Thursday, 21 August 2008

Unitarity Rules

Wednesday at Strings'08 was fairly interesting for me, as I'm obviously more attracted by the interface of string and field theory rather than by hard-core strings. In the last few years there has been a lot of commotion in the no man's land between formal string theory/supergravity and the perturbative QCD. This was nicely reviewed by Lance Dixon. This subject is suitable for a 100p review rather than a blog post, but i'll try anyway to sketch the main points.

Perturbative computations in quantum field theory are by far dominated by summing the Feynman diagrams. This is a simple and intuitive method, but it can be cumbersome at times. That is especially true in gauge or gravity theories where the Feynman diagrams propagate many spurious degrees of freedom, and it often happens that the full result is much much simpler than each individual diagram. In such a case, other methods that explore fundamental properties of quantum field theories may prove very handy. Recently, we have been witnessing a come-back of the unitarity based S-matrix approach that was conceived back in the sixties. Geoffrey Chew would be turning in his grave, were he dead.

In fact, we have learned a lot since the summer of love. One thing is the importance of the spinor helicity formalism. For each momentum vector $k_\mu$ of a massless particle one can find corresponding Weyl spinors $\lambda,\bar \lambda$: $k^2$ is the determinant of the matrix $k_\mu (\sigma^\mu)_{ab} $, so that for $k^2 = 0$ the matrix can be written as $\lambda_a \bar \lambda_b$. Similarly, the polarization vectors can be represented using spinors, and one defines the +(-) plus helicity for $\epsilon_\mu (\sigma^\mu)_{ab} \sim \bar \lambda_b (\lambda_b)$. At the end of the day, scattering amplitudes of massless particles can be written as functions of spinor invariants, which reveals unexpected hidden structures. In the Yang-Mills theory, the amplitudes with all helicities of the external gluons being the same, like (++++), are always vanishing. The same is true for the amplitudes with one helicity different than all the others, like (-+++). The simplest non-vanishing amplitude -- the ones with two (+) two (-) helicities -- are called minimal helicity-violating (MHV). There is a compact formula for tree-level MHV amplitudes with an arbitrary number of external gluon legs.

This formalism opened the way for the BCFW recursion relations. To derive these relations, one takes two spinors $\lambda_{1,2}$ corresponding to two external momenta, and continues them analytically to a complex plane, $\lambda_1 \to \lambda_1 + z \lambda_2$, $\bar \lambda_2 \to \bar \lambda_2 - z \bar \lambda_1$. This way, the scattering amplitudes become analytic functions of a fictitious complex variable z. This deformation is designed in such a way that the amplitudes have only simple poles in the z-variable, and the residues of these poles encode information about amplitudes with fewer external legs. Then one can use the usual contour integral methods to relate the amplitude to the residues of its poles, which amounts to relating n-leg amplitudes to those with n-1 legs. This is a powerful method that reduces very complicated multi-legs amplitudes to a sum of simpler (less-leg) ones.
The BCFW relations are being implemented by modern computing tools and Monte Carlos programs.

Yet another interesting development are the use of generalized unitarity relations. Loop amplitudes have branch cuts (in addition to poles), and unitarity strongly constrains what the discontinuity across the branch cut might be. Two-particle cuts are the well-known textbook techniques to relate one-loop amplitudes to tree-level ones. Recently, multi-particle cut techniques have been developed. This jack-the-ripper approach turns out to be useful in reducing general loop amplitudes to simpler building blocks. In particular, arbitrary one-loop amplitudes can be reduced to simple scalar integrals knowns as the box, the triangle and the bubble.

While all the above methods can be applied to the real-life QCD or electroweak theories, they become even more powerful when applied to highly symmetric systems like N=4 super-Yang-Mills or N=8 supergravity. Unitarity is the weapon used on the battlefield of proving perturbative finiteness of N=8 supergravity. Judging from the progress so far, the definitive answer should be known by Strings'09.

It seems that the list of applications is far from being closed. At the same time, there is a feeling that all these surprising results revealed by the unitarity methods are being obtained in a bit roundabout way. There is a fascinating conjecture spelled out in this paper that there exists a dual formulation of quantum field theory where all these surprising properties of scattering amplitudes are realized in a more straightforward way.

Slides here. You can also learn about prof. Mantis Schrimp who was the first to apply the helicity formalism in practice.


Anonymous said...

Okay, I can't resist some shameless self-promotion to tack on this nice review post: just this week (0808.2598 hep-th) we showed that one possible 'dual' formulation of the field theory is just plain old string theory: this is therefore the 'simplest' quantum field theory :)

Alejandro Rivero said...

rutger, do you mean old Dirac multiparameter multiparticle theory? Or a more modern string theory :-D ?

Anonymous said...

Euh. If I actually knew what that was I might agree.... (educate me?)

It seems *any* string theory is 'simple' for the right definition of simple :) Or in other words, almost any string theory has amazingly good UV scattering properties. If the theory you mean makes itself well-behaved (finite?) in the UV by adding infinite stacks of massive particles (I'm guessing), than it will probably have similar properties, although making this precise might be hard. Any string theory is in some limit after all a quantum field theory with an infinite number of massive fields.

All of this doesn't mean of course that the Arkani-Hamed-Cachazo-Kaplan paper isn't very interesting...