Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China

 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

Bethe-Salper Equation and its instantaneous one, Salpeter equation where We introduce the symbols

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)

and then the BS equation can be written as where the propagators can be decomposed as with

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

with contour integration over the instantaneous BS equation become Finally, the instantaneous BS equation turn to the Salpeter equation

The normalization condition is

Wave functions 1, wave functions for pseudoscalar meson and scalar 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

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

Salpeter wave fucntions Wave function for state Because 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

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,

and the normalization condition is

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

And the constraint relations with the renormalization

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

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

Wave functions for state The general form of the Salpeter wave function for state ( for equal mass system) and the constraint relations

with the renormalization condition:

Wave functions for state The general form of the Salpeter wave function for state ( for equal mass system ) and the constraint relations renormalization condition

The mixing of two states For equal mass system, because of the difference of charge conjugation quantum number, the vector states can be distinguished by the charge conjugation, 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

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,

The interaction kernel We choose the Cornell potential and the coupling constant is running in one loop

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 needed in potential model methods to move all the masses with mass shift to fitting data.

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

Mass spectra

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

Tensor splitting Fine splitting

The parameters

Decay constants Decay constants for state For pseudoscalar, the decay constant is defined as In the Bethe-Salper method, it can be calculated as

Decay constants for state The decay constant for state is defined as In the BS method, it can be calculated as

Decay constants for P-wave state Decay constant for state Decay constant for (or ) state

For the mixed state, we have use the following relation to calculating the decay constants

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.

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

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

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.

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

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

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

Then the decay amplitude become

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

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

where

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

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

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 annihilations 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.