A resurgence analysis for cubic and quartic anharmonic potentials

Gahramanov, Ilmar; Tezgin, Kemal

doi:10.1142/s0217751x17500336

“…Thinking about the resurgence structure of the current double expansion is also interesting. In some cases, non-perturbative corrections can be reproduced from the perturbative expansion [35,[83][84][85][86]. This might mean that by studying ∆ pert carefully, we might be able to extract information about the non-perturbative correction that…”

Section: Discussionmentioning

confidence: 99%

Stability Analysis of a Non-Unitary CFT

Watanabe¹

2022

Preprint

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We study instability of the lowest dimension operator (i.e., the imaginary part of its operator dimension) in the rank-Q traceless symmetric representation of the O(N ) Wilson-Fisher fixed point in D = 4 + . We find a new semi-classical bounce solution, which gives an imaginary part to the operator dimension of orderis fixed. The form of F ( Q), normalised as F (0) = 1, is also computed. This non-perturbative correction continues to give the leading effect even when Q is finite, indicating the instability of operators for any values of Q. We also observe a phase transition at Q = N +8 6 √ 3 associated with the condensation of bounces, similar to the Gross-Witten-Wadia transition.

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“…Thinking about the resurgence structure of the current double expansion is also interesting. In some cases, non-perturbative corrections can be reproduced from the perturbative expansion [35,[83][84][85][86]. This might mean that by studying ∆ pert carefully, we might be able to extract information about the non-perturbative correction that we computed.…”

Section: Jhep11(2023)042mentioning

confidence: 99%

Stability analysis of a non-unitary CFT

Watanabe

2023

J. High Energ. Phys.

0

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We study instability of lowest dimension operator (i.e., the imaginary part of its operator dimension) in the rank-Q traceless symmetric representation of the O(N) Wilson-Fisher fixed point in D = 4 + ϵ. We find a new semi-classical bounce solution, which gives an imaginary part to the operator dimension of order $$ O\left({\epsilon}^{-1/2}\exp \left[-\frac{N+8}{3\epsilon }F\left(\epsilon Q\right)\right]\right) $$ O ϵ − 1 / 2 exp − N + 8 3 ϵ F ϵQ in the double-scaling limit where $$ \epsilon Q\le \frac{N+8}{6\sqrt{3}} $$ ϵQ ≤ N + 8 6 3 is fixed. The form of F(ϵQ), normalised as F(0) = 1, is also computed. This non-perturbative correction continues to give the leading effect even when Q is finite, indicating the instability of operators for any values of Q. We also observe a phase transition at $$ \epsilon Q=\frac{N+8}{6\sqrt{3}} $$ ϵQ = N + 8 6 3 associated with the condensation of bounces, similar to the Gross-Witten-Wadia transition.

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A primer on resurgent transseries and their asymptotics

Aniceto

¹

,

Başar

²

,

Schiappa

³

2019

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The computation of observables in general interacting theories, be them quantum mechanical, field, gauge or string theories, is a non-trivial problem which in many cases can only be addressed by resorting to perturbative methods. In most physically interesting problems these perturbative expansions result in asymptotic series with zero radius of convergence. These asymptotic series then require the use of resurgence and transseries in order for the associated observables to become nonperturbatively well-defined. Resurgence encodes the complete largeorder asymptotic behaviour of the coefficients from a perturbative expansion, generically in terms of (multi) instanton sectors and for each problem in terms of its Stokes constants. Some observables arise from linear problems, and have a finite number of instanton sectors and associated Stokes constants; some other observables arise from nonlinear problems, and have an infinite number of instanton sectors and Stokes constants. By means of two very explicit examples, and with emphasis on a pedagogical style of presentation, this work aims at serving as a primer on the aforementioned resurgent, large-order asymptotics of general perturbative expansions. This includes discussions of transseries, Stokes phenomena, generalized steepest-descent methods, Borel transforms, nonlinear resonance, and alien calculus. Furthermore, resurgent properties of transseries-usually described mathematically via alien calculus-are recast in equivalent physical languages: either a "statistical mechanical" language, as motions in chains and lattices; or a "conformal field theoretical" language, with underlying Virasoro-like algebraic structures.

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A resurgence analysis for cubic and quartic anharmonic potentials

Cited by 5 publications

References 29 publications

Stability Analysis of a Non-Unitary CFT

Stability Analysis of a Non-Unitary CFT

Stability analysis of a non-unitary CFT

A primer on resurgent transseries and their asymptotics

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