Resummation of Super-Leading Logarithms

Jet cross sections at high-energy colliders exhibit intricate patterns of logarithmically enhanced higher-order corrections. In particular, so-called nonglobal logarithms emerge from soft radiation emitted off energetic partons inside jets. While this is a single-logarithmic effect at lepton colliders, at hadron colliders phase factors in the amplitudes lead to double-logarithmic corrections starting at four-loop order. This effect was discovered a long time ago, but not much is known about the higher-order behavior of these terms and their process dependence. We derive, for the first time, the all-order structure of these “super-leading logarithms” for generic $2\to l$ scattering processes at hadron colliders and resum them in closed form.

Figure 1

Action of the real-emission operator ${\overline{\mathit{R}}}_m$ and the virtual piece ${\overline{\mathit{V}}}_m$ on a hard function ${\mathcal{H}}_m$. Because of the emitted gluon (blue), the product ${\mathcal{H}}_m{\overline{\mathit{R}}}_m$ defines a hard function with ($m+1$) external legs.

Figure 2

Super-leading logarithms in quark-quark scattering summed up to four-loop (red), five-loop (blue), and infinite order (black). The solid and dashed lines refer to the color octet and singlet channel, respectively.