Study of CP Violation in $D^{\pm}\rightarrow K^{\ast}(892)^{0} \pi^{\pm} + \bar{K}^{\ast }(892)^{0}\pi^{\pm}\rightarrow K_{S,L}^{0}\pi^0 \pi^{\pm}$ Decays

Within the Standard Model, we investigate the CP violations and the $K_S^0-K_L^0$ asymmetries in $D^{\pm}\rightarrow K^{\ast}(892)^{0} \pi^{\pm} + \bar{K}^{\ast }(892)^{0}\pi^{\pm}\rightarrow K_{S,L}^{0}\pi^0 \pi^{\pm}$ decays basing on the factorization-assisted topological-amplitude (FAT) approach and the topological amplitude (TA) approach of Cheng and Chiang [Phys. Rev. D 104, 073003 (2021).]. We find that the CP violations in these decays $A_{CP}^{K_{S,L}^0}$ can exceed the order of $10^{-3}$ in the two approaches and consist of three parts: the indirect CP violations in $K^0 -\bar{K}^0$ mixing $A_{CP,K_{S,L}^0}^{mix}$, the direct CP violations in charm decays $A_{CP,K_{S,L}^0}^{dir}$ and the new CP violation effects $A_{CP,K_{S,L}^0}^{int}$, which are induced from the interference between two tree (Cabibbo-favored and doubly Cabibbo-suppressed) amplitudes with the neutral kaon mixing. The indirect CP violations in $K^0 -\bar{K}^0$ mixing play a dominant role in $A_{CP}^{K_{S,L}^0}$, the new CP violation effects have a non-negligible contribution to $A_{CP}^{K_{S,L}^0}$. We estimate the numerical results of the $K_S^0-K_L^0$ asymmetries $R_{K_{S}-K_{L}}^{D^{\pm}}$ and find that there exist a large difference between the numerical results in the FAT approach and that of the TA approach. We also find that if ones adopt the values of the decay time parameters $t_0 = 3.0 \tau_S$ and $t_1 = 10.0 \tau_S$, the new CP violation effect $A_{CP,K_{S}^0}^{int}$ would dominate the CP violation in $D^{\pm} \rightarrow K^{\ast}(892)^{0} \pi^{\pm} + \bar{K}^{\ast }(892)^{0}\pi^{\pm}\rightarrow K_{S}^0\pi^0 \pi^{\pm}$ decays and could be observed with $6.7\times 10^{6}$ and $6.5\times 10^{6}$ $D^{\pm}$ events-times-efficiency in the FAT approach and the TA approach, respectively. Our results could be tested by the LHCb, Belle II and BESIII experiments.


Introduction
The exploration of CP violation is one of the main topics in particle physics and cosmology, heavy flavor meson decays provide an ideal place to study CP violation. In the Standard Model (SM), CP violation is due to a complex parameter in the Cabibbo-Kobayashi-Maskawa (CKM) matrix. However, the strength of CP violation predicted by the Standard Model is insufficient to explain the baryon asymmetry of the unverse [1,2], so it is necessary to search for new sources of CP violation. It is important to investigate as many systems as possible, to see the correlation between different processes and understand the origin of CP violation.
CP violation in Kaon and B meson systems has been well established, but not yet in charmed meson decays. In 2019, the LHCb collaboration reported the first confirmed observation of the CP asymmetries in charm sector via measuring the difference of time-integrated CP asymmetries of D 0 → K + K − and D 0 → π + π − decays with a significance of more than 5σ [3].
Combining the LHCb results in 2014 [4], 2016 [5] and 2019 [3] leads to a result of a nonzero value of ∆A CP In recent years, there are a number of theoretical works, which concentrate on studying the CP violations in the charm sector . Charmed meson decays become one of the most important platforms for studying the CP violation and its origin.
The decays with final states including K 0 S or K 0 L can be used to study CP violation . In these decays, the indirect CP violation induced by the K 0 −K 0 mixing has a non-negligible effect, even plays a dominant role. There exist 2.8σ discrepancy observed between the BaBar measurement and the SM prediction of the CP asymmetry in the τ + → π + K Sντ decay [59][60][61], this may imply the existence of the physics beyond the SM because of the absence of the direct CP violation in this decay. However, no unambiguous conclusion can be drawn due to the large uncertainty [63], so more precise data and more reactions with final states including K 0 S or K 0 L are needed in both experiment and theory.
