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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
BS equation become

the instantaneous
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
 P ( q)  ( f1  f 2 q P  f 3 P  f 4 q P P ) 5  f 5  f 6q P  f 7 P  f 8q P P



the other 8 terms vanish because of qP  P  0



But not all the remained 8 terms are pseudoscalar, half

0
of them are scalar, so when we consider a
state, the
general form is
 P ( q)  ( f1  f 2 q P  f 3 P  f 4 q P P ) 5



And a scalar wave function which J P  0
 P ( q)  f1  f 2 q P  f 3P  f 4q P P


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 0  state, we obtain the relations:
So finally, for 0 (0 ), 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
P

J

1
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 J P  0
state is
and we have the further constraint relations
the renormalization is
Wave functions for
state
P

The general form of the Salpeter wave function for J  1
state ( 1 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 ( 2 for equal mass system)
and the constraint relations
with the renormalization condition:
Wave functions for
state
P

The general form of the Salpeter wave function for J  1
state ( 1 for equal mass system )
and the constraint relations
renormalization condition
The mixing of two 1 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 0 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

Decay constant for
(or
) state

For the mixed
state, we have use the following
relation to calculating the decay constants
Annihilation rate of quarkonium

0
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 0
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

0 
state
The transition amplitude of two-photon decay of 0 
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 nonrelativistic results.
The relativistic Salpeter wave function for 0  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

2
state
The transition amplitude of two-photon decay of 2
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 1 and P-F mixing in 2  state

S-D mixing in 1 state (example)
The wave function for 1 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 Swave 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

2
state
The wave function for 2 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.