Theory

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The only known source of both parity and charge conjugation ($CP$) violation in the Standard Model (SM) of particle physics is described by the unitary 3x3 Cabibbo-Kobayashi-Maskawa (CKM) matrix. The study of $CP$ violation is important as it is necessary for the generation of matter-antimatter asymmetry in the early Universe. The weak decays of hadrons are governed by transition amplitudes $A_f$, proportional to elements of the CKM matrix. A different decay rate of a hadron, such as the $B^+$ meson, compared to its antiparticle is possible if there exists interference of at least two quark-level transition processes.

Consider the $B\rightarrow f$ transition amplitude $A_f$, together with its $CP$ conjugate decay process with amplitude $\bar{A}_\bar{f}$ . Two types of phases can appear in these decay amplitudes. $CP$ violating phases do change sign under $CP$ conjugation. In the SM, these phases are known to appear only in interactions including the $W^{\pm}$ bosons, hence given the name weak phases. Moreover, a second type of phase can also appear, called strong phase; they are generated by $CP$ -invariant interactions and are the same in both $A_f$ and $\bar{A}_\bar{f}$. For example, consider a two particle decay

$$A_f = \|a_1\|\mathrm{e}^{i(\delta_1+\phi_1)} + \|a_2\|\mathrm{e}^{i(\delta_2+\phi_2)}$$ $$\bar{A}_\bar{f} = \|a_1\|\mathrm{e}^{i(\delta_1-\phi_1)} + \|a_2\|\mathrm{e}^{i(\delta_2-\phi_2)}$$

where $\delta$ is the strong phase, $\phi$ is the weak phase, and $a$ is the magnitude of the contribution to the amplitude $A_f$. Furthermore, the $CP$ asymmetry in the decay of a $B^+$ meson\footnote{Charge conjugation is implied throughout, unless otherwise stated.} to two charmed mesons is defined as

$$\mathcal{A}^\mathrm{CP} \equiv \frac{\Gamma(B^+\rightarrow D_{(s)}^{(*)+}\bar{D}^0) - \Gamma(B^-\rightarrow D_{(s)}^{(*)-}D^0)}{\Gamma(B^+\rightarrow D_{(s)}^{(*)+}\bar{D}^0) + \Gamma(B^-\rightarrow D_{(s)}^{(*)-}D^0)}$$

where $\Gamma$ is the decay rate of the respective channel. Therefore, by taking $|A_f|^2 = \Gamma$, the $CP$ asymmetry in $B^\pm$ decays is written in terms of strong and weak phases as

$$\mathcal{A}^\mathrm{CP} = -\frac{2|a_1 a_2|\sin(\delta_2-\delta_1)\sin(\phi_2 - \phi_1)}{|a_1|^2 + |a_2|^2 + 2|a_1a_2|\cos(\delta_2-\delta_1)\cos(\phi_2 - \phi_1)}.$$

Nonzero $CP$ asymmetries are expected in the SM due to interference between the tree-level, loop-level, and annihilation type Feynman diagrams, as illustrated in Fig. 1. However, the SM predicts a $CP$ violation of $\mathcal{O}(1\%)$. New physics contributions, e.g., supersymmetric ones , can enhance the $CP$ asymmetry in the measured $B^+$ decays by up to 10%.

The main contributions to the studied decays are from tree-level and loop-level amplitudes, shown in . This paper presents the measurement of the $CP$ asymmetry in the $B^+\rightarrow D^{(*)+}\bar{D}^0$ and $B^+\rightarrow D_s^+\bar{D}^0$ decay schemes using LHCb data from both LHC runs. The Run I data corresponds to a luminosity of 3\,fb$^{-1}$, taken from 2011 to 2012 at $\sqrt{s} = 7$ or 8 TeV. The Run II data corresponds to 6.0\,fb$^{-1}$, taken in the years 2015 to 2018 at $\sqrt{s} = 13$ TeV. The studied $D$ mesons are reconstructed from the following decay chains: $\bar{D}^0\rightarrow K^+\pi^-$, $\bar{D}^0\rightarrow K^+\pi^-\pi^+\pi^-$, $D^+\rightarrow K^-\pi^+\pi^+$, and $D^+_s\rightarrow K^+K^-\pi^+$. The $D^{*+}$ de-excites by emitting a soft pion $\pi^+_s$ in $D^{*+}\rightarrow D^0\pi^+_s$.