$D$-Wave Charmonia ${\eta }_{c2}$($1{}^1{D}_2$), ${\psi }_2$($1{}^3{D}_2$), and ${\psi }_3$($1{}^3{D}_3$) in $B_c$ Decays

Abstract

We study the semi-leptonic and non-leptonic decays of $B_c$ meson to $D$-wave charmonia, namely, ${\eta }_{c2}(1^1{D}_2)$, ${\psi }_2(1^3{D}_2)$, and ${\psi }_3(1^3{D}_3)$. In our calculations, the instantaneous Bethe-Salpeter method is applied to achieve the hadronic matrix elements. This method includes relativistic corrections which are important especially for the higher orbital excited states. For the semi-leptonic decay channels with electron as the final lepton, we get the branching ratios $\mathcal{B}[B_c\to {\eta }_{c2}e{\bar{\nu }}_e]=5.9_{+1.0}^{-0.8}\times 10^{-4}$, $\mathcal{B}[B_c\to {\psi }_{2}e{\bar{\nu }}_e]=1.5_{+0.3}^{-0.2}\times 10^{-4}$, and $\mathcal{B}[B_c\to {\psi }_{3}e{\bar{\nu }}_e]=3.5_{+0.8}^{-0.6}\times 10^{-4}$. The transition form factors, forward-backward asymmetries, and lepton spectra in these processes are also presented. For the non-leptonic decay channels, those with $\rho$ as the lighter meson have the largest branching ratios, $\mathcal{B}[B_c\to {\eta }_{c2}\rho ]=8.1_{+1.0}^{-1.0}\times 10^{-4}$, $\mathcal{B}[B_c\to {\psi }_2\rho ]=9.6_{+1.0}^{-1.0}\times 10^{-5}$, and $\mathcal{B}[B_c\to {\psi }_3\rho ]=4.1_{+0.8}^{-0.7}\times 10^{-4}$.

18 pages, 9 figures

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