bethe-salper equation and its applications guo-li wang department of physics, harbin institute of...
DESCRIPTION
Bethe-Salper Equation and its instantaneous one, Salpeter equation where We introduce the symbolsTRANSCRIPT
![Page 1: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/1.jpg)
Bethe-Salper equation and its applications Guo-Li WangDepartment of Physics, Harbin Institute of Technology, China
![Page 2: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/2.jpg)
Bethe-Salper Equation and its instantaneous one, Salpeter equation
Wave functions for different states.
The theoretical predictions of mass spectra.
Theoretical calculations of decay constants.
Theoretical calculations of annihilation rates of quarkonium.
Summaries
![Page 3: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/3.jpg)
Bethe-Salper Equation and its instantaneous one, Salpeter equation
where
We introduce the symbols
![Page 4: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/4.jpg)
then we have two Lorentz invariant variables
in the center of mass system of the meson which will turn to the usual components
Instantaneous approach is that the interaction kernel taking the simple form
We define two notations (which is 3-dimension)
![Page 5: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/5.jpg)
and
then the BS equation can be written as
where the propagators can be decomposed as
with
![Page 6: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/6.jpg)
where i=1,2 for quark and antiquark, the projection operators satisfy the relations
If we introduce the notations
Then the wave function can be separate 4 parts
![Page 7: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/7.jpg)
with contour integration over the instantaneous BS equation become
Finally, the instantaneous BS equation turn to the Salpeter equation
![Page 8: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/8.jpg)
The normalization condition is
![Page 9: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/9.jpg)
Wave functions1, wave functions for pseudoscalar meson and sc
alar meson
For scalar state, the wave function can be built with momentum P, q, mass and gamma matrix.
with the instantaneous approach, the general form can
be written as
the other 8 terms vanish because of
1 2 3 4 5 5 6 7 8( ) ( )P P P P Pq f f q f P f q P f f q f P f q P
0Pq P
![Page 10: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/10.jpg)
But not all the remained 8 terms are pseudoscalar, half of them are scalar, so when we consider a state, the general form is
And a scalar wave function which
0
1 2 3 4 5( ) ( )P P Pq f f q f P f q P
0PJ
1 2 3 4( )P P Pq f f q f P f q P
![Page 11: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/11.jpg)
Salpeter wave fucntions Wave function for stateBecause of the Salpeter equation, we have the equations which are constraints on the wave functions so for state, we obtain the relations:
So finally, for , the wave function is
0
0 (0 )
![Page 12: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/12.jpg)
To solve the full Salpeter equation, we need the positive and negative wave functions
with these wave function form as input, from Salpeter equation, we obtain two independent equations,
![Page 13: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/13.jpg)
and the normalization condition is
![Page 14: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/14.jpg)
Wave function for state
The general form for the relativistic wave function of vector state can be written as 16 terms constructed by P, q, polarization vecotr , mass and gamma matrix, because of instantaneous approximation, 8 terms become zero, so we can write the wave function as
1PJ
![Page 15: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/15.jpg)
And the constraint relations
with the renormalization
![Page 16: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/16.jpg)
Wave functions for state
The general form of the Salpeter wave function for state is
and we have the further constraint relations
the renormalization is
0PJ
![Page 17: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/17.jpg)
Wave functions for state The general form of the Salpeter wave function for
state ( for equal mass system )
and the constraint relations
the renormalization condition
1PJ 1
![Page 18: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/18.jpg)
Wave functions for stateThe general form of the Salpeter wave function for
state ( for equal mass system)
and the constraint relations
2
![Page 19: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/19.jpg)
with the renormalization condition:
![Page 20: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/20.jpg)
Wave functions for state The general form of the Salpeter wave function for
state ( for equal mass system )
and the constraint relations
renormalization condition
1PJ 1
![Page 21: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/21.jpg)
The mixing of two states For equal mass system, because of the difference of cha
rge conjugation quantum number, the vector states can be distinguished by the charge conjuga
tion, so the physical states are But for non-equal mass system, there is no the quantum
number of charge conjugation, so we can not separate these two states and they mixed to other two physical states symboled as
1
![Page 22: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/22.jpg)
where is the mixing angle, and if the heavy quark mass go to infinity, then we have the following relations
where is the corresponding mixing angle. In experiment, we have all four P wave states named
so we can obtained the mixing angle for other P waves, since we have no data till now, we choose the mixing angle for others P wave states, for examples,
![Page 23: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/23.jpg)
The interaction kernel We choose the Cornell potential
and the coupling constant is running in one loop
![Page 24: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/24.jpg)
The mass spectra The parameters
for bottomonium, we choose the value with this value, we obtained for other states,
we choose and with this value, we got there is another parameter , which is n
eeded in potential model methods to move all the masses with mass shift to fitting data.
