based on Davier-H\"ocker calculation for the
  Hadronic
contributions to the muon 
$g-2$, BNL reports a 2.6$\sigma$ deviation from the SM}

\spa


There are four possible interpretations
of this discrepancy :
\begin{itemize}

\item a statistical fluctuation with 0.9\% probability.


\item something is ``wrong'' with the experiment


\item something is ``wrong'' with the theoretical analysis of the
hadronic contribution to
the muon anomalous magnetic moment.



\item the by far most exciting possibility: 
\end{itemize}


%\Green{$\bullet$} 
\Blue{
    there is \Red{``New Physics''}~ BSM}
\begin{eqnarray}
215 \times 10^{-11} \lsim \delta a_\mu^{NP} \lsim 637\times 10^{-11}
\;\;\;\;
{\rm @ 90 \% ~CL}\nonumber
\end{eqnarray}
\begin{itemize}

\item{\Blue{$\delta a_\mu^{NP}$ is positive}}

\item{\Blue{$\delta a_\mu^{NP}$ is of the same order of the EW
      contributions, 
$\delta a_\mu^{NP} \propto \frac{G_F m_\mu^2}{4\pi^2\sqrt{2}}$}}

\end{itemize}

\Green{$\bullet$} \Blue{Why not from the Higgs sector ?}

\bc\Red{\underline{\Green{Standard model (SM)}}}\ec
\begin{eqnarray}
\delta a_\mu^{h} \propto \frac{G_F m_\mu^2}{4\pi^2\sqrt{2}}
\biggl (\frac{m_\mu^2}{m_h^2} \biggr )\nonumber 
\end{eqnarray}
\Blue{suppressed since $m_h \gsim 113.5$ GeV}

\spa

\newpage

\bc
\Red{\underline{\Green{Two Higgs Doublet Model (II) [2HDM(II)] : }}}
\ec

\Blue{ The enhancement from the Higgs sector to $a_\mu$ arises from
two  sources :}



  \Red{
\begin{eqnarray}
\{h,A\}-\mu^+-\mu^- &\propto& \tan\beta \;\;\;\;\Rightarrow \;\;\delta 
a_\mu^{h,A}
\propto \tan^2\beta \nonumber\\[5mm]
H^--\mu^+-\nu &\propto & \tan\beta \;\;\;\;\Rightarrow \;\;\delta 
a_\mu^{H^\pm} \propto \tan^2\beta \nonumber }
\end{eqnarray}

 \Blue{and \Red{\begin{eqnarray}
h-Z-Z \propto \sin(\beta-\alpha)\;\; \simeq 0 \nonumber \\ \nonumber 
\end{eqnarray}}
 then
\Red{$m_h$} can be as small as \Red{$\sim 10$ GeV}.}


%%%%%%%%%%%%%%%%%%%%%PAGE 3%%%%%%%%%%%%%%%%%%%%%%%%%%
\spa

\CTran{2HDM(II) Higgs boson corrections to $a_\mu$}



\normalsize
%%%%%%%%%%%%%%% FIGURES %%%%%%%%%%%%%%%%%
%\begin{figure}[htbp]
%\begin{center}
\begin{tabular}{lr}
\begin{picture}(15,5)(0,0)
% lower horizontal line
\ArrowLine(30,20)(10,-20)
\Text(1,2)[]{\mbox{$\mu$}}
\Vertex(30,20){2}
\DashLine(30,20)(90,20){7}
\Text(2,0.4)[]{\mbox{$h,H,A$}}
\Vertex(90,20){2}
\ArrowLine(110,-20)(90,20)
\Text(0,0)[]{\mbox{${\mu}$}}
%left skew line
\ArrowLine(60,80)(30,20)
\Text(4.3,0)[r]{\mbox{$\mu$}}
% right skew line
\ArrowLine(90,20)(60,80)
\Text(3,2)[l]{\mbox{$\mu$}}
%photon vertical line
\Photon(60,80)(60,110){3}{3}
\Text(2.5,4)[c]{\mbox{$\gamma$}}
\end{picture}
&
\begin{picture}(15,5)(5,0)
% lower horizontal line
\ArrowLine(30,20)(10,-20)
\Text(1,2)[]{\mbox{$H^-$}}
\Vertex(30,20){2}
\ArrowLine(90,20)(30,20)
\Text(2,0.4)[]{\mbox{$\nu$}}
\Vertex(90,20){2}
\ArrowLine(110,-20)(90,20)
\Text(0,0)[]{\mbox{$\mu$}}
%left skew line
\DashArrowLine(60,80)(30,20){7}
\Text(4.3,0)[r]{\mbox{$\mu$}}
% right skew line
\DashArrowLine(90,20)(60,80){7}
\Text(3,2)[l]{\mbox{$H^-$}}
%photon vertical line
\Photon(60,80)(60,110){3}{3}
\Text(2.5,4)[c]{\mbox{$\gamma$}}
\end{picture}
%
%%%%%%%%%%%%%%%%%%%%
\end{tabular}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\spa

