Monday, August 03, 2009

Arkani-Hamed et al.: all leading S-matrix singularities of N=4 SYM

Let me start with a sentence that occurs quite often on this blog, and it is no coincidence: The most interesting hep-th paper today is arguably the first one. Nima Arkani-Hamed, Freddy Cachazo, Clifford Cheung, and Jared Kaplan wrote about
a duality for the S-matrix (click).
They present a concise formula for scattering amplitudes in N=4 supersymmetric Yang-Mills up to all orders in perturbation theory and they market it as a new "weak-weak dual description" for this S-matrix. The most informative formulae are equations (9) and (10), analyzed in the rest of the paper:

This is a formula for the scattering amplitude of "n" gluons with "k" of them having the opposite chirality than others. The amplitude is written as an integral over a "kn"-dimensional space of rectangular matrices "C" (with no twistorial indices!). These matrices may be interpreted as parameters of "k"-dimensional planes embedded in an "n"-dimensional space: the space of matrices is therefore an "(n choose k)-1"-dimensional projective space (plus an irrelevant scaling variable).

The integral includes a measure which is a "cyclic" product of "n" inverse determinants of "k x k" matrices. They're multiplied by the integrand, which is a product of "k" delta-functions in a twistor "four-bosonic plus four-fermionic" space. I guess that there should exist a generalized superspace formula that unifies all the n-gluon formulae for a fixed "n" but all values of "k" into one formula that allows the helicities to be arbitrary and that includes the other superpartners as well. It may even be written somewhere in the paper already. (Well, they surprisingly find that different helicities are obtained from the same integral over different charts...)

Even now, their formula has the clear ambition to extend the BCFW twistor rules for the scattering amplitudes to all orders of perturbation theory. This obviously brings two possible question marks:
  1. whether some quantities describing the loop scattering amplitudes of N=4 SYM may be well-defined, despite the infrared divergences
  2. whether we should be surprised that only the perturbative portion of such amplitudes is being calculated, and whether the lack of nonperturbative knowledge is a serious problem.
It's not hard to see that you want the answers to be Yes, No for their paper to be physically important but even if the answers are different, the integrals they study are mathematically interesting.

Another general question is whether such a new compact formula should be considered a "theory". Well, in some generalized sense, it is surely a "theory". As soon as the amplitudes are written as integrals over a space, we have a new "physical picture", even though it could look counterintuitive.

We may always say that the matrices "C" - the planes - describe some virtual degrees of freedom in some generalized Feynman diagrams.

Incidentally, it is impossible for me not to mention my own observation, namely a possible relationship with an unknown but interesting 1996 paper by Albert Schwarz, Grassmannian and String Theory that noticed that the moduli spaces of Riemann surfaces in perturbative string theory can be embedded into an infinite-dimensional Grassmannian manifold.

Schwarz has argued that such Grassmannians are natural objects as moduli spaces and made some guesses about the mechanisms of "localization" that produce the perturbative part from his Grassmannians which were supposed to contain the full nonperturbative answer, as well as a sketch of a possible proof of S-dualities in this framework.

It just happened that an integral over a Grassmannian, i.e. a space of subspaces of another linear space, occurred in this new formula, too, somewhat confirming Schwarz's general mathematical "prediction". You are invited to think about the integration variables as some new generalized moduli of a new generalized "worldsheet" or "worldvolume" optimized for the on-shell N=4 supersymmetric gauge theory.

Further research will say whether their line of reasoning leads somewhere: Witten's original twistor paper has already seen a lot of additional insights. The future results may uncover all mysteries hiding behind the N=4 SYM and its integrability and they may become even more far-reaching if they shed light on all possible integration variables that may emerge in formulae for amplitudes anywhere in physical theories dual to quantum gravity.


  1. Why is the most interesting paper in a given day so often the first? Is it because the editors can reliably estimate "interestingness" quickly? I expect that estimating "quality" or "correctness" would take more time, so if the filter is that papers get accepted when they are sufficiently interesting OR sufficiently high-quality there will be a tendency for the quicker-accepted ones to be more interesting.

    This also raises the possibility of a negative correlation in the selected group between interest and quality -- in the same way that brains and beauty are positively correlated in real life but negatively correlated among celebrities, dual selection will tend to anti-correlate interest and quality even when those attributes are positively correlated among the entire population pre-selection.

  2. Dear Joe,

    your analysis would be interesting if it weren't plagued by a basic technical misunderstanding: the ordering of the papers on each day is not decided by editors but by the (competing) authors themselves. The order is First Comes First Goes.

    Whoever submits the first paper after the previous deadline, i.e. after 4 p.m. Boston time or whatever is the current deadline, becomes the first paper on the following day.

    So if some people want their paper to be the first one and they're the only ones with this dream, they can easily grab the title. On the other hand, there exist days when there are several authors/teams trying to achieve the same goal. ;-)

    Then it's a bloody contest for the priority! :-)