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Analysis of CP Violation in [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0].

1. 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

[A.sub.CP] ~ [[absolute value of ([V.sup.*.sub.cb][V.sub.ub])]/[absolute value of ([V.sup.*.sub.cs][V.sub.us])]][[[alpha].sub.s]/[pi]] ~ [10.sup.-4], (1)

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 [DELTA]C = 1 QCD penguin and chromomagnetic dipole operators, while such contributions can affect neither the Cabibbo-favored (CF) (c [right arrow] s[bar.d]u) nor the doubly-Cabibbo-suppressed (DCS) (c [right arrow] d[bar.s]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 quarkmass, 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 [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] in the FAT approach. Our attention will be mainly focused on the region of the phase space where two intermediate resonances, [K.sup.*][(892).sup.+] and [K.sup.*][(892).sup.-], 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 [D.sup.0] [right arrow] f, which is defined as

[mathematical expression not reproducible], (2)

can be expressed as [34]

[a.sub.f] = [a.sup.d.sub.f] + [a.sup.m.sub.f] + [a.sup.i.sub.f], (3)

where [a.sup.d.sub.f], [a.sup.m.sub.f], and [a.sup.i.sub.f] 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 [a.sup.ind] [equivalent to] [a.sup.m] + [a.sup.i] 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 [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0]. In view of the universality of the indirect CP asymmetry, we will only consider the direct CP violations of the decay [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.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 [D.sup.0] [right arrow] [K.sup.[+ or -]][K.sup.*][(892).sup.[+ or -]] are formulated via the FAT approaches. In Section 3, we study the CP asymmetries of [D.sup.0] [right arrow] [K.sup.[+ or -]][K.sup.*][(892).sup.[+ or -]] and the CP asymmetry of [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.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.

2. Decay Amplitude for [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0]

In the overlapped region of the intermediate resonances [K.sup.*][(892).sup.+] and [K.sup.*][(892).sup.-] in the phase space, the decay process [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] is dominated by two cascade decays, [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] and [D.sup.0] [right arrow] [K.sup.- ][K.sup.*][(892).sup.+] [right arrow] [K.sup.-][K.sup.+][[pi].sup.0], respectively. Consequently, the decay amplitude of [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] can be expressed as

[mathematical expression not reproducible] (4)

in the overlapped region, where [mathematical expression not reproducible] and [mathematical expression not reproducible] are the amplitudes for the two cascade decays and S is the relative strong phase. Note that nonresonance contributions have been neglected in (4).

The decay amplitude for the cascade decay [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] can be expressed as

[mathematical expression not reproducible], (5)

where [mathematical expression not reproducible] and [mathematical expression not reproducible] represent the amplitudes corresponding to the strong decay [K.sup.*-] [right arrow] [K.sup.-][[pi].sup.0] and weak decay [D.sup.0] [right arrow] [K.sup.+][K.sup.*-], respectively, [lambda] is the helicity index of [mathematical expression not reproducible] is the invariant mass square of [[pi].sup.0][K.sup.-] system, and [mathematical expression not reproducible] and rK.- are the mass and width of [K.sup.*][(892).sup.-], respectively. The decay amplitude for the cascade decay, [D.sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+] [right arrow] [K.sup.-][K.sup.+][[pi].sup.0], is the same as (5) except replacing the subscripts [K.sup.*-] and [K.sup.[+ or -]] with [K.sup.*+] and [K.sup.[+ or -]], respectively.

For the strong decays [K.sup.*][(892).sup.[+ or -]] [right arrow] [[pi].sup.0][K.sup.[+ or -]], one can express the decay amplitudes as

[mathematical expression not reproducible], (6)

where [mathematical expression not reproducible] and [mathematical expression not reproducible] represent the momentum for [[pi].sup.0] and [K.sup.[+ or -]] mesons, respectively, and [mathematical expression not reproducible] is the effective coupling constant for the strong interaction, which can be extracted from the experimental data via

[mathematical expression not reproducible], (7)

with

[mathematical expression not reproducible], (8)

and [mathematical expression not reproducible]. The isospin symmetry of the strong interaction implies that [mathematical expression not reproducible].

The decay amplitudes for the weak decays, [D.sub.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] and [D.sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+], will be handled with the aforementioned FAT approach [48,49]. The relevant topological tree and penguin diagrams for D [right arrow] PV are displayed in Figure 1, where P and V denote a light pseudoscalar and vector meson (representing [K.sup.[+ or -]] and [K.sup.*[+ or -]] in this paper), respectively.

