This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP 3 .
1. Introduction
Charge-parity (
The direct
The three-body decays of heavy
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
2. The Framework
Based on the operator product expansion, the effective weak Hamiltonian for the decay processes
The Wilson coefficient
The combinations
For the two-body decay processes of
The
One can denote the light (anti-)quark momenta
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
The shape parameter
The distribution amplitudes of vector meson (
3.
3.1. Formalism
The decay width
The longitudinal
In the vector meson dominance model [71, 72], the photon propagator is dressed by coupling to vector mesons. Based on the same mechanism,
The relative magnitude and phases between the tree and penguin operator contribution are defined as follows:
The parameter of
With the
[figure omitted; refer to PDF]
Introducing the
From the physical representation, we can obtain the decay amplitude:
From Eqs. (29), (31), (38), and (39), one has
defining [25, 76]
3.2. Calculation Details
We can decompose the decay amplitudes for the decay processes
In the pQCD, there are eight types of the leading order Feynman diagrams contributing to
[figures omitted; refer to PDF]
With the Hamiltonian Equation (1), depending on CKM matrix elements of
The tree dominant decay amplitude for
Based on the definition of Eq. (43), we can get
From above equations, the new strong phases
4. Branching Ratio of
Based on the relationship of Eqs. (23) and (28), we can calculate the decay rates for the processes of
In this case, we take into account the
5. Input Parameters
The CKM matrix, which elements are determined from experiments, can be expressed in terms of the Wolfenstein parameters
From Eqs. (63) and (64), we have
The other parameters and the corresponding references are listed in Table 1.
Table 1
Input parameters.
Parameters | Input data | References |
Fermi constant (in | [84] | |
Masses and decay widths (in MeV) | [84] | |
Decay constants (in MeV) | [65, 85, 86] | |
6. The Numerical Results of
6.1.
We have investigated the
[figure omitted; refer to PDF]
The
From Eq. (34), one can find that the
[figure omitted; refer to PDF]
We have shown that the
6.2. Branching Ratio via
In the pQCD, we calculate the value of the branching ratio via
[figure omitted; refer to PDF]
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
At the LHC, the
7. Summary and Conclusion
In this paper, we have studied the direct
The
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.
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.
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
The nonfactorizable amplitudes
The hard scale
The function
The threshold resums factor
The evolution factors
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Abstract
In perturbative QCD approach, based on the first order of isospin symmetry breaking, we study the direct
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