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\begin{document}        
\begin{titlepage}
\begin{flushright}
AMES-HET-97-5\\

\end{flushright}
\vspace{0.1in}
\begin{center}
{\Large Flavor-Changing Top Quark Decays In MSSM Without R-Parity }
\vspace{.2in}

  Jin Min Yang$^{a,b,}${\footnote{ On leave of absence 
                                from Department of Physics, 
                                   Henan Normal University, China}}, 
  Bing-Lin Young$^{a}$,  and X. Zhang$^c$

 
\vspace{.2in}
\it

$^a$     Department of Physics and Astronomy, Iowa State University,\\
         Ames, Iowa 50011, USA\\
$^b$     International Institute of Theoretical and Applied Physics,\\
         Iowa State University, Ames, Iowa 50011, USA\\
$^c$     Institute of High Energy Physics, Academia Sinica, \\
         Beijing 100039, China
\rm
\end{center}
\vspace{3cm}

\begin{center} ABSTRACT\end{center}

The flavor changing top quark decays $t\rightarrow cV $ ($V=Z,\gamma, g$)
induced by R-parity-violating couplings in the minimal 
supersymmetric standard model (MSSM) are evaluated.
We find that the decays $t \rightarrow cV$ can be  significantly enhanced 
relative
to those in the R-parity conserving SUSY model. Our results show that the
top quark FCNC decay can be as large as
$Br(t \rightarrow cg) \sim 10^{-3}$,
$Br(t \rightarrow cZ) \sim 10^{-4}$ and $Br(t \rightarrow c\gamma)
\sim 10^{-5}$, which are observable at the upgraded Tevatron and
the LHC.
\vfill

PACS: 14.65.Ha, 14.80.Ly
\end{titlepage}
\eject
\baselineskip=0.30in

%\begin{center} {\Large 1. Introduction }\end{center}

The unexpected large mass of the top quark suggest that it may be more 
sensitive to new physics than other fermions.
In the standard model (SM) the flavor changing neutral current (FCNC)
decays of the top quark, $t\rightarrow cV$, suppressed by GIM, are found 
to be far below the detectable levels at current and future colliders [1,2].
So, searching for such FCNC top decays serves as a powerful probe to 
effects of new physics, which has attracted attentions of both theorists 
and experimentalists.

 On the experimental side, the CDF [3,4] and D0 [5] collaborations 
have reported interesting bounds on these decays [4].
Undoubtedly more stringent bounds will be obtained in the 
future at the Tevatron upgrade and the LHC. 
On the theoretical side, a systematic study of the experimental
observability for these FCNC top quark decays at the Tevatron and the
LHC has been undertaken [6,7]. The results show that
the sensitivity on detecting $Br(t \rightarrow cV)$ is significant [6,7]:
\begin{eqnarray} \label{level1}
Br(t \rightarrow cZ)&      \simeq & 4\times 10^{-3} (6\times 10^{-4}),\\ 
\label{level2}
Br(t \rightarrow c\gamma) &\simeq & 4\times 10^{-4}(8\times 10^{-5}),\\
\label{level3}
Br(t \rightarrow cg)     & \simeq & 5\times 10^{-3}(1\times 10^{-3}),
\end{eqnarray}
at the upgraded Tevatron with an integrated luminosity of 10 (100) fb$^{-1}$,
and improve several fold,
\begin{eqnarray} \label{level4}
Br(t \rightarrow cZ)     & \simeq & 8\times 10^{-4} (2\times 10^{-4}),\\
\label{level5}
Br(t \rightarrow c\gamma)& \simeq & 2\times 10^{-5}(5\times 10^{-6}).
\end{eqnarray}
at the LHC with similar integrated luminosities.
Despite the interesting experimental possibility indicating in
Eqs.(1)-(5), there is so far lacking the demonstration in the 
minimal supersymmetric model (MSSM), an attractive
candidate for physics beyond the standard model, that such limits can be
 realized. In MSSM with R-parity, the 
predictions for branching ratios of these FCNC top quark 
decays were found to be below the above listed detectable 
levels[8]. In this paper we will show that in the case of R-parity 
violating MSSM [9, 10] with the existing 
bounds on the R-parity violating couplings,  $Br(t \rightarrow cV)$
can reach the detectable level at the upgraded Tevatron and the LHC. 

