\section{Introduction}

In spite of being an attractive material with excellent electronic properties \cite{ahcn09},  practical applications of graphene as in conventional semiconductor devices are still questionable due to its gapless nature. In particular, the ON/OFF current ratio is low while the saturation of current is poor in pristine graphene transistors \cite{schw10}. Many efforts of bandgap engineering in graphene \cite{yhan07,khar11,lher13,jbai10,zhan09} have been made to solve these issues. The pioneer technique proposed \cite{yhan07} is to cut 2D graphene sheets into 1D narrow nanoribons. In 2D graphene sheets, some options as Bernal-stacking of graphene on hexagonal boron nitride substrate \cite{khar11}, nitrogen-doped graphene \cite{lher13}, graphene nanomesh lattice \cite{jbai10,berr13} and Bernal-stacking bilayer graphene \cite{zhan09} have been explored. However, the possibility to open a sizable bandgap in graphene as large as those of standard semiconductors is still very unlikely. In particular, it requires a very good control of lattice geometry and edge disorder in narrow graphene nanoribbons (GNRs) \cite{quer08} and in graphene nanomesh lattices \cite{hung13}, while the bandgap opening in bilayer graphene by a perpendicular electric field may not be large enough for realistic applications \cite{fior09}. Other methods should be further verified by experiments.

\begin{figure}[!t]
\centering
\includegraphics[width=2.8in]{Fig01.pdf}
\caption{Schematic of unstrained/strained graphene junctions investigated in this work.}
\label{fig_sim1}
\end{figure}

On the other hand, graphene was experimentally demonstrated to be able to sustain a much larger strain than conventional semiconductors, making it a promising candidate for flexible electronics (see in a recent review \cite{shar13}). Indeed, strain engineering has been suggested to be an alternative approach to modulating efficiently the electronic properties of graphene nanomaterials. In particular, the bandgap has periodic oscillations in the armchair GNRs \cite{ylu210} while the spin polarization at the ribbon edges (and also the bandgap) can be modulated by the strain in the zigzag cases. In 2D graphene sheets, a finite gap can open under large strains, otherwise, it may remain close to zero but the Dirac points are displaced \cite{cocc10,per209,pere09,huan10}. Many interesting electrical, optical, and magnetic properties induced by strain in graphene have been also explored, e.g. see in \cite{bunc07,pere09,kuma12,per010,pell10,guin10,tlow10,zhai11}.

Besides, local strain is a good option to improve the electrical performance of graphene devices \cite{pere09,ylu010,fuji10,juan11,baha13}. For instance, it has been shown to enhance the ON current in a GNR tunneling FET \cite{ylu010} and to fortify the transport gap in GNR strained junctions \cite{baha13}. In a recent work \cite{hung14}, we have investigated the effects of uniaxial strain on the transport in 2D unstrained/strained graphene junctions and found that due to the strain-induced shift of Dirac points, a significant conduction gap of a few hundreds meV can open with a small strain of a few percent. This type of strained junction was then demonstrated to be an excellent candidate to improve the electronic operation of graphene transistors. It hence motivates us to further investigate the properties of this conduction gap so as to optimize the performance of graphene devices. On the one hand, the effects of strain should be, in principle, dependent on its applied direction. On the other hand, because the appearance of conduction gap is a consequence of the shift of Dirac points along the $k_y$-axis, it is predicted that this gap should also depend on the transport direction. Note that here, Oy (Ox) - axis is assumed to be perpendicular (parallel) to the transport direction. The effects of both strain and transport directions will be clarified systematically in the current work.

\section{Model and calculations}

In this work, the $\pi$-orbital tight binding model constructed in \cite{per209} is used to investigate the electronic transport through the graphene strained junctions schematized in Fig. 1. The Hamiltonian is ${H_{tb}} = \sum\nolimits_{nm} {{t_{nm}}c_n^\dag {c_m}}$ where $t_{nm}$ is the hopping energy between nearest neighbor \emph{n}th and \emph{m}th atoms. The application of a uniaxial strain of angle $\theta$ causes the following changes in the $C-C$ bond vectors:
\begin{eqnarray}
 {{\vec r}_{nm}}\left( \sigma \right) &=& \left\{ {1 + {M_s}\left( \sigma, \theta \right)} \right\}{{\vec r}_{nm}}\left( 0 \right) \\
 {M_s}\left( \sigma, \theta \right) &=& \sigma \left[ {\begin{array}{*{20}{c}}
 {{{\cos }^2}\theta  - \gamma {{\sin }^2}\theta }&{\left( {1 + \gamma } \right)\sin \theta \cos \theta }\\
 {\left( {1 + \gamma } \right)\sin \theta \cos \theta }&{{{\sin }^2}\theta  - \gamma {{\cos }^2}\theta }
 \end{array}} \right] \nonumber
\end{eqnarray}
where $\sigma$ represents the strain and $\gamma \simeq 0.165$ is the Poisson ratio \cite{blak70}. The hopping parameters are defined as $t_{nm} \left( \sigma \right) = t_0 \exp\left[-3.37\left(r_{nm} \left( \sigma \right) /r_0 - 1\right)\right]$, where the hopping energy $t_0 = -2.7$ $eV$ and the bond length $r_{nm} \left( 0 \right) \equiv r_0 = 0.142$ $nm$ in the unstrained case. Therefore, there are three different hoping parameters $t_{1,2,3}$ corresponding to three bond vectors ${\vec r}_{1,2,3}$, respectively, in the strained graphene part of the structure (see Fig. 1). Here, we assume a 1D profile of applied strain, i.e., the strain tensor is a function of position along the transport direction Ox while it is constant along the Oy-axis. The transport direction, $\phi$, and strain direction, $\theta$, are determined as schematized in Fig. 1. Based on this tight binding model, two methods described below can be used to investigate the conduction gap of the considered strained junctions.

\textbf{Green's function calculations.} First, we split the graphene sheet into the smallest possible unit cells periodically repeated along the Ox/Oy directions with the indices $p/q$, respectively (similarly, see the details in \cite{hung12}). The tight-binding Hamiltonian can therefore be expressed in the following form:
\begin{eqnarray}
{H_{tb}} = \sum\limits_{p,q} {\left( {{H_{p,q}} + \sum\limits_{{p_1},{q_1}} {{H_{p,q \to p_1,q_1}}} } \right)}
\end{eqnarray}
where $H_{p,q}$ is the Hamiltonian of cell $\{p,q\}$, and $H_{p,q \to p_1,q_1}$ denotes the coupling of cell $\{p,q\}$ to its nearest neighbor cell $\{p_1,q_1\}$. We then Fourier transform the operators in Eq. (2) as follows:
\begin{eqnarray}
 {c_{p,q}} = \frac{1}{{\sqrt {{M_{cell}}} }}\sum\limits_{{\kappa_y}} {{e^{i{q\kappa_y}}}} {{\hat c}_{p,{\kappa_y}}},
\end{eqnarray}
where $M_{cell}$ is the number of unit cells and $\kappa_y \equiv k_y L_y$ with the size $L_y$ of unit cells along the Oy direction. The Hamiltonian (2) is finally rewritten as a sum of $\kappa_y$-dependent 1D-components:
\begin{eqnarray}
{H_{tb}} &=& \sum\limits_{{\kappa_y}} {\hat H\left( {{\kappa_y}} \right)} \\
\hat H\left( {{\kappa_y}} \right) &=& \sum\limits_p {{{\hat H}_{p \to p - 1}}\left( {{\kappa_y}} \right) + {{\hat H}_p}\left( {{\kappa_y}} \right) + {{\hat H}_{p \to p + 1}}}\left( {{\kappa_y}} \right) \nonumber
\end{eqnarray}
With this Hamiltonian form, the Green's function formalism can be easily applied to compute transport quantities in the graphene strained junction with different transport directions. In particular, the conductance at zero temperature is determined as:
\begin{eqnarray}
 \mathcal{G} \left( \epsilon \right) = \frac{{e^2 W}}{{\pi h L_y}}\int\limits_{BZ} {d{\kappa_y} \mathcal{T}\left( {\epsilon, {\kappa_y}} \right)}
\end{eqnarray}
where $\mathcal{T}\left( {\epsilon,{\kappa_y}} \right)$ is the transmission probability computed from the Green's functions. The integration over $\kappa_y$ is performed in the whole first Brillouin zone. As in ref. \cite{hung13}, the gap of conductance (conduction gap) is then measured from the obtained data of conductance.

\textbf{Bandstructure analyses.} To determine the conduction gap of strained junctions, we find that another simple way based on the analysis of graphene bandstructures could be efficiently used. It is described as follows. Since the conductance is computed from Eq. (5), the appearance of conduction gap is essentially governed by the gaps of transmission probability, which is determined from the energy gaps in the unstrained and strained graphene sections. These energy gaps can be defined directly from the graphene bandstructures. Therefore, our calculation has two steps, similar to that in \cite{hung14}. From the graphene bandstructures obtained using the tight-binding Hamiltonian above, we first look for the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ for a given $\kappa_y$ of two graphene sections. The maximum of these energy gaps determines the gap $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ of transmission probability through the junction. Finally, the conduction gap $E_{cond.gap}$ is obtained by looking for the minimum value of $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ when varying $\kappa_y$ in the whole Brillouin zone.

In particular, the energy bands of strained graphene are given by
\begin{eqnarray}
 E\left( {\vec k} \right) =  \pm \left| {{t_1}{e^{i\vec k{{\vec a}_1}}} + {t_2}{e^{i\vec k{{\vec a}_2}}} + {t_3}} \right|
\end{eqnarray}
where the plus/minus sign corresponds to the conduction/valence bands, respectively. For a given direction $\phi$ of transport, in principle, the vectors $\vec L_{x,y}$ defining the sizes of unit cell along the Ox and Oy directions, respectively, can be always expressed as ${\vec L_x} = {n_1}{\vec a_1} + {n_2}{\vec a_2}$ and ${\vec L_y} = {m_1}{\vec a_1} + {m_2}{\vec a_2}$ with $\cos \phi = \frac{{{{\vec L}_x}\vec L_x^0}}{{{L_x}L_x^0}}$ and $\sin \phi = \frac{{{{\vec L}_x}\vec L_y^0}}{{{L_x}L_y^0}}$ while $\vec L_{x,y}^0 = {\vec a_1} \pm {\vec a_2}$. Note that $n_{1,2}$ and $m_{1,2}$ are integers while $\frac{{{m_1}}}{{{m_2}}} =  - \frac{{{n_1} + 2{n_2}}}{{{n_2} + 2{n_1}}}$, i.e., ${\vec L_{x}} {\vec L_{y}} = 0$. In other words, we have the following expressions
\begin{eqnarray}
{{{\vec a}_1} = \frac{{ - {m_2}{{\vec L}_x} + {n_2}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}};\,\,\,{{\vec a}_2} = \frac{{{m_1}{{\vec L}_x} - {n_1}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}}}
\end{eqnarray}
On this basis, the energy bands can be rewritten in terms of $\kappa_{x, y} = \vec k \vec L_{x,y} \left( { \equiv {k_{x,y}}{L_{x,y}}} \right)$ by substituting Eqs. (7) into Eq. (6). This new form of energy bands is finally used to compute the conduction gap of strained junctions.

As a simple example, in the case of $\phi = 0$ (armchair direction), we calculate the conduction gap as follows. First, Eq. (6) is rewritten in the form
\begin{eqnarray}
 E_{\phi = 0}\left( {\vec \kappa} \right) =  \pm \left| {{t_1}{e^{i\kappa_y/2}} + {t_2}{e^{ - i\kappa_y/2}} + {t_3}{e^{ - i\kappa_x/2}}} \right|
\end{eqnarray}
with the vectors $\vec L_{x,y} \equiv \vec L_{x,y}^0$. Using this new form, the energy gap of strained graphene for a given $\kappa_y$ is determined as
\begin{equation}
{E_{strain}^{gap}}\left( {{\kappa_y}} \right) = 2 \left| {\sqrt {{{\left( {{t_1} - {t_2}} \right)}^2} + 4{t_1}{t_2}{{\cos }^2}\frac{{{\kappa_y}}}{2}}  + {t_3}} \right|
\end{equation}
while ${E_{unstrain}^{gap}}\left( {{\kappa_y}} \right)$ is given by the same formula with $t_1$ = $t_2$ = $t_3$ $\equiv$ $t_0$. The gap of transmission probability through the junction is then determined as ${E_{junc}^{gap}}\left( {{\kappa_y}} \right) = \max \left[ {E_{unstrain}^{gap}\left( {{\kappa_y}} \right),E_{strain}^{gap}\left( {{\kappa_y}} \right)} \right]$ and, finally, the conduction gap is given by ${E_{cond.gap}} = \min \left[ {E_{junc}^{gap}\left( {{\kappa_y}} \right)} \right]$ for $\kappa_y$ in the whole Brillouin zone.

We would like to notice that the Green's function calculations and the banstructure analyses give the same results of conduction gap in the junctions where the transition region between unstrained and strained graphene sections is long enough, i.e., larger than about 5 to 6 nm. In the case of short length, as discussed in \cite{baha13,hung14}, this transition zone can have significant effects on the transmission between propagating states beyond the energy gaps and hence can slightly enlarge the gap of conductance, compared to the results obtained from the bandstructure calculations.

