Recent Developments in the Bethe-Salpeter Description of Light Mesons

Results for the light meson mass spectrum from a Bethe-Salpeter approach are presented. The results obtained in the standard framework are Poincaré covariant and compare favourably with lattice results. Using a more sophisticated scheme, the pseudoscalar, vector and 1++ (a1/ f1) axialvector charge eigenstate masses are unaltered whereas the 1+− (b1/h1) axialvector meson mass is raised. The Bethe-Salpeter equation [BSE] is the fully relativistic description of the two-body bound state problem. It has been found in the last decade that the ladder truncation of the BSE, using as input the quark propagators derived from the rainbow truncation of the Schwinger-Dyson equation [DSE], gives rise to a good description of the light flavor non-singlet pseudoscalar and vector mesons [1]. The underlying mechanism for this is chiral symmetry manifested through the flavor non-singlet axialvector Ward-Takahashi identity [AXWTI]. By ensuring that the kernels of both equations respect the AXWTI, it is shown that the pion emerges as both a bound state of massive constituents and as an almost massless Goldstone boson of the broken chiral symmetry [2]. The ladder truncation of the homogeneous BSE for quark-antiquark mesons is written (working in Euclidean space with Hermitian Dirac matrices obeying {γμ ,γν} = 2δμν ): Γ(p;P) = − 4 3 ∫ d4k (2π)4 g∆μν(p− k)γμS(k+)Γ(k;P)S(k−)γν , (1) where Γ is the Bethe-Salpeter amplitude, k+ = k + ξ P and k− = k + (ξ − 1)P with ξ = [0,1] the momentum sharing parameter between the two quarks. Invariance of the resulting observables with respect to ξ is a reflection of Poincaré covariance. The total momentum P = p+ − p− is such that the equation is solved for P2 = −M2 where M is the mass of the meson. In Eq. (1), the dressed quark propagators are the solution of the rainbow quark DSE S−1(p) = ı / p+m+ 4 3 ∫ d4k (2π)4 g∆μν(p− k)γμS(k)γν , (2) where m is the current mass parameter of the quark; the two truncations being consistent with the AXWTI. The effective interaction g∆μν(q) has the following form [3]: g∆μν(q) = tμν(q)4πD q2 ω2 exp (