Abstract
In perturbative QCD approach, based on the first order of isospin symmetry breaking, we study the direct violation in the decay of . An interesting mechanism is applied to enlarge the violating asymmetry involving the charge symmetry breaking between and . We find that the violation is large by the mixing mechanism when the invariant masses of the pairs are in the vicinity of the resonance. For the decay process of , the maximum violation can reach . Furthermore, taking mixing into account, we calculate the branching ratio for . We also discuss the possibility of observing the predicted violation asymmetry at the LHC.
1. Introduction
Charge-parity () violation is an open problem, even though it has been known in the neutral kaon systems for more than five decades [1]. The study of violation in the heavy quark systems is important to our understanding of both particle physics and the evolution of the early universe. Within the standard model (SM), violation is related to the nonzero weak complex phase angle from the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which describes the mixing of the three generations of quarks [2, 3]. Theoretical studies predicted large violation in the meson system [4–6]. In recent years, the LHCb collaboration has measured sizable direct CP asymmetries in the phase space of the three-body decay channels of and [7–9]. These processes are also valuable for studying the mechanism of multibody heavy meson decays. Hence, more attention has been focused on the nonleptonic meson three-body decay channels in searching for violation, both theoretically and experimentally.
The direct violation in the hadron decays occurs through the interference of at least two amplitudes with different weak phase and strong phase . The weak phase difference is determined by the CKM matrix elements, while the strong phase can be produced by the hadronic matrix elements and interference between the intermediate states. The hadronic matrix elements are not still well determined by the theoretical approach. The mechanism of two-body decays is still not quite clear, although many physicists are devoted to this field. Many factorization approaches have been developed to calculate the two-body hadronic decays, such as the naive factorization approach [10–13], the QCD factorization (QCDF) [14–18], perturbative QCD (pQCD) [19–21], and soft-collinear effective theory (SCET) [22–24]. Most factorization approaches are based on heavy quark expansion and light-cone expansion in which only the leading power or part of the next to leading power contributions are calculated to compare with the experiments. However, the different methods may present different strong phases so as to affect the value of the violation. Meanwhile, in order to have a large signal of violation, we need an appeal to some phenomenological mechanism to obtain a large strong phase . In Refs. [25–30], the authors studied the direct violation in hadronic (including and ) decays through the interference of tree and penguin diagrams, where - mixing was used for this purpose in the past few years and focused on the naive factorization and QCD factorization approaches. This mechanism was also applied to generalize the pQCD approach to the three-body nonleptonic decays in and , where even larger violation may be possible [31, 32]. In this paper, we will investigate direct violation of the decay process involving the same mechanism in the pQCD approach.
The three-body decays of heavy mesons are more complicated than the two-body decays as they receive both resonant and nonresonant contributions. Unlike the two-body case, to date, we still do not have effective theories for hadronic three-body decays, though attempts along the framework of pQCD and QCDF have been used in the past [33–36]. As a working starting point, we intend to study - mixing effect in three-body decays of the meson. The mixing mechanism is caused by the isospin symmetry breaking from the mixing between the and flavors [37, 38]. In Ref. [39], the authors studied the mixing and the pion form factor in the time-like region, where mixing comes from three part contributions: two from the direct coupling of the quasi-two-body decay of and and the other from the interference of mixing. Generally speaking, the amplitudes of their contributions are as follows: . and were used to obtain the (effective) mixing matrix element [40–42]. The magnitude has been determined by the pion form factor through the data from the cross section of in the and resonance region [39, 42–45]. Recently, isospin symmetry breaking was discussed by incorporating the vector meson dominance (VMD) model in the weak decay process of the meson [27, 32, 46–49]. However, one can find that mixing produces the large violation from the effect of isospin symmetry breaking in the three and four bodies decay process. Hence, in this paper, we shall follow the method of Refs. [27, 32, 46–49] to investigate the decay process of by the isospin symmetry breaking.
The remainder of this paper is organized as follows. In Sec. 2, we will present the form of the effective Hamiltonian and briefly introduce the pQCD framework and wave functions. In Sec. 3, we give the calculating formalism and details of the violation from mixing in the decay process . In Sec. 4, we calculate the branching ratio for the decay process of . In Sec. 5, we show the input parameters. We present the numerical results in Sec. 6. Summary and discussion are included in Sec. 7. The related functions defined in the text are given in the Appendix.
