Renormalization Group Improved Small-x Equation

TL;DR

The paper tackles the instability of NL BFKL corrections by formulating a renormalization group–improved small-x equation that respects collinear RG constraints. It constructs an infinite-series ω-dependent kernel whose leading and NL components reproduce known limits and uses an ω-expansion to derive a stable, scheme-resistant description of the gluon anomalous dimension and hard pomeron. Key results include a universal, t-dependent anomalous dimension that conserves momentum and two hard pomeron exponents that bound the intercept, yielding a more reliable small-x behavior that matches fixed-order results in the appropriate regime. The approach also demonstrates resilience to renormalization-scale/scheme changes and systematically estimates residual truncation errors, paving the way for improved phenomenology of two-scale processes at small x.

Abstract

We propose and analyze an improved small-x equation which incorporates exact leading and next-to-leading BFKL kernels on one hand and renormalization group constraints in the relevant collinear limits on the other. We work out in detail the recently proposed omega-expansion of the solution, derive the Green's function factorization properties and discuss both the gluon anomalous dimension and the hard pomeron. The resummed results are stable, nearly renormalization-scheme independent, and join smoothly with the fixed order perturbative regime. Two critical hard pomeron exponents are provided, which - for reasonable strong-coupling extrapolations - are argued to provide bounds on the pomeron intercept.

Figures (8)

• Figure 1: A comparison of the collinearly-enhanced (double and triple poles only) part of the NLO corrections with the full NLO corrections; $n_f=0$.
• Figure 2: $\chi(\gamma,\omega)$ as a function of $\gamma$ for various values of $\omega$, for the symmetric energy-scale $s_0=kk'$ on the left and for $s_0=k^2$ on the right. Here $n_f=0$.
• Figure 3: $\chi_{{\rm eff}}^{(u)}(\gamma,\bar{\alpha}_\mathrm{s})$ as a function of $\gamma$ for various values of $\bar{\alpha}_\mathrm{s}$, for energy-scale $s_0=k^2$ and $n_f=0$.
• Figure 4: $g_\omega(t) {\rm e}^{-t/2}$ for $\omega=0.15$, with energy-scale $s_0=k^2$. The normalization is arbitrary.
• Figure 5: The anomalous dimension in various approximations