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Color-glass condensate beyond the Gaussian approximation

Jani Penttala

TL;DR

The paper generalizes the color-glass condensate from a Gaussian MV model to a local, non-Gaussian weight W[ρ] by formulating a characteristic-function approach with w_z(σ^2). Introducing stable distributions with w_z(σ^2) ∝ μ_z^2 σ^α yields a dipole amplitude that scales as N(x,y) ∝ 1 − exp[ − ∫ d^3z ħF_R(α) |K_{xyz}|^α μ_z^2 ], producing a small-dipole behavior N(r) ∝ r^α that reduces to the Gaussian case when α = 2. The work also develops a differential-equation framework to compute higher-order Wilson-line correlators, validates it through adjoint-dipole and 4-point crosschecks, and provides practical numerical sampling strategies for the sCGC model. These results enable flexible, numerically implementable phenomenology of nuclear structure at high energy, with implications for interpreting small-x data and future Electron–Ion Collider measurements.

Abstract

In the high-energy limit, perturbative calculations in QCD are conveniently done using the dipole picture which factorizes the scattering amplitude into a perturbative part and the nonperturbative scattering off the nuclear target, described using correlators of Wilson lines. These correlators can be computed in the color-glass condensate effective field theory by using a Gaussian model for the color density of the target. In this work, we generalize the Gaussian model to a generic function that is local in the transverse coordinates and the light-cone time, and show how to compute physical Wilson-line correlators in this model. We also consider a simple model for the color density based on stable probability distributions and show that the small-dipole behavior of the dipole amplitude is modified from quadratic to a power law, where the power is given by the stability parameter of the distribution. This generalization of the Gaussian model is suitable for numerical applications in the high-energy limit and can be used in future phenomenological studies of the nuclear structure.

Color-glass condensate beyond the Gaussian approximation

TL;DR

The paper generalizes the color-glass condensate from a Gaussian MV model to a local, non-Gaussian weight W[ρ] by formulating a characteristic-function approach with w_z(σ^2). Introducing stable distributions with w_z(σ^2) ∝ μ_z^2 σ^α yields a dipole amplitude that scales as N(x,y) ∝ 1 − exp[ − ∫ d^3z ħF_R(α) |K_{xyz}|^α μ_z^2 ], producing a small-dipole behavior N(r) ∝ r^α that reduces to the Gaussian case when α = 2. The work also develops a differential-equation framework to compute higher-order Wilson-line correlators, validates it through adjoint-dipole and 4-point crosschecks, and provides practical numerical sampling strategies for the sCGC model. These results enable flexible, numerically implementable phenomenology of nuclear structure at high energy, with implications for interpreting small-x data and future Electron–Ion Collider measurements.

Abstract

In the high-energy limit, perturbative calculations in QCD are conveniently done using the dipole picture which factorizes the scattering amplitude into a perturbative part and the nonperturbative scattering off the nuclear target, described using correlators of Wilson lines. These correlators can be computed in the color-glass condensate effective field theory by using a Gaussian model for the color density of the target. In this work, we generalize the Gaussian model to a generic function that is local in the transverse coordinates and the light-cone time, and show how to compute physical Wilson-line correlators in this model. We also consider a simple model for the color density based on stable probability distributions and show that the small-dipole behavior of the dipole amplitude is modified from quadratic to a power law, where the power is given by the stability parameter of the distribution. This generalization of the Gaussian model is suitable for numerical applications in the high-energy limit and can be used in future phenomenological studies of the nuclear structure.

Paper Structure

This paper contains 30 sections, 156 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Color-glass condensate
  3. Computing Wilson-line correlators
  4. Dipole amplitude
  5. Higher-order correlators
  6. Crosscheck: adjoint dipole amplitude and the 4-point correlator
  7. From nucleons to nuclei
  8. Stable color-glass condensate
  9. Dipole amplitude
  10. Large-Nc limit
  11. Numerical implementation
  12. Discussion
  13. Computing xiR
  14. General approach for finite Nc
  15. SU(2)
  16. ...and 15 more sections

Figures (2)

  • Figure 1: The dipole amplitude in the sCGC model evaluated as a function of the dipole size $r = \abs{{\mathbf{x}} - {\mathbf{y}}}$ at zero impact parameter $b = \frac{1}{2} \abs{{\mathbf{x}} + {\mathbf{y}}} = 0$. Different figures correspond to using different probability distributions for numerical sampling. The analytical results are shown as solid lines, whereas the shaded bands correspond to the $1\sigma$ variation of the numerically sampled results.
  • Figure 2: Contours for evaluating the fractional Laplacian.