The two tree diagrams in first line of Figure 1 represent the color-favored tree diagram for D [right arrow] 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 [T.sub.P(V)] and [E.sub.P(V)].

According to these topological structures, the amplitudes of the color-favored tree diagrams [T.sub.P(V)], which are dominated by the factorizable contributions, can be parameterized as

[T.sub.P] = [[G.sub.F]/[square root of 2]][[lambda].sub.s][a.sub.2]([mu]) [f.sub.V][m.sub.V][F.sup.D[right arrow]P.sub.1] ([m.sup.2.sub.V])2([[epsilon].sup.*] x [p.sub.D]), (9)

and

[T.sub.V] = [[G.sub.F]/[square root of 2]][[lambda].sub.s][a.sub.2]([mu]) [f.sub.P][m.sub.V][A.sup.D[right arrow]V.sub.0] ([m.sup.2.sub.P])2([[epsilon].sup.*] x [p.sub.D]), (10)

respectively, where [G.sub.F] is the Fermi constant, [[lambda].sub.s] = [V.sub.us][V.sup.*.sub.cs], with [V.sub.us] and [V.sub.cs] being the CKM matrix elements, [a.sub.2]([mu]) = [c.sub.2]([mu]) + [c.sub.1]([mu])/[N.sub.c], with [c.sub.1]([mu]) and [c.sub.2]([mu]) being the scale-dependent Wilson coefficients, and the number of color [N.sub.c] = 3, [f.sub.V{P)] and [m.sub.V(P)] are the decay constant and mass of the vector (pseudoscalar) meson, respectively, [F.sup.D[right arrow]P.sub.1] and [A.sup.D[right arrow]V.sub.0] are the form factors for the transitions D [right arrow] P and D [right arrow] V, respectively, e is the polarization vector of the vector meson, and [p.sub.D] is the momentum of D meson. The scale [mu] 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]

[mu] = [square root of ([LAMBDA][m.sub.D](1 - [r.sup.2.sub.P)(1 - [r.sup.2.sub.V]))], (11)

with the mass ratios [r.sub.V(P)] = [m.sub.V(P)]/[m.sub.D], where [LAMBDA] 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

[mathematical expression not reproducible], (12)

where [m.sub.D] is the mass of D meson, [f.sub.D], [f.sub.[pi]], and [f.sub.[rho]] are the decay constants of the D, [pi], and [rho] mesons, respectively, and [[chi].sup.E.sub.q] and [[phi].sup.E.sub.q] 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 [f.sub.P][f.sub.V] over [f.sub.[pi]][f.sub.[rho]] 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 [PT.sub.P(V)], [PE.sub.P(V)], and [PA.sub.P(V)], respectively.

Since a vector meson cannot be generated from the scalar or pseudoscalar operator, the amplitude [PT.sub.P] does not include contributions from the penguin operator [O.sub.5] or [O.sub.6]. Consequently, the color-favored penguin amplitudes [PT.sub.P] and [PT.sub.V] can be expressed as

[PT.sub.P] = -[[G.sub.F]/[square root of 2]][[lambda].sub.b][a.sub.4]([mu]) [f.sub.V][m.sub.V][F.sup.D[right arrow]P.sub.1] ([m.sup.2.sub.V])2([[epsilon].sup.*] x [p.sub.D]), (13)

and

[PT.sub.V] = -[[G.sub.F]/[square root of 2]][[lambda].sub.b][[a.sub.4]([mu]) - [r.sub.[chi]][a.sub.6]([mu])] [f.sub.P][m.sub.V][A.sup.D[right arrow]V.sub.0] ([m.sup.2.sub.P]) x 2([[epsilon].sup.*] x [p.sub.D]), (14)

respectively, where [[lambda].sub.b] = [V.sub.ub][V.sup.*.sub.cb] with [V.sub.ub] and [V.sup.*.sub.cb] being the CKM matrix elements, [a.sub.4,6]([mu]) = [c.sub.4,6]([mu]) + c[3.sub.,5]([mu])/[N.sub.c], with [c.sub.3,4,5,6] being the Wilson coefficients, and [r.sub.[chi]] is a chiral factor, which takes the form

[r.sub.[chi]] = 2[m.sup.2.sub.P]/([m.sub.u] + [m.sub.q])([m.sub.q] + [m.sub.c]), (15)

with [m.sub.u(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 [c.sub.3,4,5,6] as shown in [49].