\section{Results and discussion}

\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{Fig02.pdf}
\caption{Dependence of graphene bandgap (in the unit of eV) on the applied strain and its direction: tensile (a) and compressive (b). The radius from the central point indicates the strain strength ranging from 0 (center) to 30 $\%$ (edge of maps) while the graphene lattice is superimposed to show visibly the strain direction. The orange circle corresponds to the strains of $\sigma = 23 \%$.}
\label{fig_sim2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3.4in]{Fig03.pdf}
\caption{Conductance ($G_0 = e^2W/hL_y$) as a function of energy in graphene strained junctions for $\sigma = 4 \%$ with different strain directions. The transport along the armchair direction ($\phi = 0$) is considered. The data obtained in a uniformly strained graphene is displayed for the comparison.}
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5.8in]{Fig04.pdf}
\caption{Local density of states (left panels) and corresponding transmission coefficient (right panels) for three different wave-vectors $k_y$ obtained in an unstrained/strained graphene junction of $\sigma = 4 \%$, and $\theta \equiv \phi = 0$. On the top is a schematic of graphene bandedges illustrating the strain-induced shift of Dirac points along the $k_y$-direction.}
\label{fig_sim4}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=5.6in]{Fig05.pdf}
\caption{Maps of conduction gap in unstrained/strained graphene junctions: tensile (a,c) and compressive cases (b,d). The transport is along the armchair $\phi = 0$ (a,b) and zigzag $\phi = 30^\circ$ directions (c,d). The strain strength ranges from 0 (center) to 6 $\%$ (edge of maps) in all cases.}
\label{fig_sim4}
\end{figure*}
First, we re-examine the formation of the bandgap of graphene under a uniaxial strain. From Eq. (9), it is shown that a strain-induced finite-bandgap appears only if ${E_{strain}^{gap}}\left( {{\kappa_y}} \right) > 0$ for all $k_y$ in the first Brillouin zone, i.e., ${k _y} \in \left[ { - \frac{\pi}{L_y}, \frac{\pi}{L_y}} \right]$, otherwise, the bandgap remains zero. Hence, the condition for the bandgap to be finite is either
\begin{equation*}
 \left| {{t_1} - {t_2}} \right| > \left| {{t_3}} \right|\,\,\,\,\,{\rm{OR}}\,\,\,\,\,\left| {{t_3}} \right| > \left| {{t_1} + {t_2}} \right|
\end{equation*}
and the corresponding values of bandgap are
\begin{equation*}
 {E_{gap}} = 2\left( {\left| {{t_1} - {t_2}} \right| - \left| {{t_3}} \right|} \right)\,\,\,\,\,{\rm{OR}}\,\,\,\,\,2\left( {\left| {{t_3}} \right| - \left| {{t_1} + {t_2}} \right|} \right)
\end{equation*}
This result was actually reported in \cite{per209,hase06}. We remind as displayed in Fig. 2(a) that a finite bandgap opens only for strain larger than $\sim 23 \%$ and the zigzag (not armchair) is the preferred direction for bandgap opening under a tensile strain \cite{per209}. We extend our investigation to the case of compressive strain and find  (see in Fig. 2(b)) that (i) the same gap threshold of $\sigma \simeq 23 \%$ is observed but (ii) the preferred direction to open the gap under a compressive strain is the armchair, not the zigzag as the case of tensile strain. This implies that the properties of graphene bandstructure at low energy should be qualitatively the same when applying strains of $\left\{ {\sigma ,\theta } \right\}$ and of $\left\{ {-\sigma ,\theta + 90^\circ} \right\}$. This feature can be understood by considering, for example, strains of $\left\{ {\sigma , \theta = 0} \right\}$ and of $\left\{ {-\sigma , \theta = 90^\circ} \right\}$. Indeed, these strains result in the same qualitative changes on the bond-lengths, i.e., an increased bond-length $r_3$ and reduced bond-lengths $r_{1,2}$. However, for the same strain strength, because of the exponential dependence of hoping energies on the bond-lengths, the compressive strain generally induces a larger bandgap than the tensile one, as can be seen when comparing the data displayed in Figs. 2(a) and 2(b). To conclude, we would like to emphasize that a large strain is necessary to open a bandgap in graphene. This could be an issue for practical applications, compared to the use of graphene strained junctions explored in \cite{hung14}.

We now go to explore the properties of conduction gap in the graphene strained junctions. In Fig. 3, we display the conductance as a function of energy computed from Eq. (5) using the Green's function technique. As discussed above, a small strain of a few percent (e.g., 4 $\%$ here) can not change the gapless character of graphene, i.e., there is no gap of conductance in the case of uniformly strained graphene. However, similar to that reported in \cite{hung14}, a significant conduction-gap of a few hundreds meV can open in the unstrained/strained graphene junctions. The appearance of this conduction gap, as mentioned previously, is due to the strain-induced shift of Dirac points and is explained as follows. Actually, the strain causes the lattice deformation and can result in the deformation of graphene bandstructure. Therefore, the bandedges as a function of wave-vector $k_y$ in unstrained and strained graphene can be illustrated schematically as in the top panel of Fig. 4. As one can see, the shift of Dirac points leads to the situation where there is no value of $\kappa_y$, for which the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are simultaneously equal to zero. This means that the transmission probability always shows a finite gap for any $\kappa_y$. For instance, the energy gap is zero (or small) in the unstrained (resp. strained) graphene section but finite in the strained (resp. unstrained) one in the vicinity of Dirac point $k_y = K_{unstrain}$ (resp. $K_{strain}$). Accordingly, as illustrated in the pictures of LDOS in the left panels of Fig. 4 and confirmed in the corresponding transmissions in the right panels, clear gaps of transmission are still obtained. Far from these values of $k_y$, $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are both finite (e.g., see the LDOS plotted for $k_y = K_{gap}$) and hence a finite gap of transmission also occurs. On this basis, a finite gap of conductance is achieved. More important, Fig. 3 shows that besides the strength of strain, the strain effect is also strongly dependent on the applied direction. For instance, the conduction gap takes the values of $\sim$ 295, 172 and 323 meV for $\theta = 0$, $30^\circ$ and $90^\circ$, respectively.

Below, we will discuss the properties of the conduction gap with respect to the strain, its applied direction, and the direction of transport. Note that due to the lattice symmetry, the transport directions $\phi$ and $\phi + 60^\circ$ are equivalent while the applied strain of angle $\theta$ is identical to that of $\theta + 180^\circ$. Hence, the data obtained for $\phi$ ranging from $-30^\circ$ to $30^\circ$ and $\theta  \in \left[ {0^\circ ,180^\circ } \right]$ covers the properties of conduction gap in all possible cases.

In Fig. 5, we present the maps of conduction gap with respect to the strain and its applied direction in two particular cases: the transport is either along the armchair ($\phi = 0$) or the zigzag ($\phi = 30^\circ$) directions. Both tensile and compressive strains are considered. Let us first discuss the results obtained in the armchair case. Figs. 5(a,b) show that (i) a large conduction gap up to about 500 meV can open with a strain of 6 $\%$ and (ii) again the conduction gap is strongly $\theta$-dependent, in particular, its peaks occur at $\theta = 0$ or $90^\circ$ while the gap is zero at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain. In principle, the conduction gap is larger if the shift of Dirac points in the $\kappa_y$-axis is larger, as discussed above about Figs. 3-4. We notice that the strain-induced shifts can be different for the six Dirac points of graphene \cite{kitt12} and the gap is zero when there is any Dirac point observed at the same $\kappa_y$ in the two graphene sections. From Eq. (9), we find that the Dirac points are determined by the following equations:
\begin{eqnarray*}
  {\cos}\frac{\kappa_y}{2} &=& \pm \frac{1}{2}\sqrt{\frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{{t_1}{t_2}}}}, \\
  \cos \frac{{\kappa_x}}{2} &=& \frac{{{t_1} + {t_2}}}{{\left| {{t_3}} \right|}}\cos \frac{{\kappa_y}}{2},\,\,\,\sin \frac{{\kappa_x}}{2} = \frac{{{t_2} - {t_1}}}{{\left| {{t_3}} \right|}}\sin \frac{{\kappa_y}}{2},
\end{eqnarray*}
which simplify into ${\cos}\frac{\kappa_y}{2} = \pm \frac{1}{2}$ and, respectively, $\cos \left( {\frac{{{\kappa _x}}}{2}} \right) = \mp 1$ in the unstrained case. Hence, the zero conduction gap is obtained if
\begin{equation*}
  \frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{4{t_1}{t_2}}} = \frac{1}{4}
\end{equation*}
Additionally, it is observed that the effects of a strain $\{\sigma,\theta\}$ are qualitatively similar to those of a strain $\{-\sigma,\theta+90^\circ\}$, i.e., the peaks and zero values of conduction gap are obtained at the same $\theta$ in these two situations. To understand this, we analyze the strain matrix $M_s \left(\sigma,\theta\right)$ and find that in the case of small strains studied here, there is an approximate relationship between the bond lengths under these two strains, given by \[{r \left( \sigma, \theta \right)} - {r \left( -\sigma, \theta + 90^\circ\right)} \simeq \sigma \left( {1 - \gamma } \right) r_0,\] which is $\theta$-independent for all \emph{C-C} bond vectors. It implies that there is a fixed ratio between the hopping energies $t_i \left( \sigma, \theta \right)$ and $t_i \left( -\sigma, \theta + 90^\circ\right)$ and hence there is the similar shift of Dirac points in these two cases.
\begin{figure}[!t]
\centering
\includegraphics[width=3.4in]{Fig06.pdf}
\caption{Map showing the dependence of conduction gap on the directions ($\theta,\phi$) for $\sigma = 4 \%$. The top is a diagram illustrating the rotation of Dirac points in the \emph{k}-space with the change in the transport direction $\phi$.}
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5.5in]{Fig07.pdf}
\caption{Maps of conduction gap obtained in tensile/compressive strained junctions. The transport along the armchair/zigzag directions is considered in (a,b)/(c,d), respectively. The strains $\sigma_c = -2 \%$ and $\sigma_t = 2 \%$ are applied in (a,c) while $\sigma_c = -1 \%$ and $\sigma_t = 3 \%$ in (b,d).}
\label{fig_sim4}
\end{figure*}

We now go to analyze the properties of conduction gap shown in Figs. 5(c,d) where the transport is along the zigzag direction $\phi = 30^\circ$. In fact, the conduction gap in this case can reach a value as high as that of the case of $\phi = 0$ but has different $\theta$-dependence. In particular, the conduction gap has peaks at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain, where it is zero in the case of $\phi = 0$. It is also equal to zero at $\theta = 0$ and $\theta = 90^\circ$ where the peaks of conduction gap occur in the latter case of $\phi = 0$. The relationship between these two transport directions can be explained as follows. On the one hand, based on the analyses above for $\phi = 0$, we find that for a given strength of strain, a maximum shift of Dirac points along the $k_y$-axis corresponds to a minimum along the $k_x$-one and vice versa when varying the strain direction $\theta$. On the other hand, as schematized in the top of Fig. 6 below, the change in the transport direction results in the rotation of the first Brillouin zone, i.e., the $k_x$ (resp. $k_y$) axis in the case of $\phi = 30^\circ$ is identical to the $k_y$ (resp. $k_x$) axis in the case of $\phi = 0$. These two features explain essentially the opposite $\theta$-dependence of conduction gap for $\phi = 30^\circ$, compared to the case of $\phi = 0$ as mentioned. Again, we found the same qualitative behavior of conduction gap when applying the strains of $\{\sigma,\theta\}$ and $\{-\sigma,\theta+90^\circ\}$.

Next, we investigate the conduction gap with respect to different transport directions $\phi$. We display a ($\theta,\phi$)-map of conduction gap for $\sigma = 4 \%$ in Fig. 6 and, in the top, an additional diagram illustrating the rotation of Dirac points in the $k-$space with the change in the transport direction. It is clearly shown that (i) a similar scale of conduction gap is obtained for all different transport directions, (ii) there is a smooth and continuous shift of $E_{cond.gap}-\theta$ behavior when varying $\phi$, and (iii) the same behavior of $E_{cond.gap}$ is also observed when comparing the two transport directions of $\phi$ and $\phi+30^\circ$, similarly to the comparison above between $\phi = 0^\circ$ and $30^\circ$. The data plotted in Fig. 6 additionally shows that $E_{cond.gap}$ takes the same value in both cases of $\{\phi,\theta\}$ and $\{-\phi,-\theta\}$ with a remark that the strains of $-\theta$ and $180^\circ-\theta$ are identical. Moreover, the values of $\theta$ and $\phi$, for which the conduction gap has a peak or is equal to zero, have an almost linear relationship. In particular, the relationship for conduction gap peaks is approximately given by $\theta = \theta_A - \eta_s \phi$. For tensile strains, $\eta_s$ takes the values of $\sim 1.5667$ and $1.4333$ for $\theta_A = 0$ and $90^\circ$, respectively. On the opposite, it is about $1.4333$ and $1.5667$ for $\theta_A = 0$ and $90^\circ$, respectively, for compressive strain cases. All these features are consequences of the rotation of Dirac points in the $k$-space with respect to the transport direction $\phi$ as illustrated in the diagram on the top and the lattice symmetry of graphene.

Finally, we investigate other junctions based on compressive and tensile strained graphene sections. The idea is that in this type of strained junction, the shifts of Dirac points are different in two graphene sections of different strains, which offers the possibilities to use smaller strains to achieve a similar conduction gap, compared to the case of unstrained/strained junction. In Fig. 7, we display the maps of conduction gap with respect to the directions of compressive ($\theta_c$) and tensile ($\theta_t$) strains in two cases of transport direction $\phi = 0$ (armchair) and $30^\circ$ (zigzag) for given strain strengths. Indeed, as seen in Fig. 7(a,b), with smaller strains $\left\{ {{\sigma _c},{\sigma _t}} \right\} = \left\{ { - 2\% ,2\% } \right\}$ or $\left\{ { - 1\% ,3\% } \right\}$, similar conduction gap of about 310 meV can be achieved (see Figs. 7(a,b)) while it requires a strain of 4 $\%$ in the unstrained/strained junctions discussed above. However, since the shift of Dirac points is strongly dependent on the direction of applied strains and the transport direction, the properties of conduction gap are more complicated than in the latter case. In particular, our calculations show that the preferred transport directions to achieve a large conduction gap are close to the armchair one. Otherwise, the conduction gap is generally smaller, similarly to the data for $\phi = 30^\circ$ compared to $\phi = 0$, as shown in Fig. 7. Additionally, it is shown that the preferred directions of applied strains in the case of $\phi = 0$ are close to ${\theta _c} \equiv {\theta _t} = 0$ or $90^\circ$.

