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Higher Order Effects in Non Linear Evolution from a Veto in Rapidities

G. Chachamis, M. Lublinsky, A Sabio Vera

TL;DR

The paper investigates higher-order corrections to nonlinear QCD evolution by introducing a rapidity veto that forbids close emissions. Analytically, the veto modifies the linear BFKL kernel, slowing the growth of the saturation line and yielding a reduced exponent λ; numerically, applying the veto to the BK equation shows a substantial but diminishing reduction of λ, more pronounced at fixed coupling than with running coupling. The dominant effect at phenomenological energies comes from running coupling, which preserves λ around 0.3 and maintains geometrical scaling, supporting the use of running-coupled BK as a reliable baseline. Overall, the rapidity veto provides a practical proxy for NLO corrections and clarifies how higher-order effects modify saturation dynamics.

Abstract

Higher order corrections to the Balitsky-Kovchegov equation have been estimated by introducing a rapidity veto which forbids subsequent emissions to be very close in rapidity and is known to mimic higher order corrections to the linear BFKL equation. The rapidity veto constraint has been first introduced using analytical arguments obtaining a power growth with energy, Q_s (Y) ~ exp(lambda Y), of the saturation scale of lambda ~ 0.45. Then a numerical analysis for the non-linear Balitsky-Kovchegov equation has been carried out for phenomenological rapidities: when a veto of about two units of rapidity is introduced for a fixed value of the coupling constant of alpha_s = 0.2 the saturation scale lambda decreases from ~ 0.6 to ~ 0.3, and when running coupling effects are taken into account it decreases from ~ 0.4 to ~ 0.3.

Higher Order Effects in Non Linear Evolution from a Veto in Rapidities

TL;DR

The paper investigates higher-order corrections to nonlinear QCD evolution by introducing a rapidity veto that forbids close emissions. Analytically, the veto modifies the linear BFKL kernel, slowing the growth of the saturation line and yielding a reduced exponent λ; numerically, applying the veto to the BK equation shows a substantial but diminishing reduction of λ, more pronounced at fixed coupling than with running coupling. The dominant effect at phenomenological energies comes from running coupling, which preserves λ around 0.3 and maintains geometrical scaling, supporting the use of running-coupled BK as a reliable baseline. Overall, the rapidity veto provides a practical proxy for NLO corrections and clarifies how higher-order effects modify saturation dynamics.

Abstract

Higher order corrections to the Balitsky-Kovchegov equation have been estimated by introducing a rapidity veto which forbids subsequent emissions to be very close in rapidity and is known to mimic higher order corrections to the linear BFKL equation. The rapidity veto constraint has been first introduced using analytical arguments obtaining a power growth with energy, Q_s (Y) ~ exp(lambda Y), of the saturation scale of lambda ~ 0.45. Then a numerical analysis for the non-linear Balitsky-Kovchegov equation has been carried out for phenomenological rapidities: when a veto of about two units of rapidity is introduced for a fixed value of the coupling constant of alpha_s = 0.2 the saturation scale lambda decreases from ~ 0.6 to ~ 0.3, and when running coupling effects are taken into account it decreases from ~ 0.4 to ~ 0.3.

Paper Structure

This paper contains 6 sections, 18 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The rapidity veto in BFKL and the saturation line
  3. The rapidity veto in the BKe
  4. The fixed coupling case
  5. The running coupling case
  6. Conclusions

Figures (10)

  • Figure 1: Dependence of the LO eigenvalue of the kernel on the veto upon $\nu$
  • Figure 2: The solution to Eq. (\ref{['critical_solution']}) as a function of the veto in rapidity, $\eta$.
  • Figure 3: The saturation power (Eq. (\ref{['Ls']})) as a function of the veto in rapidity, $\eta$.
  • Figure 4: Solution to BKe for different values of the veto as a function of $r$, for ${\rm Y}=10$ and $\alpha_s=0.2$.
  • Figure 5: Solution to BKe for different values of the veto as a function of $r$, for ${\rm Y}=14$ and $\alpha_s=0.2$
  • ...and 5 more figures