Tunnel Effects in Field Theory and Asymptotic Behavior of Perturbation Theories

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This year, we made progress in the analysis of the structure of the quantum theories using the valley method. The imaginary-time path-integral method is known to be effective for the treatment of the tunneling phenomena. The well-known calculation utilizing instanton or bounce solutions, however, have various limitations. In order to overcome those, the team involving the present investigator has established the "valley method".The one-dimensional asymmetric double-well potential model has been analyzed in detail this year, to yield the following results :・The non-perturbative effects due to the tunneling can be incorporated by use of valley-instanton and valley-bounces.・This effect can be completely separated from perturbative effect by an analytic continuation of the valley parameter. This analytic continuation at the same time allows calculation by using only the leading contribution of the interactions of the valley-instantons.・This analysis shows that the singularity in the valley integration leads to the evaluation of the leading term of the non-Borcl-summable divergence in perturbation series. This result differs from a widely known folklore.・Our prediction for the asymptotic behavior of the perturbative series has been confirmed to the 300-th order by direct algebraic calculation.We believe that these results lead to a knowledge of the general relation between the perturbative and non-perturbative effects in quantum theories.