Advances in High Energy Physics
Volume 2018 (2018), Article ID 7627308, 11 pages
https://doi.org/10.1155/2018/7627308

Research Article

Analysis of Violation in

Hang Zhou,¹ Bo Zheng,1,2 and Zhen-Hua Zhang¹

¹School of Nuclear Science and Technology, University of South China, Hengyang, Hunan 421001, China
²Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany

Correspondence should be addressed to Bo Zheng; zhengbo_usc@163.com and Zhen-Hua Zhang; zhangzh@usc.edu.cn

Received 21 August 2018; Accepted 3 October 2018; Published 22 November 2018

Guest Editor: Tao Luo

Copyright © 2018 Hang Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

We study the violation induced by the interference between two intermediate resonances and in the phase space of singly-Cabibbo-suppressed decay . We adopt the factorization-assisted topological approach in dealing with the decay amplitudes of . The asymmetries of two-body decays are predicted to be very tiny, which are and , respectively, for and , while the differential asymmetry of is enhanced because of the interference between the two intermediate resonances, which can reach as large as . For some NPs which have considerable impacts on the chromomagnetic dipole operator , the global asymmetries of and can be then increased to and , respectively. The regional asymmetry in the overlapped region of the phase space can be as large as .

1. 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 three-body decay channels of meson [4–7], which was followed by a number of theoretical works [8–19].

The study of violation in singly-Cabibbo-suppressed (SCS) meson decays provides an ideal test of the SM and exploration of New Physics (NP) [20–23]. 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 [24–33]. 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 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 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 two-body 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 symmetry [44–47] and the factorization-assisted topological-amplitude (FAT) approach with the inclusion of flavor 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 in the FAT approach. Our attention will be mainly focused on the region of the phase space where two intermediate resonances, and , 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 , 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 is universal and channel-independent for two-body -eigenstate. This conclusion is easy to be generalized to decay processes with three-body -eigenstate in the final state, such as . In view of the universality of the indirect asymmetry, we will only consider the direct violations of the decay 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 are formulated via the FAT approaches. In Section 3, we study the asymmetries of and the asymmetry of 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.

2. Decay Amplitude for

In the overlapped region of the intermediate resonances and in the phase space, the decay process is dominated by two cascade decays, and , respectively. Consequently, the decay amplitude of can be expressed as

in the overlapped region, where and are the amplitudes for the two cascade decays and is the relative strong phase. Note that nonresonance contributions have been neglected in (4).

The decay amplitude for the cascade decay can be expressed as

where and represent the amplitudes corresponding to the strong decay and weak decay , respectively, is the helicity index of , is the invariant mass square of system, and and are the mass and width of , respectively. The decay amplitude for the cascade decay, , is the same as (5) except replacing the subscripts and with and , respectively.

For the strong decays , one can express the decay amplitudes as

where and represent the momentum for and mesons, respectively, and is the effective coupling constant for the strong interaction, which can be extracted from the experimental data via

with

and . The isospin symmetry of the strong interaction implies that .

The decay amplitudes for the weak decays, and , will be handled with the aforementioned FAT approach [48, 49]. The relevant topological tree and penguin diagrams for are displayed in Figure 1, where and denote a light pseudoscalar and vector meson (representing and in this paper), respectively.

(a)

(b)

(c)

(d)

(e)

Figure 1: The relevant topological diagrams for with (a) the color-favored tree amplitude , (b) the -exchange amplitude , (c) the color-favored penguin amplitude , (d) the gluon-annihilation penguin amplitude , and (e) the gluon-exchange penguin amplitude .

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, , with and being the scale-dependent Wilson coefficients, and the number of color , and are the decay constant and mass of the vector (pseudoscalar) meson, respectively, and 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 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 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 or . Consequently, the color-favored penguin amplitudes and can be expressed as

and

respectively, where with and being the CKM matrix elements, , with 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 as shown in [49].

Similar to the amplitudes , the amplitudes only include the nonfactorizable contributions as well. Therefore, the amplitudes , which are dominated by and [48], can be parameterized as

For the amplitudes and , the helicity suppression does not apply to the matrix elements of , 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 [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 and 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, , , , , 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 and 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 , while is small in , 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 and 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 breaking effect have been considered.

