Musings

Rather than the ERGE of Polchinski (reviewed here), which involves a sliding UV cutoff, and applies to the Wilsonian action, we will discuss a variant ERGE, due to Wetterich, which takes the following form. Fix a UV cutoff, $\Lambda$, and a bare action, $S_\Lambda(\Phi)$. Wetterich defines a functional $\Gamma_\mu(\Phi)$ which interpolates between the bare action and the 1PI generating functional, $\Gamma(\Phi)$. $\Gamma_\mu(\Phi)$ obeys

where $R^{A B}_\mu(p^2)$ is an IR cutoff function $$\begin{aligned} R_\mu(p^2) &\to \infty,&&\quad \frac{p^2}{\mu^2} \ll 1\\ R_\mu(p^2)/p^2 &\to 0,&&\quad \frac{p^2}{\mu^2} \gg 1 \end{aligned}$$ and “$STr$”, involves a sum over fields and an integral over momenta, with an extra minus sign thrown in for fermions.

As an initial condition, we set $$\lim_{\mu\to\Lambda} \Gamma_\mu(\Phi) = S_\Lambda(\Phi)$$ and then the claim is that

the 1PI generating functional. This has a great conceptual advantage over the “Wilsonian” ERGE of Polchinski, in that $\Gamma(\Phi)$ is directly phrased in terms of physical observables. It’s a little awkward in supersymmetric theories, where we lose certain nonrenormalization theorems, which apply to the Wilsonian action. But, for present purposes, this will not be much of a drawback.

(1) is, manifestly, a 1-loop equation, but in an infinite number of couplings. Schematically, it looks like

$$\mu\frac{\partial}{\partial\mu} \Gamma_\mu = \frac{1}{2} \array{\begin{svg}<svg xmlns="http://www.w3.org/2000/svg" width="96" height="80"> <desc>1-loop diagram for the Wetterich ERGE</desc> <g fill="none" stroke="#000"> <line stroke-width="1" x1="20" y1="40" x2="1" y2="40"/> <circle stroke-width="3" cx="56" cy="40" r="36"/> <circle fill="#F00" stroke="none" cx="20" cy="40" r="8"/> </g> </svg>\end{svg}}$$ where $\begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="20" height="20"> <desc>Derivative of the IR regulator</desc> <circle fill="#F00" stroke="none" cx="10" cy="10" r="8"/> </svg> \end{svg}= \mu\frac{\partial}{\partial \mu} R_\mu$ and $\begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="40" height="20"> <desc>thick line</desc> <line fill="none" stroke="#000" stroke-width="3" x1="2" y1="14" x2="40" y2="14"/> </svg> \end{svg}=$ the (IR-regulated) full propagator associated to $\Gamma_\mu$.

To make any headway, we need to truncate it to a finite number (“$n$”) of couplings, which more-or-less amounts to defining an $l$-loop RG equation, for some $l\leq n$. But that’s just a heuristic argument. The real test is that there are contributions to $\Gamma(\Phi)$ that are invisible in perturbation theory. Can (1) capture them?

An anomalous symmetry

Consider an $SU(2)$ gauge theory with $2n$ left-handed Weyl fermions, $\psi^i$, in the fundamental representation. We will require $1 \lt n \lt 11$, where the upper limit comes from demanding asymptotic freedom, and the lower limit avoids a certain UV-sensitivity of the phenomenon to be discussed below. Our theory will also contain a scalar (“Higgs”), again in the fundamental. The scalar, $\phi$, has some symmetry-breaking potential, and we’ll denote the magnitude of the VEV of $\phi$ by $v$.

Classically, the theory has a $U(2n) = (SU(2n)\times U(1))/Z_{2n}$ symmetry, acting on the $\psi^i$. Quantum-mechanically, this is broken to $SU(2n)$. We can distinguish, in our theory, between those operators, $O$, which are invariant under the full $U(2n)$, and those, $\Psi$, which are invariant only under $SU(2n)$.

Perturbatively, all the $\Psi$ have very high dimension and are irrelevant in the infrared. Moreover consider any truncation of (1) to a finite number of operators. I could, if I chose, set the coefficient of all of the $\Psi$ to zero in $S_\Lambda$, (the initial condition for the RG flow). If I use such a $U(2n)$-invariant initial condition, (1) does not generate nonzero coefficients for the $\Psi$ along the flow.

