Asymptotic Safety

January 30, 2008

Asymptotic Safety

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Once upon a time, I wrote a blog post about the proposal by Reuter and various collaborators that quantum gravity in four dimension is controlled by a UV fixed point. Below some cutoff (at the Planck scale, if not below it), gravity is described by an effective theory. This effective description breaks down at the cutoff scale, and the theory is ill-defined unless one of two things happens

New degrees of freedom enter (as in String Theory).
The UV physics is controlled by a fixed point, so the apparently infinite number of coupling are actually not independent, but rather lie on an (IR-repulsive) trajectory emanating from the fixed point set.

The latter is known to occur for pure gravity in $2 + ϵ$ dimensions. The hope is that the same holds true in 4 dimensions. The technique used to study this is the so-called Exact Renormalization Group, truncated to some finite-dimensional space of couplings. Reuter’s original work truncated to a 2-dimensional subspace, consisting of the cosmological constant and Einstein-Hilbert term. The existence of a fixed point in that (brutal) truncation was not terribly convincing.

Since I wrote my post, there have been many followup papers, by various authors (see the recent review by Percacci), purporting to adduce further evidence, by including various other couplings, and checking to see whether the fixed point persists. For instance, this paper considers adding a polynomial (up to 6th order) in the curvature scalar. Reuter considered adding the square of the Weyl tensor.

The trouble with all of these papers is that they really don’t address the issue in a meaningful way.

The terms considered vanish on-shell (in flat space) and in conventional perturbation theory, any divergence in these terms can be absorbed by a field redefinition. The ${(Weyl)}^{2}$ term is a slight exception. But it can be rewritten as the Gauss-Bonnet density plus terms which vanish on-shell. The Gauss-Bonnet density, being a topological invariant, receives no corrections.

The first term which can receive a nontrivial renormalization in pure gravity, and hence which would actually serve as an acute test of whether the fixed point really exists, is cubic in the Riemann tensor. Goroff and Sagnotti did the perturbative computation to show that it, in fact, received a log-divergent correction at 2-loops. This is the first divergence in pure gravity; the 1-loop divergences can be absorbed by field redefinitions, by the argument of the previous paragraph.

So the first nontrivial test of the asymptotic safety proposal will come when someone computes the ERGE for

S = \int d^{4} x \sqrt{- g} (M^{4} c_{1} + \frac{M^{2}}{c_{2}} R + \frac{c_{3}}{M^{2}} R_{μ}_{ν}^{α}^{β} R_{α}_{β}^{ρ}^{σ} R_{ρ}_{σ}^{μ}^{ν})

Now, I’ve thought about doing this computation myself. But

Goroff and Sagnotti’s computation was hard. And using the ERGE approach can’t make it any easier.
It’s a pretty foregone conclusion what the result will be: there is no fixed point for any finite value of $c_{3}$ .

So maybe I should throw this out there for the readers of this blog. Anyone want to attain fame and fortune by performing the first nontrivial test of the gravitational asymptotic safety hypothesis?

Posted by distler at January 30, 2008 10:43 AM

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29 Comments & 2 Trackbacks

Re: Asymptotic Safety

Yes, I want to do this. Even if it is a foregone conclusion, it needs to be checked.
It may take me some time though, as I will have to go over the papers first and review renormalization group flow.

p.s
I am a high energy theory graduate student looking for a real problem to work on.

Posted by: Metal on January 30, 2008 8:19 PM | Permalink | Reply to this

Read the post Opera and MathML
Weblog: Musings
Excerpt: A rant.
Tracked: January 31, 2008 12:01 AM

Re: Asymptotic Safety

Anybody contemplating this calculation might actually not want to do it in the Goroff and Sagnotti brute force way but rather using the covariant methods (heat-kernel etc) of

Two loop quantum gravity.
A.E.M. van de Ven (Hamburg U. and SUNY, Stony Brook) . DESY-91-115, ITP-SB-91-52, Oct 1991. 51pp.
Published in Nucl.Phys.B378:309-366,1992.

SPIRES HEP link

Posted by: Robert on January 31, 2008 6:06 AM | Permalink | Reply to this

Re: Asymptotic Safety

Dear Jacques,

I have some questions about the Reuter program which I cannot resist asking you.

First of all, from reading your blog, here is what I think I understand. Please correct me if I am wrong.