 While this is an interesting possibility in its
own right, the recent anomalous events at HERA provide an additional
motivation for the study of the R-parity violating SUSY.
The recent HERA data, which showed excess events in deep-inelastic 
positron-proton
scattering at high-$Q^2$ and high $x$ in apparent conflict with the
Standard Model predictions [11], have been interpreted 
as evidence of R-parity breaking SUSY [12]. Hence the examination of
effects of R-parity breaking in other processes are desirable.

 In MSSM the superpotential with
 R-parity violating is given by [10]
\begin{equation}
{\cal W}_{\not \! R}=\lambda_{ijk}L_iL_jE_k^c
+\lambda_{ijk}^{\prime}L_iQ_jD_k^c
              +\lambda_{ijk}^{\prime\prime}U_i^cD_j^cD_k^c+\mu_iL_iH_2,
\end{equation}
where $L_i(Q_i)$ and $E_i(U_i,D_i)$ are the left-handed
lepton (quark) doublet and right-handed lepton (quark) singlet chiral 
superfields.
$i,j,k$ are generation indices and $c$ denotes charge conjugation. 
$H_{1,2}$ are the Higgs-doublets chiral superfields.
The  $\lambda_{ijk}$ and $\lambda^{\prime}_{ijk}$ are lepton-number 
violating couplings, $\lambda^{\prime\prime}_{ijk}$ baryon-number 
violating couplings.
$\lambda_{ijk}$ is antisymmetric in the first two
indices and $\lambda^{\prime\prime}_{ijk}$ is antisymmetric in
the last two indices. 
Phenomenologies of  R-parity violating couplings 
at various colliders have been investigated recently [13]. 
Constraints on these couplings have also been  obtained 
from various other processes including perturbative unitarity [14,15],
$n-\bar n$ oscillation [15,16], 
$\nu_e$-Majorana mass [17], neutrino-less double $\beta$ decay [18], 
charged current universality [19], $e-\mu-\tau$ universality [19],
$\nu_{\mu}-e$ scattering [19], atomic parity violation [19], 
$\nu_{\mu}$ deep-inelastic scattering [19], $K$-decay [20,21],
$\tau$-decay [22], $D$-decay [22], B-decay [23-25] and $Z$-decay at LEP
I [26,27]. 

 Although it is theoretically possible to have B-violating
and L-violating terms in the Lagrangian, the non-observation
of proton decay imposes very stringent conditions on their simultaneous
presence [15]. We, therefore, assume the existence of either  L-violating 
couplings or B-violating couplings, and investigate separately their 
effects in top quark decays.

The FCNC decays  $t\rightarrow cV$ can be
induced by either the $\lambda^{\prime}$ or $\lambda^{\prime\prime}$
couplings at the one loop level.
In terms of the four-component Dirac notation, the Lagrangian of the
L-violating $\lambda^{\prime}$ couplings and B-violating
$\lambda^{\prime\prime}$ couplings are given by
\begin{eqnarray}
{\cal L}_{\lambda^{\prime}}&=&-\lambda^{\prime}_{ijk}
\left [\tilde \nu^i_L\bar d^k_R d^j_L+\tilde d^j_L\bar d^k_R\nu^i_L
       +(\tilde d^k_R)^*(\bar \nu^i_L)^c d^j_L\right.\nonumber\\
& &\hspace{1cm} \left. -\tilde e^i_L\bar d^k_R u^j_L
       -\tilde u^j_L\bar d^k_R e^i_L
       -(\tilde d^k_R)^*(\bar e^i_L)^c u^j_L\right ]+h.c.,\\
{\cal L}_{\lambda^{\prime\prime}}&=&-\lambda^{\prime\prime}_{ijk}
\left [\tilde d^k_R(\bar u^i_L)^c d^j_L+\tilde d^j_R(\bar d^k_L)^c u^i_L
       +\tilde u^i_R(\bar d^j_L)^c d^k_L\right ]+h.c.
\end{eqnarray}
The terms proportional to $\lambda$ are not relevant to our 
discussion and will not be considered here.

