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%%%            Single Charged Higgs Boson Production in              %%%
%%%      Polarized Photon Collision and the Probe of New Physics     %%%
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%%%          Hong-Jian He,  Shinya Kanemura,  C.-P. Yuan             %%%
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\def\thisday{September, 2002} 
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\begin{document}                                                              



\preprint{{\large }

\title{Single Charged Higgs Boson Production in\\[-1.5mm]              
    Polarized Photon Collision and the Probe of New Physics
\vspace*{5mm} 
        }%
%  
\author{%
{\sc Hong-Jian He}\,$^1$\footnote{hjhe@physics.utexas.edu},~~      
{\sc Shinya Kanemura}\,$^2$\footnote{shinya.kanemura@kek.jp},~~    
{\sc C.--P. Yuan\,$^3$}\footnote{yuan@pa.msu.edu}
}
\affiliation{%
%\address{\vspace*{5mm}
\vspace*{5mm} 
$^1$Center for Particle Physics, 
University of Texas at Austin, Texas 78712, USA\\
$^2$Theory Group, KEK, Tsukuba, Ibaraki 305-0081, Japan\\
$^3$Department of Physics and Astronomy, 
Michigan State University, East Lansing, Michigan 48824, USA
\vspace*{30mm} 
}
%\maketitle

\begin{abstract}
\hspace*{-0.35cm}
We study single charged Higgs boson production
in photon-photon collision as a probe of the new dynamics of
Higgs interactions.  
This is particularly important
when the mass ($M_{H^\pm}$) of charged Higgs bosons 
($H^{\pm}$) is relatively heavy and above the kinematic limit 
of the pair production (\,$M_{H^\pm} > \sqrt{s}/2$\,). 
We analyze the cross sections of single charged Higgs boson production 
from the photon-photon fusion processes, 
$\gamma\gamma\to \tau \bar \nu H^+$ and 
$\gamma\gamma\to  b   \bar c H^+$, as 
motivated by the minimal supersymmetric 
standard model and the dynamical Topcolor model.
We find that the cross sections at such a $\gamma\gamma$ collider
can be sufficiently large even for 
\,$M_{H^\pm} > \sqrt{s}/2$\,, and is typically one to
two orders of magnitude higher than that at its parent 
$e^-e^+$ collider.
We further demonstrate that the polarized photon beams can
provide an important means to determine the chirality structure of 
Higgs Yukawa interactions with the fermions. 
%
\pacs{\,12.60.-i,\,12.15.-y,\,11.15.Ex 
\hfill   ~~ [ \thisday\, ] }

\end{abstract}

\maketitle


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\newpage
\section{Introduction} 


The Standard Model (SM) of particle physics demands a single
neutral physical Higgs scalar ($h^0$)\,\cite{Higgs} to generate masses
for all observed weak gauge bosons, quarks and leptons,
while leaving the mass of Higgs boson and all its Yukawa couplings
unpredicted.  
A charged Higgs boson ($H^\pm$) is an unambiguous 
signature of the new physics beyond the SM.
Most extensions of the SM require an extended electroweak symmetry
breaking (EWSB) sector with charged Higgs scalars as part of its
physical spectrum at the weak scale.  
The electroweak gauge interactions of $H^\pm$
are universally determined by its electric charge and weak-isospin,
while the Yukawa couplings of $H^\pm$ are model-dependent
and can initiate new production mechanisms for $H^\pm$ at high energy
colliders. Most of the underlying theories that describe the EWSB
mechanism can be categorized as either a ``supersymmetric'' (with
fundamental Higgs scalars) \cite{SUSY} or a ``dynamical'' (with
composite Higgs scalars) \cite{DSB} model.
The minimal supersymmetric SM (MSSM) \cite{MSSM} and the dynamical
Top-color model \cite{Hill} are two typical examples.
As we will show, the Yukawa couplings
associated with the third family quarks and leptons can be large
and distinguishable in these models, so that measuring the single
charged scalar production rate in the polarized photon collisions
can discriminate these models of flavor symmetry breaking.



If a charged Higgs boson could be sufficiently light, 
with mass ($M_{H^\pm}$) below $\sim 170$\,GeV, it may 
be produced from the top quark decay, 
$\, t \to H^+ b$~\cite{tbH},  at
the hadron colliders, including the Fermilab Tevatron and the  
CERN Large Hadron Collider (LHC).
For $\,M_{H^\pm} > m_t-m_b$,\, 
$H^\pm$ can be searched at the Tevatron and the LHC
from the production processes 
$g b \to H^\pm t$ \cite{gbHt}, 
$cs, cb \to H^\pm$ \cite{hy,bhy,dhy,xx}, and 
$gg$, $q\bar q$ 
$\to H^\pm W^\mp$\cite{ppHW,ppHW2}, etc.  
The associated 
production of $H^\pm t$ from $gb$ fusion is difficult to detect  
at the Tevatron because of its small rate (largely 
suppressed by the final state phase space), 
but it should be
observable at the LHC for $M_{H^\pm} \lesssim 1$\,TeV. 
The single $H^\pm$ production from $cs$ or $cb$ 
fusions is kinematically advantageous so that it can yield a sizable 
signal rate, and can be detected at colliders as long as   
the relevant Yukawa couplings are not too small\,\cite{hy,dhy}. 
The $gg \to H^\pm W^\mp$ process originates from loop corrections,
and is generally small for producing a heavy $H^\pm$
unless its rate is enhanced by $s$-channel resonants, 
such as $gg \to H^0 ({\rm or~} A^0) \to H^\pm W^\mp$.\,
Similarly, the rate of $q\bar q \to H^\pm W^\mp$ is small 
in a general two-Higgs-doublet model (2HDM).
This is because for light quarks in the initial state, this process 
can only occur at loop level, and for heavy quarks in the initial state,
this process can take place at tree level via Yukawa couplings but is 
suppressed by small parton luminositites of 
heavy quarks inside the proton (or anti-proton).
If $H^\pm$ is in a triplet representation, the
$Z$-$H^\pm$-$W^\mp$ vertex can arise from a custodial breaking term
in the tree level Lagrangian, but its strength has to be
small due to the strong experimental constraint on the $\rho$-parameter.
Hence, the production rate of $q\bar q \to Z \to H^\pm W^\mp$
cannot be large either.
At hadron colliders, the charged Higgs bosons can also be produced in 
pairs via the $s$-channel 
$q \bar q$ fusion process through the gauge interactions of 
$\gamma$-$H^+$-$H^-$ and $Z$-$H^+$-$H^-$\,\cite{ppHpHm}. 
However, the rate of
the pair production generally is much smaller than that predicted by  
the single charged Higgs boson production mechanisms  
when the mass of the charged Higgs boson increases. 



If $M_{H^\pm}$ is smaller than half of the center-of-mass
energy ($\sqrt{s}$) of a Linear Collider (LC), then $H^\pm$ may be 
copiously produced in pairs via the scattering processes
$e^-e^+ \to H^- H^+$ and $\gamma \gamma \to H^- H^+$~\cite{eeHpHm}.
The production rate of a $H^- H^+$ pair is determined by 
the electroweak gauge interactions of $H^\pm$, which 
depends only on the electric charge and weak-isospin of $H^\pm$.
When $\,M_{H^\pm} > \sqrt{s}/2$,\, 
it is no longer possible to produce the 
charged Higgs bosons in pairs. In this case, the predominant production
mechanism of the charged Higgs boson is via the 
single charged Higgs boson production processes, such as 
the loop induced process 
$e^-e^+ \to H^\pm W^\mp$~\cite{eeWH,KMO1}, 
and the tree level processes 
$e^-e^+ \to b \bar c H^+, \, \tau \bar \nu H^+$  and 
$\gamma\gamma \to b \bar c H^+, \, \tau \bar \nu H^+$\,\cite{hky}.
The production rate of the above tree level processes depends on the 
Yukawa couplings of fermions with $H^\pm$.
This makes it possible to discriminate models of flavor 
symmetry breaking by measuring the production rate of the single 
charged Higgs boson at LC.
However, as to be discussed below, at $e^+e^-$ colliders,
the cross sections of the single $H^\pm$ production processes
induced by the Yukawa couplings of fermions with $H^\pm$
are generally small because single $H^\pm$ events are  
produced via $s$-channel processes 
(with a virtual photon or $Z$ propagator). 
On the other hand, at $\gamma\gamma$ colliders, 
the single $H^\pm$ cross sections 
are enhanced by the presence of the $t$-channel 
diagrams which contain collinear poles in high energy collisions.

