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\bb mixing and a CP violation parameter in \kk mixing $\epsilon_K$ are studied in the minimal supergravity model. We solve the one-loop renormalization group equations for the minimal supersymmetric standard model (MSSM) parameters numerically in order to determine the masses and mixings of the supersymmetric particles, while all off-diagonal (generation mixing) elements and phases of Yukawa coupling matrices and those of squark mass matrices are taken into account. Applying the radiative electroweak symmetry breaking condition and phenomenological constraints including the recent measurement of the \bsg inclusive branching ratio, we obtain the allowed parameter region. We have found that the present constraints still allow a parameter region where both \bb mass splitting $\Delta M_B$ and $\epsilon_K$ are $\sim 20$\% larger than the standard model values. By explicit numerical calculations, we have also found that the complex phase of \bb mixing matrix element in this model is almost the same as the standard model value in a good accuracy in the whole allowed parameter region. It is shown that $\Delta M_B$ and $\epsilon_K$ can put useful constraints on the supersymmetry's parameter space when the Cabibbo-Kobayashi-Maskawa matrix elements are determined through the measurements of CP violations in $B$ decay with future $B$-factories.

$B^0$--$\overline{B}^0$ mixing and $\epsilon_K$ parameter in the minimal supergravity model