Stability of the Solutions of Instantaneous Bethe–salpeter Equations with Confining Interactions

For two bound-state equations derived as simplified forms of the Bethe–Salpeter equation with confining interaction, stability of all solutions is rigorously shown. 1 Motivation: Instabilities of Klein-Paradox Type The Salpeter equation is a frequently applied three-dimensional reduction of the Bethe–Salpeter formalism describing bound states within quantum field theory, derived by assuming all interactions to be instantaneous. For given interactions, encoded in its integral kernel K(p, q) depending on relative three-momenta p, q of the bound-state constituents, it can be regarded as an eigenvalue equation for the Salpeter amplitude Φ(p), with the massM of the bound state as eigenvalue. For confining interactions, however, its solutions exhibit in numerical studies 1) certain instabilities, possibly related to Klein’s paradox, causing states to decay. In view of this highly unsatisfactory state of the art, we began a systematic rigorous analysis of the conditions for stability of the energy levels predicted, for confining interactions, within this framework. We regard a bound state as stable if its mass eigenvalueM belongs to a real discrete spectrum bounded from below. On energetic grounds, any instabilities should manifest themselves first for pseudoscalar bound states; accordingly, we focus to fermion–antifermion bound states characterized by spin-parity-charge conjugation assignment J = 0. This allows to take advantage of experience gained in earlier studies 2, 3, 4, 5). We analyze three-dimensional reductions of the Bethe–Salpeter formalism for increasing complexity of the problem: the reduced Salpeter equation 6, 7), a generalization thereof, proposed in Ref. [8] and applied in Ref. [9], towards exact propagators of the bound-state constituents 10), and the full Salpeter equation. With Dirac couplings Γ, assumed to be identical for both constituents, and related potential functions VΓ(p, q), the action of some kernelK(p, q) on Φ(p) is [K(p, q)Φ(q)] = ∑ Γ VΓ(p, q) ΓΦ(q) Γ . For all interactions of harmonic-oscillator type in configuration space, the above bound-state equations simplify to ordinary differential equations, which may be cast into the form of “tractable” eigenvalue equations for Schrödinger operators, at least in the case of the reduced bound-state equations studied in Secs. 2 and 3. Our primary tool is a (well-known) theorem which states that the spectrum of a Schrödinger Hamiltonian operator with a locally bounded positive potential V rising to infinity in all directions, V (x) → +∞ for |x| → ∞, is purely discrete. 2 Reduced Salpeter Equation Approximating the propagation of both bound-state constituents by that of free particles of constant effective “constituent” massm yields the Salpeter equation Φ(p) = ∫ dq (2π) ( Λ(p) γ0 [K(p, q)Φ(q)] Λ (p) γ0 M − 2E(p) − Λ(p) γ0 [K(p, q)Φ(q)] Λ (p) γ0 M + 2E(p) )