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Conformal bootstrap for the BFKL Pomeron

G. P. Korchemsky

TL;DR

This work builds a bridge between BFKL reggeon dynamics in perturbative QCD and two-dimensional conformal field theory by showing that the interaction vertex for three BFKL states can be interpreted as a three-point function of quasiprimary operators $O_{h,ar{h}}(z,ar{z})$. The authors derive the full vertex $V(eta oeta,eta)$, including planar and suppressed nonplanar contributions, and demonstrate that in the large-$N_c$ limit the vertex satisfies the conformal bootstrap constraints encoded in operator product expansion and crossing symmetry. They construct an operator algebra for the interpolating fields and establish that the 4-point CFT correlator built from these data reproduces, up to normalization factors, the reggeon Green functions and their high-energy evolution, linking the BPZ conformal blocks to the BFKL dynamics. Although crossing symmetry is broken by the high-energy evolution, the conformal data and associativity constraints provide a consistent CFT framework for reggeon interactions and lay groundwork for incorporating unitarity corrections via higher reggeon transitions. The paper also provides explicit results for special conformal weights (e.g., $h= frac{1}{2}$) and gives closed-form planar contributions, including a precise value for the bare triple BFKL Pomeron vertex, which is valuable for Regge phenomenology.

Abstract

We calculate the interaction vertex of three BFKL states including the bare triple BFKL Pomeron coupling and discuss its relation with the correlation functions in two-dimensional conformal field theory. We construct the operator algebra of the fields interpolating the BFKL states and show that in the multi-color limit the vertex satisfies the constraints imposed by the conformal bootstrap on the structure constants of the operator product expansion in conformal field theory.

Conformal bootstrap for the BFKL Pomeron

TL;DR

This work builds a bridge between BFKL reggeon dynamics in perturbative QCD and two-dimensional conformal field theory by showing that the interaction vertex for three BFKL states can be interpreted as a three-point function of quasiprimary operators . The authors derive the full vertex , including planar and suppressed nonplanar contributions, and demonstrate that in the large- limit the vertex satisfies the conformal bootstrap constraints encoded in operator product expansion and crossing symmetry. They construct an operator algebra for the interpolating fields and establish that the 4-point CFT correlator built from these data reproduces, up to normalization factors, the reggeon Green functions and their high-energy evolution, linking the BPZ conformal blocks to the BFKL dynamics. Although crossing symmetry is broken by the high-energy evolution, the conformal data and associativity constraints provide a consistent CFT framework for reggeon interactions and lay groundwork for incorporating unitarity corrections via higher reggeon transitions. The paper also provides explicit results for special conformal weights (e.g., ) and gives closed-form planar contributions, including a precise value for the bare triple BFKL Pomeron vertex, which is valuable for Regge phenomenology.

Abstract

We calculate the interaction vertex of three BFKL states including the bare triple BFKL Pomeron coupling and discuss its relation with the correlation functions in two-dimensional conformal field theory. We construct the operator algebra of the fields interpolating the BFKL states and show that in the multi-color limit the vertex satisfies the constraints imposed by the conformal bootstrap on the structure constants of the operator product expansion in conformal field theory.

Paper Structure

This paper contains 10 sections, 100 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Transition vertex
  3. Symmetry properties of the vertex
  4. Nonplanar diagram
  5. Planar diagram
  6. Conformal bootstrap
  7. Operator algebra
  8. Crossing symmetry
  9. Conformal blocks
  10. Summary

Figures (2)

  • Figure 1: The Feynman diagrams corresponding to the planar (a) and nonplanar (b) contribution to the vertex $V(\alpha\to\beta,\gamma)$. Solid lines represent two-dimensional propagators $\frac{1}{z^h\bar{z}^{\bar{h}}}$ with the exponents $h$ and $\bar{h}$ depending on the line and the vertices do not bring any additional factors.
  • Figure 2: Graphical form of the relation (\ref{['4']}). The crossing property (\ref{['cr']}) follows from the symmetry of the diagram in the r.h.s.