TL;DR
This paper investigates how free energies on a d-dimensional sphere behave under perturbations, revealing their relation to beta functions and supporting the c, F, and a-theorems across various dimensions.
Contribution
It provides a perturbative analysis of free energies on spheres in arbitrary dimensions, connecting their derivatives to beta functions and supporting conformal invariance criteria.
Findings
Derivatives of free energies relate to beta functions.
Results support c, F, and a-theorems in multiple dimensions.
Rules out non-conformal scale invariant theories.
Abstract
We study perturbative behavior of free energies on a d-dimensional sphere S^d for theories with marginal interactions. The free energies are interpreted as the "dilaton effective action" with the dilaton having a nontrivial background vacuum expectation value. We compute the dependence of the free energies on the radius of the sphere by using dimensional regularization. It is shown that the first (second) derivative of the free energies in odd (even) dimensions with respect to the radius of the sphere are proportional to the square of the beta functions of coupling constants. The result is consistent with the c, F and a-theorems in two, three, four and six dimensions. The result is also used to rule out a large class of scale invariant theories which are not conformally invariant.
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