Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

July 28, 2007


At Strings 2007, one of the things I was curious to hear about was what the old-time supergravity experts thought about the conjectures by Zvi Bern et al about the possible finiteness of 𝒩=8\mathcal{N}=8 supergravity (see here for some previous comments on the subject). So I went around surveying their opinions. To a man (or woman), they were united in the opinion that 𝒩=8\mathcal{N}=8 supergravity would diverge. All that they disagreed about was the loop order at which the divergence would first occur.

Some said 5 loops, based on the hypothetical existence of a harmonic superspace formulation for 𝒩=8\mathcal{N}=8 supergravity. The more linearly-realized supersymmetries you have, the more powerful the convergence properties they impose. If a harmonic superspace formulation, with 24 of the 32 supersymmetries linearly realized, were to exists, this would postpone the first divergence to 5 loops.

Renata Kallosh was betting on 8 loops where, many years ago, she constructed an explicitly E 7,7E_{7,7}-invariant counterterm. Others were betting on 9 loops, due to an argument by Berkovits, to do with pure spinors.

All of them, in other words, had their own pet explanation for the improved convergence properties found by Bern et al but, depending on what they believed was responsible, this would only protect you from divergences up to some finite loop order, well beyond what’s been calculated heretofore.

Zvi was having none of this. He insisted that the extra convergence he was finding was a generic property of (even) pure gravity. And, after the conference, he and collaborators put out a paper elaborating on that claim.

A key step in their procedure is to rewrite the loop integration to be done in terms of scalar diagrams. Let l il_i be the loop momenta being integrated over, and assume that 1/l 21/l^2 and 1/(lk) 21/(l-k)^2 are propagators appearing in the integral, where kk is an external momentum, obeying k 2=0k^2=0. The one can write a factor of lkl\cdot k appearing in the numerator as lk=12(l 2(lk) 2) l\cdot k = \frac{1}{2} ( l^2 - (l-k)^2) trading this one diagram for a pair of diagrams where we’ve cancelled one of the propagators.

Consider a graph with NN external legs. At one loop, this receives contributions from nn-gon graphs with nNn\leq N.

In gauge theory, in Feynman gauge, where the interaction vertex has at most 1 derivative, an nn-gon one loop graph has 2n2n powers of the loop momentum in the denominator but only (at most) nn powers in the numerator. So graphs with a larger number of internal legs are necessarily softer in the UV. Carrying out the above trick reduces the nn-gon to an (n1)(n-1)-gon with at most n1n-1 powers of the loop momenta in the numerator. Eventually, you reduce everything to triangle and bubble graphs with the expected 3 and 2 powers of ll in the numerator. (Note that this doesn’t tell you about the actual divergence structure of the amplitude, because there are cancellations between the various scalar graphs that you produce by this procedure.)

In gravity, things work differently. Each vertex carries two derivatives, so an nn-gon one loop graph has terms which carry up to 2n2n powers of the loop momentum in the numerator, along with 2n2n powers in the denominator. So graphs with an arbitrary number of internal lines have the same naïve power counting. The reduction trades an nn-gon with 2n2n powers of ll in the numerator for an (n1)(n-1)-gon with 2n12n-1 powers of ll in the numerator. That is, it seems to make the power counting worse.

According to Bern et al, this is not quite the case: there are cancellations which result in the bubble and triangle graphs having numerators which scale at worst as l 4l^4 and l 6l^6, respectively, independent of the number of external legs.

In 𝒩=8\mathcal{N}=8 supergravity, at one loop, the coefficients of the bubble and triangle graphs vanish. Bern et al hypothesize that this continues to hold at higher loops. The “no triangle hypothesis” is that only boxes (and higher nn-gons) appear in the reduction to scalar loop integrals. This is, apparently, borne out by an explicit calculation of the 3 loop 4-point function.

The aim of their present paper is to separate those cancellations which are “due to supersymmetry” from those that are due to this, somewhat obscure, new mechanism, present even in pure gravity. The claim is that supersymmetric cancelations are responsible for reducing the power of ll in the numerator of the bubble and triangle graphs by a factor of l 𝒩l^{\mathcal{N}} for 𝒩\mathcal{N} even and l 𝒩+1l^{\mathcal{N}+1} for 𝒩\mathcal{N} odd. The rest of the reduction, they say, is due to this new mechanism.

This doesn’t come close to proving the “no triangle hypothesis.” But it does indicate something interesting going on in the structure of gravitational scattering amplitudes.

I’m still betting on my supergravity colleagues, though…

Posted by distler at July 28, 2007 6:32 PM

TrackBack URL for this Entry:

4 Comments & 0 Trackbacks

Re: Cancellations

If someone’s feeling a little evil and sadistic, one could always entask calculating the 4 point, 4 loop amplitude for N=8 SUGRA to a few eager grad students.

Posted by: Haelfix on July 31, 2007 2:04 AM | Permalink | Reply to this

Re: Cancellations

Since it is the first time I post here I think it is apropiate to point that I find your blog very interesting and usefull.

About these particular entry I would like to point you to an article from which I have had notice from a few mounths but I still don´t know what to think about it. The articles is these:

The authro cliams that by a resumation tecnologie, concretely the YSF (Yennie,Fruschi and Saura) one he can prove that conventional perturbative quantum gravity is renormalizable, which, obviously, seems to be somewhat related, in spirit, to your post.

As far as I don´t know anything about that YFS ressummation aproach (and I am busy learning more ortodoux things about QG) I can´t myself to evaluate how reliables the claims of these author are, but maybe you could know it and debunk it (if appropiate).

Posted by: Javier on August 1, 2007 2:17 AM | Permalink | Reply to this

Re: Cancellations

Has anyone a comment on how these cancellations are related to what Dirk Kreimer is talking about in his Remark on Quantum Gravity? (I heard him talk about that, but didn’t really follow the details, here.)

Posted by: Urs Schreiber on August 1, 2007 2:34 PM | Permalink | Reply to this


I haven’t really studied Kreimer’s work, but I don’t see a straightforward connection.

Bern et al are, pointedly, not talking about the overall degree of divergence of particular graphs in pure gravity. These are, as far as I can see, exactly as divergent as expected. Rather, they are talking about the degree of divergence of particular terms, when the calculation is re-expressed in terms of scalar graphs.

More broadly, they do make use of recursion relations between different NN-point tree amplitudes. And, using the cancellations (both the supersymmetric ones, and the mysterious ones being discussed here), they argue that 𝒩=8\mathcal{N}=8 supergravity should be “cut constructible” (which is to say that the loop amplitudes are determined, recursively, from the tree amplitudes).

Posted by: Jacques Distler on August 2, 2007 1:01 PM | Permalink | PGP Sig | Reply to this

Post a New Comment