{\bf L-violating Couplings}:~~Let us first consider
$t\rightarrow cV$ induced by L-violating couplings.
The relevant Feynman diagrams are shown in Fig.1.
They give rise to effective $tcV$ vertices of the form
\begin{eqnarray} \label{verz}
V^{\mu}(tcZ)&=&ie\left[\gamma^{\mu}P_LA^Z
              +ik_{\nu}\sigma^{\mu\nu}P_R B^Z \right],\\
V^{\mu}(tc\gamma)&=&ie\left[ ik_{\nu}\sigma^{\mu\nu}P_R B^{\gamma}\right ],\\
\label{verg}
V^{\mu}(tcg)&=&ig_sT^a\left[ ik_{\nu}\sigma^{\mu\nu}P_R B^g \right ],
\end{eqnarray}
where  $P_{R,L}=\frac{1}{2}(1\pm \gamma_5)$ and $k$ is the momentum 
of the vector boson.  

The form factors are given by
\begin{eqnarray}
A^Z&=&A^Z_1+A^Z_2,\\
B^V&=&B^V_1+B^V_2,(V=Z,\gamma,g),
\end{eqnarray}
where 
\begin{eqnarray}
A^Z_1&=&\frac{1}{16\pi^2}\lambda^{\prime}_{i2k}\lambda^{\prime}_{i3k}
\left \{ (v_c+a_c)B_1(M_t,M_{e^i},M_{\tilde d^k})
+\frac{1}{2} (v_e+a_e) \right.\nonumber\\
& & - (v_e+a_e)\left [2c_{24}+M_V^2(c_{12}+c_{23})\right]
                  (-p_t,p_c,M_{e^i},M_{\tilde d^k},M_{e^i})\nonumber\\
& &\left. + \xi_V\left[2c_{24}
+M_t^2\left(c_{11}-c_{12}+c_{21}-c_{23}\right)\right]
                  (-p_t,k,M_{e^i},M_{\tilde d^k},M_{\tilde d^k})\right\}\\
B^Z_1&=&\frac{1}{16\pi^2}\lambda^{\prime}_{i2k}\lambda^{\prime}_{i3k}
\left \{ (v_e+a_e)M_t \left [ c_{11}-c_{12}+c_{21}-c_{23}\right ]
                  (-p_t,p_c,M_{e^i},M_{\tilde d^k},M_{e^i}) \right.\nonumber\\
& &\left. + \xi_V M_t \left[c_{11}-c_{12}+c_{21}-c_{23}\right]
                  (-p_t,k,M_{e^i},M_{\tilde d^k},M_{\tilde d^k})\right\}\\
A^Z_2&=&\frac{1}{16\pi^2}\lambda^{\prime}_{i2k}\lambda^{\prime}_{i3k}
\left \{ (v_c+a_c)B_1(M_t,M_{d^k},M_{\tilde e^i})
+\frac{1}{2} (a_d-v_d) \right.\nonumber\\
& & - (a_d-v_d)\left[2c_{24}+M_V^2(c_{12}+c_{23})\right]
                  (-p_t,p_c,M_{d^k},M_{\tilde e^i},M_{d^k})\nonumber\\
& &\left. - \xi'_V \left[ 2c_{24}+M_t^2 (c_{11}-c_{12}+c_{21}-c_{23})\right]
                  (-p_t,k,M_{d^k},M_{\tilde e^i},M_{\tilde e^i})\right\}\\
B^Z_2&=&\frac{1}{16\pi^2}\lambda^{\prime}_{i2k}\lambda^{\prime}_{i3k}
\left\{ (a_d-v_d)M_t\left[c_{11}-c_{12}+c_{21}-c_{23}\right]
                  (-p_t,p_c,M_{d^k},M_{\tilde e^i},M_{d^k})\right.\nonumber\\
& &\left. - \xi'_V M_t\left[c_{11}-c_{12}+c_{21}-c_{23}\right]
                  (-p_t,k,M_{d^k},M_{\tilde e^i},M_{\tilde e^i})\right\},
\end{eqnarray}
where the sum over the family 
indices $i$ $(=1,2,3)$ and $k$ $(=1,2,3)$ is implied.
The momenta $p_t$ and $p_c$ are that of the top and the charm quarks.
The functions $B_1$, and $c_{ij}$
are 2- and 3-point Feynman integrals given in [28].
The functional dependences of the $c_{ij}'s$ are indicated in the bracket
following them. $M_V$ stands for the vector boson mass; $V=Z, \gamma, g$.
The constant $\xi_V(\xi'_V)= -e_d s_W/c_W\left(-(1-2s_W^2)/2 s_W c_W\right ),~
e_d(-1),~1(0)$ for the $Z$ boson, photon and gluon, respectively. 
The vector and axial-vector couplings 
$a_f$ and $v_f$ $(f=c,d,e)$ are given by
\begin{eqnarray}
v_f&=&\frac{I^f_3-2e_fs_W^2}{2s_Wc_W},\\
a_f&=&\frac{I^f_3}{2s_Wc_W},
\end{eqnarray}
where $e_f$ is the electric charge of the fermion $f$ in unit of $e$ 
and $I_3^f=\pm 1/2$ the corresponding third components of the weak isospin. 

