# Index Concordia

## What Alday & Maldacena did (I)

Posted in AdS/CFT, Gauge Theory, Paper discussion, Quantum Field Theory by Index Guy on June 3, 2008

Let us start with the work of Alday and Maldacena on scattering amplitudes of gluons at strong coupling (arXiv:0705.0303). I will summarize what I got from reading the first few pages below, but most of it is really verbatim from the preprint. For future references I will include some questions between square brackets that I need to answer. Section names headline summary of the corresponding preprint. Yes, I even summarized the “Abstract”. I know… 🙂

Abstract

In this work Alday and Maldacena compute planar gluon scattering amplitudes at strong coupling in $\mathcal{N} = 4$ SYM by using the gauge theory/gravity duality (GTGD). Their task is to find a classical string configuration whose boundary conditions are determined by the gluon momenta. Since the results are infrared (IR) divergent, one has to introduce a regulator.

Infrared divergences are associated to low energies, or long distance scales. Recall that the energy of a particle is given by $E = hc/\lambda$. Low energy means large wavelengths. Infrared divergences usually are related to massless particles (since $p^2 = 0$).

As a regulator the authors introduce the gravitational version of dimensional regularization. Then they compute the leading and sub-leading IR divergences. The authors also compute the 4-point amplitude and find agreement with the result from the Bern-Dixon-Smirnov (BDS) ansatz.

Introduction

The authors will describe methods to compute gluon scattering amplitudes at strong coupling in $\mathcal{N}=4$ SYM (this is sometimes refer to as Maximally Supersymmetric Yang-Mills theory, so we will denote it by MSYM). These amplitudes are IR divergent and hence are not good observables. The strong coupling computation is done with the GTGD that relates MSYM to a string theory on $AdS_{5}\times S^{5}$. On the string side, at strong coupling the leading order term [Leading order in what? I suppose this is in the t`Hooft coupling. Am I wrong?] is given by a classical string configuration. The final form of color-ordered, planar scattering amplitude of n-gluons at strong coupling:

$\displaystyle\mathcal{A}_{n} = \exp{\left(\displaystyle-\frac{\sqrt{\lambda}}{2\pi} A_{cl}\right)}$,

with $A_{cl}$ the area of the string world sheet (SWS) from the classical solution to the SWS equations.

Here I should insert a reminder of what does it mean for a scattering amplitude to be color-ordered and planar.

This solution depends on the gluon momenta. The coupling dependence is an overall factor. By comparing the structure of IR divergences obtained with those expected from the field theory, one is able to compute the strong coupling expression for the function that characterizes the sub-leading IR terms. Using dimensional regularization and GTGD we start with a Dp-brane and keep p arbitrary then analytically continue to $p = 3 - 2\epsilon$.

Finally the authors compute the 4-point gluon amplitude and compare their result with that obtained by BDS,

$\displaystyle\mathcal{A}_{4} = \displaystyle\mathcal{A}_{4}^{tree}\exp{\left(IR + \displaystyle\frac{f(\lambda)}{8} \left( \log{s/t}\right)^2 + const.\right)}.$

In this expression $f(\lambda)$ is directly related with the cusp anomalous dimension [What is the cusp anomalous dimension?]. The IR terms are characterized by Sudakov-like factors [What are Sudakov factors?]