We study the CP violation induced by the interference between two intermediate resonances K⁎(892)+ and K⁎(892)- in the phase space of singly-Cabibbo-suppressed decay D0→K+K-π0. We adopt the factorization-assisted topological approach in dealing with the decay amplitudes of D0→K±K⁎(892)∓. The CP asymmetries of two-body decays are predicted to be very tiny, which are (-1.27±0.25)×10-5 and (3.86±0.26)×10-5, respectively, for D0→K+K⁎(892)- and D0→K-K⁎(892)+, while the differential CP asymmetry of D0→K+K-π0 is enhanced because of the interference between the two intermediate resonances, which can reach as large as 3×10-4. For some NPs which have considerable impacts on the chromomagnetic dipole operator O8g, the global CP asymmetries of D0→K+K⁎(892)- and D0→K-K⁎(892)+ can be then increased to (0.56±0.08)×10-3 and (-0.50±0.04)×10-3, respectively. The regional CP asymmetry in the overlapped region of the phase space can be as large as (1.3±0.3)×10-3.
National Natural Science Foundation of China114470211157507711705081National Natural Science Foundation of Hunan Province2016JJ3104Innovation Group of Nuclear and Particle Physics in USCChina Scholarship Council1. Introduction
Charge-Parity (CP) violation, which was first discovered in K meson system in 1964 [1], is one of the most important phenomena in particle physics. In the Standard Model (SM), CP 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 CP violation is that the unitary phase is usually small. Nevertheless, CP 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 CP 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 CP asymmetry can be larger than the global one. In fact, such kind of enhanced CP violation has been observed in several three-body decay channels of B meson [4–7], which was followed by a number of theoretical works [8–19].
The study of CP violation in singly-Cabibbo-suppressed (SCS) D meson decays provides an ideal test of the SM and exploration of New Physics (NP) [20–23]. In the SM, CP violation is predicted to be very small in charm system. Experimental researches have shown that there is no significant CP violation so far in charmed hadron decays [24–33]. CP asymmetry in SCS D meson decay can be as small as(1)ACP~Vcb∗VubVcs∗Vusαsπ~10-4,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 ΔC=1 QCD penguin and chromomagnetic dipole operators, while such contributions can affect neither the Cabibbo-favored (CF) (c→sd¯u) nor the doubly-Cabibbo-suppressed (DCS) (c→ds¯u) decays [34]. Besides, the decays of charmed mesons offer a unique opportunity to probe CP violation in the up-type quark sector.
Several factorization approaches have been wildly used in nonleptonic B 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 B 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 B meson decays.
However, for D 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 SU(3) symmetry [44–47] and the factorization-assisted topological-amplitude (FAT) approach with the inclusion of flavor SU(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 CP violation of SCS D meson decay D0→K+K-π0 in the FAT approach. Our attention will be mainly focused on the region of the phase space where two intermediate resonances, K∗(892)+ and K∗(892)-, are overlapped. Before proceeding, it will be helpful to point out that direct CP asymmetry is hard to be isolated for decay process with CP-eigen-final-state. When the final state of the decay process is CP eigenstate, the time integrated CP violation for D0→f, which is defined as(2)af≡∫0∞ΓD0→fdt-∫0∞ΓD¯0→fdt∫0∞ΓD0→fdt+∫0∞ΓD¯0→fdt,can be expressed as [34](3)af=afd+afm+afi,where afd, afm, and afi are the CP asymmetries in decay, in mixing, and in the interference of decay and mixing, respectively. As is shown in [34, 50, 51], the indirect CP violation aind≡am+ai is universal and channel-independent for two-body CP-eigenstate. This conclusion is easy to be generalized to decay processes with three-body CP-eigenstate in the final state, such as D0→K+K-π0. In view of the universality of the indirect CP asymmetry, we will only consider the direct CP violations of the decay D0→K+K-π0 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 D0→K±K∗(892)∓ are formulated via the FAT approaches. In Section 3, we study the CP asymmetries of D0→K±K∗(892)∓ and the CP asymmetry of D0→K+K-π0 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.
In the overlapped region of the intermediate resonances K∗(892)+ and K∗(892)- in the phase space, the decay process D0→K+K-π0 is dominated by two cascade decays, D0→K+K∗(892)-→K+K-π0 and D0→K-K∗(892)+→K-K+π0, respectively. Consequently, the decay amplitude of D0→K+K-π0 can be expressed as(4)MD0→K+K-π0=MK∗++eiδMK∗-in the overlapped region, where MK∗+ and MK∗- are the amplitudes for the two cascade decays and δ is the relative strong phase. Note that nonresonance contributions have been neglected in (4).
