## September 23, 2005

### Unpleasantness

Apologies for the light blogging. My iBook (from which I do my blogging) developed a problem with the ATA Controller chip last week, and had to be sent out for repair. It is currently sitting in the Apple Repair facility in Houston, awaiting Rita… repair.

The other reason I haven’t been posting much is that I’ve had this post in preparation, and I’ve had trouble psyching myself up to post it.

In a previous post, I outlined the tough row that anyone hoping to construct a theory of quantum gravity needs to hoe. One of the possibilities I discussed is that the UV behaviour of quantum gravity might be controlled by a UV fixed point. There was an interesting discussion, in the midst of which, the work of Reuter and collaborators was mentioned. My response was a bit cursory. It’s a big body of work, and responding properly would really require a full-length post.

Wading through all these papers is not an easy task. But you can get a pretty good idea of where things are headed by looking at the first one, Nonperturbative Evolution Equation For Quantum Gravity. It’s a very pretty paper, which generalizes the so-called Exact Renormalization Group of Polchinski to (pure) gravity.

There are a bunch of technical innovations required for the gravity case: one has to work in background field gauge, deal with gauge fixing, etc. But the basic idea of the Exact RGE can be understood by looking at Polchinski’s construction for a scalar field.

Define

(1)
\array{\arrayopts{\colalign{right left}} Z(J,\Lambda)=&\int D\phi \exp\left(\int\frac{d^4p}{(2\pi)^4}\left[ -\frac{1}{2}\phi(p)\phi(-p) \left(p^2+m^2\right)K^{-1}\left(p^2/\Lambda^2\right) +J(p)\phi(-p) \right]+L(\phi,\Lambda)\right)\\ \equiv&\int D\phi \exp S(\phi,J,\Lambda) }

We assume that the source has no short-wavelength Fourier modes, $J(p) =0\, \text{unless} p^2\ll \Lambda^2$ The UV cutoff function, $K(s) =\left\{ \array{1 & s\lt 1\\ 0& s\gg 1}\right.$ and is assumed to be smooth. $m^2$ is an IR regulator and is not the location of the pole in the inverse 2-point function (or whatever). The latter is controlled, in part, by the quadratic terms in $L(\phi,\Lambda)$, which we are not being explicit about. $L(\phi,\Lambda)$ is a completely general effective Lagrangian, which varies according to

(2)
$\Lambda \frac{\partial L}{\partial \Lambda}= - \int d^4 p \frac{(2\pi)^4}{2} (p^2+m^2)^{-1} \Lambda \frac{\partial K\left(p^2/\Lambda^2\right)}{\partial\Lambda}\left(\frac{\partial L}{\partial\phi(p)}\frac{\partial L}{\partial\phi(-p)}+ \frac{\partial^2 L}{\partial\phi(p)\partial\phi(-p)}\right)$

Plugging into (1), one formally obtains \array{\arrayopts{\colalign{right left}} \Lambda \frac{d Z}{d\Lambda}=&\int d^4 p \Lambda \frac{\partial K}{\partial\Lambda}\cdot\int D\phi \frac{\partial}{\partial\phi(p)}\left[\left(\phi(p)K^{-1}\left(p^2/\Lambda^2\right) + \frac{(2\pi)^4}{2} (p^2+m^2)^{-1} \frac{\partial}{\partial\phi(-p)}\right)\exp S(\phi,J,\Lambda)\right]\\ =&0} These formal manipulations are justified because we have imposed both a UV and an IR cutoff.

(2) is the Exact RG equation. It is a coupled set of differential equations for the infinite number of couplings in $L(\phi,\Lambda)$. Graphically, the first term on the RHS of (2) (for the $\beta$-function of the $n$-point coupling $g_n(p_1,p_2,\dots,p_{n-1},-p_1-p_2-\dots-p_{n-1};\Lambda)$) is given by a tree diagram, where we sew together $g_{k+1}$ and $g_{n-k+1}$ using the “propagator” $\frac{1}{p^2+m^2}\Lambda\frac{\partial K\left(p^2/\Lambda^2\right)}{\partial\Lambda}$ The second term corresponds to a 1-loop diagram, where we tie together two of the legs of $g_{n+2}$. The $\beta$-function for each coupling in $L(\phi,\Lambda)$, is exactly given by a quadratic expression in the other couplings.

