Estimating higher order perturbative coefficients using Borel transform

A new method of estimating higher order perturbative coefficients is discussed. It exploits the rapid, asymptotic growth of perturbative coefficients and the information on the singularities in the complex Borel plane. A comparison with other methods is made in several Quantum Chromodynamics (QCD) expansions. Typeset using REVTEX 1 The ordinary perturbative expansions in weak coupling constant in quantum field theories are generally asymptotic expansions, with their perturbative coefficients growing factorially at large orders [1]. In practice, in many cases the asymptotic growth sets in quite early in perturbation, and the rapid growth of the coefficients becomes apparent already at first few orders. The asymptotic divergence of the perturbative coefficients implies singularities in the Borel plane. There are three kinds of known singularities, all on the real axis: those instanton-induced, ultraviolet renormalons, and infrared renormalons. In this paper, we show that this rapid growth of the perturbative coefficients and the information on the singularities in the Borel plane can be turned into a useful tool that allows us to estimate the unknown (N + 1)th order coefficient using the known coefficients up to order N . The method we are going to present can be most easily understood by working out an explicit example, the double-well potential in quantum mechanics. The tunneling between the two potential wells splits the degenerate perturbative ground states into a parity even, and an odd states, and the average energy E(α) of the energies of these two states has the perturbative expansion of the form E(α) = − [