Polarization and quantum entanglement effects in $B_c^{\pm }\to J\mathrm{/}\psi +{\pi }^{\pm }+{\pi }^0$ process

Motivated by the very recent observation of the $B_c^{+}\to J\mathrm{/}\psi +{\pi }^{+}+{\pi }^0$ decay using proton-proton collision data by the LHCb collaboration, we study the four-body angular distributions and the quantum entanglement effects in the $B_c^{+}\to J\mathrm{/}\psi +{\pi }^{+}+{\pi }^0$ associated with $J\mathrm{/}\psi \to {\mu }^{+}+{\mu }^{-}$. The helicity angular distributions are given in the QCD effective theory and the von Neumann entropy is obtained in $B_c^{\pm }\to J\mathrm{/}\psi (\to {\mu }^{+}{\mu }^{-})+{\rho }^{\pm }(\to {\pi }^{\pm }{\pi }^0)$ decay process.