Table 1: The magnitude of tree and penguin contributions (in unit of ) corresponding to the topological amplitudes in (19) and (20). The factors “” and “” are omitted in this table.

Decay modes

Since the form factors are inevitably model-dependent, we list in Table 2 the branching ratios of and 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 model-dependence of form factor leads to large uncertainty of the branching fraction, as large as . 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.

Table 2: Branching ratios (in unit of ) of singly-Cabibbo-suppressed decays and . Both experimental data [55–57] and theoretical predictions of FAT approach of the branching ratios are listed.

Form factors
Br()
Br()

Pole

Dipole

CLF

Exp.

3. Asymmetries for and

The direct asymmetry for the two-body decay is defined as

where represents the decay amplitude of the conjugate process , such as or . In the framework of FAT approach, we predict very small direct asymmetries of and presented in Table 3. The uncertainties induced by the model-dependence of form factor to the asymmetries of and are about and , respectively.

Table 3: asymmetries (in unit of ) of and predicted by the FAT approach with pole, dipole, and CLF models adopted. The uncertainties in this table are mainly from decay constants.

Form factors

Pole

Dipole

CLF

The differential asymmetry of the three-body decay , which is a function of the invariant mass of and , is defined as

where the invariant mass . As can be seen from (4), the differential asymmetry 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 between decays to and can be extracted from data. One should notice that is not the same as the strong phase defined in (4). The strong phase is the relative phase between the decay amplitudes of and . On the other hand, the phase is defined through

in the overlapped region of the phase space, where is the phase of the amplitude :

Therefore, neglecting the CKM suppressed penguin amplitudes, and can be related by

where are the phases in tree-level amplitudes of and are equivalent to if the penguin amplitudes are neglected. With the relation of (25), and measured by the BABAR Collaboration [56], we have .

In Figure 2, we present the differential asymmetry of in the overlapped region of and in the phase space, with . Namely, we will focus on the region of the phase space. One can see from Figure 2 that the differential asymmetry of can reach in the overlapped region, which is about 10 times larger than the asymmetries of the corresponding two-body decay channels shown in Table 3.

Figure 2: The differential asymmetry distribution of in the overlapped region of and in the phase space.

The behavior of the differential asymmetry of in Figure 2 motivates us to separate this region into four areas, area A (), area B (), area C (), and area D (). We further consider the observable of regional asymmetry in areas A, B, C, and D displayed in Table 4, which is defined by

where represents a certain region of the phase space.

Table 4: Three from factor models: the pole, dipole, and CLF models are used for the regional asymmetries (in unit of ) in the four areas, A, B, C, and D, of the phase space.

Form factors

Pole

Dipole

CLF

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 . In addition, if we focus on the right part of area A, that is, , the regional violation will be .

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 propagator by the Flatté Parametrization [60], the correction to the regional asymmetry will be about .

Since the asymmetry of is extremely suppressed, it should be more sensitive to the NP. For example, some NPs have considerable impacts on the chromomagnetic dipole operator [34, 61–66]. Consequently, the violation in SCS decays may be further enhanced. In practice, the NP contributions can be absorbed into the corresponding effective Wilson coefficient [67, 68]. For comparison, we first consider a relative small value of (as in [48, 64]) lying within the range and the global asymmetry of are no larger than . Moreover, if we follow [49] taking (while , which is extracted from measured by LHCb [69], is a quite large quantity even for the coefficients corresponding tree-level operators, however, such large contribution can be realized if some NPs effects are pulled in. For example, the up squark-gluino loops in supersymmetry (SUSY) can arise significant contributions to . More details about the squark-gluino loops and other models in SUSY can be found in [34, 62, 70–72]), the global asymmetries of and are then and , respectively.

We further display the asymmetry of in the overlapped region of and in Figures 3(a) and 3(b) for and , respectively. After taking the interference effect into account, the differential asymmetry of can be increased as large as and for and , respectively. The regional ones (in phase space of ) can reach and for and , respectively.

(a)

(b)

Figure 3: The differential asymmetry distribution of for (a) and (b) , in the overlapped region of and in the phase space.