\spa

\spa
\Large
\bc
\underline{Non-decoupling limit: \Green{$\sin(\beta-\alpha)=0$}}
\ec

\Blue{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%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 \\[1cm]
&& \simeq\frac{G_Fm_\mu^2}{4\pi^2 \sqrt{2}}
\tan^2\beta \bigg \{ \Blue{
\frac{m_\mu^2}{\mhl^2} \biggl[\ln\left(\frac{\mhl^2}{m_\mu^2}\right)
-\frac{7}{6} \biggr ] }
\Red{-\frac{m_\mu^2}{\mha^2} \biggl[
\ln\left(\frac{\mha^2}{m_\mu^2}\right) -\frac{11}{6}\biggr]
\Red{-}\frac{m_\mu^2}{6m_{H^{\pm}}^2} }\biggr \} \;. \nonumber \\
\nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
}


\Green{$\surd$} \Blue{$\delta a_\mu^H$ can be neglected since
it does not give a contribution proportional to
$\tan\beta$}

\spa

\Green{$\surd$} \Blue{$\ha$ and $\hpm$ exchange contribute a
negative value to $\amu$.}

\spa

\Green{$\surd$} \Blue{In order to explain the BNL $g-2$ we need
positive corrections. We take $\mha$ and $\mhpm$ large, $\gsim 100$ GeV.} 

\spa

\Green{$\surd$} \Blue{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.}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\spa

\Blue{Therefore :}

\Red{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\delta a_{\mu}^{\rm Higgs}\simeq 
\delta a_\mu^{{\hl}} \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 ] \;.
\nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%
}


\spa

\Blue{A light Higgs boson with a mass of
around \Red{10 GeV} and with \Red{$\tan\beta=35$} gives 
\Red{$\delta a_{\mu}^{\rm
Higgs}\simeq 280 \times 10^{-11}$}, which is within the
\Red{90\%~CL} allowed range for $\amu$.}

\spa

\bc
\underline{Decoupling limit: \Green{$\cos(\beta-\alpha)=0$}}
\ec

\spa

\Green{$\surd$} \Blue{The couplings of the light Higgs boson,
$\hl$, are (nearly) identical to those of the SM Higgs boson :
\Red{$\mhl\gsim 113.5$~GeV}.}

\spa

\Green{$\surd$} \Blue{The $\hh$ couplings to down-type fermion pairs are 
enhanced by $\tan\beta$}

\spa


\Green{$\surd$} \Blue{For : $\cos(\beta-\alpha)=0$ 
and $\mhh=\mha=\mhpm$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\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 ]\;. \nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

\spa

\Green{$\surd$} \Blue{For \Red{100~GeV $<\mha< 1000$~GeV}, and 
\Red{$30<\tan\beta <100$},
the Higgs sector contribution to \Red{$a_\mu$} ranges from about \Red{$5\times
10^{-12}$} to \Red{$5\times 10^{-14}$}, which is
three to five orders of magnitude below what
is needed to explain the BNL measurement of \Red{$a_\mu$}.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\CTran{Results}

\spa

\vspace*{2cm}

\centerline{\epsfig{figure=non_decoup.ps,height=6in,angle=90}}


%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
&& \Red{m_{\Upsilon}\lsim \mhl \lsim 2 m_B \;,} \nonumber \\
&& \Red{\sin(\beta-\alpha)\simeq 0 \;,} \nonumber \\
&& \Red{30\lsim\tan\beta\lsim 35 \;.} \nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\CTran{CESR constraints on the light $m_h$}

\spa

\Green{$\bullet$} \Blue{An experimental search 
for \Red{$\Upsilon\to\hl\gamma$} by the
CUSB Collaboration at CESR 
found no candidates.}

\Green{$\bullet$} \Blue{In addition to the non-relativistic,
tree-level prediction there are :}

\begin{itemize}

\Red{\item} \Blue{${\cal O}(\alpha_s)$ hard QCD
corrections}

\Red{\item} \Blue{relativistic corrections to the
non-relativistic treatment of the $b\bar b$ bound state}

\Red{\item} \Blue{bound state threshold corrections }

\end{itemize}

\spa

\Red{Standard Model:} \Blue{ $\mhl\lsim 5$---7~GeV can be ruled out
at $95\%$~CL but no precise upper limit due to 
the theoretical uncertainties}

\spa

\Red{2HDM :}  \Blue{\Red{$\Gamma(\Upsilon\to\hl\gamma)$} is enhanced by a
factor of \Red{$\tan^2\beta$} }

\spa

\Green{$\surd$} \Blue{ For values of $\tan\beta\gsim 10$, the CUSB data
can reliably rule out Higgs masses up to about 8~GeV.}

