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Rare fluctuations and the high-energy limit of the S-matrix in QCD

Edmond Iancu, A. H. Mueller

TL;DR

This work argues that fluctuations—though rare for most high-energy QCD processes—dominate the elastic S-matrix in the small-S regime and cannot be captured by the mean-field Kovchegov equation, which overestimates the suppression exponent by a factor of two. Through center-of-mass and general-frame analyses, the authors show that rare configurations with selectively suppressed evolution yield a larger S than Kovchegov's prediction, with a representative result $S_Y(r_0)\n igr. ext{~}\simeq e^{- rac{c}{4}ar{oldsymboleta}_s^2 (Y-Y_0)^2}S_{Y_0}(r_0)$, indicating fluctuations control the high-energy, small-$S$ limit. The findings emphasize the frame dependence of dominant configurations and establish a frame-independent mechanism whereby fluctuations set the asymptotic behavior of the $S$-matrix, with significant implications for unitarity in high-energy QCD. Overall, the paper highlights the necessity of incorporating rare fluctuations beyond mean-field approximations to correctly describe elastic scattering at very high energies.

Abstract

We argue that one cannot correctly calculate the elastic scattering S-matrix for high-energy dipole-dipole scattering, in the region where S is small, without taking fluctuations into account. The relevant fluctuations are rare and unimportant for general properties of inelastic collisions. We find that the Kovchegov equation, while giving the form of the S-matrix correctly, gives the exponential factor twice as large as the result which emerges when fluctuations are taken into account.

Rare fluctuations and the high-energy limit of the S-matrix in QCD

TL;DR

This work argues that fluctuations—though rare for most high-energy QCD processes—dominate the elastic S-matrix in the small-S regime and cannot be captured by the mean-field Kovchegov equation, which overestimates the suppression exponent by a factor of two. Through center-of-mass and general-frame analyses, the authors show that rare configurations with selectively suppressed evolution yield a larger S than Kovchegov's prediction, with a representative result , indicating fluctuations control the high-energy, small- limit. The findings emphasize the frame dependence of dominant configurations and establish a frame-independent mechanism whereby fluctuations set the asymptotic behavior of the -matrix, with significant implications for unitarity in high-energy QCD. Overall, the paper highlights the necessity of incorporating rare fluctuations beyond mean-field approximations to correctly describe elastic scattering at very high energies.

Abstract

We argue that one cannot correctly calculate the elastic scattering S-matrix for high-energy dipole-dipole scattering, in the region where S is small, without taking fluctuations into account. The relevant fluctuations are rare and unimportant for general properties of inelastic collisions. We find that the Kovchegov equation, while giving the form of the S-matrix correctly, gives the exponential factor twice as large as the result which emerges when fluctuations are taken into account.

Paper Structure

This paper contains 10 sections, 47 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Kovchegov equation
  3. Dipole-dipole scattering in the CM system
  4. The BFKL approximation
  5. The onset of unitarity corrections
  6. The collision of two Color Glass Condensates
  7. Rare vs typical fluctuations in the Kovchegov equation
  8. The dominant configuration in the center of mass frame
  9. Sampling other configurations
  10. The picture in a general frame

Figures (5)

  • Figure 1: The configuration retained by the Kovchegov equation in the frame in which the left mover has rapidity $-Y_0$.
  • Figure 2: The optimal configuration in the center--of--mass frame.
  • Figure 3: The shaded area is the domain of suppressed evolution for a parent dipole within the small unshaded rectangle in Fig. 2.
  • Figure 4: Choosing the optimal configuration in some generic frame.
  • Figure 5: The optimal configuration (above) and the configuration retained by the Kovchegov equation (below) in the frame in which $Y_2=0$.