From your older post AND the discussions in the comment section, it seems to me that Reuter et al’s renormalization group is *almost* non-perturbative. By *almost* I mean that the formulation seems non-perturbative enough, except that one needs to specify initial conditions for the flow in the UV which can (implicitly) involve the choice of a perturbative cutoff. Especially because (as you pointed out), their IR cutoff function might NOT be compatible with the use of a lattice regularization in the UV. But if one assumes the best-case scenario for the Reuter program (which is what I think you are doing in your latest post), am I wrong in thinking that there might exist *some* non-perturbative regularization scheme which might be compatible with their choice of the IR cutoff function?

My point is that if one does not believe in the non-perturbative validity of the RG evolution equation, what is the sense in talking about including the Riemann cubed coupling? Even if one ends up finding that there is a non-trivial fixed point, would it be anything more than a curiosity?

One more question: Apologies in advance if I am being naive, but even before we start with this exact renormalization group business, how can a theory with black holes at high energies[1] look like a CFT? Is there any motivation to believe that there are no black holes in strongly coupled, asymptotically flat, gravity? (Asymptotically flat, because I believe in the existence of black holes in AdS because of Hawking-Page vs. Confinement-Deconfinement in AdS/CFT.).

[1] That one can produce black holes in high energy scattering is something I have heard many times, especially from Willy, and it has always seemed reasonable to me.

Hope everybody on 9th floor is doing fine,
Chethan.

Posted by: Chethan Krishnan on January 31, 2008 9:25 AM | Permalink | Reply to this

Re: Asymptotic Safety

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But if one assumes the best-case scenario for the Reuter program (which is what I think you are doing in your latest post), am I wrong in thinking that there might exist some non-perturbative regularization scheme which might be compatible with their choice of the IR cutoff function?

That, indeed, is a very dubious point.

To get the initial condition for the RG flow, one needs to assume a compatibility between the IR cutoff (which they make explicit) and the UV cutoff (which they don’t specify). It’s clear that one can impose a perturbative UV cutoff, compatible with their IR cutoff. But if you do that, then the realm of validity of the “Exact” RGE is to sum up all orders in perturbation theory; it’s not nonperturbative. Conversely, it’s hard to imagine a nonperturbative UV cutoff which would be compatible.

I am willing to suspend disbelief on this point, and see what the calculation yields.

My point is that if one does not believe in the non-perturbative validity of the RG evolution equation, what is the sense in talking about including the Riemann cubed coupling? Even if one ends up finding that there is a non-trivial fixed point, would it be anything more than a curiosity?

Of course, there are an infinite number of coupling which receive nontrivial additive renormalizations in pure gravity. It is very very dubious that these additive renormalization all vanish simultaneously at some point in coupling-constant space.

My point is that Reuter and company have assiduously avoided including any of these coupling in their ansatz. So they really haven’t performed a nontrivial test of the hypothesis. Including this ${Riemann}^{3}$ coupling would be the first nontrivial test.

how can a theory with black holes at high energies[1] look like a CFT? Is there any motivation to believe that there are no black holes in strongly coupled, asymptotically flat, gravity?

I don’t think that’s even a question we can ask here. Recall that the fixed point is purportedly at some positive value of the cosmological constant. I assume that we are supposed to be in the part of the phase diagram where the flos stays in the regime of positive cosmological constant. (There’s the troubling matter that $Λ$ seems to flow to $∞$ as one goes to the infrared, but whatever …)

In any case, the existence of a “quantum” conformal symmetry in quantum gravity is compatible with there being a nontrivial dimensionful scale in the theory, so I don’t see a-priori why it’s incompatible with blackholes.

The bigger headache is that, since we’re definitely in de Sitter space, it’s not clear what the observables are supposed to be, and hence what any of this means.

Posted by: Jacques Distler on January 31, 2008 1:30 PM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety

Dear Jacques,

About my Q#1: Thanks for answering precisely my question.

About my Q #2: Oops, I mis-spoke. Yes, I should have said asymptotically de Sitter, not asymptotically flat. What I meant was essentially only that it is *not* asymptotically anti de Sitter.

The reason I brought up black holes was because if we are confident that they dominate high energies, you might expect an area-like entropy as opposed to a volume-like entropy which is what one would expect if the theory was *still* a QFT even at high energies. I have seen variations of arguments of this sort in mnay places I think, one specific place I just looked up is hep-th/9812237.