 
In this work, we systematically study single charged Higgs boson production 
associated with a fermion pair ($\bar{f'} f$) at photon colliders, i.e., 
$\gamma\gamma\to \bar f' f H^\pm$, 
($\bar{f'} f = bc,~{\rm or},~\tau\nu $), based on our recent proposal
in Ref.\,\cite{hky}.
Two general classes of models will be discussed to predict the signal 
event rates -- one is the weakly interacting models 
represented by the MSSM\,\cite{MSSM} 
and another is the dynamical symmetry breaking models
represented by the Top-color (TopC) model\,\cite{Hill}.
We show that the yield of a heavy charged Higgs boson 
at a $\gamma\gamma$ collider is typically one to two orders
of magnitude larger than that at an  $e^-e^+$ collider.
Furthermore, we demonstrate that
a polarized photon collider can either enhance or suppress 
the single charged Higgs boson production, depending on the 
chirality structure of the corresponding Yukawa couplings.
In the current analysis, we shall consider the center-of-mass energy of 
a \,$\gamma\gamma$\, collider to be about 80\% of 
an \,$e^-e^+$\, collider.
  

It is well known that the main motivation for building a
high-energy polarized photon collider is 
to determine the \tx{CP} property of 
the neutral Higgs bosons\,\cite{CPprop1,CPprop2,RevLC}.  
In this work, we provide another motivation 
for having a polarized photon collider -- 
to determine the chirality structure of the fermion Yukawa couplings 
with the charged Higgs boson via single charged Higgs 
boson production so as to discriminate the dynamics of flavor symmetry
breaking.



\section{Yukawa Interactions in MSSM and Top-color Model
}
%\newpage


For generality, we may define the charged Higgs Yukawa interaction as 
\beq
\label{eq:yukawa}
{\cal L}_{\rm Y} = \ov{f'} 
\left( Y_L^{f'f} P_{L} + Y_R^{f'f} P_{R} \right)
       f \,H^- + {\rm h.c.}\,, 
\eeq 
where $f$ and $f'$ represent up-type and down-type fermions, respectively, 
and  
$P_{L,R}$ are the chirality projection operators 
$ P_{L,R}= \(1\mp \gamma_5\)/2$\,.