In Ref. [51], the authors study the CP asymmetries in the D ± → K 0 S π ± decays, they show that besides the indirect CP violation due to the K 0 −K 0 mixing, a new CP violation effect induced by the interference between the Cabibbo-favored (CF) and doubly Cabibbo- diagram, which are displayed in Fig 1, the DCS channels D + → K * 0 π + and D − →K * 0 π − can occur through the color-suppressed internal W-emission tree diagram and the W-annihilation diagram, which are displayed in Fig 2. Here, all diagrams meant to have all the strong interactions included, i.e., gluon lines are included implicitly in all possible ways [75]. The effective Hamiltonian relevant to the D ± → K * 0 (K * 0 )π ± decays is given by where G F is the Fermi coupling constant, V qq is the corresponding CKM matrix element, α and β are the color indices, (qq ) V −A representsqγ µ (1 − γ 5 )q . C 1 (µ) and C 2 (µ) are the Wilson coefficients, the evolutions of these Wilson coefficients in the scale µ are given in Ref. [14]. For convenience, we duplicate these explicit expressions in Appendix A. Based on the topological amplitude approach [76,77], the decay amplitudes of the diagrams in parameterized as where the subscript T V in Eq.(3) and Eq.(4) denotes that the decay amplitude is the colorallowed external W-emission tree diagram amplitude with the D + →K * 0 and D − → K * 0 transitions, the subscript C P in Eqs.(5)- (8) represents that the decay amplitude is the color-suppressed internal W-emission tree diagram amplitude with the D ± → π ± transitions, the subscript A V in Eq. (9) and Eq.(10) denotes that the decay amplitude is the W-annihilation diagram amplitude with the s (ors) quark from the weak decay entering in theK * 0 (or K * 0 ) meson. In our calculation, we based on the results of two topological amplitude approaches: the factorization-assisted topological-amplitude (FAT) approach and the topological amplitude approach of Ref. [40] (hereinafter for brevity referred to as the TA approach of Ref. [40]). In the FAT approach, the topological amplitudes can be expressed as [14,23,78] T 0 where m K * , m π + and m D + is the mass of the meson K * 0 , π + and D + , respectively. f π + , f K * , f ρ and f D + is the decay constant of the meson π + , K * 0 , ρ and D + , respectively. ε (we denote ε ≡ ε(p K * , λ) for simplicity) is the polarization vector of the K * 0 meson, it yields the following relations [40] ε µ (p K * , λ)p µ K * = 0, The effective Wilson coefficients α V 1 and α P 2 in Eq. (11) and Eq.(12)are χ A q , φ A q , χ C P and φ C P in Eq. (13) and Eq.(16) are the non-factorizable parameters, e iSπ in Eq.(13) is a strong phase factor which is introduced for each pion involved in the non-factorizable contributions of the W-annihilation diagram amplitude. We note that the parameters χ A q , φ A q , χ C P , φ C P and S π are free and universal, they can be determined by fitting the data. µ A , µ T and µ C in Eq. (13) and Eq.(16) is, respectively, the scale for the W-annihilation diagram, the color-allowed external W-emission tree diagram and the color-suppressed internal W-emission tree diagram [23,78] with Λ represents the momentum of the soft degree of freedom in the D decays, fixed to be Λ = 0.5GeV in this work. f + (p 2 K * ) in Eq. (12) is the D ± → π ± transition form factor, which can be written as (11) is the D ± →K * 0 transition form factor, which can be written as [79][80][81] (20) with q = p D + − p K * and There exist many model and lattice calculations for D ± to π ± , K * 0 transition form factors. In this paper, we shall use the following parametrization for form-factor q 2 dependence [33,75,82,83] where for the form factor f + (q 2 ), m pole = m D * (2010) + , F (0) = 0.666 and α = 0.24, while for the form factor A 0 (q 2 ), m pole = m D + s F (0) = 0.78 and α = 0.24. In the TA approach of Ref. [40], basing on the solution (S3') of the fitting result in Table   II of Ref. [40], we can obtain the following numerical results of the topological amplitudes |A 0 V | = 0.028 ± 0.002, Figure 3: Resonant contribution to the amplitudes of D ± → π ± K 0 π 0 and D ± → π ±K0 π 0 through the intermediate states K * 0 andK * 0 , where the blob stands for a transition due to weak interactions.
