\documentstyle[aps,prd,epsfig,eqsecnum]{revtex}
\begin{document}
\draft
%\preprint
\title{Non-decoupling SUSY Quantum Effects in Higgs-Boson Production
      $p p \to b h+X$}

\author{Junjie Cao $^{a,b}$, Guangping Gao $^b$, Yuling Su $^c$, Jin Min Yang $^b$ \\\ }

\address{$^a$ CCAST(World Laboratory), P.O.Box 8730, Beijing 100080, China}
\address{$^b$ Institute of Theoretical Physics, Academia Sinica, Beijing 100080, China}
\address{$^c$ Department of Physics, Zhengzhou Institute of Light Industry, Henen 450002, China}
\date{\today}
\maketitle

\begin{abstract}
The Higgs boson production $p p ~( or ~p\bar{p} ) \to b h +X$ via 
subprocess $b g \to b h$ at hadron colliders, which is an important 
channel for testing Higgs Yukawa coupling to bottom quark, is subject
to large supersymmetric quantum corrections. We in this work
studied the SUSY-QCD corrections to this process and found such
corrections are so large in some parameter space that they must be
included in the prediction. We also studied the decoupling
behavior of the corrections. The analytic expression of the
corrections in the heavy limit of sparticle masses is derived
explicitly. We found that in some scenarios the corrections  do
not vanish in the decoupling limit. Such non-decoupling quantum
effects will play an important role in the indirect  test of
supersymmetry, especially in case that the sparticles are too
heavy (above TeV) to be directly discovered at the LHC.
\end{abstract}

\pacs{14.80.Cp, 13.85.Qk,12.60.Jv}

\section{Introduction}
\label{sec:sec1} Searching for the Higgs boson is the most
important task for the Tevatron and the LHC.  Among various
Higgs production mechanisms, those involving Higgs Yukawa coupling
to bottom quark are particularly important because in some
extensions of the Standard Model (SM) such coupling could be
considerably enhanced and thus the production rates of the
Higgs boson may be much larger than the SM predictions. 
The Minimal Supersymmetric Standard Model (MSSM)\cite{HaberKane}
is a good example of such extensions, where the coupling of the 
lightest $CP$-even Higgs boson (denoted by $h$) to the bottom quark is
proportional to $\tan{\beta}$ \cite{Gunion} and may be
significantly enhanced for large $\tan\beta$.

In the production channels of the Higgs boson via its coupling to
bottom quark , the process $p p ~( or ~p\bar{p} ) \to b h +X$ via
the subprocess $ b g \to b h$ was recently emphasized in
Ref.\cite{willenbrock}. The advantage of this process over the
production via subprocess $b \bar b \to h$ \cite{bbh}, the
dominant production channel of the Higgs boson via its coupling to
bottom quark, is that the final bottom quark can be used to reduce
backgrounds and to identify the Higgs boson production mechanism
\cite{background}.  And compared with the production via $gg, q
\bar q \to h  b \bar{b}$ \cite{ggbbh}, the production rate of $p p
~( or ~p\bar{p} ) \to b h +X$  is one order of magnitude larger.
So the production $p p ~( or ~p\bar{p} ) \to b h +X$ may be
a crucial channel for testing Higgs Yukawa coupling to bottom
quark. 

If the MSSM is indeed chosen by Nature, then the prediction of the
cross section for the production $p p ~( or ~p\bar{p} ) \to b h +X$
\cite{huang} must be renewed with the inclusion of SUSY quantum
corrections because, like the process of the charged Higgs boson
production $pp~(~or~p\bar{p}) \to t H^-+X$ \cite{gao} and the
relevant Higgs decays\cite{decay1,haber-nondecoup}, the SUSY
quantum corrections to this process may be quite large. This is
the aim of this paper. In this work we study  the one-loop
SUSY-QCD (SQCD) corrections to this process, which is believed to
be the dominant radiative corrections among various SUSY
corrections.

It is noticeable that while the calculation of SUSY quantum
corrections to the production  $pp ~( or ~p\bar{p} ) \to b h +X$
is important in its own right, the study of the behavior of these
corrections in the  limit of heavy SUSY scale may be particularly
important. As is well known, in the MSSM the Higgs boson $h$ must
be light enough to be discovered at the Tevatron or the LHC. But
the sparticles may be heavy. If  the sparticles  are too heavy
(above TeV) to be directly discovered at the LHC, the only way to
detect them is through their quantum loop effects in some
observable processes like $pp ~( or ~p\bar{p} ) \to b h +X$. If
SUSY loop effects do not decouple for large sparticle masses, the
detection of these effects will serve as a robust evidence of
SUSY.

This paper is organized as follows. In Section II we present our
strategy to calculate the one-loop SUSY-QCD corrections to  $pp ~(
or ~p\bar{p} ) \to b h +X$. In Section III, we scan the parameter
space of the MSSM to estimate the size of these SUSY corrections.
In Section IV, we study the behaviors of these corrections,
especially in the limit of heavy SUSY scale. Conclusion is given
in Section V and the detailed formula are presented in the
Appendix.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Calculations}
\label{sec:calculations} The subprocess  $gb \to b h$ occurs
through both $s$-channel and $t$-channel shown in
Fig.~\ref{fig:feyman}$(a, b)$. The spin- and color-averaged
differential cross section at tree-level is given by
\begin{equation}
\frac{d \hat{\sigma}_{g b \to b h}^{0}}{d \hat{t}} = -
\frac{\alpha_s (\mu)}{24} \left (\frac{g m_b (\mu)}{2 m_W}
\right )^2 \left( \frac{\sin{\alpha}}{\cos{\beta}} \right )^2
\frac{1}{\hat{s}^2} \frac{m_h^4+(\hat{s}+
\hat{t}-m_h^2)^2}{\hat{s} \hat{t}} , \label{tree}
\end{equation}
where $\hat{s}$ and $ \hat{t}$ are  the usual Mandelstam
variables, $\alpha_s (\mu) $ is the running strong
coupling,  and $m_b (\mu) $ is the running bottom
quark mass\footnote{It is important to use $\overline{MS}$ running mass
of bottom quark rather than the pole mass in this process since the
latter is significantly larger than the former if we set $\mu = m_h$.}\cite{bbh}.
$\alpha $ represents the mixing angel between the two CP-even Higgs
boson eigenstates and $ \cos{\beta}= v_1 /\sqrt{v_1^2+v_2^2} $
with $ v_{1, 2} $ denoting the vacuum expectation values of the
two Higgs doublets\cite{Gunion}. In Eq.(\ref{tree}), the SM
prediction of the cross section is recovered when
$|\sin{\alpha}/\cos {\beta}| =1 $\cite{willenbrock}.
Throughout the calculations we neglect
the b-quark mass except in the b-quark Yukawa couplings.
%%%%% Fig. 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hbt]
\begin{center}
\epsfig{file=fig1.ps,width=10cm}
\vspace*{.5cm}
\caption{ Feynman diagrams of $g
b \to b h $ with one-loop SUSY-QCD corrections: $(a)$ and $(b)$ are tree level diagrams;
$(c)-(e)$ are one-loop vertex diagrams for s-channel; $(f)-(h)$ are one-loop vertex diagrams for
t-channel ; $(i)-(l)$ are self-energy diagrams and $(m,n)$ are the box diagrams.} \label{fig:feyman}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The one-loop Feynman diagrams of SUSY QCD corrections are shown in
Fig.~\ref{fig:feyman}$(c)$-$(n)$. In our calculations we use
dimensional regularization to control all the ultraviolet
divergences in the virtual loop corrections and we adopt the
on-mass-shell renormalization scheme. For the renormalization of strong
coupling constant $g_s$, we employ the scheme adopted in Ref.\cite{gao}:
\begin{eqnarray} \label{gs}
\frac{\delta g_s}{g_s}=-\frac{1}{2} \delta Z_2^g,
\end{eqnarray}
where $\delta g_s$ and  $ Z_2^g$ are the renormalization
constants, defined by $g^0_s \equiv g_s+\delta g_s$ and $A^0_{\mu}
\equiv \sqrt{Z_2^g} A_{\mu}$ with $ g^0_s$ denoting bare coupling
constant and $A^0_{\mu}$ bare gluon fields (color index
suppressed). In this way, the contribution of the self-energy of
the initial gluon field is canceled out by $\delta g_s$. However,
the effects of such gluon self-energy diagrams in SUSY-QCD should
be included in the running of $g_s$ when the scale is higher than
SUSY scale \cite{alpha}.