We first consider the Yukawa sector of the MSSM, which is similar to
that of a Type-II 2HDM.
The corresponding tree-level Yukawa couplings of fermions 
with $H^\pm$ are given by 
\begin{eqnarray}
\label{eq:yukawaC}
Y_{L(0)}^{f'f} \!=\! \frac{\sqrt{2} m_{f'}}{v} V_{ff'}  \tan\!\beta, \;\;\;
Y_{R(0)}^{f'f} \!=\! \frac{\sqrt{2} m_f}{v} V_{ff'}  \cot\!\beta,  
\end{eqnarray}
where $m_{f}$ ($m_{f'}$) is  the mass of the fermion 
$f$ ($f'$), 
\,$\tan\beta = \langle H_u\rangle / \langle H_d\rangle$\,
is the ratio of the vacuum expectation values 
(${\langle H_u\rangle}$ and ${\langle H_d\rangle}$)
of two Higgs doublets with 
$v = \sqrt{{\langle H_u\rangle}^2 + {\langle H_d\rangle}^2} 
   \simeq 246$\,GeV,
and $V_{ff'}$ is the relevant 
Cabibbo-Kobayashi-Maskawa (CKM) matrix element
of the fermions $f$ and $f'$. 
The coupling constants 
$Y_{L(0)}^{f'f}$ and $Y_{R(0)}^{f'f}$ 
vary as the input parameter $\tan\beta$ changes.   
%
For instance, for the $\bar \tau$-$\nu$-$H^-$ coupling,  
$Y_{L(0)}^{\tau\nu}$ increases as $\tan\beta$ grows, and 
reaches about \,$0.20-0.51$\, for \,$\tan\beta=20-50$,\, while 
$Y_{R(0)}^{\tau\nu}$ is zero because of the absence of
right-handed Dirac neutrinos in the MSSM.
Without losing generality, we shall choose the following 
typical inputs
for our numerical analysis:
\beq
\(Y_{L(0)}^{\tau\nu},\, Y_{R(0)}^{\tau\nu}\)
\,=\, (0.3,~0)\,,
~~~~{\rm for}~ \tanb = 30\,.
\label{eq:taunuH-MSSM-def}
\eeq


The tree level $\bar b$-$c$-$H^-$ coupling 
contains a CKM suppression factor $V_{cb}\simeq 0.04$, so that 
$Y_{L(0)}^{bc}$ is around $0.03$ for $\tan\beta=50$, and  
$Y_{R(0)}^{bc}$ is less than about $2\times 10^{-4}$ for $\tan\beta > 2$. 
However, the SUSY radiative correction can significantly enhance 
the tree level $\bar b$-$c$-$H^-$ coupling.
It was shown in Ref.\,\cite{dhy} that 
the radiatively generated $\bar b$-$c$-$H^-$ coupling from 
the stop-scharm ($\ts-\cs$) mixings 
in the SUSY soft-breaking sector
can be quite sizable.
For instance, in the minimal Type-A SUSY models, the 
non-diagonal scalar trilinear $A$-term for the 
up-type squarks can be written as \cite{dhy}  
\beq
A_u ~=~
      \left\lgroup
      \bea{ccc}
      0 ~&~ 0 ~&~ 0\\
      0 ~&~ 0 ~&~ x\\
      0 ~&~ y ~&~ 1
      \eea
      \right\rgroup A \,,
\label{eq:Au}
\eeq
which generates a non-trivial $4\times4$ squark mass-matrix among
$(\cs_L,\,\cs_R,\,\ts_L,\,\ts_R)$. 
In $A_u$, the parameters $(x,\,y)$ can be naturally of  
order 1,
representing large $\ts-\cs$ mixings that are consistent with
all the known theoretical 
and experimental constraints \cite{CCBVS,FCNC}.
An exact diagonalization of
this  $4\times4$ mass-matrix results in the following mass eigenvalues:
\beq
\bea{ll}
M_{\cs1,2}^2 & = \sm0^2 \mp\f{1}{2}|\sqrt{\omega_+}-\sqrt{\omega_-}|\,, 
\\[3mm]
M_{\ts1,2}^2 & = \sm0^2 \mp\f{1}{2}|\sqrt{\omega_+}+\sqrt{\omega_-}|\,,
\eea
\label{eq:Mass}
\eeq
with 
$
M_{\ts1} < M_{\cs1} < M_{\cs2} < M_{\ts2} \,.
$
Here, $\ms_0$ is a common scalar mass in the diagonal blocks of
the squark mass-matrix, 
$~\omega_\pm = X_t^2+(x\Ah\pm y\Ah )^2\,$, 
$ X_t = \Ah - \mu\,m_t\,\cotB$ and 
$ \Ah = Av\,\sB/\sqrt{2}\,$.
In the squark mass-eigenbasis, the $\bar b$-$c$-$H^-$ 
coupling can be radiatively induced from the
vertex corrections [scharm(stop)-sbottom-gluino loop]
and the self-energy corrections [scharm(stop)-gluino loop]. 
In the Type-A models with $ x\neq 0$ and $y=0$,
including the one-loop SUSY-QCD 
corrections yields the pattern\,\cite{dhy}:
\beq
 \delta Y_L^{bc}  \neq 0\,  ~~~{\rm and}~~~
 \delta Y_R^{bc}   \simeq  0\,,
\label{eq:A1-cbH}
\eeq
for a moderate to large $\tanb$.
(As to be shown below, this pattern is opposite to that predicted 
in the dynamical Top-color model.) 
The coupling $ Y_L^{bc}$ is a function of 
the mixing parameter $x$, 
the Higgs mass $M_{H^\pm}$, 
the gluino mass $M_{\gs}$ and the relevant squark masses. 
In Fig.~\ref{fig:YRbcH}, we show  $ Y_L^{bc} $
as a function of the parameter $x$
for a typical set of SUSY inputs, 
$(m_{\tilde{g}},\,\mu,\,\ms_0 )= (300,\,300,\,600)$\,GeV,
$A=-A_b=1.75$\,TeV, and $\,\tanb=50\,$.
In this figure, 
we have also included the QCD running effects for
the tree-level Yukawa couplings, cf. Eq.\,(\ref{eq:yukawaC}).
We find that the magnitude of the total coupling $ Y_L^{bc} $ can be 
naturally in the range of $0.03-0.07$ 
for a moderate to large $\tanb$\,.
For a smaller
value of $\tan \beta$, the coupling $ Y_L^{bc} $ decreases.
For instance, for $\tanb=20$, the value of $ Y_L^{bc}$ is about 
half of that shown in the Fig.~\ref{fig:YRbcH}. 


\begin{figure}[H]
\vspace*{-12mm}
\begin{center}
\hspace*{-3mm}
\includegraphics[width=14cm,height=18cm]{fig_mssma1ff.eps}
%\includegraphics[width=9.8cm,height=16cm]{fig_mssma1ff.eps}
%{\par\centering \resizebox*{0.4\textwidth}
%{!}{\includegraphics{fig_mssma1.eps}} \par}
\end{center}
\vspace*{-8.5cm}
\caption{
The radiative $\bar b$-$c$-$H^-$ coupling
as a function of the parameter $x$
in the minimal Type-A SUSY models with $y=0$.
Here, we set $(m_{\tilde{g}},\mu,\ms_0 )= (300,\,300,\,600)$\,GeV,
and $A=-A_b=1.75$\,TeV. This result also includes the QCD running 
effect for the Born level Yukawa coupling. 
}
\label{fig:YRbcH}
\end{figure}


As discussed above, 
the SUSY radiative corrections are not suppressed 
by the small CKM matrix element $V_{cb}$. In addition to these,
there are also corrections proportional to $V_{cb}$, similar to 
those present in the production of $\phi^0 b {\bar b}$ 
($\phi^0=h^0,H^0,A^0$) with large $\tanb$~\cite{hbb,efflag}.  
This effect can be formulated by the corresponding effective
Lagrangian\,\cite{dmb},
\begin{equation}
\mathcal{L\,} ~=~
\frac{\,\sqrt{2}\, V_{cb}\,}{v}\,\frac{
~\overline{m}_{b}(\mu^{~}_{R})\tan\beta~}{1+\Delta_{b}}
\,H^{+}\overline{c_L}
\,b_{R} \,+\, {\rm h.c.} \, ,
\label{eq:efflag}  
\end{equation}
where $\mu_{R}$ is the relevant renormalization scale at which we
evaluate the bottom quark running mass $\overline{m}_{b}(\mu_{R})$ 
including the NLO QCD contributions under the
$\overline{\rm MS}$ scheme.
In the on-shell scheme, the bare mass of the bottom quark
$m_{bare}$ is equal to $m_{b}+\delta m_{b}$, where $m_b$ 
is the pole mass and  $\delta m_{b}$ the counter term.
A straightforward calculation shows that the threshold 
corrections to $\Delta_{b}$ originating from the SUSY-QCD and 
SUSY-electroweak (SUSY-EW) contributions are equal to
$-\delta m_{b}/m_{b}$.
In general, 
the SUSY-EW correction comes from loop contributions induced by 
the Yukawa and electroweak gauge interactions, where the latter 
contribution is usually smaller than the former contribution.
(Since in the generic Type-A model the trilinear term $A$ needs not
to be much smaller than \,$\mu\tanb$,\, we will not make the approximation
\,$A_b -\mu\tanb \approx -\mu\tanb$ \cite{efflag} 
in the $\Delta_{b}$ formula.)
The SUSY-QCD correction is given by
the finite contributions of sbottom-gluino loop due to
the left-right mixings in the squark-mass matrix\,\cite{dmb},  
%
\begin{eqnarray}
\left( \Delta_{b}\right) _{\mathrm{SUSY-QCD}} & =&
-\,{
\frac{~C_F\alpha_{s}(\mu_R^{~})~}{2\pi}}\,
   m_{\tilde{g}}\,M_{LR}^{b}\,
{\cal I}(m_{\tilde {b}_{1}},\,m_{\tilde{b}_{2}},\,m_{\tilde{g}})  
\,,
\label{eq:dmbQCD}
\end{eqnarray}
%
where 
\,$C_{F}=\dis\f{1}{2}\( N_{c}-\f{1}{N_c}\) = \f{4}{3}$\, 
with $N_{c}=3$,~
$\alpha_{s}\simeq 0.09$\, at the scale of 
$\mu_R^{~}=M_{H^\pm}= O(100)$\,GeV, and 
$M_{LR}^{b}=A_{b}-\mu\,\tan\beta$\,. 
The SUSY-Yukawa correction to $\Delta_{b}$ arises 
from similar loops involving the stop and charged
