# How do ideas of leading singularities and Grassmanian help in curing infrared divergences when calculating N=4 scattering amplitudes?

Broadly speaking how do ideas of leading singularities and Grassmanian help in curing infrared divergences when calculating N=4 scattering amplitudes? My understanding is that one gets infra red divergences because the external gluon momenta becomes collinear with the the loop momenta. I am confused as to what Nima's and Freddie's collaboration are doing to avoid this? If people can clarify my confusion and direct me to the appropriate literature I would be grateful.Also, is there a clear way to understand this for theories like planar QCD?

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I had the same problem - it was one of just two main problems with the motivation of this whole "twistor uprising" (the other, surviving problem for me is the relevance of the whole formalism for the off-shell gauge theory which I find important, especially for AdS/CFT etc.) - but it has been fully answered for me. The solution is as follows:

In general, the infrared divergences arise from integrals over momenta or positions - from the parts of the integral with very low momenta or very long distances.

However, in the normal formalisms and for generic QFTs, the amplitudes are written are the sum of many Feynman diagrams. Each of them integrates over different loop variables but there is no way to reconcile them. The sum is an infrared divergent but there is no "common integrand" that would be convergent.

Nevertheless, at the planar level of gauge theory, one may show that all the terms have a natural common parameterization of the infrared divergences, so the finite piece is totally well-determined. It's because the natural loop variable is what runs around the boundary of the planar diagram, if pictured as a disk, and shifting the internal loop momentum by the external momenta - which is the only ambiguity - doesn't change the form of the "finite piece" of the divergent amplitude. You may also imagine that the planar diagrams are merged into a disk diagram of string theory and you parameterize the infrared divergence in terms of $\tau$ - the length of the disk - in some Schwinger-like parameterization.

The latest two papers by Nima et al. make this point very manifest because the amplitude for a given process is given by a single integral of a convergent integrand over the Grassmannian (rather than a sum of many similar integrals over different domains). The integration variables and domain may be viewed as "pure kinematics" while the integrand (which is still a sum, but a sum of functions of the universal variables) is the "true dynamics", so in this sense, the total dynamical result is totally convergent.

This doesn't mean that the actual scattering amplitudes "become" convergent. They're still divergent, as in any conformal theory. However, the divergence is fully isolated into some kinematics and there exists a sharp quantity that may be calculated arbitrarily accurately, at any number of normal loops, and that is indisputably related to the scattering amplitudes.

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While I also find this argument convincing, I was wondering how this is related to another potential answer to this question. In QCD and other gauge theories with massless gauge bosons, one has IR divergences which cancel, but only upon resummation of perturbation theory, and only for so-called IR safe quantities (like scattering of appropriately chosen jets). If this logic works similarly in N=4, somehow the "integrand" should be then one of those IR safe quantities. Would be great to understand things in this kind of language, the integrand is a bit of a formal object. –  user566 Feb 28 '11 at 5:57
Good point, @Moshe R. Congrats to your extra letter, by the way. Well, I think that these two answers are ultimately equivalent. The point of localizing the IR divergences to the integral mean that they may be consistently canceled. Then, effectively, there must be a way to rewrite the integrals or their product so that the cancellation is manifest, point-by-point in the Grassmannian one integrates over. And vice versa: if the jets' well-defined amplitudes are totally OK, it should mean that the IR divergences for the bad quantities must be of a constrained type. –  Luboš Motl Mar 1 '11 at 15:24
However, I guess that in $N=4$ which is conformal and has no scale, there may exist no quantities that cancel all the IR divergences, or at least not "sufficiently generic" ones. But I have no proof of this expectation. –  Luboš Motl Mar 1 '11 at 15:24