The form factors $B^{\gamma}_1$,  $B^{\gamma}_2$,  $B^g_1$ and  $B^g_2$
can be obtained from $B^Z_1$ and $B^Z_2$ by the substitutions
\begin{eqnarray} 
 B^{\gamma}_1&=&B^Z_1\left\vert_{a_e\rightarrow 0, v_e\rightarrow e_e},
                                  \right.\\ 
 B^{\gamma}_2&=&B^Z_2\left\vert_{a_d\rightarrow 0, v_d\rightarrow e_d},
                                 \right.\\ 
 B^g_1&=&B^Z_1\left\vert_{a_e\rightarrow 0, v_e\rightarrow 0},
                                  \right.\\ 
 B^g_2&=&B^Z_2\left\vert_{a_d\rightarrow 0, v_d\rightarrow 1}.
                                 \right.
\end{eqnarray}

Note that the Feynman integrals $B_1$ and $c_{24}$ contain 
ultraviolet divergencies, i.e.,
\begin{eqnarray}
B_1&=&-\frac{1}{2}\Delta+\bar B_1,\\ 
c_{24}&=&\frac{1}{4}\Delta+\bar c_{24},
\end{eqnarray}
where $\bar B_1$ and $\bar c_{24}$  are finite, and
$\Delta\equiv \frac{1}{\epsilon}-\gamma_E+\log 4\pi$, 
with $\gamma_E$ being the Euler constant and $D=4-2\epsilon$ 
the space-time dimension.
We have checked that all the ultraviolet divergencies  
cancelled in the form factors as a result of renormalizability
of the theory.