\section{Conclusion}

Based on the tight binding calculations, we have investigated the effects of uniaxial strain on the transport properties of graphene strained junctions and discuss systematically the possibilities of achieving a large conduction gap with respect to the strain, its applied direction and the transport direction. It has been shown that due to the strain-induced deformation of graphene lattice and hence of graphene bandstructure, a finite conduction gap higher than 500 meV can be achieved for a strain of only 6 $\%$. Moreover, as a consequence of the shift of Dirac points along the $k_y$-axis, the conduction gap is strongly dependent not only on the strain strength but also on the direction of applied strain and the transport direction. A full picture of these properties of conduction gap has been presented and explained. The study hence could be a good guide for the use of this type of unstrained/strained graphene junction in electronic applications.

\textbf{\textit{Acknowledgment.}} This research in Hanoi is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.02-1012.42. We also acknowledges the French ANR for financial support under the projects NANOSIM-GRAPHENE (Grant no. ANR-09-NANO-016) and MIGRAQUEL (Grant no. ANR-10-BLAN-0304).


\section{Introduction}

Quantum Chromodynamics (QCD) suggests the possible
existence of hadrons with a substructure that is more complex
than the three quark baryons and the quark-antiquark mesons
of the Quark Parton Model (QPM).  Possibilities for these
so-called {\it exotic} hadrons include
pentaquark baryons ($qqq\bar{q}q$), tetraquark mesons
($q\bar{q}q\bar{q}$) and quark-gluon hybrids ($q\bar{q}g$).
Although considerable theoretical and experimental effort
has gone into identifying exotic states, the situation
remains unclear.  The interest in this subject is demonstrated
by the huge literature related to the purported observation
of the $\Theta(1535)$ strangeness=+1 pentaquark.  According to
SPIRES, the experimental paper~\cite{nakano_theta}
that claimed first observation of the $\Theta(1530)$
has received over 785 citations.

There has been some recent progress in the identification
of what may be exotic mesons.  The BaBar and Belle
$B$-factory experiments have, somewhat unexpectedly,
discovered a number of interesting charmonium-like meson states
that have defied assignment to any of the unfilled levels
of the $c\bar{c}$ spectrum and, thus, remain unclassified.
These have come to be known
collectively as the ``$XYZ$'' mesons, and
include: the experimentally well established
$X(3872)$~\cite{belle_x3872} and
 $Y(4260)$~\cite{babar_y4260}, which decay to
$\pipiJ/\psi$; the $X(3940)$~\cite{belle_x3940}, seen
in $D^*\bar{D}$~\cite{charge_conjugate},
and the $X(4160)$~\cite{belle_x4160}
seen in $D^*\bar{D}^*$;  the
$Y(3940)$~\cite{belle_y3940,babar_y3940}, seen in $\omegaJ/\psi$;
and the $Y(4350)$~\cite{babar_y4325}
and $Y(4660)$~\cite{belle_y4350} seen in
$\pi^+\pi^- \psi '$.  In addition, Belle reported observations of similar
states but with non-zero electric charge:  the
$Z(4430)$~\cite{belle_z4430}
seen in $\pi^+\psi '$ and the $Z_1(4040)$ \&
$Z_2(4240)$~\cite{belle_z4040} seen in $\pi^+\chi_{c1}$.  These
$Z$ states have not yet been confirmed by other
experiments and remain somewhat controversial~\cite{babar_z4430}.
Table~{\ref{table1}} summarizes the
abovementioned $XYZ$ candidate states
as well as some other states discussed below.

In this report I will briefly review the reasons why these states
have eluded conventional $c\bar{c}$ assignments, discuss possible
alternative interpretations, and present some evidence for similar
states in the $s$- and $b$-quark sectors.

\begin{table}[htb!]
\begin{center}
\caption{\label{table1} Summary of the candidate $XYZ$ mesons discussed
in this talk.}
\footnotesize
\begin{tabular*}{180mm}{lcccccc}
\hline\hline
state    & $M$~(MeV) &$\Gamma$~(MeV)    & $J^{PC}$ & Decay Modes        &
Production  Modes & Observed by:\\\hline
$Y_s(2175)$& $2175\pm8$&$ 58\pm26 $& $1^{--}$ & $\phi f_0(980)$     &
$e^{+} e^{-}$~(ISR), $J/\psi\rightarrow\eta Y_s(2175)$  &  BaBar, BESII, Belle\\
$X(3872)$& $3871.4\pm0.6$&$<2.3$& $1^{++}$ & $\pi^{+}\pi^{-}
J/\psi$,$\gamma J/\psi$,
$D\bar{D^*}$
 & $B\rightarrow KX(3872)$, $p\bar{p}$ & Belle, CDF, D0, BaBar\\
$Z(3930)$& $3929\pm5$&$ 29\pm10 $& $2^{++}$ & $D\bar{D}$   &
$\gamma\gamma\rightarrow Z(3940)$ & Belle \\
$X(3940)$& $3942\pm9$&$ 37\pm17 $& $0^{?+}$ & $D\bar{D^*}$ (not
$D\bar{D}$ or $\omega J/\psi$) & $e^{+} e^{-}\rightarrow J/\psi  X(3940)$ & Belle  \\
$Y(3940)$& $3943\pm17$&$ 87\pm34 $&$?^{?+}$ & $\omega J/\psi$ (not
$D\bar{D^*}$) & $B\rightarrow K Y(3940)$ & Belle, BaBar \\
$Y(4008)$& $4008^{+82}_{-49}$&$ 226^{+97}_{-80}$ &$1^{--}$& $\pi^{+}\pi^{-} J/\psi$ &
$e^{+} e^{-}$(ISR) & Belle \\
$X(4160)$& $4156\pm29$&$ 139^{+113}_{-65}$ &$0^{?+}$& $D^*\bar{D^*}$
 (not $D\bar{D}$) & $e^{+} e^{-} \rightarrow J/\psi X(4160)$  & Belle \\
$Y(4260)$& $4264\pm12$&$ 83\pm22$ &$1^{--}$&  $\pi^{+}\pi^{-} J/\psi$ & $e^{+} e^{-}$(ISR)
&BaBar, CLEO, Belle    \\
$Y(4350)$& $4361\pm13$&$ 74\pm18$ &$1^{--}$&  $\pi^{+}\pi^{-} \psi '$ & $e^{+} e^{-}$(ISR)
& BaBar, Belle  \\
$Y(4660)$& $4664\pm12$&$ 48\pm15 $ &$1^{--}$&  $\pi^{+}\pi^{-} \psi '$ & $e^{+} e^{-} $(ISR)
& Belle     \\
$Z_1(4050)$& $4051^{+24}_{-23}$&$ 82^{+51}_{-29}$ & ? &
$\pi^{\pm}\chi_{c1}$ & $B\rightarrow K Z_1^{\pm}(4050)$  & Belle \\
$Z_2(4250)$& $4248^{+185}_{-45}$&$ 177^{+320}_{-72}$ & ? &
$\pi^{\pm}\chi_{c1}$ & $B\rightarrow K
Z_2^{\pm}(4250)$  & Belle \\
$Z(4430)$& $4433\pm5$&$ 45^{+35}_{-18}$ & ? & $\pi^{\pm}\psi '$ & $B\rightarrow K
Z^{\pm}(4430)$  & Belle \\
$Y_b(10890)$  & $10,890\pm 3$ & $55\pm 9$ &   $1^{--}$ &
$\pi^{+}\pi^{-}\Upsilon(1,2,3S)$
& $e^{+} e^{-}\rightarrow Y_b$ & Belle \\
\hline\hline
\end{tabular*}%
\end{center}
\end{table}

\section{Charmonium possibilities}
\label{charmonium}

The $c\bar{c}$ charmonium meson level diagram is shown in
Fig.~\ref{fig:charmonium_spectrum}~\cite{charmonium_spectrum}. Here the
states that have already been assigned are labeled by their commonly used
symbols and measured mass values.  The solid lines indicate the
measured levels and the broken lines indicate masses derived from
QCD-motivated potential model calculations~\cite{BGS}.  If any of the
$XYZ$ mesons are to be interpreted as simple quark-antiquark states,
they must be assigned to one of the figure's unlabeled levels.
All of the states with mass below the $M=2m_D=3.73$~GeV
``open-charm'' threshold (indicated by the horizontal line in
Fig.~\ref{fig:charmonium_spectrum}) have already been identified and
have properties that are in good agreement with potential model
expectations.  In addition, all of the $1^{--}$ levels above the
open charm threshold have been assigned to peaks in
the total annihilation cross section for
$e^{+} e^{-}\rightarrow~hadrons$~\cite{bes_R-fit}.

Of the $XYZ$ states listed in Table~\ref{table1}, only the
$Z(3930)$~\cite{belle_z3930} has been convincingly assigned to a
charmonium
level;
there is general agreement that this is the ($2^3P_2$)
$\chi_{c2}^\prime$.

\begin{figure}[t]
\includegraphics[scale=0.43]{olsen_panic_fig1.eps}
\caption{The predicted and observed spectrum of $c\bar{c}$ charmonium
mesons.  Already assigned states and their experimentally
measured masses are indicated by solid bars and their
commonly used names.  The broken lines indicate
various theoretical predictions.  The horizontal line
at 3.73~GeV indicates the mass threshold for decays
to $D\bar{D}$ ``open charm'' final states.}
\label{fig:charmonium_spectrum}
\end{figure}

\subsection{The $X(3872)$}

The experimentally preferred $J^{PC}$ value for the $X(3872)$
is $1^{++}$, although $2^{-+}$ has not been conclusively ruled
out~\cite{CDF_jpc}.  The only unfilled $1^{++}$ level in
Fig.~\ref{fig:charmonium_spectrum} is the $\chi_{c1}^\prime$, the
$2^3P_1$ $c\bar{c}$ state.   As mentioned above, the $J=2$
triplet partner state
for this level has been identified as the $Z(3930)$
with a mass of $3929\pm 5$~MeV.  A
$\chi_{c1}^\prime$ assignment for the $X(3872)$ would imply
a $\chi_{c2}^{\prime}$-$\chi_{c1}^{\prime}$ mass splitting for
radial quantum number
$n=2$ ({\it i.e.} $\delta m \sim 57$~MeV) that is larger than that
for the $n=1$ splitting ($\delta m = 46$~MeV), contrary to
potential model expectations.  A bigger difficulty with this
assignment is the fact that the
$X(3872)\rightarrow\rhoJ/\psi$ discovery mode would be an isospin violating
transition that should be strongly suppressed compared to
the  $X(3872)\rightarrow \gammaJ/\psi$ mode; the latter is measured to
be much smaller than the former~\cite{x3872_2gammajpsi}.
Using the $2^{-+}$ assignment does not help; in this
case the $\pipiJ/\psi$ mode would also be isospin violating, and the
$\gammaJ/\psi$ transition, which would be a $\Delta L=2$ transition,
would be unmeasurably small, which it isn't.

\subsection{The $X(3940)$ and $X(4160)$}

The $X(3940)$ is seen in the $D\bar{D^*}$ system recoiling from
the $J/\psi$ in exclusive $ee\rtJ/\psi D\bar{D^*}$ annihilations;
the $X(4160)$ is seen in the $D^*\bar{D^*}$ system in
$J/\psi$ in $ee\rtJ/\psi D^*\bar{D^*}$.  Neither are seen in the
experimentally more accessible $D\bar{D}$ channel.  The
only known charmonium states that are seen recoiling from the $J/\psi$
in $e^{+} e^{-}\rtJ/\psi X$ processes have $J=0$. This, plus the
absence of the $D\bar{D}$ mode,
provides circumstantial evidence that favors $J^{PC}=0^{-+}$ assignments
for both states, which for charmonium would be the $\eta_c^{\prime\prime}$
and $\eta_c^{\prime\prime\prime}$.  Such an assignment has difficulty
with the measured masses: the predicted $\eta_c^{\prime\prime}$ mass
is about 4050~MeV, over 100~MeV too high for the $X(3940)$; the
predicted $\eta_c^{\prime\prime\prime}$ is around 4400~MeV, more
than 200~MeV higher than the $X(4160)$,

\subsection{The $Y(3940)$}

The $Y(3940)$ was first seen by Belle
as a near-threshold peak in the $\omegaJ/\psi$
invariant mass spectrum in exclusive $B\rightarrow K\omegaJ/\psi$
decays~\cite{belle_y3940}.  It was subsequently confirmed
by BaBar~\cite{babar_y3940}, although there remain some
($\sim 2\sigma$) discrepancies between the Belle \& BaBar
measurements  of the mass and width.

It is unlikely
that the $Y(3940)$ (seen in $\omegaJ/\psi$)  and the
$X(3940)$ (seen in $D\bar{D^*}$) are different
decay modes of the same state.  Belle has searched for
$Y(3940)\rightarrow D\bar{D^*}$ in $B\rightarrow K D\bar{D^*}$
decays and finds a 90\%~CL lower limit of
$\mathcal{B}(Y\rightarrow\omegaJ/\psi)/
\mathcal{B}(Y\rightarrow D\bar{D^*})>0.71$~\cite{belle_y3940_ddstr}
that contradicts a 90\%~CL upper limit from a search for
$X(3940)\rightarrow\omegaJ/\psi$ in $e^{+} e^{-}\rightarrow J/\psi\jp\omega$ annihilations:
$\mathcal{B}(X\rightarrow\omegaJ/\psi)/
\mathcal{B}(X\rightarrow D\bar{D^*})<0.58$~\cite{belle_x3940}.

