Abstract
In the QCD factorization (QCDF) approach we study the direct CP violation in via the mixing mechanism. We find that the CP violation can be enhanced by double mixing when the masses of the pairs are in the vicinity of the resonance, and the maximum CP violation can reach 28%. We also compare the results from the naive factorization and the QCD factorization.
1. Introduction
CP violation is an extensive research topic in recent years. In standard model (SM), violation is related to the weak complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2]. In the past few years more attention has been focused on the decays of meson system both theoretically and experimentally. Recently, the large violation was found by the LHCb Collaboration in the three-body decay channels of and [3, 4]. Hence, the theoretical mechanism for the three- or four-body decays becomes more and more interesting. In this paper, we focus on the interference from intermediate and mesons in the four-body decay.
It is known that the naive factorization [5, 6], the QCD factorization (QCDF) [7–9], the perturbative QCD (PQCD) [10–12], and the soft-collinear effective theory (SCET) [13, 14] are the most extensive approaches for calculating the hadronic matrix elements. These factorization approaches present different methods for dealing with the hadronic matrix elements in the leading power of ( is the -quark mass). Direct violation occurs through the interference of two amplitudes with different weak phases and strong phases. The weak phase difference is directly determined by the CKM matrix elements, while the strong phase is usually difficult to control from a theoretical approach. The meson decay amplitude involves the hadronic matrix elements whose computation is not trivial. Different methods may present different strong phases. Meanwhile, we can also obtain a large strong phase difference by some phenomenological mechanism. mixing has been used for this purpose in the past few years [15–25]. In this paper, we will investigate the violation via double mixing in the QCDF approach.
In the QCDF approach, at the rest frame of the heavy meson, meson can decay into two light mesons with large momenta. In the heavy-quark limit, QCD corrections can be calculated for the nonleptonic two-body meson decays. The decay amplitude can be obtained at the next-to-leading power in and the leading power in . In the QCD factorization, there is cancellation of the scale and renormalizaion scheme dependence between the Wilson coefficients and the hadronic matrix elements. However, this does not happen in the naive factorization. The hadronic matrix elements can be expressed in terms of form factors and meson light-cone distribution amplitudes including strong interaction corrections.
The remainder of this paper is organized as follows. In Section 2 we present the form of the effective Hamiltonian. In Section 3 we give the calculating formalism of violation from mixing in . Input parameters are presented in Section 4. We present the numerical rusults in Section 5. Summary and discussion are included in Section 6.
2. The Effective Hamiltonian
With the operator product expansion, the effective weak Hamiltonian can be written as [26]: where represents the Fermi constant, are the Wilson coefficients, and , are the CKM matrix elements. The operators have the following forms: where and are color indices, and are the tree operators, are QCD penguin operators which are isosinglets, and arise from electroweak penguin operators which have both isospin and components. and are the electromagnetic and chromomagnetic dipole operators, are the electric charges of the quarks, and is implied.
For the decay channel , neglecting power corrections of order , the transition matrix element of an operator in the weak effective Hamiltonian is given by [8, 9]: Here denotes ( represent and mesons) form factor, and is the light-cone distribution amplitude for the quark-antiquark Fock state of mesons and . and are hard-scattering functions, which are perturbatively calculable. The hard-scattering kernels and light-cone distribution amplitudes (LCDA) depend on the factorization scale and the renormalization scheme. denote the and masses, respectively.
We match the effective weak Hamiltonian onto a transition operator, the matrix element is given by with [8, 9]: where denotes the contribution from vertex correction, penguin amplitude, and spectator scattering in terms of the operators ; refers to annihilation terms contribution by operators . is the helicity of the final state.
The flavor operators are defined in [8, 9] as follows: where is the number of colors, the upper (lower) signs apply when is odd (even), and . It is understood that the superscript “” is to be omitted for . The quantities account for one-loop vertex corrections, for hard spectator interactions, and for penguin contractions. is given by
The coefficients of the flavor operators can be expressed in terms of the coefficients . We will present the form in the following section. Using the unitarity relation we can get where the sums extend over and denotes the spectator antiquark.
