## February 21, 2008

### ERGE

The discussion engendered by my recent post on of the application of the Exact Renormalization Group to gravity, was rather illuminating. But I think I should return to a point emphasized in my first post on the subject, namely that the ERGE , at least when truncated (which is the only recourse for practical calculations), is not fully nonperturbative. Instead, it provides a (very useful) resummation of the loop expansion.

In this post, I will discuss three examples where, as far as I can tell, the ERGE fails to capture important physics, which just happens to be invisible to all orders in the loop expansion.

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

(1)$\mu\frac{\partial}{\partial \mu} \Gamma_\mu(\Phi) = \frac{1}{2} STr\left[ {\left(R^{A B}_\mu + \frac{\delta^2 \Gamma_\mu}{\delta\Phi_A\delta\Phi_B}\right)}^{-1} \mu\frac{\partial}{\partial \mu}R^{B A}_\mu\right]$

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

(2)$\lim_{\mu\to0} \Gamma_\mu(\Phi) = \Gamma(\Phi)$

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

$\mu\frac{\partial}{\partial\mu} \Gamma_\mu = \frac{1}{2} \array{\begin{svg}\end{svg}}$ where $\begin{svg} \end{svg}= \mu\frac{\partial}{\partial \mu} R_\mu$ and $\begin{svg} \end{svg}=$ 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

(3)$v^{4-3n} e^{-8\pi^2/g^2(v)} \psi^{2n} + \text{h.c.}$

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:

(4)$W_{\text{eff}} = (N_f - N_c) \Lambda_{\text{QCD}}^{\frac{3N_c-N_f}{N_f - N_c}} det(M)^{-\frac{1}{N_f -N_c}}$

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.

1 I’d like to thank Roberto Percacci for pointing out this reference.

2 Despite this success, I think it’s still true that most accurate determinations of the critical exponents at Wilson-Fisher are obtained by more conventional techniques.

Posted by distler at February 21, 2008 11:54 PM

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### Re: ERGE

Hi,

I’m having difficulties reading all the equations in this article. For example, the equation after (2) in the introductory paragraph stops after the 1/2, and has no number. I haven’t had problems like this before. Any suggestions? (I’m running Firefox on a Mac and have installed all the fonts for MathML.)

Although I want to be able to read everything before making detailed comments, I would like to draw your attention to http://arxiv.org/abs/hep-th/9410141, in which the derivative expansion of the ERG is applied to 2-D Scalar Field Theory. 10 fixed-points are found, and about 100 quantities are computed to at least reasonable accuracy, as compared with CFT. This surely shows that this technique does much more than just resum perturbation theory.

I’d also like to mention, since I haven’t noticed it anywhere explicitly on this blog (and it might be of interest), that Polchinski’s equation and Wetterich’s equation are related by a Legendre transform.

Posted by: Oliver Rosten on February 22, 2008 1:22 PM | Permalink | Reply to this

### Browser woes

Sorry you’re having browser trouble. For what it’s worth, I use Mozilla on a Mac, too. My best current advice is contained on this wiki page.

The current nightly builds of Firefox actually work great with the STIX fonts (much better than the yucky Mathematica fonts). As soon as this bug gets fixed, I’ll heartily recommend using a nightly build (and will update the instructions on the Wiki page, appropriately).

Although I want to be able to read everything before making detailed comments, I would like to draw your attention to Tim Morris’s paper, in which the derivative expansion of the ERG is applied to 2-D Scalar Field Theory. 10 fixed-points are found, and about 100 quantities are computed to at least reasonable accuracy, as compared with CFT.

Tim’s paper is very beautiful.

It would be unfair to say that the ERGE is “just” a resummation of perturbation theory, but I think it’s also true that there are cases (such as the ones I’ve listed) where it fails to capture some essential nonperturbative information.

I think we need an adjective that means “more than just perturbative, but not fully nonperturbative.”

Posted by: Jacques Distler on February 22, 2008 3:11 PM | Permalink | PGP Sig | Reply to this

### Re: Browser woes

Thanks for the tips, but I just can’t get it to work.