2. The Framework
Based on the operator product expansion, the effective weak Hamiltonian for the decay processes can be expressed as [50] where represents the Fermi constant, () are the Wilson coefficients, and , , , and are the CKM matrix elements. The operators have the following forms: where and are the SU(3) color indices, is the electric charge of quark in the unit of , and the sum extend over or quarks. In Eq. (11) and are the tree operators, – are the QCD penguin operators, and – are the operators associated with electroweak penguin diagrams.
The Wilson coefficient in Eq. (1) describes the coupling strength for a given operator and summarizes the physical contributions from scales higher than [51]. They are calculable perturbatively with the renormalization group improved perturbation theory. Usually, the scale is chosen to be of order for meson decays. Since we work in the leading order of perturbative QCD (), it is consistent to use the leading order Wilson coefficients. So, we use the numerical values of as follow [19, 21]:
The combinations – of the Wilson coefficients are defined as usual [52–55]: where the upper (lower) sign applies, when is odd (even).
For the two-body decay processes of , we denote the emitted or annihilated meson as , while the recoiling meson is . The meson ( or ) and the final state meson () move along the direction of and in the light-cone coordinates, respectively. The decay amplitude can be expressed as the convolution of the wave functions , , and and the hard scattering kernel in the pQCD. The pQCD factorization theorem has been developed for the two-body nonleptonic heavy meson decays, based on the formalism of Botts, Lepage, Brodsky, and Sterman [56–59]. The basic idea of the pQCD approach is that it takes into account the transverse momentum of the valence quarks in the hadrons which results in the Sudakov factor in the decay amplitude. Then, the decay channels of are conceptually written as the following: where is the momentum of light quark in each meson. denotes the trace over Dirac structure and color indices. is the short distance Wilson coefficients at the hard scale . The meson wave functions and , including all nonperturbative contribution during the hadronization of mesons, can be extracted from the experimental data or other nonperturbative methods. The hard kernel describes the four quark operator and the spectator quark connected by a hard gluon, which can be perturbatively calculated including all possible Feynman diagrams of the factorizable and nonfactorizable contributions without endpoint singularity.
The and mesons are treated as a light-light system. At the meson rest frame, they are moving very fast. We define the ratios , , and . In the limits , , and , one can drop the terms of proportional to , , and safely. The symbols , , and refer to the meson momentum, the meson momentum, and the final state meson momentum, respectively. The momenta of the participating mesons in the rest frame of the meson can be written as:
One can denote the light (anti-)quark momenta , , and for the initial meson and the final mesons and , respectively. We can choose where , , and are the momentum fractions. , , and refer to the transverse momentum of the quark, respectively. To extract the helicity amplitudes, we parameterize the following longitudinal polarization vectors of the and as the following: which satisfy the orthogonality relationship of , and the normalization of . The transverse polarization vectors can be adopted directly as
Within the pQCD framework, both the initial and the final state meson wave functions and distribution amplitudes are important as nonperturbative input parameters. For the meson, the wave function of the meson can be expressed as where the distribution amplitude is shown in Refs. [60–62]:
The shape parameter is a free parameter and is a normalization factor. Based on the studies of the light-cone sum rule, lattice QCD or be fitted to the measurements with good precision [63], we take for the meson. The normalization factor depends on the values of the shape parameter and decay constant , which is defined through the normalization relation .
The distribution amplitudes of vector meson (=, , or ), , , , , , and , can be written in the following forms [64, 65]: where . Here, is the decay constant of the vector meson with longitudinal (transverse) polarization. The Gegenbauer polynomials can be defined as [66, 67]
3. Violation in Decay Process
3.1. Formalism
The decay width for the processes of is given by where is the absolute value of the three-momentum of the final state mesons. The decay amplitude which is decided by QCD dynamics will be calculated later in the pQCD factorization approach. The superscript denotes the helicity states of the two vector mesons with the longitudinal (transverse) components L(T). The amplitude for the decays can be decomposed as follows [67–70]: where is the polarization vector of the vector meson. The amplitude ( refers to the three kinds of polarizations, longitudinal (L), normal (N), and transverse (T)) can be written as where , , and are the Lorentz-invariant amplitudes. and are the masses of the vector mesons and , respectively.
The longitudinal and transverse of helicity amplitudes can be expressed where and are the penguin level and tree level helicity amplitudes of the decay process from the three kinds of polarizations, respectively. The helicity summation satisfies the relation
In the vector meson dominance model [71, 72], the photon propagator is dressed by coupling to vector mesons. Based on the same mechanism, mixing was proposed and later gradually applied to meson physics [29, 39, 47, 73]. According to the effective Hamiltonian, the amplitude () for the three-body decay process () can be written as follows [47]: where and are the Hamiltonian for the tree and penguin operators, respectively.
