Analysis of $CP$ violation in $ D^0 \to K^+ K^- \pi^0 $

We study the $\mathit{CP}$ violation induced by the interference between two intermediate resonances $K^*(892)^+$ and $K^*(892)^-$ in the phase space of singly-Cabibbo-suppressed decay $D^0 \to K^+K^-\pi^0$. We adopt the factorization-assisted topological approach in dealing with the decay amplitudes of $D^0 \to K^\pm K^*(892)^\mp$. The $\mathit{CP}$ asymmetries of two-body decays are predicted to be very tiny, which are $(-1.27 \pm 0.25) \times 10^{-5}$ and $(3.86 \pm 0.26) \times 10^{-5}$ respectively for $D^0 \to K^+ K^*(892)^-$ and $D^0 \to K^- K^*(892)^+$. While the differential $\mathit{CP}$ asymmetry of $D^0 \to K^+K^-\pi^0$ is enhanced because of the interference between the two intermediate resonances, which can reach as large as $3\times10^{-4}$. For some NPs which have considerable impacts on the chromomagnetic dipole operator $O_{8g}$, the global $\mathit{CP}$ asymmetries of $D^0 \to K^+ K^*(892)^- $ and $D^0 \to K^- K^*(892)^+ $ can be then increased to $(0.56\pm0.08)\times10^{-3}$ and $(-0.50\pm0.04)\times10^{-3}$, respectively. The regional $\mathit{CP}$ asymmetry in the overlapped region of the phase space can be as large as $( 1.3\pm 0.3)\times10^{-3}$.


Introduction
Charge-Parity () violation, which was first discovered in  meson system in 1964 [1], is one of the most important phenomena in particle physics.In the Standard Model (SM),  violation originates from the weak phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [2,3] and the unitary phases which usually arise from strong interactions.One reason for the smallness of  violation is that the unitary phase is usually small.Nevertheless,  violation can be enhanced in three-body decays of heavy hadrons, when the corresponding decay amplitudes are dominated by overlapped intermediate resonances in certain regions of phase space.Owing to the overlapping, a regional  asymmetry can be generated by a relative strong phase between amplitudes corresponding to different resonances.This relative strong phase has nonperturbative origin.As a result, the regional  asymmetry can be larger than the global one.In fact, such kind of enhanced  violation has been observed in several threebody decay channels of  meson [4][5][6][7], which was followed by a number of theoretical works [8][9][10][11][12][13][14][15][16][17][18][19].
The study of  violation in singly-Cabibbo-suppressed (SCS)  meson decays provides an ideal test of the SM and exploration of New Physics (NP) [20][21][22][23].In the SM,  violation is predicted to be very small in charm system.Experimental researches have shown that there is no significant  violation so far in charmed hadron decays [24][25][26][27][28][29][30][31][32][33]. asymmetry in SCS  meson decay can be as small as or even less, due to the suppression of the penguin diagrams by the CKM matrix as well as the smallness of Wilson coefficients in penguin amplitudes.The SCS decays are sensitive to new contributions to the Δ = 1 QCD penguin and chromomagnetic dipole operators, while such contributions can affect neither the Cabibbo-favored (CF) ( → ) nor the doubly-Cabibbo-suppressed (DCS) ( → ) decays [34].
Besides, the decays of charmed mesons offer a unique opportunity to probe  violation in the up-type quark sector.Several factorization approaches have been wildly used in nonleptonic  decays.In the naive factorization approach [35,36], the hadronic matrix elements were expressed as a product of a heavy to light transition form factor and a decay constant.Based on Heavy Quark Effect Theory, it is shown 2 Advances in High Energy Physics in the QCD factorization approach that the corrections to the hadronic matrix elements can be expressed in terms of short-distance coefficients and meson light-cone distribution amplitudes [37,38].Alternative factorization approach based on QCD factorization is often applied in study of quasi twobody hadronic  decays [19,39,40], where they introduced unitary meson-meson form factors, from the perspective of unitarity, for the final state interactions.Other QCD-inspired approaches, such as the perturbative approach (pQCD) [41] and the soft-collinear effective theory (SCET) [42], are also wildly used in  meson decays.