The decay amplitude for the cascade decay D0→K+K∗(892)-→K+K-π0 can be expressed as(5)MK∗-=∑λMK∗-→K-π0λ·MD0→K∗-K+λsπ0K--mK∗-2+imK∗-ΓK∗-,where MK∗-→K-π0λ and MD0→K+K∗-λ represent the amplitudes corresponding to the strong decay K∗-→K-π0 and weak decay D0→K+K∗-, respectively, λ is the helicity index of K∗-, sπ0K- is the invariant mass square of π0K- system, and mK∗- and ΓK∗- are the mass and width of K∗(892)-, respectively. The decay amplitude for the cascade decay, D0→K-K∗(892)+→K-K+π0, is the same as (5) except replacing the subscripts K∗- and K± with K∗+ and K∓, respectively.
For the strong decays K∗(892)±→π0K±, one can express the decay amplitudes as(6)MK∗±→π0K±=gK∗±K±π0pπ0-pK±·εK∗±p,λ,where pπ0 and pK± represent the momentum for π0 and K± mesons, respectively, and gK∗±K±π0 is the effective coupling constant for the strong interaction, which can be extracted from the experimental data via(7)gK∗±K±π02=6πmK∗±2ΓK∗±→K±π0λK∗±3,with(8)λK∗±=12mK∗±mK∗±2-mπ0+mK±2·mK∗±2-mπ0-mK±2,and ΓK∗±→K±π0=Br(K∗±→K±π0)·ΓK∗±. The isospin symmetry of the strong interaction implies that ΓK∗±→K±π0≃(1/3)ΓK∗±.
The decay amplitudes for the weak decays, D0→K+K∗(892)- and D0→K-K∗(892)+, will be handled with the aforementioned FAT approach [48, 49]. The relevant topological tree and penguin diagrams for D→PV are displayed in Figure 1, where P and V denote a light pseudoscalar and vector meson (representing K± and K∗± in this paper), respectively.
The relevant topological diagrams for D→PV with (a) the color-favored tree amplitude TP(V), (b) the W-exchange amplitude EP(V), (c) the color-favored penguin amplitude PTP(V), (d) the gluon-annihilation penguin amplitude PEP(V), and (e) the gluon-exchange penguin amplitude PAP(V).
TP(V)
EP(V)
PTP(V)
PEP(V)
PAP(V)
The two tree diagrams in first line of Figure 1 represent the color-favored tree diagram for D→P(V) transition and the W-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 TP(V) and EP(V).
According to these topological structures, the amplitudes of the color-favored tree diagrams TP(V), which are dominated by the factorizable contributions, can be parameterized as(9)TP=GF2λsa2μfVmVF1D→PmV22ε∗·pD,and(10)TV=GF2λsa2μfPmVA0D→VmP22ε∗·pD,respectively, where GF is the Fermi constant, λs=VusVcs∗, with Vus and Vcs being the CKM matrix elements, a2(μ)=c2(μ)+c1(μ)/Nc, with c1(μ) and c2(μ) being the scale-dependent Wilson coefficients, and the number of color Nc=3, fV(P) and mV(P) are the decay constant and mass of the vector (pseudoscalar) meson, respectively, F1D→P and A0D→V are the form factors for the transitions D→P and D→V, respectively, ε is the polarization vector of the vector meson, and pD is the momentum of D meson. The scale μ 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](11)μ=ΛmD1-rP21-rV2,with the mass ratios rV(P)=mV(P)/mD, where Λ represents the soft degrees of freedom in the D meson, which is a free parameter.
For the W-exchange amplitudes, since the factorizable contributions to these amplitudes are helicity-suppressed, only the nonfactorizable contributions need to be considered. Therefore, the W-exchange amplitudes are parameterized as(12)EP,Vq=GF2λsc2μχqEeiϕqEfDmDfPfVfπfρε∗·pD,where mD is the mass of D meson, fD, fπ, and fρ are the decay constants of the D, π, and ρ mesons, respectively, and χqE and ϕqE characterize the strengths and the strong phases of the corresponding amplitudes, with q=u,d,s representing the strongly produced q quark pair. The ratio of fPfV over fπfρ indicates that the flavor SU(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 PTP(V), PEP(V), and PAP(V), respectively.
Since a vector meson cannot be generated from the scalar or pseudoscalar operator, the amplitude PTP does not include contributions from the penguin operator O5 or O6. Consequently, the color-favored penguin amplitudes PTP and PTV can be expressed as(13)PTP=-GF2λba4μfVmVF1D→PmV22ε∗·pD,and(14)PTV=-GF2λba4μ-rχa6μfPmVA0D→VmP22ε∗·pD,respectively, where λb=VubVcb∗ with Vub and Vcb∗ being the CKM matrix elements, a4,6(μ)=c4,6(μ)+c3,5(μ)/Nc, with c3,4,5,6 being the Wilson coefficients, and rχ is a chiral factor, which takes the form(15)rχ=2mP2mu+mqmq+mc,with mu(c,q) being the masses of u(c,q) quark. Note that the quark-loop corrections and the chromomagnetic-penguin contribution are also absorbed into c3,4,5,6 as shown in [49].
