On the use of the proximity force approximation for deriving limits to short-range gravitational-like interactions from sphere-plane Casimir force experiments

TL;DR

This paper critically examines the use of the proximity force approximation in deriving limits on short-range Yukawa-like forces from sphere-plane Casimir experiments, highlighting potential overestimations and uncertainties.

Contribution

It clarifies the validity of different forms of the proximity force approximation and their impact on the accuracy of constraints on hypothetical short-range forces.

Findings

The traditional PFA overestimates Yukawa forces in experimental settings.

The newer PFA form is equivalent to exact two-body interaction integration for infinite planar geometries.

Finite size effects introduce additional systematic errors in force limit estimations.

Abstract

We discuss the role of the proximity force approximation in deriving limits to the existence of Yukawian forces - predicted in the submillimeter range by many unification models - from Casimir force experiments using the sphere-plane geometry. Two forms of this approximation are discussed, the first used in most analyses of the residuals from the Casimir force experiments performed so far, and the second recently discussed in this context in R. Decca et al. [Phys. Rev. D 79, 124021 (2009)]. We show that the former form of the proximity force approximation overestimates the expected Yukawa force and that the relative deviation from the exact Yukawa force is of the same order of magnitude, in the realistic experimental settings, as the relative deviation expected between the exact Casimir force and the Casimir force evaluated in the proximity force approximation. This implies both a systematic shift making the actual limits to the Yukawa force weaker than claimed so far, and a degree of uncertainty in the alpha-lambda plane related to the handling of the various approximations used in the theory for both the Casimir and the Yukawa forces. We further argue that the recently discussed form for the proximity force approximation is equivalent, for a geometry made of a generic object interacting with an infinite planar slab, to the usual exact integration of any additive two-body interaction, without any need to invoke approximation schemes. If the planar slab is of finite size, an additional source of systematic error arises due to the breaking of the planar translational invariance of the system, and we finally discuss to what extent this may affect limits obtained on power-law and Yukawa forces.