Scattering amplitudes à la Alday-Maldacena (I)
I would like to summarize the work of Luis Fernando Alday and Juan Maldacena on gluon scattering amplitudes at strong coupling via AdS/CFT methods. I will follow mostly (i.e. verbatim) these two preprints from the arXiv:
The AdS/CFT correspondence states that four-dimensional Super Yang-Mills theory (MSYM) is equivalent to type IIB string theory on . This correspondence tells us how the YM coupling, the number of colors and the radius of compactification are related:
The string coupling goes as the inverse of the number of colors. So in the limit of large N, the strings will not split or join. Hence the theory can be described by a non-linear sigma model [What exactly is a non-linear sigma model?] For large YM coupling, the sigma model is weakly coupled.
In order to define the scattering amplitudes one introduces an infra-red regulator. The authors use a D-brane as such regulator, localized in the fifth dimension (the “radial” dimension). In Poincaré coordinates the spacetime metric looks like:
The D-brane regulator is placed at some and extends along the other 4 dimensions. Our asymptotics states will be open strings that end on this D-brane. These are the strings that will scatter.
We will define the proper momentum of the string as
The momentum that is conjugate to the four spacetime coordinates corresponds to . This momentum will play the role of the gauge theory momentum and will be kept fix one removes the IR regulator. Due to the warping of the metric, the proper momentum is very large. Previusly Gross and Mende [Nuclear Physics B] considered a similar situation but in flat space. Their main result was the amplitude being dominated by a saddle point of the classical action. Extrapolating this to anti-de Sitter space, we wish to find the saddle point of the classical action with strings in AdS.
At tree-level we will consider a worldsheet with the topology of a disk with vertex operator inserted along its boundary. These insertions will correspond to the external states. By fixing the ordering of the vertex operator we obtain a color-ordered amplitude in the gauge theory.
What about the boundary conditions for this worldsheet? Near the vertex operators the momenta of the external states fixes the form of the solution. Since the open strings are attached to the D-brane, at the boundary we have
The authors introduce “T-dual” coordinates that allow the boundary conditions to be stated more simply. These coordinates are defined by
with . Now the boundary conditions are written in terms of “winding” in y:
Upon defining one obtains metric that has the AdS form (in terms of y and r). The boundary of the worldsheet is found at so
The worldsheet will end on a line that is constructed by concatenating segments that have as length. Since we will consider gluon, which are massless, the segments will be lightlike. Momentum conservation closes the line.
Very important point:
The worldsheet ends on this special line because we have used the “T-dual” coordinates.
When one removes the IR regulator the boundary of the worldsheet collapses towards the boundary of the T-dual space at .
According to Alday and Maldacena, the leading exponential behavior of the scattering amplitude is given by the area of the minimum surface that ends on the lightlike segments mentioned above.