The Pomeron and Gauge/String Duality

Abstract

The traditional description of high-energy small-angle scattering in QCD has two components -- a soft Pomeron Regge pole for the tensor glueball, and a hard BFKL Pomeron in leading order at weak coupling. On the basis of gauge/string duality, we present a coherent treatment of the Pomeron. In large-N QCD-like theories, we use curved-space string-theory to describe simultaneously both the BFKL regime and the classic Regge regime. The problem reduces to finding the spectrum of a single j-plane Schrodinger operator. For ultraviolet-conformal theories, the spectrum exhibits a set of Regge trajectories at positive t, and a leading j-plane cut for negative t, the cross-over point being model-dependent. For theories with logarithmically-running couplings, one instead finds a discrete spectrum of poles at all t, where the Regge trajectories at positive t continuously become a set of slowly-varying and closely-spaced poles at negative t. Our results agree with expectations for the BFKL Pomeron at negative t, and with the expected glueball spectrum at positive t, but provide a framework in which they are unified. Effects beyond the single Pomeron exchange are briefly discussed.

68 pages, uses JHEP3.cls, utphys.bst; references added, typos corrected, and clarifying remarks added

References in corpus (55)

• Gauge Theory Correlators from Non-Critical String Theory
• Large N Field Theories, String Theory and Gravity
• Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi$SB-Resolution of Naked Singularities
• The glueball spectrum from an anisotropic lattice study
• BFKL pomeron in the next-to-leading approximation
• Hard Scattering and Gauge/String Duality
• Total cross sections
• Bulk vs. Boundary Dynamics in Anti-de Sitter Spacetime
• The String Dual of a Confining Four-Dimensional Gauge Theory
• Deep Inelastic Scattering and Gauge/String Duality
• Three-loop universal anomalous dimension of the Wilson operators in N=4 SUSY Yang-Mills model
• D-brane dynamics
• Unitarity and the BFKL Pomeron
• Liouville Field Theory -- A decade after the revolution
• Small-x physics in perturbative QCD
• A resummation of large sub-leading corrections at small x
• NLO corrections to the BFKL equation in QCD and in supersymmetric gauge theories
• The QCD Pomeron with Optimal Renormalization
• The Large N Limit of $\mathcal{N}=2,1$ Field Theories from Threebranes in F-theory
• Glueball Spectrum for QCD from AdS Supergravity Duality
• Light-Front Hadron Dynamics and AdS/CFT Correspondence
• Glueball Mass Spectrum From Supergravity
• String Theory and Quark Confinement
• Strings, Black Holes and Lorentz Contraction
• Light-cone Superstring in AdS Space-time
• High energy scattering and the AdS/CFT correspondence
• Evaluation Of Glueball Masses From Supergravity
• Glueball Regge trajectories and the Pomeron -- a lattice study --
• Spin-Two Glueballs, Positive Energy Theorems and the AdS/CFT Correspondence
• Plasma-balls in large N gauge theories and localized black holes
• Glueball Regge trajectories from gauge/string duality and the Pomeron
• POMERON FLUX RENORMALIZATION IN SOFT AND HARD DIFFRACTION
• Irreducible part of the next-to-leading BFKL kernel
• The Effect of Higher Order Corrections to the BFKL Equation on the Perturbative Pomeron
• High energy QCD scattering, the shape of gravity on an IR brane, and the Froissart bound
• A collinear model for small-x physics
• Regge Trajectories Revisited in the Gauge/String Correspondence
• DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory
• Diffractive Production of Vector Mesons at Large t
• Glueball Regge Trajectories
• String Fluctuations, AdS/CFT and the Soft Pomeron Intercept
• Hard Scattering in the M-Theory Dual for the QCD String
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• Heisenberg saturation of the Froissart bound from AdS-CFT
• Discrete Spectrum of the Graviton in the $Ad{S}^5$ Black Hole Background
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• Quantized tension: Stringy amplitudes with Regge poles and parton behavior
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• The soft Pomeron from AdS-CFT
• High energy amplitude as an admixture of "soft" and "hard" Pomerons
• Quark-Antiquark Regge Trajectories in Large N_c QCD
• ${\gamma }^{\ast }{\gamma }^{\ast }$ Scattering via Secondary Reggeon Exchange in QCD
• Diffractive Production at Collider Energies and Factorization

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