Spin-dependent fermion potentials from mixed tensor couplings of massive spin-1 and spin-2 bosons

Abstract: Searches for weak spin-dependent forces are commonly interpreted in terms of a basis of sixteen rotationally invariant two-fermion potentials $V_{1-16}$, including potentials which violate parity (P) and time-reversal (T) invariance. We derive finite-range coordinate-space potentials generated by a massive spin-1 boson with vector, axial-vector, tensor, and pseudotensor couplings to Dirac fermions. In addition to the familiar vector--vector, vector--axial-vector, and axial-vector--axial-vector interactions, we obtain the mixed vector--tensor, vector--pseudotensor, axial-vector--tensor, and axial-vector--pseudotensor potentials, together with tensor--tensor, tensor--pseudotensor, and pseudotensor--pseudotensor terms. We retain the contact terms needed in atomic-scale applications and give compact expressions that include both orderings of inequivalent fermion vertices. We then extend the analysis to a massive spin-2 mediator described by a symmetric Fierz--Pauli field. Besides the conventional coupling to the conserved fermion energy--momentum tensor, we consider a symmetric axial-tensor current. Its nonconservation for massive fermions makes the longitudinal spin-2 helicities contribute to axial-tensor--axial-tensor exchange. The leading minimal spin-2 interactions are either $P,T$ even or $P$ odd and $T$ even; in particular, they do not generate a $T$-odd potential. Heavy-mediator contact limits and the distinction between the $M\to0$ massive-spin-2 limit and a genuine massless graviton are also discussed. We map the new interactions onto the sixteen-potential Dobrescu--Mocioiu basis and use this mapping to obtain numerical translations of existing limits for both spin-1 and spin-2 coupling products in the small- and large-mediator-mass regimes, together with representative limits at finite mediator masses.