PACS: 11.10.St; 03.65.Ge; 03.65.Pm; 12.39.Pn

1.Introduction

The Bethe-Salpeter equation ^{1} is the appropriate tool to deal with bound-state problems within the framework of relativistic quantum field theory. The nature of the Bethe-Salpeter equation renders it difficult to deal with. It is a four dimensional integral equation, and it requires the full propagators for constituents as well as their interaction kernel from the very beginning. Therefore, simplification is necessary, for example, the ladder approximation and replacing full propagators with free ones. Besides, more simplifications are often used in practice when the Bethe-Salpeter equation is applied to different problems, for instance, massless exchanging boson ^{2}, massless bound state ^{3}, massless constituents ^{4}, large constituent’s mass ^{5} and so on.

Due to various practical reasons, the Salpeter equation^{6}, one of the three-dimensional reductions of the Bethe-Salpeter equation, is frequently used in practice. From the Salpeter wave function and Salpeter equation, it can be found that the *L* = *j* wave components play dominant role in the wave function for state with parity *L* is the relative orbital angular momentum quantum number, *j* is the spin of the bound state. These features can be shown more obviously in the limit. In addition, when taking the limit the *L* = *j* - 1 wave components and *L* = *j* +1 wave components will decouple for state with parity

This paper is organized as follows. In Sec. 2, the Salpeter equation is briefly reviewed. In Sec. 3, the reduction of the Salpeter equation for the massless constituents is presented. The conclusion is given in Sec. 4.

2. Salpeter equation

Due to the problems in actual applications ^{4}, the reductions of the Bethe-Salpeter equation are highly desirable ^{1}^{,}^{16}^{,}^{11}^{,}^{12}^{,}^{13}. It has been shown ^{14} that there exist infinite versions of the reduced Bethe-Salpeter equation. The Salpeter equation is the most famous one, which is based on the instantaneous approximation ^{6}. In this paper, the Salpeter equation is employed.

Let us briefly review the Salpeter equation in this section. The Salpeter equation for a fermion-antifermion bound state reads in covariant form ^{10}

where *M* is the mass of the bound state, *P* is the bound-state momentum, and *p* is the relative momentum. *p* and parallel to *P*, *p* and perpendicular to *P*^{5}^{,}^{11}^{,}^{15}

In the rest frame of the bound state with momentum

The projection operators are written in covariant form as

with the properties

In case of massless constituents,

Applying the energy projectors

together with the constraints on the Salpeter wave function

where

Let the bound state be normalized as

where

3. Salpeter equation for massless constituents

In this section, we will consider the Salpeter wave function and Salpeter equation for bound state with massless constituents. In the ultrarelativistic limit

3.1. State 0^{+}

For state

where *i* = 1,2,5,6, and when g i are to be integrated,

Applying the constraints (7) on the Salpeter wave function (9) yields

The normalization condition reads

For simplicity, taken as an example, the interaction kernel takes the form

where V_{s} is scalar function corresponding to interactions of the scalar type, V_{0} is the time-component Lorentz-vector part, and V_{v} is the Lorentz-vector part. Other types of interaction kernels can be treated in similar way. The coupled equations for 0^{+} are obtained from Eqs. (6) and (9)

In the limit, state 0^{+} is P wave state, the P wave components

3.2. State 0^{-}

For state 0^{-}, the Salpeter wave function reads

The constraints on the Salpeter wave function (14) are

The normalization condition is

The coupled equations read

In the limit, state 0^{-} is S wave state, therefore, the S wave components

It is interesting that for both psudoscalar state 0^{-} and scalar state 0^{+}, the constraints and the normalization conditions are the same, see Eqs. (10), (11), (15) and (16). In case of the limit, state 0^{+} is P wave state and state 0^{-} is S wave state. But in case of the massless constituents, the wave components are the same for both state 0^{-} and state 0^{+}, the S wave component _{s} will destroy the degeneracy.

3.3. State with parity Ƞ_{p} = (-1)^{j}

For state with parity

where *i* = 1,2,3,4,5,6,7,8. g 3 , g 4 are pure *L* = *j* -1 wave components, and *L* = *j* wave components. In the limit, ^{9}, while

The constraints on the Salpeter wave function (18) read in the ultrarelativistic limit

In the limit, for state with parity *L* = *j* -1 wave components *L* = *j*+1 wave components *L* = *j* wave components

Using Eqs. (8) and (18), the normalization condition can be obtained

where

The definition of

where

In Eq. (22), there are two sets of coupled equations in which *i.e*., the *L* = *j* - 1 wave components g 3 and g 4 and the *L* = *j* + 1 wave components *L* = *j* - 1 wave components and the *L* = *j* + 1 wave components are decoupled, and the *L* = *j* wave components are small terms. But in the ultrarelativistic limit, the *L* = *j* wave components

The constraints (19) rule out the exotic states with parity ^{8}^{,}^{9}^{)}
*i.e*., such states cannot be constructed by the Salpeter equation, which is consistent with the results obtained by L-S coupling analysis in the limit.

3.4. State with parity Ƞ_{P} = (-1)*
^{j+1}
*

For state with parity

where *L* = *j* wave component ^{9}. While

The constraints on the Salpeter wave function (24) read

which are the same as the constraints for state with parity (-1)*
^{j}
* , see Eq. (19).

In case of the ultrarelativistic limit, the normalization condition reads

which is the same as Eq. (20).

The coupled equations read

Eqs. (22) and (27) are different only in the sign of V s term.

In the limit, the constraints on the Salpeter wave functions, the normalization conditions and the spectra for the states with parity ^{10}. But in the ultrarelativistic limit, the constraints on the Salpeter wave functions and the normalization conditions are the same for states with different parity, see Eqs. (19), (20), (25) and (26). Moreover, we can obtain from Eqs. (13), (17), (22) and (27) that there are degenerate doubles with the same spin but with different parity if the interaction is vector or time-component of vector. And the scalar interaction will destroy this degeneracy. These results maybe be only of academic interest. Nevertheless, it is instructive to pursue the insight of the bound states in the ultrarelativistic limit.

4. Conclusion

In this paper, we have presented the reduction of the Salpeter equation for the massless constituents. It is shown that *L* = *j* and *L* = *j* + 1 wave components play main roles for both states with different parity in the ultrarelativistic limit while in the limit, *L* = *j* wave components are large terms for