Chethan.

Posted by: Chethan on January 31, 2008 2:28 PM | Permalink | Reply to this

Re: Asymptotic Safety

The reason I brought up black holes was because if we are confident that they dominate high energies, you might expect an area-like entropy as opposed to a volume-like entropy which is what one would expect

There are funny statements in Reuter et al about a reduction in the effective dimensionality of spacetime as one approaches the fixed point, which may accord with what you are saying.

But this is all handwaving. I’d like a clean gauge-invariant statement, which is something hard-to-come-by in de Sitter spacetimes.

What’s even less believable is the other side of the critical RG trajectory, where the running cosmological constant flips sign as you flow to the IR.

I’m pretty sure I know what quantum gravity is in anti-de Sitter spacetimes. If pure gravity exists in 3+1 AdS (I have reasons to believe it does not), then I’m hard-pressed to imagine how it could be described by this asymptotic safety hypothesis.

(I presume, for this reason, that those who believe in asymptotic safety don’t believe in AdS/CFT.)

Posted by: Jacques Distler on January 31, 2008 6:51 PM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety

One would think they could perform a proof of concept by using a more well known nonrenormalizable field theory that has a known nontrivial fixed point (say derived from a lattice analysis), and then applying the same procedure (truncation + Polchinksi’s ERG) and analyzing the strong coupling regime.

2+epsilon gravity doesn’t suffice since it seems to me the fixed point shows up at weak coupling.

Off the top of my head, I can’t think of a good candidate, but perhaps some exist in condensed matter or somesuch.

Posted by: Haelfix on February 2, 2008 2:01 AM | Permalink | Reply to this

Re: Asymptotic Safety

The Exact RG has been very successful at finding nonperturbative fixed points, particularly in scalar field theory. There is a nice review by Bagnuls and Bervillier, hep-th/0002034, which includes a discussion of how to find the Wilson-Fisher fixed point in three dimensions.

The most successful truncation scheme for doing such studies (in scalar field theory) is the derivative expansion, whereby the momentum dependence of the vertices is truncated, but an infinite number of interactions are retained. This is in contrast to an expansion in powers of the field, where the full momentum dependence of some finite number of vertices is kept. Field expansions are, in general, not reliable and can lead to the `discovery’ of spurious fixed points. This is not surprising as the approximation only makes sense if the field is not fluctuating very much, and one might reasonably expect the opposite to be true in the nonperturbative regime of interest.

The derivative expansion, on the other hand, is generally qualitatively reliable and quantitatively reasonable, even at lowest order [the Local Potential Approximation (LPA)]. From a computational point of view, retaining an infinite number of interactions is a less brutal way of treating the non-linear ERG equation than the alternative.

Regarding the approach of Reuter et al., where the truncation is severe, one must be very wary of the possibility that the claimed fixed point is an artefact of the approximation scheme. That said, it’s existence does seem to have a certain stability, as discussed by the authors (though the issue of gauge invariance is perhaps down-played), and is surely intriguing.

Posted by: Oliver on February 2, 2008 11:58 AM | Permalink | Reply to this

Re: Asymptotic Safety

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Adding terms like a polynomial in the curvature scalar is not a robust test of anything, since such terms can be eliminated by a field redefinition.

The first robust test will be to add a term that is

not a topological invariant,
cannot be eliminated by a local field redefintion,
receives a nontrivial renormalization in perturbation theory.

So far, all of the papers seem to have carefully avoided including such terms.

If the fixed point were to persist in the presence of such terms, that would, indeed, be intriguing.

Posted by: Jacques Distler on February 2, 2008 1:41 PM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety

Dear Jacques

thank you for your thoughts and comments.

> So far, all of the papers seem to have carefully avoided including such terms.

Well, people do what they can. It seems reasonable to start from the simplest truncation
and then progressively add more complicated terms.
Be assured that nobody has purposefully avoided the Goroff Sagnotti term.
Our reason for not having done it is the same as yours - technical complexity.

An easier calculation that would achieve the same goal would be to consider gravity with
curvature squared terms coupled to matter.
As discussed in ‘t Hooft and Veltman’s classic paper, in this theory the (one loop)
curvature squared divergences cannot be eliminated by field redefinitions.

