$CP$ violation for $B_c^{+}\to D_{(s)}^{+}{\pi }^{+}{\pi }^{-}$ in perturbative QCD

In the perturbative QCD (PQCD) approach, we study the direct $CP$ violation in the decay processes of $B_c^{+}\to D_{(s)}^{+}{\rho }^0(\omega )\to D_{(s)}^{+}{\pi }^{+}{\pi }^{-}$ via the $\rho -\omega$ mixing mechanism. We find that the $CP$ violation can be enhanced by $\rho -\omega$ mixing when the invariant masses of the ${\pi }^{+}{\pi }^{-}$ pairs are in the vicinity of the $\omega$ resonance. For the decay process of $B_c^{+}\to D^{+}{\rho }^0(\omega )\to D^{+}{\pi }^{+}{\pi }^{-}$, the maximum $CP$ violation can reach 7.33%.

Figure 1

The $CP$-violating asymmetry, $A_{cp}$, as a function of $\sqrt{s}$ for different CKM matrix elements. The dashed line, dotted line, and solid line correspond to the maximum, middle, and minimum CKM matrix elements for the decay channel of $B_c^{+}\to D^{+}{\pi }^{+}{\pi }^{-}$, respectively.

Figure 2

The $CP$-violating asymmetry, $A_{cp}$, as a function of $\sqrt{s}$ for different CKM matrix elements. The dashed line, dotted line, and solid line correspond to the maximum, middle, and minimum CKM matrix elements for the decay channel of $B_c^{+}\to D_s^{+}{\pi }^{+}{\pi }^{-}$, respectively.

Figure 3

Here, $\sin \delta$ as a function of $\sqrt{s}$ for different CKM matrix elements. The dashed line, dotted line, and solid line correspond to the maximum, middle, and minimum CKM matrix elements for the decay channel of $B_c^{+}\to D^{+}{\pi }^{+}{\pi }^{-}$, respectively.

Figure 4

Here, $\sin \delta$ as a function of $\sqrt{s}$ for different CKM matrix elements. The dashed line, dotted line, and solid line correspond to the maximum, middle, and minimum CKM matrix elements for the decay channel of $B_c^{+}\to D_s^{+}{\pi }^{+}{\pi }^{-}$, respectively.

Figure 5

Here, $r$ as a function of $\sqrt{s}$ for different CKM matrix elements. The dashed line, dotted line, and solid line correspond to the maximum, middle, and minimum CKM matrix elements for the decay channel of $B_c^{+}\to D^{+}{\pi }^{+}{\pi }^{-}$, respectively.

Figure 6

Here, $r$ as a function of $\sqrt{s}$ for different CKM matrix elements. The dashed line, dotted line, and solid line correspond to the maximum, middle, and minimum CKM matrix elements for the decay channel of $B_c^{+}\to D_s^{+}{\pi }^{+}{\pi }^{-}$, respectively.