Index Concordia

Scattering amplitudes à la Alday-Maldacena (II)

Posted in AdS/CFT, Gauge Theory, String Theory by Index Guy on June 20, 2008

Continuing what we started before, I would like to discuss the four-point scattering amplitude using AdS/CFT methods.

We will follow Alday and Maldacena in their calculation of the four-point amplitude of gluons with momentum k_{i}^{m}. We define the Mandelstam variables:

s = -(k_{1} + k_{2})^2   and   t = -(k_{2} + k_{3})^2.

As mentioned previously, the worldsheet will end on a (closed) line constructed with lightlike segments, each with a length that depends on the particle’s momentum. The minimizing of the area is better understood when using the Nambu-Goto string action. The authors use Poincaré coordinates: 4 spacetime coordinates y^m and the radial coordinate r. We will fix one of the spacetime coordinates (a spatial one) to vanish and parameterize the worldsheet surface by using the remaining two spatial coordinate. Then we are left with two yet-undetermined fields (the time and the radial coordinates). The NG action reduces to:

S \propto \displaystyle\int dy_{1} dy_{2} \frac{\sqrt{1 + (\partial r)^2 + (\partial y_{0})^2 - (\partial_{1} r \partial_{2} y_{0} - \partial_{2} r \partial_{1} y_{0})^2 }}{r^2}.

First the authors considered the case where the two Mandelstam invariants are the same. The projection of the lightlike polygon is now a square. Since the sides are given by the momenta, the length of the diagonals of the square will correspond to the square root of the Mandelstam invariants. Because we have scale invariance, we can change the size of the square and not change the problem. So the authors choose a square with vertices when the coordinates are \pm 1. The boundary conditions are:

r(\pm 1, y_{2}) = 0 = r(y_{1}, \pm 1)   ,   y_{0}(\pm 1, y_{2}) = \pm y_{2}   and   y_{0}(y_{1}, \pm 1) = \pm y_{1} .

The authors then claim that the solution near each of the cusps is similar to that worked out by Kruczenski [arXiv]. To be continued…

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