Similar to the amplitudes EP,V, the amplitudes PE only include the nonfactorizable contributions as well. Therefore, the amplitudes PEP,V, which are dominated by O4 and O6 [48], can be parameterized as(16)PEP,Vq=-GF2λbc4μ-c6μχqEeiϕqEfDmDfPfVfπfρε∗·pD.
For the amplitudes PAP and PAV, the helicity suppression does not apply to the matrix elements of O5,6, so the factorizable contributions exist. In the pole resonance model [54], after applying the Fierz transformation and the factorization hypothesis, the amplitudes PAP and PAV can be expressed as(17)PAPq=-GF2λb-2a6μ2gS1mD2-mP∗2fP∗mP∗0fDmD2mc+c3μχqAeiϕqAfDmDfPfVfπfρε∗·pD,and(18)PAVq=-GF2λb-2a6μ-2gS1mD2-mP∗2fP∗mP∗0fDmD2mc+c3μχqAeiϕqAfDmDfPfVfπfρε∗·pD,respectively, where gS is an effective strong coupling constant obtained from strong decays, e.g., ρ→ππ, K∗→Kπ, and ϕ→KK, and is set as gS=4.5 [54] in this work, mP∗ and fP∗ are the mass and decay constant of the pole resonant pseudoscalar meson P∗, respectively, and χqA and ϕqA are the strengths and the strong phases of the corresponding amplitudes.
From Figure 1, the decay amplitudes of D0→K+K∗(892)- and D0→K-K∗(892)+ in the FAT approach can be easily written down(19)MD0→K+K∗-λ=TK∗-+EK+u+PTK∗-+PEK∗-s+PEK+u+PAK∗-s,and(20)MD0→K-K∗+λ=TK-+EK∗+u+PTK-+PEK-s+PEK∗+u+PAK-s,respectively, where λ is the helicity of the polarization vector ε(p,λ). In the FAT approach, the fitted nonperturbative parameters, χq,sE, ϕq,sE, χq,sA, ϕq,sA, 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 D0→K+K∗(892)- and D0→K-K∗(892)+ by using the global fitted parameters for D→PV in [49]. One can see from Table 1 that the penguin contributions are greatly suppressed. PT is dominant in the penguin contributions of D0→K-K∗(892)+, while PT is small in D0→K+K∗(892)-, which is even smaller than the amplitude PA. This difference is because of the chirally enhanced factor contained in (14) while not in (13). The very small PE do not receive the contributions from the quark-loop and chromomagnetic penguins, since these two contributions to c4 and c6 are canceled with each other in (16). Besides, the relations PEVs=PEPs, PEVu=PEPu, and PEVs≠PEVu can be read from Table 1; this is because that the isospin symmetry and the flavor SU(3) breaking effect have been considered.
The magnitude of tree and penguin contributions (in unit of 10-3) corresponding to the topological amplitudes in (19) and (20). The factors “GF/2λs(ε∗⋅pD)” and “-GF/2λb(ε∗⋅pD)” are omitted in this table.
Decay modes
TK∗-
EK+u
PTK∗-
PEK∗-s
PEK+u
PAK∗-s
D0→K+K∗(892)-
0.23
-0.02+0.15i
3.83+4.32i
0.96-0.03i
0.13-0.81i
6.73+8.22i
TK-
EK∗+u
PTK-
PEK-s
PEK∗+u
PAK-s
D0→K-K∗(892)+
0.44
-0.02+0.15i
-23.3-19.3i
0.96-0.03i
0.13-0.81i
-8.53-5.53i
Since the form factors are inevitably model-dependent, we list in Table 2 the branching ratios of D0→K+K∗(892)- and D0→K-K∗(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 model-dependence 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.
Branching ratios (in unit of 10-3) of singly-Cabibbo-suppressed decays D0→K+K∗(892)- and D0→K-K∗(892)+. Both experimental data [55–57] and theoretical predictions of FAT approach of the branching ratios are listed.
The direct CP asymmetry for the two-body decay D→PV is defined as(21)ACPD→PV=MD→PV2-MD¯→P¯V¯2MD→PV2+MD¯→P¯V¯2,where MD¯→P¯V¯ represents the decay amplitude of the CP conjugate process D¯→P¯V¯, such as D¯0→K+K∗(892)- or D¯0→K-K∗(892)+. In the framework of FAT approach, we predict very small direct CP asymmetries of D0→K+K∗(892)- and D0→K-K∗(892)+ presented in Table 3. The uncertainties induced by the model-dependence of form factor to the CP asymmetries of D0→K+K∗(892)- and D0→K-K∗(892)+ are about 30% and 10%, respectively.
CP asymmetries (in unit of 10-5) of D0→K+K∗(892)- and D0→K-K∗(892)+ predicted by the FAT approach with pole, dipole, and CLF models adopted. The uncertainties in this table are mainly from decay constants.