I’ve somewhat belaboured the derivation of this “Exact” RGE to make two points very clear

1. This really is a coupled set of differential equations in an infinite number of variables. Except for the Gaussian fixed point, $g_2(p,-p)\neq 0,\quad g_n=0\,\, \forall n\gt 0$ the fixed-point sets (though they are finite dimensional) do not line up nicely, so that only a finite number of coupling in $L_*(\phi)$ are nonzero.
2. Despite the adjective “Exact”, (2) is very much a perturbative equation. The formal manipulations, by which we arrived at it, were justified to the extent that introducing the cutoff function, $K\left(p^2/\Lambda^2\right)$ really does provide a UV regulator. This is true in perturbation theory, where it provides a cutoff on the loop-momentum integrals for all Feynman diagrams1. It is, most certainly, not true nonperturbatively2.

What’s “exact” about (2) is not that we have somehow magically learned something about the nonperturbative behaviour of the theory by summing some tree and 1-loop Feynman diagrams. What’s exact is that, whereas in the usual formulations of the perturbative RGE the $\beta$-function receives contributions at all loop-orders, here the perturbative $\beta$-function is given exactly by 1-loop. The cost of this “simplification” is that we must consider, simultaneously, the $\beta$-functions for the full infinite set of couplings in the effective Lagrangian.

So, anyone reading the title of Reuter’s first paper should have been very nervous.

Einstein gravity in $d=2+\epsilon$ dimensions has a fixed-point at weak coupling. As Reuter reviews, this is successfully reproduced by the “exact” RGE. This is not surprising, as perturbation theory should be good for small $\epsilon$.

His subsequent papers go on to study a purported strong coupling fixed point in $d=4$ dimensions.

Technically, of course, no one can do anything with the full set of coupled equations, (2). So Reuter truncates to a finite number of couplings (cosmological constant + Einstein-Hilbert term), and studies the truncated set of equations. As I emphasized in my previous post, there’s no reason to expect that the infinite set of higher couplings are negligible in the UV. Reuter and collaborators perform various tests (throwing in an $R^2$ coupling, and seeing its effect) to attempt to justify neglecting these other couplings. I could discuss the details of that. I could re-emphasize that including matter will, almost certainly, destroy whatever fixed point you found in the pure gravity theory. I could …

But this post has gone on long enough already. Most fundamentally, there’s really no reason to trust the perturbatively “exact” RGE for strong coupling (where Reuter et al wish to use it). So I won’t detain you, Dear Reader, any further.

1 The Feynman rules for (1) use vertices constructed from $L(\phi,\Lambda)$ and $\frac{K\left(p^2/\Lambda^2\right)}{p^2+m^2}$ as propagators.

2 This point is not merely a pedantic one. There are nonperturbative contributions to the running of the coupling constants in field theory. This is seen most vividly in supersymmetric theories, where the superpotential (and, beyond one-loop, the holomorphic gauge coupling) are unrenormalized to all orders in perturbation theory, but can (and do) receive nonperturbative corrections.

Posted by distler at September 23, 2005 3:26 PM

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### Re: Unpleasantness

Jacques,

did you perhaps have a chance to take a look at the Pilati (1982) paper as well? I do not want to consume your valuable time, but I find this paper interesting because it uses the fact that gravity is somewhat different from other field theories, the strong-couling limit G = infinite is equivalent to the limit c = 0. In other words, different points in space decouple and the resulting model reduces to quantum mechanics and can be solved exactly. One can then try (and I think Pilati has actually done this) to do perturbation theory for finite c/G.

Posted by: Wolfgang on September 27, 2005 11:01 AM | Permalink | Reply to this

### Time sinks

Sorry, I never got around to it.

Nothing personal (you, after all, weren’t the one who suggested it), but I feel I wasted altogether too much time looking into the much-ballyhooed work of Reuter et al. (I, certainly, was influenced by the enthusiastic reception it has received in some quarters.)