Possible charmonium assignments for the $Y(3940)$ are the
$\eta_c^{\prime\prime}$ ($0^{-+}$) --- although its mass
is a little low --- and the $\chi_{c0}^{\prime}$ ($0^{++}$)
for which its mass is too high.   The primary
difficulty with a charmonium assignment for the $Y(3940)$ is
its large partial width to $\omegaJ/\psi$, which reasonable
estimates put above 1~MeV~\cite{godfrey_olsen}
and which may in fact be quite a bit higher.  This is
well above the measured partial widths for any of the
observed hadronic transitions between charmonium states.

\subsection{The $J^{PC}=1^{--}$ $Y$ states}

The $Y(4260)$ was first seen by BaBar as a peak in the
$\pipiJ/\psi$ mass spectrum in the initial-state-radiation
(ISR)  process $e^{+} e^{-}\rightarrow\gamma_{ISR}\pipiJ/\psi$.
The $Y(4350)$ and $Y(4660)$ are seen in the $\pi^{+}\pi^{-}\psi '$ mass
spectrum in the $ee\rightarrow\pi^{+}\pi^{-}\psi '$ ISR process.  (Belle also sees
a broad $Y(4008)$ peak in $\pipiJ/\psi$~\cite{belle_y4008}, but this
has not been confirmed by BaBar~\cite{babar_y4008}.)  Since these
states are produced via ISR, their $J^{PC}$ has to be $1^{--}$.

There are no unassigned $1^{--}$ slots for any of these states in
the $M<4.4$~GeV spectrum of Fig.~\ref{fig:charmonium_spectrum}.
Moreover, no hint of any of them in seen in any of the
$D^{(*)}\bar{D^{(*)}}$ channels~\cite{galina}. This implies
that their $\pipiJ/\psi(\psi ')$ decay widths must be quite large.
In the case of the $Y(4260)$, the $\pipiJ/\psi$ has been
established to be more that 1.6~MeV~\cite{mo_xiaohu}. This is much too
large for charmonium, where allowed $\pipiJ/\psi$ transitions have
measured partial widths of 100~keV or less.

\subsection{The charged $Z$ particles}

Belle reported a peak with a $\sim 6.5\sigma$ statistical significance
near 4430~MeV in  the $\pi^{\pm}\psi '$ channel
in exclusive  $B\rightarrow K\pi^{\pm}\psi '$ decays~\cite{belle_z4430}.
A peak at the observed $\pi\psi '$ invariant mass value
cannot be produced by reflections from the $K\pi$ system.
However, BaBar did not confirm this peak, finding at most a
signal of $\sim 1.7\sigma$ significance~\cite{babar_z4430}.
A subsequent full Dalitz-plot analysis
(see Fig.~\ref{fig:z_projections}  left)
of the Belle $B\rightarrow
K\pi^{\pm}\psi '$ sample confirms their
original mass and significance determinations~\cite{chistov_qwg}.
 Belle also reported two peaks with greater than $5\sigma$
statistical significance in the $\pi^{\pm}\chi_{c1}$ channel,
the $Z_1(4050)$ \& $Z_2(4250)$, in exclusive
$B\rightarrow K\pi^{\pm}\chi_{c1}$ decays, again from a  Dalitz-plot
analysis (Fig.~\ref{fig:z_projections} right).
If these peaks are interpreted as meson states, they must have a
minimal tetraquark $c\bar{c} u\bar{d}$
substructure and there are no possible charmonium
or charmonium hybrid assignments.

\begin{figure}[h]
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.40]{olsen_panic_fig2a.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.55]{olsen_panic_fig2b.eps}
\end{minipage}
\caption{\label{fig:z_projections} {\bf Left:} ($B\rightarrow K\pi^{\pm}\psi '$)
The points with errors are data and the histograms show fits
to a Dalitz-plot projection that has the $K^*$ bands removed with \&
without the inclusion of the $Z(4430)$ resonance in the
$\pi^{\pm}\psi '$ channel.
{\bf Right:} ($B\rightarrow K\pi^{\pm}\chi_{c1}$)
A similar Dalitz-plot projection for data \& fits with \& without
the inclusion of two resonances in the $\pi^{\pm}\chi_{c1}$ channel.
}
\end{figure}

\section{Exotic possibilities}
\label{exotic}
In this section I discuss the possible interpretations of the
$XYZ$ peaks as $c\bar{c}q\bar{q}$ tetraquark states or $c\bar{c}$-gluon
hybrid states.

\subsection{Tetraquarks}
Two very distinct types of tetraquark mesons have been proposed:
{\it molecular states}, which are relatively
loosely bound structures comprised of deuteron-like
mesons-antimeson bound states~\cite{molecules},
and {\it diquark-diantiquark mesons} in which the two quarks form
an anticolor triplet state that binds tightly to a
color triplet that is formed from the two antiquarks~\cite{maiani}.  These
two types of structures have very different phenomenologies.

\subsubsection{Molecules}
A molecular state is expected to have a mass that is
slightly below  the sum of the masses of its
meson-antimeson constituents and exhibit large
isospin violations.  The $X(3872)$, with a mass
that is within errors of the $m_D + m_{D^*}$ mass
threshold and has decay rates to $\pipiJ/\psi$ and
$\pi^{+}\pi^{-}\pi^0J/\psi$ that are nearly equal~\cite{belle_x23pijpsi}.
Thus, this is a nearly ideal candidate for a $D\bar{D^*}$ molecular
state, either real~\cite{tornqvist2004,swanson,voloshin,braaten}
or virtual~\cite{hanhart}.  On the other hand, its proximity
to the $D\bar{D^*}$ threshold has also led to speculation
that it is some kind of a threshold effect~\cite{bugg,chao,zhang}.
In theses latter schemes, mixing with the $\chi_{c1}^{\prime}$
charmonium state can play an important role.

A new piece of experimental information, the significance of
which has yet to be commented on by any theorist, is a study
of $X(3872)$ production in exclusive $B\rightarrow K\pi X(3872)$
decays~\cite{belle_kpix3872}.
The left panel of Fig.~\ref{fig:mkpi} shows the $K\pi$ invariant
mass distribution, where it is evident that non-resonant
$K\pi$ production dominates, and the $K^*(890)$ contribution
is small and of marginal significance.  This is in contrast to what is
seen in all other
$B\rightarrow K\pi$+charmonium decays in which the $K^*(890)$ contribution
dominates; for example,
the right panel of Fig.~\ref{fig:mkpi} shows the
$M(K\pi)$ distribution for $B\rightarrow K^-\pi^+\chi_{c1}$
decays~\cite{belle_kpichic1}, where a prominent $K^*(890)$ signal
is clearly evident.

Not all of the $XYZ$ states fit the molecule picture.
For example the $X(3940)$, $Y(3940)$ and $Y(4660)$ are not
near any $D^{(*)}\bar{D^{(*)}}$ mass threshold. (Note that
$\pi$-exchange, the dominant binding term in molecular models,
is absent in $D_s^{(*)}\bar{D_s^{(*)}}$ systems.)

\begin{figure}[h]
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.3]{olsen_panic_fig3a.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.3]{olsen_panic_fig3b.eps}
\end{minipage}
\caption{\label{fig:mkpi} {\bf Left:}
The $M(K\pi)$ distribution for $B\rightarrow K^-\pi^+ X(3872)$ decays.
The lower curves show backgrounds from
non-resonant and $K^*(890)$ $K\pi$ systems. {\bf Right:}
The $M(K\pi)$ distribution for $B\rightarrow K^-\pi^+ \chi_{c1}$  decays.
}
\end{figure}

\subsubsection{Diquark-diantiquarks}
Essentially all of the observed $XYZ$ states can be accommodated
by the diquark-diantiquark model.    However, in this picture, each
of the assigned state is expected to have an associated flavor-$SU(3)$
multiplet of states.   One prediction of this model is that there
should be two $X(3872)$ states --- $X_u = cu\bar{c}\bar{u}$ and
$X_d = cd\bar{c}\bar{d}$ --- with a mass difference of $8\pm 3$~MeV.
No evidence for such a pairing has been
found~\cite{belle_y3940_ddstr,belle_kpix3872}.
In addition, in this model one expects
a charged isospin partner of the $X(3872)$ to be produced
in $B$ decays.  BaBar searched for such an
$X^+(3872)\rightarrow\rho^+J/\psi$ state in neutral $B$
meson decays and and set an upper limit that is well below
isospin-based expectations~\cite{babar_xplus3872}.  No isospin
partners of any of the other $XYZ$ states have been reported.

\subsection{Hybrids}
The lattice QCD expectation for the mass of the lowest-lying
charmonium hybrid is around 4.3~GeV and the relevant open-charm
threshold is 4.29~GeV, the $D^{**}\bar{D}$ mass threshold, where
$D^{**}$  denotes  the lowest mass $P$-wave charmed meson with mass
2.42~GeV.  Since the $Y(4260)$ mass is near the LQCD value and
below the $D^{**}\bar{D}$ mass threshold --- which would explain its
relatively strong decay rate to $\pipiJ/\psi$ as opposed to open
charm states --- a charmonium hybrid interpretation is attractive.
However, the $Y(4260)$ is broad and $D^{**}\bar{D}$ decays are
accessible from its high mass side, but there is no sign of a lineshape
distortion that might be expected when a dominant new decay
channel opens up.  Moreover, the $Y(4350)$ \& $Y(4660)$ are both
well above all $D^{**}\bar{D}$ thresholds and no open charm decays
have been seen.   So, while the hybrid interpretation might work for
the $Y(4260)$, it does not seem to apply to the other $1^{--}$
ISR-produced states.

\section{Evidence for $XYZ$-like states in the $s$- and $b$-quark
sectors}
An obvious question is whether or not there are counterpart states
in the $s$- and $b$-quark sectors.  Recent results
suggest that there are.  In a BaBar study of the ISR process
$e^{+} e^{-}\rightarrow\gamma_{ISR}\pi^{+}\pi^{-}\phi$, where the $\pi^{+}\pi^{-}$ comes from
$f_0(980)\rightarrow\pi^{+}\pi^{-}$, a distinct $\pi^{+}\pi^{-}\phi$ mass peak is seen
at 2175~MeV~\cite{babar_y2175}.  This peak was confirmed in an ISR
measurement by Belle~\cite{belle_y2175} (left panel of Fig.~\ref{fig:yb})
and seen in $J/\psi\rightarrow\eta f_0\phi$
decays by BES~\cite{bes_y2175}.  Although a conventional $s\bar{s}$
assignment cannot be ruled out~\cite{mlyan_y2175}, this state has
properties similar to what one would expect for an $s$-quark sector
counterpart of the $Y(4260)$.

The Belle group recently reported measurements of the energy dependence
of the $e^{+} e^{-}\rightarrow\pi^{+}\pi^{-}\Upsilon(nS)~(n=1,~2~\&~3)$ cross section around
$E_{cm}\sim 10.9$~GeV and found peaks in all three channels at
10.899~GeV (right panel of Fig.~\ref{fig:yb})~\cite{belle_y10899}.
The peak mass
and width valuse are quite distinct from those of
 the nearby $\Upsilon(5S)$ bottomonium ($b\bar{b})$
state, and the cross section values are more than two-orders-of-magnitude
above expectations for a conventional $b\bar{b}$ system.  One
interpretation for this peak is that it is a $b$-quark sector
equivalent of the $1^{--}$ $Y$ states seen in the $c$-quark
sector~\cite{weishu}.

\begin{figure}[h]
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.35]{olsen_panic_fig4a.epsi}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{75mm}
\includegraphics[scale=0.35]{olsen_panic_fig4b.eps}
\end{minipage}
\caption{\label{fig:yb} {\bf Left:}
The $M(f_0(980)\phi))$ distribution for $e^{+} e^{-}\rightarrow \gamma_{ISR} f_0\phi$.
{\bf Right:} the energy dependence of $e^{+} e^{-}\rightarrow \pi^{+}\pi^{-}
\Upsilon(nS)~(n=1,~2~\&~3)$
near $\sqrt{s}=10.9$~GeV. In both cases the data are from Belle.}
\end{figure}

\section{Summary}
There is a growing body of evidence for a new type of hadron spectroscopy
involving pairs of $c$-quarks that neither fits well to classic Quark
Parton Model expectations nor QCD-motivated extensions.  A recurring
feature of these new state are large partial widths for decays to
charmonium plus light hadrons.  There is some evidence for similar
structures in the $s$- and $b$-quark sectors.

\section{Acknowledgments}
I thank Professor Tserruya and the PANIC organizing committee for their
gracious hospitality.  I also thank Avraham~Gal
and my colleagues Haibo~Li, Roman~Mizuk,
Chengping~Shen and Bruce~Yabsley
for their assistance in the preparation of this manuscript.


\section*{Abstract}
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\newcommand{\sl Phys.~Rep.}{\sl Phys.~Rep.}
\newcommand{\sl Phys.~Rev.}{\sl Phys.~Rev.}
\newcommand{\sl Phys.~Rev.~Lett.}{\sl Phys.~Rev.~Lett.}
\newcommand{\sl Nucl.~Phys.}{\sl Nucl.~Phys.}
\newcommand{\sl Prog.~Theor.~Phys.}{\sl Prog.~Theor.~Phys.}
\newcommand{\sl Sov.~J.~Nucl.~Phys.}{\sl Sov.~J.~Nucl.~Phys.}
\newcommand{\sl Z.~Phys.}{\sl Z.~Phys.}
\newcommand{\it ibid.}{\it ibid.}

\begin{document}
\baselineskip 18pt

\begin{titlepage}

\preprintnumber{ICRR-Report-317-94-12 \\ April 1994}
\vspace*{\titlesep}
\begin{center}
{\LARGE\bf Flavor mixing in the gluino coupling \\
  and the nucleon decay}\\
\vspace*{\titlesep}
{\large Toru {\sc Goto}, Takeshi {\sc Nihei}} and
{\large Jiro {\sc Arafune}}\\
\vspace*{\titlesep}
{\it Institute for Cosmic Ray Research, University of Tokyo,\\
  Midori-cho, Tanashi-shi, Tokyo 188 JAPAN}\\
\end{center}
\vspace*{\titlesep}
\begin{abstract}
Flavor mixing in the quark-squark-gluino coupling is studied for the
minimal SU(5) SUGRA-GUT model and applied to evaluation of the nucleon
lifetime.
All off-diagonal (generation mixing) elements of Yukawa coupling
matrices and of squark/slepton mass matrices are included in solving
numerically one-loop renormalization group equations for MSSM
parameters, and the parameter region consistent with the radiative
electroweak symmetry breaking condition is searched.
It is shown that the flavor mixing in the gluino coupling for a large
$\tan\beta$ is of the same order of magnitude as the corresponding
Kobayashi-Maskawa matrix element in both up-type and down-type sector.
There exist parameter regions where the nucleon decay amplitudes for
charged lepton modes are dominated by the gluino dressing process,
while for all the examined regions the neutrino mode amplitudes are
dominated by the wino dressing over the gluino dressing.
\end{abstract}

\end{titlepage}

\section{Introduction}
\label{sec:introduction}

SU(5) supersymmetric (SUSY) grand unified theory (GUT) is an
attractive candidate for the unified theory of strong and electroweak
interactions.
The analyses of the gauge coupling unification \cite{LL-ABF-M} suggest
the validity of the minimal supersymmetric standard model (MSSM)
just above the electroweak scale $\sim 100 \ {\rm GeV}\ $ and the unification
of SU(3)$\times$SU(2)$\times$U(1) gauge group into a simple SU(5) at
the GUT scale $M_X \sim 10^{16} \ {\rm GeV}\ $.

