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A similar question about the difference between 1PI and Wilsonian effective actions was asked and answered here. Now I ask, when are they the same? Particularly, Seiberg says here (Pg 6, Sec 2.3) that the two actions are identical when we are considering non-interacting, massless particles, which is often the case for Higgs or confining phases.

Is there a formal proof of this statement or some more elaboration of this point? This seems to hold generally, without needing any SUSY context (the rest of the lecture is on SUSY, anyway).

AngryBach
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  • The issue described by Seiberg sounds very reminiscent of the situation in large deviation analysis, in which the "rate function" is equal to the Legendre(-Frenchel) transform (LT) of the cumulant generating function (CGF) when the CGF is analytic. When the CGF is non-analytic the LT is the convex hull of the actual rate function. This is similar to how the Gibbs free energy looks like the convex hull of the Landau Free energy in thermodynamics, which are analogous to the 1PI action and Wilson effective action in field theory. So an answer to your Q may be 'when the CGF is analytic'. – bbrink Jun 17 '21 at 12:38

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