here, we note that the values of |T 0 V |, |C 0 P | and |A 0 V | are obtained by the products of the values of the corresponding topological amplitudes in Table II of Ref. [40] and √ 2/G F , the values of Table II of Ref. [40]. In the overlapped region of the K * 0 andK * 0 resonances, the decay amplitudes of the cascade decays D ± → K * 0 π ± → K 0 π 0 π ± and D ± →K * 0 π ± →K 0 π 0 π ± , which are depicted in Fig 3, can be written as A(D + →K * 0 π + →K 0 π 0 π + ) = with the Lagrangian [40] and the relativistic Breit-Wigner line shape for K * 0 where Γ K * (s) is the mass dependent width of K * 0 with q K 0 denotes the c.m. momentum of K 0 in the rest frame of K * , q 0 K 0 is the value of q K 0 when s is equal to m 2 K * , where the function f is In Eq.(32), Γ 0 K * is the nominal total width of K * with Γ 0 with r BW ≈ 4.0GeV −1 .
Using the Lagrangian in Eq.(30), one obtains where g K * 0 →K 0 π 0 is the coupling of K * 0 to K 0 π 0 , which can be extracted from Substituting Eq. (7), Eq.(9), Eq.(31) and Eq.(36) into Eq.(26), we can obtain the decay amplitude of the cascade decay In order to account for the off-shell effect of K * , we follow Ref. [40,84] to add a form factor F ( √ s, m K * ) into the above equation, the form factor F ( √ s, m K * ) can be parameterized as with the cutoff Λ not far from the mass of the resonance K * , where β = 1.0 ± 0.2 and Λ QCD = 0.25GeV.
From Eq. (14) and Eq.(15), we obtain in the rest frame of K 0 and π 0 . Substituting the above equation into Eq.(39), we obtain Similarly, we can obtain the following amplitudes 2.2 the effect of the K 0 −K 0 mixing Now, we proceed to study the time evolution of the initially-pure K 0 (K 0 ) states. In the K 0 −K 0 system, the two mass eigenstates, K 0 S of mass m S and width Γ S and K 0 L of mass m L and width Γ L , are linear combinations of the flavor eigenstates K 0 andK 0 . Under the assumption of CPT invariance, these mass eigenstates can be expressed as [85] K 0 where p and q are complex mixing parameters. CP conservation requires both p = q = √ 2/2.
The mass and width eigenstates, K 0 S,L , may also be described with the popular notations where the complex parameter signifies deviation of the mass eigenstates from the CP eigenstates. The parameters p and q can be expressed in terms of Combining Eq. (47), Eq. (48), Eq.(51) and neglecting the tiny direct CP asymmetry in the The time-evolved states of the K 0 −K 0 system can be expressed by the mass eigenstates Using Eq.(54) and Eq.(55), the time-dependent amplitudes of the cascade decays D ± → K * 0 π ± +K * 0 π ± → K 0 (t)π 0 π ± +K 0 (t)π 0 π ± → f K 0 π 0 π ± (hereinafter for brevity referred to as can be written as where f K 0 denotes the final state from the decay of the K 0 orK 0 meson.

respectively. They have the following forms
For convenience, we introduce the following substitutions where r sf , r wf and r f are positive numbers, r f denotes the magnitude of the ratio of the DCS amplitude to the CF amplitude, δ and φ is the strong phase difference and the weak phase difference, respectively. Making use of Eqs. (40), (43), (44), (56) and (59) and performing integration over phase space, we can obtain where we use the following substitutions p 2 0 and p 2 1 in Eq.(60) is the lower bound and the upper bound of p 2 K * , respectively. In order to select the K * event and suppress the background, we adopt p 2 0 = (m K * − 3Γ 0 K * ) 2 and p 2 1 = (m K * + 3Γ 0 K * ) 2 in our calculation, where m K * and Γ 0 K * is the mass and decay width of the K * resonance, respectively.