One of the features of MSSM/SUSY-GUT is the existence of soft SUSY
breaking.
It gives quarks (or leptons) and their superpartners different mass
matrices in the generation (flavor) space.
In results, due to the discrepancy between the mass diagonalizing
bases of quarks and those of squarks, a generation mixing occurs in
the quark-squark-gaugino coupling (gluino coupling, in particular),
which may give considerable contributions to the nucleon decay, flavor
changing neutral currents (FCNC) and other various phenomena in
SUSY-GUT.
The flavor mixing in the gaugino coupling plays an important role in
the nucleon decay, since its amplitude is dominated by the dimension
five interaction followed by the gaugino ``dressing'' process
\cite{SY-W} in the minimal SU(5) SUSY-GUT model.
However, only a simplified treatment is made in the previous analyses
of the nucleon decay \cite{HMY,MATS,NCA,ENR}, where the diagonal
(in the generation space) gluino coupling is assumed leading to the
negligible contribution from the gluino dressing process.

The purpose of the present paper is to study the flavor mixing in the
gaugino coupling extensively and evaluate the contribution of the
gluino dressing process to the nucleon decay amplitude.
Based on the minimal SU(5) supergravity (SUGRA-) GUT \cite{N-LN},
we assume that the soft SUSY breaking parameters are ``universal'' at
the GUT scale \cite{BFS-CAN-HLW}.
We include all off-diagonal elements of Yukawa coupling matrices and
of squark/slepton mass matrices in solving numerically one-loop
renormalization group equations (RGEs)%
\footnote{Two-loop RGEs for soft SUSY breaking parameters are obtained
  recently \protect\cite{Y-MH}, which will be important for more
  accurate analysis.}
for all MSSM parameters \cite{IKKT-APW,BKS} with the universal
boundary conditions.
We then evaluate the effective potential for the Higgs fields at the
electroweak scale to find a consistent SU(2)$\times$U(1) breaking
minimum in accordance with the radiative electroweak symmetry breaking
scenario \cite{IKKT-APW}.
The obtained mass matrices of all particles are diagonalized to
evaluate the flavor mixing in the gaugino couplings.
The mass spectrum and the mixing are then used to calculate the
nucleon decay amplitudes for various decay modes.
It is found that the flavor mixing in the gluino coupling depends on
$\tan\beta$ (ratio of vacuum expectation values of the Higgs doublets)
and is roughly of the same order of magnitude as the corresponding
Kobayashi-Maskawa matrix element.
As for the nucleon decay, we find the gluino dressing process
dominates the amplitude for the decay modes containing a charged
lepton (and a meson) if $\tan\beta$ is large and if the gluino mass is
much smaller than the squark masses.
Note that the flavor mixing in the gaugino couplings is studied
previously in a systematic analysis of FCNC \cite{BKS} with a
semi-analytic solution of the RGEs for small $\tan\beta$ (top Yukawa
coupling $\gg$ bottom Yukawa coupling).
On the contrary, our numerical method cover the whole range of
$\tan\beta$, since all Yukawa couplings are taken into account.

The remaining part of this paper is organized as follows.
After introducing in the next section the minimal SU(5) SUGRA-GUT
model which we consider, we formulate the flavor mixing in the gaugino
couplings in Sec.~\ref{sec:mixing}.
In Sec.~\ref{sec:decay} the nucleon decay amplitudes are obtained with
a careful treatment of the flavor mixing.
The outline of the numerical calculation and the results are presented
in Sec.~\ref{sec:calculations} and our conclusions are summarized in
Sec.~\ref{sec:conclusion}.

\section{Minimal SU(5) SUGRA-GUT model}
\label{sec:model}

Minimal SU(5) SUSY-GUT model contains three generations of matter
multiplets with \repr{10} and \repr{\ol{5}} representations,
$\Psi_i^{AB}$ and $\Phi_{jA}$ respectively, where suffices
$A,B=1,2,\cdots,5$ are SU(5) indices and $i,j=1,2,3$ are the
generation labels, and three kinds of Higgs multiplets with \repr{5},
\repr{\ol{5}} and \repr{24} representations, $H_5^A$, $H_{\ol{5}A}$
and $\Sigma^A_B$ respectively.
SU(5) and R-parity invariant superpotential $W_{\rm GUT}$ at the GUT
scale is written as
\begin{eqnarray}
  W_{\rm GUT}(M_X) &=& f^{ij} \Psi_i^{AB} \Phi_{jA} H_{\ol{5}B}
        + \frac{1}{8} g^{ij} \epsilon_{ABCDE}
          \Psi_i^{AB} \Psi_j^{CD} H_5^E
\nonumber\\
    & & + \lambda H_{\ol{5}A}
          \left( \Sigma^A_B + M \delta^A_B \right) H_5^B
        + W_\Sigma \left( \Sigma \right) ~.
  \label{GUTsuperpotential}
\end{eqnarray}
Here, $\epsilon_{ABCDE}$ is the totally antisymmetric constant,
$f^{ij}$, $g^{ij}=g^{ji}$ and $\lambda$ are dimensionless couplings,
$M$ is a mass parameter and $W_\Sigma$ is a self-interaction
superpotential for $\Sigma^A_B$.
SU(5) symmetry is spontaneously broken down to SU(3) $\times$ SU(2)
$\times$ U(1) with the nonvanishing vacuum expectation value of the
adjoint Higgs $\Sigma^A_B$.
Below the GUT scale, the model is reduced to MSSM with effective
higher dimensional operators, which are obtained by integrating out
the superheavy particles in Eq.~(\ref{GUTsuperpotential}).
The effective superpotential is then written as
\begin{eqnarray}
  W_{\rm eff} &=& W_{\rm MSSM} + W_5 + O(M_X^{-2}) ~,
  \nonumber\\
  W_{\rm MSSM} &=& f_D^{ij} Q_i^{a\alpha} D_{ja} H_{1\alpha}
      + f_L^{ij} \epsilon^{\alpha\beta} E_i L_{j\alpha} H_{1\beta}
      + g_U^{ij} \epsilon_{\alpha\beta} Q_i^{a\alpha} U_{ja} H_2^\beta
      + \mu H_{1\alpha} H_2^\alpha ~,
  \nonumber\\
  W_5 &=& \frac{1}{M_C} \left\{
        \frac{1}{2} C_L^{ijkl} \epsilon_{abc} \epsilon_{\alpha\beta}
               Q_k^{a\alpha} Q_l^{b\beta} Q_i^{c\gamma} L_{j\gamma}
      + C_R^{ijkl} \epsilon^{abc} U_{ia} D_{jc} E_k U_{lb} \right.
  \nonumber\\
  & & ~~~~~~~+ ( \mbox{baryon number/lepton number
                       conserving terms} ) \biggr\} ~.
  \label{superpotential}
\end{eqnarray}
Here, $Q$, $U$ and $E$ are chiral superfields which contain
left-handed quark doublet, right-handed up-type quark and right-handed
charged lepton respectively, and are embedded in $\Psi$ (to be
specified in Eq.~(\ref{Embedding}));
 $D$ and $L$, which are embedded in $\Phi$, contain right-handed
 down-type quark and left-handed lepton doublet respectively;
$H_1$ and $H_2$ are Higgs doublets embedded in $H_{\ol{5}}$ and $H_5$,
respectively.
The suffices $a,b,c=1,2,3$ are SU(3) indices and $\alpha,\beta=1,2$
are SU(2) indices.
$M_C$ is the colored Higgs mass which is assumed to be $O(M_X)$, while
the supersymmetric mass of Higgs doublet $\mu$ is of the order of the
$Z$ boson mass $m_Z$.
This discrepancy is owing to a tree level fine-tuning in the GUT
superpotential.
At the GUT scale, $f_D$ and $f_L$ are unified and $C_L$ and $C_R$ are
written in terms of the Yukawa coupling constants (see
Sec.~\ref{sec:decay}).
Baryon number (and lepton number) violating terms in $W_5$ give
dominant contributions to the nucleon decay in this model.

In addition to the supersymmetric Lagrangian to be derived from
(\ref{superpotential}), the following soft SUSY breaking terms are
included:
\begin{eqnarray}
  -{\cal L}_{\rm soft} &=&
       \left(m_Q^2\right)_i^j \tilde{q}^{\dagger i} \tilde{q}_j
     + \left(m_D^2\right)_i^j \tilde{d}^{\dagger i} \tilde{d}_j
     + \left(m_U^2\right)_i^j \tilde{u}^{\dagger i} \tilde{u}_j
  \nonumber\\
  && + \left(m_L^2\right)_i^j \tilde{l}^{\dagger i} \tilde{l}_j
     + \left(m_E^2\right)_i^j \tilde{e}^{\dagger i} \tilde{e}_j
     + \Delta_1^2 h_1^\dagger h_1 + \Delta_2^2 h_2^\dagger h_2
  \nonumber\\
  && + \left\{   A_D^{ij} \tilde{q}_i \tilde{d}_j h_1
               + A_L^{ij} \tilde{e}_i \tilde{l}_j h_1
               + A_U^{ij} \tilde{q}_i \tilde{u}_j h_2
               - B\mu h_1 h_2 + {\rm h.~c.} \right\}
  \nonumber\\
  && + \frac{1}{2} \left\{   M_1 \tilde{B} \tilde{B}
                           + M_2 \tilde{W} \tilde{W}
                           + M_3 \tilde{G} \tilde{G}
                           + {\rm h.~c.} \right\} ~,
  \label{softbreaking}
\end{eqnarray}
where $\tilde{q}_i$, $\tilde{d}_i$, $\tilde{u}_i$, $\tilde{e}_i$,
$\tilde{l}_i$, $h_1$ and $h_2$ are scalar components of $Q_i$, $D_i$,
$U_i$, $E_i$, $L_i$, $H_1$ and $H_2$, respectively, and $\tilde{B}$,
$\tilde{W}$ and $\tilde{G}$ are U(1), SU(2) and SU(3) gauge fermion
fields (bino, wino and gluino), respectively.
SU(2) and SU(3) suffices are omitted in (\ref{softbreaking}) for
simplicity.
We assume that the soft SUSY breaking parameters satisfy simple
relations at the GUT scale:
\begin{eqnarray}
  \left( m_Q^2 \right)_i^j &=& \left( m_D^2 \right)_i^j ~=~
  \left( m_U^2 \right)_i^j ~=~ \left( m_L^2 \right)_i^j ~=~
  \left( m_E^2 \right)_i^j ~\equiv~ m_0^2\ \delta_i^j ~,
  \nonumber\\
  \Delta_1^2 &=& \Delta_2^2 ~=~ m_0^2 ~,
  \nonumber\\
  A_D^{ij} &=& f_{DX}^{ij} A_X m_0 ~, ~~
  A_L^{ij} ~=~ f_{LX}^{ij} A_X m_0 ~, ~~
  A_U^{ij} ~=~ g_{UX}^{ij} A_X m_0 ~,
  \nonumber\\
  M_1 &=& M_2 ~=~ M_3 ~\equiv~ M_{gX} ~,
  \label{boundaryconditions}
\end{eqnarray}
where the suffix ``$X$'' stands for the value at the GUT scale.
The boundary conditions (\ref{boundaryconditions}) are due to the
minimal SUGRA model, where local SUSY is spontaneously broken in the
hidden sector which couples to the observable sector (SUSY-GUT, in the
present case) only gravitationally, and hence universal soft SUSY
breaking terms are induced in the observable sector \cite{BFS-CAN-HLW}.

Below the GUT scale, radiative corrections modify all parameters in
the superpotential (\ref{superpotential}) and the soft SUSY breaking
terms (\ref{softbreaking}), as well as three gauge coupling constants
$g_1$, $g_2$ and $g_3$ for U(1), SU(2) and SU(3), respectively.
The evolution of the parameters are described by the RGEs \cite{BKS}.
According to the radiative SU(2) $\times$ U(1) breaking scenario
\cite{IKKT-APW}, we numerically solve the RGEs down to the electroweak scale
$m_Z$ and evaluate the effective potential for the neutral Higgs
fields:
\begin{eqnarray}
  V({\rm Higgs}) &=& V_0 + V_1 ~,
  \nonumber\\
  V_0 &=&   \left( \mu^2 + \Delta_1^2 \right) |h_1|^2
          + \left( \mu^2 + \Delta_2^2 \right) |h_2|^2
          - \left( B\mu h_1 h_2 + {\rm h.~c.} \right)
  \nonumber\\
    & & + \frac{g_1^2 + g_2^2}{8} \left( |h_1|^2 - |h_2|^2 \right)^2 ~,
  \nonumber\\
  V_1 &=& \frac{1}{64\pi^2}
          \ {\rm Str}\  {\cal M}^4 \left( \log \frac{{\cal M}^2}{m_Z^2} -
                                 \frac{3}{2} \right) ~,
  \label{Higgspotential}
\end{eqnarray}
where $\ {\rm Str}\ $ means the supertrace and ${\cal M}$ includes all
(s)quark and (s)lepton masses.
Then the electroweak symmetry breaking condition
\begin{eqnarray}
  \vev{h_1} &=& v \cos \beta ~, ~~
  \vev{h_2} ~=~ v \sin \beta ~,
  \label{vev}
  \\
  m_Z^2 &=& \frac{g_2^2}{2\cos^2\theta_W} v^2 ~,
  \nonumber
\end{eqnarray}
is imposed.