Form factors
ACP(D0→K+K∗(892)-)
ACP(D0→K-K∗(892)+)
Pole
-1.45±0.25
3.60±0.23
Dipole
-1.63±0.26
3.70±0.24
CLF
-1.27±0.25
3.86±0.26
The differential CP asymmetry of the three-body decay D0→K+K-π0, which is a function of the invariant mass of sπ0K+ and sπ0K-, is defined as(22)ACPD0→K+K-π0sπ0K+,sπ0K-=MD0→K+K-π02-MD¯0→K-K+π02MD0→K+K-π02+MD¯0→K-K+π02,where the invariant mass sπ0K±=(pπ0+pK±)2. As can be seen from (4), the differential CP asymmetry ACPD0→K+K-π0 depends on the relative strong phase δ, 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 δ from data. By using a Dalitz plot technique [55, 58, 59], the phase difference δexp between D0 decays to K+K∗(892)- and K-K∗(892)+ can be extracted from data. One should notice that δexp is not the same as the strong phase δ defined in (4). The strong phase δ is the relative phase between the decay amplitudes of D0→K+K∗(892)- and D0→K-K∗(892)+. On the other hand, the phase δexp is defined through(23)MD0→K+K-π0=MK∗++eiδexpMK∗-eiδK∗+in the overlapped region of the phase space, where δK∗± is the phase of the amplitude MK∗±:(24)MK∗±=MK∗±eiδK∗±.Therefore, neglecting the CKM suppressed penguin amplitudes, δexp and δ can be related by(25)δexp-δ≈δK∗-K+-δK∗+K-,where δK∗∓K±=arg(TK∗∓+EK±u) are the phases in tree-level amplitudes of D0→K±K∗(892)∓ and are equivalent to δK∗∓ if the penguin amplitudes are neglected. With the relation of (25), and δexp=-35.5∘±4.1∘ measured by the BABAR Collaboration [56], we have δ≈-51.85∘±4.1∘.
In Figure 2, we present the differential CP asymmetry of D0→K+K-π0 in the overlapped region of K∗(892)- and K∗(892)+ in the phase space, with δ=-51.85∘. Namely, we will focus on the region mK∗-2ΓK∗<sπ0K-,sπ0K+<mK∗+2ΓK∗ of the phase space. One can see from Figure 2 that the differential CP asymmetry of D0→K+K-π0 can reach 3.0×10-4 in the overlapped region, which is about 10 times larger than the CP asymmetries of the corresponding two-body decay channels shown in Table 3.
The differential CP asymmetry distribution of D0→K+K-π0 in the overlapped region of K∗(892)- and K∗(892)+ in the phase space.
The behavior of the differential CP asymmetry of D0→K+K-π0 in Figure 2 motivates us to separate this region into four areas, area A (mK∗<sπ0K-<mK∗+2ΓK∗,mK∗-2ΓK∗<sπ0K+<mK∗), area B (mK∗<sπ0K-<mK∗+2ΓK∗,mK∗<sπ0K+<mK∗+2ΓK∗), area C (mK∗-2ΓK∗<sπ0K-<mK∗,mK∗-2ΓK∗<sπ0K+<mK∗), and area D (mK∗-2ΓK∗<sπ0K-<mK∗,mK∗<sπ0K+<mK∗+2ΓK∗). We further consider the observable of regional CP asymmetry in areas A, B, C, and D displayed in Table 4, which is defined by(26)ACPΩ=∫ΩMtot2-M¯tot2dsπ0K-sπ0K+∫ΩMtot2+M¯tot2dsπ0K-sπ0K+,where Ω represents a certain region of the phase space.
Three from factor models: the pole, dipole, and CLF models are used for the regional CP asymmetries (in unit of 10-4) in the four areas, A, B, C, and D, of the phase space.
Form factors
ACPA
ACPB
ACPC
ACPD
ACPAll
Pole
0.87±0.11
0.42±0.08
0.39±0.07
-0.30±0.08
0.33±0.05
Dipole
0.87±0.11
0.41±0.08
0.38±0.07
-0.30±0.08
0.32±0.05
CLF
0.84±0.10
0.45±0.08
0.42±0.07
-0.25±0.08
0.36±0.06
Comparing with the CP asymmetries of two-body decays, the regional CP 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 δexp. In addition, if we focus on the right part of area A, that is, mK∗<sπ0K-<mK∗+2ΓK∗,mK∗-ΓK∗<sπ0K+<mK∗, the regional CP 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 CP asymmetry. For example, if we replace the Breit-Wigner propagator by the Flatté Parametrization [60], the correction to the regional CP asymmetry will be about 1%.