\spa

\Green{$\surd$} \Blue{ For large $\tan\beta\simeq 30$
 values, the $\tan^2\beta$
factor should dwarf the theoretical
uncertainties!}

\Green{
\bc
\begin{eqnarray}m_h \gsim m_\Upsilon\nonumber\end{eqnarray}
\ec 
}


[See \Red{red} region in the previous figure]


%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%

\CTran{LEP constraints on the light $m_h$}

\spa


\Green{$\bullet$} \Blue{A second bound on $\mhl$ can be 
derived from the non-observation of 
Higgs bosons at LEP via the process
 $e^+e^-\to \hl f\bar f$ ($f=b$, $\tau$).}

\spa

\spa

\spa


\centerline{\epsfig{figure=delphi.ps,height=5in}}


\spa

\spa

\sl{From J.~Kurowska, O.~Grajek and P.~Zalewski  [DELPHI Collaboration],
%``Search for Yukawa production of a light neutral Higgs at LEP 1,''
CERN-OPEN-99-385.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\spa

\spa

\vspace*{2cm}

\centerline{\epsfig{figure=aleph.ps,height=5in}}

\spa

\Blue{Similar for $m_h$}
\spa

\sl{From J.B.~de Vivie and P.~Janot [ALEPH Collaboration], 
%``Search for a Light
%Higgs Boson in the Yukawa Process,'' 
PA13-027 contribution to the
International Conference on High Energy Physics, Warsaw, Poland,
25--31 July 1996.


\spa
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%

\spa 

\spa

\Green{$\bullet$} \Blue{Other constraints :}

\begin{itemize}

\item \Blue{the one-loop process \Red{$Z\to\hl\gamma$}}

\item \Blue{precision EW observables}

\end{itemize}

\Red{$\rightarrow$} \Blue{weaker bounds on $m_h-\tan\beta$
 than CESR+LEP ones}

\spa

\spa

\spa

\centerline{\underline{\Red{One noteworthy consequence of $\mhl\sim
      10$~ GeV}}}

\spa 

\Green{$\bullet$} \Blue{the possibility of mixing
between the $\hl$ and the $0^{++}$ $b\bar b$ bound states
$\chi\ls{b0}(1P)$ and $\chi\ls{b0}(2P)$}


%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
{\Gamma(\chi\ls{b0}\to\tau^+\tau^-)\over\Gamma(\chi\ls{b0}\to~{\rm hadrons})}
\simeq {2.5\times 10^{-7}~{\rm GeV}^2\over
(m_\chi-\mhl)^2}\tan^4\beta \nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%


\Green{$\surd$} \Blue{Due to the large $\tan^4\beta$ enhancement, 
the predicted branching ratio for 
$\chi\ls{b0}\to\tau^+\tau^-$ can be substantial.}

\spa

\Green{$\surd$} \Blue{Particle Data Group  provides no data on possible 
decay modes of the $\chi\ls{b0}$ other than the radiative decays, 
$\chi\ls{b0}\to\Upsilon\gamma$, $\Upsilon'\gamma$.}


%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%
\CTran{Final Results and Conclusions}

\spa

\Green{$\bullet$} \Blue{Combined CESR+LEP+BNL(DH) in 2HDM(II):}

%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
&& \Green{\bullet}~\Blue{m_{\Upsilon}\lsim \mhl \lsim 2 m_B \;,} \nonumber \\
&& \Green{\bullet}~\Blue{\sin(\beta-\alpha)\simeq 0 \;,} \nonumber \\
&& \Green{\bullet}~\Blue{30\lsim\tan\beta\lsim 35 \;.} \nonumber
\end{eqnarray}
%%%%%%%%%%%%%%%%%%

\Green{$\bullet$} \Blue{In addition, 
$\hh$, $\ha$ and $\hpm$ must be sufficiently heavy to
satisfy the LEP experimental constraints :}
\begin{eqnarray}
\Green{\surd}~\Blue{\mhh &\gsim &  113.5~{\rm GeV}} \nonumber \\
\Green{\surd}~\Blue{\mha &\gsim & 80~{\rm GeV}}\nonumber \\
\Green{\surd}~\Blue{\mhpm &\gsim & 78.7~{\rm GeV}} \nonumber
\end{eqnarray}

\spa

\underline{\Red{These results can be confirmed or ruled out by :}}

\spa

\Green{$\bullet$} \Blue{a more complete
analysis by the LEP Collaborations of their data in search of
\Red{$e^+e^-\to \hl f\bar f$ ($f=b$ or $\tau$)}}

\spa

\Green{$\bullet$} \Blue{further improvement of the BNL 
measurement of the muon anomalous magnetic moment $a_\mu$}

\spa

\Green{$\bullet$} \Blue{further improvement of the theoretical
calculations on the hadronic contributions to 
the muon  anomalous magnetic moment.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\spa

\spa

\vspace*{2cm}

\centerline{\epsfig{figure=sinbma.ps,height=5in}}



\end{document}





