Universal renormalons in principal chiral models

Abstract

Perturbative expansions in many physical systems yield 'only' asymptotic series which are not even Borel resummable. Interestingly, the corresponding ambiguities point to nonperturbative physics. We numerically verify this renormalon mechanism for the first time in two-dimensional sigma models, that, like four-dimensional gauge theories, are asymptotically free and generate a strong scale through dimensional transmutation. We perturbatively expand the energy through a numerical version of stochastic quantization. In contrast to the first energy coefficients, the high order coefficients are independent on the rank of the model. Technically, they require a sophisticated analysis of finite volume effects and the continuum limit of the discretized model. Although the individual coefficients do not grow factorially (yet), but rather decrease strongly, the ratio of consecutive coefficients clearly obey the renormalon asymptotics.

A new renormalon in two dimensions

“…In asymptotically free theories and in QED, renormalons are believed to control the large order behavior of perturbation theory. Evidence for this was found in [11,12] in the case of Yang-Mills theory, and in [13][14][15][16] for integrable two-dimensional theories.…”

Resurgence in the O(4) sigma model

“…The resurgence program has been pushed forward for several quantum field theories including supersymmetric theories [11][12][13][14][15][16], various quantities in the maximally supersymmetric 4D gauge theory in the large N limit [17][18][19], the simple φ 4 theory in 2D, [20,21]. Factorial growth can be seen in lattice simulations [22]. Much is known in the large N , [23,24] and also semiclassically [25][26][27].…”

Renormalons in integrable field theories

“…Given the subtleties of renormalon physics, it is useful to look at simple field theories where one has more analytic control. For example, renormalons in the non-linear sigma model at large N were analyzed in some detail in [21][22][23][24][25], and very recently evidence for the dominance of the leading IR renormalon was obtained numerically in [26], in the case of the principal chiral field (PCF) [27]. Many asymptotically free theories in two dimensions turn out to be integrable, i.e.…”