4. Discussion and Conclusion

In this work, we studied violations in via the FAT approach. The violations in two-body decay processes and are very small, which are and , respectively. Our discussion shows that the violation can be enhanced by the interference effect in three-body decay . The differential asymmetry can reach when the interference effect is taken into account, while the regional one can be as large as .

Besides, since the chromomagnetic dipole operator is sensitive to some NPs, the inclusion of this kind of NPs will lead to a much larger global asymmetries of and , which are and , respectively, while the regional asymmetry of can be also increased to when considering the interference effect in the phase space. Since the of asymmetry is attributed to the large , 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 needed for testing such kind of asymmetries, which is about . This could be observed in the future experiments at Belle II [73, 74], while the current largest yields are about at BABAR and Belle [75, 76] and at BESIII [77].

Appendix

Some Useful Formulas and Input Parameters

(1) Effective Hamiltonian and Wilson Coefficients. The weak effective Hamiltonian for SCS meson decays, based on the Operator Product Expansion (OPE) and Heavy Quark Effective Theory (HQET), can be expressed as [78]

where is the Fermi constant, , is the Wilson coefficient, and , , , and are four-fermion operators which are constructed from different combinations of quark fields. The four-fermion operators take the following form:

where and are color indices and . Among all these operators, and are tree operators, are QCD penguin operators, and is chromomagnetic dipole 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.

The Wilson coefficients used in this paper are evaluated at , which can be found in [48].

(2) CKM Matrix. We use the Wolfenstein parameterization for the CKM matrix elements, which up to order read [79, 80]

where , and are the Wolfenstein parameters, which satisfy following relation:

Numerical values of Wolfenstein parameters which have been used in this work are as follows:

(3) Decay Constants and Form Factors. In (17) and (18), the pole resonance model was employed for the matrix element in the annihilation diagrams. By considering angular momentum conservation at weak vertex and all conservation laws are preserved at strong vertex, the matrix element is therefore dominated by a pseudoscalar resonance [54],

where is a strong coupling constant and and are the mass and decay constant of the pseudoscalar resonance . Therefore, and are the dominant resonances for the final states of , which can be expressed as flavor mixing of and ,

where is the mixing angle and and are defined by

The decay constants of and are defined by

where

According to [81, 82], the decay constants of and can be expressed as

where and [81], and the mixing angle [83]. Other decay constants used in this paper are listed in Table 5.

Table 5: The meson decay constants used in this paper (MeV) [57, 84].

The transition form factors and , 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 Table 6):

Table 6: The parameters of transitions form factors in (A.12).

Form factor

0.69
0.78

1.04
1.05

0.44
0.23

(4) Decay Rate. The decay width takes the form

where represents the center of mass (c.m.) 3-momentum of each meson in the final state and is given by

is the corresponding decay amplitude.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was partially supported by National Natural Science Foundation of China (Project Nos. 11447021, 11575077, and 11705081), National Natural Science Foundation of Hunan Province (Project No. 2016JJ3104), the Innovation Group of Nuclear and Particle Physics in USC, and the China Scholarship Council.

References

J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, “Evidence for the 2π Decay of the Meson,” Physical Review Letters, vol. 13, no. 4, pp. 138–140, 1964.

M. Kobayashi and T. Maskawa, “CP-violation in the renormalizable theory of weak interaction,” Progress of Theoretical and Experimental Physics, vol. 49, pp. 652–657, 1973.

N. Cabibbo, “Unitary symmetry and leptonic decays,” Physical Review Letters, vol. 10, no. 12, pp. 531–533, 1963.

R. Aaij, LHCb Collaboration et al., “Measurement of CP Violation in the Phase Space of B±→K±π⁺π⁻ and B±→K±K⁺K⁻ Decays,” Physical Review Letters, vol. 111, no. 2, Article ID 101801, 2017.

R. Aaij, LHCb Collaboration et al., “Measurement of CP Violation in the Phase Space of B±→K⁺K⁻π± and B±→π⁺π⁻π± Decays,” Physical Review Letters, vol. 112, no. 2, Article ID 011801, 2014.

R. Aaij, LHCb Collaboration et al., “Measurements of CP violation in the three-body phase space of charmless B± decays,” Physical Review D, vol. 90, Article ID 112004, 2014.