I think that, as Jan says in his paper, the issue as to how exact the ERG is comes down to the truncation. If one could solve the ERG exactly, all nonperturbative information would be included. Given that this can’t be done, a truncation must be performed, and the nature of the truncation will determine whether the resulting equations are sensitive to certain effects. I don’t think any of the examples you give contain effects that the ERG is inherently blind to, but successfully picking them out might be a subtle business.

So, perhaps we can agree that truncations of the Exact RG are exact in the same sense that the contents of Basil Faulty’s tinned fruit salad were fresh when they were picked.

Posted by: Oliver Rosten on February 23, 2008 5:13 AM | Permalink | Reply to this

### Re: ERGE

Forget about nonperturbative corrections, does the truncation + the RG capture all the strictly UV perturbative information thats been tossed out?

I’m sort of inclined to think that it captures a little bit of both, but neither exactly, unless you get really lucky.

The Argyres-Douglas fixed point example further confuses me, b/c naively I would have thought this sort of thing would do well in capturing IR nonperturbative effects.

Posted by: haelfix on February 23, 2008 9:32 AM | Permalink | Reply to this

### Re: ERGE

The Argyres-Douglas fixed point example further confuses me, b/c naively I would have thought this sort of thing would do well in capturing IR nonperturbative effects.

Each of the three examples should confuse you, then. They are each examples of (successively more intricate) nonperturbative IR effects that are invisible to all orders in the loop expansion.

Posted by: Jacques Distler on February 23, 2008 10:17 AM | Permalink | PGP Sig | Reply to this

### Re: ERGE

Yes I understand that, and I follow the reasoning in the first two examples. The third I do not pretend to understand other than the first three or four lines. Why are there extra massless nonperturbative degrees of freedom there from those seemingly arbitrary curves?

Posted by: haelfix on February 23, 2008 12:14 PM | Permalink | Reply to this

### Re: ERGE

It’s worth rereading Seiberg and Witten, then.

At a generic point in the moduli space, the $SU(3)$ gauge symmetry is spontaneously broken to $U(1)\times U(1)$. Thus the theory contains ‘t Hooft-Polyakov monopoles (and dyons). These are massive, and hence decouple from the far-IR physics.

But, in complex codimension 1 (along the aforementioned curves), some monopole or dyon becomes massless.

Dualizing the appropriate $U(1)$, the massless electrically charged hypermultiplet drives the dual gauge coupling to zero, and the IR fixed point is Gaussian (but in a way that is completely non-obvious from the original $SU(3)$ gauge theory variables).

In complex codimension 2 (i.e., at special points in the moduli space), yet more complicated things ensue. In particular, at the points just mentioned, the IR fixed point is nontrivial.

A panoply of similar fixed points are known in a variety of supersymmetric gauge theories. And a few of the critical exponents, in each case, are known exactly.

Posted by: Jacques Distler on February 23, 2008 1:07 PM | Permalink | PGP Sig | Reply to this

### Re: ERGE

Ahh ok, makes sense now. I remember the SU(2) case and haven’t studied much of the generalizations to other gauge groups.

So, let me see if I get this right, SU(3) breaks down to the maximal torus U(1)*U(1) and unlike the SU(2) case, mutually *nonlocal* dyons become simultaneously massless at the AD points in the quantum moduli space. Exactly there, the theory becomes conformally invariant. The classical theory has a guage symmetry at those singularities, but the quantum theory does not.

I see why it would be extremely hard for the ERGE to capture this at any finite approximation, thats pretty pathological.

Posted by: Haelfix on February 25, 2008 8:42 AM | Permalink | Reply to this

### Re: ERGE

…thats pretty pathological.

It’s not pathological; it’s generic.

Pretty much every $\mathcal{N}=2$ gauge theory (including $SU(2)$ with matter) has such points somewhere in its moduli space. The same is true of $\mathcal{N}=1$ theories with Coulomb branches and probably $\mathcal{N}=0$ as well, though there we currently have no analytic control.