Similarly, we can derive the decay width for the 2.3 the decay widths for the D ± → K * 0 π ± +K * 0 π ± → K 0 S,L π 0 π ± decays In experiment, the K 0 S state is defined via a final state π + π − with m ππ ≈ m S and a time difference between the D ± decay and the K 0 S decay [59,86,87]. By taking into account these experimental features, the partial decay width for the D + → K * 0 π + +K * 0 π + → K 0 S π 0 π + (hereinafter for brevity referred to as D + → K * π + → K 0 S π 0 π + ) decay can be defined as where t 0 = 0.1τ S and t 1 = 2τ S ∼ 20τ S with τ S is the K 0 S lifetime, we adopt t 1 = 10τ S in our calculation. Combining Eq.(52), Eq.(57), Eq.(58), Eq.(60) and Eq. (65), we can obtain with where The terms in the square brackets of Eq.(67)- (69) are related to the effect of the K 0 S decay, the effect of the K 0 L decay and their interference, respectively. From Particle Data Group [85],we can obtain Similarly, we can derive the decay width for the In experiment, the K 0 L state is defined via a large time difference between the D ± decay and the K 0 L decay, so the K 0 L states mostly decay outside the detector [88]. Basing on these experimental features, the partial decay width for the D + → K * 0 π + +K * 0 π + → K 0 L π 0 π + (hereinafter for brevity referred to as D + → K * π + → K 0 L π 0 π + ) decay can be defined as where t 2 ≥ 100τ S . Using Eq.(53), Eq.(57), Eq.(58), Eq.(60) and Eq.(75), we can derive with where Using the result from Particle Data Group [85]: with t 2 ≥ 100/Γ S , so the last two terms in the square brackets of Eq.
In the D ± → K * 0 π ± +K * 0 π ± → K 0 S,L π 0 π ± decays, the time-independent CP violation observables are defined as Substituting Eq. (73) and Eq. (74) into Eq. (84), we can derive where A mix CP,K 0 S denotes the CP violation in kaon mixing [51,59], the two terms in the square bracket of Eq.(86) correspond to the pure K 0 S term and the K 0 L − K 0 S interference term, respectively. The K 0 L − K 0 S interference term, which is a function of t 0 and t 1 , is as important as the pure K 0 S term [59]. A dir CP,K 0 S denotes the direct CP asymmetry induced by the interference between the tree level CF and DCS amplitudes. A int CP,K 0 S represents a new CP violating effect, which relates to the following expression i.e., this new CP violating effect arises from the interference between two tree (CF and DCS) amplitudes with the neutral kaon mixing [51,89,90]. Here, we also note that the K 0 L has a large contribution to the new CP violating effect, as shown in Eq. (88). In our calculation, we adopt t 0 = 0.1τ S and t 1 = 10τ S . In addition, we will discuss the impact of the choice of t 0 on A mix Similarly, substituting Eq.(82) and Eq.(83) into Eq.(84), we can derive the expression for CP asymmetry in the D ± → K * 0 π ± +K * 0 π ± → K 0 L π 0 π ± decays where A mix , one can find that all CP violation effects in the D ± → K * 0 π ± +K * 0 π ± → K 0 L π 0 π ± decays receive no contribution from the K 0 L − K 0 S interference and are independent of the decay time t 2 .

K 0
S − K 0 L asymmetries in the D ± → K * 0 π ± +K * 0 π ± → K 0 S,L π 0 π ± decays The K 0 S − K 0 L asymmetries in the D meson decays are induced by the interference between the CF and DCS amplitudes, which was first pointed out by Bigi and Yamamoto [91]. The determination on the K 0 S − K 0 L asymmetries in the D meson decays can be useful to study the DCS processes and understand the dynamics of charm decay [78,92]. In the D ± → K * 0 π ± + K * 0 π ± → K 0 S,L π 0 π ± decays, the K 0 S − K 0 L asymmetries are defined by Using Eq.(73), Eq.(82) and Eq.(94), we can obtain with From the above equation, we can see that the main contribution to R D + K S −K L come from the pure K 0 S and K 0 L decay, the contribution from the K 0 L is small because of the suppression of the parameter . Similarly, combining Eq.(74), Eq. (83) and Eq.(95), we can derive the expression for K 0 with According to the definition of the weak phase difference in Eq.(59), we have sin φ = O(10 −3 ) and cos φ ≈ 1. Hence as a good approximation, cos(φ ± δ) ≈ cos δ and sin(φ ± δ) ≈ ± sin δ.