J. H. A. Nogueira, S. Amato, A. Austregesilo et al., “Summary of the 2015 LHCb workshop on multi-body decays of D and B mesons,” https://arxiv.org/abs/1605.03889.

Z.-H. Zhang, X.-H. Guo, and Y.-D. Yang, “CP violation in B±→π±π⁺π⁻ in the region with low invariant mass of one π⁺π⁻ pair,” Physical Review D, vol. 87, Article ID 076007, 2013.

I. Bediaga, T. Frederico, and O. Lourenço, “CP violation and CPT invariance in B± decays with final state interactions,” Physical Review D, vol. 89, Article ID 094013, 2014.

H.-Y. Cheng and C.-K. Chua, “Branching fractions and direct CP violation in charmless three-body decays of B mesons,” Physical Review D, vol. 88, Article ID 114014, 2013.

Z.-H. Zhang, X.-H. Guo, and Y.-D. Yang, “CP violation induced by the interference of scalar and vector resonances in three-body decays of bottom mesons,” https://arxiv.org/abs/1308.5242.

B. Bhattacharya, M. Gronau, and J. L. Rosner, “CP asymmetries in three-body decays to charged pions and kaons,” Physics Letters B, vol. 726, no. 1-3, pp. 337–343, 2013.

D. Xu, G.-N. Li, and X.-G. He, “Large SU(3) Breaking Effects and CP Violation in B⁺ Decays Into Three Charged Octet Pseudoscalar Mesons,” International Journal of Modern Physics A, vol. 29, Article ID 1450011, 2014.

W.-F. Wang, H.-C. Hu, H.-N. Li, and C.-D. Lü, “Direct CP asymmetries of three-body B decays in perturbative QCD,” Physical Review D, vol. 89, Article ID 074031, 2014.

Z.-H. Zhang, C. Wang, and X.-H. Guo, “Possible large CP violation in three-body decays of heavy baryon,” Physics Letters B, vol. 751, pp. 430–433, 2015.

C. Wang, Z.-H. Zhang, Z.-Y. Wang, and X.-H. Guo, “Localized direct CP violation in B± → ρ⁰(ω)π± → π⁺π⁻π±,” The European Physical Journal C, vol. 75, p. 536, 2015.

J. H. A. Nogueira, I. Bediaga, A. B. R. Cavalcante, T. Frederico, and O. Lourenço, “CP violation: Dalitz interference, CPT, and final state interactions,” Physical Review D, vol. 92, Article ID 054010, 2015.

J. Dedonder, A. Furman, R. Kamiński, L. Leśniak, and B. Loiseau, “S-, P- and D-wave ππ final state interactions and CP violation in B± → π±π∓π± decays,” Acta Physica Polonica B, vol. 42, no. 9, Article ID 2013, 2011.

B. El-Bennich, A. Furman, R. Kamiński, L. Leśniak, and B. Loiseau, “Interference between f₀(980) and ρ(770)⁰ resonances in B → π⁺π⁻K decays,” Physical Review D, vol. 74, Article ID 114009, 2006.

I. Bigi and A. Sanda, “On D⁰ mixing and CP violation,” Physics Letters B, vol. 171, no. 2-3, pp. 320–324, 1986.

G. Blaylock, A. Seiden, and Y. Nir, “The role of CP violation in D⁰ mixing,” Physics Letters B, vol. 335, no. 3, pp. 555–560, 1995.

S. Bergmann, Y. Grossman, Z. Ligeti, Y. Nir, and A. A. Petrov, “Lessons from CLEO and FOCUS measurements of D⁰- mixing parameters,” Physics Letters B, vol. 486, no. 3-4, pp. 418–425, 2000.

U. Nierste and S. Schacht, “Neutral D→KK∗ decays as discovery channels for charm CP violation,” Physical Review Letters, vol. 119, Article ID 251801, 2017.

G. Bonvicini, CLEO Collaboration et al., “Search for CP violation in D⁰→π⁰, D⁰→π⁰π⁰ and D⁰→ decays,” Physical Review D, vol. 63, Article ID 071101, 2001.

J. M. Link, “Search for CP Violation in the decays D⁺→KSπ⁺ and D⁺→KSK⁺,” Physical Review Letters, vol. 88, Article ID 041602, 2002.