Since the CP asymmetry of D0→K+K-π0 is extremely suppressed, it should be more sensitive to the NP. For example, some NPs have considerable impacts on the chromomagnetic dipole operator O8g [34, 61–66]. Consequently, the CP violation in SCS decays may be further enhanced. In practice, the NP contributions can be absorbed into the corresponding effective Wilson coefficient c8geff [67, 68]. For comparison, we first consider a relative small value of c8geff (as in [48, 64]) lying within the range (0,1) and the global CP asymmetry of D0→K∗(892)±K∓ are no larger than 5×10-5. Moreover, if we follow [49] taking c8geff≈10 (while c8geff=10, which is extracted from ΔACP 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 c8g. More details about the squark-gluino loops and other models in SUSY can be found in [34, 62, 70–72]), the global CP asymmetries of D0→K+K∗(892)- and D0→K-K∗(892)+ are then (0.56±0.08)×10-3 and (-0.50±0.04)×10-3, respectively.
We further display the CP asymmetry of D0→K+K-π0 in the overlapped region of K∗(892)- and K∗(892)+ in Figures 3(a) and 3(b) for c8geff=1 and c8geff=10, respectively. After taking the interference effect into account, the differential CP asymmetry of D0→K+K-π0 can be increased as large as 5.5×10-4 and 2.8×10-3 for c8geff=1 and c8geff=10, respectively. The regional ones (in phase space of 0.74GeV<sπ0K-<0.81GeV,0.84<sπ0K+<mK∗+2ΓK∗ ) can reach (2.7±0.5)×10-4 and (1.3±0.3)×10-3 for c8geff=1 and c8geff=10, respectively.
The differential CP asymmetry distribution of D0→K+K-π0 for (a) c8geff=1 and (b) c8geff=10, in the overlapped region of K∗(892)- and K∗(892)+ in the phase space.
4. Discussion and Conclusion
In this work, we studied CP violations in D0→K∗(892)±K∓→K+K-π0 via the FAT approach. The CP violations in two-body decay processes D0→K+K∗(892)- and D0→K-K∗(892)+ are very small, which are (-1.27±0.25)×10-5 and (3.86±0.26)×10-5, respectively. Our discussion shows that the CP violation can be enhanced by the interference effect in three-body decay D0→K+K-π0. The differential CP asymmetry can reach 3.0×10-4 when the interference effect is taken into account, while the regional one can be as large as (1.09±0.16)×10-4.
Besides, since the chromomagnetic dipole operator O8g is sensitive to some NPs, the inclusion of this kind of NPs will lead to a much larger global CP asymmetries of D0→K+K∗(892)- and D0→K-K∗(892)+, which are (0.56±0.08)×10-3 and (-0.50±0.04)×10-3, respectively, while the regional CP asymmetry of D0→K+K-π0 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 CP asymmetry is attributed to the large c8geff, which is almost impossible for the SM to generate such large contribution, it will indicate NP if such CP violation is observed. Here, we roughly estimate the number of D0D¯0 needed for testing such kind of asymmetries, which is about 1/Br1/ACP2~109. This could be observed in the future experiments at Belle II [73, 74], while the current largest D0D¯0 yields are about 108 at BABAR and Belle [75, 76] and 107 at BESIII [77].
AppendixSome Useful Formulas and Input Parameters
(1) Effective Hamiltonian and Wilson Coefficients. The weak effective Hamiltonian for SCS D meson decays, based on the Operator Product Expansion (OPE) and Heavy Quark Effective Theory (HQET), can be expressed as [78](A.1)Heff=GF2∑q=d,sλqc1O1q+c2O2q-λb∑i=36ciOi+c8gO8g+h.c.,where GF is the Fermi constant, λq=VuqVcq∗, ci(i=1,…,6) is the Wilson coefficient, and O1q, O2q, Oi(i=1,…,6), and O8g are four-fermion operators which are constructed from different combinations of quark fields. The four-fermion operators take the following form:(A.2)O1q=u¯αγμ1-γ5qβq¯βγμ1-γ5cα,O2q=u¯γμ1-γ5qq¯γμ1-γ5c,O3=u¯γμ1-γ5c∑q′q¯′γμ1-γ5q′,O4=u¯αγμ1-γ5cβ∑q′q¯′βγμ1-γ5qα′,O5=u¯γμ1-γ5c∑q′q¯′γμ1+γ5q′,O6=u¯αγμ1-γ5cβ∑q′q¯β′γμ1+γ5qα′,O8g=-gs8π2mcu¯σμν1+γ5Gμνc,where α and β are color indices and q′=u,d,s. Among all these operators, O1q and O2q are tree operators, O3-O6 are QCD penguin operators, and O8g 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 μ=1GeV, which can be found in [48].
(2) CKM Matrix. We use the Wolfenstein parameterization for the CKM matrix elements, which up to order O(λ8) read [79, 80](A.3)Vus=λ-12A2λ7ρ2+η2,Vcs=1-12λ2-18λ41+4A2-116λ61-4A2+16A2ρ+iη-1128λ85-8A2+16A4,Vub=Aλ3ρ-iη,Vcb=Aλ2-12A3λ8ρ2+η2,where A,ρ,η, and λ are the Wolfenstein parameters, which satisfy following relation:(A.4)ρ+iη=1-A2λ4ρ¯+iη¯1-λ21-A2λ4ρ¯+iη¯.Numerical values of Wolfenstein parameters which have been used in this work are as follows:(A.5)λ=0.22548-0.00034+0.00068,A=0.810-0.024+0.018,ρ¯=0.145-0.007+0.013,η¯=0.343-0.012+0.011.