Therefore, the determinations of R D + K S −K L and R D − K S −K L are useful to understand the strong phase difference between the DCS and CF amplitudes [78].
In order to see physics more transparently, we use the Wolfenstein parametrization of the CKM matrix elements, which imaginary part satisfy the unitarity relation to order λ 5 [85, 99-101]  Table 1: The values of the branching ratios for the D ± → K * π ± → K 0 S,L π 0 π ± decays in the FAT approach and the TA approach of Ref. [40].
observables the FAT approach the TA approach of Ref. [40] B By substituting the values of the parameters listed above into Eqs. (73), (74), (82) and (83), we can obtain the numerical values of the branching ratios, which are shown in Table 1.
Here, the results in the last two lines of Table 1 are the averaged branching ratios of the decay and its charge conjugate. The results given in Table 1 are consistent with the experimental measurement of B(D + → K * π + → K 0 S π 0 π + ) = (2.64 ± 0.32) × 10 −3 from BESIII [85,103]. We also note that the reasons for the differences between the results of the FAT approach and that of the TA approach of Ref. [40] are the small values of cos δ and | (C 0 P + T 0 V ) | 2 in the TA approach of Ref. [40].

The numerical results of the CP asymmetries
Now, we move on to calculate the numerical results of the CP asymmetries in D ± → K * 0 π ± + K * 0 π ± → K 0 S,L π 0 π ± decays. By substituting the values of the parameters in Eqs.(100), (102) and (104) into Eqs.(85)- (88) and Eqs.(90)-(93), we can obtain the numerical results of the CP asymmetries in D ± → K * 0 π ± +K * 0 π ± → K 0 S,L π 0 π ± decays, which are shown in Table 2. From these numerical values, we can obtain the following points: L ) is more than twice of that from the pure K 0 S decay term Re( ) in A mix CP,K 0 S , and they interfere destructively.
2. The direct CP asymmetry A dir CP,K 0 S suffer from both the r wf and sin φ suppression, thus its numerical value is small. Table 2: The values of the CP asymmetries in the D ± → K * π ± → K 0 S,L π 0 π ± decays in the FAT approach and the TA approach of Ref. [40].
observables the FAT approach the TA approach of Ref. [40] A mix 3. The value of r sf and sin δ vary from 2.49 to 2.97 and from −0.91 to −0.57 in the integral interval of p 2 K * in the FAT approach, respectively. In the TA approach of Ref. [40], the value of r sf and sin δ is 2.42 and −0.99, respectively, so the new CP violation effect A int CP,K 0 S only suffer from the r wf suppression relative to the indirect CP violation in K 0 −K 0 mixing, as shown in Eq. (86) and Eq. (88). Moreover, the pure K 0 S decay term Im( ) and the K 0 , all these reasons result in a non-negligible contribution of the new CP violation effect to the CP asymmetry in D ± → K * 0 π ± +K * 0 π ± → K 0 S π 0 π ± decays.
4. The value of r f and cos δ vary from 0.13 to 0.16 and from 0.42 to 0.82 in the integral interval of p 2 K * in the FAT approach, respectively, however, the value of r f and cos δ is 0.13 and −0.13 in the TA approach of Ref. [40], respectively. 5. Basing on the numerical values of sinδ and cos δ in the FAT approach and the TA approach of Ref. [40] and according to the expressions for CP asymmetries in Eqs.(85)-(88) and Eqs.(90)-(93), we can derive that the large value of | sin δ| and the negative value of cos δ in the TA approach of Ref. [40] result in the differences between the numerical values of the CP asymmetries in the FAT approach and that in the TA approach of Ref. [40].
