CP violation analysis of $D^{0}\to M^{0}K\to M^{0}(\pi^{\pm}\ell^{\mp}\nu)$

The first case we focus on is $D^{0}(t_{1})\to\pi^{0}K(t_{2})\to\pi^{0}(\pi^{-}\ell^{+}\nu_{\ell})$, with its decay process depicted in the left panel of Fig. 1, where the spectator meson $\pi^{0}$ is omitted. The amplitudes in the first-level decay $D\to\pi^{0}K$ are interconnected through several parameters

where $\omega_{1,2}$ are the weak phases, $r$ is the magnitude ratio, and $\delta_{2}$ is the strong phase. Under the U-spin symmetry hypothesis, we have approximately $A(D^{0}\to\pi^{0}K^{0})/A(D^{0}\to\pi^{0}\bar{K}^{0})$ = $-\tan^{2}{\theta_{C}}$ CLEO:2007rhw ; Rosner:2006bw , where $\theta_{C}$ is the Cabibbo angle, which is also accepted by the factorization-assisted topological amplitude approach Li:2012cfa ; Qin:2013tje ; Qin:2021tve . Therefore, we adopt $r=\absolutevalue{(V_{cd}^{\star}V_{us})/(V_{ud}V_{cs}^{\star})}$ and $\delta_{2}=0$ for the numerical analysis, with the value of $r$ being provided in Table 1. We neglect the direct CP violation in $K^{0}$ decays, i.e., $\innerproduct{\pi^{-}\ell^{+}\nu_{\ell}}{K^{0}}=\innerproduct{\pi^{+}\ell^{-}\bar{\nu}_{\ell}}{\bar{K}^{0}}$. The CP violation associated with coefficients $C_{+}(t_{2})$ and $C_{-}(t_{2})$ is induced by the interference between amplitudes corresponding to all possible direct decay of the $D^{0}$ meson, $D^{0}\to K^{0}(\bar{K}^{0})$, and by the interference between amplitudes corresponding to all oscillation-mediated paths, $D^{0}\to\bar{D}^{0}\to K^{0}(\bar{K}^{0})$, respectively. They are calculated to be

where the sum of $C_{+,1}(t_{2})$ and $C_{+,2}(t_{2})$ equals to $C_{+}(t_{2})$. Based on the input parameter values in Table 1, we obtain $\sin{(\omega_{2}-\phi_{K})}\approx 3.7\times 10^{-3}$, $\sin{(\omega_{1}-\omega_{2}-\phi_{D})}\approx-6.4\times 10^{-4}$, $\sin{(\omega_{1}-\phi_{D}-\phi_{K})}\approx-3.1\times 10^{-3}$, $\sin{(\omega_{1}-2\omega_{2}-\phi_{D}+\phi_{K})}\approx 4.4\times 10^{-3}$ and $r^{2}\approx 2.8\times 10^{-3}$, which are of order $\mathcal{O}(10^{-3})$ or smaller. When multiplied with ${\rm{Re}}(\epsilon_{K})\sim 10^{-3}$ and ${\rm{Re}}(\epsilon_{D})\sim 10^{-4}$, their overall contributions are negligible. The contributions from $C_{+},C_{-}$, as well as the double-mixing CP violation terms $S_{h,\rm{dm}}$ and $S_{n,\rm{dm}}$ within $S_{h}$ and $S_{n}$ in Eq. (7), with coefficients of order $\mathcal{O}(10^{-5})$ or smaller, where the coefficients refer to the parts excluding time-evolution factors such as $\absolutevalue{g_{-,K}(t_{2})}^{2}$, are neglected, as these contributions are at least one order smaller than the leading contributions in $C_{+}$, $C_{-}$, $S_{h,{\rm{dm}}}$ and $S_{n,{\rm{dm}}}$. Therefore, the CP violation induced by $D^{0}\to\bar{D}^{0}\to\bar{K}^{0}\to K^{0}$, as well as the CP violation induced by the interference between $D^{0}\to K^{0}$ and $D^{0}\to\bar{D}^{0}\to\bar{K}^{0}\to K^{0}$ are neglected. For the non-double-mixing CP violation terms in $S_{h}$ and $S_{n}$, given their minimal contributions, no approximations are made to allow for a full analysis of their behavior. In Eq. (7), only the leading terms in the denominators are retained, with terms involving the aforementioned small quantities omitted.