The relative magnitude and phases between the tree and penguin operator contribution are defined as follows: where and are the strong and weak phase differences, respectively. The weak phase difference can be expressed as a combination of the CKM matrix elements, and it is for the transition. The parameter is the absolute value of the ratio of the tree and penguin amplitudes:
The parameter of violating asymmetry, , can be written as where represent the tree level helicity amplitudes of the decay process from , , and of Eq. (27), respectively. refer to the absolute value of the ratio of the tree and penguin amplitude for the three kinds of polarizations, respectively. are the relative strong phases between the tree and penguin operator contributions from the three kinds of the helicity amplitudes, respectively. We can see explicitly from Eq. (34) that both weak and strong phase differences are needed to produce violation. In order to obtain a large signal for the direct violation, we intend to apply the mixing mechanism, which leads to large strong phase differences in hadron decays.
With the mixing mechanism, the process of the decay is shown in Figure 1. In the isospin representation, the decay amplitude in Figure 1 can be written as [31, 39, 48, 74]
Introducing the [31, 39, 48, 74], we have identified the physical amplitudes as
From the physical representation, we can obtain the decay amplitude: where corrections are neglected and , = or , and = or are used. So, we can get =. is the effective mixing amplitude which also effectively includes the direct coupling . At the first order of the isospin violation, we have the following tree and penguin amplitudes when the invariant mass of pair is near the resonance mass [26, 47]: where and are the tree (penguin) level helicity amplitudes for and , respectively. The amplitudes , , , and can be found in Sec. 3.2. is the coupling constant for the decay process . , , and (= or ) are the inverse propagator, mass, and decay width of the vector meson , respectively. can be expressed as with being the invariant masses of the pairs. The mixing parameters are [75]
From Eqs. (29), (31), (38), and (39), one has
defining [25, 76] where , , and are the strong phases of the decay process from the three kinds of polarizations, respectively. One finds the following expression from Eqs. (42) and (43):
, , and will be calculated in the perturbative QCD approach. In order to obtain the violating asymmetry in Eq. (34), , sin, and cos are needed. is determined by the CKM matrix elements. In the Wolfenstein parametrization [77], the weak phase comes from . One has where the same result has been used for transition from Ref. [28, 78].
3.2. Calculation Details
We can decompose the decay amplitudes for the decay processes in terms of the tree and penguin contributions depending on the CKM matrix elements of and . From Eqs. (34), (42), and (43), in the leading order to obtain the formulas of the violation, we need to calculate the amplitudes , , , and in the perturbative QCD approach. The relevant function can be found in the Appendix from the perturbative QCD approach.
In the pQCD, there are eight types of the leading order Feynman diagrams contributing to decays, which are shown in Figure 2. The first row is for the emission-type diagrams, where the first two diagrams in Figures 2(a) and 2(b) are called factorizable emission diagrams and the last two diagrams in Figures 2(c) and 2(d) are called nonfactorizable emission diagrams [68, 79]. The second row is for the annihilation-type diagrams, where the first two diagrams in Figures 2(e) and 2(f) are called factorizable annihilation diagrams and the last two diagrams in Figures 2(g) and 2(h) are called nonfactorizable annihilation diagrams [62, 80]. The relevant decay amplitudes can be easily obtained by these hard gluon exchange diagrams, and the Lorenz structures of the mesons wave functions. Through calculating these diagrams, the formulas of or are similar to those of and [79, 81]. We just need to replace some corresponding Wilson coefficients, wave functions, and corresponding parameters.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
With the Hamiltonian Equation (1), depending on CKM matrix elements of and , the tree dominant decay amplitudes for in pQCD can be written as where the superscript denote the different helicity amplitudes , , and . The longitudinal and transverse of helicity amplitudes satisfy relationship from Eq. (27). The amplitudes of the tree and penguin diagrams can be written as and , respectively. The formula for the tree level amplitude is where refers to the decay constant of meson. The penguin level amplitudes are expressed in the following:
The tree dominant decay amplitude for can be written as where and refer to the tree and penguin amplitude, respectively. We can give the tree level contribution in the following: where refers to the decay constant of meson. The penguin level contribution is given as follows:
Based on the definition of Eq. (43), we can get where
From above equations, the new strong phases , , and are obtained from the tree and penguin diagram contributions by the interference. Substituting Eqs. (53), (54), and (55) into (44), we can obtain total strong phase in the framework of the pQCD. Then in combination with Eqs. (45) and (46), the violating asymmetry can be obtained.
4. Branching Ratio of
Based on the relationship of Eqs. (23) and (28), we can calculate the decay rates for the processes of by using the following expression: where is the c.m. momentum of the product particle and are the helicity amplitudes.
