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Anti-collinear resummation in JIMWLK evolution in the linear regime

Alex Kovner, Michael Lublinsky, Maxim Nefedov, Vladimir Skokov

TL;DR

The paper develops and tests an all-orders anti-collinear resummation of the JIMWLK kernel in the linear BFKL limit at fixed coupling, producing a closed-form resummed momentum-space kernel and a modified characteristic function $\chi(n,\gamma)$. It demonstrates that the LO pole at $\gamma=1$ is removed, yielding a finite $\chi(1)$ whose value depends on the quark flavor content, and analyzes consistency with the NLO BFKL eigenvalue and all-poles resummations. The work also shows how DGLAP-like anti-collinear effects enter the resummed kernel, derives the target-Bjorken limit behavior, and examines subleading corrections and scale-choice dependencies, including a remedy via a smooth scale. These results advance understanding of high-energy QCD evolution in the dilute regime and set the stage for extending the resummation to running coupling and nonlinear saturation, as well as to non-forward processes.

Abstract

The recently-proposed resummation procedure for anti-collinear logarithms in the JIMWLK kernel~\cite{Kovner:2023vsy} is studied in the linear (BFKL) regime in the fixed-coupling approximation. Simple closed form expressions for the resummed momentum space kernel and characteristic function $χ(n,γ)$ are found. We find that the anti-collinear pole in the leading order characteristic function at $γ=1$ disappears, and instead $χ(γ=1)=\frac{12}{11}\fracπ{α_sN_c}$ for $n_F=0$. Comparison with the known NLO BFKL eigenvalue, with the target-Bjorken limit ($Q_T\gg Q_P$) of the $γ^*(Q_P)+γ^*(Q_T)$-scattering amplitude and with the ``all-poles'' resummation prescription are presented.

Anti-collinear resummation in JIMWLK evolution in the linear regime

TL;DR

The paper develops and tests an all-orders anti-collinear resummation of the JIMWLK kernel in the linear BFKL limit at fixed coupling, producing a closed-form resummed momentum-space kernel and a modified characteristic function . It demonstrates that the LO pole at is removed, yielding a finite whose value depends on the quark flavor content, and analyzes consistency with the NLO BFKL eigenvalue and all-poles resummations. The work also shows how DGLAP-like anti-collinear effects enter the resummed kernel, derives the target-Bjorken limit behavior, and examines subleading corrections and scale-choice dependencies, including a remedy via a smooth scale. These results advance understanding of high-energy QCD evolution in the dilute regime and set the stage for extending the resummation to running coupling and nonlinear saturation, as well as to non-forward processes.

Abstract

The recently-proposed resummation procedure for anti-collinear logarithms in the JIMWLK kernel~\cite{Kovner:2023vsy} is studied in the linear (BFKL) regime in the fixed-coupling approximation. Simple closed form expressions for the resummed momentum space kernel and characteristic function are found. We find that the anti-collinear pole in the leading order characteristic function at disappears, and instead for . Comparison with the known NLO BFKL eigenvalue, with the target-Bjorken limit () of the -scattering amplitude and with the ``all-poles'' resummation prescription are presented.
Paper Structure (24 sections, 152 equations, 8 figures)

This paper contains 24 sections, 152 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Anti-collinear resummation in JIMWLK
  3. The resummation setup
  4. Resummed BFKL kernel
  5. Resummed BFKL kernel in coordinate space
  6. Transforming the kernel to momentum space
  7. Linearized resummation equations
  8. Coordinate space
  9. Second order resummation equations in momentum space
  10. LLA DGLAP effects in the resummed characteristic function
  11. LL anti-collinear resummation in the characteristic function
  12. Characteristic function at $\gamma\to 1$
  13. Comparison with the NLO BFKL eigenvalue
  14. Target-Bjorken limit of the $\gamma^*\gamma^*$-scattering
  15. Comparison with the "all-poles" resummation prescription
  16. ...and 9 more sections

Figures (8)

  • Figure 1: Solid lines -- the scale choice function (\ref{['eq:scale choice-smooth']}) as function of $X^2/Y^2$ at fixed $X^2Y^2$ for several values of $\lambda$. Dashed line (marked as "max") -- the same plot but for the function (\ref{['eq:scale choice']}). In the inset, the neighbourhood of the point $X^2/Y^2=1$ is zoomed-in.
  • Figure 2: The plot of resummed characteristic function (\ref{['eq:chi-anti-coll']}) for $n=0$.
  • Figure 3: Panel (a) -- factorisation for the imaginary part of the $\gamma^*\gamma^*$-scattering amplitude in the Regge limit. Dashed lines denote Reggeised gluon ($\alpha^a(\boldsymbol{x})$) exchanges; Panel (b) -- plots of the photon impact factor "wave functions" of Eq. (\ref{['eq:gamma-WF-def']}) as a function of $\tau=\boldsymbol{k}^2/Q^2$, solid lines -- exact expressions (\ref{['eq:phiL-exact']}) and (\ref{['eq:phiT-exact']}), dashed lines -- approximate expressions (\ref{['eq:appr-phi']}).
  • Figure 4: Dots -- numerical results for the function $h_2[Q_{\star}](\gamma)$ (eqns. (\ref{['eq:h2-1-coord']})+(\ref{['eq:h2-2-coord']})) with the default scale choice (\ref{['eq:scale choice']}) and smooth scale choice (\ref{['eq:scale choice-smooth']}) for several values of $\lambda$. Solid line -- function $h_2(\gamma)$ from Eq. (\ref{['eq:h2(gamma)']}), dashed line -- the LLA function $h_2^{\text{(LLA)}}(\gamma)=1/(1-\gamma)$.
  • Figure 5: Solid lines -- plots of functions $\pi_{ij}(p)$ defined in Eq. (\ref{['eq:pi_ij-def']}), dashed lines -- asymptotic expressions (\ref{['eq:pi_gg-asy']}) -- (\ref{['eq:pib_gq-pi_qg-asy']}).
  • ...and 3 more figures