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In the last 5 years there has been a silent revolution in QFT called the unitarity method and the Maximum Violating Helicity (MVH) Amplitudes that basically consist an alternative way to obtain the same amplitudes that you would obtain by the Lagrangian formalisms and the Feynmann diagrams without breaking the bank in supercluster computing time.

Now, there is little known about why/how this method works in the theoretical aspect. Are there interesting things that might result from this in the theoretical domain, or it just happens to be an extremely convenient calculation tool? What are the main insights that the existence of these identities give us?

If you feel that a summary of the state of the art is all that can be offered for now, feel free to give that as an answer.

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3 Answers 3

The MHV ideas are concerned, typically, with scattering amplitudes of gluons in Yang Mills theories. Most of the foundational work has been done with $\mathcal{N}=4$ supersymmetric Yang Mills theory, though I believe there have been extensions beyond this.

The problem addressed is that you have n gluons meeting at a vertex, some incoming, some outgoing and you want to calculate the scattering amplitude. The assumption is that we wish to look at very high energy processes, so the gluons can be effectively treated as massless. Therefore, to specify an incoming or outgoing gluon we need only give its (null) momentum and its helicity (we should also need color, but this can be effectively put to one side by means of color ordering).

Computing the amplitude the traditional way using Feynman diagrams rapidly becomes intractible as the number of gluons considered increases (see slides 4 and 5 for the five gluon case in this talk by Zvi Bern). The answer was conjectured (I think) by Parke and Taylor and was an extremely simple formula:

$$A(1+, 2+, ....j-,...k-,...n+) = \frac{\langle i,j \rangle^4}{\prod_{k=1}^{n}\langle k, k+1 \rangle}\delta^4(\sum_{k=1}^{n}\lambda^A_k{\tilde{\lambda}}^{A'}_k)$$

Here the lambdas are Weyl (two) spinors. A null vector $p^a$ can be written as a product of an unprimed and complex conjugate primed spinor. $p^a = \lambda^A \bar{\lambda}^{A'}$. Here the notation $\langle i,j \rangle := \epsilon_{AB}\lambda_i^A \lambda_j^B$ is used, where $\epsilon_{AB}$ is the antisymmetric two - spinor. The formula gives the amplitude for n gluons, two of which (j and k in this example) have the opposite helicity from the others. (The cases where none or one of the helicities is different have vanishing amplitudes).

The Parke Taylor formula was verified, for smallish numbers of gluons by direct calculation.

If you go away from the MHV criterion (but stay at tree level), and have, say, 3+ and 3- helicity gluons scattering, it came as a great surprise, that there was STILL a way to write the scattering amplitude in a neat and concise form. This came about through the use of the BCFW recursion relations. Here, the MHV amplitudes are treated as building blocks and connected together in various ways.

The interesting thing about these scattering amplitude formulas is that they exhibit lots of symmetries. They have a conformal symmetry - the conformal group acts on them in a natural way. They also have "dual conformal symmetry" - if you take the gluon momenta, they add up to zero (obviously). If you represent this by placing the momentum vectors head-to-tail in momentum space and label the points where the vectors meet, then, there is a conformal group action on THIS space too. So the big question is - where are all these symmetries coming from?

It would be instructive if the problems could be reformulated in a way that makes these symmetries explicit from the outset. One clue about this was given when Witten reformulated the MHV amplitude problem in twistor space. When this was done, for the tree level amplitudes, the twistors corresponding to the gluons are found to lie on a line ($\mathbb{C}P^1$) in twistor space. As is well known in twistor lore, lines in twistor space are parametrized by points in spacetime. Here the point in spacetime is the scattering vertex. Other scattering amplitudes, including ones with loops, corresponded to the cases where the gluons corresponded to twistors lying on curves of other genus in twistor space. In this context, a key thing about twistor space is that it has a natural conformal group action. So performing computations on twistor space is going to make conformal symmetry manifest at each step, and you're naturally going to end up with conformally symmetric amplitudes.

For a reasonably up to date picture of these "twistor-inspired" reformulations of the scattering problems see here (I only understand about 10% of it).

