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NLO evolution of color dipoles in N=4 SYM

Ian Balitsky, Giovanni A. Chirilli

TL;DR

This work advances the understanding of high-energy gauge theory dynamics by computing the NLO evolution of color dipoles in N=4 SYM and isolating conformal (Möbius) invariant contributions. By introducing composite conformal dipole operators with a conformal rapidity cutoff, the authors construct a Möbius-invariant NLO BK kernel and demonstrate its equivalence to forward NLO BFKL in the N=4 setting. The analysis incorporates gluon, scalar, and gluino loops, and extends to the fundamental representation and to QCD by appropriate substitutions, highlighting how conformal structure can be preserved or restored at NLO. The results have implications for building conformally consistent high-energy effective descriptions and for connecting BK evolution with BFKL in strongly coupled and supersymmetric contexts.

Abstract

High-energy behavior of amplitudes in a gauge theory can be reformulated in terms of the evolution of Wilson-line operators. In the leading logarithmic approximation it is given by the conformally invariant BK equation for the evolution of color dipoles. In QCD, the next-to-leading order BK equation has both conformal and non-conformal parts, the latter providing the running of the coupling constant. To separate the conformally invariant effects from the running-coupling effects, we calculate the NLO evolution of the color dipoles in the conformal ${\cal N}$=4 SYM theory. We define the "composite dipole operators" with the rapidity cutoff preserving conformal invariance. The resulting Möbius invariant kernel for these operators agrees with the forward NLO BFKL calculation of Ref. 1.

NLO evolution of color dipoles in N=4 SYM

TL;DR

This work advances the understanding of high-energy gauge theory dynamics by computing the NLO evolution of color dipoles in N=4 SYM and isolating conformal (Möbius) invariant contributions. By introducing composite conformal dipole operators with a conformal rapidity cutoff, the authors construct a Möbius-invariant NLO BK kernel and demonstrate its equivalence to forward NLO BFKL in the N=4 setting. The analysis incorporates gluon, scalar, and gluino loops, and extends to the fundamental representation and to QCD by appropriate substitutions, highlighting how conformal structure can be preserved or restored at NLO. The results have implications for building conformally consistent high-energy effective descriptions and for connecting BK evolution with BFKL in strongly coupled and supersymmetric contexts.

Abstract

High-energy behavior of amplitudes in a gauge theory can be reformulated in terms of the evolution of Wilson-line operators. In the leading logarithmic approximation it is given by the conformally invariant BK equation for the evolution of color dipoles. In QCD, the next-to-leading order BK equation has both conformal and non-conformal parts, the latter providing the running of the coupling constant. To separate the conformally invariant effects from the running-coupling effects, we calculate the NLO evolution of the color dipoles in the conformal =4 SYM theory. We define the "composite dipole operators" with the rapidity cutoff preserving conformal invariance. The resulting Möbius invariant kernel for these operators agrees with the forward NLO BFKL calculation of Ref. 1.

Paper Structure

This paper contains 21 sections, 135 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Derivation of the BK equation
  3. Calculation of the NLO BK kernel in ${\cal N}=4$ SYM
  4. Gluon contribution to NLO BK
  5. Contribution of scalar particles
  6. Diagrams with two scalar-shockwave intersections
  7. Scalar loop
  8. Gluino contribution
  9. The N=4 kernel
  10. Conformal dipole and conformal NLO kernel
  11. Comparison to NLO BFKL
  12. Evolution equation in the fundamental representation
  13. In ${\cal N}=4$ SYM
  14. In QCD
  15. Conclusion
  16. ...and 6 more sections

Figures (8)

  • Figure 1: Leading-order diagrams for the small-$x$ evolution of color dipole. Gauge links are denoted by dotted lines.
  • Figure 2: Leading-order diagrams proportional to the original dipole.
  • Figure 3: Diagrams with the scalar loop bisected by the shock wave.
  • Figure 4: Diagrams with bare scalar loop.
  • Figure 5: Diagrams without scalar-shockwave intersection.
  • ...and 3 more figures