\section{Flavor mixing in the gluino coupling}
\label{sec:mixing}

In order to discuss the flavor mixing in the gluino coupling, we have
to diagonalize the mass matrices for quarks and squarks.
Throughout the calculation hereafter, we choose the basis in the
generation space for the superfields so that the Yukawa coupling
constants for up-type quarks and leptons should be diagonalized {\em
  at the electroweak scale}.
The Yukawa terms in (\ref{superpotential}) are then written as
\begin{equation}
  W_{\rm Yukawa}(m_Z) =
         \hat{f}_D^{kj}
         \left( V_{\rm KM}^\dagger \right)_k^{~i} Q_i D_j H_1
       + \hat{f}_L^{ij} E_i L_j H_1
       + \hat{g}_U^{ij} Q_i U_j H_2 ~,
  \label{YukawaZ}
\end{equation}
where the notation ``$\hat{~~}$'' stands for a diagonal matrix and
$V_{\rm KM}$ is the Kobayashi-Maskawa matrix.
All eigenvalues of $\hat{f}_D$, $\hat{f}_L$ and $\hat{g}_U$ are taken
to be real positive.
Since this choice of the basis is different from that in the GUT
superpotential (\ref{GUTsuperpotential}), a re-diagonalization of the
Yukawa couplings at the GUT scale is needed in order to find the
unification condition of $f_D$ and $f_L$ and the relation between the
Yukawa coupling constants and the dimension-five coupling constants
$C_{L,R}$.
$W_{\rm Yukawa}$ at the GUT scale is diagonalized with appropriate
unitary matrices $U_Q^{(f)}$, $U_D$, $U_E$, $U_L$, $U_Q^{(g)}$ and
$U_U$:
\begin{subequations}
\begin{eqnarray}
  W_{\rm Yukawa}(M_X) &=& f_{DX}^{ij} Q_i D_j H_1
                        + f_{LX}^{ij} E_i L_j H_1
                        + g_{UX}^{ij} Q_i U_j H_2
  \label{YukawaX}
  \\
    &=& \hat{f}_{DX}^{kl} \left( U_Q^{(f)} \right)_k^{~i}
                          \left( U_D \right)_l^{~j} Q_i D_j H_1
      + \hat{f}_{LX}^{kl} \left( U_E \right)_k^{~i}
                          \left( U_L \right)_l^{~j} E_i L_j H_1
  \nonumber\\
  & & + \hat{g}_{UX}^{kl} \left( U_Q^{(g)} \right)_k^{~i}
                          \left( U_U \right)_l^{~j} Q_i U_j H_2 ~.
\end{eqnarray}
\end{subequations}
The unification condition of $f_D$ and $f_L$ is then written as%
\footnote{The ``unification'' of $\hat{f}_{DX}$ and $\hat{f}_{LX}$,
  with quark masses given in Ref.~\protect\cite{GL}, however, is not
  so satisfactory numerically in the first and the second generations
  as the gauge coupling unification.
  We ignore the difference in the present calculation, since we may
  still have ambiguities of the renormalization effect in the very low
  energy region.}
\begin{equation}
  \hat{f}_{DX}^{ij} = \hat{f}_{LX}^{ij} ~.
  \label{YukawaUnification}
\end{equation}
The matter multiplets are accommodated into the SU(5) multiptets as
\begin{eqnarray}
  \Psi_i &\Leftarrow& \left\{ Q_i
          ,~ \left( U_Q^{(g)\dagger} P^\dagger U_U \right)_i^{~j} U_j
          ,~ \left( U_Q^{(f)\dagger} U_E \right)_i^{~j} E_j \right\} ~,
  \nonumber\\
  \Phi_i &\Leftarrow& \left\{ D_i
          ,~ \left( U_D^\dagger U_L \right)_i^{~j} L_j \right\} ~,
  \label{Embedding}
\end{eqnarray}
and the GUT Yukawa coupling constants in Eq.~(\ref{GUTsuperpotential})
are expressed with those in Eq.~(\ref{YukawaX}) as
\begin{eqnarray}
  f^{ij} &=& f_{DX}^{ij} ~,
  \nonumber\\
  g^{ij} &=& g_{UX}^{ik}
             \left( U_U^\dagger P U_Q^{(g)} \right)_k^{~j} ~,
  \label{GUTYukawa}
\end{eqnarray}
where $P$ is a diagonal phase matrix which cannot be absorbed by field
redefinitions in the colored Higgs coupling \cite{EGN}.

The origin of the flavor mixing in the gluino coupling lies in the
difference between the mass basis for quarks and that for squarks.
The mass matrix for up-type squarks is expressed as
\begin{eqnarray}
  -{\cal L}(\mbox{s-up mass}) &=&
       ( \tilde{q}_{u} ,~ \tilde{u}^{\dagger} )
       {\cal M}_{\tilde{u}}^2
        \left( \begin{array}{c}
                 \tilde{q}_{u}^{\dagger} \\ \tilde{u}
               \end{array}\right) ~,
  \nonumber\\
  &=&  ( \tilde{q}_{ui} ,~ \tilde{u}^{\dagger i} )
       \left( \begin{array}{cc}
                \left( m_{LL}^2 \right)^i_j &
                \left( m_{LR}^2 \right)^{ij} \\
                \left( m_{RL}^2 \right)_{ij} &
                \left( m_{RR}^2 \right)_i^j
              \end{array}\right)
              \left( \begin{array}{c}
                       \tilde{q}_{u}^{\dagger j} \\ \tilde{u}_j
                     \end{array}\right) ~,
  \nonumber\\
  \left( m_{LL}^2 \right)^i_j &=&  \left( M_U M_U^\dagger \right)^i_j
    + \left( m_Q^2 \right)^i_j
    + m_W^2 \cos 2\beta \left(   \frac{1}{2}
                               - \frac{1}{6}\tan^2 \theta_W \right)
      \delta^i_j ~,
  \nonumber\\
  \left( m_{RR}^2 \right)_i^j &=& \left( M_U^\dagger M_U \right)_i^j
    + \left( m_U^2 \right)_i^j
    + m_W^2 \cos 2\beta \left( \frac{2}{3}\tan^2 \theta_W \right)
      \delta_i^j   ~,
  \nonumber\\
  \left( m_{LR}^2 \right)^{ij} &=& \mu M_U^{ij} \cot\beta
    + A_U^{ij} v \sin\beta ~,
  \nonumber \\
  m_{RL} &=& m_{LR}^{\dagger} ~,
  \label{squarkmass}
\end{eqnarray}
where $M_U$ is the up-type quark mass matrix $M_U^{ij} = g_U^{ij} v
\sin\beta$ and $\tilde{q}_u$ is the up-type component of the SU(2)
doublet $\tilde{q}$.
The squark mass matrix ${\cal M}_{\tilde{u}}^2$ is not diagonalized
with the quark mass basis (\ref{YukawaZ}) since off-diagonal elements
are induced in the soft SUSY breaking parameter matrices due to the
renormalization effect.
Squark mass basis is obtained by diagonalizing (\ref{squarkmass}) with
a 6$\times$6 unitary matrix $\tilde{U}_U$:
\begin{eqnarray}
  \tilde{u}'_I &=& \left( \tilde{U}_U \right)_I^J \tilde{u}_J ~,
  ~~ I ~=~ 1,\ 2,\ \cdots,\ 6 ~,
  \nonumber\\
  \tilde{u}_I &=&
    \left\{ \begin{array}{lcl}
               \tilde{q}_{uI} & \mbox{for} & I = 1,\ 2,\ 3 \\
               \tilde{u}_{I-3} & \mbox{for} & I = 4,\ 5,\ 6
            \end{array} \right. ~,
  \nonumber\\
  \tilde{U}_U^\dagger {\cal M}_{\tilde{u}}^2 \tilde{U}_U &=&
     \mbox{diagonal} ~,
  \label{DiagonalizeSquark}
\end{eqnarray}
where $\tilde{u}'_I$ is the mass eigenstate of up-type squark.
We define the numbering of $\tilde{u}'_I$ such that the mixing of
$\tilde{u}_I$ is the largest in $\tilde{u}'_I$.
Accordingly we call $\tilde{u}'_1$, $\tilde{u}'_2$, $\cdots$,
$\tilde{u}'_6$ as $\tilde{u}_L$, $\tilde{c}_L$, $\tilde{t}_L$,
$\tilde{u}_R$, $\tilde{c}_R$ and $\tilde{t}_R$ respectively in the
later discussions.
The mass bases of down-type squarks and charged sleptons are obtained
in the same way with 6$\times$6 unitary matrices $\tilde{U}_D$ and
$\tilde{U}_E$, respectively.
Notice that no generation mixing occurs in the lepton/slepton sector
since the right-handed (s)neutrino does not exist in the minimal
model; the nonvanishing off-diagonal elements of the slepton mass
matrix are left-right mixing components only.
Consequently, quark-squark-gluino coupling is written as
\begin{eqnarray}
  {\cal L}_{\rm int}(\mbox{gluino}) &=&
    -i \sqrt{2} g_3 \left\{ \tilde{d}'^{\dagger I}
                            \left(
                            \left( \tilde{U}_D \right)_I^k
                            \left( V_{\rm KM} \right)_k^j
                            \tilde{G} d_{Lj} +
                            \left( \tilde{U}_D \right)_I^{j+3}
                            \ol{\tilde{G}} \ol{d}_{Rj}
                            \right)
                    \right.
  \nonumber\\
    & & ~~~~~~~~~ + \left.  \tilde{u}'^{\dagger I}
                            \left(
                            \left( \tilde{U}_U \right)_I^j
                            \tilde{G} u_{Lj} +
                            \left( \tilde{U}_U \right)_I^{j+3}
                            \ol{\tilde{G}} \ol{u}_{Rj}
                            \right)
                   \right\} + {\rm h.~c.}
  \nonumber\\
    &=&
    -i \sqrt{2} g_3 \left\{ \tilde{d}'^{\dagger I}
                            \left(
                            \left( \tilde{U}'_D \right)_I^j
                            \tilde{G} d_{Lj} +
                            \left( \tilde{U}_D \right)_I^{j+3}
                            \ol{\tilde{G}} \ol{d}_{Rj}
                            \right)
                    \right.
  \nonumber\\
    & & ~~~~~~~~~ + \left.  \tilde{u}'^{\dagger I}
                            \left(
                            \left( \tilde{U}_U \right)_I^j
                            \tilde{G} u_{Lj} +
                            \left( \tilde{U}_U \right)_I^{j+3}
                            \ol{\tilde{G}} \ol{u}_{Rj}
                            \right)
                   \right\} + {\rm h.~c.} ~,
  \label{GluinoCoupling}
\end{eqnarray}
with the definition of $\tilde{U}'_D$ as
\begin{equation}
  \left( \tilde{U}'_D \right)_I^j \equiv
    \left( \tilde{U}_D \right)_I^k \left( V_{\rm KM} \right)_k^j ~.
  \label{primedU}
\end{equation}
$u_{Li}$ and $d_{Li}$ in (\ref{GluinoCoupling}) are left-handed quarks
of mass eigenstates which compose the SU(2) doublet as
\begin{equation}
  Q_i \ni \left( \begin{array}{c}
                   u_{Li} \\
                   \left( V_{\rm KM} \right)_i^j d_{Lj}
                 \end{array}
          \right) ~,
  \label{Doublet}
\end{equation}
and $u_{Ri}$ and $d_{Ri}$ are the fermion components of $U_i$ and
$D_i$, respectively.
Similar flavor mixing formulae are obtained for other gaugino (wino
and bino) coupling terms.

\section{Nucleon decay with dimension five operators}
\label{sec:decay}

As mentioned in Sec.~\ref{sec:model}, the nucleon decay amplitude in
the minimal SU(5) SUSY-GUT model is dominated by the dimension five
operators \cite{SY-W} induced by colored higgsino/Higgs exchanges.
Since the dimension five operators are made from two fermion
(quark/lepton) and two boson (squark/slepton) component fields,
effective baryon number violating four-fermion operators are generated
by one loop ``dressing'' diagrams which involve gauginos or higgsinos
(see Fig.~\ref{fig:diagrams}).
In the present calculation, only the gluino dressing and the charged
wino dressing diagrams are included;
contributions from higgsino dressing diagrams are negligibly small due
to the small Yukawa couplings of light quarks ($u,d,s$), compared to
the SU(2) gauge coupling $g_2$;
neutral wino and bino coupling have the same flavor mixing structure
as that in the gluino coupling, hence their contributions are smaller
than that from the gluino dressing.

