We compute the radial action for massive particles along hyperboliclike geodesics in various spherically symmetric spacetimes: the standard $4D$ Schwarzschild spacetime, its $d$-dimensional generalization known as Schwarzschild-Tangherlini solutions and for the D3-brane spacetimes, showing useful resummation properties in terms of special (hypergeometric, Fox-Wright) functions in the eikonal limit and generalizing previous results valid for null geodesics. As a consequence, the scattering angle can be resummed too, and we explicitly display the resummed expressions. In addition, in the more interesting situation of a $4D$ Schwarzschild black hole spacetime, following the approach of the quantum Seiberg-Witten curves to the radial equation associated with a massive scalar field, we show that the quantum $a$ cycle (or, equivalently, the “renormalized angular momentum”) is simply related to the $\pi$ part of the radial action also in this massive case, providing fully resummed expressions. Finally, we display the explicit, expanded-form expression of the dual $a_D$ cycle, for which, however, no resummed expressions have been derived yet.

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• General relativity