The one-loop renormalized amplitude for $g b \to b h$ can be
written as
\begin{eqnarray}
\delta M &=& \frac{i g_{s}^{3}  T^{a} }{16 \pi^2} \frac{g
m_b} {2 m_W} \overline{u} (p_2)  ( C_1 \gamma^{\mu}P_{L} + C_2
\gamma^{\mu}P_{R} + C_3 \gamma^{\mu} \not{k} P_{L} + C_4
\gamma^{\mu} \not{k} P_{R} + C_5 p_1^{\mu} P_{L} + C_6
p_1^{\mu}P_{R}  \nonumber \\
& & + C_7 p_1^{\mu} \not{k} P_{L} + C_8 p_1^{\mu} \not{k} P_{R} +
C_9 p_2^{\mu} P_{L} + C_{10} p_2^{\mu}P_{R} + C_{11} p_2^{\mu}
\not{k} P_{L} + C_{12} p_2^{\mu} \not{k} P_{R}) u (p_1)
\epsilon_{\mu} (k) ,
\end{eqnarray}
where $P_{L,R}\equiv (1\mp \gamma_5)/2$, $T^a\equiv \lambda^a/2$ with  $\lambda^a$
being the Gell-Mann matrices, and
$k$, $p_1$ and $p_2$ are the momentum of the incoming gluon, $b$-quark and
the outgoing $b$-quark, respectively.
$g_s$ and $m_b$ should be understood
as the running coupling and mass as in Eq.(\ref{tree})
The coefficients $C_{i}$ arise from the loops and are given explicitly
in Appendix A. We have checked that all the ultraviolet divergences
canceled as a result of renormalizability of the MSSM.

The differential cross section of $ g b \to b h $ up to
one-loop level can be expressed in the following form
\begin{eqnarray}
\frac{d \hat{\sigma}_{b g \to b h}}{d \hat{t}} =\frac{d
\hat{\sigma}_{b g \to b h}^{0}}{d \hat{t}} + \frac{d
\hat{\sigma}_{b g \to b h}^{loop}}{d \hat{t}} ,
\end{eqnarray}
where the first term is given by
Eq.(\ref{tree}) and the second term can be written as
\begin{eqnarray}
\frac{d \hat{\sigma}_{b g \to b h}^{loop}}{d \hat{t}} & = &
-\frac{\alpha_s}{48} \frac{\alpha_s}{4 \pi} \left (
\frac{\sin{\alpha}}{\cos{\beta}} \right )^2  \left (\frac{g m_b}{
2 m_W} \right )^2 \frac{1}{\hat{s}^2} \nonumber \\
& & \times \left [ 2 (C_3+C_4) m_h^2+ (C_5 + C_6)\frac{m_h^2 (\hat{s}+
\hat{t} -m_h^2)}{\hat{t}} + (C_9+ C_{10}) \frac{-(\hat{s}+ \hat{t}
-m_h^2)^2}{\hat{s}} \right ] .  \label{loop}
\end{eqnarray}

The cross section for the parton process  $g b\rightarrow b h$ is
then given by
\begin{equation}
\hat{\sigma}(\hat s) =\int_{\hat{t}_{min}}^{\hat{t}_{max}} {\rm d}
\hat{t} \ \frac{d \hat{\sigma}_{b g \to b h}}{d \hat{t}}
\label{cross}
\end{equation}
where $ \hat{t}_{max} = 0 $ and $ \hat{t}_{min}= -\hat{s}+ m_h^2 $
with $m_h$ denoting the Higgs mass. In order to avoid collinear
divergence in Eq.(\ref{cross}) and to enable the outgoing $b$-jet
to be tagged by silicon vertex detector at Tevatron and LHC , we
require the transverse momentum of the outgoing $b$-jet to be
larger than $15$ GeV.

The total hadronic cross section for $pp({\rm or}~p\bar{p}) \to b
h +X$ can be obtained by folding the subprocess cross section
$\hat{\sigma}$ with the parton luminosity
\begin{equation}
\sigma(s)=\int_{\tau_0}^1 \!d\tau\, \frac{dL}{d\tau}\, \hat\sigma
(\hat s=s\tau) ,  \label{cross1}
\end{equation}
where $\tau_0=m_h^2/s$ and $s$ denotes the $p p({\rm
or}~p\bar{p})$ squared center-of-mass energy. $dL/d\tau$ is the
parton luminosity given by
\begin{equation}
\frac{dL}{d\tau}=\int^1_{\tau} \frac{dx}{x}[f^p_g(x,Q)
f^{p}_b(\tau/x,Q)+(g\leftrightarrow b)].
\end{equation}
Here $f^p_b$ and $f^p_g$ are the bottom quark and gluon
distribution functions in a proton, respectively. In our numerical
calculation, we use the CTEQ5L parton distribution
functions~\cite{pm} with $Q= \sqrt{\hat s}/2$.