The partial widths for $t\rightarrow cV$  are given by
\begin{eqnarray}\label{widthz}
\Gamma(t\rightarrow cZ)&=&\frac{\alpha}{8M_t^3}(M_t^2-M_Z^2)^2
\left \{ \left (\frac{M_t^2}{M_Z^2}+2\right ) \left \vert A^Z\right \vert^2
-6M_t{\rm Re}\left [ A^Z(B^Z)^* \right ]\right. \nonumber\\
& & \hfill \left. +(2M_t^2+M_Z^2)\left \vert B^Z\right \vert^2 \right \},~~~\\
\Gamma(t\rightarrow c\gamma)&=&\frac{ \alpha}{4} M_t^3\vert B^{\gamma}\vert^2,
                \\ \label{widthg}
\Gamma(t\rightarrow cg)&=&\frac{ \alpha_s}{3} M_t^3\vert B^g\vert ^2,
\end{eqnarray}
where we neglected the charm quark mass.
The branching ratios $Br(t\rightarrow cV)$ are defined as
\begin{equation}
Br(t\rightarrow cV)=\frac{\Gamma(t\rightarrow cV)}{\Gamma(t\rightarrow W^+b)},
\end{equation}
where 
\begin{equation}
\Gamma(t\rightarrow W^+b)=\frac{G_F}{\sqrt 2} \frac{M_t^3}{8\pi}
\left[1-\frac{M_W^2}{M_t^2} \right ]^2\left[1+2\frac{M_W^2}{M_t^2}\right ].
\end{equation}
\vspace{.5cm}

{\bf B-violating Couplings}:~~This is simpler than the case of the L-violating
coupling with the relevant Feynman diagrams shown in Fig.2.   
The induced effective $tcV$ vertices have forms similar to 
Eqs.(\ref{verz}-\ref{verg})
\begin{eqnarray}
\tilde V^{\mu}(tcZ)&=&ie\left[\gamma^{\mu}P_LF^Z_1
              +ik_{\nu}\sigma^{\mu\nu}P_R F^Z_2 \right],\\
\tilde V^{\mu}(tc\gamma)&=&ie
  \left[ ik_{\nu}\sigma^{\mu\nu}P_R F^{\gamma}_2\right ],\\
\tilde V^{\mu}(tcg)&=&ig_sT^a\left[ ik_{\nu}\sigma^{\mu\nu}P_R F^g_2 \right ],
\end{eqnarray}
where the form factors are given by
\begin{eqnarray}
F^Z_1&=&\frac{1}{16\pi^2}\lambda^{\prime\prime}_{2jk}
\lambda^{\prime\prime}_{3jk}
\left \{ (v_c+a_c)B_1(M_t,M_{d^j},M_{\tilde d^k})\right.\nonumber\\
& & + (v_d+a_d)\left[\frac{1}{2}-2c_{24}-M_V^2(c_{12}+c_{23})\right]
                  (-p_t,p_c,M_{d^j},M_{\tilde d^k},M_{d^j})\nonumber\\
& &\left. - \xi_V\left[2c_{24}+M_t^2(c_{11}-c_{12}+c_{21}-c_{23})\right]
                  (-p_t,k,M_{d^j},M_{\tilde d^k},M_{\tilde d^k})\right\}\\
F^Z_2&=&\frac{1}{16\pi^2}\lambda^{\prime\prime}_{2jk}
\lambda^{\prime\prime}_{3jk}
\left \{ (v_d+a_d)M_t \left [c_{11}-c_{12}+c_{21}-c_{23}\right ]
                  (-p_t,p_c,M_{d^j},M_{\tilde d^k},M_{d^j})\right.\nonumber\\
 & &\left. - \xi_VM_t \left[c_{11}-c_{12}+c_{21}-c_{23}\right]
                  (-p_t,k,M_{d^j},M_{\tilde d^k},M_{\tilde d^k})\right\}\\
F^{\gamma}_2&=&F^Z_2\left\vert_{a_d\rightarrow 0, v_d\rightarrow e_d},
                                  \right.\\ 
F^g_2&=&F^Z_2\left\vert_{a_d\rightarrow 0, v_d\rightarrow -1,
                                 \xi_V\rightarrow -\xi_V},
                                  \right. 
\end{eqnarray}
where sum over the family indices $j$ $(=1,2,3)$ and $k$ $(=1,2,3)$ 
is implied. Again we have verified that all the ultraviolet divergencies  
cancelled in the form factors.

The partial decay widths can be obtained from Eqs.(\ref{widthz}-\ref{widthg})
by the substitution $A^V\rightarrow F^V_1$ and $B^V\rightarrow F^V_2$
$(V=Z,\gamma,g)$.