Next time, I’ll be a little less trusting, and will have more time left over to look at other papers.

Posted by: Jacques Distler on September 27, 2005 11:57 PM | Permalink | PGP Sig | Reply to this

### Re: Unpleasantness

Is it possible to tell `in advance’ whether beta-functions for a given field theory will have non-perturbative corrections (in the sense you mean in your footnote)?

(I should think about this more, but what I’m getting at is that for a gauge theory you are usually looking for beta functions expanded around zero coupling. Is it then the existence of classical instanton solutions that leads to non-perturbative contributions to beta-functions?)

Posted by: boreds on September 28, 2005 5:11 AM | Permalink | Reply to this

### When?

“Generically,” the $\beta$-function will always have nonperturbative contributions. Near weak coupling, in a theory where it also receives peturbative contributions, the nonperturbative contributions may be hard to see. In my footnote, I suggested looking at supersymmetric theories (where, for some couplings, the perturbative contributions vanish).

When one says that “$N=4$ SYM is a finite theory,” this is an assertion not just about the vanishing of the perturbative $\beta$-function. It is a statement about the vanishing of nonperturbative contributions as well. (This, stronger, statement is certainly required by S-duality.)

Posted by: Jacques Distler on September 28, 2005 9:30 AM | Permalink | PGP Sig | Reply to this

### Re: When?

OK, thanks, but I’m still missing something.

I am imagining the typical $e^{-1/g^2}$ type dependence, which of course has no perturbation series around $g=0$.

Should I expect functions of this kind or similar to appear in the $\beta$-functions of any field theory, in principle? Regardless of the existence of any classical instanton-type solution, I mean.

(I’m not right now certain which equations I want an instanton solution of…since the classical Lagrangian shouldn’t be so relevant, but anyway…perhaps that’s the origin of my confusion.)

Posted by: boreds on September 28, 2005 10:08 AM | Permalink | Reply to this

### Re: When?

In many (supersymmetric) gauge theories, the leading nonperturbative effects at weak coupling are of this form. But there are also cases where one gets stronger effects, of order $e^{-1/g^2 N}$.

I’m not sure what general statement I can make.

Posted by: Jacques Distler on September 28, 2005 10:25 AM | Permalink | PGP Sig | Reply to this

### Re: When?

I think I understand what you are saying in the case of susy gauge theories. Though maybe I am missing your point about the stronger effects.

My question might be (marginally) clearer for non-gauge theories. Suppose I have a scalar field theory, with no finite-euclidean-action classical solution. Should I still expect non-perturbative contributions to $\beta$-functions for the couplings in this theory?

Sorry, this is all a bit general and vague—but I’m basically confused about whether non-perturbative contributions of the sort you mean are always directly related to the existence of instanton-type classical solutions (which I guess do exist for the susy gauge theories) or something different.

Posted by: boreds on September 28, 2005 10:55 AM | Permalink | Reply to this

### Re: When?

Suppose I have a scalar field theory, with no finite-euclidean-action classical solution. Should I still expect non-perturbative contributions to $\beta$-functions for the couplings in this theory?

Yes, I expect you should. I don’t have much of an intuition as to what they should look like, though.

For a scalar field theory, you can certainly attempt to directly compute using the Renormalization Group of Wilson and Kadanoff (which is, by contrast, a nonperturbative RG, as it is derived using a nonperturbative regulator – the lattice).

Posted by: Jacques Distler on September 28, 2005 11:30 PM | Permalink | PGP Sig | Reply to this

### Re: When?

(silly questions)

Is is possible to have non-perturbative corrections which are not of the form $e^{-1/g^2}$ ? In general, is there anything to restrict the functional form of non-perturbative corrections?

Can a non-perturbative correction have a functional form of, say, $e^{-1/g^1000}$, $\sin (sin (1/(g^20 + 1000)) )$, $ln (tan(g))$, or for that matter, a Laurent type series in general?

Posted by: JC on September 29, 2005 12:35 PM | Permalink | Reply to this

### What?

We want something that vanishes at $g=0$, is smooth (maybe only $n$-times differentiable, but probably smooth) at $g=0$, but non-analytic there.