The study of the process $p p ( {\rm or} ~p\bar{p}) \to b h + X$
in the SM \cite{willenbrock} showed that the tree-level cross
section is of $O(1) $ fb for Tevatron with $\sqrt{s} =2$ TeV and
$O(100) $ fb for LHC with $ \sqrt{s} =14$ TeV, and the QCD
correction enhances the cross section by $ 20\% $ to $ 60\% $. For
convenience to show the main feature of this process in the MSSM,
we define the relative SUSY-QCD correction as
\begin{eqnarray}
\Delta_{SQCD}&=&\frac{\sigma_{MSSM}-\sigma^{0}_{MSSM}}{\sigma^{0}_{MSSM}},
\end{eqnarray}
where $\sigma^{0}_{MSSM}$  is the cross section at tree-level and
$\sigma_{MSSM}$ is that with the SUSY-QCD corrections.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical results}
\label{sec:parameters}

Before performing numerical calculations, we take a look at the
relevant parameters involved.
For the SM parameters, we took $m_W=80.448$ GeV, $m_Z=91.187$ GeV,
$m_t=176$ GeV, $m_b(m_b)= 4.2$ GeV  \cite{Groom} $
\sin^2 \theta_W =0.223$ and $\alpha_s(m_Z) = 0.118$ . We used the
one-loop running coupling constant $\alpha_s(\mu)$ and mass
$m_b (\mu) $ with $\mu=m_h$ .

For the SUSY parameters, apart from gluino mass, the mass
parameters of sbottoms are involved. The sbottom squared-mass
matrix is ~\cite{Gunion}
\begin{equation}
M_{\tilde b}^2 =\left(\begin{array}{cc}
m_{{\tilde b}_L}^2& m_b X_b\\
 m_b X_b & m_{{\tilde b}_R}^2
    \end{array} \right),
\end{equation}
where
\begin{eqnarray}
m_{{\tilde b}_L}^2 &=& m_{\tilde Q}^2+m_b^2 + m_Z^2(I_3^b
-Q_b \sin^2\theta_W)\cos(2\beta), \\
m_{{\tilde b}_R}^2 &=& m_{\tilde D}^2+m_b^2
+ m_Z^2 Q_b \sin^2\theta_W\cos(2\beta),\\
X_b&=& A_b-\mu\tan\beta.  \label{smass1}
\end{eqnarray}
Here $m_{\tilde Q}^2$ and $m_{\tilde{D}}^2$ are soft-breaking mass
terms for left-handed squark doublet $\tilde Q$ and right-handed
down squark $\tilde D$, respectively. $A_b $ is the coefficient of the
trilinear term $H_1 \tilde Q \tilde D$  in soft-breaking terms and
$\mu$ the bilinear coupling of the two Higgs doublet in the
superpotential.  $ I_3^b =-1/2 $ and $ Q_b =-1/3 $ are the isospin
and electric charge of the b-quark, respectively. This mass square
matrix can be diagonalized by a unitary rotation
\begin{eqnarray}
  \left(  \begin{array}{c} \tilde b_L \\ \tilde b_R \end{array} \right )
    =\left (
             \begin{array}{cc}
            \cos\theta_b       &\sin\theta_b\\
           -\sin\theta_b       &\cos\theta_b\\
           \end{array} \right )
\left(  \begin{array}{c} \tilde b_1 \\ \tilde b_2 \end{array} \right ),
\end{eqnarray}
and consequently $\theta_b$ and the masses of physical
sbottoms $ \tilde{b}_{1,2} $  can be expressed as
\begin{eqnarray}
\tan{2 \theta_b} &=& \frac{2 m_b X_b}{(m_{\tilde{b}_L}^2
-m_{\tilde{b}_R}^2)}, \label{theta} \\
m_{\tilde{b}_1}^2&= &m_{\tilde{b}_L}^2 \cos^2{\theta_b} -2 m_b X_b
\cos{\theta_b} \sin{\theta_b} + m_{\tilde{b}_R}^2 \sin^2 \theta_b ,\\
m_{\tilde{b}_2}^2&= &m_{\tilde{b}_L}^2 \sin^2{\theta_b} + 2 m_b
X_b \cos{\theta_b} \sin{\theta_b} + m_{\tilde{b}_R}^2 \cos^2{\theta_b} .
 \end{eqnarray}


From Eq.(\ref{tree},\ref{loop},\ref{cross}), we know that the cross section
also depends on the Higgs mass, $\alpha$ and $\beta$, which
can be determined at tree level by $\tan{\beta}$ and
the CP-odd Higgs mass $m_A$ \cite{Gunion}. Noticing the fact that
both the mass and the mixing angle receive large radiative
corrections when SUSY scale is high above $m_t$ \cite{Haber},
we use the loop-corrected relations of Higgs masses
and mixing angle \cite{Carena1,Carena2} in the computation of cross section.
In our calculation, we use the program SUBHPOLE2 \cite{Carena1},
where two loop leading-log effects of the MSSM are incorporated in
Higgs masses and the mixing angel, to generate
$m_h$ and $\alpha$ needed for our computation.
The input parameters for this program are the mass parameters
in scalar-top and scalar-bottom sector, $m_A$, $\tan \beta$
and the largest chargino mass $m_{\tilde{\chi}}$.

We found that the usage of the loop-corrected relations of
Higgs masses and mixing angle is indeed necessary.
Comparing with the results obtained by using tree-level relations
for Higgs masses and mixing angel, the absolute value of
$\Delta_{SQCD}$ by using  the loop-corrected relations
is generally enhanced by $30\%$ to $200 \%$.
We also checked that this conclusion is also applied to SUSY-QCD
correction to Higgs partial width $\Gamma (h \to b \bar{b})$.

Now we know the relevant parameters are
\begin{equation} \label{para}
m_{\tilde{Q}}, m_{\tilde{U}}, m_{\tilde{D}}, A_{t,b}, m_{\tilde{g}},
m_{\tilde{\chi}}, \mu, m_A, \tan{\beta} .
\end{equation}
To show the main features of SUSY effects on  $ pp({\rm
or}~p\bar{p}) \to b h +X$, we performed a scan over this
ten-dimensional parameter space. In our scan we make no
assumptions about the relations among these parameters to keep our
result model-independent, but restrict the parameters with mass
dimension to be less than $2$ TeV. In addition, we consider the
following experimental constraints:
\begin{itemize}
\item[{\rm(1)}]
   $\mu>0$ and a large $\tan\beta$ in the range $5\le\tan\beta\le 50$, which seems
   favored by the muon $g-2$ measurement~\cite{Brown01}.
\item[{\rm(2)}]
   The LEP and CDF lower mass bounds on Higgs, gluino, stop, sbottom and chargino~\cite{LEP,PDG00}
\begin{eqnarray}
m_h \geq 114~GeV, m_{{\tilde t}_1}\geq 86.4~GeV,~ m_{{\tilde
b}_1}\geq 75.0~GeV, ~ m_{\tilde{g}}\geq 190~GeV, ~
m_{\tilde{\chi}} \geq 67.7~GeV .\label{constrain}
\end{eqnarray}
\end{itemize}
%%%%%%% Fig.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\epsfig{file=fig2.ps,width=8cm} \caption{The scatter plot in the
plane of  $\sigma^{0}_{MSSM}/\sigma_{SM}^{0}$
versus $m_A$ for different $\tan{\beta}$. } \label{fig:scan1}
\end{center}
\end{figure}
\vspace*{-1cm}
%%%%%%% Fig. 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\epsfig{file=fig3.ps,width=8cm} \caption{The scatter plot in the
plane of $\Delta_{SQCD}$ versus $m_A$ for the LHC. } \label{fig:scan2}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Fig.\ref{fig:scan1} and Fig.\ref{fig:scan2} show the
dependence of  $\sigma^{0}_{MSSM}/\sigma_{SM}^{0}$
and $\Delta_{SQCD}$ on $m_A $ for various
$\tan{\beta}$. From Fig.\ref{fig:scan1} one
sees that at tree-level the MSSM predictions may be significant
larger than that of the SM for a light $m_A $; while for a heavy $m_A$
the MSSM predictions approach to the SM value. This character was first
noticed in \cite{haber-decoup} and, as a result of this character,
distinguishing the lightest MSSM Higgs boson in the large $m_A$ limit
from the SM Higgs boson will be very difficult.
When SQCD corrections are added, this character remains unchanged
if sbottoms is not too light (see following discussions).
From Fig.\ref{fig:scan1} one also finds that there exists the
possibility (although very rare) that the MSSM cross section
is suppressed to be below the SM value \cite{Gordon}.
In this case, the SUSY-QCD corrections will play a more
important role in Higgs phenomenology at colliders \cite{Carena3}.

