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Onium-Onium scattering at fixed impact parameter: exact equivalence between the color dipole model and the BFKL Pomeron

H. Navelet, S. Wallon

TL;DR

The paper establishes an exact leading-order equivalence between the color-dipole model and the BFKL Pomeron for onium–onium scattering at fixed impact parameter by exploiting conformal invariance in the dipole cascade and the dipole–dipole kernel. It derives the dipole density and the elementary cross-section in a conformal basis, obtains an exact amplitude that matches the BFKL result, and demonstrates frame invariance. The analysis yields explicit asymptotic dipole distributions and a consistent total cross-section, aligning with numerical simulations and providing a robust analytic handle on high-energy QCD dynamics. The work further extends the dipole approach to onium–electron DIS via k_T-factorization, delivering predictions for structure functions and their ratios in the small-x regime, thereby reinforcing the dipole picture as a powerful perturbative tool with potential for unitarization studies and broader phenomenological applications.

Abstract

We compute the onium-onium scattering amplitude at fixed impact parameter in the framework of the perturbative QCD dipole model. Relying on conformal properties of the dipole cascade and of the elementary dipole-dipole scattering amplitude, we obtain an exact result for this onium-onium scattering amplitude, which is proven to be identical to the BFKL result, and which exhibits the frame invariance of the calculation. The asymptotic expression for this amplitude and for the dipole distribution in an onium at fixed impact parameter agree with previous numerical simulations. We show how it is possible to describe onium-$e^{\pm}$ deep inelastic scattering in the dipole model, relying on k_T-factorization properties. The elementary scattering amplitudes involved in the various processes are computed using eikonal techniques.

Onium-Onium scattering at fixed impact parameter: exact equivalence between the color dipole model and the BFKL Pomeron

TL;DR

The paper establishes an exact leading-order equivalence between the color-dipole model and the BFKL Pomeron for onium–onium scattering at fixed impact parameter by exploiting conformal invariance in the dipole cascade and the dipole–dipole kernel. It derives the dipole density and the elementary cross-section in a conformal basis, obtains an exact amplitude that matches the BFKL result, and demonstrates frame invariance. The analysis yields explicit asymptotic dipole distributions and a consistent total cross-section, aligning with numerical simulations and providing a robust analytic handle on high-energy QCD dynamics. The work further extends the dipole approach to onium–electron DIS via k_T-factorization, delivering predictions for structure functions and their ratios in the small-x regime, thereby reinforcing the dipole picture as a powerful perturbative tool with potential for unitarization studies and broader phenomenological applications.

Abstract

We compute the onium-onium scattering amplitude at fixed impact parameter in the framework of the perturbative QCD dipole model. Relying on conformal properties of the dipole cascade and of the elementary dipole-dipole scattering amplitude, we obtain an exact result for this onium-onium scattering amplitude, which is proven to be identical to the BFKL result, and which exhibits the frame invariance of the calculation. The asymptotic expression for this amplitude and for the dipole distribution in an onium at fixed impact parameter agree with previous numerical simulations. We show how it is possible to describe onium- deep inelastic scattering in the dipole model, relying on k_T-factorization properties. The elementary scattering amplitudes involved in the various processes are computed using eikonal techniques.

Paper Structure

This paper contains 16 sections, 199 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Onium-onium cross-section at fixed impact parameter
  3. The large rapidity dipole density at fixed impact parameter
  4. Exact result for the onium-onium cross-section at fixed impact parameter and equivalence with BFKL Pomeron
  5. Calculation of the onium-onium total cross-section
  6. Electron-Onium Deep Inelastic Scattering
  7. Conclusion
  8. Appendices
  9. Calculation of $\sigma_{DD}$ using eikonal methods
  10. Calculation of $\sigma_{DD}(\underline{x}, \underline{x}',\underline{b}-\underline{b}')$
  11. Calculation of $\hat{\sigma}_{g d}$ using eikonal methods
  12. Calculation of the onium-onium cross section in the laboratory frame of one onium
  13. Approximate calculation of $F^{(1)}$ in the asymptotic regime
  14. Expression of $E^{0,\nu}_q$ in terms of Bessel functions
  15. Properties of the three-points correlation functions $E^{n,\nu}$ and $E^{n,\nu}_q$
  16. ...and 1 more sections

Figures (7)

  • Figure 1: Onium-Onium scattering at leading order.
  • Figure 2: $k_T$-factorization and dipole model applied to $e^{\pm}-onium$ deep inelastic scattering.
  • Figure 3: Contribution to the dipole-dipole scattering.
  • Figure 4: Contributions to the elementary dipole-dipole scattering.
  • Figure 5: Contributions to the elementary dipole-dipole scattering with non zero exchanged momentum.
  • ...and 2 more figures