Exact saddles and finite-T Picard-Lefschetz contour integrals over quasi-zero modes encode the full resurgent structure and yield non-perturbative splittings for every energy level in the double well.
Critical Points at Infinity, Non-Gaussian Saddles, and Bions
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abstract
It has been argued that many non-perturbative phenomena in quantum mechanics (QM) and quantum field theory (QFT) are determined by complex field configurations, and that these contributions should be understood in terms of of Picard-Lefschetz theory. In this work we compute the contribution from non-BPS multi-instanton configurations, such as instanton-anti-instanton $[{I}\bar{I}]$ pairs, and argue that these contributions should be interpreted as exact critical points at infinity. The Lefschetz thimbles associated with such critical points have a specific structure arising from the presence of non-Gaussian, quasi-zero mode (QZM), directions. When fermion degrees of freedom are present, as in supersymmetric theories, the effective bosonic potential can be written as the sum of a classical and a quantum potential. We show that in this case the semi-classical contribution of the critical point at infinity vanishes, but there is a non-trivial contribution that arises from its associated non-Gaussian QZM-thimble. This approach resolves several puzzles in the literature concerning the semi-classical contribution of correlated $[{I}\bar{I}]$ pairs. It has the surprising consequence that the configurations that dominate the expansion of observables, and the critical points that define the Lefschetz thimble decomposition need not be the same, a feature not present in the traditional Picard-Lefschetz approach.
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Exact WKB analysis produces median-summed spectra and an algebraic equation for the exceptional point of PT-symmetry breaking in the inverted triple-well system.
Renormalons can be understood as saddle points of the 1-loop effective action in toy models, enabled by the quantum scale anomaly.
Presents explicit trans-series calculations for the double-well spectrum via exact WKB and path integral approaches up to four-instanton level.
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Beyond the Dilute Instanton Gas: Resurgence with Exact Saddles in the Double Well
Exact saddles and finite-T Picard-Lefschetz contour integrals over quasi-zero modes encode the full resurgent structure and yield non-perturbative splittings for every energy level in the double well.
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Exact WKB analysis of inverted triple-well: resonance, PT-symmetry breaking, and resurgence
Exact WKB analysis produces median-summed spectra and an algebraic equation for the exceptional point of PT-symmetry breaking in the inverted triple-well system.
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Renormalons as Saddle Points
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The Double Well Done Doubly-Well
Presents explicit trans-series calculations for the double-well spectrum via exact WKB and path integral approaches up to four-instanton level.