 the energy curves (in λ or αZ) are not continuous. For small values of λ, one can show that the eigenvalues correspond to the following behavior : On the other hand, for B = π 2, another state with no correspondence with the spectrum for αZ < 1 2 appears. Its energy is given, from Eq. (2.15), by the equation and has the value E/M ≃ −0.049.
We now turn to the question of a possible instability of the system, described by the KG equation (2.1) with a pointlike attractive Coulomb interaction, with respect to the spontaneous pair creation. This would correspond to the existence of a critical value of To see whether such a solution of Eq. (2.12) exists for small values of λ, it is advantageous to transform the third term of Eq. (2.12) by using the following formula [23] : (2.21) One finds, using Eq. (2.12) for large values of |k| and Euler's constant γ = −ψ(1) = 0.57721..., that the critical value of Z c is given by corresponds (for small λ) to the boundary case B = π/2 between the two domains B < π/2 and B > π/2.
For the KG equation, the sign of the KG norm unambiguously distinguishes between particle and possible antiparticle bound states. We have calculated numerically the norm of the states of E ≃ −M wave functions and established that they are indeed particle states.
We further show in Fig. 3 the radial wave function corresponding to the lowest state for αZ = 1 and B = 1.58 (the dot-dashed curve in Fig. 2). It, of course, displays an infinite number of nodes consistent with the behaviour (2.9), but the outermost of these nodes occurs at a distance representing a rather small fraction (here of the order of one tenth) of the "radius" of the state. Globally, the wave function has a "nodeless" structure, typical of a ground state wavefunction.
We thus find that the stability property with respect to pair creation of a Coulomb source with vanishing radius and charge Z > 137/2 crucially depends on the choice of the self-adjoint extension characterized by the constant B. In principle, this constant should be determined by comparison with experimental data. However, the case of physical interest corresponds here to a finite radius R of the source and has been extensively discussed in the literature [24]. We shall not pursue further the investigation of this academic problem.
Let us simply note that the Coulomb field of a charge with radius R satisfying MR << 1 is supercritical for αZ > αZ c = 1/2 [8], so that the choice B > π/2 would be mandatory to reproduce finite radii results.

III. STRONGLY COUPLED POSITRONIUM
We now consider the problem of two particles of equal mass m and opposite charges with spin one-half, in mutual electromagnetic interaction. Within RQCD the 1 S 0 states of the system with the Todorov choice of the interaction [13] are described in the c.m. frame by the radial equation [8,9] (see also the appendix) : W being the c.m. energy of the two-body system. Equation. From now on, we shall consider α as a free parameter.
where the scalar product is now defined by : The conserved norm N corresponding to Eq. (3.6) is given by : For α > 1/2, the solutions of Eq. (3.1), vanishing at r = ∞, are given by : As r → 0, the wave function ϕ(r) exhibits the behavior described in Eqs.
Only a few states corresponding to Eq. and examine the behavior of the wave function near the origin; it is given by Eq. (2.8) : In order to display the main qualitative features of this type of solution, we consider, in the following, large values of |η|,while keeping λ small. Then, the functions Γ( can be expanded in terms of iλ : [ǫ(η) is the sign of η,] leading in turn to the following behavior of W k,µ : The wave function ϕ has the same formal behavior as in Eqs. (2.9) or (2.11), with β defined as (4.5) Notice that β, and hence B, is complex.
In order to study the orthogonality conditions for these states, we observe that when tachyons exist, there is a doubling of states : for each "eigenvalue" iη, the value −iη is also a solution. The rules of constructing scalar products for such states have been studied in the two papers of Ref. [25]. For a state with complex "eigenvalue" iη one also considers the state with the complex conjugate "eigenvalue" −iη, called the associated vector. While the norm of a state with complex eigenvalue is zero, its scalar product with its associated vector is nonzero [25]. Therefore, the following rule should be adopted for the choice of admissible states. Define admissible states as those corresponding to a definite sign of η in the "eigenvalues" iη. Then, the corresponding adjoint states in the scalar product are the associated vectors (with "eigenvalues" −iη); these appear there complex conjugated.
The analysis of Case can then be repeated for the tachyonic states. Once the same complex value of B is chosen for the tachyonic solutions, then these will satisfy among themselves the orthogonality conditions (3.5) and might constitute admissible states. [Notice that associated vectors cannot be considered as admissible states, for, according to Eq. (4.5), they would not have the same value of B; they appear only as the adjoints of admissible states.] However, the orthogonality condition (3.5) fails when considered between a tachyonic state and a "normal" state (with W 2 > 0 and W > 0), because the coefficient B cannot be chosen the same for both types of solution (it must be complex for the former and real for the latter). This is the reason why tachyons are rejected from the Hilbert space of physical states.
Therefore, we end up with the conclusion that the physical spectrum is free of tachyons and its ground state is the zero energy state. Within these circumstances, the limiting value α = 1 2 + ǫ (ǫ = +0) plays a particular role for a physical interpretation of the theory. In this limit, the oscillating behaviors of the type (2.9) or (2.11) disappear from the wave functions and, according to Eq. (3.18), all the low mass states shrink to a single state with zero mass. A definite mass gap appears between the zero mass ground state and the other massive states of the spectrum.
To study in more detail the properties of the corresponding wave functions, it is preferable to reanalyze Eq. (3.1) for the particular value α = 1 2 + ǫ. The normalizable solutions are the Whittaker functions W k,0, which behave near the origin as In order that the corresponding solutions be the limits of those found for α > 1/2 [Eqs. where c 0 is a dimensionless normalization constant and Ψ (also denoted by U in the literature) is the confluent hypergeometric function which behaves at infinity with a power law [21]; its dominant behavior near the origin is given by The solution (5.7) is a nodeless function and represents the ground state of the spectrum.
[The function Ψ(a, c; x) does not have positive zeros for a and c real and either a > 0 or a − c + 1 > 0. Also, it can be checked that the formal tachyonic solutions, found in Sec. IV, are absent in the present case.] It has a distribution-like behavior, due to the fact that it is defined by the limiting procedure W 0 → 0 : taking the limit W 0 → 0, while keeping r fixed, shows that the wave function is actually peaked at the origin (recall that the complete wave function is ϕ/r).
To compute the normalization constant of this wave function (as well as of the others), it is necessary to reconstruct the whole sixteen-component spinor wave function and to use its relationship with the Bethe−Salpeter wave function, which ultimately fixes the normalization coefficients. Some details of these calculations can be found in the appendix. One finds for the norm N of Eq. (3.17) the expression : where W is the mass of the bound state.
As to the massless state, after using the change of variable (5.5) in Eq. (3.7), we find that the dominant contribution for small W comes from the third term in the integral. This leads to the following behavior of the normalization constant c 0 of the wave function ϕ 0 (y) [Eq. (5.7)] for small W : We now turn to the calculation of the coupling constants of the bound states to the axial vector current. These are defined as < 0|j R µ5 (0)|P > = P µ F, (5.11) where P is the four-momentum of the pseudoscalar state and j R µ5 is the renormalized axial vector current. In general, in the absence of anomalies, the axial vector current undergoes only a finite multiplicative renormalization by radiative corrections [26]. This feature is, however, the result of compensating contributions from propagator and vertex renormalizations (with factors Z 2 and Z −1 A, respectively). More explicitly, one has where j µ5 is the unrenormalized current, with Z 2 Z