higgsinos \,$\widetilde{H}_{1,2}$\,, 
%
\begin{eqnarray}
\left( \Delta_{b}\right) _{\mathrm{SUSY-Yukawa}} 
& =& + \frac{~m_t^2~}{~8\pi^2 v^2~} \f{\mu}{\tanb} M_{LR}^{t}\,
{\cal I}(m_{\tilde {t}_{1}},\,m_{\tilde{t}_{2}},\,\mu)  \, ,
\label{eq:dmbEW}
\end{eqnarray}
%
where 
\,$y_b=\dis\f{\sqrt{2}m_{b}}{v}\,\tan\beta$,~ 
$y_t=\dis\f{\sqrt{2}m_t}{v}\,\cot\beta$,\,
and \,$M_{LR}^{t}=A_{t}-\mu\cot\beta$.\, 
In the above formulas, we have defined
\begin{eqnarray}
{\cal I}({m_1},\,{m_2},\,{m_3}) &  =& 
-\frac{\dis ~m_{1}^2\,m_{2}^2\ln\frac{m_1^2}{m_2^2}
+m_{2}^2\,m_{3}^2\ln\frac{m_2^2}{m_3^2} + m_3^2\,m_1^2
\ln\frac{m_3^2}{m_1^2} ~}
{(m_1^2-m_2^2)\,(m_2^2-m_3^2)\,(m_3^2-m_1^2)}\, ,
\end{eqnarray}
which, in the special case of  
\,$m_1=m_2=m_3\equiv M$,\,
equals to \,$\dis\f{1}{2M^2}$\,.






With the sample values of the SUSY-parameters 
given in the caption of Fig.\,\ref{fig:YRbcH}, 
$\Delta_{b}$ is found to be about $0.17$,
among which, \,$0.20$\, comes from 
the SUSY-QCD contribution,  $0.00011$ from
SUSY-Yukawa contribution, and \,$-0.022$\, from
the electroweak gauge contribution\footnote{
The electroweak gauge contribution depends also on other
supersymmetry (SUSY) parameters \cite{dmb}. Here, $M_2$ is taken to be 
$300$\,GeV, but higher values of $M_2$ will make the electroweak 
gauge contribution even smaller due to the decoupling feature.
}.
Hence,  $\Delta_{b}$ yields a factor
of $1/(1+0.17)\simeq 0.85$ suppression in the $b$-$c$-$H^+$ coupling 
as compared to the 
QCD-improved Born level coupling 
(which is about $0.03$ for a 300\,GeV 
charged Higgs boson), 
and the coupling of $H^+$-$\overline{c_L}$-$b_R$ in
Eq.\,(\ref{eq:efflag})
is about $0.026$ for this set of SUSY parameters.
In other words, the threshold correction 
due to the SUSY-QCD and SUSY-EW contributions to $Y_L^{bc}$ is
$(0.85-1)\times 0.03\simeq -0.0045$, which 
is not significant in the current case.
(When  SUSY $\mu$ parameter flips sign, the threshold correction 
from $1/(1+\Delta_b)$ becomes an enhancement rather than
suppression factor.)
The additional contribution to $Y_L^{bc}$ arising from the 
$\ts-\cs$  mixing can be read out from Fig.\,\ref{fig:YRbcH} after 
subtracting the strength of the QCD-improved Born level coupling. 
For instance, using the same set of SUSY parameters described above,
the radiative correction from
$\ts-\cs$ mixings with $x=0.44$ enhances the $ Y_L^{bc}$ coupling by 
an amount of $0.02$ ($\simeq 0.05-0.03$) for $M_{H^\pm}=300$\,GeV.
Therefore, the coupling of $Y_L^{bc}$, after including the 
QCD-improved Born level coupling $(0.03)$, the radiative correction from 
$\ts-\cs$ mixings $(0.02)$, and the threshold correction 
due to the SUSY-QCD and SUSY-EW contributions $(-0.0045)$, is about
\,$0.046\,(\simeq 0.05)$\, for the sample SUSY parameters we have chosen.
Hence, without losing generality, 
in the following numerical analysis, we choose 
\beq
( Y_L^{bc},\, Y_R^{bc}) = (0.05,\,0) \,
\label{eq:bcH-MSSM-def}
\eeq 
as the sample couplings for the MSSM with natural $\ts-\cs$ mixings,
which correspond to the Type-A SUSY models with 
$x=O(1)$ and $y=0$ as defined in Ref.\,\cite{dhy}.
(The total decay width of $H^\pm$ will be evaluated for $\tanb=50$.)
It is worth to mention that 
the sample flavor-changing $b$-$c$-$H^\pm$ coupling
(\ref{eq:bcH-MSSM-def}) is about a factor-6 smaller than the
sample $\tau$-$\nu$-$H^\pm$ tree-level coupling
(\ref{eq:taunuH-MSSM-def}).



We then consider the dynamical Top-color (TopC) model \cite{Hill},
which is strongly motivated by the experimental fact 
that the observed large top quark mass 
($m_t\simeq \dis\f{v}{\sqrt{2}}\simeq 174$\,GeV) 
is right at the weak scale,
distinguishing the top quark from all other SM fermions. 
This scenario explains the 
top quark mass from the $\langle \bar{t}t \rangle$ condensation
via the strong $SU(3)_{\rm tc}$ TopC interaction at the TeV scale.
The associated strong tilting $U(1)$ force is attractive in the
$\langle \bar{t}t \rangle$ channel and repulsive in the
$\langle \bar{b}b \rangle$ channel, so that the bottom quark
mainly acquires its mass from the TopC instanton contribution
\cite{Hill}.
This model predicts three relatively light physical 
top-pions $(\pi_t^0,\,\pi^\pm )$.
The Yukawa interactions of these top-pions 
with the third family quarks are given by the Lagrangian,
\beq
\bea{l}
\dis\f{m_t\tanb}{v}\hspace*{-1.1mm}\left[
i{K_{UR}^{tt}}
{K_{UL}^{tt}}^{\hspace*{-1.3mm}\ast}\overline{t_L}t_R\pi_t^0
\hspace*{-0.7mm}+\hspace*{-0.8mm}\sq2
{K_{UR}^{tt}}{K_{DL}^{bb}}^{\hspace*{-1.3mm}\ast}
\overline{b_L}t_R\pi_t^- +
 \right.
\\[3.3mm]
\hspace*{1.3cm}~~\left.
i{K_{UR}^{tc}}
 {K_{UL}^{tt}}^{\hspace*{-1.3mm}\ast}\overline{t_L}c_R\pi_t^0
\hspace*{-0.7mm}+\hspace*{-0.8mm}\sq2
{K_{UR}^{tc}} {K_{DL}^{bb}}^{\hspace*{-1.3mm}\ast}
\overline{b_L}c_R\pi_t^-
\hspace*{-0.7mm}+\hspace*{-0.7mm}{\rm h.c.}  \right],
\eea
\label{eq:Ltoppi}
\eeq
where {\small $\tanb = \sqrt{(v/v_t)^2-1}$} and
the top-pion decay constant
$v_t\simeq O(60-100)$~GeV.
The rotation matrices $K_{UL,R}$ and $K_{DL,R}$ are needed 
for diagonalizing the up- and down-quark mass matrices
$M_U$ and $M_D$, i.e.,
{\small $~K_{UL}^\dag M_U K_{UR} = M_U^{\rm dia}~$} and
{\small $~K_{DL}^\dag M_D K_{DR} = M_D^{\rm dia}$},~
from which the CKM matrix is defined as
\,{\small $V=K_{UL}^\dag K_{DL}$}\,.\,  
As shown in Ref.\,\cite{hy}, to yield a realistic form of 
the CKM matrix $V$ (such as
the Wolfenstein-parametrization), the TopC model 
generally has the following features:
\beq
\bea{l}
K_{UR}^{tt}\simeq 0.99\hspace*{-0.5mm}-\hspace*{-0.5mm}0.94~,~~~
K_{UR}^{tc}\lesssim 0.11\hspace*{-0.5mm}-\hspace*{-0.5mm}0.33~,
\\[3.3mm]
K_{UL}^{tt} \simeq K_{DL}^{bb} \simeq 1 ~,
\eea
\label{eq:KURtc}
\eeq
which suggests that the $t_R$-$c_R$ transition
can be naturally around $10-30\%$. 
Combining Eqs.~(\ref{eq:Ltoppi}) and (\ref{eq:KURtc}), 
we can deduce the Yukawa couplings of fermions with
the charged top-pion
(also called charged Higgs boson throughout this paper) as 
\beq
\bea{l}
Y_L^{bt} \,=\, Y_L^{bc} \,=\, 0\,,          
\\[3.3mm]
Y_R^{bt} \,\simeq \, \dis\f{\sqrt{2}m_t}{v}\tanb \,,~~~~~
Y_R^{bc} \,\simeq \, Y_R^{bt} K_{UR}^{tc} \,. 
\eea
\eeq
Thus, taking a typical value of $\tanb $ to be $3$ and 
a conservative input for the \,$t_R-c_R$\, mixing
$ K_{UR}^{tc}$ to be $0.1 $ in the TopC model,
we obtain   
\beq
Y_R^{bt} \simeq 3\,, 
~~~~{\rm and}~~~~ 
\(Y_L^{bc},\,Y_R^{bc}\) 
= (0,\, 0.3) \,,
\label{eq:bcH-TopC-def}
\eeq
which will be used as the sample 
TopC parameters for our numerical analysis.
We note that in contrast to the radiative 
coupling  of 
the charged Higgs boson predicted in the Type-A SUSY model 
with $y=0$ 
%[cf. Eq.\,(\ref{eq:A1-cbH})],  
(in which $Y_L^{bc} \neq 0 $ and $\, Y_R^{bc}  \simeq  0$, i.e.,
 mainly left-handed),
the charged top-pions only have a right-handed coupling. 
This feature of the TopC
 is also opposite to the tree-level $\tau$-$\nu$-$H^\pm$
coupling (which is purely left-handed) 
predicted in the MSSM [cf. Eq.\,(\ref{eq:taunuH-MSSM-def})].
As we will demonstrate below, this feature 
makes it possible to discriminate the dynamical TopC model from the 
MSSM or a Type-II 2HDM 
by measuring the production rates of single charged Higgs 
boson at polarized photon colliders.
Finally, we note that apart form the opposite chirality structures 
of the $H^\pm$ Yukawa interactions,  the magnitude of the
sample Top-color $b$-$c$-$H^\pm$ coupling chosen in
(\ref{eq:bcH-TopC-def}) is the same as that of 
the sample $\tau$-$\nu$-$H^\pm$ coupling
(\ref{eq:taunuH-MSSM-def}).


