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 equation6, 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 ηP=(-1)j+1, and the L=j±1 wave components are main terms for state with parity ηP=(-1)j, where 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 (-1)j, and for state with parity (-1)j+1 the spin singlet and spin triplet will decouple, too. In the limit of vanishing masses of the two bound-state constituents, the Salpeter wave function and Salpeter equation will be simplified in this paper. It maybe be not only of academic interest but also instructive and useful to obtain the features of the Salpeter equation in the ultrarelativistic limit, which are different from the features obtained in the limit.
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
ψP(p⊥)=Λ1+(p⊥)⧸P^Γ(p⊥)⧸P^Λ2-(-p⊥)M-ω1-ω2-Λ1-(p⊥)⧸P^Γ(p⊥)⧸P^Λ2+(-p⊥)M+ω1+ω2,
1
where M is the mass of the bound state, P is the bound-state momentum, and p is the relative momentum. p∥ is the longitudinal part of p and parallel to P, p⊥ is the transverse part of p and perpendicular to P5,11,15
P^=PM, M=P2, pl=p⋅P^, p=p∥+p⊥,p∥=plP^, p⊥=p-plP^, d4p=dpld3p⊥.
2
In the rest frame of the bound state with momentum P=(M,0), pl=p0, p‖=(p0,0) and p⊥=(0,p). Γ(p⊥) is defined as
Γp⊥=∫d3p´⊥2π3V(p⊥,p´⊥)ψP(p´⊥)
3
The projection operators are written in covariant form as
Λi±(p⊥)=ωi±Hi(p⊥)2ωi,Hi(p⊥)=⧸P^(mi-⧸p⊥),ωi=mi2+ϖ2,ϖ=-p⊥2
4
with the properties
Λi∓(p⊥)Λi±(p⊥)=0,Λi+(p⊥)+Λi-(p⊥)=1,Λi±(p⊥)Λi±(p⊥)=Λi±(p⊥),Hi(p⊥)Λi±(p⊥)=±ωiΛi±(p⊥).
5
In case of massless constituents, m1=m2=0, ω1=ω2
=ϖ, Λ1∓(p⊥)=Λ2∓(p⊥).
Applying the energy projectors Λ1±(p⊥) from the left hand side and Λ2±(-p⊥) from the right hand side to the Salpeter equation (1) leads to
(M-ω1-ω2)ψP+-(p⊥)=Λ1+(p⊥)⧸P^Γ(p⊥)⧸P^Λ2-(-p⊥),(M+ω1+ω2)ψP-+(p⊥)=-Λ1-(p⊥)⧸P^Γ(p⊥)⧸P^Λ2+(-p⊥)
6
together with the constraints on the Salpeter wave function
ψP++p⊥=ψP--P⊥=0,
7
where ψP±±(p⊥)=Λ1±(p⊥)ψP(p⊥)Λ2±(-p⊥).
Let the bound state be normalized as ⟨P|P'⟩=(2π)32P0δ(P-P'). Then the explicit form of the normalization condition for the Salpeter wave function reads
∫d3p⊥(2π)3Tr{⧸P^ψ¯(p⊥)⧸P^Λ1+(p⊥)ψ(p⊥)Λ2-(-p⊥)-⧸P^ψ¯(p⊥)⧸P^Λ1-(p⊥)ψ(p⊥)Λ2+(-p⊥)}=2M,
8
where ψ¯(p⊥)=γ0ψ†(p⊥)γ0.
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 m1=m2=0, the constraints on the Salpeter wave function as well as the Salpeter equation will be different from that obtained in the limit.
3.1. State 0+
For state 0+, the Salpeter wave function reads
ψ0+(p⊥)=g1+⧸P^g2+⧸p^⊥g5+⧸p^⊥⧸P^g6,
9
where p^⊥μ=p⊥μ/ϖ. When the scalar functions gi are not to be integrated, gi≡gi(ϖ), i = 1,2,5,6, and when g i are to be integrated, gi≡gi(ϖ'), ϖ'=-p⊥'2. g1 and g2 are S wave components, while g5 and g6 are P wave components.
