Resurgence in the O(4) sigma model

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.

Resurgence and 1/N Expansion in Integrable Field Theories

“…The other possibility is that the phenomenon we have found is generic in theories with renormalons. The detailed study performed for the O(4) sigma model in [25,26] validates standard resurgence expectations and seems to favor the first possibility. Clearly, additional studies are necessary in order to clarify this fundamental issue.…”

“…We find a new, explicit solution for F (h) at leading order in 1/N from TBA, and for the choice of charges used in [34]. The exact answer can be understood as the median resummation of a non-trivial trans-series which can be obtained analytically and has an infinite number of IR renormalon singularities (in contrast to the solution of [46,47], which has a single IR renormalon singularity, and similar to the numerical results obtained in [25,26] for the O(4) NLSM). Therefore, in this case the large N limit provides an explicit, analytic, yet non-trivial example of resurgence and median resummation in a model with infinitely many IR renormalon corrections.…”

“…In addition, a powerful method developed in [41,42] makes it possible to extract the perturbative series for the free energy at very high orders. This has led to many quantitative studies of renormalon physics and resurgence in relativistic [25,26,43] and non-relativistic [22,23,44,45] integrable quantum field theories.…”

“…In a recent tour de force, Abbott and collaborators [25,26] were able to obtain detailed information on the trans-series expansion for the normalized energy density in the O(4) sigma model, which is nothing but the PCF model we are studying at N = 2. By extrapolating numerical results to analytic results, they obtained an expression very similar to (5.53), and they gave evidence that the exact answer can be recovered by median resummation of the trans-series.…”

“…There is growing evidence that many exact quantities in quantum theory can be obtained as Borel resummations of these trans-series. These include energy levels in quantum mechanics [13][14][15][16][17][18], 1/N expansions in matrix models [19,20], and perturbative expansions in some quantum field theories [21][22][23][24][25][26][27].…”