Scalar propagator in a rotating thermal medium: Consistency checks and applications
We derive the scalar propagator in a thermal medium under global rotation without restricting the spacetime points to a common comoving trajectory. The resulting propagator retains an explicit dependence on the radial coordinates, reflecting the lack of translational invariance in the transverse plane, and therefore cannot in general be formulated solely in momentum space. We show that the usual free Feynman propagator is exactly recovered in the nonrotating limit, providing a first consistency check of the formulation. As a nontrivial test, we apply the propagator to the interacting $\lambda\phi^4$ theory at finite temperature and rotation. The one-loop scalar self-energy obtained within the Matsubara formalism correctly separates into vacuum and medium contributions and reproduces the standard finite-temperature result when $\Omega\to0$. We subsequently construct the effective potential including screening effects through ring resummation and, for $\Omega/T\ll1$, evaluate the rotational corrections up to $\mathcal{O}(\Omega^4)$. Within this model, the resulting effective potential exhibits the expected thermal restoration of the spontaneously broken $\mathbb{Z}_2$ symmetry, while global rotation favors symmetry restoration and enhances this effect as the transverse size of the system increases. These results provide complementary consistency checks of the rotating scalar propagator and highlight the need to retain its explicit position-space structure in more general perturbative calculations.