Instantaneous Bethe–salpeter Equation: (semi-)analytical Solution

The Bethe–Salpeter equation for bound states of a fermion–antifermion pair in the instantaneous approximation for the involved interaction kernel is converted into an equivalent matrix eigenvalue problem with explicitly (algebraically) given matrices. PACS numbers : 11.10.St, 03.65.Ge ∗ E-mail address : [email protected] ‡ E-mail address : [email protected] † E-mail address : [email protected] INSTANTANEOUS BETHE–SALPETER EQUATION: (SEMI-)ANALYTICAL SOLUTION Wolfgang LUCHA Institute for High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18, A-1050 Vienna, Austria E-mail: [email protected] Khin MAUNG MAUNG Department of Physics, Hampton University, Hampton, VA 23668 E-mail: [email protected] Franz F. SCHÖBERL Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria E-mail: [email protected] The Bethe–Salpeter equation for bound states of a fermion–antifermion pair in the instantaneous approximation for the involved interaction kernel is converted into an equivalent matrix eigenvalue problem with explicitly (algebraically) given matrices. 1 The Instantaneous Bethe–Salpeter Equation (IBSE) For a system of massless fermion and antifermion forming bound states with the “pion-like” spin, parity, and charge conjugation quantum numbers J = 0, the (homogeneous) Bethe–Salpeter equation, in free-propagator approximation and instantaneous approximation for the involved interaction kernel, reads for a time-component Lorentz vector interaction (i.e., the Dirac structure γ⊗γ) 2 kΨ2(k) + ∞ ∫ 0 dk k (2π)2 V0(k, k ′)Ψ2(k ′) = M Ψ1(k) ,