Resurgence in sine-Gordon quantum mechanics: exact agreement between multi-instantons and uniform WKB

Tatsuhiro Misumi, Muneto Nitta, Norisuke Sakai

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34 Citations (Scopus)


Abstract: We compute multi-instanton amplitudes in the sine-Gordon quantum mechanics (periodic cosine potential) by integrating out quasi-moduli parameters corresponding to separations of instantons and anti-instantons. We propose an extension of Bogomolnyi-Zinn-Justin prescription for multi-instanton configurations and an appropriate subtraction scheme. We obtain the multi-instanton contributions to the energy eigenvalue of the lowest band at the zeroth order of the coupling constant. For the configurations with only instantons (anti-instantons), we obtain unambiguous results. For those with both instantons and anti-instantons, we obtain results with imaginary parts, which depend on the path of analytic continuation. We show that the imaginary parts of the multi-instanton amplitudes precisely cancel the imaginary parts of the Borel resummation of the perturbation series, and verify that our results completely agree with those based on the uniform-WKB calculations, thus confirming the resurgence structure: divergent perturbation series combined with the nonperturbative multi-instanton contributions conspire to give unambiguous results. We also study the neutral bion contributions in the ℂPN−1 model on ℝ1×S1 with a small circumference, taking account of the relative phase moduli between the fractional instanton and anti-instanton. We find that the sign of the interaction potential depends on the relative phase moduli, and that both the real and imaginary parts resulting from quasi-moduli integral of the neutral bion get quantitative corrections compared to the sine-Gordon quantum mechanics.

Original languageEnglish
Article number157
JournalJournal of High Energy Physics
Issue number9
Publication statusPublished - 2015 Sept 30


  • Field Theories in Lower Dimensions
  • Nonperturbative Effects
  • Sigma Models
  • Solitons Monopoles and Instantons

ASJC Scopus subject areas

  • Nuclear and High Energy Physics


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