Abstract
We analyze the free energy of the integrable two dimensional O(4) sigma model in a magnetic field. We use Volin’s method to extract high number (2000) of perturbative coefficients with very high precision. The factorial growth of these coefficients are regulated by switching to the Borel transform, where we perform several asymptotic analysis. High precision data allowed to identify Stokes constants and alien derivatives with exact expressions. These reveal a nice resurgence structure which enables to formulate the first few terms of the ambiguity free trans-series. We check these results against the direct numerical solution of the exact integral equation and find complete agreement.
Highlights
Perturbation theory has proved to be a useful tool in calculating physical processes for the electromagnetic and weak interactions, but has had only a limited success for their strong counterpart
The factorial growth of these coefficients are regulated by switching to the Borel transform, where we perform several asymptotic analysis
Perturbation theory in QCD is expected to be asymptotic, with coefficients growing factorially, see e.g [1, 2]. This factorial growth can be traced to the proliferation of Feynman diagrams [3, 4], or to integrals in specific renormalon diagrams, when loop momenta lie in various IR and UV domains [5]
Summary
Perturbation theory has proved to be a useful tool in calculating physical processes for the electromagnetic and weak interactions, but has had only a limited success for their strong counterpart. Perturbation theory in QCD is expected to be asymptotic, with coefficients growing factorially, see e.g [1, 2] This factorial growth can be traced to the proliferation of Feynman diagrams [3, 4], or to integrals in specific renormalon diagrams, when loop momenta lie in various IR and UV domains [5]. It is a signal of non-perturbative contributions, which usually originate from non-trivial saddle points in the path integral. The tools for doing this are known as resurgence theory [6,7,8]
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