But ‘t Hooft calculated the coefficient, in $\Gamma(\Phi)$, of the lowest dimension such operator. Ignoring stupid numerical factors of $2$ and $\pi$, it looks like

where $g^2(v)$ is the running gauge coupling evaluated at the $SU(2)$-breaking scale, and $\psi^{2n}$ is schematic for $$\epsilon_{i_1 j_1 ... i_n j_n} \epsilon^{\alpha_1 \beta_1} ... \epsilon^{\alpha_n \beta_n} \epsilon_{a_1 b_1} ... \epsilon_{a_n b_n} \psi^{i_1 a_1}_{\alpha_1} \psi^{j_1 b_1}_{\beta_1} ... \psi^{i_n a_n}_{\alpha_n} \psi^{j_n b_n}_{\beta_n}$$

This is a small effect if the gauge coupling is weak at the scale $v$, but it is definitely nonzero. While the anomaly, which breaks $U(2n)\to SU(2n)$, is visible at 1-loop, no $U(2n)$-violating operators are induced in $\Gamma(\Phi)$ to any order in perturbation theory. (3) is a 1-instanton effect.

Having said that, it seems obvious how to “fix” the problem. The space of gauge field configurations is not connected. Perhaps $\Gamma_\mu$, as defined, is only receiving contributions from the topologically-trivial sector and one needs to add “by hand” contributions from the $n$-instanton sector, “$\Gamma_\mu^{(n)}$”. Jan Pawlowski does this¹, where $\Gamma^{(\pm 1)}_\mu$ is defined to receive contributions from instantons of scale size $1/\Lambda \lt \rho \lt 1/\mu$. All very well, but the result doesn’t satisfy (3) and, besides, instantons do not exhaust the range of nonperturbative effects in field theory.

$\mathcal{N}=1$ Super QCD

Consider an $\mathcal{N}=1$ supersymmetric $SU(N_c)$ gauge theory, with $N_f$ chiral multiplets, $Q_i$, in the fundamental and $N_f$ chiral multiplets, $\tilde{Q}_i$. To all orders in perturbation theory, this theory has a moduli space of vacua, characterized by the vacuum expectation value of the gauge-invariant scalar operator $$M_{i j} = \tilde{Q}_i Q_j$$ For $N_f \lt N_c$, this degeneracy is lifted because a superpotential is generated nonperturbatively:

Again, there’s a classical $U(1)$ symmetry that forbids (4) from being generated at any order of perturbation theory. (In the Wilsonian action, there’s a theorem that the superpotential cannot be corrected at any order in perturbation theory. Here, we are dealing with the 1PI generating functional, and it is possible to generate, in perturbation theory, nonlocal terms that look like superpotential terms. With some care, they can be cleanly distinguished from “genuine” superpotential terms. But, in this particular case, we don’t need to be that careful.)

For $N_f= N_c$, the moduli space of vacua persists in the full nonperturbative theory, but its topology is different from that seen in perturbation theory. For $N_c \lt N_f \lt 3 N_c$, even more interesting physics obtains.

In each of these cases, there are important nonperturbative effects that are invisible in the loop-expansion, and hence in any truncated version of the ERGE. Only for $N_f = N_c-1$ can those effects be ascribed to instantons.

An Argyres-Douglas Fixed Point

One of the successful applications of the ERGE has been in studying the Wilson-Fisher fixed point in 3 dimensional scalar field theory. But what about fixed points whose existence is is somewhat more … ah … nonperturbative in nature?

Consider an $\mathcal{N}=2$ supersymmetric pure $SU(3)$ gauge theory. This has a 2-complex dimensional moduli space of vacua, parametrized by the vacuum expectation values of $$u = \tfrac{1}{2} tr \phi^2,\quad v = \tfrac{1}{3}tr \phi^3$$ where $\phi$ is the complex scalar in the adjoint of $SU(3)$.

At a generic point in this moduli space, the infrared physics is Gaussian: a pure $\mathcal{N}=2$ supersymmetric $U(1)\times U(1)$ gauge theory. But along a pair of complex curves, $$\Delta_\pm = \{ 4 u^3 - 27 (v\pm 2\Lambda^3_3)^3= 0\}$$ where $\Lambda_3$ is the dimensionful scale of the $SU(3)$, there are extra massless nonperturbative degrees of freedom. The resulting theory is still Gaussian, but understanding these extra degrees of freedom (or, conversely, understanding the divergence of $\Gamma(\Phi)$ if we fail to include them) strikes me as an incredible challenge for the ERGE.

But there is more.

At $$u=0,\quad v= \mp 2\lambda^3_3$$ the infrared physics is not Gaussian. Instead, we have a nontrivial RG fixed point, with nontrivial critical exponents, some of which are known exactly.

Computing the critical exponents at the Wilson-Fisher fixed point is all very nice, but how about the critical exponents at one of these Argyres-Douglas points?

Implications

These examples were carefully chosen to contain important effects which happen to vanish to all orders in the loop expansion. This put in sharp relief that the word “Exact” in the phrase “Exact Renormalization Group” is something of a misnomer.

This lesson is particularly relevant to the gravity case, where we are trying to use the “E”RGE to define the UV behaviour of the theory. It’s in the UV where the nonperturbative effects of quantum gravity (whatever they are) will, surely be important.