\begin{figure}[H]
%\vspace*{-8mm}
\begin{center}
\hspace*{-3mm}
%\includegraphics[width=12cm,height=12cm]{0312totwid.eps}
\includegraphics[width=11.5cm,height=11cm]{prd_fig2.eps}
\end{center}
\vspace*{-5mm}
\caption{The total decay widths of $H^+$ 
predicted by the models discussed in the text.
}
\label{fig:width}
\end{figure}
\begin{figure}[H]
\vspace*{-8mm}
\begin{center}
\hspace*{-3mm}
%\includegraphics[width=12cm,height=12cm]{0312br.eps}
\includegraphics[width=11.5cm,height=11cm]{prd_fig3.eps}
\end{center}
\vspace*{-5mm}
\caption{The relevant decay branching ratios of $H^+$ 
predicted by the models discussed in the text.
}
\label{fig:branch}
\end{figure}




\section{
$H^\pm$ Production in ${\gamma}{\gamma}$ 
Collision as a Probe of New Physics}


We calculate the cross section of 
$\gamma\gamma\to \bar f' f H^+$ 
using the helicity amplitude method for 
$f'f=b \bar c \,$  or $\tau\bar\nu$.
For the $b \bar c$ channel, we will consider 
both the MSSM (with stop-scharm mixings)  
and the TopC model using the sample parameters 
listed in Eqs.~(\ref{eq:bcH-MSSM-def}) and (\ref{eq:bcH-TopC-def}),
respectively.
For the $\tau\bar\nu$ channel, we will 
consider the MSSM with the sample parameters given in 
Eq.~(\ref{eq:taunuH-MSSM-def}). 
The cross sections for other values
of couplings, different from our sample inputs, can 
be deduced by a proper rescaling.
In order to predict the event rate
of $\,\gamma\gamma\to \bar f' f H^+$,\, we need to specify the 
total decay width $\Gamma_{H^+}$ for $H^\pm$, from which the
decay branching ratio of \,$ H^\pm \to f' f$\, can be calculated.
For simplicity, we shall only include 
the quark and lepton decay modes of $H^\pm$ to evaluate 
$\Gamma_{H^+}$.
Its bosonic decay modes are not included because 
their contributions are generally small and strongly depend on the 
other parameters of the model. For example, 
in the MSSM, the partial decay width of $H^\pm \to W^\pm h^0$ 
also depends on the neutral Higgs boson 
mixing angle $\alpha$ and the light \tx{CP}-even Higgs boson mass
$m_{h}$. However, we find that 
it is generally small, especially when
 $M_{H^\pm}$ becomes large which corresponds to the decoupling limit. 
We will also neglect all the loop-induced decay modes such as 
$H^\pm \to W^\pm Z$ \cite{hwz},  
and assume that the relevant sparticles 
are relatively heavy so that the SUSY decay channels of $H^\pm$ 
are not kinematically accessible.
Finally, 
in the TopC model, only the dominant $tb$ and $cb$ decay modes 
are included in the calculation of \,$\Gamma_{H^+}$\,.
For the later analysis and discussion, 
we show the predicted total decay widths and the relevant
decay branching ratios of $H^+$ in Figs.\,\ref{fig:width} 
and \ref{fig:branch} as the Higgs mass  $M_{H^\pm}$ varies. 



In our numerical analyses, the dominant QCD corrections are included 
in the Yukawa couplings by using the running quark masses. 
For instance, at the 100\,GeV scale, the running masses of 
the bottom and charm quarks 
are $m_b=2.9$ GeV and $m_c=0.6$ GeV, respectively.




\subsection{$bc H^\pm$ Production}


Using the default parameters of the models as 
described in Section\,II, we calculate 
the total cross sections of 
$\gamma\gamma \to b \bar c H^+$ and 
$e^+e^- \to b \bar c H^+$ as a function of $M_{H^\pm}$.
The result for the TopC model 
is shown in Fig.\,\ref{fig:bch_topc_tot}, where,  
for comparison, we have taken the center-of-mass energy of 
the $\gamma\gamma$ collider to be 0.8 times of that of
the $e^-e^+$ collider.
The result for the MSSM with stop-scharm mixings 
can be easily obtained from  Fig.\,\ref{fig:bch_topc_tot} by 
rescaling the $y$-axis (i.e. the cross sections) by a factor of 
\,$(0.3/0.05)^2=36$\, when \,$M_{H^\pm} > \sqrt{s}/2$\,.\,
For \,$M_{H^\pm} < \sqrt{s}/2$,\, where the pair production mechanism
dominates, the actual rate also depends on the 
decay branching ratio Br$(H^- \to b {\bar c})$
and the total decay width $\Gamma_{H^+}$ in the MSSM.
For completeness, we also show the result for the MSSM in 
Fig.\,\ref{fig:bch_mssm_tot}, which is qualitatively similar to
Fig.\,\ref{fig:bch_topc_tot} except near the boundary of
the available phase space for pair production, i.e. when
$M_{H^\pm} \sim \sqrt{s}/2$.
This is because the total decay width of $H^\pm$ 
in the TopC model is much larger than that in 
the Type-A SUSY mode. For instance, the $\Gamma_{H^+}$
of the charged Higgs boson with a mass $200$\,GeV ($400$\,GeV)  
is about $7$\,GeV ($143$\,GeV) in the TopC model 
[cf. Eq.\,(\ref{eq:bcH-TopC-def})], and
$1.5$\,GeV ($13$\,GeV) in the Type-A SUSY model
[cf. Eq.\,(\ref{eq:bcH-MSSM-def})].
The branching ratios for the 
decay mode \,$H^+ \to c {\bar b}$\, predicted in these two models
 are $\,0.15~ (0.015)$\, and 
     $\,0.02~ (0.0046)$,\, respectively.
 

\begin{figure}[H]
\vspace*{-9mm}
\begin{center}
\includegraphics[width=12cm,height=10cm]{ee_feyn.eps}
\end{center}
\vspace*{-12mm}
\caption{ 
The complete set of Feynman diagrams for $e^-e^+ \to b \bar c H^+$.
}
\label{fig:eefyn}
\end{figure}


A few discussions on the feature of the results shown in 
Fig.\,\ref{fig:bch_topc_tot} are in order.
(The same discussions also apply to Fig.\,\ref{fig:bch_mssm_tot}.)
For \,$M_{H^\pm} < \sqrt{s}/2$\,,\,  
the charged Higgs pair production is kinematically allowed. 
In this case, the production cross section for 
$\gamma\gamma \to b \bar c H^+$ 
( and  $e^-e^+ \to b \bar c H^+$) is dominated by the contribution 
from the pair production diagrams with the produced $H^-$ decaying 
into a $b \bar c$ pair. 
Hence, its rate is proportional to the decay branching ratio
Br$(H^- \to b \bar c)$.
As shown in the figure, there is a {\it kink} structure when 
$M_{H^\pm}$ is around 180\,GeV. That is caused by the change in 
Br$(H^- \to b \bar {c})$ when the decay channel 
 $H^- \to b \bar {t}$ becomes available.
Furthermore, for \,$M_{H^\pm} < \sqrt{s}/2$,\, the cross section 
in \,$\gamma\gamma$\, collisions is 
typically an order of magnitude larger 
than that in \,$e^-e^+$\, collisions.
 


\begin{figure}
\begin{center}
\includegraphics[width=16cm,height=16cm]{aa_feyn.eps}
\end{center}
\vspace*{-5mm}
\caption{ 
The complete set of Feynman diagrams for $\gamma\gamma \to b \bar c H^+$.
}
\label{fig:aafyn}
\end{figure}



It is evident that the cross section of
 \,$\gamma\gamma \to b \bar c H^+$\, 
is larger than that of \,$e^+e^- \to b \bar c H^+$\, in 
the whole $M_{H^\pm}$ region. 
For \,$M_{H^\pm} > \sqrt{s}/2$,\, where 
the pair production is not kinematically allowed,
 the difference between these two cross sections becomes much larger 
 (two to three orders of magnitude) for a larger 
$M_{H^\pm}$ value.  
To understand the cause of this difference, we have to examine
the Feynman diagrams that contribute to the scattering processes
\,$e^-e^+ \to b \bar c H^+$\,
and
\,$\gamma\gamma \to b \bar c H^+$ .\,
The complete set of Feynman diagrams for the above processes 
are depicted in 
Figs.\,\ref{fig:eefyn} and \ref{fig:aafyn}, respectively.
In the former process, all the Feynman diagrams contain an $s$-channel 
propagator which is either a virtual photon or a virtual $Z$ boson.
Therefore, when $M_{H^\pm}$ increases for a fixed $\sqrt{s}$, the 
cross section decreases rapidly.
On the contrary, in the latter process, when $M_{H^\pm} > \sqrt{s}/2$,
the dominant contribution arises from the fusion diagram 
$\gamma\gamma \to (c \bar c) (b \bar b) 
\to  b \bar c H^+$, whose contribution is enhanced by the 
two collinear poles (in a $t$-channel diagram) 
generated from  $\gamma \to c \bar c$ and 
$\gamma \to b \bar b$ in high energy collisions.
Since the collinear enhancement takes the form of 
$\,\ln (M_{H^\pm}/m_q)\,$, 
with $m_q$ being the bottom or charm quark mass, 
the cross section of $\gamma\gamma \to b \bar c H^+$ 
does not vary much as $M_{H^\pm}$ increases until it is close
to \,$\sqrt{s}$\,.