However, for  meson decays, such QCD-inspired factorization approaches may not be reliable since the charm quark mass, which is just above 1 GeV, is not heavy enough for the heavy quark expansion [43,44].For this reason, several model-independent approaches for the charm meson decay amplitudes have been proposed, such as the flavor topological diagram approach based on the flavor (3) symmetry [44][45][46][47] and the factorization-assisted topological-amplitude (FAT) approach with the inclusion of flavor (3) breaking effect [48,49].One motivation of these aforementioned approaches is to identify as complete as possible the dominant sources of nonperturbative dynamics in the hadronic matrix elements.
In this paper, we study the  violation of SCS  meson decay  0 →  +  −  0 in the FAT approach.Our attention will be mainly focused on the region of the phase space where two intermediate resonances,  * (892) + and  * (892) − , are overlapped.Before proceeding, it will be helpful to point out that direct  asymmetry is hard to be isolated for decay process with -eigen-final-state.When the final state of the decay process is  eigenstate, the time integrated  violation for  0 → , which is defined as can be expressed as [34] where    ,    , and    are the  asymmetries in decay, in mixing, and in the interference of decay and mixing, respectively.As is shown in [34,50,51], the indirect  violation  ind ≡   +   is universal and channel-independent for twobody -eigenstate.This conclusion is easy to be generalized to decay processes with three-body -eigenstate in the final state, such as  0 →  +  −  0 .In view of the universality of the indirect  asymmetry, we will only consider the direct  violations of the decay  0 →  +  −  0 throughout this paper.
The remainder of this paper is organized as follows.In Section 2, we present the decay amplitudes for various decay channels, where the decay amplitudes of  0 →  ±  * (892) ∓ are formulated via the FAT approaches.In Section 3, we study the  asymmetries of  0 →  ±  * (892) ∓ and the  asymmetry of  0 →  +  −  0 induced by the interference between different resonances in the phase space.Discussions and conclusions are given in Section 4. We list some useful formulas and input parameters in the Appendix.
The two tree diagrams in first line of Figure 1 represent the color-favored tree diagram for  → () transition and the -exchange diagram with the pseudoscalar (vector) meson containing the antiquark from the weak vertex, respectively.The amplitudes of these two diagrams will be, respectively, denoted as  () and  () .
According to these topological structures, the amplitudes of the color-favored tree diagrams  () , which are dominated by the factorizable contributions, can be parameterized as and respectively, where   is the Fermi constant,   =    *  , with   and   being the CKM matrix elements,  2 () =  2 () +  1 ()/  , with  1 () and  2 () being the scale-dependent Wilson coefficients, and the number of color   = 3,  () and  () are the decay constant and mass of the vector (pseudoscalar) meson, respectively,  → 1 and  → 0 are the form factors for the transitions  →  and  → , respectively,  is the polarization vector of the vector meson, and   is the momentum of  meson.The scale  of Wilson coefficients is set to energy release in individual decay channels [52,53], which depends on masses of initial and final states and is defined as [48,49] with the mass ratios  () =  () /  , where Λ represents the soft degrees of freedom in the  meson, which is a free parameter.
For the -exchange amplitudes, since the factorizable contributions to these amplitudes are helicity-suppressed, only the nonfactorizable contributions need to be considered.Therefore, the -exchange amplitudes are parameterized as where   is the mass of  meson,   ,   , and   are the decay constants of the , , and  mesons, respectively, and    and    characterize the strengths and the strong phases of the corresponding amplitudes, with  = , ,  representing the strongly produced  quark pair.The ratio of     over     indicates that the flavor (3) breaking effects have been taken into account from the decay constants.
The penguin diagrams shown in the second line of Figure 1 represent the color-favored, the gluon-annihilation, and the gluon-exchange penguin diagrams, respectively, whose amplitudes will be denoted as  () ,  () , and  () , respectively.