Next we need to change the annihilation part into the following form [8, 9]: where , , and will be given in the following section.
3. CP Violation in
3.1. Formalism
The ( and are the polarization vectors (momenta) of and , resp.); decay rate is written as where refers to the c.m. momentum. is the helicity amplitude for each helicity of the final state. The decay amplitude, , can be decomposed into three components , , and according to the helicity of the final state. With the helicity summation, we can get
In the vector meson dominance model [27], the photon propagator is dressed by coupling to vector mesons. Based on the same mechanism, mixing was proposed [28, 29]. The formalism for violation in the decay of a bottom hadron, , will be reviewed in the following. The amplitude for , , can be written as where and are the Hamiltonians for the tree and penguin operators, respectively. We define the relative magnitude and phases between these two contributions as follows: where and are strong and weak phase differences, respectively. The weak phase difference arises from the appropriate combination of the CKM matrix elements: . The parameter is the absolute value of the ratio of tree and penguin amplitudes: The amplitude for is Then, the violating asymmetry, , can be written as where and represent the tree-level helicity amplitudes. We can see explicitly from (16) that both weak and strong phase differences are needed to produce violation. mixing has the dual advantages that the strong phase difference is large and well-known [15, 16]. In this scenario one has where or is the tree amplitude and is the penguin amplitude for producing a vector meson, . or is the tree annihilation amplitude and is the penguin annihilation amplitude. is the coupling for , is the effective mixing amplitude, and is from the inverse propagator of the vector meson : with being the invariant mass of the pair. The direct is effectively absorbed into , leading to the explicit dependence of [30, 31]. Making the expansion , the mixing parameters were determined in the fit of Gardner and O’Connell [32]: , , and . In practice, the effect of the derivative term is negligible. From (16) and (18), one has Defining where , , and are strong phases, one finds the following expression from (21): , , and will be calculated in the QCD factorization approach in the next section. With (25), we can obtain and . In order to get the violating asymmetry, , in (16), and are needed. is determined by the CKM matrix elements. In the Wolfenstein parametrization [33, 34], one has
3.2. The Calculation Details
The nonfactorizable corrections are included in the coefficients which contain vertex corrections and hard spectator interactions and which contain annihilation contributions.
In the QCD factorization approach, associated with the coefficient can be written as follows (helicity indices are neglected) [8, 9]: where we have used the following notation: with and referring to the transverse decay constant and decay constant of the vector meson, respectively.
The flavor operators include short-distance nonfactorizable corrections such as vertex corrections, hard spectator interactions, and Penguin terms. These contribution and annihilation part are given by [8, 9].
3.3. The Calculation of CP Violation
In order to obtain the violation of in (16), we calculate the amplitudes , , , , , , , and in (18) and (19) in the QCDF approach, which are tree-level and penguin-level amplitudes. The decay amplitudes for the process are in the QCD factorization as follows: where
From (22), one can get where
In a similar way, with the aid of the Fierz identities, we can evaluate the penguin operator contributions and . From (23), we have where
Form (24), we have where
4. Input Parameters
In the numerical calculations, we should input distribution amplitudes and the CKM matrix elements in the Wolfenstein parametrization. For the CKM matrix elements, which are determined from experiments, we use the results in [35]: where
The general expressions of the helicity-dependent amplitudes can be simplified by considering the asymptotic distribution amplitudes for , :
Power corrections in QCDF always involve endpoint divergences which produce some uncertainties. The endpoint divergence in the annihilation and hard spectator scattering diagrams is parameterized as with the unknown real parameters and [8, 9]. For simplicity, we will assume that and are helicity-independent: and .
5. Numerical Results
In the numerical results, we find that for the decay channel we are considering the violation can be enhanced via mixing when the invariant mass of is in the vicinity of the resonance. The uncertainties of the CKM matrix elements mainly come from and . In our numerical results, we let and vary between the limiting values. We find that the results are not sensitive to the values of and . Hence, the numerical results are shown in Figures 1, 2, and 3 with the central parameter values of CKM matrix elements. From the numerical results, it is found that there is a maximum violating parameter value, , when the masses of the pairs are in the vicinity of the resonance. In Figure 1, one can find that the maximum violating parameter reaches in the case of ().
