Weak Annihilation Contribution to Angular Observables in $B_c^{+}\to D^{\ast +}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ Decays

We analyze the rare semileptonic decays $B_c^{+}\to D^{\ast +}(\to P_1{P}_2){\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$, with $P_1{P}_2=D^{+}{\pi }^0$ or $D^0{\pi }^{+}$, and $\mathrm{\ell }=\mu ,\tau$. We focus on the impact of weak annihilation contributions alongside penguin, box, and long-distance effects. Using the effective Hamiltonian for $b\to d{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ transitions and $B_c\to D^{\ast }$ form factors from covariant confined quark model inputs, we compute the differential branching ratios, forward-backward asymmetry, longitudinal helicity fraction of the $D^{\ast }$, and various normalized angular coefficients. The results of the observables show that weak annihilation effects are sizable, particularly at low $q^2$, significantly modifying several observables and shifting zero-crossings. Resonance effects dominate at high $q^2$, restricting reliable analysis windows. We conclude that the inclusion of weak annihilation is essential for precise Standard Model predictions and for isolating possible New Physics effects in $B_c^{+}\to D^{\ast +}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ decays.