Abstract
AbstractThis paper considers the $1/\epsilon$ problem, which is the divergent behavior of the ground-state energy of the asymmetric potential in quantum mechanics, which is calculated with semi-classical expansion and the resurgence technique. Using the resolvent method, it is shown that including not only one complex bion but a multi-complex bion and multi-bounce contributions solves this problem. This result indicates the importance of summing all possible saddle-point contributions and also the relationship between the exact Wentzel–Kramers–Brillouin approximation and path-integral formalism.
Highlights
For many quantum theories, perturbative expansion is used to compute physical quantities
The papers [17] [22] discuss supersymmetric quantum mechanics and show Borel ambiguity corresponding to the perturbative expansion around the vacuum is cancelled by a complex bion3
We showed the 1/ǫ problem in the tilted double well potential is solved by including multi-complex bion and multi-bounce contributions
Summary
Perturbative expansion is used to compute physical quantities. The papers [17] [22] discuss supersymmetric quantum mechanics and show Borel ambiguity corresponding to the perturbative expansion around the vacuum is cancelled by a complex bion3 It explains the nonperturbative shift of the ground state energy (dynamical SUSY breaking). Resurgence theory claims the Borel smbiguity from a perturbative expansion is cancelled by considering the contribution of other saddle points In this system, there are classical solutions called the complex bion [22]: xcb(t) x1. In the case of symmetric double well, the nonperturbative contribution of the ground state energy is coming from one-instanton.[27] So even if we remove the singularity by hand, the result is still incorrect This is the 1/ǫ problem in deformed SUSY quantum system. The problem does occur in the case of tilted double well, but is known to occur in CP N and sine-Gordon model. [20][28][29]
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