Fig.\ref{fig:scan2} shows four main characters of SUSY-QCD
corrections. The first one is the correction size
$|\Delta_{SQCD}|$ may be quite large. The second one is
$|\Delta_{SQCD}|$ can be enhanced by large $\tan{\beta}$. The
third one is that for $m_A$ lighter than $500$ GeV $\Delta_{SQCD}$
tends to be negative. The last one is that for large $m_A$, $\Delta_{SQCD}$ may be
positive and the maximum reach of $\Delta_{SQCD}$ seems to be
independent of the value of $m_A$. These features can be well
explained by using the main results of SUSY-QCD corrections to
vertex $h b \bar{b}$\cite{haber-nondecoup}. For large SUSY mass
parameters which are satisfied in most cases of our scan,
$\Delta_{SQCD} $ can be expressed in the approximate form
\begin{eqnarray}
\Delta_{SQCD} \sim C_1 \frac{M_{EW}^2}{M_A^2} +C_2
\frac{M_{EW}^2}{M_{\tilde{b}}^2} , \label{explain}
\end{eqnarray}
where  $M_{EW} $ and $ M_{\tilde{b}} $ denote electro-weak scale
and typical mass of sbottom, respectively.  $C_{1} $ and $C_2 $
are functions of $m_{\tilde{b}_1}$, $m_{\tilde{b}_2}$,
$m_{\tilde{g}}$, $\mu $ and $A_b$, but independent of $m_A$.  It
is found that in general $C_1$ is negative and $C_2 $ is positive
and either $C_1$ or both $C_1$ and $C_2$ are enhanced by large
$\tan{\beta}$. For a light $m_A$, the first term of the RHS in
Eq.(\ref{explain}) is dominant and hence $\Delta_{SQCD}$ tends to
be negative. While for a large $m_A$ and light $m_{\tilde{b}}$,
the second term is dominant and $\Delta_{SQCD}$ tends to be
positive \footnote{It should be noted that experimental bound on
Higgs mass requires a rather heavy scalar top-quark. If we demand the
sbottom masses to be comparable with stop masses, the
contributions to $\Delta_{SQCD}$ from the second term in
Eq.(\ref{explain}) can be neglected and $\Delta_{SQCD}$ tends to
be negative.}.

From our scan of $\Delta_{SQCD}$ we noticed  that in some corners
of parameter space the SUSY-QCD loop contributions to the process are comparable
or even larger than the tree level contributions. In such cases,
it is important to resume the leading $\tan{\beta}$ contributions
to all orders of perturbation theory by using an effective Lagrangian
approach (for relevant techniques, see \cite{technique}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Scenario Analysis}
\label{sec:decouple}

From above analysis, we know that ten parameters are involved in
our calculation. To show the explicit characters of SUSY-QCD
corrections, we must make some assumptions about these parameters.
Since we are interested in the non-decoupling property of SUSY-QCD
correction in the large limit of SUSY mass parameters, we consider
four scenarios presented in Ref.\cite{haber-nondecoup} where the
non-decoupling property of SUSY-QCD correction to the vertex $h b
\bar{b} $ is analyzed \footnote{One difference between our
analysis and Ref.\cite{haber-nondecoup} is that in scenario B and
D we also require $A_{t,b}$ to be large since large $A_{t,b}$ is
favored by Higgs mass bound. This does not change the decoupling
behavior drastically. }. In fact we found from our numerical
calculations that $\Delta_{SQCD}(p p \to b h +X)$ is correlated
with $\Delta_{SQCD}(h \to b \bar{b})$ when the correction size is
larger than about $4 \%$, which indicates that the dominant
corrections in our production process is from the vertex of
$hb\bar b$. So some behaviors of our results can be well explained
by the formula presented in Ref.\cite{haber-nondecoup} . To better
understand the feature of each scenario, we present these formula
in this paper, but in practical calculations we use the precise
one-loop expressions of $ \Delta_{SQCD}$ to get precise numerical
results. There are two main differences between our calculations
and those in \cite{haber-nondecoup}. One is that in
\cite{haber-nondecoup} the tree-level  relations for Higgs masses
and mixing angle were used. But in our calculations we use the
loop-corrected relations for Higgs masses and mixing angle. As we
discussed earlier, using the loop-corrected relations leads to a
significantly larger correction than that by using the tree-level
relations. The other difference that in our analysis  we
considered the experimental bounds in Eq.(\ref{constrain}).