But let me try to understand this business of field redefinitions better.
I thought that the argument for looking only at the Riemann cube (or better Weyl cube)
term holds only in a perturbative setting, where you take the Hilbert action
and treat everything else as an infinitesimal perturbation.
The terms that can be eliminated by field redefinitions are those that vanish on shell,
and the equations that are used in this argument are Einstein’s equations in vacuum.
So, perturbatively, everything that contains the Ricci tensor can be eliminated up to
terms of higher order.

If you want to do a nonperturbative calculation and treat the other terms on the same
footing as the Hilbert term, you would have to do this test using the full field equations.
It gets very complicated very quickly.
Are you saying that all the terms that can be eliminated using the perturbative argument
can also be eliminated in the nonperturbative sense?
I am aware of the transformation that will rid us of the R-squared and Ricci-squared terms
(though at a cost - see below) but what about the Ricci-squared-Weyl and Ricci-Weyl-squared?

> Adding terms like a polynomial in the curvature scalar is not a robust test of anything,
> since such terms can be eliminated by a field redefinition.

This is indeed another example where there is an explicitly known finite (as opposed to
infinitesimal) field redefinition that can be used to “eliminate” some terms, in this
case all higher powers of R.
But in doing this you generate infinitely many new terms in a scalar potential and you have
not actually reduced the number of couplings.
You have just converted one theory into an equivalent one.
And since there is no a priori argument that in this equivalent formulation there must be a
nontrivial fixed point, finding one in an f(R) theory is definitely not an empty statement.

Some of the other issues that were raised here are discussed in a faq page I have set up at
www.percacci.it/roberto/physics/as/

Best regards

Roberto

Posted by: Roberto Percacci on February 4, 2008 12:28 PM | Permalink | Reply to this

Re: Asymptotic Safety

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An easier calculation that would achieve the same goal would be to consider gravity with curvature squared terms coupled to matter. As discussed in ‘t Hooft and Veltman’s classic paper, in this theory the (one loop) curvature squared divergences cannot be eliminated by field redefinitions.

Yes, that would also serve as a good test, and would probably be easier than including the Goroff-Sagnotti term in pure gravity.

The terms that can be eliminated by field redefinitions are those that vanish on shell, and the equations that are used in this argument are Einstein’s equations in vacuum. So, perturbatively, everything that contains the Ricci tensor can be eliminated up to terms of higher order.

If you want to do a nonperturbative calculation and treat the other terms on the same footing as the Hilbert term, you would have to do this test using the full field equations.

It gets very complicated very quickly.

The equations of motion for $f (R)$ gravity are more complicated. But they still admit Einstein spaces ( $R_{μ ν} \propto g_{μ ν}$ ) as solutions. And, indeed, you expand about an appropriate Einstein space in your ‘nonperturbative’ treatment.

By a field redefinition, you can alter the coefficients of $(R - c)^{n}$ , just as you could in the ‘perturbative’ case.

Perhaps I’m missing something …

But in doing this you generate infinitely many new terms in a scalar potential and you have not actually reduced the number of couplings. You have just converted one theory into an equivalent one.

Could you elaborate (or give a reference)? I’m not sure what “scalar potential” we’re talking about, here.

Posted by: Jacques Distler on February 4, 2008 3:36 PM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety

> The equations of motion for f(R) gravity are more complicated. But they still admit Einstein spaces as
>solutions. And, indeed, you expand about an appropriate Einstein space in your ‘nonperturbative’ treatment.

When you say that something vanishes on shell it means that it vanishes
for every solution of the field equations, not just some solution.

The use of a particular background to compute some beta function is
a mathematical trick to extract specific terms from a functional trace.
The resulting beta functions would come out the same for any background.

Anyway, the issue here is not how we compute the beta function,
but whether, or how, one can eliminate certain terms from the action.

> Could you elaborate (or give a reference)? I’m not sure what “scalar potential” we’re talking about, here.

I am referring to the fact that a theory with Lagrangian f(R) is equivalent
via field redefinitions to a metric-scalar theory where the action contains
the Hilbert term, a canonical scalar kinetic term and a scalar potential.
See for example equations 3-6 of astro-ph/0307338.
For the R-squared and Ricci-squared terms see hep-th/9601082.