From the large-order growth of perturbation theory, one can put a bound on how fast these non-analytic terms must vanish as $g\to 0$.

More than that, I can’t say in general.

Posted by: Jacques Distler on September 29, 2005 4:31 PM | Permalink | PGP Sig | Reply to this

### Re: Unpleasantness

Your discussion here is focused on the Polchinski form of the ERGE, but the work of Reuter is phrased in terms of the effective action form of the RGEs. Looking at various review articles (hep-th/0002034 by Bagnuls and Bervillier, hep-ph/0005122 by Berges, Tetradis, and Wetterich, hep-th/0110026 by Polonyi) the claim seems to be that this form really is nonperturbative.

The idea is simple: define a family of effective actions,

(1)$\Gamma_k[J] = -W_k[J] + \int d^d x J(x)\phi(x) - \Delta S_k[\phi],$

where

(2)$W_k[J] = log \int D\phi \exp\left(-S[phi] - \Delta S_k [\phi] + \int d^d x J(x)\phi(x)\right).$

Now $\Delta S_0 = 0$ and by construction $\Gamma_0[J]$ is the full quantum effective action. The term $\Delta S_k$ is constructed as an IR cutoff, which at large $k$ shuts off quantum fluctuations and causes $\Gamma_k[J]$ to approach the classical action. There is a simple flow equation describing the evolution of $\Gamma_k$ between the classical and quantum effective action.

This is a very different point of view from the Polchinski approach; one is not demanding that the partition function be constant, but is instead looking at a family of different actions interpolating between the known classical action and the unknown quantum action. It certainly seems like a nonperturbative construction. Of course one must worry about whether the IR cutoff term is doing anything pathological, and about whether the truncation is introducing large errors, but your basic objection to the Polchinski ERGE doesn’t seem to apply in this case.

Am I missing something? I’m perfectly willing to admit that I might just be dense, but I would like to see what is wrong with the claim that this effective action ERGE is nonperturbative if it is in fact wrong.

Thanks,

A Confused Student

Posted by: confused on April 16, 2006 1:34 PM | Permalink | Reply to this

### Cutoffs

It doesn’t matter whether one is studying a Wilsonian RG (sliding UV cutoff) or a Gell-Mann/Low RG (sliding renormalization scale).

Everything revolves around whether the cutoff regulates feynman diagrams (e.g., a momentum cutoff) or regulates the theory nonperturbatively (as in a lattice cutoff).

I haven’t looked at the papers you cited, but Reuter et al are using a perturbative cutoff (a higher-covariant derivative regularization, which is the covariant version of a momentum cutoff).

If you know of a covariant, nonperturbative cutoff, people would be most grateful to hear about it.

Posted by: Jacques Distler on April 16, 2006 4:19 PM | Permalink | PGP Sig | Reply to this

### Re: Cutoffs

I’m afraid I still don’t understand. The momentum-dependent term we add here serves as an IR cutoff, not a UV cutoff. It seems like you should be able to UV-regularize however you like and send the UV cutoff to infinity.

The IR cutoff is just the term $\Delta S_k[\phi]$. The “flow equation” is just what you get from taking a derivative with respect to $k$. Since the only term this can affect is the cutoff term $\Delta S_k[\phi]$, it produces a universal equation for how the effective action depends on $k$ and the cutoff. There isn’t any particular physical meaning to $\Gamma_k$ for various $k$, since they are effective actions for other theories; all you care about is that $\Gamma_0$ is the effective action for the theory you’re interested in, and $\Gamma_{\Lambda}$ is the classical action (you can send $\Lambda \rightarrow \infty$). This is by construction of $\Delta S_k$; only the limits of small and large $k$ really matter.

Assuming the nonperturbative dynamics of the theory are captured by the path integral, the only place where I see an assumption entering this derivation is when we assume we can move the derivative with respect to $k$ inside the path integral.

So, to me this seems to be on just as valid a footing as the usual derivation of the Schwinger-Dyson equations (which, I think everyone would agree, are the fully nonperturbative equations of motion) from the path integral.