(1) {\em Scenario A:}~~ All SUSY breaking parameters and $\mu$ are
of the same size (collectively denoted by $M_S$ ), i.e.,
\begin{eqnarray}
m_{\tilde Q} \sim m_{\tilde U} \sim m_{\tilde D} \sim A_b \sim A_t
\sim m_{\tilde g}\sim \mu  \sim M_S .
\end{eqnarray}
In this case, Higgs mass bound requires squark masses to be much
larger than $m_{EW} $ and $\Delta_{SQCD} $ can be well
approximated by
\begin{eqnarray}
\Delta_{SQCD} \simeq \frac{2 \alpha_s}{3 \pi} \left [ - (
\tan{\beta}+\cot{\alpha}) -\cot{\alpha} (\frac{m_h^2}{12 M_S^2}+
\frac{m_b^2 \tan^2{\beta}}{2 M_S^2}) + \frac{\tan{\beta}}{3}
\frac{\cos{\beta} \sin(\alpha +\beta)}{\sin{\alpha}}
\frac{m_Z^2}{M_S^2} \right ] , \label{approx}
\end{eqnarray}
where the first term in the RHS corresponds to the first term in
Eq.(\ref{explain})\footnote{In MSSM, tree level relation for Higgs
masses and mixing angle predicts the following relation :
$\cot{\alpha} = -\tan{\beta} -\frac{2 m_Z^2}{m_A^2} \tan{\beta}
\cos{2 \beta} +O(\frac{m_Z^4}{m_A^4}) $ and at loop level, the gap
between $\cot{\alpha} $ and $-\tan{\beta} $ is generally
enlarged.}  and the rest corresponds to the second term in
Eq.(\ref{explain}). For  $ m_A $ of several hundred GeV, the
first term is dominant and as a result, $ \Delta_{SQCD} $  depends
weakly on $M_S $ via $\cot{\alpha}$. The striking features of this
scenario are that for very large $ M_S $, $\Delta_{SQCD} $
approaches a nonzero constant, and this non-decoupling behavior of
SUSY-QCD corrections is enhanced  by large $\tan{\beta} $. These
features are illustrated in Fig.\ref{fig:cas1A}.
From Fig.\ref{fig:cas1A} one finds that
SUSY-QCD correction in this scenario is negatively large for a light $
m_A $ and a large $\tan{\beta} $ and thus it can reduce the cross
section severely.
%%%%%% Fig.4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\epsfig{file=fig4.ps,width=15cm}
\caption{$\Delta_{SQCD}$ versus $M_S$ in Scenario A for the LHC.}
\label{fig:cas1A}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now we turn to the cross section of the process $p p ( {\rm or}
~p\bar{p}) \to b h + X$. Including $O(\alpha_s) $ corrections, it
can be expressed as $ \sigma_{MSSM}=\sigma^{0}_{MSSM} (1+
\Delta_{SQCD} +\Delta_{QCD}) $, where $\Delta_{QCD} $ represents
ordinary QCD corrections and it generally depends on the Higgs
mass, collider energy and detection cuts \cite{willenbrock}. For
LHC with $\sqrt{s} =14 $ TeV, $\Delta_{QCD} $ ranges from $20\% $
to $ 40\% $ and for Tevatron with $\sqrt{s} =2 $ TeV, it varies
from $50 \% $ to $ 60 \% $. Since the cross section in the LHC is
much larger than that at Tevatron, we present our results only for
the LHC. To reduce the dependence of our result on $\Delta_{QCD}$,
we will show the ratio of $\sigma_{MSSM}$ and $\sigma_{SM}$
\begin{eqnarray}
\frac{\sigma_{MSSM}}{\sigma_{SM}}=\frac{1+ \Delta_{SQCD}
+\Delta_{QCD}}{1+ \Delta_{QCD}} \frac{\sigma^{0}_{MSSM}}{\sigma^{0}_{SM}}
=\frac{1+ \Delta_{SQCD}+\Delta_{QCD}}{1+ \Delta_{QCD}} \left (
\frac{\sin{\alpha}}{\cos{\beta}} \right )^2
\end{eqnarray}
and fix $\Delta_{QCD} =30 \% $ in our following discussion. The
dependence of $\sigma_{MSSM}/\sigma_{SM}$ on $M_S $ is plotted in
Fig.\ref{fig:acas1A}. We see that for $ m_A $ of several hundred
GeV, although  $\sigma_{MSSM}$ may subject to negatively large
SUSY-QCD corrections, a large enhancement over the SM prediction
can still be expected. This large enhancement shows a very weak
dependence on $M_S$. So we can conclude that up to the
next-leading order, a light $ m_A $ is still able to make the
MSSM cross section larger than the SM prediction. This conclusion
is valid to other scenarios discussed below.
%%%%%%%%%% Fig.5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\epsfig{file=fig5.ps,width=15cm}
\caption{ $\sigma_{MSSM}/\sigma_{SM}$ versus $m_A $ in Scenario A
          for the LHC. }
\label{fig:acas1A}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(2) {\em Scenario B:}~~
    $M_{\tilde Q}$ $M_{\tilde U}$, $M_{\tilde D}$ and $A_{t, b} $
(collectively denoted as $M_S $) are far larger than $\mu $,
$m_{\tilde{g}} $ and $m_A $ (collectively denoted as $M $ ), i.e.,
\begin{eqnarray}
M_{\tilde Q, \tilde U, \tilde D} \sim A_{t, b}  \sim  M_S  \gg m_{\tilde g} \sim \mu
\sim m_A \sim M .
\end{eqnarray}
In this scenario $\Delta_{SQCD} $ can be roughly expressed as
\begin{eqnarray}
\Delta_{SQCD} &\simeq & \frac{2 \alpha_s}{3 \pi} \left [ \frac{-2 M^2}{M_S^2} (\tan{\beta}+\cot{\alpha} )
- \frac{m_h^2 M^2}{ 6 M_S^4} (\frac{M_S}{M}+\cot{\alpha}) \right . \nonumber \\
& & \left . -\frac{m_Z^2}{2 M_S^2}
\frac{\cos{\beta} \sin{(\alpha+\beta)}}{\sin{\alpha}} (1- (\frac{M_S}{M}-\tan{\beta}) \frac{2 M^2 }{M_S^2} ) \right ].
\end{eqnarray}
From this expression we see that in the large $M_S$ limit, the SQCD corrections decouple rapidly
as $M^2/M_S^2 $ and the decoupling behavior is delayed by large $\tan{\beta} $.
The characters of this scenario are shown in Fig.\ref{fig:casB}.
%%%% Fig.6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\epsfig{file=fig6.ps,width=15cm} \caption{ Dependence of
$\Delta_{SQCD}$ and $\sigma_{MSSM}/\sigma_{SM}$ on $M_S$ in Scenario B
for the LHC. } \label{fig:casB}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(3) {\em Scenario C:}~~ Only the gluino mass is much larger than
other SUSY parameters (collectively denoted as $ M_S $):
\begin{eqnarray}
m_{\tilde{g}} \gg M_{\tilde Q, \tilde U, \tilde D} \sim A_{t, b} \sim \mu
\sim m_A \sim M_S .
\end{eqnarray}
In this scenario the Higgs mass bound requires $M_S $ to be much
larger than electroweak scale and $\Delta_{SQCD}$ can be
written as
\begin{eqnarray}
\Delta_{SQCD} &\simeq & \frac{2 \alpha_s}{3 \pi} \left [
\frac{2 M_S}{M_{\tilde{g}}} (\tan{\beta}+ \cot{\alpha})
(1- \log{\frac{M_{\tilde{g}}^2}{M_S^2}} ) -
\frac{M_S \cot{\alpha}}{3 M_{\tilde{g}}} \frac{m_h^2}{M_S^2} \right .
\nonumber \\
&& \left . +
\frac{M_S \tan{\beta}}{M_{\tilde{g}}} \frac{m_Z^2}{M_S^2}
\frac{\cos{\beta} \sin{(\alpha+\beta)}}{\sin{\alpha}} -
\frac{m_b^2 \tan^2{\beta} \cot{\alpha}}{M_{\tilde{g}} M_S} \right ] .
\end{eqnarray}
The main character of this scenario is that as gluino mass gets
large, $\Delta_{SQCD}$ drops very slowly like
$\frac{1}{m_{\tilde{g}}} \log{\frac{m_{\tilde{g}}^2}{M_S^2}}$ and
$\Delta_{SQCD}$ is enhanced by large $\tan{\beta}$.
%%%% Fig.7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\epsfig{file=fig7.ps,width=15cm} \caption{ Dependence of
$\Delta_{SQCD}$ and $\sigma_{MSSM}/\sigma_{SM}$ on gluino mass in Scenario C
for the LHC. } \label{fig:casC}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 In Fig.\ref{fig:casC}  we show the dependence of $\Delta_{SQCD} $ and
$\sigma_{MSSM}/\sigma_{SM}$ on gluino mass. Note that in this
scenario we found that $\tan{\beta}=8$ cannot satisfy the
experimental bounds in Eq.(\ref{constrain}) for $M_S =600$ GeV. In
Fig.\ref{fig:casC} $\sigma_{MSSM}/\sigma_{SM}$ is significantly
smaller than those in Scenarios A and B. The reason is here a
large $m_A $ is chosen.

