A new renormalon in two dimensions
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Springer
Received: February Revised: June Accepted: June Published: July
27, 10, 24, 29,
2020 2020 2020 2020
Marcos Mari˜ no and Tom´ as Reis D´epartement de Physique Th´eorique et Section de Math´ematiques, Universit´e de Gen`eve, Gen`eve, CH-1211 Switzerland
E-mail: [email protected], [email protected] Abstract: According to standard lore, perturbative series of super-renormalizable theories have only instanton singularities. In this paper we show that two-dimensional scalar theories with a spontaneously broken O(N ) symmetry at the classical level, which are super-renormalizable, have an IR renormalon singularity at large N . Since perturbative expansions in these theories are made around the “false vacuum” in which the global symmetry is broken, this singularity can be regarded as a manifestation of the non-perturbative absence of Goldstone bosons. We conjecture that the Borel singularity in the ground state energy of the Lieb-Liniger model is a non-relativistic manifestation of this phenomenon. We also provide en passant a detailed perturbative calculation of the Lieb-Liniger energy up to two-loops, and we check that it agrees with the prediction of the Bethe ansatz. Keywords: 1/N Expansion, Field Theories in Lower Dimensions, Nonperturbative Effects, Renormalization Regularization and Renormalons ArXiv ePrint: 1912.06228
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)216
JHEP07(2020)216
A new renormalon in two dimensions
Contents 1 Introduction
1 3 3 8 16 20
3 On the Lieb-Liniger model
22
4 Conclusions
28
1
Introduction
The study of the large order behavior of perturbative series in quantum theory has provided an efficient window on non-perturbative phenomena. In quantum mechanics, it was found in [1, 2] and in many subsequent works that this behavior is controlled by instantons, and is due to the factorial growth in the number of Feynman diagrams [3] (see e.g. [4] for a textbook introduction). In quantum field theory, the situation is more complicated, since in many theories one can find specific diagrams which grow factorially with the loop order after integration over the momenta [5–9]. These diagrams are usually called renormalon diagrams (see [10] for an extensive review). They lead to singularities in the Borel plane of the coupling constant which, following [10], we will call renormalon singularities, or renormalons for short. Depending on the region in momenta which leads to the factorial growth, one has UV or IR renormalons. 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 [13–16] for integrable two-dimensional theories. It is often stated in the literature that renormalons, as their name indicate, are typical of renormalizable field theories, while super-renormalizable field theories only have instanton singularities (see e.g. [7, 17, 18]). A typical example of the latter
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