![Page 25: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/25.jpg)
Though we considered the relativistic corrections for wave functions, but we choose a very simple interaction kernel, so we can not fit data using same values for all the states, we chose different values of V0 shown here
![Page 26: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/26.jpg)
Mass spectra
![Page 27: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/27.jpg)
![Page 28: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/28.jpg)
![Page 29: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/29.jpg)
Mass spectra of the bottomonium We have used different values of , because of the simple interaction, in this part, we still use the earlier kernel, but with some perturbative corrections, we followed the work of S. Titard and F. J. Yndurain, PRD51(1995)6348.
Hyperfine splitting
LS splitting
![Page 30: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/30.jpg)
Tensor splitting
Fine splitting
![Page 31: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/31.jpg)
The parameters
![Page 32: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/32.jpg)
![Page 33: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/33.jpg)
![Page 34: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/34.jpg)
![Page 35: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/35.jpg)
![Page 36: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/36.jpg)
![Page 37: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/37.jpg)
![Page 38: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/38.jpg)
Decay constants Decay constants for state For pseudoscalar, the decay constant is defined as
In the Bethe-Salper method, it can be calculated as
0
![Page 39: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/39.jpg)
![Page 40: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/40.jpg)
![Page 41: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/41.jpg)
Decay constants for state The decay constant for state is defined as
In the BS method, it can be calculated as
![Page 42: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/42.jpg)
Decay constants for P-wave state Decay constant for state
Decay constant for (or ) state
![Page 43: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/43.jpg)
Decay constant for (or ) state
For the mixed state, we have use the following relation to calculating the decay constants
![Page 44: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/44.jpg)
![Page 45: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/45.jpg)
![Page 46: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/46.jpg)
Annihilation rate of quarkonium Decay rate of state The annihilation rate of quarkonium is related to the wave function, so it can helpful to understand the formalism of inter-quark interactions, and can be a sensitive test of the potential model.
0
![Page 47: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/47.jpg)
The transition amplitude of two-photon decay of state can be written as
Beause , and the symmetry, there is a good approximation , then the amplitude become
where the wave function of pseudoscalar meson
0
![Page 48: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/48.jpg)
Finally, the decay width is obtained, and it can simply written as
The two gluon decay width can be easily obtained with a simple replacement in the photon decay width formula
so the decay width is
![Page 49: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/49.jpg)
![Page 50: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/50.jpg)
Decay rate of state The transition amplitude of two-photon decay of
state can be written as
where the wave function is
and the full width can be estimated by the two-gluon decay.
0
0
![Page 51: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/51.jpg)
![Page 52: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/52.jpg)
![Page 53: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/53.jpg)
The differences of the relativistic results and non-relativistic results.
The relativistic Salpeter wave function for state
and the renormalization condition is
The non-relativistic wave function
and the renormalization function is
0
![Page 54: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/54.jpg)
![Page 55: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/55.jpg)
So the relativistic corrections for P wave is large even the state is a heavy one Compare with S wave, the relativistic corrections
are larger for P wave, this conclusion can be seen easily by the wave functions
![Page 56: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/56.jpg)
Decay rate of state The transition amplitude of two-photon decay of
state can be written as
where the wave function can be written as
with the normalization condition
2
2
![Page 57: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/57.jpg)
Then the decay amplitude become
![Page 58: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/58.jpg)
![Page 59: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/59.jpg)
![Page 60: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/60.jpg)
S-D mixing in and P-F mixing in state S-D mixing in state (example)
The wave function for state in rectangular
coordinate is
1
1
2
1
![Page 61: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/61.jpg)
![Page 62: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/62.jpg)
![Page 63: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/63.jpg)
![Page 64: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/64.jpg)
We can see from the figures, for 1S and 2S states, the terms of f5 and f6 are S-wave, which are dominant, the terms of f3 and f4 are D-wave, which are very small. But for 1D, all the terms are D-wave dominant, and the S-wave come out from the D-wave, which can be see clearly below.
For S-wave dominant state, we can set f5= -f6=f and f3=f4=0, and in spherical polar coordinate, the wave function can be written as
![Page 65: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/65.jpg)
where
![Page 66: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/66.jpg)
For D-wave dominant state, we can set f3= f4=f and f5=f6=0, and in spherical polar coordinate, the wave function can be written as
![Page 67: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/67.jpg)
P-F mixing in state The wave function for state can be written as
For 1P and 2P states, the terms of f5 and f6 are P-wave, which are dominant, the terms of f3 and f4 are F-wave, which are very small. But for 1F, all the terms are F-wave dominant, and the P-wave come out from the F-wave
2
2
![Page 68: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/68.jpg)
![Page 69: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/69.jpg)
![Page 70: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/70.jpg)
![Page 71: Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China](https://reader030.vdocument.in/reader030/viewer/2022013013/5a4d1b3c7f8b9ab05999ef69/html5/thumbnails/71.jpg)
Summaries
The different forms of Salpeter wave function are given. The full Salpeter equations are solved for the low states, l
=0,l=1. The mass spectra for heavy mesons are calculated by BS
method. As simple applications, the decay constants and annihilati
ons of quarkonium are calculated by BS method. The relativistic corrections for the process which involved
a P-wave state are large, even it is heavy quarkonium.