According to the numerical results of the CP asymmetries in D ± → K * 0 π ± +K * 0 π ± → K 0 S,L π 0 π ± decays, we can estimate that how many D ± events-times-efficiency are needed to establish the CP asymmetries to three standard deviations (3σ). When the CP violations are observed at three standard deviations (3σ) level, the numbers of D ± events-times-efficiency needed read [104][105][106] where f K 0 S and f K 0 L denotes π + π − and π + π − π 0 , respectively. Combining Eq.(101), Eq.(105) and the numerical results of the branching ratios and the CP asymmetries in Table 1 and Table 2, we can obtain the FAT approach, (5.5 ∼ 6.7) × 10 5 , the TA approach of Ref. [40].
Similarly, substituting Eq.(101) and the numerical results of the branching ratios and the CP asymmetries in Table 1 and Table 2 into Eq. (105), we have Basing on these numerical values, we can obtain the following points: 1. From the Eqs.(96)-(99), we can see that the K 0 S − K 0 L asymmetries R D ± K S −K L only suffer from the r wf suppression, so they have a large value, which indicate that there exist a large difference between the branching ratios of D ± → K * 0 π ± +K * 0 π ± → K 0 S π 0 π ± and the branching ratios of D ± → K * 0 π ± +K * 0 π ± → K 0 L π 0 π ± .
2. The numerical results of R D ± K S −K L in the FAT approach are many times (about 5 times for R D + K S −K L and about 6 times for R D − K S −K L ) larger than that in the TA approach of Ref. [40], moreover, the signs of R D ± K S −K L in these two approaches are opposite to each other, the reason is that the values of cos δ are different in these two approaches. In addition, the in the TA approach of Ref. [40].
3. The measurement of R D ± K S −K L can help to discriminate the FAT approach and the TA approach of Ref. [40].
In the same way as the CP asymmetries in D ± → K * 0 π ± +K * 0 π ± → K 0 S,L π 0 π ± decays, the numbers of D ± events-times-efficiency needed for observing the K 0 S − K 0 L asymmetries at three standard deviations (3σ) level are Using the numerical results of the branching ratios in Table 1 Similarly, using the numerical results of the branching ratios in Table 1, Eq.(109) and Eq.(110), we have the TA approach of Ref. [40]. (112)

the observation of the new CP violation effect
In this section, we will study to observe the new CP violation effect in the D ± → K * 0 π ± + K * 0 π ± → K 0 S π 0 π ± decays. As discussed in section 3.1, the CP violation in the D ± → K * 0 π ± + K * 0 π ± → K 0 S π 0 π ± decays A . Moreover, the CP violation in the D ± → K * 0 π ± +K * 0 π ± → K 0 S π 0 π ± decays is dominated by the indirect CP violation in K 0 −K 0 mixing, which is shown in Table 2, all these make the observation of the new CP violation effect more difficulty. Now, it is important to note the following features of the three parts of the CP violation in the D ± → K * 0 π ± +K * 0 π ± → K 0 S π 0 π ± decays: 2. The K 0 L − K 0 S interference term is the function of the decay time parameters t 0 and t 1 , we adopt t 0 = 0.1τ S and t 1 = 10τ S in our above calculation.
3. As discussed in section 4.2, the contributions from the K 0 would dominate the CP violation in the D ± → K * 0 π ± +K * 0 π ± → K 0 S π 0 π ± decays, the observation of the new CP violation effect become possible.
According the Eqs.(85)-(88), we calculate the dependence of A mix CP on the selection of t 0 in the FAT approach and the TA approach of Ref. [40], which is shown in Fig. 4. Here, we note that we still adopt t 1 = 10τ S in the calculations. It can be seen from CP on the selection of t 0 with t 1 = 10/Γ S : (a)in the FAT approach, (b)in the TA approach of Ref. [40].
plays a dominant pole in the CP violation in the D ± → K * 0 π ± +K * 0 π ± → K 0 S π 0 π ± decays A K 0 S CP at some values of t 0 . For example, when t 0 = 3.0τ S , we have in the the FAT approach and in the TA approach of Ref. [40]. Obviously, if we adopt t 0 = 3.0τ S and t 1 = 10.0τ S , the new CP violation effect A int CP,K 0 S is possible to be observed.
