Quasi-two-body decays $B_c \to K^{*} h \to K \pi h $ in the perturbative QCD

TL;DR

This paper calculates branching ratios and CP violations for quasi-two-body B_c decays involving K* and D mesons using perturbative QCD, providing predictions testable at LHCb with detailed distributions.

Contribution

It introduces two-meson distribution amplitudes in PQCD for B_c decays, extending previous two-body calculations to quasi-two-body processes with detailed CP violation analysis.

Findings

Largest branching ratios up to 10^{-6} for certain decays.

Predicted CP violation of about -14.6% for specific channels.

Results consistent with previous PQCD two-body calculations.

Abstract

In this work we study the quasi-two-body decays $B_c \to \ K^{*} h \to K \pi h (h = D, D_s, K, \pi, \eta, \eta')$ in the perturbative QCD (PQCD) approach. The two-meson distribution amplitudes (DAs) $\Phi^{\text{P-wave}}_{K\pi}$ are introduced to describe the final state interactions of the K \pi pair, which involve the time-like form factors F_{K\pi}(s) parameterized by the relativistic Breit-Wigner function and the Gegenbauer polynomials. We calculate the branching ratios for these quasi-two-body decays, from which one can obtain the branching raios for the corresponding two-body decays under the narrow width approximation relation. We find that $B^+_c\to K^{*+}D^0$ and $B^+_c\to K^{*0}D^+$ have the largest branching ratios, which can reach up to $10^{-6}$, while the branching ratios for other two-body decays are very small and only about $10^{-8}\sim10^{-7}$. As we expected that the branching ratios of the pure annihilation decays are usually small, while in our considered such type of decays, the channel $B_c^+ \to \bar K^{*0}K^{+}$ has the largest branching ratio, which is near $10^{-6}$. These results are consistent with the previously PQCD calculations obtained in the two-body framework, which can be tested by the future LHCb experiments. For the decays $B_c^+ \to K^{*+} D^{0}\to K^{0}\pi^+D^{0} , B_c^+ \to K^{*0}D^{+}\to K^{+}\pi^-D^{+}$ and $B_c^+ \to \bar K^{*0}D_s^{+}\to K^{-}\pi^+D_s^{+}$, we calculate their direct CP violations and find that $A_{CP}(B_c^+ \to K^{*+}D^{0}\to K^{0}\pi^+D^{0})=(-14.6_{-1.12}^{+9.19})\%$ is the largest one, which is possible measured by the present LHCb experiments. For the pure annihilation type decays, there is no CP violations because only the tree operators are involved. Furthermore, we also give the differential distributions of the branching ratios and the direct CP violations for the decays $B_c\to K^* D_{(s)}\to K \pi D_{(s)}$.