Posted by: Roberto Percacci on February 5, 2008 10:21 AM | Permalink | Reply to this

Re: Asymptotic Safety

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This may be a bit tangential, but I’m extremely wary of the “equivalence” between f(R) theories and tensor-scalar theories. Sure, they can be thought of as equivalent at the level of equations of motion, but generically you’re talking about two different boundary value problems (or, equivalently, initial value problems).

If I consider a gravity theory with Lagrangian $R + R^{2}$ , there simply isn’t a well defined BV problem where only the metric is held fixed at the boundary. But in the scalar-tensor theory one appears to have a BV problem where a metric (and not its normal derivative) is fixed at the boundary, and some condition is also put on the scalar.

Do these two theories really end up being equivalent, and not just at the level of the equations of motion? I guess I should look at the paper claiming equivalence at the level of the path integral.

Posted by: Robert McNees on February 5, 2008 12:31 PM | Permalink | Reply to this

Re: Asymptotic Safety

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When you say that something vanishes on shell it means that it vanishes for every solution of the field equations, not just some solution.

Of course.

I’m certainly not claiming that $f (R)$ -gravity is equivalent to Einstein-Hilbert.

What I am saying (which I believe is correct) is that, when you compute the ERG functional $β$ -function for $f (R)$ -gravity, you expand in fluctuations about an Einstein space, and the computation is effectively the same as the computation in Einstein-Hilbert, with a shifted value for the effective $Λ$ and $G_{N}$ .

So, if you had found a zero of the $β$ -function before, you expect it to persist in $f (R)$ -gravity.

Posted by: Jacques Distler on February 5, 2008 1:51 PM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety

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I wrote:

… the computation is effectively the same as the computation in Einstein-Hilbert, with a shifted value for the effective $Λ$ and $G_{N}$ .

So, if you had found a zero of the $β$ -function before, you expect it to persist in $f (R)$ -gravity.

I should amend that. $R^{2}$ is an exception. That requires a separate calculation. If, however, you find a fixed point for

S = \int d^{4} x \sqrt{- g} [g_{0} M^{4} + g_{1} M^{2} R + g_{2} R^{2}]

you are guaranteed to find a fixed point in the theory where you add additional polynomial in $R^{n}$ , $n \geq 3$ .

Posted by: Jacques Distler on February 7, 2008 2:00 AM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety

Jacques, I am not sure that I understand what you are saying. We agree that f(R) gravity cannot be reduced to Einstein-Hilbert: a generic change in the higher couplings cannot be compensated by a field redefinition.

But then you propose to exploit the special properties of the background on which we do our calculation. Fair enough. We do our calculations on spheres. On a sphere the variation of the action of f(R) gravity is proportional to (2f(R)-Rf’(R)) times the trace of the variation of the metric. If we want to remain within the chosen class of backgrounds the only possible variations of the metric are global rescalings, such that the trace of the variation of the metric is a constant. By writing f as a Taylor series one can easily see that using such a variation of the metric you can compensate a change in any given coupling, except for the coefficient of R squared.

I assume that you must have been reasoning along similar lines. But here I lose you. Such global rescalings are only a one parameter group: you must change all the couplings together in a specific way. As a result, even on a sphere, you can use them to fix the value of only one coupling, not infinitely many. So maybe you had something else in mind?

Posted by: Roberto Percacci on February 7, 2008 12:29 PM | Permalink | Reply to this

Re: Asymptotic Safety

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But here I lose you. Such global rescalings are only a one parameter group: you must change all the couplings together in a specific way. As a result, even on a sphere, you can use them to fix the value of only one coupling, not infinitely many. So maybe you had something else in mind?

I am making a rather mundane observation.

The crucial ingredient in the ERGE involves expanding the effective action to quadratic order in fluctuations about your chosen background. Since, for an Einstein space, $R = const$ , this inverse propagator, for $S =$ an $n^{th}$ order polynomial in $R$ , has the same functional form as the one constructed for $S$ a quadratic polynomial in $R$ (but with shifted coefficients).

Posted by: Jacques Distler on February 8, 2008 1:09 AM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety

True but not enough to draw your conclusion. In the calculation of the beta functions you need to keep track of the functional dependence of the inverse propagator on R, which is treated as an external parameter. There is no way that you can obtain with an R-squared truncation the same R-dependence that you have with higher truncations, irrespective of any shift in the couplings.