Applying the constraints (7) on the Salpeter wave function (9) yields
g2=g5=0
10
The normalization condition reads
∫d3P⊥(2π)34g1g6=M
11
For simplicity, taken as an example, the interaction kernel takes the form
VP⊥-P´⊥=VS+γ0⊗γ0V0+γμ⊗γμVv
12
where Vs is scalar function corresponding to interactions of the scalar type, V0 is the time-component Lorentz-vector part, and Vv 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)
Mg1=2ϖg6-∫d3p⊥'(2π)3(V0-Vs)p^⊥⋅p^⊥'g6,Mg6=2ϖg1+∫d3p⊥'(2π)3(V0+Vs+4Vv)g1.
13
In the limit, state 0+ is P wave state, the P wave components g5 and g6 are large terms, while the S wave components g1 and g2 are mall and are relativistic corrections to g5 and g 6 . But in the ultrarelativistic limit, g2=g5=0, g1 and g6 are large terms, and the coupled Eqs. (13) are on g 1 and g 6 . In the ultrarelativistic limit, the energy projection operator Λi±(p⊥)=(1∓⧸P^⧸p⊥)/2 and Eqs. (6), (7) choose the g1 and g6 as large terms. In the limit, we can find that g1 and g2 are chosen as main terms by inspecting the energy projection operator Λi±(p⊥)=(1±⧸P^)/2, the Salpeter wave function (9) and the Salpeter Eq. (6).
3.2. State 0-
For state 0-, the Salpeter wave function reads
ψ0-(p⊥)=γ5#[g1+⧸Pg2+⧸p⊥g5+⧸p⊥⧸Pg6#].
14
The constraints on the Salpeter wave function (14) are
g2=g5=0.
15
The normalization condition is
∫d3P⊥(2π)34g1g6=M
16
The coupled equations read
Mg1=2ϖg6-∫d3p'⊥(2π)3(V0+Vs)p^⊥⋅p^'⊥g6,Mg6=2ϖg1+∫d3p'⊥(2π)3(V0-Vs+4Vv)g1.
17
In the limit, state 0- is S wave state, therefore, the S wave components g1 and g2 are large terms, while the P wave components g5 and g6 are small and corrections to the S wave components. Different from the case of the limit, g1 and g6 are large terms in the ultrarelativistic limit.
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 g1 and the P wave component g6 are large. Moreover, if the interaction is vector or time component of vector, Eqs. (13) and (17) imply the existence of the degeneracy of the spectra, but the scalar interaction Vs will destroy the degeneracy.
3.3. State with parity Ƞp = (-1)j
For state with parity ηP=(-1)j ( j>0), the general form of the Salpeter wave function reads
ψj(p⊥)=ϵμ1⋯μjp^⊥μ2⋯p⊥μj[p^⊥μ1(g1+⧸P^g2)+γ⊥μ1(g3+⧸P^g4)+(p^⊥μ1⧸p^⊥+j2j+1γμ1)×(g5+⧸Pg6)+σμ1νp⊥ν(g7+⧸Pg8)],
18
where γ‖μ=⧸P^P^μ, γ⊥μ=γμ-γ∥μ, σμν=[γ⊥μ,γ⊥ν], gi≡gi(ϖ), i = 1,2,3,4,5,6,7,8. g 3 , g 4 are pure L = j -1 wave components, and g5, g6 are pure L=j+1 states. g1, g2, g7 and g8 are L = j wave components. In the limit, g3, g4, g5 and g6 are main terms 9, while g1, g2, g7, and g8 are small terms, which are relativistic corrections in wave function.
The constraints on the Salpeter wave function (18) read in the ultrarelativistic limit
g2=g7=0, g4=j2j+1g6, g3=j+12j+1g5.
19
In the limit, for state with parity (-1)j, the L = j -1 wave components g3, g4 and the L = j+1 wave components g5, g6 are large. But in case of massless constituents, L = j wave components g1 and g8 are large which are small in the limit.
Using Eqs. (8) and (18), the normalization condition can be obtained
∫d3p⊥(2π)34S1jg1g6-S1j+S2jg5g8=M,
20
where
S1j=∑Pμ1⋯μjν1⋯νjp^⊥μ1⋯p^⊥μjp^⊥ν1⋯p^⊥νj,S2j=∑Pμ1⋯μjν1⋯νjgμ1ν1p^⊥μ2⋯p^⊥μjp^⊥ν2⋯p^⊥νj.