Posted by: Jacques Distler on February 25, 2008 9:43 AM | Permalink | PGP Sig | Reply to this

### Re: ERGE

does the truncation + the RG capture all the strictly UV perturbative information thats been tossed out?

I’m not exactly sure what you mean by this. There is nothing to stop one doing perturbation theory with the ERG - this is just one particular truncation.

If you perform a truncation which captures non-perturbative information, then you’re right that it won’t exactly reproduce all of perturbation theory, at any finite order in the nonperturbative truncation scheme (if it did, presumably everyone would be using it). Indeed, the non-perturbative truncation scheme may not converge (which is not necessarily as bad as it sounds - low order results can be good in certain regimes) and this will be visible in perturbation theory: for example, perturbative contributions to the β-function, as computed using certain versions of the derivative expansion, do not converge (but they do for other versions).

This is a subtle issue, relating to the choice of flow equation (Wilson / Polchinski or the Legendre transformed version) and cutoff function. It is clearly discussed, as so many things in this field are, in another of Tim Morris’s papers.

Posted by: Oliver Rosten on February 23, 2008 11:09 AM | Permalink | Reply to this

### Re: ERGE

“If you perform a truncation which captures non-perturbative information, then you’re right that it won’t exactly reproduce all of perturbation theory”

If I chop off a series, afaiu one is not even presumably in the same universality class. Applying say the Wilson Fischer RG scheme to the ensuing theory helps capture some of the nonperturbative effects that straight forward perturbation theory misses (at least some of the IR stuff). Good! But since we are using this to define the full UV behaviour of the original untruncated theory, I don’t understand why all those terms that have been tossed out couldn’t simply give you massive corrections.

Posted by: haelfix on February 23, 2008 12:32 PM | Permalink | Reply to this

### Re: ERGE

Hi,

I think that, to answer this, two different notions need to be disentangled. They are fixed points of the ERG and, in Wilson’s terminology, the renormalized trajectories (RTs) emanating from them.

Fixed points are at the heart of finding renormalizable theories. To see this, consider starting with some bare action, at the bare scale ${\Lambda }_{0}$. Now integrate out degrees of freedom between the bare scale and the effective scale, $\Lambda$. Renormalizable theories are those for which the bare scale can be safely sent to infinity (nonperturbatively). A particular class of renormalizable theories immediately follow from fixed points of the ERG. This is because fixed points correspond to conformal field theories. Conformal field theories are scale independent and are therefore independent of ${\Lambda }_{0}$, which can obviously be sent safely to infinity.

So, given some theory (and by this I mean that the field content and symmetries have been specified, but the interactions are to be determined) the first thing one might try to do is to find the fixed points. In scalar field theory in 3D, there is the Gaussian fixed point (GFP) and Wilson-Fisher fixed-point (WFFP).

Having found a fixed point, we now linearize the ERG equation about the fixed point. The resulting equation allows us to classify operators according to whether they are relevant or irrelevant (for marginal operators, we must go to the next order in the expansion about the fixed point, to determine their behaviour). Trajectories along irrelevant directions flow into the fixed point under the ERG flow; trajectories along relevant directions flow out of a fixed point.

So, there are now two things to discuss. First, consider the surface spanned by the irrelevant operators. This is the critical surface. The theories on the critical surface are in the same universality class, which is determined by the fixed point. Secondly, consider the trajectories emanating from the fixed point. It turns out to be easy to show, nonperturbatively, that these trajectories have no explicit dependence on ${\Lambda }_{0}$; again, then, ${\Lambda }_{0}$ can be sent to infinity and this is why trajectories emanating from a fixed point are called renormalized trajectories.

If I understand what you are trying to say, you are thinking of a trajectory which emanates from the Gaussian fixed point and flows towards the WFFP. This trajectory, as with all RTs, is defined through the choice of UV fixed point (Gaussian, in this case) and by the integration constants associated with the relevant directions of the GFP. Just because this trajectory happens to end up near the WFFP in the IR does not mean that we use the WFFP to define the UV behaviour. This is the wrong way round.