Since a vector meson cannot be generated from the scalar or pseudoscalar operator, the amplitude   does not include contributions from the penguin operator  5 or  6 .Consequently, the color-favored penguin amplitudes   and   can be expressed as and respectively, where   =    *  with   and  *  being the CKM matrix elements,  4,6 () =  4,6 () +  3,5 ()/  , with  3,4,5,6 being the Wilson coefficients, and   is a chiral factor, which takes the form with  (,) being the masses of (, ) quark.Note that the quark-loop corrections and the chromomagnetic-penguin contribution are also absorbed into  3,4,5,6 as shown in [49].Similar to the amplitudes  , , the amplitudes  only include the nonfactorizable contributions as well.Therefore, the amplitudes  , , which are dominated by  4 and  6 [48], can be parameterized as For the amplitudes   and   , the helicity suppression does not apply to the matrix elements of  5,6 , so the factorizable contributions exist.In the pole resonance model [54], after applying the Fierz transformation and the factorization hypothesis, the amplitudes   and   can be expressed as and respectively, where   is an effective strong coupling constant obtained from strong decays, e.g.,  → ,  * → , and  → , and is set as   = 4.5 [54] in this work,   * and   * are the mass and decay constant of the pole resonant pseudoscalar meson  * , respectively, and    and    are the strengths and the strong phases of the corresponding amplitudes.
From Figure 1, the decay amplitudes of  0 →  +  * (892) − and  0 →  −  * (892) + in the FAT approach can be easily written down and respectively, where  is the helicity of the polarization vector (, ).In the FAT approach, the fitted nonperturbative parameters,   , ,   , ,   ,s ,   , , are assumed to be universal and can be determined by the data [49].
In Table 1, we list the magnitude of each topological amplitude for  0 →  +  * (892) − and  0 →  −  * (892) + by using the global fitted parameters for  →  in [49].One can see from Table 1 that the penguin contributions are greatly suppressed. is dominant in the penguin contributions of  0 →  −  * (892) + , while  is small in  0 →  +  * (892) − , which is even smaller than the amplitude .This difference is because of the chirally enhanced factor contained in (14) while not in (13).The very small  do not receive the contributions from the quark-loop and chromomagnetic penguins, since these two contributions to c 4 and  6 are canceled with each other in (16).Besides, the relations    =    ,    =    , and    ̸ =    can be read from Table 1; this is because that the isospin symmetry and the flavor (3) breaking effect have been considered.
Since the form factors are inevitably model-dependent, we list in Table 2 the branching ratios of  0 →  +  * (892) − and  0 →  −  * (892) + predicted by the FAT approach, by various form factor models.The pole, dipole, and covariant light-front (CLF) models are adopted.The uncertainties in Table 2 mainly come from decay constants.The CLF model agrees well with the data for both decay channels, and other models are also consistent with the data.However, the modeldependence of form factor leads to large uncertainty of the branching fraction, as large as 20%.Because of the smallness of the Wilson coefficients and the CKM-suppression of the penguin amplitudes, the branching ratios are dominated by the tree amplitudes.Therefore, there is no much difference for the branching ratios whether we consider the penguin amplitudes or not.

𝐶𝑃 Asymmetries for 𝐷
The direct  asymmetry for the two-body decay  →  is defined as where M → represents the decay amplitude of the  conjugate process  → , such as  0 →  +  * (892) − or  0 →  −  * (892) + .In the framework of FAT approach, The differential  asymmetry of the three-body decay  0 →  +  −  0 , which is a function of the invariant mass of   0  + and   0  − , is defined as where the invariant mass   0  ± = (  0 +   ± ) 2 .As can be seen from ( 4), the differential  asymmetry   0 → +  −  0  depends on the relative strong phase , which is impossible to be calculated theoretically because of its nonperturbative origin.Despite this, we can still acquire some information of this relative strong phase  from data.By using a Dalitz plot technique [55,58,59], the phase difference  exp between  0 decays to  +  * (892) − and  −  * (892) + can be extracted from data.One should notice that  exp is not the same as the strong phase  defined in (4).The strong phase  is the relative phase between the decay amplitudes of  0 →  +  * (892) − and  0 →  −  * (892) + .On the other hand, the phase  exp is defined through in the overlapped region of the phase space, where   * ± is the phase of the amplitude M  * ± : Therefore, neglecting the CKM suppressed penguin amplitudes,  exp and  can be related by where   * ∓  ± = arg(  * ∓ +    ± ) are the phases in tree-level amplitudes of  0 →  ±  * (892) ∓ and are equivalent to   * ∓ if the penguin amplitudes are neglected.With the relation of ( 25), and  exp = −35.5 ∘ ± 4.1 ∘ measured by the BABAR Collaboration [56], we have  ≈ −51.85 ∘ ± 4.1 ∘ .