From the above discussions we conclude that a photon-photon collider is 
superior to an electron-positron collider for detecting a 
heavy charged Higgs boson.  
Moreover, a polarized photon collider can determine the 
chirality structure of the fermion Yukawa couplings with the 
charged Higgs boson via single charged Higgs production. 
This point is illustrated as follows.
First, let us consider the case that $M_{H^\pm} > \sqrt{s}/2$.
As noted above, in this case, the production 
cross section is dominated by the fusion 
diagram $\gamma\gamma \to (c \bar c) (b \bar b) 
\to  b \bar c H^+$. In the TopC model, because 
$Y_L^{bc}=0$ (and $Y_R^{bc} \neq 0$), it corresponds to   
$\gamma\gamma \to (c_R \ov{c_R}) (b_L \ov{b_L}) 
\to  b_L \ov{c_R} H^+$. 
On the other hand, in the MSSM with stop-scharm mixings and 
large $\tanb$, $Y_R^{bc} \sim 0$ (and $Y_L^{bc} \neq 0$), 
it becomes 
$\gamma\gamma \to (c_L \ov{c_L}) (b_R \ov{b_R}) 
\to  b_R \ov{c_L} H^+$. 
Therefore, we expect that if both photon beams are 
right-handedly polarized (i.e. $\gamma_R^{~} \gamma_R^{~}$), then
a TopC charged Higgs boson (i.e. top-pion) can be 
copiously produced, while a MSSM charged Higgs boson 
(with a large $\tanb$) is highly suppressed.
To detect a MSSM charged Higgs boson, both photon beams 
have to be left-handedly polarized 
(i.e. $\gamma_L^{~} \gamma_L^{~}$).
This is supported by an exact calculation whose results
are shown in Figs.\,\ref{fig:bch_topc_pol} 
and \ref{fig:bch_topc_pol_2} for the TopC model
at two different collider energies.
A similar feature also holds for the MSSM after interchanging 
the label of $RR$ and $LL$ in those figures,  
which can be verified in Figs.\,\ref{fig:bch_mssm_pol} and
\ref{fig:bch_mssm_pol_2}.
 
\begin{figure}[H]
\begin{center}
\includegraphics[width=13cm,height=11cm]{prd_fig4.eps}
%{\par\centering \resizebox*{0.4\textwidth}
%{!}{\includegraphics{aa.vs.ee_bch.eps}} \par}
\end{center}
\caption{Cross sections of $\gamma\gamma\to b \bar c H^+$ (solid curve) 
         and $e^+e^-\to b \bar c H^+$ (dashed curve) for the TopC model
[cf. Eq.\,(\ref{eq:bcH-TopC-def})]
         with unpolarized photon beams  
         at $\sqrt{s}=400$ GeV and $800$ GeV. 
}
\label{fig:bch_topc_tot}
\end{figure}


The feature of the polarized photon cross sections 
for $M_{H^\pm} < \sqrt{s}/2$ can be understood from examining
the production process $\gamma\gamma \to H^+H^-$. 
The helicity amplitudes for the  $H^+H^-$ pair production   
in polarized photon collisions can be computed as 
%
%\begin{eqnarray}
%\label{eq:pair}
%&&M(\gamma_{\lambda_1}^{} \gamma_{\lambda_2}^{} \to H^+H^-) \equiv 
%  2 e^2  \lambda_1 \lambda_2 \sin^2 
%\Theta  \left( \frac{s}{4} -M_{H^\pm}^2  \right) \nonumber\\
%&&      \left\{  
%          \frac{1}{t - M_{H^\pm}^2} \right. 
% \left. + \frac{1}{u - m_{H^\pm}^2} \right\}
%  + 2 e^2 \left( \lambda_1 \lambda_2 + 1 \right),      
%\end{eqnarray}
%
\begin{eqnarray}
\label{eq:pair}
M(\gamma_{\lambda_1}^{~} \gamma_{\lambda_2}^{~} \to H^+H^-) 
&=& 
   2 e^2  \lambda_1 \lambda_2   
\frac{1 - \xi^2}{~1 - \xi^2 \cos^2 \Theta~ } 
  +  e^2 \left( 1 - \lambda_1 \lambda_2 \right),      
\end{eqnarray}
where the degree of polarization of the initial state photons, 
$\lambda_{1}$ and $\lambda_{2}$, can take 
the value of either $-1$ or $+1$, 
corresponding to a left-handedly ($L$) and right-handedly ($R$) 
polarized photon beam, respectively;  
\,$\Theta$\, is the scattering angle of $H^+$ in the center-of-mass
frame; and  $\, \xi=\sqrt{ 1 - 4 M_{H^\pm}^2 / s }$\,.\,
In the massless limit, i.e., when $M_{H^\pm} \to 0$, the above result 
reduces to  
\,$M(\gamma_{\lambda_1}^{~}\gamma_{\lambda_2}^{~}\to H^+H^-) \simeq 
 e^2 \left( 1 - \lambda_1 \lambda_2 \right)$\,.\, 
Let us denote
$\,\sigma_{ \lambda_{1} \lambda_{2}}^{\rm pair}$\, as the cross section 
of \,$\gamma_{\lambda_1}^{~}\gamma_{\lambda_2}^{~}\to H^+H^-$\,.\,
We find that $\sigma_{LR}^{\rm pair}=\sigma_{RL}^{\rm pair}$, and they
dominate the total cross section when 
\,$ M_{H^\pm}^2 \ll s$\,,\, while 
$\sigma^{\rm pair}_{LL}$ and $\sigma_{RR}^{\rm pair}$ 
are equal and approach zero as \,$M_{H^\pm} \to 0$.\,
Since for \,$M_{H^\pm} < \sqrt{s}/2$\,  the bulk part of the 
cross section of \,$\gamma\gamma \to b \bar c H^+$\,
comes from  
\,$\sigma(\gamma\gamma\to H^+H^-) 
   \times {\rm Br}(H^- \to b \bar c )$,\,
the $LL$ and $RR$ cross sections are smaller than the
$LR$ ($=RL$) cross sections as $M_{H^\pm}$ 
decreases, cf. Fig.\,\ref{fig:bch_topc_pol}.


\begin{figure}
\begin{center}
\includegraphics[width=13cm,height=11cm]{prd_fig5.eps}
%{\par\centering \resizebox*{0.4\textwidth}
%{!}{\includegraphics{aa.vs.ee_bch.eps}} \par}
\end{center}
\caption{Same as Fig.~\ref{fig:bch_topc_tot}, but for the MSSM with 
stop-scharm mixings, i.e. Type-A SUSY model
 [cf. Eq.\,(\ref{eq:bcH-MSSM-def})].
}
\label{fig:bch_mssm_tot}
\end{figure}


