When does the gluon reggeize?

Received: August When does the gluon reggeize? Simon Caron-Huot 0 1 2 3 Open Access 0 1 2 3 c The Authors. 0 1 2 3 0 School of Natural Sciences, Institute for Advanced Study 1 Blegdamsvej 17 , Copenhagen 2100 , Denmark 2 Niels Bohr International Academy and Discovery Center 3 Einstein Drive , Princeton, NJ 08540 , U.S.A We propose the eikonal approximation as a simple and reliable tool to analyze relativistic high-energy processes, provided that the necessary subtleties are accounted for. An important subtlety is the need to include eikonal phases for a rapidity-dependent collection of particles, as embodied by the Balitsky-JIMWLK rapidity evolution equation. In the first part of this paper, we review how the phenomenon of gluon reggeization and the BFKL equations can be understood simply (but not too simply) in the eikonal approach. We also work out some previously overlooked implications of BFKL dynamics, including the observation that starting from four loops it is incompatible with a recent conjecture regarding the structure of infrared divergences. In the second part of this paper, we propose that in the strict planar limit the theory can be developed to all orders in the coupling with no reference at all to the concept of reggeized gluon. Rather, one can work directly with a finite, process-dependent, number of Wilson lines. We demonstrate consistency of this proposal by an exact computation in N=4 super Yang-Mills, which shows that in processes mediated with two Wilson lines the reggeized gluon appears in the weak coupling limit as a resonance whose width is proportional to the coupling. We also provide a precise operator definition of Lipatov's integrable spin chain, which is manifestly integrable at any value of the coupling as a result of the duality between scattering amplitudes and Wilson loops in this theory. Scattering Amplitudes; Resummation; 1/N Expansion 1 Introduction 1.1 Relativistic eikonal approximation 2 Review of eikonal approximation and Balitsky-JIMWLK equation The Balitsky-JIMWLK equation Linearization and gluon reggeization: a pedestrian approach The hermitian inner product and structure at higher loops The one-loop Balitsky-JIMWLK equation from hermiticity Simplifications in the planar limit Dipole evolution in the planar limit Higher-point correlators Bootstrap relations and the Odderon intercept The elastic amplitude to next-to-leading logarithm accuracy General structure of the amplitude The Regge cut contribution 4.3 Implications for infrared divergences 5 Multi-Regge limit of n-point amplitudes and OPE Shockwave formalism OPE coefficient for gluon emission The Regge cuts in the five- and six-point amplitudes The remainder function in planar N = 4 SYM The four-gluon amplitude The six-gluon amplitude Direct derivation of the exact bootstrap relation Higher-point amplitudes and zig-zag operators Wilson loop duality and the integrable SL(2, C) spin chain Summary and outlook A Evolution equation in Fourier space and connection with BFKL B The anomalous dual conformal charges C Derivation and self-duality of the one-loop SL(2, C) spin chain High-energy processes subject to the strong interactions have received continuous attention from the theory community over the past decades. Some of the most intriguing questions, historically and presently, involve processes with large spreads in rapidity. One example is the total hadronic cross-section [1, 2], and, by extension, the physics of the elastic amplitude at small angles as well as the single- and double-diffractive amplitudes. With todays experimental program, which also includes proton-ion and ion-ion collisions where saturation effects have been argued to play an important role [3, 4], the demands placed on the theory community become particularly strong. A most natural tool to analyze high-energy processes with small angular deflection is the eikonal approximation. This approximation is well known in the context of nonrelativistic systems [5], where it amounts to neglecting a projectiles deflection and simply dress each classical trajectory by a phase factor. These trajectories are labelled by a twodimensional impact parameter. The method is naturally adapted to gauge theories, and in this context the eikonal approximation is generally understood as the replacement of a fast or heavy particle by a Wilson line following its classical trajectory. These Wilson lines, for example, form an essential ingredient of heavy quark effective theory [6] and soft-collinear effective theory [7]. For ultrarelativistic forward scattering, a simple question demonstrates that a single Wilson line cannot be the final answer. The reason is that the wavefunction of a relativistic particle necessarily contains a large number of virtual particles, which, at high energies, can be easily liberated. For all intents and purposes these virtual particles are as real as the original one. Which trajectory should be dressed? Any relativisti (...truncated)