Binding energies in the $0^{-+}$ and $1^{--}$ fermion--antifermion channels from the ladder Bethe-Salpeter and Schrödinger equations

Abstract

In this work, we calculate the binding energies of an equal-mass fermion--antifermion system in the $0^{-+}$ and $1^{--}$ channels from the ladder Bethe--Salpeter (BS) equation with scalar, pseudoscalar, and vector exchange. The results are compared with those of the Schrödinger equation with the corresponding leading static interaction in two regimes: pointlike massless exchange at weak coupling and finite-mass exchange with monopole form factors at stronger coupling. For the BS calculation, we use two independent numerical methods based on momentum-space integral equations and coordinate-space differential equations. The differential-equation method gives estimated relative binding-energy errors below $10^{-9}$ in both regimes. For massless vector exchange in Coulomb gauge, the numerical results are consistent with the uncrossed-ladder perturbative spectrum through the $O(mα^5)$ terms. The results for massless scalar exchange and vector exchange in Feynman gauge are consistent with the common Wick--Cutkosky expansion through order $mα^3$. For finite exchange masses, the differences from the leading static Schrödinger spectra at the same masses, couplings, and form factors depend strongly on the exchange and channel. Scalar exchange gives nearly degenerate channels with much weaker binding than the Schrödinger reference. In contrast, vector exchange with only the metric term retained gives stronger binding than the Schrödinger reference in $0^{-+}$ and weaker binding in $1^{--}$. Notably, retaining the full Proca numerator substantially increases the vector binding in both channels. Pseudoscalar exchange also supports bound states and favors $1^{--}$, although its leading static central interaction vanishes.