It is important to point out that the complete set of 
Feynman diagrams have to be included to calculate 
\,$\sigma(\gamma\gamma \to b \bar c H^+)$\, even when 
\,$M_{H^\pm} < \sqrt{s}/2$\,  because of the requirement of
 gauge invariance.
To study the effect of the additional Feynman diagrams, other
than those contributing to the $H^+H^-$ pair production from 
\,$\gamma\gamma\to H^+H^-(\to b \bar c )$,\,
one can examine the {\it single} charged Higgs boson rate 
in this regime with the requirement that the invariant mass of 
\,$b \bar c$,\,  denoted as \,$M_{b\bar c}$,\, satisfies 
the following condition:
%
\begin{eqnarray}
\label{eq:kin-cut}
|M_{b \bar c} & - & M_{H^\pm}|  ~>~   \Delta M_{b\bar c} \, ,
\qquad {\rm with} 
\nonumber\\[3.5mm]
\Delta M_{b\bar c} & = &
\min \left[  25\,{\rm GeV},~
\max\left[ 1.18 M_{c\bar b} {\f{\,2 \delta m}{m}},\, \Gamma_{H^+} \right] 
\right] \, ,
\nonumber\\[3mm]
%{\rm and} 
\qquad \qquad
{{\,\delta m\,} \over m} & = & {0.5 \over \sqrt{M_{b\bar c}/2\,}\, }
\, ,
\end{eqnarray}
%
where $\,\dis\f{\,\delta m\,}{m}$\,   denotes 
the mass resolution of the detector for observing the final state 
$b$ and ${\bar c}$ jets originated from the decay of \,$H^-$\,.\,  
For instance, in Fig.\,\ref{fig:bch_topc_pol} the
set of dashed-lines are the polarized cross sections after 
imposing the above kinematical cut.
With this cut, the total rate reduces by about one order of 
magnitude for \,$M_{H^\pm} < \sqrt{s}/2$\,.\, 
(However, this kinematical cut hardly changes 
the event rate when \,$M_{H^\pm} > \sqrt{s}/2$\,.)
The effect of this kinematic cut on the 
$RR$ and $LL$ rates are significantly different in the 
low $M_{H^\pm}$ region. It implies that the $H^+H^-$ pair 
production diagrams cannot be the whole production mechanism, 
otherwise,
we would expect the rates of $RR$ and $LL$ be always equal 
due to the parity invariance of the QED theory.
Again, a similar feature also holds for the MSSM after 
interchanging the labels of $LL$ and $RR$.


Before closing this section, we remark that in the MSSM a heavy charged
Higgs boson $H^+$ can also be produced associated with a ${\bar c} s$
pair, whose production rate can be obtained by rescaling the cross
sections in Fig.\,\ref{fig:bch_mssm_tot} by the factor
$$
\(Y_{L(0)}^{sc}/Y_{L}^{bc}\)^2 
~=~ 1.3 \, (\tanb)^2 \times 10^{-4} 
$$
for \,$M_{H^\pm} > \sqrt{s}/2$\,.\,
Here, \,$Y_{L(0)}^{sc}=\dis\f{\sqrt{2}m_s}{v} \tanb\,$,\, and
the running mass of the strange quark at the scale of
$100$\,GeV is taken to be $m_s \simeq 0.1$\,GeV.\,
Hence, for $\tanb=30$, the production rate of 
$\,scH^\pm$\, is down by a factor of 100\,.
%
\begin{figure}
\begin{center}
%\vspace*{-3mm}
\includegraphics[width=13cm,height=11cm]{prd_fig6.eps}
%\includegraphics[width=8.1cm,height=7cm]{may13tc1000cut.eps}%
%{\par\centering \resizebox*{0.4\textwidth}
%{!}{\includegraphics{fig2.eps}} \par}
\end{center}
\vspace*{-6mm}
\caption{Cross sections of 
$\gamma_{\lambda_1}\gamma_{\lambda_2} \to b {\bar c} H^+$ 
         at $\sqrt{s}=800$ GeV in  
polarized photon collisions for the TopC model            
[cf. Eq.\,(\ref{eq:bcH-TopC-def})].         
Solid curves are the results without any kinematical cut, and 
 Dashed curves are the results with the kinematical cut 
 specified in the text [cf. Eq.\,(\ref{eq:kin-cut})]. 
%% $|M_{b\bar c} -  M_{H^\pm}| > {\Gamma_{H^+}} $.
}
\label{fig:bch_topc_pol}
\end{figure}
%
\begin{figure}[H]
\begin{center}
\includegraphics[width=13cm,height=11cm]{prd_fig7.eps}
\end{center}
\vspace*{-6mm}
\caption{Same as Fig.~\ref{fig:bch_topc_pol}, but for 
$\sqrt{s}=400$ GeV. 
}
\label{fig:bch_topc_pol_2}
\end{figure}



\subsection{$\tau\nu H^\pm$ Production}


In the MSSM with a large $\tanb$ value, the 
cross section of \,$\gamma\gamma\to\tau^- {\bar \nu} H^+$\,
can be quite sizable.
For the sample parameters chosen in Eq.\,(\ref{eq:taunuH-MSSM-def}),
its cross sections are shown in Fig.\,\ref{fig:tnh_tot} for 
various linear colliders with unpolarized collider beams.
(Our results are consistent with the calculation in Refs.~\cite{KMO2,MO9}.) 
Recall that we have chosen the sample parameters of the models so that
the Yukawa coupling of $\tau$-$\nu$-$H^+$ in the MSSM and 
that of $b$-$c$-$H^+$ in the TopC model have the same magnitude 
but opposite chiralities, as shown in Eqs.\,(\ref{eq:taunuH-MSSM-def})
and (\ref{eq:bcH-TopC-def}).
The gross feature of this figure is similar to Fig.\,\ref{fig:bch_topc_pol}.
However, a close examination reveals that the cross section 
of $\gamma\gamma\to\tau^- {\bar \nu} H^+$ is smaller than that 
of $\gamma\gamma \to b \bar c H^+$ at a fixed $M_{H^\pm}$  
for $M_{H^\pm} > \sqrt{s}/2$.
For instance, for a 700\,GeV charged Higgs boson,  
$\sigma(\gamma\gamma\to\tau^- {\bar \nu} H^+) \sim 0.02 \, {\rm fb}$
and
$\sigma(\gamma\gamma\to b \bar c H^+) \sim 0.3 \, {\rm fb}$.
This difference can again be understood by examining the 
Feynman diagrams. 
In the scattering \,$\gamma\gamma\to b \bar c H^+$,\, 
the total cross section is dominated by the fusion diagram 
\,$\gamma\gamma \to (c \bar c) (b \bar b) 
\to  b \bar c H^+$\, for  \,$M_{H^\pm} > \sqrt{s}/2$\,.\,
The contribution of this diagram is enhanced by 
two collinear poles (in a $t$-channel diagram) 
generated from  $\gamma \to c \bar c$ and 
$\gamma \to b \bar b$ in high energy collisions.
However, in the scattering \,$\gamma\gamma\to\tau^- {\bar \nu} H^+$,\,
the dominant contribution in the large mass region comes from  
the sub-diagram $\gamma \ov{\tau_R} \to H^+ \ov{\nu_L}$, and  
contains only one collinear pole (in a $t$-channel diagram) 
generated from  $\gamma \to \tau \bar \tau$ in high energy collisions.
This is because photon does not couple to neutrinos.
Hence, the production rate of \,$\tau\nu H^+$\, 
is not as large as that of \,$b \bar c H^+$,\, 
even when the relevant 
Yukawa couplings are of the same magnitude
in both production channels.
%
\begin{figure}[H]
\begin{center}
\includegraphics[width=13cm,height=9.5cm]{prd_fig8.eps}
\end{center}
\vspace*{-7mm}
\caption{Cross sections of 
$\gamma_{\lambda_1}\gamma_{\lambda_2} \to b {\bar c} H^+$ 
         at $\sqrt{s}=800$ GeV in  
polarized photon collisions for the Type-A SUSY model            
[cf. Eq.\,(\ref{eq:bcH-MSSM-def})].         
Solid curves are the results without any kinematical cut, and 
 Dashed curves are the results with the kinematical cut 
 specified in the text [cf. Eq.\,(\ref{eq:kin-cut})]. 
%% $|M_{b\bar c} -  M_{H^\pm}| > {\Gamma_{H^+}} $.
}
\label{fig:bch_mssm_pol}
\end{figure}