In this case, we take into account the mixing contribution to the branching ratio, since we are working on the first order of isospin violation. The derivation is straightforward, and we can explicitly express the branching ratio for the processes [28, 82]: where is the lifetime of the meson and take into account the helicity amplitudes of the meson and meson contribution involved in the tree and penguin diagrams.
5. Input Parameters
The CKM matrix, which elements are determined from experiments, can be expressed in terms of the Wolfenstein parameters , , , and [77, 83]: where corrections are neglected. The latest values for the parameters in the CKM matrix are [84] where
From Eqs. (63) and (64), we have
The other parameters and the corresponding references are listed in Table 1.
6. The Numerical Results of Violation and Branching Ratio
6.1. Violation via Mixing in
We have investigated the violating asymmetry, , for the of the three-body decay process in the perturbative QCD. The numerical results of the violating asymmetry are shown for the decay process in Figure 3. It is found that the violation can be enhanced via mixing for the decay channel when the invariant mass of pair is in the vicinity of the resonance within perturbative QCD scheme.
The violating asymmetry depends on the weak phase difference from CKM matrix elements and the strong phase difference in the Eq. (34). The CKM matrix elements, which relate to , , , and , are given in Eq. (63). The uncertainties due to the CKM matrix elements are mostly from and since is well determined. Hence, we take the central value of in Eq. (65). In the numerical calculations for the decay process, we use , , and which vary among the limiting values. The numerical results are shown from Figure 3 with the different parameter values of CKM matrix elements. The solid line, dotted line, and dashed line correspond to the maximum, middle, and minimum CKM matrix element for the decay channel of , respectively. We find that the numerical results of the violation are not sensitive to the CKM matrix elements for the different values of and . In Figure 3, we show the plot of violation as a function of in the perturbative QCD. From the figure, one can see the violation parameter is dependent on and changes rapidly by the mixing mechanism when the invariant mass of pair is in the vicinity of the resonance. From the numerical results, it is found that the violating asymmetry is large and ranges from - to via the mixing mechanism for the process. The maximum violating parameter can reach - for the decay channel of in the case of (, ). This error corresponds to the CKM parameters.
From Eq. (34), one can find that the violating parameter is related to and sin. In Figures 4 and 5, we show the plots of ( and ) and ( and ) as a function of , respectively. We can see that the mixing mechanism produces a large ( and ) in the vicinity of the resonance. As can be seen from Figure 4, the plots vary sharply in the cases of , and in the range of the resonance. Meanwhile, and change weakly compared with the . It can be seen from Figure 5 that , , and change more rapidly when the pairs in the vicinity of the resonance. Since amplitude is a small quantity, it contributes so little to the transverse and of the helicity amplitudes in which they are almost equal in Eq. (27). So, we can see that the two curves sin () and () of Figure 4 are the different in the region of resonance but almost the same in other regions.
We have shown that the mixing does enhance the direct violating asymmetry and provide a mechanism for large violation in the perturbative QCD factorization scheme. In other words, it is important to see whether it is possible to observe this large violating asymmetry in the experiments. This depends on the branching ratio for the decay channel of . We will study this problem in the next section.
6.2. Branching Ratio via Mixing in
In the pQCD, we calculate the value of the branching ratio via mixing mechanism for the decay channel . The numerical result is shown for the decay process in Figure 6. Based on a reasonable parameter range, we obtain the maximum branching ratio of as (), which is consistent with the result in [81, 87]. The error comes from CKM parameters. On the one hand, although we calculate the branching ratio due to mixing in the pQCD factorization scheme, we find that the contribution of mixing to the branching ratio of is small and can be neglected. This is because branching ratio is proportional to the contribution of in Eq. (59). The square of the mixing parameter is about GeV, which leads to large suppression for the contribution of the branching ratio of and the effect of the width of the resonances from in Eq. (60). On the other hand, the mixing mechanism produces new strong phase differences. In decay, we take into account the mixing contribution to the branching ratio, and we find that the two decays interfere with each other to generate a new resonance region around 1.1 GeV. By expanding the factor from Eq. (60), the denominator produces a new propagator . This is the reason why the dips of the curves in Figure 6 does not appear in the region of resonance but around 1.1 GeV .
