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High Energy Scattering in the AdS/CFT correspondence

J. Penedones, M. S. Costa, L. Cornalba, R. Schiappa

The AdS/CFT correspondence opens the fascinating possibility of having a non-perturbative formulation of string theory, and therefore of a quantum theory of gravity, based on non-gravitational gauge theories defined over flat spacetime. On the other hand, it gives hope of understanding strong coupling phenomena of gauge theories, such as confinement and the existence of a mass gap in QCD, in terms of a gravitational theory in a curved background with one higher dimension.
So far, the exploration of the conjectured AdS/CFT duality has been essentially limited to the planar limit, which corresponds to zero string coupling in AdS. In this research program we intend to study string interaction effects in AdS/CFT and therefore test the correspondence outside the planar limit. From the gauge theory point of view, these correspond to non--planar contributions in the 1/N expansion. In string theory, these effects are very important since they are responsible for decay processes of excited strings into smaller ones, and for non--trivial scattering of strings.

The understanding of the worldsheet theory describing free strings in AdS is still incomplete, despite the many recent advances. Thus, direct  computations of string scattering amplitudes as worldsheet correlation functions are presently out of reach. In this research program, we overcome this obstacle by focusing on the particle limit, where  string theory reduces to a gravitational theory, and on high energy scattering processes, where the gravitational loop expansion can be resummed using eikonal methods.
Our first step was the generalization of the eikonal approximation in flat space to AdS. This was achieved with a very clear physical picture of semi--classical interaction between null geodesics. Since AdS has a timelike boundary, it is effectively a gravitational box and the word scattering is just colloquial. In fact, the eikonal phase shift in AdS simply determines the interaction energy between the two colliding high energetic strings. Therefore, using eikonal methods, we were able to determine the energy shift of specific two string states in AdS, which, by the AdS/CFT correspondence, is the anomalous dimension of the dual double trace primary operators. Although the predicted anomalous dimension depends on the type of interaction in AdS, for high energy strings scattering at large impact parameters, the graviton contribution dominates and the result yields a universal prediction for CFT's with gravitational AdS duals.
We have also studied the leading string corrections to the eikonal amplitude in AdS. As in flat space, the leading string effects in high energy scattering can be understood as the exchange of particles of all spins lying in the leading Regge trajectory, resulting in an effective reggeon interaction of spin approximately 2 for large string tension. The appropriate treatment of these string effects required an extension of Regge theory to conformal field theories which is quite natural in our formalism.

Within this context of high energy scattering in the AdS/CFT correspondence, there are many open questions which we are know studying:

  • Up to now we have always considered large 't Hooft coupling. It is natural to ask if one can determine the anomalous dimensions of large dimension and spin composite operators using perturbative techniques on the field theory side of the duality. Indeed, it is possible to compute the four point function in the eikonal kinematical regime at weak 't Hooft coupling using BFKL techniques. At the moment, we are relating our formalism to that of BFKL, describing hard pomeron exchange at weak coupling, including the non--trivial transverse dependence relevant at non--vanishing momentum transfer.
  • At small enough impact parameters, there is the possibility of the incoming strings change their internal state due to the tidal forces of the interaction field produced by them. This type of inelastic scattering in flat space was described by promoting the eikonal phase shift to an operator acting on the internal states of the incoming strings. The phase shift  in AdS will then become an operator acting on two-string states, which will include both an orbital part as well as a contribution from the internal excitation of the two scattering strings. In AdS, one could try to determine such an eikonal phase shift operator acting on the space of  states of two small strings moving along null geodesics, by studying string propagation in an AdS gravitational shock wave background. Given the relation of anomalous dimension and phase shift, this would be a generalization, to double trace operators, of the dilatation operator which has played a crucial role in analyzing the spectrum of single trace states in N=4 SYM theory.
  • In the prototypical example of N=4 SYM, our results are valid only for impact parameters much larger than the radius of the transverse 5-sphere. Indeed, one expects corrections due to massive KK modes of the graviton to be relevant below this value. These corrections are computable with an extension of our methods, which includes the transvers 5-sphere in the transverse space.



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