You claim that the derivation of the flow equation is in fact just valid perturbatively. You seem to have spent more time looking into this than I have, so I think you’re probably right, I just don’t understand why. The derivation looks very simple and slick and I expect it’s hiding something I don’t understand.

I haven’t looked at the formulation in the Reuter papers; section 2 of hep-ph/0005122 gives what seems to me a simple account of this story.

Thanks again!

Posted by: confused on April 16, 2006 7:32 PM | Permalink | Reply to this

### Re: Cutoffs

I have partially de-confused myself, so let me amend what I said above.

These flow equation studies begin with some initial condition $\Gamma_{\Lambda}$ defined at some UV cutoff $\Lambda$. As you point out, defining such a cutoff is nontrivial, and this tends to get glossed over.

Thinking in a Wilsonian way, one has to integrate out modes above $\Lambda$ to define such an object to begin with. Then the flow equation tells you how to integrate out modes below $\Lambda$.

So it seems to me that one misses the effects of integrating out high-momentum modes. For instance, UV renormalons are entirely invisible in this approach. That is bad!

On the other hand, I think this really is a nonperturbative technique for integrating out low-momentum modes, and this is why, for instance, flow equation studies have been able to see chiral symmetry breaking in NJL models and in QCD.

So, aside from the obvious errors associated with the truncation, which are not well-understood, one also has to be careful about errors associated with the choice of initial condition, e.g. UV renormalons. On the other hand, nonperturbative IR effects are potentially approximated well in this framework.

How’s that? Am I still wrong?

Posted by: confused on April 17, 2006 7:46 PM | Permalink | Reply to this

### Re: Cutoffs

It’s taken me some time to digest what they are doing (you did point me to a 179 page paper!). But I think my conclusions are similar to yours. In fact, I would go a little further.

The assumption that the initial condition for their flow equations is given by $\lim_{k^2\to \Lambda^2} \Gamma_k(\phi) = S_\Lambda$ where $S_\Lambda$ is the Wilsonian action at UV cutoff $\Lambda$, is predicated on a certain compatibility between their IR cutoff function and the (implicit, but unspecified) UV cutoff.

If you actually look at a well-defined nonperturbative UV cutoff, such as the lattice (their equation 2.13), this is clearly not the case. With a lattice regulator, momentum space is a torus (of radius $\Lambda$), because the fields are only defined at lattice points (i.e., the transformation from momentum to position space is given by Fourier series, rather than Fourier transform).

Their “IR” cutoff function, $R_k(q^2)$ is a non-periodic function of momentum, $q^\mu$, and hence is not a function on the torus.

Since $R_k(q^2)\to 0$ for $q^2\gg k^2$, this mistake is minor for $k^2\ll \Lambda^2$. We can correct their function to a periodic one. We might even be able to argue that this introduces only minor corrections to their flow equation.

However, allowing $k^2 \to \Lambda^2$, the departure from their assumptions about the IR cutoff function become significant. And I think their arguments go out the window.

So, indeed, for a variety of reasons, one has doubts about the UV behaviour of their flow equations.

That may be OK for their purposes, where the UV theory is known to be under control. But if we are trying to apply this technology to quantum gravity, where we don’t know that that the UV theory is under control (indeed, that’s what we are trying to show), we ought to be very worried.

Posted by: Jacques Distler on April 17, 2006 10:44 PM | Permalink | PGP Sig | Reply to this

### Re: Cutoffs

Thanks for helping to clear this up for me. I am in complete agreement that attempts to apply this to gravity are rather dubious. On the other hand the numerical results obtained along these lines for theories that are known to make sense in the UV seem to be reasonable, so I’m somewhat optimistic that one can make sense of the approximations made in those cases. (I’m only dimly aware of attempts to build nonperturbative regulators that do not rely on the lattice, but it doesn’t seem a priori clear that any nonperturbative regulator will place the same sorts of restrictions on $R_k(q^2)$.)

Thanks again!

Posted by: confused on April 18, 2006 7:57 AM | Permalink | Reply to this