{\bf Numerical Results}:~~ With the analytical formula for 
$Br(t \rightarrow c V)$, we present the numerical results below.
We take
$M_t=175$ GeV, $m_Z=91.187$ GeV, $m_W=80.3$ GeV, 
$G_F=1.16639\times 10^{-5}$(GeV)$^{-2}$, $\alpha=1/128$,
$\alpha_s=0.108$, and neglect the masses of charged lepton $e^i$ $(i=1,2,3)$,
down-type quarks $d^k$ $(k=1,2,3)$, and the charm quark. 

The decay rates increase with 
the relevant $\lambda^{\prime}$ or $\lambda^{\prime\prime}$  couplings 
and decrease with the sparticle masses.
Since there are no cancellation effects between the diagrams
of different generation sparticles, we assume for simplicity   
that the masses of all charged sleptons
are degenerate, so are the masses of all down-type squarks.

To calculate the maximum value of the $Br(t\rightarrow c V)$ in the presence of
the L-violating terms, we use the following limits 
 on the L-violating 
couplings (obtained for the squark mass of 100 GeV), 
\begin{eqnarray}\label{eq34}
\vert \lambda^{\prime}_{kij}\vert&<&0.012,~(k,j=1,2,3; i=2),\\
\label{eq40}
\vert \lambda^{\prime}_{13j}\vert&<&0.16,~(j=1,2),\\
\label{eq41}
\vert \lambda^{\prime}_{133}\vert&<&0.001,\\
\label{eq42}
\vert \lambda^{\prime}_{23j}\vert&<&0.16,~(j=1,2,3),\\
\label{eq43}
\vert \lambda^{\prime}_{33j}\vert&<&0.26,~(j=1,2,3).
\end{eqnarray}
The first set of constraints Eq.(\ref{eq34}) come from the decay 
$K\rightarrow \pi \nu\nu$ with FCNC processes in the down quark sector [20];
the second and fourth set, Eqs.(\ref{eq40}) and (\ref{eq42}), 
from the semileptonic decays of the B-meson [25]; the third constraint,
Eq.(\ref{eq41}), from the Majorana mass of the electron type neutrino [17];
and the last set, Eq.(\ref{eq43}), from the leptonic decay modes of 
the $Z$ [26].

 There are also the following constraints on the products
of the $\lambda^{\prime}$ couplings
\begin{eqnarray}\label{eq39}
&\lambda^{\prime}_{13i}\lambda^{\prime}_{12i},&
\lambda^{\prime}_{23j}\lambda^{\prime}_{22j}<1.1\times 10^{-3}, 
~(i=1,2; j=1,2,3),\\
& \lambda^{\prime}_{in2}\lambda^{\prime}_{jn1}& <10^{-5},~(i,j,n=1,2,3),\\
& \lambda^{\prime}_{121}\lambda^{\prime}_{222},&~
\lambda^{\prime}_{122}\lambda^{\prime}_{221},~
\lambda^{\prime}_{131}\lambda^{\prime}_{232},
~\lambda^{\prime}_{132}\lambda^{\prime}_{231}<10^{-7},
\end{eqnarray}
where the first set of constraints Eq.(\ref{eq39}) are derived in Ref. [25] 
and the others derived in Ref. [21].

Using the upper limits of the relevant L-violating couplings, 
we find the maximum values of the branching fractions to be
\begin{eqnarray} \label{tczL}
Br(t \rightarrow cZ)     & \leq & 10^{-9}, \\ 
Br(t \rightarrow c\gamma)& \leq & 10^{-10}, \\ \label{tcgL}
Br(t \rightarrow cg)     & \leq & 10^{-8}.
\end{eqnarray}
Thus the contributions of the L-violating couplings to 
$t \rightarrow cV$ are too small to be detected at both the Tevatron
and LHC. 