However, we could consider a completely different trajectory which starts from the WFFP and flows out along the single relevant direction. In this case, the WFFP defines our UV theory.

One of the things which makes the ERG so useful is that there are approximation schemes (trucations) which preserve the fixed point structure and the existence of the associated RTs. Consequently, renormalizability is preserved (and very straightforwardly) - something which is not guaranteed in some generic approach to computing in QFT.

These issues are beautifully discussed in another of Tim Morris’s papers. There is a longer discussion in the paper of Bagnuls and Bervillier, mentioned above by Jacques.

Posted by: Oliver Rosten on February 25, 2008 6:42 AM | Permalink | Reply to this

### Re: ERGE

Hi Oliver. On page 11-12 of Morris’s paper, he discusses the truncation approximation. Generically all im saying is exactly what he says there. In the gravity case, its hard to know how much accuracy you’ve lost by the approximation or whether you’ve simply probed a spurious fp absent some other information from some other method.

Its interesting that scalar field theory provides an exception and provides such excellent numerical agreement though.

Posted by: haelfix on February 25, 2008 8:28 AM | Permalink | Reply to this

### Re: ERGE

The point is that not all truncations are equally effective at reliably finding FPs. Morris discusses (i) a truncation of the effective action so that it contains only a few operators (ii) the derivative expansion which is a different truncation. Even in scalar field theory, truncations of type (i) are (usually) unreliable.

For the types of truncation that are currently used in gravity I agree that one should worry as to whether the fixed point found is spurious. But this does not mean that there doesn’t exist, in principle, a more robust truncation.

Posted by: Oliver Rosten on February 25, 2008 9:10 AM | Permalink | Reply to this

### Re: ERGE

Its interesting that scalar field theory provides an exception and provides such excellent numerical agreement though.

I don’t think so.

In all of these example (the Wilson-Fisher fixed point in 3D, the conformal minimal models in 2D), the nontrivial fixed points are obtained as relevant perturbations of the Gaussian fixed point.

You can get excellent numerical accuracy for the critical exponents by resumming the conventional perturbative calculations of the anomalous dimensions.

We’ve already discussed the Wilson-Fisher case at some length. In the 2D case, Zamolodchikov identified the universality class of the $p$th conformal minimal model ($p=3,4,5,\dots$),

(1)$S_E = \int d^2 x \left(\tfrac{1}{2}\partial^\mu\phi\partial_\mu\phi +g \phi^{2p-2}\right)$

and he identified a subset of the scaling operators

(2)$:\phi^k: = \begin{cases}O_{(k+1,k+1)},& k=0,1,\dots, p-2\\ O_{(k-p+2,k-p+3)},& k=p-1,p, \dots, 2p-4\end{cases}$

and tells you, in principle, how to identify the rest (e.g., $O_{(1,3)} = :\partial^\mu\phi\partial_\mu\phi:$). The exact scaling dimensions are

(3)$d_{O_{(n,m)} } = \frac{{(p n - (p+1) m)}^2-1}{2p(p+1)}$

I looked around a bit to see whether anyone had tried to reproduce these by resumming the perturbative calculation (as is done, so successfully, for Wilson-Fischer). I have a dim recollection of seeing this done, but I can’t imagine anyone putting a stupendous amount of effort into reproducing an approximation to (3).

Posted by: Jacques Distler on February 25, 2008 11:47 AM | Permalink | PGP Sig | Reply to this

### Re: ERGE

I just want to clarify a point.
When ERGE is applied to 2, 3 or 4 dim scalar theory, its aim is to see if it is able to reproduce already well known results. It is kind of test of its strenght.
Imagine addressing (2 dim) Ising model with ERGE. We could spend a lot of time in working out a solution, however that will fail if compared to the exact one. I believe this will always happen in 2 and 3 dim.
The crucial point is that ERGE has something to tell about gravity and in general about theories we marked as non-renormalizable.
Anyway it is important to try to reproduce as much as possible, as you stressed.

Posted by: noneoftheabove on February 28, 2008 3:38 AM | Permalink | Reply to this