T. Aaltonen, CDF Collaboration et al., “Measurement of CP-violating asymmetries in D⁰→π⁺π⁻ and D⁰→K⁺K⁻ decays at CDF,” Physical Review D, vol. 85, Article ID 012009, 2012.

R. Cenci, “Mixing and CP Violation in Charm Decays at BABAR,” in Proceedings of the 7th International Workshop on the CKM Unitarity Triangle (CKM 2012), Cincinnati, Ohio, USA, 2012.

J. P. Lees, BABAR Collaboration et al., “Search for CP violation in the decays D±→K±, →K±, and →π±,” Physical Review D, vol. 87, Article ID 052012, 2013.

M. Starič, A. Abdesselam, I. Adachi et al., “Measurement of D⁰-⁰ mixing and search for CP violation in D⁰ → K⁺K⁻, π⁺π⁻ decays with the full Belle data set,” Physics Letter B, vol. 753, pp. 412–418, 2016.

R. Aaij, R. Aaij, B. Adeva et al., “Measurement of CP asymmetries in D±→η′π± and →η′π± decays,” Physics Letters B, vol. 771, pp. 21–30, 2017.

R. Aaij, LHCb Collaboration et al., “Measurements of charm mixing and CP violation using D⁰→K±π∓ decays,” Physical Review D, vol. 95, Article ID 052004, 2017.

LHCb Collaboration, “Search for CP violation in the phase space of D⁰ → π⁺π⁻π⁺π⁻ decays,” Physics Letters B, vol. 769, pp. 345–356, 2017.

V. Bhardwaj, “Latest Charm Mixing and CP results from B-factories,” in Proceedings of the 9th International Workshop on the CKM Unitarity Triangle (CKM2016), vol. 139, Mumbai, India, 2017.

Y. Grossman, A. L. Kagan, Y. Nir et al., “New physics and CP violation in singly Cabibbo suppressed Ddecays,” Physical Review D, vol. 75, Article ID 036008, 2007.

J. D. Bjorken, “Topics in B-physics,” Nuclear Physics B (Proceedings Supplements), vol. 11, no. C, pp. 325–341, 1989.

M. J. Dugan and B. Grinstein, “QCD basis for factorization in decays of heavy mesons,” Physics Letters B, vol. 255, no. 4, pp. 583–588, 1991.

M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, “QCD factorization for decays: strong phases and CP violation in the heavy quark limit,” Physical Review Letters, vol. 83, no. 10, pp. 1914–1917, 1999.

M. Beneke and M. Neubert, “QCD factorization for B→PP and B→PV decays,” Nuclear Physics B, vol. 675, no. 1-2, pp. 333–415, 2003.

D. R. Boito, J. Dedonder, B. El-Bennich, O. Leitner, and B. Loiseau, “Scalar resonances in a unitary,” Physical Review D, vol. 96, Article ID 113003, 2017.

A. Furmana, R. Kamińskia, L. Leśniaka, and B. Loiseaub, “Long-distance effects and final state interactions in B→ππK and B→KK decays,” Physics Letters B, vol. 622, pp. 207–217, 2005.

Y.-Y. Keum, H.-N. Li, A. I. Sanda et al., “Penguin enhancement and Kπ decays in perturbative QCD,” Physical Review D, vol. 63, Article ID 054008, 2001.

C. W. Bauer, D. Pirjol, I. Z. Rothstein, and I. W. Stewart, “B→M₁M₂: Factorization, charming penguins, strong phases, and polarization,” Physical Review D, vol. 70, Article ID 054015, 2004.

B. Loiseau, “Theory overview on amplitude analyses with charm decays,” in Proceedings of the 8th International Workshop on Charm Physics (Charm 2016), Bologna, Italy, 2017.

H.-Y. Cheng and C.-W. Chiang, “Direct CP violation in two-body hadronic charmed meson decays,” Physical Review D, vol. 85, no. 3, Article ID 034036, 2012, Erratum: [Physical Review D, vol. 85, Article ID 079903, 2012].

L.-L. Chau, “Quark mixing in weak interactions,” Physics Reports, vol. 95, no. 1, pp. 1–94, 1983.

