Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 065, 33 pages      arXiv:2302.08363      https://doi.org/10.3842/SIGMA.2025.065

On the Structure of Trans-Series in Quantum Field Theory

Marcos Mariño a, Ramon Miravitllas ab and Tomás Reis cd
a) Département de Physique Théorique et Section de Mathématiques, Université de Genève, Genève, CH-1211, Switzerland
b) HUN-REN Wigner Research Centre for Physics, Konkoly-Thege Miklós u. 29-33, 1121 Budapest, Hungary
c) SISSA, 34136 Trieste, Italy
d) INFN, Sezione di Trieste, 34127 Trieste, Italy

Received March 06, 2025, in final form July 22, 2025; Published online August 01, 2025

Abstract
Many observables in quantum field theory can be expressed in terms of trans-series, in which one adds to the perturbative series a typically infinite sum of exponentially small corrections, due to instantons or to renormalons. Even after Borel resummation of the series in the coupling constant, one has to sum this infinite series of small exponential corrections. It has been argued that this leads to a new divergence, which is sometimes called the divergence of the OPE. We show that, in some interesting examples in quantum field theory, the series of small exponential corrections is convergent, order by order in the coupling constant. In particular, we give numerical evidence for this convergence property in the case of the free energy of integrable asymptotically free theories, which has been intensively studied recently in the framework of resurgence. Our results indicate that, in these examples, the Borel resummed trans-series leads to a well defined function, and there are no further divergences.

Key words: resurgence; quantum field theory; trans-series; asymptotic series.

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