Similar to the amplitudes [E.sub.P,V], the amplitudes PE only include the nonfactorizable contributions as well. Therefore, the amplitudes [PE.sub.P,V], which are dominated by [O.sub.4] and [O.sub.6] [48], can be parameterized as

[mathematical expression not reproducible]. (16)

For the amplitudes [PA.sub.P] and [PA.sub.V], the helicity suppression does not apply to the matrix elements of [O.sub.5,6], so the factorizable contributions exist. In the pole resonance model [54], after applying the Fierz transformation and the factorization hypothesis, the amplitudes [PA.sub.P] and [PA.sub.V] can be expressed as

[mathematical expression not reproducible], (17)

and

[mathematical expression not reproducible], (18)

respectively, where [g.sub.S] is an effective strong coupling constant obtained from strong decays, e.g., [rho] [right arrow] [pi][pi], [K.sup.*] [right arrow] K[pi], and [phi] [right arrow] KK, and is set as [g.sub.S] = 4.5 [54] in this work, [mathematical expression not reproducible] and [mathematical expression not reproducible] are the mass and decay constant of the pole resonant pseudoscalar meson [P.sup.*], respectively, [[chi].sup.A.sub.q] and [[phi].sup.A.sub.q] and are the strengths and the strong phases of the corresponding amplitudes.

From Figure 1, the decay amplitudes of [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] and [D.sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+] in the FAT approach can be easily written down

[mathematical expression not reproducible], (19)

and

[mathematical expression not reproducible], (20)

respectively, where [lambda] is the helicity of the polarization vector [epsilon](p, [lambda]). In the FAT approach, the fitted nonperturbative parameters, [[chi].sup.E.sub.q,s], [[phi].sup.E.sub.q,s], [[chi].sup.A.sub.q,s], [[phi].sup.A.sub.q,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 [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] and [D.sup.0] [right arrow] [K.sup.- ][K.sup.*][(892).sup.+] by using the global fitted parameters for D [right arrow] PV in [49]. One can see from Table 1 that the penguin contributions are greatly suppressed. PT is dominant in the penguin contributions of [D.sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+], while PT is small in [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-], 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 [c.sub.4] and [c.sub.6] are canceled with each other in (16). Besides, the relations [PE.sup.s.sub.V] = [PE.sup.s.sub.P], [PE.sup.u.sub.V] = [PE.sup.u.sub.P], and [PE.sup.s.sub.V] [not equal to] [PE.sup.u.sub.V] can be read from Table 1; this is because that the isospin symmetry and the flavor SU(3) breaking effect have been considered.

Since the form factors are inevitably model-dependent, we list in Table 2 the branching ratios of [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] and [D.sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+] 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.

3. CP Asymmetries for [D.sup.0] [right arrow] [K.sup.[+ or -]][K.sup.*][(892).sup.[+ or -]] and [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0]

The direct CP asymmetry for the two-body decay D [right arrow] PV is defined as

[mathematical expression not reproducible], (21)

where [M.sub.[bar.D][right arrow][bar.PV]] represents the decay amplitude of the CP conjugate process [bar.D] [right arrow] [bar.PV], such as [[bar.D].sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] or [[bar.D].sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+]. In the framework of FAT approach, we predict very small direct CP asymmetries of [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] and [D.sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+] presented in Table 3. The uncertainties induced by the model-dependence of form factor to the CP asymmetries of [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] and [D.sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+] are about 30% and 10%, respectively.