B. Bhattacharya, M. Gronau, and J. L. Rosner, “Publisher’s Note:,” Physical Review D, vol. 85, Article ID 054014, 2012.

H.-Y. Cheng, C.-W. Chiang, and A.-L. Kuo, “Global analysis of two-body D→VP decays within the framework of flavor symmetry,” Physical Review D, vol. 93, Article ID 114010, 2016.

H.-N. Li, C.-D. Lu, F.-S. Yu et al., “Branching ratios and direct CP asymmetries in D→PP decays,” Physical Review D, vol. 86, Article ID 036012, 2012.

Q. Qin, H.-N. Li, C.-D. Lü, and F.-S. Yu, “Branching ratios and direct CP asymmetries in D→PV decays,” Physical Review D, vol. 89, Article ID 054006, 2014.

B. Aubert (BABAR Collaboration) et al., “Limits on D⁰− Mixing and CP Violation from the Ratio of Lifetimes for Decay to K⁻π⁺, K⁻K⁺, and π⁻π⁺,” Physical Review Letters, vol. 91, Article ID 121801, 2003.

K. Abe (Belle Collaboration) et al., “Measurement of the D⁰ lifetime difference using D⁰→ Kπ/KK decays,” in Proceedings of the 21st International Symposium on Lepton and Photon Interactions at High Energies, pp. 11–16, Batavia, ILL, USA, 2003, https://arxiv.org/abs/hep-ex/0308034.

Y.-Y. Keum, H.-N. Li, and A. I. Sandac, “Fat penguins and imaginary penguins in perturbative QCD,” Physics Letters B, vol. 504, pp. 6–14, 2001.

C.-D. Lü, K. Ukai, and M.-Z. Yang, “Branching ratio and CP violation of B→ππ decays in the perturbative QCD approach,” Physical Review D, vol. 63, Article ID 074009, 2001.

F.-S. Yu, X.-X. Wang, and C.-D. Lü, “Nonleptonic two-body decays of charmed mesons,” Physical Review D, vol. 84, Article ID 074019, 2011.

C. Cawlfield (CLEO Collaboration) et al., “Measurement of interfering K*⁺K⁻and K*⁻K⁺amplitudes in the decay D⁰→K⁺K⁻π⁰,” Physical Review D, vol. 74, Article ID 031108, 2006.

B. Aubert, L. L. Zhang, S. Chen et al., “Amplitude analysis of the decay D⁰→K⁻K⁺π⁰,” Physical Review D, vol. 76, Article ID 011102, 2007.

C. Patrignani, K. Agashe, G. Aielli et al., “Review of Particle Physics,” Chinese Physics C, vol. 40, no. 10, Article ID 100001, 2016.

J. L. Rosner and D. A. Suprun, “Measuring the relative strong phase in D⁰→K*⁺K⁻and D⁰→K*⁻K⁺ decays,” Physical Review D, vol. 68, Article ID 054010, 2003.

I. Bediaga, I. I. Bigi, A. Gomes, G. Guerrer, J. Miranda, and A. C. dos Reis, “On a CP anisotropy measurement in the Dalitz plot,” Physical Review D, vol. 80, no. 9, Article ID 096006, 2009.

S. M. Flatte, “Coupled-channel analysis of the πη and K systems near K threshold,” Physics Letters B, vol. 63, no. 2, pp. 224–227, 1976.

M. Golden and B. Grinstein, “Enhanced CP violations in hadronic charm decays,” Physics Letters B, vol. 222, no. 3-4, pp. 501–506, 1989.

G. F. Giudice, G. Isidori, and P. Paradisi, “Direct CP violation in charm and flavor mixing beyond the SM,” Journal of High Energy Physics, vol. 1204, Article ID 060, 2012.

M. Gronau, “New physics in singly Cabibbo-suppressed D decays,” Physics Letters B, vol. 738, pp. 136–139, 2014.

J. Brod, A. L. Kagan, and J. Zupan, “Size of direct CP violation in singly Cabibbo-suppressed D decays,” Physical Review D, vol. 86, Article ID 014023, 2012.

Y. Grossman, A. L. Kagan, and J. Zupan, “Testing for new physics in singly Cabibbo suppressed D decays,” Physical Review D, vol. 85, Article ID 114036, 2012.