Consider a similar issue in the more familiar context of a scalar field theory with a generic even potential V(ψ^2). The inverse propagator has the form

-∂∂+2V’+4ψ^2V”

If you truncate the potential at order ψ^4, in d=3 the ERGE will give you (an approximation to) the Wilson Fisher fixed point. This is rather crude, and, as Oliver was saying, a lot of work has gone into better truncations.

Translated into this context, your “mundane observation” is: for constant scalar background, the form of the inverse propagator for any potential can be reproduced by that of a quartic potential, for some suitable couplings. This is true but not very useful. What matters is the functional dependence on the scalar. A higher polynomial potential will give rise to higher powers of ψ in the inverse propagator and such information is essential in extracting the beta functions of the higher couplings. You certainly cannot conclude that “if there is a fixed point for the quartic potential there must be a fixed point for any polynomial potential”.

Posted by: Roberto Percacci on February 8, 2008 12:05 PM | Permalink | Reply to this

Re: Asymptotic Safety

$MathML-enabled post (click for more details).$

A higher polynomial potential will give rise to higher powers of $ψ$ in the inverse propagator and such information is essential in extracting the beta functions of the higher couplings.

Of course you need to compute the $β$ -functions for the higher couplings. But in $d = 3$ , we know what the result will be: they’re all irrelevant.

(For those unfamiliar with this story, $d = 3$ scalar field theory (on which we impose a $ℤ_{2} : ϕ \to - ϕ$ symmetry) has a Gaussian fixed point, at which $ϕ^{2}$ and $ϕ^{4}$ are relevant, and $ϕ^{6}$ is marginally irrelevant. If you perturb away¹ from the Gaussian fixed point, you can flow to another fixed point, the Wilson-Fisher fixed point, which has only a single relevant direction.)

You certainly cannot conclude that “if there is a fixed point for the quartic potential there must be a fixed point for any polynomial potential”.

But you can. Adding $ϕ^{6}$ or $ϕ^{8}$ , etc, does not alter the basic structure of the RG flow (but see below).

And much was known about the Wilson-Fischer fixed point before the ERGE techniques came along.

This is rather crude, and, as Oliver was saying, a lot of work has gone into better truncations.

Now, it’s true that conventional perturbative techniques do better at computing the critical exponents at the Wilson-Fischer fixed point if you include also a $ϕ^{6}$ term. The reason (at least, as I have understood it) is that $ϕ^{6}$ is marginally irrelevant, and so flows to zero rather slowly.

But the ERGE calculations do just fine in the truncation where you omit all of the higher-order polynomials. Most of the improvements (or so is my impression) come from things like optimizing the cutoff function used, rather than from adding more terms to the effective action.

¹ To reach the W-F fixed point, you want to deform in the direction of

m^{2} < 0

and

λ_{ϕ^{4}} > 0

. And, to complicate matters, you need to tune a parameter to suppress the direction that’s irrelevant at Wilson-Fisher.

Posted by: Jacques Distler on February 9, 2008 10:42 PM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety

Hi,

Roberto is absolutely correct when he says that

You certainly cannot conclude that “if there is a fixed point for the quartic potential there must be a fixed point for any polynomial potential”.

It does, just so happen, that a truncation based on a field expansion correctly identifies the Wilson-Fisher fixed point but this is, in some sense, a fluke. The series for the critical exponents are not convergent, and a spurious fixed point can be found.

I also think that the irrelevance of the 6pt, 8pt vertices etc. can be a bit of a red-herring. This statement is true in the vicinity of the Gaussian fixed point and it allows one to draw a very powerful conclusion: namely, that all scale dependence of the action along a Renormalized Trajectory (RT) emanating from the Gaussian fixed point is carried by the two point coupling, m, the four point coupling, λ, and the anomalous dimension, γ (I’m assuming that all quantities have been rescaled to dimensionless, using the effective scale). However, this does not mean that higher point couplings are not generated along the flow, nor that these couplings won’t become important at some scale. All that one can say is that (remarkably) all of these couplings can be written in the form f(m,λ,γ).

Even in the vicinity of the Gaussian fixed point, the determination of the exact scale dependence of the Wilsonian effective action is a nonperturbative problem, amounting to computing the `perfect action’. Whilst a perturbative calculation will give an excellent approximation, in this regime, there is generally no reason to trust it, away from here.