The double-mixing CP violation terms $S_{h,\rm{dm}}$ and $S_{n,\rm{dm}}$ are calculated as

They are induced by the interference between $D^{0}\to\bar{K}^{0}\to K^{0}$ and $D^{0}\to\bar{D}^{0}\to K^{0}$, while contributions from the interference between $D^{0}\to K^{0}$ and $D^{0}\to\bar{D}^{0}\to\bar{K}^{0}\to K^{0}$ are neglected due to suppression by terms of $r^{2}$. The remaining components of $S_{n}(t_{2})$ and $S_{h}(t_{2})$ are given by

which are induced by the interference between $D^{0}\to\bar{D}^{0}\to K^{0}$ and $D^{0}\to K^{0}$, as well as the interference between $D^{0}\to\bar{K}^{0}\to K^{0}$ and $D^{0}\to\bar{D}^{0}\to\bar{K}^{0}\to K^{0}$. The terms in the denominator of Eq. (7) are given by

It is important to note that, according to Eq. (7), the CP violation corresponding to each contribution is defined as

To provide a more intuitive analysis, we performed a numerical analysis of the contributions to the total CP violation in this decay channel using the values listed in Table 1 as input parameters. The results are shown in Fig. 2. The left panel illustrates the two-dimensional time-dependent total CP violation, with its peak value exceeding $5\times 10^{-3}$, which occurs when $t_{2}$ is large. After integrating $A_{CP}(t_{1},t_{2})$ in Eq. (7) with respect to $t_{2}$ from 0 to $\tau_{K}$, where $\tau_{K}\equiv 1/\Gamma_{K}=2/(\Gamma_{K_{S}}+\Gamma_{K_{L}}),$ the resulting CP violation as a function of $t_{1}$ is depicted in the middle panel. Among the contributions to the total CP violation, $A_{C_{+}}$, which is proportional to $\absolutevalue{g_{+,D}(t_{1})}^{2}$, is the dominant term. The contribution to $A_{C_{+}}$ can be divided into two components. The first ($A_{C_{+,1}}$) refers to Eq. (20) where $C_{+}(t_{2})$ is replaced by $C_{+,1}(t_{2})$. This contribution arises from CP violation induced by the interference between $D^{0}\to K^{0}$ and $D^{0}\to\bar{K}^{0}\to K^{0}$, which is associated with the interference between the Cabibbo-favored and doubly-Cabibbo-suppressed amplitudes involving $K^{0}-\bar{K}^{0}$ mixing Yu:2017oky . The second one ($A_{C_{+,2}}$) refers to Eq. (20) where $C_{+}(t_{2})$ is replaced by $C_{+,2}(t_{2})$, and originates from CP violation directly induced by $D^{0}\to\bar{K}^{0}\to K^{0}$. The latter contribution is approximately five times larger than the former, with both of the order $\mathcal{O}(10^{-3})$. The dominant contribution, $A_{C_{+,2}}$, remains almost time-independent, as the denominator in Eq. (7) is primarily proportional to $\absolutevalue{g_{+,D}(t_{1})}^{2}$, resulting in the time-dependent factor associated with $t_{1}$ canceling out. Similarly, the sub-leading contribution, $A_{C_{+,1}}$, exhibits negligible time dependence for the same reason. The double-mixing CP violation $A_{h,\rm{dm}}$ increases linearly with time $t_{1}$, reaching $\mathcal{O}(10^{-4})$, which is one order of magnitude smaller than the sub-leading term. The double-mixing CP violation $A_{n,\rm{dm}}$ and other terms $A_{\rm{others}}$ are exceedingly small, on the order of $\mathcal{O}(10^{-6})$ and $\mathcal{O}(10^{-7})$, respectively. Integrating $t_{1}$ from 0 to $5\tau_{D}$, the right panel shows that when $t_{2}<0.1\tau_{K}$, the contributions from $A_{C_{+,1}}$ and $A_{C_{+,2}}$ are comparable ($\sim 10^{-3}$), with the former slightly larger than the latter. Subsequently, $A_{C_{+,2}}$ increases and then remains constant, while $A_{C_{+,1}}$ decreases and stabilizes. In the stable state, $A_{C_{+,2}}$ is approximately five times larger than $A_{C_{+,1}}$. The double-mixing CP violation $A_{h,\rm{dm}}$ is one order smaller than the two dominant contributions, but still two orders larger than $A_{n,\rm{dm}}$. This discrepancy arises because, although the magnitude of the coefficients in front of $\sin{x_{K}\Gamma_{K}t_{2}}$ and $\sinh{y_{K}\Gamma_{K}t_{2}}$ are nearly identical in both $A_{h,dm}$ and $A_{n,dm}$, the two terms have the same sign in $A_{h,dm}$, while they have opposite signs in $A_{n,dm}$. Both $A_{C_{-}}(\sim 10^{-7})$ and the other contributions, $A_{\rm{others}}(\sim 10^{-5})$, decrease over time.