G. Isidori, J. F. Kamenik, Z. Ligetie, and G. Perez, “Implications of the LHCb evidence for charm CP violation,” Physics Letters B, vol. 711, no. 1, pp. 46–51, 2012.

M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, “QCD factorization in B→πK, ππ decays and extraction of Wolfenstein parameters,” Nuclear Physics B, vol. 606, no. 1-2, pp. 245–321, 2001.

H.-n. Li, S. Mishima, and A. I. Sanda, “Resolution to the B→πK puzzle,” Physical Review D, vol. 72, Article ID 114005, 2005.

R. Aaij (LHCb Collaboration) et al., “Evidence for CP violation in time-integrated D⁰→h⁻h⁺ decay rates,” Physical Review Letters, vol. 108, Article ID 111602, 2012.

F. Gabbiani, E. Gabrielli, A. Masiero, and L. Silvestrini, “A complete analysis of FCNC and CP constraints in general SUSY extensions of the standard model,” Nuclear Physics B, vol. 477, no. 2, pp. 321–352, 1996.

E. Gabrielli, A. Masiero, and L. Silvestrini, “Flavour changing neutral currents and CP violating processes in generalized supersymmetric theories,” Physics Letters B, vol. 374, no. 1-3, pp. 80–86, 1996.

J. S. Hagelin, S. Kelley, and T. Tanaka, “Supersymmetric flavor-changing neutral currents: exact amplitudes and phenomenological analysis,” Nuclear Physics B, vol. 415, no. 2, pp. 293–331, 1994.

G. De Pietro, “Charm physics prospects at Belle II,” in Proceedings of the European Physical Society Conference on High Energy Physics (EPS-HEP2017), vol. 314, Venice, Italy, 2017.

T. Abe, I. Adachi, K. Adamczyk et al., “Belle II Technical Design Report,” https://arxiv.org/abs/1011.0352.

J. P. Lees, V. Poireau, V. Tisserand et al., “Measurement of the D⁰→π⁻e⁺νe differential decay branching fraction as a function of q² and study of form factor parametrizations,” Physical Review D, vol. 91, Article ID 052022, 2015.

N. K. Nisar, G. B. Mohanty, K. Trabelsi et al., “Search for the rare decay D⁰→γγ at Belle,” Physical Review D, vol. 93, Article ID 051102, 2016.

M. Ablikim (BESIII Collaboration) et al., “Search for D⁰→γγ and improved measurement of the branching fraction for D⁰→π⁰π⁰,” Physicla Review D, vol. 91, no. 11, Article ID 112015, 2015.

G. Buchalla, A. J. Buras, and M. E. Lautenbacher, “Weak decays beyond leading logarithms,” Reviews of Modern Physics, vol. 68, no. 4, pp. 1125–1244, 1996.

A. J. Buras, M. E. Lautenbacher, and G. Ostermaier, “Waiting for the top quark mass, K⁺→π⁺, mixing, and CP asymmetries in B decays,” Physical Review D, vol. 50, Article ID 3433, 1994.

J. Charles, A. Höcker, and H. Lacker, “CP violation and the CKM matrix: assessing the impact of the asymmetric B factories,” The European Physical Journal C, vol. 41, no. 1, pp. 1–131, 2005.

T. Feldmann, P. Kroll, and B. Stech, “Mixing and decay constants of pseudoscalar mesons,” Physical Review D, vol. 58, Article ID 114006, 1998.

T. Feldmann, P. Kroll, and B. Stech, “Mixing and decay constants of pseudoscalar mesons: the sequel,” Physics Letters B, vol. 449, no. 3-4, pp. 339–346, 1999.

F. Ambrosino, A. Antonelli, and M. Antonelli, “A global fit to determine the pseudoscalar mixing angle and the gluonium content of the η' meson,” Journal of High Energy Physics, vol. 907, Article ID 105, 2009.

P. Ball, G. W. Jones, and R. Zwicky, “B→Vγ beyond QCD factorization,” Physical Review D, vol. 75, Article ID 054004, 2007.

H.-Y. Cheng, C.-K. Chua, and C.-W. Hwang, “Covariant light-front approach for s-wave and p-wave mesons: Its application to decay constants and form factors,” Physical Review D, vol. 69, Article ID 074025, 2004.