With the B-violating couplings, $\lambda^{\prime\prime}$,
the top rare decay rates can be drastically enhanced.
Regarding the bounds of the relevant 
B-violating couplings the perturbative unitarity gives[14,15]
\begin{equation}\label{e20}
\lambda^{\prime\prime}_{3jk}<1.25,~~ \lambda^{\prime\prime}_{2jk}<1.25.
\end{equation}
The upper bounds for $\lambda^{\prime\prime}_{3jk}$ have also been
derived from $R_l\equiv \Gamma_h/\Gamma_l$ [27]. For squark mass of 100 GeV, 
$R_l$ yields an upper bound of 1.46 and 1.83 at $2\sigma$ 
and $3\sigma$, respectively. 

There is an additional constraint on $\lambda_{3jk}^{\prime\prime}$ from the
exotic top quark decay
$t\rightarrow \bar d_L^j+\bar {\tilde d_R^k}$ 
for squark mass lighter than that of the top quark.
For the top mass of 175 GeV, we obtain
\begin{equation}
R_t\equiv\frac{\Gamma(t\rightarrow \bar d_L^j+\bar {\tilde d_R^k})}
{\Gamma(t\rightarrow W+b)}=1.12\left (\lambda''_{3jk}\right )^2\left [ 1-
 \left(\frac{M_{\tilde d^k_R}}{175 {\rm GeV}}\right )^2\right ]^2.
\end{equation}
The ${\tilde d_R^k}$ will decay into $d_R$ and the lightest neutralino
( and gluino if kinematically allowed), 
as well as quark pairs induced by the B-violating terms.
 These decay modes, which can enhance the total fraction of
hadronic decays of the top quark, alter the ratio of $t \bar t$ 
events expected in dilepton channel.
The number of dilepton events expected in the presence of the decay
$t\rightarrow \bar d_L^j+\bar {\tilde d_R^k}$ and that
in the SM is given by $R(f)\equiv (1-f)^2$, where 
f = $Br (t\rightarrow \bar d_L^j+\bar {\tilde d_R^k} ) $.
The production cross section measured by CDF with 110 pb$^{-1}$ is
$\sigma[t\bar t]_{\rm exp}=8.3^{+4.3}_{-3.3}$ pb in the dilepton channel[29],
while the SM expectation for top mass of 175 GeV is
$\sigma[t\bar t]_{\rm QCD}=5.5^{+0.1}_{-0.4}$ pb [30]. By requiring 
$R(f)$ to lie within the measured range of $\sigma[t\bar t]_{\rm exp}/ 
\sigma[t\bar t]_{\rm QCD}$, we can get the bounds on the relevant 
$\lambda''$ couplings. The $2\sigma$ bound from dilepton channel is 
found to be 
\begin{equation}
 \left (\lambda''_{3jk}\right )^2\left [ 1-
 \left(\frac{M_{\tilde d^k_R}}{175 {\rm GeV}}\right )^2\right ]^2<0.71.
\end{equation}
For $M_{\tilde d^k_R}= 100$ GeV, we have $\lambda_{3jk}^{\prime\prime}<1.25$,
which is the same as the unitarity bound. Constraints on
 $\lambda_{3jk}^{\prime\prime}$ from the experimental data of $t \bar t$
in other channels are weaker.


From Eq.(\ref{e20}) for the relevant couplings, 
we obtain the branching ratios of the FCNC decay of the top quark.
The results are plotted in Fig.3 as a function of squark mass.
From Fig.3 we have  
\begin{eqnarray} \label{tczB}
Br(t \rightarrow cZ)     & \approx & 8.4\times 10^{-4}, \\ 
Br(t \rightarrow c\gamma)& \approx & 2.0\times 10^{-5}, \\ \label{tcgB}
Br(t \rightarrow cg)     & \approx & 3.3\times 10^{-3}.
\end{eqnarray}
for squark mass no greater than 170 GeV.
We conclude from eqs.(\ref{level1})-(\ref{level5}) that the 
contribution of B-violating couplings to the decay $t \rightarrow cV$ 
can be observable at the upgraded Tevatron and LHC with an integrated 
luminosity of 100 fb$^{-1}$ for not too heavy squarks.

