Calculations of the dominant long-range, spin-independent contributions to the interaction energy between two nonrelativistic Dirac fermions from double-boson exchange of spin-0 and spin-1 bosons with spin-dependent couplings

Various theories beyond the Standard Model predict new particles with masses in the sub-eV range with very weak couplings to ordinary matter which can possess spin-dependent couplings to electrons and nucleons. Present laboratory constraints on exotic spin-dependent interactions with pseudoscalar and axial couplings for exchange boson masses between meV and eV are very poor compared to constraints on spin-independent interactions in the same mass range arising from spin-0 and spin-1 boson exchange. It is therefore interesting to analyze in a general way how one can use the strong experimental bounds on spin-independent interactions to also constrain spin-dependent interactions by considering higher-order exchange processes. The exchange of a pair of bosons between two fermions with spin-dependent couplings will possess contributions which flip spins twice and thereby generate a polarization-independent interaction energy which can add coherently between two unpolarized objects. In this paper we derive the dominant long-range contributions to the interaction energy between two nonrelativistic spin-1/2 Dirac fermions from double exchange of spin-0 and spin-1 bosons proportional to couplings of the form $g_P^{4}$, $g_S^{2}g_P^{2}$, and $g_V^{2}g_A^{2}$. Our results for $g_P^{4}$ are in agreement with previous calculations that have appeared in the literature. We demonstrate the usefulness of this analysis to constrain spin-dependent couplings by presenting the results of a reanalysis of data from a short-range gravity experiment to derive an improved constraint on $(g^N_{P})^2$, the pseudoscalar coupling for nucleons, in the range between $40$ and $200~\mu$m of about a factor of 5 compared to previous limits. The spin-independent contribution from 2-boson exchange with axial-vector couplings requires special treatment and will be explored in another paper.