However, the method mentioned above has a drawback: if we adopt t 0 = 3.0τ S and t 1 = 10.0τ S , we would lost a lot of the D ± → K * 0 π ± +K * 0 π ± → K 0 S π 0 π ± event, The reason is that the decay of a K 0 S meson to final state π + π − occurs mainly at time less than 5τ S and the decay-rate of K 0 S meson decrease rapidly with time. The event selection efficiency of t 0 = 3.0τ S and t 1 = 10.0τ S can be written as substituting Eq.(60) and Eq.(64) into Eq.(121) and using the values of the parameters in Eqs. (100), (102) and (104), we can obtain the numerical result of t 0 where the above result is the averaged efficiency of the decay and its charge conjugate. So, if the CP violations in Eq.(116) and Eq.(120) are observed at three standard deviations (3σ) level, the numbers of D ± events-times-efficiency needed read substituting Eq. (101), Eq.(116), Eq.(120), Eq.(122) and the numerical results of the branching ratios in Table 1 into Eq.(123), we can obtain the FAT approach, (5.2 ∼ 6.5) × 10 6 , the TA approach of Ref. [40]. (124) where f is the selection efficiency in experiment, it don't contain t 0 . In a word, if ones adopt the scenario t 0 = 3.0τ S and t 1 = 10.0τ S and want to observe the new CP violation effect A int CP,K 0 S in D ± → K * 0 π ± +K * 0 π ± → K 0 S π 0 π ± decays, the number of D ± events-times-efficiency needed is (5.1 ∼ 6.7)×10 6 and (5.2 ∼ 6.5)×10 6 in the FAT approach and the TA approach of Ref. [40], respectively.
because the K 0 S − K 0 L asymmetries R D ± K S −K L only suffer from the r wf suppression, so they have a large value, which indicate that there exist a large difference between the branching ratios of D ± → K * 0 π ± +K * 0 π ± → K 0 S π 0 π ± and the branching ratios of D ± → K * 0 π ± +K * 0 π ± → K 0 L π 0 π ± . In addition, Because the values of cos δ are different in the FAT approach and the TA approach of Ref. [40], the numerical results of R D ± K S −K L in the FAT approach are many times (about 5 times for R D + K S −K L and about 6 times for R D − K S −K L ) larger than that in the TA approach of Ref. [40], moreover, the signs of R D ± K S −K L in these two approaches are opposite to each other. Basing on the FAT approach, we estimate that the range of the numbers of D ± events-timesefficiency needed for observing the K 0 S − K 0 L asymmetries at three standard deviations (3σ) level is from 0.8 × 10 4 to 1.0 × 10 4 both for the D + → K * 0 π + +K * 0 π + → K 0 S,L π 0 π + decays and for the D − → K * 0 π − +K * 0 π − → K 0 S,L π 0 π − decays. In the the TA approach of Ref. [40], we derive that the range of the numbers of D ± events-times-efficiency needed for observing the K 0 S − K 0 L asymmetries at three standard deviations (3σ) level is 3.8 × 10 4 ∼ 7.8 × 10 4 for the D + → K * 0 π + +K * 0 π + → K 0 S,L π 0 π + decays and 0.5 × 10 5 ∼ 1.2 × 10 5 for the D − → K * 0 π − +K * 0 π − → K 0 S,L π 0 π − decays. We also investigate the possibility to observe the new CP violation effect A int CP,K 0 S in the D ± → K * 0 π ± +K * 0 π ± → K 0 S π 0 π ± decays in the FAT approach and the TA approach of Ref. [40]. We find that the new CP violation effect can dominate the CP violation in the D ± → K * 0 π ± +K * 0 π ± → K 0 S π 0 π ± decays when the scenario with t 0 = 3.0τ S and t 1 = 10.0τ S is adopted. However, the observation of the new CP violation effect A int CP,K 0 S in the above mentioned scenario is at the expense of the loss of the event selection efficiency. If the clean signal of the new CP violation effect A int CP,K 0 S is established, the number of D ± events-timesefficiency needed is 6.7 × 10 6 and 6.5 × 10 6 in the FAT approach and the TA approach of Ref. [40], respectively. for µ < m c .