Zeroth law of black hole thermodynamics for higher derivative Proca theories

We consider diffeomorphism invariant theories of gravity with arbitrary higher derivative corrections to the two derivative Einstein’s theory in the presence of Proca fields—vector fields without <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>U</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>1</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> gauge invariance. We prove that, in such theories, the surface gravity remains constant across the Killing horizon of stationary black hole solutions, without requiring the existence of a bifurcation surface. Our analysis is carried out within an effective field theory framework, where higher derivative contributions are treated perturbatively around the two derivative theory. The proof relies on specific components of the coupled gravity—Proca field equations together with constraints implied by the near horizon boost symmetry of stationary black hole geometries, analyzed using boost-weight arguments. Implementing these arguments in the presence of Proca fields introduces new subtleties beyond those encountered in the pure gravity case. We provide a systematic resolution of these subtleties and demonstrate that the surface gravity remains constant to all orders in the perturbative expansion, thereby establishing the zeroth law. Finally, we outline how this framework can be generalized to broader classes of vector theories with four propagating degrees of freedom, which exhibit qualitative differences from both the Proca and gauge invariant vector theories.