From (16) one can find that the violating parameter is related to and . In Figure 2, we show the plot of ( and ) as a function of . We can see that the mixing mechanism produces a large ( and ) at the resonance. As can be seen from Figure 2, the plots vary sharply in the cases of and . Meanwhile, changes weakly compared with the and . It can be seen from Figure 3 that and change more rapidly than when the masses of the pairs are in the vicinity of the resonance. The helicity amplitudes mainly come from the contributions of and . mixing presents a large strong phase which gives a small effect on the amplitudes. Hence, associated with and is not sensitive to .
In paper [23], we studied the enhanced violation for the decay channel in the naive factorization. Since nonfactorizable contribution cannot be calculated in the naive factorization, was treated as an effective parameter. We found that the violating asymmetry was large and ranges from to via the mixing mechanism strongly depending on the value when the invariant mass of the pair is in the vicinity of the resonance. However, the maximum violation can only reach via double mixing in the QCD factorization. The naive factorization scheme has been shown to be the leading order result in the framework of QCD factorization when the radiative QCD corrections and the order effects are neglected. The QCD factorization can evaluate systematically corrections to the results from the naive factorization. The distinction between the naive factorization and the QCDF mainly comes from the strong phases of the QCD corrections. In the calculating process, we find that the annihilation contributions in QCDF which introduce the unknown parameters are small. Hence, the uncertainties of the results from the QCDF become small.
6. Summary and Conclusions
In this paper, we studied the violation for the decay process due to the interference of mixing in the QCDF approach. This process induces two interferences. It was found that the violation can be enhanced at the region of resonance. As a result, the maximum violation could reach . mixing is small due to the isospin violation. However, the mixing can produce a large strong phase, , in (21). This is because when the invariant masses of the pairs are in the vicinity of , , and it becomes comparable with in (21). In other words, mixing becomes important in the vicinity of . This is the reason why we can see large CP violation in the vicinity of . Beyond the interference region, the noticeable values of CP violation are caused by the strong phases provided by the Wilson coefficients.
The LHC experiments are designed with the center-of-mass energy 14 TeV and the luminosity . The heavy quark physics is one of the main topics of LHC experiments. In particular, LHCb detector is designed to make precise studies on asymmetries and rare decays of b-hadron systems. Recently, the LHCb Collaboration found clear evidence for direct violation in some three-body decay channels in charmless decays of meson. Large violation is observed in and in the regions GeV2 and GeV2 [3]. LHCb experiment may collect data in the region of the invariant masses of associated with the resonance for detecting our prediction of violation.
In our calculations there are some uncertainties. The QCD factorization scheme provides a framework in which we can evaluate systematically corrections to the results obtained in the naive factorization scheme. However, when we take into account the nonfactorizable and chirally enhanced hard-scattering spectator and annihilation contributions which appear at orders and , respectively, the involvement of the twist-3 hadronic distribution amplitudes leads to logarithmical divergence coming from the endpoint integrals. This brings large uncertainties in the predictions of the violating asymmetries in the QCD factorization scheme. Furthermore, in addition to the model dependence appearing in the factorized hadronic matrix elements just as in the naive factorization scheme, we cannot avoid the model dependence and process dependence of the hard-scattering spectator and annihilation contributions due to their dependence on the hadronic distribution amplitudes and dependence on different processes. Such dependence will also appear if one tries to include other corrections and even higher order corrections. This leads to uncertainity of our results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by National Natural Science Foundation of China (Project nos. 11147003, 11175020, 11275025, 11347124, and U1204115), Plan For Scientific Innovation Talent of Henan University of Technology (Project no. 2012CXRC17), the Key Project (Project no. 14A140001) for Science and Technology of the Education Department Henan Province, and the Fundamental Research Funds (Project no. 2014YWQN06) for the Henan Provincial Colleges and Universities.