Wednesday, July 04, 2012

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:\[

{\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.


  1. Lubos...Great post again!!!! I am happy to say it today as well. Could you post something about to what extend we could think the 125-126 GeV thing announced today could or not (being open minded) be good things for SUSY/compositeness/technicolor and more exotics theories? I know you are an expert on SUSY/(super)strings but it could be very interesting to show in a broader post WHY this mass is critical and good/bad for wide categories of models.
    The next step: to measure width, charge, spin, decay modes in single specific channels (Cross section), coupling to gauge bosons and fermions (the coupling to the top quark is very remarkable too). The hardest stuff: self-coupling and invisible width, since LHC is a dirty hadronic machine, and likely these two measurements will have to wait for the ILC and the muon collider.
    And the good stuff, the Higgs sector/portal of the SM is going to be unveiled at last.
    Best regards.

  2. Lindau Nobelists' meeting... the SciAm journalist writes: " [Giaever] derided the Nobel committees for awarding Al Gore and R.K. Pachauri a
    peace prize, and *called agreement with the evidence of climate change a 'religion'*."
    Nice, objective reporting... worthy of the Ministry of Truth in Orwell's novel '1984'!

  3. What's so important to report about celebrating of the Higgs in other countries, what do these people have to do with it after all? You should have come to CERN, like as a couple of your fellow bloggers did come from across the ocean just for the event, to take pictures of Higgs, Englert et al, and themselves -- you'll be loosing out with your focus on the IAS!

  4. Dear anon, I am sorry but I completely disagree. I have personally met Peter Higgs and other founders of the Higgs mechanism. It's great but this event isn't really about the personal worshiping of one person or five people.

    Moreover, with all my respect to Peter Higgs, I think that there are dozens of better physicists at the Princeton's IAS.

    Also, I think it's a sort of a silly superstition that one really has to be physically present in that room. People are traveling too much, for too bad reasons. I would find it a painful kitsch if the IAS physicists had to travel to CERN because of this event. It's not really the place that matters; it's the information about the confirmation of the particle which is spectacular.

  5. Wow, the Higgs discovery is already spurring us on some unknown reality already... we have to stay tuned, I like that. ;-)

  6. Ha ha, has Nima lost a bet or why did he buy the champagne ...?
    As I can read nice blogs and celebrate, chat, and joke around with cool people it feels as if I am somewhere in the middle of the events :-D. So I dont need to fly to CERN (they would not let me in anyway :-P).

    After reading the abstract of the paper and Lumo`s explanations I was asking myself too if this new simplified way to calculate the scattering amplitudes is somehow related to the twistor revolution Nima talked about before seing the corresponding paragraph farther down, LOL :-D.

    I like this article even though I`m far away from understanding the formula of course ... ;-)

  7. Haha yeah like you I'm also a f***wit and just love this blog for the gossip and... what are twistors? a dance? Oooohh let's twist again like Penrose did a few decades ago, let's twist again, like we did last year... :D

  8. I'm lucky that Lumo is able to write about these things so that even I - who had no such 'technical intuition' before "twistor" appeared in the text - often get a sense of being nicely informed by TRF-articles of this kind.

  9. Dear Dilaton, I am pretty sure that Nima hasn't lost a Higgs bet. ;-) He's been one of the most clear and modern people when it comes to formulating the reasons why it has to be there. So he may have bought it because he's cool or because he won a bet - but I am afraid that no one would bet against the Higgs at the IAS. ;-)

  10. Wasn't Peter Higgs in Melbourne watching the outcome at the Conference there?

  11. Peter Higgs was at CERN ;-) a few meters from the CERN directors etc. In some sense, this had to be a disappointment for the organizers of the Melbourne conference because the results could or should have been announced over there. But it's still far enough and many CERN people would prefer to see the announcement at home...

  12. I recall one of your old articles.
    One of your objections to twistor methods was this:
    "The real power of twistors emerges when the twistors are applied to scale-invariant or conformal physical systems such as the N=4 gauge theory in d=4. Gravity is not conformal - it has a priviliged distance scale (the Planck length). The only gravity that is conformal is conformal gravity ;-), whose Lagrangian is essentially the squared Weyl tensor - and conformal gravity is not a physically appealing theory because of the ghosts and other defects"

    The very first paragraph after the abstract of this paper mentions the 'infinity twistor' which breaks conformal symmetry. Is this infinity twistor a good answer to your old objection?

  13. Dear Synchronize, a great point but I think it's not the answer to my objection.

    I think that my objection is still an argument why one shouldn't expect a compact twistor-based formula for the full M-theory (or M-theory on T^7) amplitude: M-theory just isn't scale-invariant as Newton's constant is dimensionful.

    This objection is circumvented if one only claims to have a formula for the tree-level amplitudes, i.e. the classical approximation. The tree-level amplitudes and the classical SUGRA may be said to be scale-invariant: the equations of motion are scale-invariant and the scattering amplitudes scale as simple powers of the scaling parameter if you change all the distances by a factor. So that's where twistor formulae - only natural in the case of a simple behavior under scale invariance - may be powerful.

    The infinity twistor isn't really a tool to generalize the methods to any scale-non-invariant theory. It's just a bookkeeping device. Yes, it breaks the conformal symmetry in some way but exactly because we don't want to admit this infinity twistor to mix with other twistors linked to the external particles in arbitrarily ways, it follows that the infinity twistor is at most a straight factor in all the expressions, so the scale invariance is only violated in a simple way, by giving the amplitudes a scaling weight.

    But the full M-theory amplitudes don't have any well-defined scaling dimension. They're complicated sums of terms proportional to different powers of the energies etc.

    I often forget about the simple and clear arguments above when I am impressed by formulae such as Cachazo and Skinner's one. And I dream about finding the full generalization including the quantum M-theory corrections to the amplitudes. But I would really bet it's not possible to use this formalism beyond the tree level.