Investigation of $B_{c}^{ + } \to D_{s}^{ + }\mathop {\bar {D}}\nolimits^0 $ and $D_{s}^{ + }{{D}^{0}}$ Decays Including Factorization Approaches and Final State Interaction Effects

In this paper the decay of $B_{c}^{ + }$ meson, consisting of two b and c heavy quarks, into the $D_{s}^{ + }{{\bar {D}}^{0}}$ and $D_{s}^{ + }{{D}^{0}}$ mesons are studied in two stages. In the first step, the QCD factorization (QCDF) approach is considered in the initial evaluation, the result of calculations are (in units of ${{10}^{{ - 6}}}$): $\mathcal{B}{{(B_{c}^{ + } \to D_{s}^{ + }{{\bar {D}}^{0}})}_{{{\text{QCDF}}}}} = 3.52 \pm 0.82$ and $\mathcal{B}{{(B_{c}^{ + } \to D_{s}^{ + }{{D}^{0}})}_{{{\text{QCDF}}}}} = 11.96 \pm 2.27$. While the available experimental result for these decays are (in units of ${{10}^{{ - 6}}}$) $\frac{{{{f}_{c}}}}{{{{f}_{u}}}}\mathcal{B}(B_{c}^{ + } \to D_{s}^{ + }{{\bar {D}}^{0}}) = 2.70 \pm 3.34,\,\,\frac{{{{f}_{c}}}}{{{{f}_{u}}}}\mathcal{B}(B_{c}^{ + } \to D_{s}^{ + }{{D}^{0}}) = - 3.42 \pm 2.36,$ by applying the upper limit of the theoretical value of ${{{{f}_{c}}} \mathord{\left/ {\vphantom {{{{f}_{c}}} {{{f}_{u}}}}} \right. \kern-0em} {{{f}_{u}}}}$ that span the range of $[0.4,1.2]$%, the results for QCDF approach become (in units of ${{10}^{{ - 8}}}$) $\frac{{{{f}_{c}}}}{{{{f}_{u}}}}\mathcal{B}{{(B_{c}^{ + } \to D_{s}^{ + }{{\bar {D}}^{0}})}_{{{\text{QCDF}}}}} = 4.22 \pm 0.98,\,\,\,\,\frac{{{{f}_{c}}}}{{{{f}_{u}}}}\mathcal{B}{{(B_{c}^{ + } \to D_{s}^{ + }{{D}^{0}})}_{{{\text{QCDF}}}}} = 14.35 \pm 2.72.$ The results for QCDF approach are about ${{10}^{2}}$ times smaller than experimental one. Therefore, it is decided to calculate the theoretical branching ratio by applying the final state interaction (FSI) through the t(crossed and uncrossed)-channels. The FSI effects are very sensitive to the changes in the phenomenological parameter $\eta $. This parameter appears in the FSI form factors that increase strong interaction share. In most calculation changing two units in this parameter, makes the final result multiply in the branching ratio, therefore the decision to use FSI is not unexpected. In this study there are nineteen intermediate states for $B_{c}^{ + } \to D_{s}^{ + }{{\bar {D}}^{0}}$ decay and four middle states for $B_{c}^{ + } \to D_{s}^{ + }{{D}^{0}}$ decay in which the contribution of each one is calculated and summed in the final amplitudes. By choosing the value of the $\eta $ according to the mass of the exchange meson, as $\eta = 3$ for exchange meson of $D{\text{*}}$ (or $D$) and ${{{{f}_{c}}} \mathord{\left/ {\vphantom {{{{f}_{c}}} {{{f}_{u}}}}} \right. \kern-0em} {{{f}_{u}}}} = 1.2\% $ the obtained results in FSI are (in units of ${{10}^{{ - 6}}}$) $\frac{{{{f}_{c}}}}{{{{f}_{u}}}}\mathcal{B}{{(B_{c}^{ + } \to D_{s}^{ + }{{\bar {D}}^{0}})}_{{{\text{FSI}}}}} = 2.57 \pm 0.54,\,\,\,\,\frac{{{{f}_{c}}}}{{{{f}_{u}}}}\mathcal{B}{{(B_{c}^{ + } \to D_{s}^{ + }{{D}^{0}})}_{{{\text{FSI}}}}} = 9.44 \pm 1.52,$ that are in very good agreement with the experimental results.