On the use of the proximity force approximation for deriving limits to
short-range gravitational-like interactions from sphere-plane Casimir force
experiments
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.Comment: 11 page, 5 figure