(3) Decay Constants and Form Factors. In (17) and (18), the pole resonance model was employed for the matrix element PVq¯1q20 in the annihilation diagrams. By considering angular momentum conservation at weak vertex and all conservation laws are preserved at strong vertex, the matrix element PVq¯1q20 is therefore dominated by a pseudoscalar resonance [54],(A.6)PVq¯1q20=PV∣P∗P∗q¯1q20=gP∗PVmP∗mD2-mP∗2fP∗,where gP∗PV is a strong coupling constant and mP∗ and fP∗ are the mass and decay constant of the pseudoscalar resonance P∗. Therefore, η and η′ are the dominant resonances for the final states of K∗±K∓, which can be expressed as flavor mixing of ηq and ηs,(A.7)ηη′=cosϕ-sinϕsinϕcosϕηqηswhere ϕ is the mixing angle and ηq and ηs are defined by(A.8)ηq=12uu¯+dd¯,ηs=ss¯.The decay constants of η and η′ are defined by(A.9)0u¯γμγ5uηp=ifηupμ,0u¯γμγ5uη′p=ifη′upμ,0d¯γμγ5dηp=ifηdpμ,0d¯γμγ5dη′p=ifη′dpμ,0s¯γμγ5sηp=ifηspμ,0s¯γμγ5sη′p=ifη′spμ,where(A.10)fηu=fηd=12fηq,fη′u=fη′d=12fη′q.According to [81, 82], the decay constants of η and η′ can be expressed as(A.11)fηq=fqcosϕ,fη′q=fqsinϕ,fηs=-fssinϕ,fη′s=fscosϕ,where fq=(1.07±0.02)fπ and fs=(1.34±0.02)fπ [81], and the mixing angle ϕ=(40.4±0.6)∘ [83]. Other decay constants used in this paper are listed in Table 5.
The meson decay constants used in this paper (MeV) [57, 84].
fK∗
fρ
fK
fπ
fD
220(5)
216(3)
156(0.4)
130(1.7)
208(10)
The transition form factors A0D0→K∗- and F1D0→K-, 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):(A.12)Fq2=F01-aq2/mD2+bq2/mD22.
The parameters of D→K∗,K transitions form factors in (A.12).
Form factor
A0D→K∗
F1D→K
F(0)
0.69
0.78
a
1.04
1.05
b
0.44
0.23
(4) Decay Rate. The decay width takes the form(A.13)ΓD→KK∗=p138πmK∗2MD→KK∗ε∗·pD2,where p1 represents the center of mass (c.m.) 3-momentum of each meson in the final state and is given by(A.14)p1=mD2-mK∗+mK2mD2-mK∗-mK22mD.M 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.
ChristensonJ. H.CroninJ. W.FitchV. L.TurlayR.Evidence for the 2π Decay of the K20 MesonKobayashiM.MaskawaT.CP-violation in the renormalizable theory of weak interactionCabibboN.Unitary symmetry and leptonic decaysAaijR.LHCb CollaborationMeasurement of CP Violation in the Phase Space of B^{±}→K^{±}π^{+}π^{−} and B^{±}→K^{±}K^{+}K^{−} DecaysAaijR.LHCb CollaborationMeasurement of CP Violation in the Phase Space of B^{±}→K^{+}K^{−}π^{±} and B^{±}→π^{+}π^{−}π^{±} DecaysAaijR.LHCb CollaborationMeasurements of CP violation in the three-body phase space of charmless B^{±} decaysNogueiraJ. H. A.AmatoS.AustregesiloA.Summary of the 2015 LHCb workshop on multi-body decays of D and B mesonshttps://arxiv.org/abs/1605.03889ZhangZ.-H.GuoX.-H.YangY.-D.CP violation in B^{±}→π^{±}π^{+}π^{−} in the region with low invariant mass of one π^{+}π^{−} pairBediagaI.FredericoT.LourençoO.CP violation and CPT invariance in B^{±} decays with final state interactionsChengH.-Y.ChuaC.-K.Branching fractions and direct CP violation in charmless three-body decays of B mesonsZhangZ.-H.GuoX.-H.YangY.-D.CP violation induced by the interference of scalar and vector resonances in three-body decays of bottom mesonshttps://arxiv.org/abs/1308.5242BhattacharyaB.GronauM.RosnerJ. L.CP asymmetries in three-body B± decays to charged pions and kaonsXuD.LiG.-N.HeX.-G.Large SU(3) Breaking Effects and CP Violation in B^{+} Decays Into Three Charged Octet Pseudoscalar MesonsWangW.-F.HuH.-C.LiH.-N.LüC.-D.Direct CP asymmetries of three-body B decays in perturbative QCDZhangZ.-H.WangC.GuoX.-H.Possible large CP violation in three-body decays of heavy baryonWangC.ZhangZ.-H.WangZ.-Y.GuoX.-H.Localized direct CP violation in B^{±} → ρ^{0}(ω)π^{±} → π^{+}π^{−}π^{±}NogueiraJ. H. A.BediagaI.CavalcanteA. B. R.FredericoT.LourençoO.CP violation: Dalitz interference, CPT, and final state interactionsDedonderJ.FurmanA.KamińskiR.LeśniakL.LoiseauB.S-, P- and D-wave ππ final state interactions and CP violation in B^{±} → π^{±}π^{