(4) {\em Scenario D:} ~~One of the sbottoms and $A_{t,b} $ become heavy while
 other mass parameters (denoted as M)  are fixed. We choose
\begin{eqnarray}
M_{\tilde D} \sim M_{\tilde U} \sim A_{t,b} \gg M_{\tilde Q} \sim m_{\tilde{g}}
\sim \mu \sim M \gg M_{EW}
\end{eqnarray}
or equally
\begin{eqnarray}
m_{\tilde{b}_2} \sim A_{t,b} \gg m_{\tilde{b}_1} \sim m_{\tilde{g}} \sim \mu \sim M \gg M_{EW}.
\end{eqnarray}
$\Delta_{SQCD} $ in this scenario can be well approximated by
\begin{eqnarray}
\Delta_{SQCD} & \simeq & \frac{2 \alpha_s}{3 \pi} \left [
\frac{2 M^2}{m_{\tilde{b}_2}^2} (\tan{\beta}+ \cot{\alpha})
\left (1+ \log\frac{M^2}{m_{\tilde{b}_2}^2} \right ) + \frac{m_Z^2}{m_{\tilde{b}_2}^2}
\frac{\cos{\beta} \sin{(\alpha+\beta)}}{\sin{\alpha}} (-1+\frac{2}{3} s_W^2)
(\frac{m_{\tilde{b}_2}}{m_{\tilde{b}_1}} -\tan{\beta} ) \right ].
\end{eqnarray}
%%%%% Fig.8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\epsfig{file=fig8.ps,width=15cm} \caption{ Dependence of
$\Delta_{SQCD}$ and $\sigma_{MSSM}/\sigma_{SM}$ on $M_{\tilde D}$
in Scenario D for the LHC. } \label{fig:casD}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The main feature of this scenario is $\Delta_{SQCD} $ decouples
like $ \frac{1}{m_{\tilde{b}_2}^2}
\log\frac{M^2}{m_{\tilde{b}_2}^2} $ and this decoupling behavior
is delayed by large $ \tan{\beta} $.  The dependence of
$\Delta_{SQCD} $ and  $\sigma_{MSSM}/\sigma_{SM}$ on $m_{\tilde
D}$ ($\sim m_{\tilde{b}_2}$) are shown in Fig.\ref{fig:casD}.
Comparing with results in scenario B, we see $\Delta_{SQCD}$
decrease more slowly as $ m_{\tilde{D}} $ becomes heavy.

Before we end this section, we want to emphasis four points. The
first is that only when MSSM parameters $\mu $ and $ m_{\tilde{g}}
$ are comparable with or larger than the masses of bottom squark
can the non-decoupling behavior of SUSY-QCD corrections occur.
The fundamental reason for such non-decoupling behavior is
that the couplings like $h\tilde{b}_i \tilde{b}_j$ are proportional
to SUSY mass parameters. The second is that in our analysis we
assumed $m_{\tilde{g}}, \mu > 0  $ and, as a result, all four
scenarios predict negative results of $\Delta_{SQCD}$. In the
anomaly-mediated SUSY breaking scenario\cite{anomal}, a negative
$m_{\tilde{g}}$ is predicted and in this case, the sign of
$\Delta_{SQCD}$ may be reversed.  The third points is that using
loop-corrected relations of Higgs masses and mixing angle may
enhance the values of  $|\Delta_{SQCD}|$, compared with the
results by using the tree-level relations of Higgs masses and
mixing angle. Numerically speaking, the magnitude of the enhancement
can reach $80 \%$ for scenario A and may be larger for other
scenarios. The last point we want to emphasize is that in case of
a light $ m_A $, although the SUSY-QCD corrections tend to reduce
the cross section severely, the MSSM cross section can still be
much larger than the SM.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
\label{sec:conclusions}

In this work, we studied the SUSY-QCD quantum effects in the
Higgs production $ p p ~( or ~p\bar{p} ) \to b h +X $ in the
framework of the MSSM. We found that for a light $m_A$ and
large $\tan{\beta}$, the corrections can be quite large and
cannot be neglected. We also observed that it is necessary to
use the loop-corrected relations for Higgs masses and
mixing angle in such calculations. We performed a detailed
analysis on the decoupling behaviors of the corrections.
We presented the analytic expressions of the corrections
in the decoupling limits. We found that in some cases
the SUSY-QCD corrections do not decouple or decouple very
slowly. In case that the sparticles are too heavy (above TeV)
to be directly discovered at the LHC, measuring their
non-decoupling loop effects in some observable
processes like $pp ~( or ~p\bar{p} ) \to b h +X$ will
be the only way to detect the sparticles.

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

\section*{Acknowledgment}
We thank Tao Han for discussions.
This work is supported in part by a grant of Chinese Academy of Science
for Outstanding Young Scholars.

