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
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
. 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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. I’ve dropped a numerical factor in the second sine, but this is a heuristic discussion anyway. The L and E represent the `baseline’ (distance the neutrinos travel) and the neutrino energy. A similar expression occurs for three neutrino mixing, such as between the three light neutrinos that we’ve come to know and love since 1998.
while oscillations go as /E&bg=ffffff&fg=29303b&s=0&c=20201002)
, this L/E dependence can shed some light about the nature of the electron excess. One proposal is to actually just move the MiniBooNE detector to 200m. Since this would be closer, it would only take one year to accumulate data equivalent to the entire seven year run so far.
. Motivated by semi-classical evaluation of path integral, we search for finite action solutions to the Euclidean equations of motion, 
. In order to have a finite action, we need field potential 
to be pure gauge at the boundary 
, i.e. 
.)=k&bg=ffffff&fg=29303b&s=0&c=20201002)
, k is usually called the charge of the instanton. For the original action to be captured by this number k, we need to have self-dual or anti self-dual field strength.
for SU(2) group can be given by _{\nu}}{(x-X)^2((x-X)^2+\rho^2)}\bar{\eta_{\mu\nu}}^i(g\sigma^i g^{-1})&bg=ffffff&fg=29303b&s=0&c=20201002)
where 
are coordinate parameter 
is scale parameter and together with the three generator of the group, we have 8 parameters called collective coordinates. 
is just some matrix to interwine the group index 
with the space index 
and a given group SU(2), an interesting question is how many independent solutions we have. The number is usually counted by given a solution 
, known as zero modes, 
is the index for this solution space, usually called moduli space.
or 4 supersymmetry in 
, it’s better to promote the instanton to be a string in 6 dimentiona or a 5 brane in 10 dimensions respectively. The details of how to solve the equation will be beyond the scope of this introduction, and We’ll refer the reader to the original lecture notes by David Tong.&bg=ffffff&fg=29303b&s=0&c=20201002)
, which we call the vielbein. This translates to `many legs’ in German. One will often also hear the term vierbein meaning `four legs,’ or `funfbein’ meaning `five legs’ depending on what dimensionality of spacetime one is working with. The index 
labels the different basis vectors.
components, where 
. The choice of this label is intentional, the components of this basis map directly (say, by exponentiation) to the manifold itself, so these really are indices relative to the basis on the manifold. We can thus write a particular basis vector of the tangent space at 
as &bg=ffffff&fg=29303b&s=0&c=20201002)
. How many basis vectors are there for the tangent space? There are  e_{a\nu}(x) = g_{\mu\nu}(x)&bg=ffffff&fg=29303b&s=0&c=20201002)
,
, not $\mu=3$) pointing in one direction (&bg=ffffff&fg=29303b&s=0&c=20201002)
) or another (&bg=ffffff&fg=29303b&s=0&c=20201002)
) relative to the manifold’s basis so long as the two directions are related by a Lorentz transformation. Thus we have an &bg=ffffff&fg=29303b&s=0&c=20201002)
symmetry (or whatever symmetry applies to the manifold). Further, we could have made this arbitrary choice independently for each point in spacetime. This means that the symmetry is local, i.e. it is a gauge symmetry. Indeed, think back to handy definitions of gauge symmetries in QFT: this is an overall redundancy in how we describe our system, it’s a `non-physical’ degree of freedom that needs to be `modded out’ when describing physical dynamics.&bg=ffffff&fg=29303b&s=0&c=20201002)
. From the point of view of Riemannian geometry this is just the connection, so we can alternately call this creature the spin connection. Note that this is all different from the (local) diffeomorphism symmetry of general relativity, for which we have the Christoffel connection.&bg=ffffff&fg=29303b&s=0&c=20201002)
) can now be recast as a vector field with tangent-space indices (V^\mu(x)&bg=ffffff&fg=29303b&s=0&c=20201002)
). By requiring that both objects have the same covariant derivative, we get the constraint = 0&bg=ffffff&fg=29303b&s=0&c=20201002)
. = \partial_\mu e^a_\nu - \Gamma^\lambda_{\mu\nu}e^a_\nu + \omega_{\mu\phantom{a}b}^{\phantom\mu a}e^b_\nu&bg=ffffff&fg=29303b&s=0&c=20201002)
.![\omega^{ab}_\mu = \frac 12 g^{\rho\nu}e^{[a}_{\phantom{a}\rho}\partial_{\nu}e^{b]}_{\phantom{b]}\nu}+ \frac 14 g^{\rho\nu}g^{\tau\sigma}e^{[a}_{\phantom{[a}\rho}e^{b]}_{\phantom{b]}\tau}\partial_{[\sigma}e^c_{\phantom{c}\nu]}e^d_\mu\eta_{cd}](https://s0.wp.com/latex.php?latex=\omega^{ab}_\mu = \frac 12 g^{\rho\nu}e^{[a}_{\phantom{a}\rho}\partial_{\nu}e^{b]}_{\phantom{b]}\nu}+ \frac 14 g^{\rho\nu}g^{\tau\sigma}e^{[a}_{\phantom{[a}\rho}e^{b]}_{\phantom{b]}\tau}\partial_{[\sigma}e^c_{\phantom{c}\nu]}e^d_\mu\eta_{cd}&bg=ffffff&fg=29303b&s=0&c=20201002)
, = e^a_\mu(x)\gamma_a&bg=ffffff&fg=29303b&s=0&c=20201002)
, where the right-hand side contains a flat-space (constant) gamma matrix with its index converted to a spacetime index via the position-dependent vielbein, resulting in a spacetime gamma matrix that is also position dependent (left-hand side). One can see that indeed the spacetime gamma matrices satisfy the Clifford algebra with the curved space metric, ,\Gamma_\nu(y)\} = 2g_{\mu\nu}(x)&bg=ffffff&fg=29303b&s=0&c=20201002)
.
theory or massless QCD. We shall further restrict to dimensionless observables, such as the ratio of cross sections. Let’s call a general observable &bg=ffffff&fg=29303b&s=0&c=20201002)
, where we have inserted a dependence on the energy 
with foresight that such things renormalize with energy scale.&bg=ffffff&fg=29303b&s=0&c=20201002)
which depends only on two massive variables 
and which is
only, =f(x/y)&bg=ffffff&fg=29303b&s=0&c=20201002)
.
, then &bg=ffffff&fg=29303b&s=0&c=20201002)
,
is the usual beta function calculated in perturbation theory. It is dimensionless and is uniquely defined by the theory. Another way one can define 
constant } \equiv K(\rho)&bg=ffffff&fg=29303b&s=0&c=20201002)
.
,
we see that )&bg=ffffff&fg=29303b&s=0&c=20201002)
. Thus we see finally that the integral of the  + K(\rho(\mu)) = K(\rho(Q))&bg=ffffff&fg=29303b&s=0&c=20201002)
.
is an integral that comes from the )&bg=ffffff&fg=29303b&s=0&c=20201002)
vanishes for some 
, then we can write our observable in terms of this scale, = K^{-1}(\log(Q/\Lambda))&bg=ffffff&fg=29303b&s=0&c=20201002)
.
is fixed for a particular theory. This latter quantity is rather interesting because even though it is an intrinsic property of the black box, it is not predicted by the black box, it must be fixed by explicit measurement of an observable. 
High Energy PhDs 
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