We also computed the production cross section 
\,$\sigma(\gamma_{\lambda_1}^{} \gamma_{\lambda_2}^{} \to\tau\nu H^+)$\, 
in the polarized photon-photon collisions, and 
the results are shown in Figs.\,\ref{fig:tnh_mssm_pol} 
and \ref{fig:tnh_mssm_pol_2}.
As expected, $LL$ rate is the dominant one when 
$M_{H^\pm} > \sqrt{s}/2$\,,\, because the Yukawa couplings
\,$Y_R^{\tau\nu} = 0$\, and \,$Y_L^{\tau\nu} \neq 0$\,.\,
The {\it single} charged Higgs boson production rate 
for \,$M_{H^\pm} < \sqrt{s}/2$\, is also calculated by imposing the 
kinematical cut:
%
\begin{eqnarray}
\label{eq:kin-cut-tau}
|M_{\tau \bar \nu} & - &  M_{H^\pm}|  ~>~  \Delta M_{\tau \bar \nu} \, , 
\qquad {\rm with} 
\nonumber\\[2mm]
\Delta M_{\tau \bar \nu} & =  &
\min\left[ 25\,{\rm GeV},\,
\max\left[ 1.18 M_{\tau \bar \nu} \f{\,2 \delta m\,}{m},\,
\Gamma_{H^+} \right] 
\right] \, ,
\nonumber\\[1mm]
\qquad \qquad 
{\delta m \over m} & = & {0.5 \over ~\sqrt{M_{\tau \bar \nu}/2\,}~} \, , 
\end{eqnarray}
  and the result is shown in Figs.\,\ref{fig:tnh_mssm_pol} 
and \ref{fig:tnh_mssm_pol_2}.
(In reality, $M_{\tau \bar \nu}$ should be replaced by, for instance, 
 the transverse mass of the $\tau \bar \nu$ pair.)
For our choice of parameters in Eq.\,(\ref{eq:taunuH-MSSM-def}),
${\Gamma_{H^+}}$ is about 
$0.54$\,GeV ($4.7$\,GeV) for a Higgs mass 
$200$\,GeV ($400$\,GeV), and
correspondingly,  
\,${\rm Br}(H^- \to \tau^- {\bar \nu})$\, is about 
$0.69$ $(0.16)$.
%
\begin{figure}
\begin{center}
\vspace*{-5mm}
\includegraphics[width=13cm,height=9.2cm]{prd_fig9.eps}
\vspace*{-7mm}
\end{center}
\caption{Same as Fig.~\ref{fig:bch_mssm_pol}, but for 
$\sqrt{s}=400$\,GeV. 
}
\label{fig:bch_mssm_pol_2}
\end{figure}











\vspace*{-6mm}
\begin{figure}
\begin{center}
\includegraphics[width=13cm,height=8.5cm]{prd_fig10.eps}%
%\includegraphics[width=8.1cm,height=7cm]{nf7.eps}%
%{\par\centering \resizebox*{0.4\textwidth}
%{!}{\includegraphics{aa.vs.ee_tnh.eps}} \par}
\end{center}
\vspace*{-7mm}
\caption{Cross sections of $\gamma\gamma\to\tau^- {\bar \nu} H^+$ 
         (solid curve) 
         and $e^+e^-\to \tau \bar \nu H^+$ (dashed curve)
         for the MSSM [cf. Eq.\,(\ref{eq:taunuH-MSSM-def})]
         with unpolarized beams  
         at $\sqrt{s}=400$ GeV and $800$ GeV.
}
\label{fig:tnh_tot}
\end{figure}
%
\begin{figure}
\begin{center}
\includegraphics[width=14cm,height=8.5cm]{prd_fig11.eps}%
\end{center}
\vspace*{-5mm}
%{\par\centering \resizebox*{0.4\textwidth}
%{!}{\includegraphics{fig2.eps}} \par}
\caption{Cross sections of 
$\gamma_{\lambda_1}\gamma_{\lambda_1} \to \tau {\bar \nu} H^+$ 
         at $\sqrt{s}=800$ GeV in  
polarized photon collisions for the MSSM             
[cf. Eq.\,(\ref{eq:taunuH-MSSM-def})].         
Solid curves are the results without any kinematical cut, and 
 Dashed curves are the results with the kinematical cut 
 specified in the text [cf. Eq.\,(\ref{eq:kin-cut-tau})]. 
%% $|M_{\tau \bar \nu} -  M_{H^\pm}| > {\Gamma_{H^+}} $.
}
\label{fig:tnh_mssm_pol}
\end{figure}





\begin{figure}
\begin{center}
\includegraphics[width=14cm,height=10cm]{prd_fig12.eps}
\end{center}
\vspace*{-3mm}
\caption{Same as Fig.~\ref{fig:tnh_mssm_pol}, but for 
$\sqrt{s}=400$ GeV. 
}
\label{fig:tnh_mssm_pol_2}
\end{figure}



\vspace*{5mm}
\section{Conclusions}

In this work,
we have studied the single charged scalar production at polarized
photon colliders via the fusion processes 
\,$\gamma \gamma \to b \bar c H^+$\, and 
\,$\gamma \gamma \to \tau \nu H^+$.\, 
For the \,$b {\bar c} H^+$\, production, 
we consider the flavor mixing couplings
of \,$b$-$c$-$H^\pm$\, generated  
from the natural stop-scharm mixings in the MSSM, and 
from the generic mixings of the right-handed top 
and charm quarks in the dynamical Top-color model.
For the $\tau {\bar \nu} H^+$ production, 
we consider the MSSM with a moderate to large $\tanb$. 
We find that  
the production rate of $H^+$ in the \,$\gamma\gamma$\, collisions 
is much larger than that in the \,$e^-e^+$\, collision.
(Needless to say that the production rate of $H^-$ is the same as
$H^+$.) 
Some of the results are shown in
Figs.\,\ref{fig:bch_topc_tot}, \ref{fig:bch_mssm_tot}
and \ref{fig:tnh_tot}.
For \,$M_{H^+}> \sqrt{s}/2$,\, 
the cross section  of   \,$\gamma \gamma \to \tau {\bar \nu} H^+$\, 
is smaller than that of \,$\gamma \gamma \to b {\bar c} H^+$\, 
even when the corresponding Yukawa couplings are of 
the same size. 
This is because in high energy collisions there is only one
collinear pole 
$\[\,\gamma \gamma \to (\tau {\bar \tau}) \gamma 
\to \tau {\bar \nu}  H^+\,\]$
in the scattering  
\,$\gamma \gamma \to\tau {\bar \nu} H^+$,\, 
but two collinear poles 
$\[\,\gamma\gamma \to (c \bar c) (b \bar b) \to  b \bar c H^+\,\]$  
in 
\,$\gamma \gamma \to b \bar c H^+$.\,
The same reason also explains why in the large $M_{H^+}$ 
region the \,$e^+e^-$\, rate is smaller than 
the \,$\gamma \gamma$\, rate by at least one 
to two orders of magnitude, 
since the \,$e^+e^-$\, processes contain only $s$-channel diagrams 
and cannot generate any collinear enhancement factor to  
the single charged Higgs boson production rate.
Furthermore, we show that it is possible to measure 
the Yukawa couplings $Y_L$ and $Y_R$, separately, 
at photon-photon colliders
by properly choosing the polarization states of the 
incoming photon beams.
This unique feature of the photon colliders can 
be used to discriminate new dynamics of the 
flavor symmetry breaking.


Given our results of the cross sections,
it is trivial to deduce the signal event rates as long as
the integrated luminosity of the collider is known. 
According to the reports of the LC Working Groups in
Refs.\,\cite{TESLA} and \cite{JLC},
the integrated luminosity can reach about $500\,\ifb$
at a 500\,GeV LC, and $1000\,\ifb$ at an  1\,TeV LC.
Hence, we conclude that a polarized photon-photon collider is not only 
useful for determining the \tx{CP} property of a neutral Higgs boson, 
but also important for detecting a heavy charged Higgs boson
and determining the chirality structure of the 
corresponding fermion Yukawa interactions with the
charged Higgs boson.



\vspace*{5mm}
\noindent
{\bf Acknowledgments}~~~\\
We thank Gordon L. Kane for valuable discussions 
on the SUSY flavor mixings.
SK would like to thank Stefano Moretti for useful discussions 
and for comparing part of our results with his calculation. 
This work was supported in part by the NSF grant 
and DOE grant DEFG0393ER40757.



\newpage
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%\end{narrowtext}
\end{document}