The differential CP asymmetry of the three-body decay [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0], which is a function of the invariant mass of [mathematical expression not reproducible] and [mathematical expression not reproducible], is defined as

[mathematical expression not reproducible], (22)

where the invariant mass [mathematical expression not reproducible]. As can be seen from (4), the differential CP asymmetry [mathematical expression not reproducible] depends on the relative strong phase S, 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 [delta] from data. By using a Dalitz plot technique [55,58,59], the phase difference [[delta].sup.exp] between [D.sup.0] decays to [K.sup.+][K.sup.*][(892).sup.-] and [K.sup.-][K.sup.*][(892).sup.+] can be extracted from data. One should notice that [[delta].sup.exp] is not the same as the strong phase [delta] defined in (4). The strong phase [delta] is the relative phase between the decay amplitudes of [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] and [D.sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+]. On the other hand, the phase [[delta].sup.exp] is defined through

[mathematical expression not reproducible] (23)

in the overlapped region of the phase space, where [mathematical expression not reproducible] is the phase of the amplitude [mathematical expression not reproducible]:

[mathematical expression not reproducible]. (24)

Therefore, neglecting the CKM suppressed penguin amplitudes, [[delta].sup.exp] and [delta] can be related by

[mathematical expression not reproducible], (25)

where [mathematical expression not reproducible] are the phases in tree-level amplitudes of [D.sup.0] [right arrow] [K.sup.[+ or -]][K.sup.*][(892).sup.+] and are equivalent to [mathematical expression not reproducible] if the penguin amplitudes are neglected. With the relation of (25), and [[delta].sup.exp] = -35.5[degrees] [+ or -] 4.1[degrees] measured by the BABAR Collaboration [56], we have [delta] [approximately equal to] -51.85[degrees] [+ or -] 4.1[degrees].

In Figure 2, we present the differential CP asymmetry of [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] in the overlapped region of [K.sup.*][(892).sup.-] and [K.sup.*][(892).sup.+] in the phase space, with [delta] = -51.85[degrees]. Namely, we will focus on the region [mathematical expression not reproducible] of the phase space. One can see from Figure 2 that the differential CP asymmetry of [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] can reach 3.0 x [10.sup.-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 behavior of the differential CP asymmetry of [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] in Figure 2 motivates us to separate this region into four areas, area A [mathematical expression not reproducible], area B [mathematical expression not reproducible], area C [mathematical expression not reproducible], and area [mathematical expression not reproducible]. We further consider the observable of regional CP asymmetry in areas A, B, C, and D displayed in Table 4, which is defined by

[mathematical expression not reproducible], (26)

where [OMEGA] represents a certain region of the phase space.

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 [[delta].sup.exp]. In addition, if we focus on the right part of area A, that is, [mathematical expression not reproducible], the regional CP violation will be (1.09 [+ or -] 0.16) x [10.sup.-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 Flatte Parametrization [60], the correction to the regional CP asymmetry will be about 1%.

Since the CP asymmetry of [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] is extremely suppressed, it should be more sensitive to the NP. For example, some NPs have considerable impacts on the chromomagnetic dipole operator [O.sub.8g] [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 [c.sup.eff.sub.8g] [67,68]. For comparison, we first consider a relative small value of [c.sup.eff.sub.8g] (as in [48,64]) lying within the range (0,1) and the global CP asymmetry of [D.sup.0] [right arrow] [K.sup.*][(892).sup.[+ or -]][K.sup.[+ or -]] are no larger than 5 x [10.sup.-5]. Moreover, if we follow [49] taking [c.sup.eff.sub.8g] [approximately equal to] 10 (while [c.sup.eff.sub.8g] = 10, which is extracted from [DELTA][A.sub.CP] 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 [c.sub.8g]. 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 [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] and [D.sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+] are then (0.56 [+ or -] 0.08) x [10.sup.-3] and (-0.50 [+ or -] 0.04) x [10.sup.-3], respectively.

We further display the CP asymmetry of [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] in the overlapped region of [K.sup.*][(892).sup.-] and [K.sup.*][(892).sup.+] in Figures 3(a) and 3(b) for [c.sup.eff.sub.8g] = 1 and [c.sup.eff.sub.8g] = 10, respectively. After taking the interference effect into account, the differential CP asymmetry of [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] can be increased as large as 5.5 x [10.sup.-4] and 2.8 x [10.sup.-3] for [c.sup.eff.sub.8g] = 1 and [c.sup.eff.sub.8g] = 10, respectively. The regional ones (in phase space of [mathematical expression not reproducible]) can reach (2.7 [+ or -] 0.5) x [10.sup.-4] and (1.3 [+ or -] 0.3) x [10.sup.-3] for [c.sup.eff.sub.8g] = 1 and [c.sup.eff.sub.8g] = 10, respectively.

