Preprint
Remarks about Dyson's instability in the large-N limit
ArXiv.org
Cornell University
03/10/2012
DOI: 10.48550/arxiv.1203.2256
Abstract
There are known examples of perturbative expansions in the 't Hooft coupling lt with a finite radius of convergence. This seems to contradict Dyson's argument suggesting that the instability at negative coupling implies a zero radius of convergence. Using the example of the linear sigma model in three dimensions, we discuss to which extent the two points of view are compatible. We show that a saddle point persists for negative values of lt until a critical value -|lt_c| is reached. A numerical study of the perturbative series for the renormalized mass confirms an expected singularity of the form (lt +|lt_c|)^1/2. However, for -|lt_c|< lt <0, the effective potential does not exist if phi^2 >phi^2_{max}(lt) and not at all if lt<-|lt_c|. We show that phi^2_{max}(lt) propto 1/|lt | for small negative lt. The finite radius of convergence can be justified if the effective theory is defined with a large field cutoff phi^2_{max}(lt) which provides a quantitative measure of the departure from the original model considered.
Details
- Title: Subtitle
- Remarks about Dyson's instability in the large-N limit
- Creators
- Y Meurice - KITP and U. of Iowa
- Resource Type
- Preprint
- Publication Details
- ArXiv.org
- Publisher
- Cornell University
- DOI
- 10.48550/arxiv.1203.2256
- ISSN
- 2331-8422
- Number of pages
- 6
- Comment
- 6 pages, 8 figures, uses revtex, expanded version with a more complete discussion of "Dyson's large-N paradox", refs. added Report number: NSF-KITP-12-027
- Language
- English
- Date posted
- 03/10/2012
- Academic Unit
- Physics and Astronomy
- Record Identifier
- 9984442023602771
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