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Instantons, renormalons and the theta angle in integrable sigma models
by Marcos Mariño, Ramon Miravitllas, Tomás Reis
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|Marcos Mariño · Tomas Reis
Some sigma models which admit a theta angle are integrable at both $\vartheta=0$ and $\vartheta=\pi$. This includes the well-known $O(3)$ sigma model and two families of coset sigma models studied by Fendley. We consider the ground state energy of these models in the presence of a magnetic field, which can be computed with the Bethe ansatz. We obtain explicit results for its non-perturbative corrections and we study the effect of the theta angle on them. We show that imaginary, exponentially small corrections due to renormalons remain unchanged, while instanton corrections change sign, as expected. We find in addition corrections due to renormalons which also change sign as we turn on the theta angle. Based on these results we present an explicit non-perturbative formula for the topological susceptibility of the $O(3)$ sigma model in the presence of a magnetic field, in the weak coupling limit.
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- Cite as: Anonymous, Report on arXiv:2205.04495v2, delivered 2023-07-17, doi: 10.21468/SciPost.Report.7519
In this work the authors exploit Bethe ansatz methods to compute the free energy in various integrable $2$d qfts. In particular, they find transseries expansions for the free energy of the $O(3)$ (and other integrable coset models) at $\theta=0$ and $\theta = \pi$.
Most of the analysis closely follows the methods introduced in a previous work by the same authors, while the novelty here regards the change in NP corrections as the theta angles varies between $0$ and $\pi$.
Although the key discussions in section 3 and 4 are rather technical, I found the paper to be clear and well-written, certainly deserving publication with Scipost.
My main issue with the paper lies with the use of the nomenclature "renormalons", to describe certain non-perturbative corrections found in the weak-coupling regime.
The present results are derived via Bethe ansatz equations and do not rely on any diagrammatica of sort. Although I do agree with the authors that such corrections are "renormalons" in nature, when compared to other "instanton" corrections, in that they survive in the large-$N$ limit, this naming seems very misleading.
I think a better way of addressing the problem is the following question: do these "renormalons" non-perturbative terms the authors have found come from finite-action semi-classical solutions to the (possibly complexified) classical field equations of the $O(3)$ sigma model in a magnetic field background?
In the authors calculation, the magnetic field, h, plays a crucial role in establishing the weak-coupling expansion, i.e. $h/m \to \infty$ with $m$ the mass-gap. In the very same limit, $ h\to \infty$, the O(3) sigma-model path-integral, with or without theta-angle, should be amenable to a reliable semi-classical expansion.
Can the authors find semi-classical solutions to the complexified field equations and responsible for these "renormalons" non-perturbative terms? Alternatively, do the authors have any argument, besides large-$N$, for why there should not be any semi-classical object responsible for such corrections?
I would like the authors to comment on this.
- Cite as: Anonymous, Report on arXiv:2205.04495v2, delivered 2023-06-04, doi: 10.21468/SciPost.Report.7301
The authors compute the free energy in several series of 2d sigma models by exploiting their (conjectured) integrability. The aim is to understand the contributions of renormalons and instantons, and they do a rather intricate calcualtion to this end. While the techniques are mainly old, they successfully surmount several substantial technical obstacles to push the techniques further. In particular, they do not rely on deforming the problem, as this seems to obscure the renormalon contribution.
The question they address is interesting and the calculation meaningful. I thus recommend publication in SciPost. I do think the authors should give more explanation of a key conceptual point. I'm sure some of the below issues are addressed in the authors' other papers, but they need to say a little more here.
Namely, they need to give more detail on how renormalons appear in the the sigma model with Lie-group symmetry. but (the authors say) "do not seem to" in the deformed theory. I presume the free energy is still continuous, but is it not analytic in the deformation parameter? Or are they saying that it is not continuous?
Moreover, the old papers by Zamolodchikov et al associate the change in sign between theta=0 and pi with instantons, but at the end of section 4.3 the authors seem to be saying otherwise. How do they know this is the renormalon and not a previously missed instanton contribution?
Another issue worth mentioning is what happens in the O(3) case for varying theta continuously. Obviously, they're not going to sovle this question, but a little speculation in light of their results would be good.
Finally, a minor but useful point: the Zamolodchikov brothers adopted the convention that Alexander goes by A.B., while Alexei goes by Al. B. Most of the references here are to Alexei's papers, and so should be fixed accordingly.