4. Discussion and Conclusion

In this work, we studied CP violations in [D.sup.0] [right arrow] [K.sup.*][(892).sup.[+ or -]][K.sup.[+ or -]] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] via the FAT approach. The CP violations in two-body decay processes [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] and [D.sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+] are very small, which are (-1.27 [+ or -] 0.25) x [10.sup.-5] and (3.86 [+ or -] 0.26) x [10.sup.-5], respectively. Our discussion shows that the CP violation can be enhanced by the interference effect in three-body decay [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0]. The differential CP asymmetry can reach 3.0 x [10.sup.-4] when the interference effect is taken into account, while the regional one can be as large as (1.09 [+ or -] 0.16) x [10.sup.-4].

Besides, since the chromomagnetic dipole operator [O.sub.8g] is sensitive to some NPs, the inclusion of this kind of NPs will lead to a much larger global CP asymmetries of [D.sup.0] [right arrow] [K.sup.+][K.sup.*][(892).sup.-] and [D.sup.0] [right arrow] [K.sup.-][K.sup.*][(892).sup.+], which are (0.56 [+ or -] 0.08) x [10.sup.-3] and (-0.50 [+ or -] 0.04) x [10.sup.-3], respectively, while the regional CP asymmetry of [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] can be also increased to (1.3 [+ or -] 0.3) x [10.sup.-3] when considering the interference effect in the phase space. Since the O([10.sup.-3]) of CP asymmetry is attributed to the large [c.sup.eff.sub.8g], 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 [D.sup.0][[bar.D].sup.0] needed for testing such kind of asymmetries, which is about (1/Br)(1/[A.sup.2.sub.CP]) ~ [10.sup.9]. This could be observed in the future experiments at Belle II [73, 74], while the current largest [D.sup.0][[bar.D].sup.0] yields are about [10.sup.8] at BABAR and Belle [75, 76] and [10.sup.7] at BESIII [77].

https://doi.org/10.1155/2018/7627308

Appendix

Some 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]

[mathematical expression not reproducible], (A.1)

where [G.sub.F] is the Fermi constant, [[lambda].sub.q] = [V.sub.uq][V.sup.*.sub.cq], [c.sub.i](i = 1,...,6) is the Wilson coefficient, and [O.sup.q.sub.1], [O.sup.q.sub.2], [O.sub.i] (i = 1,...,6), and [O.sub.8g] are four-fermion operators which are constructed from different combinations of quark fields. The four-fermion operators take the following form:

[mathematical expression not reproducible], (A.2)

where [alpha] and [beta] are color indices and q' = u, d, s. Among all these operators, [O.sup.q.sub.1] and [O.sup.q.sub.2] are tree operators, [O.sub.3] - [O.sub.6] are QCD penguin operators, and [O.sub.8g] 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 [mu] = lGeV, which can be found in [48].

(2) CKM Matrix. We use the Wolfenstein parameterization for the CKM matrix elements, which up to order O([[lambda].sup.8]) read [79, 80]

[mathematical expression not reproducible], (A.3)

where A, [rho], [eta], and [lambda] are the Wolfenstein parameters, which satisfy following relation:

[rho] + i[eta] = [square root of (1 - [A.sup.2][[lambda].sup.4])] ([bar.[rho]] + [bar.i[eta]])/[square root of (1 - [[lambda].sup.2])][1 - [A.sup.2][[lambda].sup.4]([bar.[rho]] + [bar.i[eta]])]. (A.4)

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

[mathematical expression not reproducible]. (A.5)

(3) Decay Constants and Form Factors. In (17) and (18), the pole resonance model was employed for the matrix element <PV[absolute value of ([[bar.q].sub.1][q.sub.2])]0> in the annihilation diagrams. By considering angular momentum conservation at weak vertex and all conservation laws are preserved at strong vertex, the matrix element <PV[absolute value of ([[bar.q].sub.1][q.sub.2])]0> is therefore dominated by a pseudoscalar resonance [54],

[mathematical expression not reproducible], (A.6)

where [mathematical expression not reproducible] is a strong coupling constant and [mathematical expression not reproducible], are the mass and decay constant of the pseudoscalar resonance [P.sup.*]. Therefore, [eta] and [eta]' are the dominant resonances for the final states of [K.sup.*[+ or -]][K.sup.[+ or -]], which can be expressed as flavor mixing of [[eta].sub.q] and [[eta].sub.s] ,

[mathematical expression not reproducible] (A.7)

where [phi] is the mixing angle and [[eta].sub.q] and [[eta].sub.s] are defined by

