An improvement of Watson’s theorem on Borel summability

Abstract

Watson’s theorem, which gives sufficient conditions for Borel summability, is not optimal. Watson assumes analyticity and uniform asymptotic expansion in a sector ‖argz‖<π/2+ε, ‖z‖<R, with ε≳0; in fact, only the circular region Re(1/z) ≳1/R is required. In particular, one can take ε=0. This improved theorem gives a necessary and sufficient characterization of a large class of Borel‐summable functions. I apply it to the perturbation expansion in the φ₂⁴ quantum field theory.