21
The definition of Pμ1⋯μjν1⋯νj in Eq. (21) is in appendix. Using Eqs. (6) and (18), the coupled equations can be obtained
Mg1=2ϖg6-∫d3p´⊥p(2π)3T1jS1j(V0-Vs)p^⊥⋅p´^⊥g6,Mg6=2ϖg1+∫d3p´⊥(2π)3T1jS1j(V0+Vs+4Vv)g1,Mg5=2ϖg8-∫d3p´⊥(2π)3(p^⊥∙p´^⊥T3j-T5j)(S1j+S2j)×(V0+Vs+2Vv)g8,Mg8=2ϖg5+∫d3p´⊥(2π)3(p^⊥⋅p´^⊥T1j+T2j+T3j+T4j)(S1j+S2j)×(V0-Vs+2Vv)g5,
22
where
T1j=∑Pμ1⋯μjν1⋯νjp^⊥μ1⋯p^⊥μjp'^⊥ν1⋯p'^⊥νj,T2j=∑Pμ1⋯μjν1⋯νjp'^⊥μ1p'^⊥ν1p^⊥μ2⋯p^⊥μjp'^⊥ν2⋯p'^⊥νj,T3j=∑Pμ1⋯μjν1⋯νjgμ1ν1p^⊥μ2⋯p^⊥μjp'^⊥ν2⋯p'^⊥νj.T4j=∑Pμ1⋯μjν1⋯νjp^⊥μ1p^⊥ν1p^⊥μ2⋯p^⊥μjp'^⊥ν2⋯p'^⊥νj,T5j=∑Pμ1⋯μjν1⋯νjp'^⊥μ1p^⊥ν1p^⊥μ2⋯p^⊥μjp'^⊥ν2⋯p'^⊥νj.
23
In Eq. (22), there are two sets of coupled equations in which g1 and g6 are coupled, and g5 and g8 are coupled, but g1, g6 and g5, g8 are decoupled. When not considering the limit or ultrarelativistic limit, the obtained coupled equations are on g3, g4, g5 and g6, which are coupled to each other, i.e., the L = j - 1 wave components g 3 and g 4 and the L = j + 1 wave components g5 and g6 are coupled. In the limit, the 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 g1 and g8 become large.
The constraints (19) rule out the exotic states with parity ηP=(-1)j and charge-conjugation parity ηP=(-1)j+1 , 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 ȠP = (-1)
j+1
For state with parity ηP=(-1)j+1 ( j>0), the general form of the Salpeter wave function reads
ψj(p⊥)=γ5ϵμ1⋯μjp^⊥μ2⋯p⊥μj[p^⊥μ1(g1+⧸P^g2)+γμ1(g3+⧸P^g4)+(p^⊥μ1⧸p^⊥+j2j+1γμ1)×(g5+⧸Pg6)+σμ1νp⊥ν(g7+⧸Pg8)],
24
where g1, g2, g7, and g8 are main terms which are pure L = j wave component 9. While g3, g4, g5 and g6 are small terms, which are relativistic corrections in wave function.
The constraints on the Salpeter wave function (24) read
g2=g7=0,g4=-j2j+1g6,g3=j+12j+1g5.
25
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
∫d3p⊥(2π)34S1jg1g6-(S1j+S2j)g5g8=M,
26
which is the same as Eq. (20).
The coupled equations read
Mg1=2ϖg6-∫d3p´⊥(2π)3T1jS1j(V0+Vs)p^⊥⋅p´^⊥g6,Mg6=2ϖg1+∫d3p´⊥(2π)3T1jS1j(V0-Vs+4Vv)g1,Mg5=2ϖg8-∫d3p´⊥(2π)3(p^⊥⋅p´^⊥T3j-T5j)(S1j+S2j)×(V0-Vs+2Vv)g8,Mg8=2ϖg5+∫d3p´⊥(2π)3(p^⊥⋅p´^⊥ +T1j+T2j+T3j+T4j)(S1j+S2j)×(V0+Vs+2Vv)g5.
27
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 (-1)j and (-1)j+1 are different 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.
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