<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ζ</mml:mi></mml:math>-function regularization of one-loop effective potentials in anti-de Sitter spacetime

Abstract

We use the $\ensuremath{\zeta}$-function technique to calculate the one-loop effective potential for a scalar field in anti-de Sitter (AdS) space. The $\ensuremath{\zeta}$ function is computed exactly on the four-dimensional hyperbolic space ${H}^{4}$, the Euclidean section appropriate for AdS space. The structure of the ultraviolet divergences is shown to agree with previous calculations where Pauli-Villars or some version of dimensional regularization was used. The finite part of the effective potential is given explicitly by an integral over a variable related to the spectrum of the Laplace-Beltrami operator on ${H}^{4}$.