In SUSY gauge theories there’s a big distinction between the Wilsonian and the 1PI effective actions. Seiberg makes a big distinction between the two during his lectures (e.g. see the discussion that arose during his explanation of Seiberg Duality in the SIS07 school). This isn’t explained in any of the usual QFT textbooks, so I figured it was worth writing a little note that at least collects some referenes.

The most critical application of the distinction is manifested in the beta function for supersymmetric gauge theories. The difference between the 1PI and Wilsonian effective actions ends up being the difference between the 1-loop exact beta function of and the NSVZ “exact to all orders” beta function that includes multiple loops. For a discussion of this, most paper point to Shifman and Vainshtein’s paper, “Solution of the anomaly puzzle inSUSY gauge theories and the Wilson operator expansion.” [doi:10.1016/0550-3213(86)90451-7]. It’s worth noting that Arkani-Hamed and Murayama further clarified this ambiguity in terms of the holomorphic versus the canonical gauge coupling in, “Holomorphy, Rescaling Anomalies, and Exact beta Functions in SUSY Gauge Theories,” [hep-th/9707133].

The distinction between the two is roughly this:

  • The Wilsonian effective action is given by setting a scale \mu and integrating out all modes whose mass or momentum are larger than this scale. This quantity has no IR subtleties because IR divergences are cut off. To be explicit, the Wilsonian action is a theory with a cutoff. It is a theory where couplings run according to the Wilsonian RG flow, i.e. it is a theory that we still have to treat quantum mechanically. We still have to perform the path integral.
  • The 1PI effective action is the quantity appearing in ithe generating functional of 1PI diagrams, usually called \Gamma. This quantity is formally defined including all virtual contributions coming from loops so that the tree-level diagrams are exact. (Of course we end up having to calculate in a loop expansion.) The one-loop zero-momentum contribution is the Coleman-Weinberg potential. The 1PI effective action is the quantity that we deal with when we Legendre transform the action with respect to sources and classical background fields. The 1PI effective action is meant to be classical in the sense that all quantum effects are accounted for. Because it takes into account all virtual modes, it is sensitive to the problems of massless particles. Thus the 1PI effective action can have IR divergences, i.e. it is non-analytic. It can get factors of log p coming from massless particles running in loops. Seiberg says a good example of this is the chiral Lagrangian for pions.

Further references not linked to above:

  • Bilal, “(Non) Gauge Invariance of Wilsonian Effective Actions in (SUSY) Gauge Theories: A Critical Discussion.” [0705.0362].
  • Burgess, “An Introduction to EFT.” [hep-th/0701053]. An excellent pedagogical explanation of the Wilsonian vs 1PI and how they are connected. A real pleasure to read.
  • Seiberg, “Naturalness vs. SUSY Non-renormalization.” [hep-ph/9309335]. Mentions the distinction.
  • Polchinski, “Renormalization and effective Lagrangian.” [doi:10.1016/0550-3213(84)90287-6] Only mentions Wilsonian effective action, but still a nice pedagogical read.
  • Tim Hollowood’s renormalization notes (http://pyweb.swan.ac.uk/~hollowood/) are always worth looking at.
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