The top quark rare decays, if not observed at the upgraded Tevatron and LHC, 
can be used to put strong constraints on the 
B-violating couplings. For instance, 
if the decay  $t \rightarrow cg$ is not observed 
at the upgraded Tevatron, $t \rightarrow cZ$ and 
$t \rightarrow c\gamma$  not observed at the LHC, both with 
an integrated luminosity of 100 fb$^{-1}$, 
a further constraint on the product of the
B-violating couplings, i.e.,
\begin{equation}
\Lambda''\equiv \lambda^{\prime\prime}_{212} \lambda^{\prime\prime}_{312}
                +\lambda^{\prime\prime}_{213} \lambda^{\prime\prime}_{313}
                +\lambda^{\prime\prime}_{223} \lambda^{\prime\prime}_{323},
\end{equation}
can be obtained.
In Fig.4 we plot $\Lambda''$ versus the degenerate squark mass.
The solid, dashed and dotted lines correspond to 
$Br(t \rightarrow cg)=1\times 10^{-3}$,
$Br(t \rightarrow cZ)=2\times 10^{-4}$ and 
$Br(t \rightarrow c\gamma)=5\times 10^{-6}$, respectively.
The region above the solid line corresponding to 
$Br(t \rightarrow cg)>1\times 10^{-3}$ will be excluded
if the decay  $t \rightarrow cg$ is not observed 
at the upgraded Tevatron.
The region above the dashed (dotted) line corresponds to 
$Br(t \rightarrow cZ)>2\times 10^{-4}
\left (Br(t \rightarrow c\gamma)>5\times 10^{-6}\right )$ and 
will be excluded if the decay  $t \rightarrow cZ(t \rightarrow c\gamma)$ 
is not observed  at the LHC.  The corresponding value of $\Lambda''$
can be read off from the figure. For example, for squark mass of 150 GeV, 
the upgraded Tevatron will probe the $\Lambda''$ down to 2.3, while  
the present upper limit from
requirement of perturbative unitarity
is $\Lambda''=4.7$. 
\vspace{1cm}

\begin{center}{\Large  Acknowledgement}\end{center}
We would like to thank  A. Datta, T. Han and C. Kao
for discussions.
This work was supported in part by the U.S. Department of Energy, Division
of High Energy Physics, under Grant No. DE-FG02-94ER40817 and DE-FG02-92ER40730.
XZ was also supported in part by National Natural Science Foundation of China
and JMY acknowledge the partial support provided by the Henan Distinguished
Young Scholars Fund.
\vspace{1cm}


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\begin{center}Figure Captions \end{center}

Fig. 1 Feynman diagrams for $t\rightarrow cV $ ($V=Z,\gamma, g$
for quarks and squarks; $V=Z,\gamma$ for leptons and sleptons) 
induced by L-violating couplings. The blobs denote L-violating vertex.

Fig. 2 Feynman diagrams for $t\rightarrow cV $ ($V=Z,\gamma, g$
for quarks and squarks; $V=Z,\gamma$ for leptons and sleptons)
induced by B-violating couplings.
The blobs denote B-violating vertex.

Fig. 3 The contribution of B-violating couplings to branching ratios 
as a function of squark mass, assuming the relevant couplings to take
their upper bounds derived from perturbative unitarity.

Fig. 4 The product of the B-violating couplings
$\Lambda''\equiv \lambda^{\prime\prime}_{212} \lambda^{\prime\prime}_{312}
                +\lambda^{\prime\prime}_{213} \lambda^{\prime\prime}_{313}
                +\lambda^{\prime\prime}_{223} \lambda^{\prime\prime}_{323}$
versus squark mass for given values of branching ratios.
The solid, dashed and dotted lines correspond to 
$Br(t \rightarrow cg)=1\times 10^{-3}$,
$Br(t \rightarrow cZ)=2\times 10^{-4}$ and 
$Br(t \rightarrow c\gamma)=5\times 10^{-6}$, respectively.

\end{document}

