Bethe-Salper equation and its applications Guo-Li WangDepartment 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 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

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

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 )

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

1PJ

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

0PJ

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

Wave functions for stateThe general form of the Salpeter wave function for

state ( for equal mass system)

and the constraint relations

2

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

1PJ 1

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

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 n

eeded 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

0

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

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.

0

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

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.

0

0

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

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

2

2

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

1

1

2

1

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

2

2

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.