Relativistic corrections to the form factors of $B_c$ into $S$-wave charmonium

We investigate the form factors of $B_c$ meson into $S$-wave charmonium within the nonrelativistic QCD effective theory and obtain the next-to-leading order relativistic corrections to the form factors, where both the $B_c$ meson and the charmonium are treated as the nonrelativistic bound states. Treating the charm quark as a light quark in the limit $m_c/m_b\to 0$, some form factors are identical at the maximum recoil point, which are consistent with the predictions in the heavy-quark effective theory and the large-energy effective theory. Considering that the branching ratios of $B_c^{+}\to J/\psi D_s^{+}$ and $B_c^{+}\to J/\psi D_s^{*+}$ have been measured by the LHCb and ATLAS Collaborations recently, we employ the form factors of $B_c$ meson into $S$-wave charmonium at the next-to-leading order accuracy to these two decay channels and obtain more precise predictions of their decay rates. Numerical results indicate that the factorizable diagrams dominate the contribution in these two channels, while the color-suppressed and the annihilation diagrams contribute less than 10 percent. Our results are consistent with the LHCb and ATLAS data.

Figure 1

Feynman diagrams for the form factors of $B_c$ into $S$-wave charmonium, where “$\oplus$” denotes certain current operators.

Figure 2

Typical Feynman diagrams for $B_c^{+}\to J/\psi +D_s^{+(*)}$, where two “$\oplus$” denote four-fermion weak interaction operators. There are four types of topologies: (a) factorizable diagrams; (b)nonfactorizable diagrams; (c) color-suppressed diagrams; (d) annihilation diagrams.