High Energy Bounds on Soft N=4 SYM Amplitudes from AdS/CFT

Using the AdS/CFT correspondence, we study the high-energy behavior of colorless dipole elastic scattering amplitudes in N=4 SYM gauge theory through the Wilson loop correlator formalism and Euclidean to Minkowskian analytic continuation. The purely elastic behavior obtained at large impact-parameter L, through duality from disconnected AdS_5 minimal surfaces beyond the Gross-Ooguri transition point, is combined with unitarity and analyticity constraints in the central region. In this way we obtain an absolute bound on the high-energy behavior of the forward scattering amplitude due to the graviton interaction between minimal surfaces in the bulk. The dominant "Pomeron" intercept is bounded by alpha less than or equal to 11/7 using the AdS/CFT constraint of a weak gravitational field in the bulk. Assuming the elastic eikonal approximation in a larger impact-parameter range gives alpha between 4/3 and 11/7. The actual intercept becomes 4/3 if one assumes the elastic eikonal approximation within its maximally allowed range L larger than exp{Y/3}, where Y is the total rapidity. Subleading AdS/CFT contributions at large impact-parameter due to the other d=10 supergravity fields are obtained. A divergence in the real part of the tachyonic KK scalar is cured by analyticity but signals the need for a theoretical completion of the AdS/CFT scheme.Comment: 25 pages, 3 eps figure

On the running coupling constant in QCD

We try to review the main current ideas and points of view on the running coupling constant in QCD. We begin by recalling briefly the classic analysis based on the Renormalization Group with some emphasis on the exact solutions of the RG equation for a given number of loops, in comparison with the usual approximate expressions. We give particular attention to the problem of eliminating the unphysical Landau singularities, and of defining a coupling that remains significant at the infrared scales. We consider various proposals of couplings directly related to the quark-antiquark potential or to other physical quantities (effective charges) and discuss optimization in the choice of the scale parameter and of the RS. Our main focus is, however, on dispersive methods, their application, their relation with non-perturbative effects. We try also to summarize the main results obtained by Lattice simulations in various MOM schemes. We conclude briefly recalling the traditional comparison with the experimental data.Comment: 75 pages, 8 figures. Corrected typos, added references, replaced 1 figure. Accepted for publication in Progress in Particle and Nuclear Physic

Knot theory and quantum gravity in loop space: a primer

These notes summarize the lectures delivered in the V Mexican School of Particle Physics, at the University of Guanajuato. We give a survey of the application of Ashtekar's variables to the quantization of General Relativity in four dimensions with special emphasis on the application of techniques of analytic knot theory to the loop representation. We discuss the role that the Jones Polynomial plays as a generator of nondegenerate quantum states of the gravitational field.Comment: 44 pages. v2: figures added, also available as PDF at http://www.phys.psu.edu/~pullin/primerfigs.pd

Analytic resummation for the quark form factor in QCD

The quark form factor is known to exponentiate within the framework of dimensionally regularized perturbative QCD. The logarithm of the form factor is expressed in terms of integrals over the scale of the running coupling. I show that these integrals can be evaluated explicitly and expressed in terms of renormalization group invariant analytic functions of the coupling and of the space--time dimension, to any order in renormalized perturbation theory. Explicit expressions are given up two loops. To this order, all the infrared and collinear singularities in the logarithm of the form factor resum to a single pole in epsilon, whose residue is determined at one loop, plus powers of logarithms of epsilon. This behavior is conjectured to extend to all loops.Comment: 24 pages, LaTeX, no figure

High-energy hadron-hadron (dipole-dipole) scattering from lattice QCD

In this paper the problem of high-energy hadron-hadron (dipole-dipole) scattering is approached (for the first time) from the point of view of lattice QCD, by means of Monte Carlo numerical simulations. In the first part, we give a brief review of how high-energy scattering amplitudes can be reconstructed, using a functional-integral approach, in terms of certain correlation functions of two Wilson loops and we also briefly recall some relevant analyticity and crossing-symmetry properties of these loop-loop correlation functions, when going from Euclidean to Minkowskian theory. In the second part, we shall see how these (Euclidean) loop-loop correlation functions can be evaluated in lattice QCD and we shall compare our numerical results with some nonperturbative analytical estimates that appeared in the literature, discussing in particular the question of the analytic continuation from Euclidean to Minkowskian theory and its relation to the still unsolved problem of the asymptotic s-dependence of the hadron-hadron total cross sections.Comment: Revised version (to be published in Phys. Rev. D) with new comments in section 4, a new figure [Fig. 6], two new references in Refs. [3] and [34], and some other minor changes; 27 pages, 17 figure