Resummation of non-global logarithms and the BFKL equation

TL;DR

The paper introduces a color density matrix sigma[U] to capture soft wide-angle radiation and systematically resum non-global logarithms to all orders. It derives a universal evolution kernel K through a two-loop analysis built from renormalized soft currents, and shows that K matches the BFKL Hamiltonian under a conformal transformation, reproducing the Balitsky-JIMWLK framework in conformal theories with beta-function corrections in QCD. The authors prove all-orders exponentiation of infrared divergences within this formalism and present a concrete all-loop structure for K in terms of finite building blocks, outlining how to extend to three loops. They also compare results to N=4 SYM and discuss running-coupling effects, providing a pathway to practical numerical implementations and potential phenomenological applications to jet observables sensitive to non-global logs.

Abstract

We consider a `color density matrix' in gauge theory. We argue that it systematically resums large logarithms originating from wide-angle soft radiation, sometimes referred to as non-global logarithms, to all logarithmic orders. We calculate its anomalous dimension at leading- and next-to-leading order. Combined with a conformal transformation known to relate this problem to shockwave scattering in the Regge limit, this is used to rederive the next-to-leading order Balitsky-Fadin-Kuraev-Lipatov equation (including its nonlinear generalization, the so-called Balitsky-JIMWLK equation), finding perfect agreement with the literature. Exponentiation of divergences to all logarithmic orders is demonstrated. The possibility of obtaining the evolution equation (and BFKL) to three-loop is discussed.

Figures (8)

• Figure 1: Color density matrix. For each colored final state, an independent color rotation is applied between the amplitude and its complex conjugate.
• Figure 2: Minimal example of a non-global observable: the total cross-section to produce particles inside a given potato-shaped allowed region $R$, allowing only a small total energy $E_{\rm out}$ outside of it. In the limit $E_{\rm out}\to 0$, large logarithms need to be resummed, which suppress the cross-section: the effective excluded region grows as the veto suppresses near-boundary radiation.
• Figure 3: (a) Scattering in the Regge limit. The thin shock is the Lorentz-contracted target. (b) Branching of soft gluons. To connect the pictures one 'folds' along the target and sends it to infinity.
• Figure 4: One-loop evolution of the density matrix. (a) Real emission of one soft gluon. This adds one $U$-matrix (cf. eq. (\ref{['Gamma0_real']})). (b) Virtual correction.
• Figure 5: Building block for next-to-leading order computation: amplitude for two soft particles. Solid lines are eikonal Wilson lines. (a) Two soft gluons. The non-abelian part of the first graph gives a connected contribution. (b) Two soft fermions or scalars.