Hadronic lensing

We introduce an analytic approach to study gravitational lensing in the presence of a distribution of hadrons. The situation is analogous to the propagation of photons in a medium with a nontrivial Cooper-pair condensate, where the photon acquires an effective mass term that may depend on the coordinates if the condensate is not homogeneous. As a result, photons generally do not follow null geodesics in the hadronic medium. In this setup, hadrons are described by the nonlinear sigma model minimally coupled to Maxwell theory. The modified Raychaudhuri equation, including hadronic corrections, is derived, along with the integral curves of probe photons in the eikonal approximation. These results are consistent with the theory of gravitational lensing in plasma media, with the advantage that transport properties, such as the refractive index, can be expressed analytically in terms of the hadronic density without assuming a phenomenological modeling thereof. As an example, we study the hadronic lensing produced by an analytic black hole sourced by superfluid pionic vortices, and we obtain the hadronic correction to the deflection angle in the weak-field limit.