Getting to your question - about why these methods work - well, according to Nima Arkani-Hamed, it's because the conventional route via Feynman diagrams forces manifest spacetime locality and unitarity at each step in the calculation. This comes at the expense of making the conformal invariance manifest. A more natural way for these gluon scattering calculations to be performed is to make manifest the (dual super-) conformal invariance during the calculation steps, and just check that unitarity is respected by the result. Indeed his view is that this is indicative that, ultimately, when we find the right way to formulate physics, the spacetime picture will be emergent rather than fundamental, hence his slogan "spacetime is doomed".

I believe these techniques (MHV vertices/BCFW recursion relations) have even been used to compute backgrounds for the LHC experiments, so they're of much more than just academic interest. The feeling is that if we understood better WHY they work, we'd have a big arrow pointing towards a useful reformulation of physics, and this arrow is pointing away from spacetime as the framework of choice.

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What are the BCFW recursion relations? Yep Prof. Strassler said that these things go into certain programs used to do for the LHC here. What would happen with ST in such a reformulation of physics? Maybe I could ask a new question ... :-P –  Dilaton Feb 20 '13 at 13:25
@Dilaton Actually, Lubos already answered that one! physics.stackexchange.com/questions/37972/… ps thanks for the Strasser link - I hadn't read that one. –  twistor59 Feb 20 '13 at 13:36
Thanks for the link, twistor :-). Concerning Prof. Strassler's, which I like, you better dont read the comments, too many of them are not fun at all :-/ –  Dilaton Feb 20 '13 at 13:46

To add to @twistor59's nice answer...

There has been progress over the last few years, with the most recent and comprehensive paper being: Scattering Amplitudes and the Positive Grassmannian

The updated picture for now seems to fundamentally depend on permutations and mathematical objects called Grassmannians Gr(k,n) (the set of all k-planes in n-dimensions).

The fundamental interactions in YM (represented in terms of plabic graphs, NOT Feynman diagrams) consist of two types of 3-particle vertices, which respectively permute the interacting particles in a clockwise and counter-clockwise manner. Building up graphs from these vertices makes graphs corresponding to "decorated" permutations. These amplitudes can be canonically formulated in twistor space. Enforcing consistency conditions (such as momentum conservation) and integrating over the Grassmannian in twistor space gives the partial amplitude.

As for the applications of these developments, apart from calculational improvements, it seems like they might bring about a better understanding of the structure of Yang-Mills theories. For example, the superconformal and dual-superconformal symmetry become manifest as just $SL(4)$ action on the twistor and dual-twistor variables, which makes the symmetry of the theory less mysterious.

Note: The long paper seems to explain a few things and open up many more interesting questions. I'm still trying to understand the contents and what I've said above reflects my current (incomplete) understanding.

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first of all: the unitarity method is not new. It's been known since the 60s basically and goes back to work by Cutkowsky and has been extended in work by Bern, Dixon, Dunbar, and Kowoser in the 90s. The idea is that if you cut an amplitude in two all particles will be exchanged in the cut channel. It's basically just a statement about probablities summing to unity - so it's no big magic. Now, there is an extension to this called generalized unitarity which involves cutting amplitudes several times. Cutting basically means replacing loop propagators by delta-functions so that in the end you replace a loop amplitude on the cut by products of tree amplitudes which are known and much simpler than the actual loop amplitude. This is a pretty nice way to calculate loop amplitude and has for instance been used in the recent effort to show that maximal supergravity is finite. While unitarity cuts of Cutkowsky do as i said have a physical interpretation generalized unitarity not so much. At least none that I know of.

Secondly, MHV amplitudes are just one class of amplitudes: those with two negative helicities and the rest positve. These have been known and proven in the 80s by Parke, Taylor, Berends and Giele. For a full answer you have to consider also all other helicity configurations. At tree level the computation can be down via BCFW recursion mentioned above whereas at loop level one has to use other means. E.g. generalized unitarity.

Now why are these amplitudes so simple? In a way it's like Feynman (off-shell) graphs are the worng system of coordinates. On-shell coordinates like spinors/twistors seem to be the rigt ones. This is like the problem of the earth moving around the sun. You could do everything in cartesian coordinates but it will look horrible in between. If you take the symmetry of the problem into account all the computations will become much easier. In a sense now spinors/twistors that the 'symmetry' into account best.


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