  Modeling the underlying pairing mechanism of charge carriers in strongly
correlated electrons, starting from a microscopic theory, is among the central
challenges of condensed-matter physics. Hereby, the key task is to understand
what causes the appearance of superconductivity at comparatively high
temperatures upon hole doping an antiferromagnetic (AFM) Mott insulator.
Recently, it has been proposed that at strong coupling and low doping, the
fundamental one- and two-hole meson-type constituents -- magnetic polarons and
bipolaronic pairs -- likely realize an emergent Feshbach resonance producing
near-resonant $d_{x^2-y^2}$ interactions between charge carriers. Here, we
provide detailed calculations of the proposed scenario by describing the open
and closed meson scattering channels in the $t$-$t'$-$J$ model using a
truncated basis method. After integrating out the closed channel constituted by
bipolaronic pairs, we find $d_{x^2-y^2}$ attractive interactions between open
channel magnetic polarons. The closed form of the derived interactions allows
us analyze the resonant pairing interactions and we find enhanced (suppressed)
attraction for hole (electron) doping in our model. The formalism we introduce
provides a framework to analyze the implications of a possible Feshbach
scenario, e.g. in the context of BEC-BCS crossover, and establishes a
foundation to test quantitative aspects of the proposed Feshbach pairing
mechanisms in doped antiferromagnets.