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



\section{Appendix A}

Before presenting the explicit form of $C_i$s, we define the
following abbreviations:
\begin{eqnarray}
\hat{s}&=& (p_1+k)^2,\ \ \ \ \ \ \hat{t}= (k-p_2)^2,     \\
a_{1,2}&=&\frac{1}{\sqrt{2}} (\sin{\theta_b} \mp \cos{\theta_b}),\
\ \ \ \ \ \ \ \ \ b_{1,2}=\frac{1}{\sqrt{2}} (\cos{\theta_b} \pm
\sin{\theta_b}),    \\
A_{I}^L&=& (a_I-b_I)^2, \ \ \ \ A_{I}^R=(a_I + b_I)^2,\ \ \ \
A_I=a_I^2-b_I^2,\ \ \ \ (I=1,2),   \\
A_{IJ}^L&=& a_I a_J+b_I b_J+ a_I b_J+ b_I a_J, \ \ \ \ \
A_{IJ}^R=a_I
a_J+b_I b_J- a_I b_J - b_I a_J,     \\
B_{IJ}^L&=&a_I a_J-b_I b_J - a_I b_J+ b_I a_J, \ \ \ \ \
B_{IJ}^R=a_I a_J - b_I b_J + a_I b_J - b_I a_J,    \\
cc_1 &=& -\frac{m_Z}{\cos{\theta_W}} \left (\frac{1}{2}
-\frac{1}{3} \sin^2{\theta_W} \right ) \sin(\alpha+\beta),
  \\ cc_2& = & -\frac{m_Z}{\cos{\theta_W}} \frac{1}{3}
\sin^2{\theta_W} \sin(\alpha+\beta), \ \ \ \
cc_3 = \frac{m_b}{2 m_W \cos{\beta}} (A_b \sin{\alpha} + \mu \cos{\alpha}),    \\
Q_{11}&=&cc1 \cos^2\theta_b +cc2 \sin^2 \theta_b +2 cc3
\sin\theta_b \cos \theta_b,   \\
Q_{12}&=&(cc2-cc1) \sin{\theta_b}\cos{\theta_b}
+ cc3 (\cos^2\theta_b -\sin^2\theta_b),     \\
Q_{21}&=&(cc2-cc1) \sin{\theta_b}\cos{\theta_b} + cc3
(\cos^2\theta_b -\sin^2\theta_b),   \\
Q_{22}&= &cc1 \sin^2\theta_b +cc2 \cos^2 \theta_b - 2 cc3
\sin\theta_b \cos \theta_b,     \\
B^{I}_i&= & B_i (p, m_{\tilde{g}}, m_{\tilde{b}_I}) |_{p^2=m_b^2},\\
B^{s\ I}_i&=& B_i (p, m_{\tilde{g}}, m_{\tilde{b}_I})|_{p^2=\hat{s}},
\ \ \ \ \ \ B^{t\ I}_i = B_i (p, m_{\tilde{g}},
m_{\tilde{b}_I}) |_{p^2=\hat{t}},   \\
C^{a\ I}_{ij}&=&C_{ij}(p_1,k,m_{\tilde{b}_I}, m_{\tilde{g}},
m_{\tilde{g}}), \ \ \ \ C^{b\ I}_{ij}
=C_{ij}(-p_1,-k,m_{\tilde{g}}, m_{\tilde{b}_I},
m_{\tilde{b}_I} ),   \\
C^{c\ I}_{ij}&=&C_{ij}(-p_2,k,m_{\tilde{b}_I}, m_{\tilde{g}},
m_{\tilde{g}}), \ \ \ \ C^{d\ I}_{ij} =C_{ij}(-p_2,k,m_{\tilde{g}},
m_{\tilde{b}_I}, m_{\tilde{b}_I} ),   \\
C^{e\ IJ}_{ij}&=&C_{ij}(-p_2,-p_h, m_{\tilde{g}}, m_{\tilde{b}_I},
m_{\tilde{b}_J}), \ \ \ \ C^{f\ IJ}_{ij} =C_{ij}(-p_1,p_h,
m_{\tilde{g}}, m_{\tilde{b}_J}, m_{\tilde{b}_I} ),   \\
D^{g\ IJ}_{ij}&=&D_{ij} (-p_1,-k, p_h, m_{\tilde{g}},
m_{\tilde{b}_J}, m_{\tilde{b}_J}, m_{\tilde{b}_I}), \ \ \ \ D^{h\
IJ}_{ij}=D_{ij} (-p_1, p_h, p_2, m_{\tilde{g}}, m_{\tilde{b}_J},
m_{\tilde{b}_I}, m_{\tilde{g}}),
\end{eqnarray}
where $B_i$, $C_{ij} $ and $D_{ij} $ are loop functions defined in
\cite{hooft}.

After $ \frac{g_s^2}{16 \pi^2} $ is factored out, the
renormalization constant of b-quark can be expressed as
\begin{eqnarray}
\delta Z_L & = & C_F \sum_{I=1}^2 A_I^L B_1^I,\ \ \ \ \ \delta Z_R
 =  C_F \sum_{I=1}^2 A_I^R B_1^I,     \\
\frac{\delta m_b}{m_b}& = & C_F \sum_{I=1}^2
(\frac{m_{\tilde{g}}}{m_b} A_I B_0^I -\frac{1}{2} A_I^L B_1^I
-\frac{1}{2} A_I^R B_1^I) ,
\end{eqnarray}
where $C_F=4/3 $. The contributions of the self-energy diagrams of $b$-quark propagator
can be written as
\begin{eqnarray}
\Sigma_L^s&=&C_F \sum_{I=1}^2 A_I^L B_1^{s\ I} - \delta Z_L, \ \ \
\ \Sigma_L^t=C_F \sum_{I=1}^2 A_I^L B_1^{t\ I} - \delta Z_L, \\
\Sigma_R^s&=&C_F \sum_{I=1}^2 A_I^R B_1^{s\ I} - \delta Z_R, \ \ \
\ \Sigma_R^t=C_F \sum_{I=1}^2 A_I^R B_1^{t\ I} - \delta Z_R.
\end{eqnarray}

