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Register a new userOn the Eikonal Approximation in AdS Space
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Richard C. Brower
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We explore the eikonal approximation to graviton exchange in AdS_5 space, as relevant to scattering in gauge theories. We restrict ourselves to the regime where conformal invariance of the dual gauge theory holds, and to large t Hooft coupling where the computation involves pure gravity. We give a heuristic argument, a direct loop computation, and a shock wave derivation. The scalar propagator in AdS_3 plays a key role, indicating that even at strong coupling, two-dimensional conformal invariance controls high-energy four-dimensional gauge-theory scattering.
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We consider the eikonal phase associated with the gravitational scattering of a highly energetic light particle off a very heavy object in AdS spacetime. A simple expression for this phase follows from the WKB approximation to the scattering amplitude and has been computed to all orders in the ratio of the impact parameter to the Schwarzschild radius of the heavy particle. The eikonal phase is related to the deflection angle by the usual stationary phase relation. We consider the flat space limit and observe that for sufficiently small impact parameters (or angular momenta) the eikonal phase develops a large imaginary part; the inelastic cross-section is exactly the classical absorption cross-section of the black hole. We also consider a double scaling limit where the momentum becomes null simultaneously with the asymptotically AdS black hole becoming very large. In the dual CFT this limit retains contributions from all leading twist multi stress tensor operators, which are universal with respect to the addition of higher derivative terms to the gravitational lagrangian. We compute the eikonal phase and the associated Lyapunov exponent in the double scaling limit.
The eikonal phase which determines the Regge limit of the gravitational scattering amplitude of a light particle off a heavy one in Minkowski spacetimes admits an expansion in the ratio of the Schwarzschild radius of the heavy particle to the impact parameter. Such an eikonal phase in AdS spacetimes of any dimensionality has been computed to all orders and reduces to the corresponding Minkowski result when both the impact parameter and the Schwarzschild radius are much smaller than the AdS radius. The leading term in the AdS eikonal phase can be reproduced in the dual CFT by a single stress tensor conformal block, but the subleading term is a result of an infinite sum of the double stress tensor contributions. We provide a closed form expression for the OPE coefficients of the leading twist double stress tensors in four spacetime dimensions and perform the sum to compute the corresponding lightcone behavior of a heavy-heavy-light-light CFT correlator. The resulting compact expression passes a few nontrivial independent checks. In particular, it agrees with the subleading eikonal phase at large impact parameter.
E-CDCC and DEA, two eikonal-based reaction models are compared to CDCC at low energy (e.g. 20AMeV) to study their behaviour in the regime at which the eikonal approximation is supposed to fail. We confirm that these models lack the Coulomb deflection of the projectile by the target. We show that a hybrid model, built on the CDCC framework at low angular momenta and the eikonal approximation at larger angular momenta gives a perfect agreement with CDCC. An empirical shift in impact parameter can also be used reliably to simulate this missing Coulomb deflection.
We formulate eikonal approximation to the calculation of high energy scattering amplitude in the frame where both colliding objects are very energetic. We express the eikonal scattering matrix in terms of the color charge densities of the colliding objects. The calculation is performed in the Hamiltonian formalism. We also show that the appearance of the longitudinal electric and magnetic fields immediately following the collision is fully taken into account in the eikonal approximation.
The new data of the elastic scattering of $^{6}$He+$^{12}$C at about 40 MeV/nucleon are analyzed in the eikonal approximation. The $^{6}$He+$^{12}$C phase-shift function is evaluated completely without any {it ad hoc} assumption by a Monte Carlo integration, which makes it possible to use a realistic 6-nucleon wave function for a halo nucleus $^{6}$He. The effect of the breakup of $^6$He on the elastic differential cross sections as well as the optical potential is studied at different energies from 40 to 800 MeV/nucleon. PACS number(s): 24.10.-i; 21.60.Ka; 25.60.Bx; 25.10.+s Keywords: Eikonal; Glauber; Monte Carlo; Halo; Breakup
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