Adding ϕ^6 or ϕ^8, etc, does not alter the basic structure of the RG flow (but see below).

If one were really able to compute the action exactly, in the Wilsonian framework, it wouldn’t make sense to talk about adding a coupling to the action. The action along an RT is determined, in principle, by the flow equation, given

The choice of fixed point;
The integration constants associated with the relevant / marginally relevant directions.

As it is, adding couplings to an action, in this context, implicitly implies a truncation and such a procedure could well have a profound effect on the (approximation to the) flow.

Posted by: Oliver Rosten on February 10, 2008 3:51 PM | Permalink | Reply to this

Re: Asymptotic Safety

Is there any analytic statement about the bound on the error as you move away from your fixed point whilst adding different couplings? I mean,I would think the divergence structure would manifest itself pretty clearly.

Posted by: Haelfix on February 10, 2008 4:50 PM | Permalink | Reply to this

Re: Asymptotic Safety

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The series for the critical exponents are not convergent …

Perturbation theory is never convergent.

Last I checked, the most accurate determination of the critical exponents at the Wilson-Fisher fixed point came from doing a conventional multiloop computation of the anomalous dimensions (ie., working about the Gaussian fixed point), and then using Padé approximants to extrapolate the results.

and a spurious fixed point can be found.

This whole discussion has been about the fact that truncations of the ERGE can lead to the appearance of spurious fixed points.

I have argued that the fixed point found in the ERGE analysis of pure gravity is probably spurious, and suggested the operator(s) which should be added to the truncation to test this.

I did explain why I think that adding $R^{n}$ couplings to the truncation does not provide an acute test of the persistence of the fixed point.

Roberto was the one who suggested the similarity to adding $ϕ^{2 n}$ terms to the truncation of the ERGE in $d = 3$ scalar field theory.

In the latter case, there are lots of ways to argue that adding $ϕ^{2 n}$ terms to the truncation will not affect the W-F fixed point.

You seem to want to argue that none of these arguments are valid because all kinds ‘o crazy stuff can happen along RG trajectories. I would respond that, if you believe the behaviour to be sufficiently wild, then none of the usual families of truncations (the field expansion, the derivative expansion) can be expected to be valid. And no amount of “evidence” amassed in the context of one of those families of truncations can be persuasive.

Posted by: Jacques Distler on February 11, 2008 12:17 AM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety

Even if you did use the Goroff and Sagnatti coupling and found a fixed point, couldn’t you just argue the same thing and say it could just be a spurious point merely an artifact of the approximation scheme? It is after all just the first of an infinite amount of bad terms (never mind the large N additivity divergences)

So back to the premise, under what circumstances (divergence structure) of a field theory can the truncation +ERGE be expected to provide an adequate flow sufficiently accurate enough so thta people can trust the fixed point? Ultimately it should be a statement about the error bound I would think

Posted by: Haelfix on February 11, 2008 3:55 AM | Permalink | Reply to this

Re: Asymptotic Safety

Haelfix: I am inclined to believe that if a fixed point exists there should be a way of understanding it. But I think the question you raise is premature. Things may well get stuck at the next test, so for the time being we should just keep humbly collecting evidence.
Concerning analytic bounds on errors, I am not aware of any such bounds that could be applied here, but maybe other people have more to say on this.

Jacques: Please, do not push my Wilson-Fisher analogy beyond its narrow limits. My statement was referring only to the inverse propagator and the beta functions that come from it via the ERGE. In the scalar case in d=3 there may be other arguments that allow you to say that once you have a fixed point for a quartic potential there should also be one for a higher polynomial, but such arguments do not exist in the case of gravity. So there is no shortcut to doing the actual calculation of the beta functions and looking for their zeroes. In this sense, I repeat, testing polynomials in R is nontrivial.

Whether you consider the result of such calculation a “robust” or “acute” test is a matter of taste and semantics. On february 2nd you gave a precise definition of what you mean by a “robust” test. I am happy to agree with you that our calculation of polynomials in R was not a “robust” test in your sense, because it does not meet your third criterion.