The dimension five coupling constants $C_L$ and $C_R$ of
(\ref{superpotential}) at the GUT scale are written in terms of the
Yukawa coupling constants (see (\ref{Embedding}) and
(\ref{GUTYukawa})):
\begin{eqnarray}
  C_{LX}^{ijkl} &=& f_{DX}^{im} \left( U_D^\dagger U_L \right)_m^j
                    g_{UX}^{kn} \left( U_U^\dagger P
                                       U_Q^{(g)} \right)_n^l ~,
  \nonumber\\
  C_{RX}^{ijkl} &=& f_{DX}^{mj} \left( U_Q^{(g)\dagger} P^\dagger
                                       U_U \right)_m^i
                    g_{UX}^{nl} \left( U_Q^{(f)\dagger}
                                       U_E \right)_n^k ~.
  \label{GUTDim5}
\end{eqnarray}
Note that this relation with the index ``$X$'' removed does not hold
true at the electroweak scale.
The effective baryon number violating four-fermion operators at the
electroweak scale are written as
\begin{eqnarray}
  {\cal L}_{\rm eff} (\Delta B = \pm 1) &=&
        \left(   \tilde{C}_{\nu}^{ijkl}(\tilde{G})
               + \tilde{C}_{\nu}^{ijkl}(\tilde{W}) \right)
        \epsilon_{abc} (u_{Lk}^a d_{Ll}^b) (d_{Li}^c \nu_{Lj})
  \nonumber\\
  & & + \left(   \tilde{C}_{e}^{ijkl}(\tilde{G})
               + \tilde{C}_{e}^{ijkl}(\tilde{W}) \right)
        \epsilon_{abc} (u_{Lk}^a d_{Ll}^b) (u_{Li}^c e_{Lj})
  \nonumber\\
  & & + (\mbox{right-handed quark/lepton}) + \mbox{h.~c.} ~,
  \label{4Fermi}
\end{eqnarray}
where $\tilde{C}_{\nu,e}$ are calculated as follows with use of the
numerical values of $C_L$ and $C_R$ at the electroweak scale to be obtained
through their RGEs, squark/slepton mass eigenvalues and the mixing
matrices in the gaugino couplings%
\footnote{Contributions from $C_R$'s and higgsino/neutralino dressings
  are estimated to be small and neglected in the present calculations.
  }:
\begin{eqnarray}
  \tilde{C}_{\nu}^{ijkl}(\tilde{G}) &=& -\frac{4 g_3^2}{3M_C}
    \left\{   \left( C_L^{ijmn} - C_L^{njmi} \right)
              \left( \tilde{U}_U^\dagger \right)_m^I
              \left( \tilde{U}_D^\dagger \right)_n^J
              F_{\tilde{G}}( \tilde{u}'_I, \tilde{d}'_J )
              \left( \tilde{U}_U \right)_I^k
              \left( \tilde{U}'_D \right)_J^l
    \right.
  \nonumber\\
  & & ~~~~~~~~
    \left. - \left( C_L^{mjkn} - C_L^{njkm} \right)
             \left( \tilde{U}_D^\dagger \right)_m^I
             \left( \tilde{U}_D^\dagger \right)_n^J
             F_{\tilde{G}}( \tilde{d}'_I, \tilde{d}'_J )
             \left( \tilde{U}'_D \right)_I^i
             \left( \tilde{U}'_D \right)_J^l
    \right\} ~,
  \nonumber\\
  \tilde{C}_{\nu}^{ijkl}(\tilde{W}) &=& -\frac{g_2^2}{M_C}
    \left\{  \left( C_L^{ijmn} - C_L^{njmi} \right)
             \left( \tilde{U}_U^\dagger \right)_m^I
             \left( \tilde{U}_D^\dagger \right)_n^J
             F_{\tilde{W}}( \tilde{u}'_I, \tilde{d}'_J )
             \left( \tilde{U}'_U \right)_I^k
             \left( \tilde{U}_D \right)_J^l
    \right.
  \nonumber\\
  & & ~~~~~~~~
    \left. + \left( C_L^{mnkl} - C_L^{knml} \right)
             \left( \tilde{U}_U^\dagger \right)_m^I
             \left( \tilde{U}_E^\dagger \right)_n^J
             F_{\tilde{W}}( \tilde{u}'_I, \tilde{e}'_J )
             \left( \tilde{U}'_U \right)_I^i
             \left( \tilde{U}_E \right)_J^j
    \right\} ~,
  \nonumber\\
  \tilde{C}_{e}^{ijkl}(\tilde{G}) &=& -\frac{4 g_3^2}{3M_C}
    \left\{  \left( C_L^{ijmn} - C_L^{mjin} \right)
             \left( \tilde{U}_U^\dagger \right)_m^I
             \left( \tilde{U}_D^\dagger \right)_n^J
             F_{\tilde{G}}( \tilde{u}'_I, \tilde{d}'_J )
             \left( \tilde{U}_U \right)_I^k
             \left( \tilde{U}'_D \right)_J^l
    \right.
  \nonumber\\
  & & ~~~~~~~~
    \left. - \left( C_L^{mjnl} - C_L^{njml} \right)
             \left( \tilde{U}_U^\dagger \right)_m^I
             \left( \tilde{U}_U^\dagger \right)_n^J
             F_{\tilde{G}}( \tilde{u}'_I, \tilde{u}'_J )
             \left( \tilde{U}_U \right)_I^i
             \left( \tilde{U}_U \right)_J^k
    \right\} ~,
  \nonumber\\
  \tilde{C}_{e}^{ijkl}(\tilde{W}) &=& -\frac{g_2^2}{M_C}
    \left\{  \left( C_L^{ijmn} - C_L^{mjin} \right)
             \left( \tilde{U}_U^\dagger \right)_m^I
             \left( \tilde{U}_D^\dagger \right)_n^J
             F_{\tilde{W}}( \tilde{u}'_I, \tilde{d}'_J )
             \left( \tilde{U}'_U \right)_I^l
             \left( \tilde{U}_D \right)_J^k
    \right.
  \nonumber\\
  & & ~~~~~~~~
    \left. + \left( C_L^{mjkl} - C_L^{ljkm} \right)
             \left( \tilde{U}_D^\dagger \right)_m^I
             F_{\tilde{W}}( \tilde{d}'_I, \tilde{\nu}'_j )
             \left( \tilde{U}_D \right)_I^i
    \right\} ~.
  \label{4FermiCoupling}
\end{eqnarray}
Here, $( \tilde{U}'_D )_I^i$ is defined in (\ref{primedU}) and
$( \tilde{U}'_U )_I^i = ( \tilde{U}_U )_I^j (V_{\rm KM})_j^i$.
$F_{\tilde{G}}$ and $F_{\tilde{W}}$ are obtained by the loop integral
\cite{HMY,MATS,NCA}:
\begin{eqnarray}
  F_{\tilde{G}}( \tilde{f}_1, \tilde{f}_2 ) &=&
    \tilde{F}( m_{\tilde{f}_1}, m_{\tilde{f}_2}; M_3 ) ~,
  \nonumber\\
  F_{\tilde{W}}( \tilde{f}_1, \tilde{f}_2 ) &=&
    \left( U_{-} \right)_1^\alpha
    \tilde{F}( m_{\tilde{f}_1}, m_{\tilde{f}_2}; M_{\pm}^\alpha )
    \left( U_{+}^{\dagger} \right)_\alpha^1 ~,
  \label{Floop}
  \\
    \tilde{F}( m_1, m_2; M ) &=&
    \frac{1}{16\pi^2} \frac{M}{m_1^2 - m_2^2}
    \left(   \frac{m_1^2}{m_1^2 - M^2}\log\frac{m_1^2}{M^2}
           - \frac{m_2^2}{m_2^2 - M^2}\log\frac{m_2^2}{M^2}
    \right) ~,
  \nonumber
\end{eqnarray}
where $U_{-}$, $U_{+}$ are 2$\times$2 unitary matrices which diagonalize
the chargino mass matrix and $M_{\pm}^\alpha$ ($\alpha = 1, 2$) are its
eigenvalues:
\begin{eqnarray}
  M(\mbox{chargino}) &=& \left( \begin{array}{cc}
                                  M_2 & \sqrt{2} m_W \sin\beta \\
                                  -\sqrt{2} m_W \cos\beta & -\mu
                                \end{array} \right)
  \nonumber\\
  &=& U_{-} \left( \begin{array}{cc}
                     M_{\pm}^1 & 0 \\
                     0 & M_{\pm}^2
                   \end{array} \right) U_{+}^\dagger ~.
  \label{CharginoMass}
\end{eqnarray}
The low energy QCD correction between $m_Z$ and $1 \ {\rm GeV}\ $ is taken into
account in order to evaluate the four fermion operators in the next
section.
The quark Lagrangian at $\sim 1 \ {\rm GeV}\ $ is then converted to the hadron
chiral Lagrangian with $\Delta B = \pm 1$ terms \cite{HMY,CD-CWH} with
use of the matrix element
\begin{equation}
  \vev{ 0 | \epsilon_{abc} (d_L^a u_L^b) u_L^c | p } = \beta_p N_L ~,
  \label{Hadron}
\end{equation}
where $N_L$ is a left-handed proton wave function; it enables us to
evaluate the partial lifetimes of the nucleon decay.

\section{Numerical calculations}
\label{sec:calculations}

According to the framework described in the previous sections, we
calculate the flavor (and left-right) mixing in gaugino couplings and
the nucleon partial lifetimes in a five-dimensional parameter space
$\{m_{\rm top},~ \tan\beta,~ m_0,~ M_{gX},~ \tilde{A}_X\}$, where a
dimensionful $A$ parameter is defined as $\tilde{A}_X \equiv A_X m_0$.
Actual calculations are made in the following procedure.
At first, $m_{\rm top}$ and $\tan\beta$ (at the electroweak scale) are fixed.
Using the numerical values of light quark masses and the
Kobayashi-Maskawa mixing angles given in literatures \cite{GL,PDG}
with the above fixed $m_{\rm top}$ and $\tan\beta$, we evaluate the
Yukawa coupling constants at the electroweak scale (\ref{YukawaZ}).
QCD corrections below the electroweak scale for quark masses other than
$m_{\rm top}$ are included at the one-loop level.
Next, the RGEs for the dimensionless parameters {\it i.e.\ }, the gauge coupling
constants and the Yukawa coupling constants are solved upward to the
GUT scale with the boundary conditions at the electroweak scale.
At the GUT scale, the Yukawa coupling constants are re-diagonalized to
obtain the boundary conditions for the dimension-five coupling
constants (\ref{GUTDim5}).
Then the RGEs for the soft SUSY breaking parameters and dimension-five
coupling constants are solved downward with the boundary conditions
(\ref{boundaryconditions}) and (\ref{GUTDim5}).
Since the RGEs are linear for the dimensionful parameters, all soft
SUSY breaking parameters at the electroweak scale are written as linear
combinations of the initial parameters $(m_0,~ M_{gX},~ \tilde{A}_X)$
\cite{MATS}:
\begin{eqnarray}
  \tilde{m}^2_I (m_Z) &=& c_{1I} m_0^2 + c_{2I} M_{gX}^2 +
                  c_{3I} \tilde{A}_X^2 + c_{4I} M_{gX} \tilde{A}_X ~,
  \nonumber\\
  \tilde{M}_J (m_Z) &=& d_{1J} M_{gX} + d_{2J} \tilde{A}_X ~,
  \label{LinearCombinations}
\end{eqnarray}
where $\tilde{m}^2_I$ and $\tilde{M}_J$ are collective notations for
the soft SUSY breaking parameters of mass dimension two
($m^2_{Q,D,U,L,E}$, $\Delta^2_{1,2}$) and one ($M_{1,2,3}$,
$A_{D,U,L}$), respectively.
The coefficients $c$'s and $d$'s are implicit functions of the gauge
couplings and the Yukawa couplings and are determined numerically by
solving the RGEs with four cases of boundary conditions $(m_0,~
M_{gX},~ \tilde{A}_X) = (1,~0,~0)$, $(0,~1,~0)$, $(0,~0,~1)$ and
$(0,~1,~1)$.
Once the coefficients are obtained, the values of soft SUSY breaking
parameters at the electroweak scale for given $(m_0,~ M_{gX},~ \tilde{A}_X)$
are evaluated with the formulae (\ref{LinearCombinations}), and it is
easy to scan the three-dimensional parameter space $\{m_0,~ M_{gX},~
\tilde{A}_X\}$ for fixed $m_{\rm top}$ and $\tan\beta$ with this
method.
The next step is to evaluate the remaining two parameters $\mu$ and
$B$ with the electroelectroweak SU(2) $\times$ U(1) symmetry breaking
condition.
The requirement that the minimum of the Higgs potential
(\ref{Higgspotential}) gives the vacuum expectation values (\ref{vev})
leads to
\begin{eqnarray}
  \mu^2 &=& \frac{\Delta_2^2 - \Delta_1^2}{2\cos 2\beta}
          - \frac{\Delta_1^2 + \Delta_2^2}{2} - \frac{1}{2}m_Z^2
  \nonumber\\
       && - \frac{1}{v\cos 2\beta} \left.
            \left(   \frac{\partial V_1}{\partial h_1} \cos\beta
                   - \frac{\partial V_1}{\partial h_2} \sin\beta
            \right)
            \right|_{\mathop{}^{\scriptstyle h_1 = v\cos\beta}
                              _{\scriptstyle h_2 = v\sin\beta} } ~,
  \label{minimum1}
  \\
  B\mu &=& \left( \mu^2 + \frac{\Delta_1^2 + \Delta_2^2}{2}
           \right) \sin 2\beta
  \nonumber\\
      && + \frac{1}{v} \left.
           \left(   \frac{\partial V_1}{\partial h_1} \sin\beta
                  + \frac{\partial V_1}{\partial h_2} \cos\beta
           \right)
           \right|_{\mathop{}^{\scriptstyle h_1 = v\cos\beta}
                             _{\scriptstyle h_2 = v\sin\beta} } ~.
  \label{minimum2}
\end{eqnarray}
Notice that the solution of the equation (\ref{minimum1}) for
$\mu$ cannot be written in a simple formula since the one-loop part of
the Higgs potential $V_1$ depends on $\mu$.
We solve (\ref{minimum1}) numerically for both signs of $\mu$ and then
calculate $B$ with (\ref{minimum2}).
Since all the MSSM parameters and the dimension-five coupling
constants at the electroweak scale for a given parameter set $(m_{\rm top},~
\tan\beta,~ m_0,~ M_{gX},~ \tilde{A}_X)$ are thus determined, the mass
spectrum of all superparticles and the mixing matrices in the gaugino
couplings are obtained by diagonalizing their mass matrices.

