One-loop effective potential of the Higgs field on the Schwarzschild background

A one-loop effective potential of the Higgs field on the Schwarzschild background is derived in the framework of a toy model: a SO(N) scalar multiplet interacting with the gauge fields, the SO(N) gauge symmetry being broken by the Higgs mechanism. As expected, the potential depends on the space point and results in a mass shift of all massive particles near a black hole. It is shown that the obtained potential is generally covariant, depends on the space point through the metric component g(00) in the adapted coordinates, and has the same form for an arbitrary static, spherically symmetric background. Some properties of this potential are investigated. In particular, if the conformal symmetry holds valid for massless particles on the given background, there exist only two possible scenarios depending on the sign of an arbitrary constant arising from the regularization procedure: the masses of all massive particles grow infinitely, when they approach the black hole horizon, or the gauge symmetry is restored at a finite distance from the horizon and all particles become massless. If the conformal symmetry is spoiled, an additional term in the effective potential appears and the intermediate regime arises. Several normalization conditions fixing the undefined constants are proposed, and estimations for the mass shifts are given in these cases. It should be mentioned that the use of the one-loop approximation becomes questionable in the region where the one-loop effective potential acquires large values. So, in that region, we can believe in the obtained results only to a certain extent.