∓}π^{±} decaysEl-BennichB.FurmanA.KamińskiR.LeśniakL.LoiseauB.Interference between f_{0}(980) and ρ(770)^{0} resonances in B → π^{+}π^{−}K decaysBigiI.SandaA.On D^{0}D-0 mixing and CP violationBlaylockG.SeidenA.NirY.The role of CP violation in D^{0}D-0 mixingBergmannS.GrossmanY.LigetiZ.NirY.PetrovA. A.Lessons from CLEO and FOCUS measurements of D^{0}-D-0 mixing parametersNiersteU.SchachtS.Neutral D→KK^{∗} decays as discovery channels for charm CP violationBonviciniG.CLEO CollaborationSearch for CP violation in D^{0}→KSoπ^{0}, D^{0}→π^{0}π^{0} and D^{0}→KSoKSo decaysLinkJ. M.Search for CP Violation in the decays D^{+}→K_{S}π^{+} and D^{+}→K_{S}K^{+}AaltonenT.CDF CollaborationMeasurement of CP-violating asymmetries in D^{0}→π^{+}π^{−} and D^{0}→K^{+}K^{−} decays at CDFCenciR.Mixing and CP Violation in Charm Decays at BABARProceedings of the 7th International Workshop on the CKM Unitarity Triangle (CKM 2012)2012Ohio, USACincinnatiLeesJ. P.BABAR CollaborationSearch for CP violation in the decays D^{±}→KSoK^{±}, Ds±→KSoK^{±}, and Ds±→KSoπ^{±}StaričM.AbdesselamA.AdachiI.Measurement of D^{0}-D-^{0} mixing and search for CP violation in D^{0} → K^{+}K^{-}, π^{+}π^{-} decays with the full Belle data setAaijR.AaijR.AdevaB.Measurement of CP asymmetries in D^{±}→η′π^{±} and Ds±→η′π^{±} decaysAaijR.LHCb CollaborationMeasurements of charm mixing and CP violation using D^{0}→K^{±}π^{
∓} decaysLHCb CollaborationSearch for CP violation in the phase space of D^{0} → π^{+}π^{-}π^{+}π^{-} decaysBhardwajV.Latest Charm Mixing and CP results from B-factories139Proceedings of the 9th International Workshop on the CKM Unitarity Triangle (CKM2016)2017Mumbai, IndiaGrossmanY.KaganA. L.NirY.New physics and CP violation in singly Cabibbo suppressed DdecaysBjorkenJ. D.Topics in B-physicsDuganM. J.GrinsteinB.QCD basis for factorization in decays of heavy mesonsBenekeM.BuchallaG.NeubertM.SachrajdaC. T.QCD factorization for B→ππ decays: strong phases and CP violation in the heavy quark limitBenekeM.NeubertM.QCD factorization for B→PP and B→PV decaysBoitoD. R.DedonderJ.El-BennichB.LeitnerO.LoiseauB.Scalar resonances in a unitaryFurmanaA.KamińskiaR.LeśniakaL.LoiseaubB.Long-distance effects and final state interactions in B→ππK and B→KK-K decaysKeumY.-Y.LiH.-N.SandaA. I.Penguin enhancement and B→Kπ decays in perturbative QCDBauerC. W.PirjolD.RothsteinI. Z.StewartI. W.B→M_{1}M_{2}: Factorization, charming penguins, strong phases, and polarizationLoiseauB.Theory overview on amplitude analyses with charm decaysProceedings of the 8th International Workshop on Charm Physics (Charm 2016)2017Bologna, ItalyChengH.-Y.ChiangC.-W.Direct CP violation in two-body hadronic charmed meson decaysChauL.-L.Quark mixing in weak interactionsBhattacharyaB.GronauM.RosnerJ. L.Publisher’s Note:ChengH.-Y.ChiangC.-W.KuoA.-L.Global analysis of two-body D→VP decays within the framework of flavor symmetryLiH.-N.LuC.-D.YuF.-S.Branching ratios and direct CP asymmetries in D→PP decaysQinQ.LiH.-N.LüC.-D.YuF.-S.Branching ratios and direct CP asymmetries in D→PV decaysAubert (BABAR Collaboration)B.Limits on D^{0}−D-0 Mixing and CP Violation from the Ratio of Lifetimes for Decay to K^{−}π^{+}, K^{−}K^{+}, and π^{−}π^{+}Abe (Belle Collaboration)K.Measurement of the D^{0}D-0 lifetime difference using D^{0}→ Kπ/KK decaysProceedings of the 21st International Symposium on Lepton and Photon Interactions at High Energies2003Batavia, ILL, USA1116https://arxiv.org/abs/hep-ex/0308034KeumY.