SciPost Submission Page
EFT Asymptotics: the Growth of Operator Degeneracy
by Tom Melia, Sridip Pal
This is not the current version.
|As Contributors:||Sridip Pal|
|Arxiv Link:||https://arxiv.org/abs/2010.08560v1 (pdf)|
|Date submitted:||2020-10-27 17:48|
|Submitted by:||Pal, Sridip|
|Submitted to:||SciPost Physics|
We establish formulae for the asymptotic growth (with respect to the scaling dimension) of the number of operators in effective field theory, or equivalently the number of $S$-matrix elements, in arbitrary spacetime dimensions and with generic field content. This we achieve by generalising a theorem due to Meinardus and applying it to Hilbert series---partition functions for the degeneracy of (subsets of) operators. Although our formulae are asymptotic, numerical experiments reveal remarkable agreement with exact results at very low orders in the EFT expansion, including for complicated phenomenological theories such as the standard model EFT. Our methods also reveal phase transition-like behaviour in Hilbert series. We discuss prospects for tightening the bounds and providing rigorous errors to the growth of operator degeneracy, and of extending the analytic study and utility of Hilbert series to EFT.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2021-1-18 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2010.08560v1, delivered 2021-01-18, doi: 10.21468/SciPost.Report.2426
I will not be able to accept this paper as it stands, with all the good will.
The authors start with a bad definition in equation 2.1 of a diverging quantity.
The PE is defined on a function which vanishes when all its variables are set to 0, and they fail to make sure of it.
Consequently, the PE of some function must start with 1, and again the authors fail to make sure of this.
There are many ways of working with finite expressions and there is a lot of literature which demonstrate how to do this.
The fact that the mistake happens at the very beginning makes it very hard to proceed, as it is based on a wrong definition of a basic function that is used throughout the paper.
From there the text goes downhill, as the authors are fixing this mistake with some “tricks” that are not acceptable.
One should remember that the PE is a combinatorial function that has no reason to diverge at any point of the computation.
In fact, as the authors are interested in a large order behavior, the origin is not so crucial, but this does not mean that they should start with a bad definition. One needs a minimal amount of modifications to keep the computations mathematically viable and with avoiding such sloppiness. After all this is not a path integral with infinitely many degrees of freedom.
Anyhow, if the authors want my approval, they will need to remove the ill definitions and to present their computations in a divergence less manner. Once this is fixed, I am happy to have another look at the paper.
Anonymous Report 1 on 2020-12-7 (Contributed Report)
- Cite as: Anonymous, Report on arXiv:2010.08560v1, delivered 2020-12-07, doi: 10.21468/SciPost.Report.2259
1) Broad in terms of applications
2) Compact analytic formulae to compare with numerical results
1) The manuscript is echnical and addressed principally to experts in the field
2) Applications a bit vague
Using methods based on Hilbert series, the authors study the asymptotic growth of the number of operators in Effective Field Theories (EFTs) with different field content (with and without spin) and in different dimensions.
The connection between the mathematical technique employed in the paper and the underlying physics is interesting. The result in terms of analytic formulae fills an important gap in a field that was (at his frontier) defined by numerical progress.
I feel that, moreover, the manuscript is addressing the more subtle question of whether Hilbert series are useful at all, beyond checking numerical calculations. The authors provide a number of interesting potential applications, though I feel these could have been given more weight and be used to shape the manuscript.
All in all the article meets the SciPost criteria in terms of interest. I nevertheless have a few questions and comments that I would like the authors to address :
1) The manuscript gives quite some importance to the notion of phase transitions (these are mentioned in the abstract, introduction and conclusions), while they occupy only a quarter of a page in the main text. What are the physics implications of this? In what space should it be considered a phase transition?
2) Can the authors be more precise (in the introduction) as to what concrete physics application these analytic results could have?
3) The authors introduce a non-rigorous trick in section 3 to compute the asymptotics, is this necessary? One of the motivations is that it is fast, but I could not understand why this would make such detour worth.
4) Mainardus theorem is never explicitly reported in the manuscript : is it really a theorem or simply a technique? In what sense it differes from Cardy’s approach?
In addition to the changes triggered by the comments in my report, I suggest the following small changes:
1) Below eq 1.1 mention that the PE is the generating functional
2) The number of partitions is introduced (very pedagogically) in section 3.2, while it is already used and discussed in section 2.2. Perhaps parts of section 3.2 could appear earlier.
3) I think ref 71 should be cited in the main text.