[[eta].sub.q] = 1/[square root of 2](u[bar.u] + d[bar.d]), [[eta].sub.s] = s[bar.s]. (A.8)

The decay constants of [eta] and [eta]' are defined by

[mathematical expression not reproducible], (A.9)

where

[f.sup.u.sub.[eta]] = [f.sup.d.sub.[eta]] = [1/[square root of 2]][f.sup.q.sub.[eta]], [f.sup.u.sub.[eta]'] = [f.sup.d.sub.[eta]'] = [1/[square root of 2]][f.sup.q.sub.[eta]']. (A.10)

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

[f.sup.q.sub.[eta]] = [f.sub.q] cos [phi], [f.sup.q.sub.[eta]'] = [f.sub.q] sin [phi], [f.sup.s.sub.[eta]] = -[f.sub.s] sin [phi], [f.sup.s.sub.[eta]'] = [f.sub.s] cos [phi], (A.11)

where [f.sub.q] = (1.07 [+ or -] 0.02) [f.sub.[pi]] and [f.sub.s] = (1.34 [+ or -] 0.02) [f.sub.[pi]] [81], and the mixing angle [phi] = (40.4 [+ or -] 0.6)[degrees] [83]. Other decay constants used in this paper are listed in Table 5.

The transition form factors [mathematical expression not reproducible], 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):

F([q.sup.2]) = F(0)/1 - a([q.sup.2]/[m.sup.2.sub.D]) + b[([q.sup.2]/[m.sup.2.sub.D]).sup.2]. (A.12)

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

[mathematical expression not reproducible], (A.13)

where [p.sub.1] represents the center of mass (c.m.) 3-momentum of each meson in the final state and is given by

[mathematical expression not reproducible]. (A.14)

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.

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Hang Zhou, (1) Bo Zheng (iD), (1, 2) and Zhen-Hua Zhang (iD) (1)

(1) School of Nuclear Science and Technology, University of South China, Hengyang, Hunan 421001, China

(2) Helmholtz 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

Caption: Figure 1: The relevant topological diagrams for D [right arrow] PV with (a) the color-favored tree amplitude [T.sub.P(V)], (b) the W-exchange amplitude [E.sub.P(V)], (c) the color-favored penguin amplitude [PT.sub.P(V)], (d) the gluon-annihilation penguin amplitude [PE.sub.P(V)], and (e) the gluon-exchange penguin amplitude [PA.sub.P(V)].

Caption: Figure 2: The differential CP asymmetry distribution of [D.sup.0] [right arrow] [K.sup.+][K.sup.-][[pi].sup.0] in the overlapped region of [K.sup.*][(892).sup.-] and [K.sup.*][(892).sup.+] in the phase space.

Caption: Figure 3: The differential CP asymmetry distribution of [D.sup.0] [right arrow] [K.sup.+][K.sup.-] [[pi].sup.0] for (a) [c.sup.eff.sub.8g] = 1 and (b) [c.sup.effs.sub.8g] = 10, in the overlapped region of [K.sup.*][(892).sup.-] and [K.sup.*][(892).sup.+] in the phase space.
Table 1: The magnitude of tree and penguin contributions (in unit of
[10.sup.-3]) corresponding to the topological amplitudes in (19) and
(20). The factors "([G.sub.F]/[square root of 2])[[lambda].sub.s]
([[epsilon].sup.*] x [p.sub.D])" and "-([G.sub.F]/[square root of 2])
[[lambda].sub.b]([[epsilon].sup.*] x [p.sub.D])" are omitted in this
table.

Decay modes                         [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.+]   0.23
[K.sup.*][(892).sup.-]

                                    [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.-]   0.44
[K.sup.*][(892).sup.+]

Decay modes                         [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.+]   -0.02 + 0.15i
[K.sup.*][(892).sup.-]

                                    [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.-]   -0.02 + 0.15i
[K.sup.*][(892).sup.+]

Decay modes                         [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.+]   3.83 + 4.32i
[K.sup.*][(892).sup.-]

                                    [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.-]   -23.3 - 19.3i
[K.sup.*][(892).sup.+]

Decay modes                         [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.+]   0.96 - 0.03i
[K.sup.*][(892).sup.-]

                                    [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.-]   0.96 - 0.03i
[K.sup.*][(892).sup.+]

