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Higher-Order Corrections to Instantons
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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
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