Supported by the National Natural Science Foundation of China (11705047, U1632109, 11547014, 11475055) and Open Research Program of Large Research Infrastructures (2017), Chinese Academy of Sciences
The
Article funded by SCOAP^{3}
The exclusive
Phenomenologically, the non-leptonic
It is well known that according to the quark model assignments, spin-triplet charmonium states with different orbital angular momentum
In this paper, we will investigate the
This paper is organized as follows. The theoretical framework and the amplitudes for the
The
Hence, some relevant energy scales are introduced theoretically, such as the infrared confinement scale Λ_{QCD} of the strong interactions, the mass
The renormalization scale
The participation of the strong interaction greatly complicates the theoretical calculation of HMEs for the non-leptonic
In the heavy quark limit, the light quark from the bottom quark decay is assumed to fly quickly away from the interaction point at near the speed of light. The light-cone dynamics can be used to describe the relativistic system. The relations between the four-dimensional space-time coordinates (
The wave functions and/or distribution amplitudes (DAs) are the essential ingredient in the master pQCD formula of Eq. (
According to the twist classification in Refs. [
In our calculation, the expressions for the DAs involved are listed as follows [
The DAs of Eqs. (
A distinguishing feature of the DAs in Eqs. (
(color online) The normalized DAs of
Some properties of the psion resonances are collected in Table
Some properties of the psion resonances [
meson | mass/MeV | Γ/keV | Γ |
|||
---|---|---|---|---|---|---|
3686.097 ± 0.025 | 296 ± 8 | (7.89 ± 0.17) × 10^{−3} | 2.34 ± 0.04 | 358.8 ± 3.1 | 0.227 | |
3773.13 ± 0.35 | (27.2 ± 1.0) × 10^{3} | (9.6 ± 0.7) × 10^{−6} | 0.262 ± 0.018 | 121.2 ± 4.2 | 0.225 | |
4039 ± 1 | (80 ± 10) × 10^{3} | (1.07 ± 0.16) × 10^{−5} | 0.86 ± 0.07 | 225.4 ± 9.4 | 0.220 | |
ψ (4160) | 4191 ± 5 | (70 ± 10) × 10^{3} | (6.9 ± 3.3) × 10^{−6} | 0.48 ± 0.22 | 170.9 ± 45.2 | 0.217 |
Within the pQCD framework, the Feynman diagrams for the
(color online) The Feynman diagrams for the
After a direct calculation, the amplitudes for the
In addition, the amplitudes for the
In the rest frame of the
The numerical values of the input parameters are listed in Tables
The numerical values of the input parameters.
CKM parameters^{1)} | ||
---|---|---|
mass of the particles | ||
decay constants | ||
Gegenbauer moments at the scale of |
||
The branching ratios for the
final states | unit | data [ |
|||
---|---|---|---|---|---|
10^{−4} | 6.26 ± 0.24 | ||||
( |
( |
( |
|||
10^{−5} | 2.44 ± 0.30 | ||||
( |
( |
( |
|||
10^{−4} | 4.9 ± 1.3 | ||||
( |
( |
( |
|||
10^{−6} | --- | ||||
( |
( |
( |
|||
10^{−4} | 6.7 ± 1.4 | ||||
( |
( |
( |
|||
10^{−5} | --- | ||||
( |
( |
( |
|||
10^{−4} | --- | ||||
( |
( |
( |
|||
10^{−6} | --- | ||||
( |
( |
( |
The branching ratios for the
final states | unit | data [ |
||
---|---|---|---|---|
10^{−4} | < 1.3 | |||
( |
( |
|||
10^{−6} | --- | |||
( |
( |
|||
10^{−4} | 5.1 ± 2.7 | |||
( |
( |
|||
10^{−6} | --- | |||
( |
( |
|||
10^{−4} | --- | |||
( |
( |
|||
10^{−5} | --- | |||
( |
( |
|||
10^{−4} | --- | |||
( |
( |
|||
10^{−5} | --- | |||
( |
( |
(1) It has been shown in Refs. [
(2) The
(3) The
(color online) The branching ratios (vertical axis) versus the
(4) The excited psions with a large mass will certainly carry a large portion of energy in the
(color online) The percentage contribution to branching ratio for the
(5) Because of the large mass of the excited psions, the phase space for the
(6) There are lots of theoretical uncertainties, especially from
The color-suppressed non-leptonic
The symbols
The symbols
The numbers in parentheses indicate the approximate masses of the particles in the unit of MeV. The dominant components of the particles
The possible values of the 2
The relation between the CKM parameters (
The charmonium systems are usually assumed to be non-relativistic, and their wave functions can be obtained from the solutions of the time-independent} Schrödinger equation. Here, we will take the conventional notation to specify the
The function