New results on small-x resummation for splitting functions

Abstract

We revisit the basic steps necessary to obtain next-to-leading-logarithmic accurate small-$x$ results for the DGLAP splitting functions, and their implementations within the HELL framework. We derive new analytical all-order results for the leading-logarithmic $gg$ anomalous dimension, the $qg$ and $gg$ finite Green functions, and most importantly for the $qg$ anomalous dimension, which allows us to arrive for the first time at a properly resummed $qg$ splitting kernel. We use these results as cornerstones of a new implementation of small-$x$ splitting-function resummation which is more solid and numerically better behaved with respect to those available thus far. All of these novelties are included in the upcoming 4.0 version of HELL.

Figures (13)

• Figure 1: Momentum fractions probed at high-energy colliders as a function of $\tau = M/\sqrt{s}$ for various rapidities. On the secondary axis on the top, the value of $M$ for $\sqrt{s} = 20$ TeV is indicated as reference. Note that the largest accessible rapidity is determined by $\tau$, with $|y| < 1/2\log(1/\tau)$, as marked by the black dashed line.
• Figure 2: Plot of the relative difference $R_n$, eq. \ref{['RABFcoeff']}, between exact and numerical $h_k$ coefficients in logarithmic scale as a function of $n$. The blue (red) points feature $h_n(\text{exact}) > 0$ ($h_n(\text{exact}) < 0$).
• Figure 3: The function $h_{qg}(1/s)$ in the complex $s$ plane (left) and $h_{qg}(z)$ in the complex $z$ plane (right).
• Figure 4: Comparison of $\Delta_3P_{qg}(\alpha_s,x)$ computed with different combinations of the ingredients and procedure. Each plot corresponds to a different value of $\alpha_s$, from 0.2 to 0.35 in steps of 0.05.
• Figure 5: Variations of the Mellin inversion path (left-hand panel) and of the order of the Padé approximant (right-hand panel). The dashed green curve is the same in both plots.