We define the absolute branching ratio for $D^{0}(t_{1})\to\pi^{0}K(t_{2})\to\pi^{0}(\pi^{-}\ell^{+}\nu_{\ell})$ as

where the branching ratios $\mathcal{B}(D^{0}\to\pi^{0}K^{0}_{S})$ and $\mathcal{B}(K^{0}_{S}\to\pi^{+}\pi^{-})$ are 1.24% and 69.2% ParticleDataGroup:2024cfk , respectively. $A(t_{2})$ represents the fraction of $K^{0}_{S}$ decays that have occurred by time $t_{2}$, as defined in Eq. (44). The results of the integration over the two time parameters, $t_{1}$ and $t_{2}$, are shown in Table 2. The absolute branching ratio is sensitive to the integration range of the time parameter $t_{2}$ for $K$ meson, but nearly insensitive to the integration range of the time parameter $t_{1}$ for $D$ meson. The absolute branching ratio increases as the integration range of $t_{2}$ grows.

In contrast to the previous decay channel, the semi-leptonic final state of the neutral kaon is changed from $\pi^{-}\ell^{+}\nu_{\ell}$ to $\pi^{+}\ell^{-}\bar{\nu}_{\ell}$. The decay process for this channel is illustrated in the right panel of Fig. 1. By substituting Eq. (8) into Eq. (7), the time-dependent contributions to the overall CP violation as a function of $t_{2}$ are given by

Similar to the previous decay channel, terms with coefficients of $\mathcal{O}(10^{-5})$ or smaller are neglected, and thus the CP violation induced by $D^{0}\to K^{0}\to\bar{K}^{0}$ and $D^{0}\to\bar{D}^{0}\to\bar{K}^{0}$, as well as the CP violation incuded by the interference between $D^{0}\to\bar{D}^{0}\to\bar{K}^{0}$ and $D^{0}\to K^{0}\to\bar{K}^{0}$ are omitted. After disregarding negligible terms, $C_{+}(t_{2})$ originates from the interference between $D^{0}\to K^{0}\to\bar{K}^{0}$ and $D^{0}\to\bar{K}^{0}$, which is related to the effect discussed in Yu:2017oky , while $C_{-}(t_{2})$ arises from two distinct contributions. The first contribution is associated with interference between $D^{0}\to\bar{D}^{0}\to\bar{K}^{0}$ and $D^{0}\to\bar{D}^{0}\to K^{0}\to\bar{K}^{0}$. The second contribution is induced by the path $D^{0}\to\bar{D}^{0}\to K^{0}\to\bar{K}^{0}$. The $t_{2}$-dependent factors $S_{n,\rm{dm}}(t_{2})$ and $S_{h,\rm{dm}}(t_{2})$, which are associated with the double-mixing CP violation, are primarily induced by the interference between the paths $D^{0}\to\bar{D}^{0}\to K^{0}\to\bar{K}^{0}$ and $D^{0}\to\bar{K}^{0}$. In these terms, the double-mixing of $D^{0}$ and $K^{0}$ manifests in the expression through the combination of the mixing angles $\phi_{D}+\phi_{K}$. The terms in the denominator of Eq. (7) are given by