Ultimately I think we agree on everything and our difference lies just in how much weight we are willing to give to perturbative evidence. You seem to think that the ERGE approach will stumble in the same place where perturbation theory did. I do not say that this is unreasonable. After all most of the solid understanding we have of the world comes from perturbation theory. On the other hand, from whatever little experience I have of playing with the ERGE, I see no reason to expect that the Weyl cube term should play any special role. I may be wrong of course. Hopefully time will tell.

Since I think that I made my points clear and that there cannot be much gain in repeating myself, I am probably not going to make other entries here, at least not until new facts emerge. For the time being let me say that I have found this blog an instructive experience.

Posted by: Roberto Percacci on February 11, 2008 5:43 PM | Permalink | Reply to this

Re: Asymptotic Safety

Since I think that I made my points clear and that there cannot be much gain in repeating myself, I am probably not going to make other entries here, at least not until new facts emerge. For the time being let me say that I have found this blog an instructive experience.

Well, I’d like to thank you for stopping by, and taking the time to comment. I found this discussion very helpful. And I’m sure many of my readers did, as well.

Posted by: Jacques Distler on February 11, 2008 5:56 PM | Permalink | PGP Sig | Reply to this

Re: Asymptotic Safety

Yes, thank you for clearing things up, I’ve learned quite a bit and I think I understand you’re position better as a result.

Cheers

Posted by: Haelfix on February 11, 2008 6:04 PM | Permalink | Reply to this

Field redefinitions

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Let us take an action $S [ψ]$ with a coupling $c$ . If we vary the coupling constant $c$ , then the variation of the action is $δ c \frac{\partial S [ψ]}{\partial c}$ . If this variation can be undone by a field redefinition $ψ \to ψ + δ ψ$ , the we must have

(1)

δ c \frac{\partial S [ψ]}{\partial c} = \int d^{4} x \frac{δ S [ψ]}{δ ψ (x)} δ ψ (x) .

Now, in the case of gravitation,

(2)

S = \frac{1}{16 π G} \int d^{4} x \sqrt{- g} (f (R) - 2 Λ) .

Take $f (R) = R + c R^{2}$ , for example. In order to be able to undo the constant $c$ above, we should, if I’m not mistaken, find a transformation for the metric, $g_{μ ν} \to g_{μ ν} + δ α_{μ ν}$ such that

(3)

\int d^{4} x \sqrt{- g} [- δ c R^{2} + (1 + 2 c R) R_{μ ν} δ α^{μ ν} - \frac{1}{2} (R + g R^{2} - Λ) g_{μ ν} δ α^{μ ν}] = 0 .

Now, I haven’t tried very hard, but I couldn’t find a solution for $δ α^{μ ν}$ . I have’t looked at the ${Weyl}^{2}$ ter, yet.

Posted by: Sidious Lord on February 5, 2008 2:53 PM | Permalink | Reply to this

Re: Field redefinitions

> Now, I haven’t tried very hard, but I couldn’t find a solution for δα μν.

Before you try harder: in the variation of R squared you are forgetting a term 2cR(trace δ Ricci). (It’s not a total derivative.)

For someone who is looking for a fixed point, the fewer couplings one has to consider the better. So I would be delighted if somebody told me that c can be eliminated in this way but I don’t think it’s possible.

Posted by: Roberto Percacci on February 6, 2008 4:02 AM | Permalink | Reply to this

Re: Field redefinitions

$MathML-enabled post (click for more details).$

Before you try harder: in the variation of R squared you are forgetting a term 2cR(trace δ Ricci). (It’s not a total derivative.)

I agree. In the Einstein-Hilbert action variation there also is a term $g^{μ ν} δ R_{μ ν}$ , but this is a divergence and doesn’t contribute to the equations of motion.

However, in this case it appears multiplying a curvature scalar, and the result is not a divergence anymore. Thank you for this correction.

So the correct equation (correcting another typo where I wrote $g$ instead of $c$ ) is

(1)

\int d^{4} x \sqrt{- g} [- δ c R^{2} + (1 + 2 c R) R_{μ ν} δ α^{μ ν} + 2 c R g^{μ ν} δ R_{μ ν} - \frac{1}{2} (R + c R^{2} - Λ) g_{μ ν} δ α^{μ ν}] = 0 .

Now, the variation $δ R_{μ ν}$ will involve the Riemann tensor, so things will start to look a bit messy.

Posted by: Lord Sidious on February 6, 2008 12:29 PM | Permalink | Reply to this

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January 30, 2008