$C_i$ appeared in Eq.(\ref{loop}) are given by
\begin{eqnarray}
C_3 &= & \sum_{I=1}^2 \{ -\frac{3}{2} (\hat{s} C_{12}^{a\ I} +
\hat{s} C_{23}^{a\ I}+ 2 C_{24}^{a\ I} -1/2 - m_{\tilde{g}}^2
C_{0}^{a\ I}) A_I^L/\hat{s}+ \frac{1}{3}  C_{24}^{b\ I} A_I^L/\hat{s} \nonumber  \\
& &  -\frac{3}{2} (\hat{t} C_{12}^{c\ I} + \hat{t} C_{23}^{c\ I}+
2 C_{24}^{c\ I} -1/2 - m_{\tilde{g}}^2 C_{0}^{c\ I})
A_I^R/\hat{t}+ \frac{1}{3}  C_{24}^{d\ I} A_I^R /\hat{t} \} \nonumber\\
&& -\delta Z_L/\hat{s}-\delta
Z_R/\hat{t}-\Sigma_L^s/\hat{s}-\Sigma_R^t/\hat{t}-(\frac{1}{2}
\delta Z_L+\frac{1}{2} \delta Z_R +\frac{\delta m_b}{m_b} )
(\frac{1}{\hat{s}}+\frac{1}{\hat{t}}) \nonumber  \\
&&+ \frac{2 m_W \cos{\beta}}{m_b \sin{\alpha}} \sum_{I,J=1}^2
Q_{IJ} \{ \frac{4}{3} m_{\tilde{g}} C_{0}^{e\ IJ} B_{IJ}^L
/\hat{s}+\frac{4}{3} m_{\tilde{g}} C_{0}^{f\ IJ} B_{IJ}^L
/\hat{t}+ \frac{3}{2} m_{\tilde{g}} D_{0}^{h\ IJ} B_{IJ}^L \},\\
C_4 &= & \sum_{I=1}^2 \{ -\frac{3}{2} (\hat{s} C_{12}^{a\ I} +
\hat{s} C_{23}^{a\ I}+ 2 C_{24}^{a\ I} -1/2 - m_{\tilde{g}}^2
C_{0}^{a\ I}) A_I^R/\hat{s}+ \frac{1}{3}  C_{24}^{b\ I} A_I^R/\hat{s}\nonumber   \\
& &  -\frac{3}{2} (\hat{t} C_{12}^{c\ I} + \hat{t} C_{23}^{c\ I}+
2 C_{24}^{c\ I} -1/2 - m_{\tilde{g}}^2 C_{0}^{c\ I})
A_I^L/\hat{t}+ \frac{1}{3}  C_{24}^{d\ I} A_I^L /\hat{t} \} \nonumber\\
&& -\delta Z_R/\hat{s}-\delta
Z_L/\hat{t}-\Sigma_R^s/\hat{s}-\Sigma_L^t/\hat{t}-(\frac{1}{2}
\delta Z_L+\frac{1}{2} \delta Z_R +\frac{\delta m_b}{m_b} )
(\frac{1}{\hat{s}}+\frac{1}{\hat{t}})   \nonumber \\
&&+ \frac{2 m_W \cos{\beta}}{m_b \sin{\alpha}} \sum_{I,J=1}^2
Q_{IJ} \{ \frac{4}{3} m_{\tilde{g}} C_{0}^{e\ IJ} B_{IJ}^R
/\hat{s}+\frac{4}{3} m_{\tilde{g}} C_{0}^{f\ IJ} B_{IJ}^R
/\hat{t}+ \frac{3}{2} m_{\tilde{g}} D_{0}^{h\ IJ} B_{IJ}^R \}, \\
C_5 &= & \sum_{I=1}^2 \{ -\frac{3}{2} (-4 C_{24}^{a\ I} +1 + 2
m_{\tilde{g}}^2 C_{0}^{a\ I}) A_I^L/\hat{s}- \frac{1}{3} ( \hat{s}
C_{12}^{b\ I}+\hat{s} C_{23}^{b\ I}+ 2 C_{24}^{b\ I} )  A_I^L/\hat{s} \} \nonumber  \\
&& + 2 \delta Z_L/\hat{s}+ 2 \Sigma_L^s/\hat{s}+( \delta Z_L+
\delta Z_R +2 \frac{\delta m_b}{m_b} )/\hat{s}+ \frac{2 m_W
\cos{\beta}}{m_b \sin{\alpha}} \sum_{I,J=1}^2 Q_{IJ} \{
-\frac{8}{3} m_{\tilde{g}} C_{0}^{e\ IJ} B_{IJ}^L /\hat{s} \nonumber \\
&& +\frac{1}{3} m_{\tilde{g}} (D_{0}^{g\ IJ}+
D_{11}^{g\ IJ}-D_{13}^{g\ IJ} ) B_{IJ}^L + 3 m_{\tilde{g}}
(D_{11}^{h\ IJ}- D_{12}^{h\ IJ}) B_{IJ}^L  \},\\
C_6 &= & \sum_{I=1}^2 \{ -\frac{3}{2} (-4 C_{24}^{a\ I} +1 + 2
m_{\tilde{g}}^2 C_{0}^{a\ I}) A_I^R/\hat{s}- \frac{1}{3} ( \hat{s}
C_{12}^{b\ I}+\hat{s} C_{23}^{b\ I}+ 2 C_{24}^{b\ I} )  A_I^R/\hat{s} \} \nonumber  \\
&& + 2 \delta Z_R/\hat{s}+ 2 \Sigma_R^s/\hat{s}+( \delta Z_L+
\delta Z_R +2 \frac{\delta m_b}{m_b} )/\hat{s}+ \frac{2 m_W
\cos{\beta}}{m_b \sin{\alpha}} \sum_{I,J=1}^2 Q_{IJ} \{
-\frac{8}{3} m_{\tilde{g}} C_{0}^{e\ IJ} B_{IJ}^R /\hat{s} \nonumber\\
&& +\frac{1}{3} m_{\tilde{g}} (D_{0}^{g\ IJ}+
D_{11}^{g\ IJ}-D_{13}^{g\ IJ} ) B_{IJ}^R + 3 m_{\tilde{g}}
(D_{11}^{h\ IJ}- D_{12}^{h\ IJ}) B_{IJ}^R  \},
  \\
C_9 &= & \sum_{I=1}^2 \{ -\frac{3}{2} (-4 C_{24}^{c\ I} +1 + 2
m_{\tilde{g}}^2 C_{0}^{c\ I}) A_I^R/\hat{t}- \frac{1}{3} ( \hat{t}
C_{12}^{d\ I}+\hat{t} C_{23}^{d\ I}+ 2 C_{24}^{d\ I} )  A_I^R
/\hat{t} \} \nonumber  \\
&& + 2 \delta Z_R/\hat{t}+ 2 \Sigma_R^t/\hat{t}+( \delta Z_L+
\delta Z_R +2 \frac{\delta m_b}{m_b} )/\hat{t}+ \frac{2 m_W
\cos{\beta}}{m_b \sin{\alpha}} \sum_{I,J=1}^2 Q_{IJ} \{
-\frac{8}{3} m_{\tilde{g}} C_{0}^{f\ IJ} B_{IJ}^L /\hat{t} \nonumber\\
&& +\frac{1}{3} m_{\tilde{g}} D_{13}^{g\ IJ} B_{IJ}^L
+ 3 m_{\tilde{g}} (D_{12}^{h\ IJ}- D_{13}^{h\ IJ}) B_{IJ}^L  \}, \\
C_{10} &= & \sum_{I=1}^2 \{ -\frac{3}{2} (-4 C_{24}^{c\ I} +1 + 2
m_{\tilde{g}}^2 C_{0}^{c\ I}) A_I^L/\hat{t}- \frac{1}{3} ( \hat{t}
C_{12}^{d\ I}+\hat{t} C_{23}^{d\ I}+ 2 C_{24}^{d\ I} )  A_I^L
/\hat{t} \}  \nonumber \\
&& + 2 \delta Z_L/\hat{t}+ 2 \Sigma_L^t/\hat{t}+( \delta Z_L+
\delta Z_R +2 \frac{\delta m_b}{m_b} )/\hat{t}+ \frac{2 m_W
\cos{\beta}}{m_b \sin{\alpha}} \sum_{I,J=1}^2 Q_{IJ} \{
-\frac{8}{3} m_{\tilde{g}} C_{0}^{f\ IJ} B_{IJ}^R /\hat{t}
\nonumber \\ && +\frac{1}{3} m_{\tilde{g}} D_{13}^{g\ IJ} B_{IJ}^R
+ 3 m_{\tilde{g}} (D_{12}^{h\ IJ}- D_{13}^{h\ IJ}) B_{IJ}^R  \}.
\end{eqnarray}

Since we have neglect the mass of b-quark throughout this paper,
$C_{1,2,7,8,11,12}$ are irrelevant to our result and we do not
present their explicit forms here.

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

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\end{thebibliography}\endgroup

\end{document}

