Higher-Order Corrections to Instantons.pdf

a r X i v : m a t h - p h / 0 1 0 3 0 1 0 v 1 9 M a r 2 0 0 1 SPhT T01/25 math-ph/0103010 Higher–Order Corrections to Instantons Ulrich D. Jentschura? and Jean Zinn-Justin? ? Laboratoire Kastler–Brossel, Unite? Mixte de Recherche du CNRS No. C8552, Universite? Pierre et Marie Curie, Case 74, 4 pl. Jussieu, F-75252 Paris Cedex 05, France ? Commissariat a? l’E?nergie Atomique, Service de Physique The?orique, F-91191 Gif-Sur-Yvette Cedex, France Email: ulj@spectro.jussieu.fr, zinn@spht.saclay.cea.fr Abstract The energy levels of the double-well potential receive, be- yond perturbation theory, contributions which are non-analytic in the coupling strength; these are related to instanton effects. For example, the separation between the energies of odd- and even-parity states is given at leading order by the one-instanton contribution. However to determine the energies more accurately multi-instanton configu- rations have also to be taken into account. We investigate here the two-instanton contributions. First we calculate analytically higher- order corrections to multi-instanton effects. We then verify that the difference betweeen numerically determined energy eigenvalues, and the generalized Borel sum of the perturbation series can be described to very high accuracy by two-instanton contributions. We also cal- culate higher-order corrections to the leading factorial growth of the perturbative coefficients and show that these are consistent with an- alytic results for the two-instanton effect and with exact data for the first 200 perturbative coefficients. PACS numbers 11.15.Bt, 11.10.Jj Keywords General properties of perturbation theory; Asymptotic problems and properties 1 In theories with degenerate minima, the energy eigenvalues of the states at nonvanishing coupling g 6= 0 cannot in principle be obtained by analytic continuation from the unperturbed situation at vanishing coupling g = 0 because a potential with degenerate minima introduces a degeneracy in the sp