Another Look at the Weak-Field Limit of Generalized Hybrid Metric-Palatini Gravity

We investigate the weak-field regime of generalized hybrid metric-Palatini theories, described by a generic function $f(R,\mathcal{R})$, where $R$ is the metric Ricci scalar and $\mathcal{R}$ is constructed from an independent torsionless connection. Linearizing the field equations about Minkowski spacetime, we show, without using the scalar-tensor representation, that the theory propagates the usual massless spin-2 mode and two massive scalar modes, with an effective gravitational coupling. The absence of tachyonic and ghostlike instabilities at the linearized level, together with the nondegeneracy of the scalar sector, is shown to impose algebraic restrictions on the derivatives of $f(R,\mathcal R)$ evaluated on the Minkowski background, which generalize previously obtained conditions. The Newtonian limit for an extended static source is derived, yielding a gravitational potential with two Yukawa corrections whose amplitudes are fixed by the scalar residues, while finite-size effects are encoded in source-dependent form factors. We determine the conditions under which the usual Newtonian limit is recovered and derive the effective post-Newtonian parameter $\gamma_\Sigma$ governing light propagation. Finally, we compute the radial epicyclic frequency and the corresponding anomalous periapsis advance, and compare it with planetary precession data to constrain the parameters of a viable hierarchical scalar-mass regime.