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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.

The CDF multi-muon anomaly has been an experimental curiosity for a few months now, but it seems to have taken a back seat to PAMELA/ATIC for exciting experimental directions’ in particle phenomenology. [Of course, it’s still doing better than the LHC…]

The end of 2008 for hep-ph‘ists was marked by three interesting leptonic signals. The CDF multi-muon anomaly, PAMELA/ATIC, and the MiniBooNE excess. Of these three, PAMELA/ATIC have gotten the lion’s share of papers — but there have been rumblings that the Fermi/GLAST preliminary results are favoring a vanilla’ astrophysical explanation. There have been a couple of notable papers attempting to explain PAMELA/ATIC along these lines (0812.4457, 0902.0376) along these lines. As for MiniBooNE, not very much has been said about the excess in low-energy electrons. I hope to be able to learn a bit more about this before I blog about it.

The original multi-muon paper is quite a read (and the associated initial model-building attempt), and indeed produced an interesting response’ from a theorist (much of which is an excellent starting point for multi-muon model-building), which in-turn produced a response on Tommaso’s blog… which eventually turned a bit ugly in the comments section. Anyway, the best armchair’ reading on the multi-muon anomaly is still Tommaso’s set of notes: part 0, part 1, part 2, part 3, part 4. An excellent theory-side discussion can be found at Resonaances.

There have been a handful of model attempts by theorists since the above discussions. General remarks on the hidden-valley context and how to start thinking about this signal can be found in the aforementioned paper 0811.1560. A connection between CDF multi-muons and the cosmic ray lepton excess was presented in 0812.4240. A very recent paper also attacks CDF + PAMELA with a hidden valley scalar, 0902.2145. As usual any map from possible new signals to variants of the MSSM is surjective (though never one-to-one), so it’s no surprise that people have found a singlet extension to the MSSM to fit the CDF anomaly in 0812.1167. An exploration of what can we still squeeze out of the Tevatron’ comes from Fermilab, which explains that a very heavy t’ could not only be found at the Tevatron, but could explain the CDF anomaly, 0902.0792.

There are some very respectable theorists trying their hand at multi-muon model building, though there generally seems to be some reluctance from the community as a whole to devote much effort towards it. Maybe people are holding their breath for direct production of new physics at the LHC, or are otherwise convinced that the thing to do right now is construct theories of dark matter since we know dark matter must eventually show up in a particle experiment.

For me, the threshold for jumping into the field head-first was waiting to hear what the D0 collaboration had to say about this. According to rumors, however, it seems like the Tevatron’s other detector won’t have anything to say since it won’t be doing this analysis. From what I understand, this comes from the way that the D0 collaboration skims their data. (What does skim’ mean?) Rumor has it that it’s very difficult for them to do the same analysis that CDF did, and they’ve decided that (1) the likelihood of new physics is so unlikely that it’s not worth their effort to jump in and try to get in on the glory, and (2) the signal is so absurd that it’s not even worth their effort to do the analysis to disprove their friendly rivals at CDF. Not being an experimentalist I can’t comment on the validity or rationale for this — if it is indeed true — but as a phenomenologist I’m smacking my head.

If the multi-muon signal pans out, it could be an experimental discovery that would launch a thousand theorists (Helen of Troy reference indended). If not, a cross check with D0 would have definitively (to the extent that anything is definite in science) put the issue to rest. There are some people in the CDF collaboration who are really convinced by their analysis, and I hope that there will be an opportunity in the near future to cross-check those results at another detector.