In Figure 2, we present the differential  asymmetry of  0 →  +  −  0 in the overlapped region of  * (892) − and  * (892) + in the phase space, with  = −51.85∘ .Namely, we will focus on the region   * − 2Γ  * < √  0  − , √  0  + <   * + 2Γ  * of the phase space.One can see from Figure 2 that the differential  asymmetry of  0 →  +  −  0 can reach 3.0 × 10 −4 in the overlapped region, which is about 10 times larger than the  asymmetries of the corresponding two-body decay channels shown in Table 3.
The behavior of the differential  asymmetry of  0 →  +  −  0 in Figure 2 motivates us to separate this region into four areas, area A ( We further consider the observable of regional  asymmetry in areas A, B, C, and D displayed in Table 4, which is defined by where Ω represents a certain region of the phase space. Comparing with the  asymmetries of two-body decays, the regional  asymmetries, from Table 4, are less sensitive to the models we have used.We would like to use only the CLF model for the following discussion.The uncertainties in Table 4 come from decay constants as well as the relative phase  exp .In addition, if we focus on the right part of area A, that is,   * < √  0  − <   * + 2Γ  * ,   * − Γ  * < √  0  + <   * , the regional  violation will be (1.09± 0.16) × 10 −4 .
The energy dependence of the propagator of the intermediate resonances can lead to a small correction to  asymmetry.For example, if we replace the Breit-Wigner Table 3:  asymmetries (in unit of 10 −5 ) of  0 →  +  * (892) − and  0 →  −  * (892) + predicted by the FAT approach with pole, dipole, and CLF models adopted.The uncertainties in this table are mainly from decay constants.propagator by the Flatté Parametrization [60], the correction to the regional  asymmetry will be about 1%.
Besides, since the chromomagnetic dipole operator  8 is sensitive to some NPs, the inclusion of this kind of NPs will lead to a much larger global  asymmetries of  0 →  +  * (892) − and  0 →  −  * (892) + , which are (0.56 ± 0.08) × 10 −3 and (−0.50 ± 0.04) × 10 −3 , respectively, while the regional  asymmetry of  0 →  +  −  0 can be also increased to (1.3 ± 0.3) × 10 −3 when considering the interference effect in the phase space.Since the O(10 −3 ) of Table 4: Three from factor models: the pole, dipole, and CLF models are used for the regional  asymmetries (in unit of 10 −4 ) in the four areas, A, B, C, and D, of the phase space. asymmetry is attributed to the large  eff 8 , which is almost impossible for the SM to generate such large contribution, it will indicate NP if such  violation is observed.Here, we roughly estimate the number of  0  0 needed for testing such kind of asymmetries, which is about (1/)(1/ 2  ) ∼ 10 9 .This could be observed in the future experiments at Belle II [73,74], while the current largest  0  0 yields are about 10 8 at BABAR and Belle [75,76] and 10 7 at BESIII [77].

Appendix Some Useful Formulas and Input Parameters
where  and  are color indices and   = , , .Among all these operators,  The Wilson coefficients used in this paper are evaluated at  = 1GeV, which can be found in [48].
( ) CKM Matrix.We use the Wolfenstein parameterization for the CKM matrix elements, which up to order O( 8 ) read [79,80] where , , , and  are the Wolfenstein parameters, which satisfy following relation: , based on the relativistic covariant light-front quark model [85], are expressed as a momentum-dependent, 3-parameter form (the parameters can be found in  M is the corresponding decay amplitude.

𝑞 1 and 𝑂 𝑞 2
are tree operators,  3 −  6 are QCD penguin operators, and  8 is chromomagnetic dipole Advances in High Energy Physics operator.The electroweak penguin operators are neglected in practice.One should notice that SCS decays receive contributions from all aforementioned operators while only tree operators can contribute to CF decays and DCS decays.