Compact formula for all tree \(\NNN=8\) SUGRA amplitudes

Update: A few weeks later, a proof of the formula explained below was posted on the arXiv.

Even though many of us expected the Higgs discovery today, it's been a very intense day and it's not yet over.



IAS director and my ex-co-author Robbert Dijkgraaf starts a Higgs celebration at Princeton. The champagne was paid for by Nima Arkani-Hamed. ;-) Via Graham Farmello

I was told that 30 people at the Institute for Advanced Studies in Princeton, New Jersey gathered in the rooms previously occupied by Albert Einstein and similar colleagues and started to watch the CERN seminars at 3 am East Coast Time. (Some of them may have attended because of the champagne only, however.) I know that many of you stayed up, too. But it's here: a new particle that so far exactly matches all the predictions for the Standard Model Higgs boson was discovered at 5 sigma by ATLAS and independently at additional 5 sigma by CMS.

I had to do lots of computer-related things linked to the Higgs (including e-mails with IBANs needed for a bet I have just won) and the same is true for many of you but it doesn't mean that nothing else happened on the 2012 Independence Day. Congratulations to the U.S. readers, by the way. (I left the job to create a more detailed congratulatory message to the Americans to my president who's paid for such things.) Advanced formal theory hasn't been sleeping at all, either.

Freddy Cachazo and David Skinner of the Perimeter Institute (I know Freddy from Harvard) just published a new remarkable paper claiming to have a compact formula for all tree-level amplitudes in the maximally supersymmetric, \(\NNN=8\) supergravity in four dimensions, a descendant of the eleven-dimensional supergravity.




It's "only" the tree-level amplitudes – those encoding the evolution of waves in the classical supergravity (no loops: that's where the SUGRA theories become ill, anyway) – but you should still try to appreciate how shocking it is to have a probably correct compact formula for the full result. The supergravity theory has lots of fields and they interact via lots of interactions. In fact, the Lagrangian is non-polynomial and the expression \(\sqrt{-g}\), the square root of the determinant of the metric (which is really a rather complicated object if you think about it), is really just the simplest factor in all the terms.

Now, you must take this Lagrangian that may be expanded to infinitely many terms that are polynomial in many different types of fields. And you must draw all possible tree-level diagrams that connect \(n\) external lines (an arbitrary number of them!) representing the particles we scatter. There are many ways how to draw such trees – we may merge some of the pairs of external lines before others – and there are many types of vertices we may use. We finally sum lots diagrams, translate each propagator in each diagram to the complicated structures (you know what a propagator is and how tough it becomes for bosons with gauge invariances), and then you try to sum up everything to announce the result.

The result can't possibly have a simple form, can it? Well, Cachazo and Skinner just presented their new paper

Gravity from Rational Curves

where they argue that the scattering amplitude for \(n\) particles in the maximally supersymmetric supergravity is actually given by this compact formula:\[

\eq{
{\mathcal M}_n &= \sum_{d=0}^{\infty}\int \frac{
\prod_{a=0}^d \dd^{4|8} {\mathcal Z}_a
}{{\rm vol}({\rm GL}(2,\CC))} \det'(\Phi) \det'(\tilde\Phi)\times\\
&\times \prod_{i=1}^n \dd^2 \sigma_i \delta^2 (\lambda_i-\lambda(\sigma_i))
\exp [\![ \mu(\sigma_i)\tilde\lambda_i ]\!]
}

\] Wow, even the first draft of this equation I reproduced in \(\LaTeX\) seems to match the original with no errors so it must be right. ;-)

The integral goes over all degree \(d\) curves in a twistor space with 4 bosonic and 8 fermionic coordinates and one integrates over a "world sheet" associated with each external particle. Some of the symbols, like the objects in the determinant, need some extra explanations and depend on various twistor-related products and I don't want to copy the whole paper.

But in principle, everything may be explained and the formula could be expanded to an expression that only uses symbols you have known before. It's a simple sum over one integer of a product over \(d\) and/or \(n\) expressions and the determinants are really the most complicated objects in the expression. When the integrals are actually evaluated, everything boils down to some residues and rational functions and you reproduce the many diverse Feynman diagrams with their propagators, it seems.

Because we integrate over (curved) complex curves, this implements Witten's decade-old "connected prescription" – previously understood in the case of the \(\NNN=4\) gauge theory only – for the case of the \(\NNN=8\) supergravity. This has been believed to be much more subtle and displaying much less simplification (note that the number of bosonic and fermionic twistor coordinates don't match which is a problem, just like what a child could guess) but it may seem today that a simple answer does exist, after all.

From one viewpoint, I tend to think that these are just some horrible non-transparent technical identities whose understanding is about pure maths and one doesn't learn any conceptual physical insights. On the other hand, the simplicity of the expressions and the novel character of the objects it involves suggests that there is a certain conglomerate of insights that we simply have to internalize if we want to comprehend supersymmetric gauge theories and supergravity theories at a deeper level.

So I feel that the role of such twistor rules resembles the role of Feynman's path integral. In some way, it's just an equivalent way to describe quantum mechanics. You could live without them; you could do everything in an operator approach. However, such an anti-path-integral attitude would exhibit a certain intellectual bias (preventing one from seeing many things clearly) and your approach could be getting extremely obscure for many questions (especially those with gauge symmetries, topologically nontrivial and nonperturbative contributions etc.). So whether you dismiss them as pure mathematical tricks, you simply should be interested in Feynman's path integrals if you're a theoretical physicist.

At this level, the twistor miracles could play a similar role. I would still love to know "why" all these simplified forms exist. And if it is fundamentally right, what is the generalization of the formulae that produce the exact M-theory scattering amplitude? There is a lot of particular marvelous technical progress in these corners of knowledge but what is the big picture?

And that is the question (if you allow me to replace Bill O'Reilly by Hamlet, at least once).

P.S.: For obvious reasons, there are lots of papers related to the Higgs on the hep-ph arXiv today. For example, a paper tries to explain the "emerging" excess of the diphoton decays and suppression of the di-tau events by saying that the Higgs boson is one that belongs to the Minimal Supersymmetric Standard Model with some extra comments. The paper seems to contradict some lore that the MSSM should suppress the diphoton rate and not increase it. I will surely return to these matters rather soon – I can't tell you the exact reason why it is so, however, but be ready to see that these seemingly irrelevant deviations – if confirmed – could actually hold answers to some big questions in physics. ;-)

Too little time to describe things like a new Linda Nobel prize meeting where they talk about cosmology, Ivar Giaever explains why the global warming doctrine is bogus, and other things.