Remarks about Dyson's instability in the large-N limit

Details Remarks about Dyson's instability in the large-N limit Remarks about Dyson's instability in the large-N limit

2012-03-10

arxiv.org/abs/1203.2256v2

Physics

High Energy Physics

High Energy Physics - Theory

11 pages, 7 figures, uses revtex, refs. added, minor changes

Scientific paper

There are known examples of perturbative expansions in the 't Hooft coupling lambdat with a finite radius of convergence. This seems to contradict Dyson's argument stating that the instability at negative coupling implies a zero radius of convergence. Using the example of the linear sigma model in three dimensions, we resolve this paradox. We show that a saddle point persists for negative values of lambdat until it reaches a critical value -|lambdat_c| and that the radius of convergence of the perturbative series is |lambdat_c|. On the other hand, an explicit construction shows that for -|\lambdat_c|< lambdat <0, the effective potential does not exist for large values of the field and not at all if lambdat <-|lambdat_c|. This could be relevant to understand de Sitter instabilities and complex RG flows in gauge theories.

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