-Y.LiH.-N.SandacA. I.Fat penguins and imaginary penguins in perturbative QCDLüC.-D.UkaiK.YangM.-Z.Branching ratio and CP violation of B→ππ decays in the perturbative QCD approachYuF.-S.WangX.-X.LüC.-D.Nonleptonic two-body decays of charmed mesonsCawlfield (CLEO Collaboration)C.Measurement of interfering K*^{+}K^{−}and K*^{−}K^{+}amplitudes in the decay D^{0}→K^{+}K^{−}π^{0}AubertB.ZhangL. L.ChenS.Amplitude analysis of the decay D^{0}→K^{−}K^{+}π^{0}PatrignaniC.AgasheK.AielliG.Review of Particle PhysicsRosnerJ. L.SuprunD. A.Measuring the relative strong phase in D^{0}→K*^{+}K^{−}and D^{0}→K*^{−}K^{+} decaysBediagaI.BigiI. I.GomesA.GuerrerG.MirandaJ.dos ReisA. C.On a CP anisotropy measurement in the Dalitz plotFlatteS. M.Coupled-channel analysis of the πη and KK- systems near KK- thresholdGoldenM.GrinsteinB.Enhanced CP violations in hadronic charm decaysGiudiceG. F.IsidoriG.ParadisiP.Direct CP violation in charm and flavor mixing beyond the SMGronauM.New physics in singly Cabibbo-suppressed D decaysBrodJ.KaganA. L.ZupanJ.Size of direct CP violation in singly Cabibbo-suppressed D decaysGrossmanY.KaganA. L.ZupanJ.Testing for new physics in singly Cabibbo suppressed D decaysIsidoriG.KamenikJ. F.LigetieZ.PerezG.Implications of the LHCb evidence for charm CP violationBenekeM.BuchallaG.NeubertM.SachrajdaC. T.QCD factorization in B→πK, ππ decays and extraction of Wolfenstein parametersLiH.-n.MishimaS.SandaA. I.Resolution to the B→πK puzzleAaij (LHCb Collaboration)R.Evidence for CP violation in time-integrated D^{0}→h^{-}h^{+} decay ratesGabbianiF.GabrielliE.MasieroA.SilvestriniL.A complete analysis of FCNC and CP constraints in general SUSY extensions of the standard modelGabrielliE.MasieroA.SilvestriniL.Flavour changing neutral currents and CP violating processes in generalized supersymmetric theoriesHagelinJ. S.KelleyS.TanakaT.Supersymmetric flavor-changing neutral currents: exact amplitudes and phenomenological analysisDe PietroG.Charm physics prospects at Belle II314Proceedings of the European Physical Society Conference on High Energy Physics (EPS-HEP2017)2017Venice, Italy66210.22323/1.314.0662AbeT.AdachiI.AdamczykK.Belle II Technical Design Reporthttps://arxiv.org/abs/1011.0352LeesJ. P.PoireauV.TisserandV.Measurement of the D^{0}→π^{−}e^{+}ν_{e} differential decay branching fraction as a function of q^{2} and study of form factor parametrizationsNisarN. K.MohantyG. B.TrabelsiK.Search for the rare decay D^{0}→γγ at BelleAblikim (BESIII Collaboration)M.Search for D^{0}→γγ and improved measurement of the branching fraction for D^{0}→π^{0}π^{0}BuchallaG.BurasA. J.LautenbacherM. E.Weak decays beyond leading logarithmsBurasA. J.LautenbacherM. E.OstermaierG.Waiting for the top quark mass, K^{+}→π^{+}νν-, Bs0-B-s0 mixing, and CP asymmetries in B decaysCharlesJ.HöckerA.LackerH.CP violation and the CKM matrix: assessing the impact of the asymmetric B factoriesFeldmannT.KrollP.StechB.Mixing and decay constants of pseudoscalar mesonsFeldmannT.KrollP.StechB.Mixing and decay constants of pseudoscalar mesons: the sequelAmbrosinoF.AntonelliA.AntonelliM.A global fit to determine the pseudoscalar mixing angle and the gluonium content of the η' mesonBallP.JonesG. W.ZwickyR.B→Vγ beyond QCD factorizationChengH.-Y.ChuaC.-K.HwangC.-W.Covariant light-front approach for s-wave and p-wave mesons: Its application to decay constants and form factors