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Constraints on beta functions in field theories
by Han Ma, Sung-Sik Lee
This is not the current version.
|As Contributors:||Han Ma|
|Arxiv Link:||https://arxiv.org/abs/2009.11880v3 (pdf)|
|Date submitted:||2021-01-04 17:42|
|Submitted by:||Ma, Han|
|Submitted to:||SciPost Physics|
The $\beta$-functions describe how couplings run under the renormalization group flow in field theories. In general, all couplings allowed by symmetry and locality are generated under the renormalization group flow, and the exact renormalization group flow takes place in the infinite dimensional space of couplings. In this paper, we show that the renormalization group flow is highly constrained so that the $\beta$-functions defined in a measure zero subspace of couplings completely determine the $\beta$-functions in the entire space of couplings. We provide a quantum renormalization group-based algorithm for reconstructing the full $\beta$-functions from the $\beta$-functions defined in the subspace. The general prescription is applied to two simple examples.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2021-5-17 (Invited Report)
I believe the paper requires a significant rewriting. Please see the report attached
Anonymous Report 1 on 2021-5-11 (Invited Report)
1. This explains reasonably well Lee's ideas about a quantum renormalisation group and attempts some advance on these ideas.
1. The ideas rest on formal manipulations. It is far from clear that these manipulations are well defined outside the simple models considered.
2. It is not clear to me that this paper adds anything substantive to previous papers the senior author has already released on the subject.
The central claim of the paper (that the beta-functions for all symmetric operators in a general quantum field theory can be reconstructed from beta-functions in a measure zero subspace of such conjugate couplings) is not surprising, since the continuum limit of such quantum field theories are parametrised by typically a finite dimensional space of marginally relevant couplings. At least within perturbation theory, there are well established procedures for determining the form of the beta-functions for all other operators in terms of these couplings -- crucially after appropriate care is taken to define the regularisation and renormalisation of those operators. What the authors actually do is both weaker (they are typically left to deal with a subspace still containing an infinite number of couplings) and nowhere near as well-defined. To achieve their result they cast the Polchinski version of the exact renormalisation group flow equation into a kind of functional Schrodinger equation where the dependence on sources for all operators outside the subspace, is finessed into a functional generalisation of a Schrodinger wave function. No algorithm is presented for solving this functional Schrodinger equation, or justification for thinking this is easier than working with the original quantum field theory (most likely it is not) and it is far from clear that this reformulation is in general well defined (again most likely it is not). Instead in examples, the authors make Gaussian-like ansatze for the beta functions inside this subspace, equations (93) and (109). Only with these simple ansatze can they proceed to `reconstruct' the full theory but in these now over-simplified models.