2. Maung-Norbury-Kahana equation

In this section, the MNK equation is reviewed and the spinless MNK equation is derived. To discuss the bound states, e+e-, μ+μ- and μ+e-, the MNK equation with the Coulomb potential is needed. The logarithmic singularity in the momentum-space Coulomb potential is removed by the Landé subtraction method.

2.1. Reduction of the Bethe-Salpeter equation

The Bethe-Salpeter equation in momentum space reads ^1,6

where

In order to have a correct one-body limit in three-dimensional reduced equations, the Wrightmann-Gordon choice ¹³ of η1 and η2 should be applied,

where s = P ². In Eq.(1), SiF(pi) are the full fermion propagators. We will approximate the full propagators SiF(pi) by free propagators ⁵

where m₁ and m₂ are interpreted as effective masses for the fermion and antifermion.

We introduce components of the relative momentum p=p∥+p⊥ parallel and perpendicular to the bound-state momentum P by ^11,14,15,16

where p∥ is the longitudinal part and p⊥ is the transversepart. In the rest frame of the bound state with momentum P=(M,0),pl=p0,p‖=(p0,0) and p⊥=(0,p). The projection operators can be written in covariant form

with the properties

In this paper the covariant instantaneous approximation is employed ¹⁴, in which the approximated kernel is independent of the change of the longitudinal component of the relative momentum,

It is a good approximation for a system composed of heavy and light constituents or of two heavy constituents which can move relativistically as a whole. It will reduce to the instantaneous approximation in the rest frame of the bound state.

Introduce the notation for later convenience

where ψP(p⊥) is the Salpeter wave function. Using Eqs. (4), (6), (8) and (9), the Bethe-Salpeter equation (1) becomes

where

and

In Refs. ⁷ and ⁸, g0(P,p) is given as

where f(ι) is defined as

In the above equation, ι is the parameter describing the relative virtuality of two components in bound state. When ι=1, the constituent 1 is on-shell and constituent 2 are off-shell arbitrarily; vice versa, when ι=-1, the constituent 2 is on-shell with another constituent’s virtuality arbitrary. For the MNK equation,

Eq. (13) can be simplified as

where

and

If constituent 1 takes positive energy as 0≤ι≤1 and constituent 2 takes positive energy as -1≤ι<0, pl+ should be

After integrating over pl, we have from Eq. (16)

From Eqs. (11), (16) and (20), we have

Using Eqs. (9), (19) and (21), Eq. (10) reduces to the MNK equation

The MNK equation has been understood physically meaningful: when masses of constituents are not equal but comparable, this kind of choice of ι [Eq. (15)] promises that the heavier particle is less virtual while the lighter massive particles is further off-mass-shell. For the bound states composed of equally massive constituents, the constituents will be put equally off-mass-shell.

Assuming

we have from Eq. (22)

Neglecting any reference to the spin degrees of freedom of the involved bound-state constituents, we have the spinless MNK equation from Eq. (24)

where ι is in Eq. (15). Eq. (25) describes the semirelativistic bound states composed to two spinless constituents which are virtual according to Eqs. (14) and (15). Following the approaches in Refs. ⁶ and ²⁴, the spin-independent terms and spin-dependent terms can be obtained from Eq. (24).

2.2. Landé subtraction method

In this paper, the Coulomb potential is considered. The Coulomb potential reads in the momentum space

where α is the fine structure constant. The partial wave expansion of the spinless MNK equation (25) is expressed as

where n is the principal quantum number, l is the orbital angular quantum number. f (M, p) reads

Vl(p,p') is the partial wave expanded Coulomb potential,

where Q _l (z) is the Legendre polynomial of the second kind,

The Coulomb potential has the logarithmic singularity at point p' = p, and the singularity comes from Q ₀(z).

Applying the Landé subtraction method ^{17,18,19,20,21} to cancel out the singularity, the singular equation (27) becomes

where z' = 1, P_l (1) = 1,. In the above calculation, we have used the identity

3.1. Eigenvalue integral equation

The eigenvalue integral equation (31) can be written formally as

Due to the complicated form of g(M,p), Eq. (33) cannot be solved directly. Rewrite the above equation as ²²

where M̃ is a trial value. If M̃=M, ε will be equal to zero. The eigenvalue equation (34) can be solved by standard method.

3.2. Gauss-Legendre quadrature rule

Rewrite the subtracted integral equation (31) in the form of Eq. (34), then apply the Gauss-Legendre quadrature rule to the regular integral ²¹. Finally, a matrix equation can be obtained from Eq. (31) by employing the Nyström method and it can be solved easily.

At first, we map the semi-infinite interval [0,∞) onto some standard finite interval [a, b] which we take to be [-1,1). In this paper, we may take the rational transformation,

where ξ is a numerical parameter providing additional control of the rate of convergence. Then we have

The Gauss-Legendre quadrature formula for regular integral reads

where

In Eq. (38), prime stands for the derivative.

3.3. Numerical results and discussions

By employing the methods discussed above, the spinless MNK equation [Eq. (25)] with the Coulomb potential is solved numerically and the numerical results are listed in Tables I, II and III. The input parameters are N = 180, ξ=miα where m_i is the mass of the lighter constituent. For comparison of the eigenvalues obtained in this paper with the eigenvalues obtained in Ref. ¹⁷, the electron mass m_e = 0.51099906 MeV/c², the muon mass mμ=105.658389MeV/c² and fine structure constant α=1/137.0359895 are used ²³.

In the spinless MNK equation (25), the virtuality parameter ι is expressed in Eq. (15). As -1<ι<1, constituents are virtual. As ι=±1, regardless whether the constituents are light or heavy, the spinless MNK equation (25) reduces to the spinless Salpeter equation ¹⁷ in which both of constituents are on their mass shell. The spinless Salpeter equation is a well-defined standard approximation to the Bethe-Salpeter equation and a relativistic extension of the nonrelativistic Schrödinger equation. By comparing the binding energies of the Schrödinger equation and that of the spinless Salpeter equation, we can obtain the relativistic effects because the constituents are also put on-mass-shell in the Schrödinger equation as in the spinless Salpeter equation. From Tables I, II and III, we can see that the relativistic corrections are the differences in energies which occur after a few decimal places.

The spinless MNK equation includes not only the relativistic effects but also the virtuality effects. By comparing the eigenvalues of the spinless MNK equation and that of the spinless Salpeter equation, we can obtain the virtuality effects. The eigenvalues of the spinless MNK equation are smaller than that of the spinless Salpeter equation and the Schrödinger equation, see Tables I, II and III. It means that the virtuality effect of constituents results in stronger binding. For the positronium, the virtuality effect is about of the same order as the relativistic effects. For the muonium, the virtuality effect is smaller than the relativistic effect. The data show that the virtuality effect varies with the virtuality parameter ι. For more general cases, the relation between the virtuality effect and ι will become complex ²².

The spinless MNK equation (25) describes the bound states composed of the spinless virtual constituents. By employing the approaches applied in Refs. ⁶ and ²⁴, the spin-independent terms and spin-dependent terms can be obtained from Eq. (24). Then spin effects can be included according to the discussed problems.