Decay modes                         [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.+]   0.13 - 0.81i
[K.sup.*][(892).sup.-]

                                    [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.-]   0.13 - 0.81i
[K.sup.*][(892).sup.+]

Decay modes                         [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.+]   6.73 + 8.22i
[K.sup.*][(892).sup.-]

                                    [mathematical expression
                                    not reproducible]

[D.sup.0] [right arrow] [K.sup.-]   -8.53 - 5.53i
[K.sup.*][(892).sup.+]

Table 2: Branching ratios (in unit of [10.sup.-3]) of singly-Cabibbo-
suppressed decays [D.sup.0] [right arrow] [K.sup.+][K.sup.*]
[(892).sup.-] and [D.sup.0] [right arrow] [K.sup.-][K.sup.*]
[(892).sup.+]. Both experimental data [55-57] and theoretical
predictions of FAT approach of the branching ratios are listed.

Form         Br([D.sup.0] [right arrow]
factors   [K.sup.+][K.sup.*][(892).sup.-])

Pole             1.57 [+ or -] 0.04
Dipole           1.69 [+ or -] 0.04
CLF              1.45 [+ or -] 0.04
Exp.             1.56 [+ or -] 0.12

Form         Br([D.sup.0] [right arrow]
factors   [K.sup.+][K.sup.*][(892).sup.-])

Pole             3.73 [+ or -] 0.17
Dipole           4.02 [+ or -] 0.19
CLF              4.44 [+ or -] 0.20
Exp.             4.38 [+ or -] 0.21

Table 3: CP asymmetries (in unit of [10.sup.-5]) of [D.sup.0]
[right arrow] [K.sup.+][K.sup.*][(892).sup.-] and [D.sup.0]
[right arrow] [K.sup.-][K.sup.*][(892).sup.+] predicted by the FAT
approach with pole, dipole, and CLF models adopted. The uncertainties
in this table are mainly from decay constants.

Form      [A.sub.CP]([D.sup.0] [right arrow]
factors     [K.sup.+][K.sup.*][(892).sup.-])

Pole              -1.45 [+ or -] 0.25
Dipole            -1.63 [+ or -] 0.26
CLF               -1.27 [+ or -] 0.25

Form      [A.sub.CP]([D.sup.0] [right arrow]
factors     [K.sup.-][K.sup.*][(892).sup.+])

Pole               3.60 [+ or -] 0.23
Dipole             3.70 [+ or -] 0.24
CLF                3.86 [+ or -] 0.26

Table 4: Three from factor models: the pole, dipole, and CLF models are
used for the regional CP asymmetries (in unit of [10.sup.-4]) in the
four areas, A, B, C, and D, of the phase space.

Form       [A.sup.A.sub.CP]     [A.sup.B.sub.CP]     [A.sup.C.sub.CP]
factors

Pole      0.87 [+ or -] 0.11   0.42 [+ or -] 0.08   0.39 [+ or -] 0.07
Dipole    0.87 [+ or -] 0.11   0.41 [+ or -] 0.08   0.38 [+ or -] 0.07
CLF       0.84 [+ or -] 0.10   0.45 [+ or -] 0.08   0.42 [+ or -] 0.07

Form       [A.sup.D.sub.CP]     [A.sup.All.sub.CP]
factors

Pole      -0.30 [+ or -] 0.08   0.33 [+ or -] 0.05
Dipole    -0.30 [+ or -] 0.08   0.32 [+ or -] 0.05
CLF       -0.25 [+ or -] 0.08   0.36 [+ or -] 0.06

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

[mathematical expression   [f.sub.[rho]]   [f.sub.K]   [f.sub.[pi]]
not reproducible]

220(5)                         216(3)      156(0.4)      130(1.7)

[mathematical expression   [f.sub.D]
not reproducible]

220(5)                      208(10)

Table 6: The parameters of D [right arrow] [K.sup.*], K transitions
form factors in (A.12).

Form     [mathematical expression   [F.sup.D[right arrow]K.sub.1]
factor       not reproducible]

F(0)               0.69                          0.78
a                  1.04                          1.05
b                  0.44                          0.23
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Title Annotation:Research Article; Charge-Parity
Author:Zhou, Hang; Zheng, Bo; Zhang, Zhen-Hua
Publication:Advances in High Energy Physics
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
Words:9563
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