The Large Hadron Collider (LHC) is a proton-proton collider that has started at the European Organization for Nuclear Research (CERN). With the designed center-of-mass energy 14 TeV and luminosity , the LHC provides a high energy frontier at TeV-level scale and an opportunity to further improve the consistency test for the CKM matrix. LHCb is a dedicated heavy flavor physics experiments and one of the main projects of LHC. Its main goal is to search for indirect evidence of new physics in violation and rare decays in the interactions of beauty and charm hadrons systems, by looking for the effects of new particles in decay processes that are precisely predicted in the SM. Such studies can help us to comprehend the matter-antimatter asymmetry of the universe. Recently, the LHCb collaboration found clear evidence for direct violation in some three-body decay channels of meson. Large violation is obtained for the decay channels of in the localized phase spaces region GeV and GeV [7, 88]. A zoom of the invariant mass from the decay process is shown the region GeV zone in Ref. [88]. In addition, the branching ratio of is probed in the invariant mass range MeV/ [89]. In the next years, we expect the LHCb collaboration to collect date for detecting our prediction of violation from the decay process when the invariant mass of is in the vicinity of the resonance.
At the LHC, the hadrons come from collisions. The possible asymmetry between the numbers of the hadrons and those of their antiparticles has been studied by using the intrinsic heavy quark model and the Lund string fragmentation model [90, 91]. It has been shown that this asymmetry can only reach values of a few percent. In the following discussion, we will ignore this small asymmetry and give the numbers of pairs needed for observing our prediction of the violating asymmetries. These numbers depend on both the magnitudes of the violating asymmetries and the branching ratios of heavy hadron decays which are model dependent. For one-standard deviation (1) signature and three-standard deviation (3) signature and the numbers of pairs, we need [92–94] where is the violation in the process of . Now, we can estimate the possibility to observe violation. The branching ratio for is of the order , and then, the number is 106 for signature and for signature, theoretically, in order to achieve the current experiments on hadrons, which can only provide about pairs. Therefore, it is very possible to observe the large violation for when the invariant masses of pairs are in the vicinity of the resonance in experiments at the LHC.
7. Summary and Conclusion
In this paper, we have studied the direct violation for the decay process of in perturbative QCD. It has been found that, by using mixing, the violation can be enhanced at the area of resonance. There is the resonance effect via mixing which can produce large strong phase in this decay process. As a result, one can find that the maximum violation can reach - when the invariant mass of the pair is in the vicinity of the resonance. Furthermore, taking mixing into account, we have calculated the branching ratio of the decays of . We have also given the numbers of pairs required for observing our prediction of the violating asymmetries at the LHC experiments.
The mixing is a small effect duo to the isospin violation. One can estimate the contributions by comparing the two terms on the right-hand side of Eqs. (38) and (39). However, when the invariant mass squared of the pair is in the vicinity of the omega, , it becomes comparable with . In other words, the mixing becomes important in the vicinity of the omega. This is also the reason why we only see large violation in the vicinity of . At the same time, the mixing parameter has determined the magnitude, and dependence of the effective mixing matrix element fits to data in the vicinity of the resonance. We can make the Taylor expansion for at and ignore the dependence of in practice. It will cause a large resonance near invariant masses of the pairs, and the effect will be negligible in distant regions. In the mechanism, we will ignore the contribution of the final state interaction between the pions in the resonant regions associated with and in the decay process of .
In our calculation, there are some uncertainties. The major uncertainties come from the input parameters. In particular, these include the CKM matrix element, the particle mass, the perturbative QCD approach, and the hadronic parameters (decay constants, the wave functions, the shape parameters, etc.). We expect that our predictions will provide useful guidance for future experiments.
Appendix
Related Functions Defined in the Text
In this appendix, we present explicit expressions of the factorizable and nonfactorizable amplitudes in the perturbative QCD [19–21, 60]. The factorizable amplitudes , , and () are written as with the color factor , represents the corresponding Wilson coefficients for the specific decay channels and and refer to the decay constants of ( or ) and mesons. In the above functions, and , with and being the masses of the initial and final states.
The nonfactorizable amplitudes , , , , and () are written as
The hard scale are chosen as the maximum of the virtuality of the internal momentum transition in the hard amplitudes, including :
The function , coming from the Fourier transform of hard part H, are written as where and are the Bessel function with .
The threshold resums factor follows the parameterized where the parameter [60].
The evolution factors and entering in the expressions for the matrix elements are given by in which the Sudakov exponents are defined as where is the anomalous dimension of the quark. The explicit form for the function is where the variables are defined by and the coefficients and are where is the number of the quark flavors and is the Euler constant.
Data Availability
This manuscript has no associated data or the data will not be deposited (no data were used to support this study).
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work (https://arxiv.org/abs/2102.07984) was supported by the National Natural Science Foundation of China (Project Number 11605041) and the Research Foundation of the young core teacher from Henan Province.