Musings

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 $l_i$ be the loop momenta being integrated over, and assume that $1/l^2$ and $1/(l-k)^2$ are propagators appearing in the integral, where $k$ is an external momentum, obeying $k^2=0$. The one can write a factor of $l\cdot k$ appearing in the numerator as $$l\cdot k = \frac{1}{2} ( l^2 - (l-k)^2)$$ trading this one diagram for a pair of diagrams where we’ve cancelled one of the propagators.

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

In gauge theory, in Feynman gauge, where the interaction vertex has at most 1 derivative, an $n$-gon one loop graph has $2n$ powers of the loop momentum in the denominator but only (at most) $n$ 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 $n$-gon to an $(n-1)$-gon with at most $n-1$ 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 $l$ 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 $n$-gon one loop graph has terms which carry up to $2n$ powers of the loop momentum in the numerator, along with $2n$ powers in the denominator. So graphs with an arbitrary number of internal lines have the same naïve power counting. The reduction trades an $n$-gon with $2n$ powers of $l$ in the numerator for an $(n-1)$-gon with $2n-1$ powers of $l$ 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 $l^4$ and $l^6$, respectively, independent of the number of external legs.

In $\mathcal{N}=8$ 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 $n$-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 $l$ in the numerator of the bubble and triangle graphs by a factor of $l^{\mathcal{N}}$ for $\mathcal{N}$ even and $l^{\mathcal{N}+1}$ for $\mathcal{N}$ 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…