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Renormalons at Low x

Eugene Levin

TL;DR

Renormalons at Low x investigates incorporating a running QCD coupling into the BFKL framework and quantifying infrared and ultraviolet renormalon effects on small-x evolution. The authors develop a running-α_s BFKL equation via bootstrap-inspired kernel reconstruction, analyze the first-order corrections, and apply Borel summation to renormalon-induced uncertainties, identifying a shadowing-correction mechanism that absorbs IR ambiguities and a nonperturbative √(Λ^2/Q^2) term from UV renormalons. They show that running α_s slows the BFKL growth (smaller intercept ω0) and derive a Green-function solution with an Airy-function structure, including an ω_00 spectrum governing the asymptotic x-dependence. A reduced, numerically tractable evolution equation is proposed, predicting a slower rise of the gluon structure function at small x and clarifying the perturbative domain and GLAP matching in this regime.

Abstract

The role of infrared and ultraviolet renormalons are discussed in context of leading log(1/x) approximation of perturbative QCD. We generalize the BFKL equation for the case of running coupling QCD constant and show that the uncertainties related to the contribution of infrared renormalons turn out to be smaller than the shadowing correction to the total cross section. The contribution of infrared and ultraviolet renormalons to the BFKL equation are studied, and the solution of the BFKL equation with running coupling constant is discussed.

Renormalons at Low x

TL;DR

Renormalons at Low x investigates incorporating a running QCD coupling into the BFKL framework and quantifying infrared and ultraviolet renormalon effects on small-x evolution. The authors develop a running-α_s BFKL equation via bootstrap-inspired kernel reconstruction, analyze the first-order corrections, and apply Borel summation to renormalon-induced uncertainties, identifying a shadowing-correction mechanism that absorbs IR ambiguities and a nonperturbative √(Λ^2/Q^2) term from UV renormalons. They show that running α_s slows the BFKL growth (smaller intercept ω0) and derive a Green-function solution with an Airy-function structure, including an ω_00 spectrum governing the asymptotic x-dependence. A reduced, numerically tractable evolution equation is proposed, predicting a slower rise of the gluon structure function at small x and clarifying the perturbative domain and GLAP matching in this regime.

Abstract

The role of infrared and ultraviolet renormalons are discussed in context of leading log(1/x) approximation of perturbative QCD. We generalize the BFKL equation for the case of running coupling QCD constant and show that the uncertainties related to the contribution of infrared renormalons turn out to be smaller than the shadowing correction to the total cross section. The contribution of infrared and ultraviolet renormalons to the BFKL equation are studied, and the solution of the BFKL equation with running coupling constant is discussed.

Paper Structure

This paper contains 25 sections, 180 equations.

Table of Contents

  1. Introduction.
  2. The BFKL equation with running QCD coupling constant.
  3. The BFKL equation in the lowest order of $\alpha_{\rm S}$.
  4. The main property of the BFKL equation.
  5. Eigenfunctions of the BFKL equation.
  6. The general solution of the BFKL equation.
  7. Anomalous dimension from the BFKL equation.
  8. The bootstrap property of the BFKL equation.
  9. The BFKL equation with running $\alpha_{\rm S}$ in the lowest order.
  10. The BFKL equation with running $\alpha_{\rm S}$ from the bootstrap equation.
  11. The matching with the GLAP evolution equation at $x \,\rightarrow\,0$.
  12. The first correction to the BFKL equation due to running $\alpha_{\rm S}$.
  13. General formula.
  14. Simplification of ${\bf \tilde{K}_r }{\bf(\, \gamma( \omega_1 )}; { \bf \alpha_{\rm S}( q'^2_t )\, )}$.
  15. Infrared Renormalons.
  16. ...and 10 more sections