We investigate the parameter space $\{m_0,~ M_{gX},~ A_X\}$ within the
range $10 \ {\rm GeV}\  \leq m_0 \leq 10 \ {\rm TeV}\ $, $10 \ {\rm GeV}\  \leq M_{gX} \leq 10
\ {\rm TeV}\ $ and $-5 \leq A_X \leq +5$ for each combination of $m_{\rm top}$
= 120, 150 or 180 GeV and $\tan\beta = 2$, 10, 30 or 50.
For $m_{\rm top}$ = 180 GeV and $\tan\beta = 2$, the top Yukawa
coupling diverges below the GUT scale when solving the RGEs.
For $\tan\beta = 50$, no consistent radiative breaking solution is
found for  any $m_{\rm top}$.
Figs.~\ref{fig:mixingu2} -- \ref{fig:mixings3} are histograms for the
specified off-diagonal elements of the gluino coupling matrices
$\tilde{U}_U$ and $\tilde{U}'_D$ for $m_{\rm top}$ = 150 GeV.
The magnitudes of the generation mixing in the right-right and
left-right sectors are small compared to the corresponding left-left
elements.
For a small $\tan\beta$, the left-left elements of $\tilde{U}'_D$ are
approximately equal to the Kobayashi-Maskawa matrix elements:
$(\tilde{U}'_D)_2^1 \approx V_{cd}$, $(\tilde{U}'_D)_3^1 \approx
V_{td}$, {\it etc.\ } in most of the parameter space, while the corresponding
off-diagonal elements of $\tilde{U}_U$ are small: the mass matrices of
up-type quarks, up-type squarks and down-type squarks are diagonalized
in the same basis.
This agrees with the conclusion of Ref.~\cite{BKS} where those mixing
matrices are semi-analytically obtained with an assumption that the
bottom Yukawa coupling is much smaller than the top Yukawa coupling,
which is applicable for small $\tan\beta$.
On the other hand, for a large $\tan\beta$, we find that nonvanishing
off-diagonal elements in the left-left sector of $\tilde{U}_U$ arises
with the same order of magnitudes as $V_{\rm KM}$, which contributes
significantly to the charged lepton decay modes (see below).
Off-diagonal elements of $\tilde{U}'_D$ are smaller than those for
small $\tan\beta$.
The qualitative behavior of the mixing matrices is rather independent
of the different values of $m_{\rm top}$.

Using the obtained values of superpartner masses and mixing matrices,
we evaluate $\tilde{C}$'s in (\ref{4Fermi}) with the formula
(\ref{4FermiCoupling}).
We then take into account of the low energy QCD correction to
$\tilde{C}$'s at the one-loop level%
\footnote{This corresponds to taking $A_{\rm L} \approx 0.46$, where
  $A_{\rm L}$ is the low energy QCD factor used in literatures
  \cite{HMY,MATS,NCA,ENR,AN}. It is argued the two-loop analysis
  gives $A_{\rm L}\approx 0.28$ in Ref.~\cite{AN} }.
We take the chiral Lagrangian factors given in Ref.~\cite{HMY} to
derive amplitudes of various decay modes from (\ref{4Fermi}).
A large uncertainty of the nucleon lifetime comes from the numerical
values of $\beta_p$ and $M_C$.
$\beta_p$ is calculated with various methods \cite{GKSMPT-BEHS}, which
give
\begin{displaymath}
  0.003 \mbox{GeV}^3 \leq \beta_p \leq 0.03 \mbox{GeV}^3 ~,
\end{displaymath}
and Ref.~\cite{HMY,HY} shows
\begin{displaymath}
  2.2 \times 10^{13} \mbox{GeV} \leq M_C \leq
  2.3 \times 10^{17} \mbox{GeV} ~.
\end{displaymath}
Here, we take a small value of $\beta_p$ = 0.003 GeV$^3$ and a large
value of $M_C = 10^{17}$ GeV so that we have a longer nucleon lifetime
for the safety of the later arguments.
Fig.~\ref{fig:lifetime} shows the partial lifetime for each nucleon
decay mode with fixed $m_{\rm top}$ = 150 GeV and $\tan\beta = 2$.
The range of each lifetime comes mainly from the ranges of the soft
SUSY breaking parameters.
As can be seen in the figure, $K \ol{\nu}$ decay modes are dominant
and most severely constrained by the experiments \cite{Kamioka,IMB} as
\begin{displaymath}
  \begin{array}{lcrr}
  \tau( p \rightarrow K^+ \ol{\nu} ) & \geq & 1.0 \times 10^{32} ~
  {\rm yrs} & ~~~~~ {\rm (KAMIOKANDE)} ~, \\
  \tau( p \rightarrow K^+ \ol{\nu} ) & \geq & 6.2 \times 10^{31} ~
  {\rm yrs} & ~~~~~ {\rm (IMB)} ~, \\
  \tau( n \rightarrow K^0 \ol{\nu} ) & \geq & 8.6 \times 10^{31} ~
  {\rm yrs} & ~~~~~ {\rm (KAMIOKANDE)} ~, \\
  \tau( n \rightarrow K^0 \ol{\nu} ) & \geq & 1.5 \times 10^{31} ~
  {\rm yrs} & ~~~~~ {\rm (IMB)} ~.
  \end{array}
\end{displaymath}
Our main concern is to study the contribution of the gluino dressing
diagrams.
To do that, we compare $\tau$(wino) with $\tau$(total), where
$\tau$(wino) is a partial lifetime calculated
with only the wino dressing diagrams taken into account and
$\tau$(total) is that calculated with both wino and gluino dressing
diagrams combined.
The results for $m_{\rm top}$ = 150 GeV are presented in
Figs.~\ref{fig:ratio02} -- \ref{fig:ratio30}.
The nucleon decay amplitude is dominated by the wino dressing diagrams
if $\tau (\mbox{wino}) / \tau(\mbox{total}) \approx 1$, while it is
dominated by the gluino dressing diagrams if $\tau (\mbox{wino}) /
\tau(\mbox{total}) \gg 1$.
There occur cancellations between the wino dressing contributions and
gluino dressing contributions in $\tau (\mbox{wino}) /
\tau(\mbox{total}) \ll 1$ region.
The gluino contributions are small for any modes in the small
$\tan\beta$ case.
For large $\tan\beta$, however, gluino dominant region is realized in
the charged lepton modes.
This is brought about by the following two reasons:
(1) in the simplified analyses based on an assumption of the diagonal
gluino coupling, the gluino dressing diagrams contribute only to the
$K \ol{\nu}$ modes due to the color antisymmetry in the dimension-five
coupling \cite{NCA,ENR}.
In the present case, the gluino coupling is not diagonal any more and
hence the gluino dressing processes contribute to the decay modes
other than $K \ol{\nu}$ modes.
The nonvanishing off-diagonal gluino coupling in the {\em up sector\ }
for large $\tan\beta$ significantly contributes to the charged lepton
modes.
(2) furthermore, if the squark masses are much larger than the gaugino
masses, the amplitude from the gluino dressing diagrams is enhanced by
a factor of $(\alpha_3(m_Z)/\alpha_2(m_Z))^2$ compared to that from the
wino dressing diagrams, since $\tilde{F}$ in Eq.~(\ref{Floop}) is
asymptotically
\begin{equation}
  \tilde{F}(m,m;M) \sim \frac{1}{16\pi^2} \frac{M}{m^2}
  ~~~~~ {\rm for} ~ m \gg M ~,
  \label{asymptoticF}
\end{equation}
and $M_3(m_Z)/M_2(m_Z) = \alpha_3(m_Z)/\alpha_2(m_Z)$ because of the
GUT relation of gaugino masses.
Since (\ref{asymptoticF}) gives an overall suppression of the nucleon
decay amplitude for $m \gg M$ (see Fig.~\ref{fig:md1-M2}), the
dominant gluino contribution in Figs.~\ref{fig:ratio10} and
\ref{fig:ratio30} is realized in the long lifetime region (see
Fig.~\ref{fig:lifetimevsratio}).

Translating the scanned parameters $(m_0,~ M_{gX},~ A_X)$ into the
MSSM parameters $(m_{\tilde{d}_L},~ M_2,~ \mu)$ where we take
$m_{\tilde{d}_L}$, the down-type squark mass of the first generation
as the typical squark mass, we plot the calculated points in the MSSM
parameter space for $m_{\rm top}$ = 150 GeV and $\tan\beta = 2$ in
Figs.~\ref{fig:mu-M2} and \ref{fig:md1-M2}.
The region A in Fig.~\ref{fig:mu-M2} is excluded by LEP constraints on
charginos and neutralinos \cite{LEP}:
\begin{eqnarray}
  m_{\chi^\pm} &>& 45 \ {\rm GeV}\  ~,
  \nonumber\\
  \Gamma ( Z \rightarrow \chi \chi ) &<& 22 \ {\rm MeV}\  ~,
  \nonumber\\
  B ( Z \rightarrow \chi \chi' ) &<& 5 \times 10^{-5} ~,
  \nonumber\\
  B ( Z \rightarrow \chi' \chi' ) &<& 5 \times 10^{-5} ~,
  \nonumber
  \label{lep}
\end{eqnarray}
where $\chi^\pm$ is a chargino, $\chi$ is the lightest neutralino and
$\chi'$ is a heavier neutralino.
No solution with radiative SU(2)$\times$U(1) breaking is found in
region B, which is a forbidden region.
Points plotted with small dots are excluded due to the present lower
bound for the proton lifetime $\tau( p \rightarrow K^+ \ol{\nu} ) >
10^{32}$ yrs, giving the constraint of $|\mu| \gsim$ 300 GeV.
This constraint for $\mu$ is roughly unchanged for different $m_{\rm
  top}$ and/or $\tan\beta$.
Fig.~\ref{fig:md1-M2} shows the squark mass bound $m_{\tilde{d}_L}
\gsim 400 \ {\rm GeV}\ $.
If the lower bound for the proton lifetime is raised to $\tau( p
\rightarrow K^+ \ol{\nu} ) > 10^{33}$ yrs with the near future
experiment at Super-KAMIOKANDE, most of the parameter region with
$m_{\tilde{d}_L} \lsim$ 1 TeV will be excluded.
Lower bounds for other first and second generation squark masses are
found similar to that for $m_{\tilde{d}_L}$, while the bound for the
third generation squarks is lower in general due to the
renormalization effect and the left-right mixing.
Since the nucleon lifetime is approximately proportional to
$(\tan\beta)^{-2}$ \cite{HMY}, the lower bound for the squark mass is
raised for larger $\tan\beta$.
In fact, $\tau( p \rightarrow K^+ \ol{\nu} ) > 10^{32}$ yrs implies
$m_{\tilde{d}_L} \gsim 1 \ {\rm TeV}\ $ for $\tan\beta = 30$.

\section{Conclusion}
\label{sec:conclusion}

In this paper we have made a systematic analysis of the flavor mixing
in the gaugino couplings within the framework of the minimal SUGRA-GUT.
We have solved the one-loop RGEs for all MSSM parameters including
off-diagonal Higgs coupling matrices with five input parameters,
namely $(m_{\rm top},~ \tan\beta)$ at the electroweak scale $m_Z$ and
$(m_0,~ M_{gX},~ A_X)$ at the GUT scale $M_X$, and we have numerically
obtained full mass spectra and mixing matrices, which satisfy the
radiative electroweak symmetry breaking condition.
For a small $\tan\beta$ ($\tan\beta = 2$), we have obtained a result
consistent with the semi-analytic study \cite{BKS}, in which the top
Yukawa coupling is assumed to be much larger than other Yukawa
couplings: the left-left sector of the generation mixing matrix in the
down-type quark-squark-gluino coupling is approximately equal to the
Kobayashi-Maskawa matrix, while the off-diagonal mixing matrix
elements in the up-type gluino coupling are small.
On the other hand, for large $\tan\beta = 10$ and 30 where the bottom
(and tau, for extremely large $\tan\beta$) Yukawa coupling is not
negligibly small compared with the top Yukawa coupling, we have found
that nonvanishing generation mixing in the up-type gluino coupling
occurs with the magnitudes comparable to the corresponding
Kobayashi-Maskawa matrix elements.
The generation mixing in the down-type gluino coupling is also changed
considerably.

We have applied the generation mixing to the calculation of nucleon
decay widths to study the contributions from the gluino dressing
diagrams compared with the wino dressing diagrams.
In result, it is found that the gluino dressing diagrams give the
dominant contribution to the decay mode containing a charged lepton if
$\tan\beta \gg 1$ and $M_3 \ll m_{\tilde{q}}$ (typical squark mass).
For the charged lepton modes with small $\tan\beta$, or the (anti-)
neutrino emission modes with any $\tan\beta$, the gluino contributions
are relatively small.
In those cases, the contributions from the gluino dressing are at most
of the same order of magnitude as the wino dressing contributions.
We have scanned the MSSM parameter space to find allowed regions with
the present constraints given by the nucleon decay experiments and the
accelerator experiments%
\footnote{In addition, cosmological constraint will be given by the
  analyses of the relic abundance of the lightest superprticle
  \protect\cite{MY-DN-KM}.}.
The latter excludes the parameter region of small superpartner masses,
and the former gives a strict bound to the masses of first and second
generation squarks.
We argue that the whole parameter region with $m_{\tilde{d}_L} \lsim 1
\ {\rm TeV}\ $ in the minimal SU(5) SUGRA-GUT model can be tested by
Super-KAMIOKANDE.

Our method of calculations and the numerical result itself are
adaptable to the analyses of FCNC in the minimal SUGRA model, which
will be discussed elsewhere.

\subsection*{Acknowledgment}

One of the authors (T.~G.) would like to thank J.~Hisano for helpful
discussions.

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