Summing Large-N Towers in Colour Flow Evolution

TL;DR

The paper develops a systematic framework for soft-gluon evolution in the colour-flow basis for multi-jet final states by deriving the general one-loop soft anomalous dimension matrix for arbitrary parton numbers and proposing a controlled large-N expansion (N^dLC) to resum color-suppressed contributions. It provides explicit leading-color (LC) and progressively subleading (NLC, NNLC, N3LC) exponentiation formulas, including detailed constructions of R-functions and Q-polynomials that enable through-N^3LC accuracy. Numerical tests in a simple 2→2 QCD setup demonstrate rapid convergence, with N3LC achieving sub-permille agreement and practical applicability to high-multiplicity scenarios. The work lays groundwork for improved colour-aware parton showers and analytic resummation in multi-jet observables, and it integrates with colour-flow-based matrix-element frameworks through the CVolver library.

Abstract

We consider soft gluon evolution in the colour flow basis. We give explicit expressions for the colour structure of the (one-loop) soft anomalous dimension matrix for an arbitrary number of partons, and show how the successive exponentiation of classes of large-N contributions can be achieved to provide a systematic expansion of the evolution in terms of colour supressed contributions.

Figures (5)

• Figure 1: An illustration of the colour basis chosen for the case of two colour flows. Connected lines correspond to Kronecker-$\delta$ symbols in the space of (anti-)fundamental representation indices.
• Figure 2: Illustration of the non-diagonal contributions to colour charge products acting on colour flow basis tensors. Note that the 'singlet' operators are entirely equivalent to the 'swapping' ones.
• Figure 3: Real and imaginary parts of a diagonal evolution matrix element for quark-quark scattering at $s=100\ {\rm GeV}^2$, $\mu^2=25\ {\rm GeV}^2$ as a function of the momentum transfer $|t|$, comparing the exact results to various approximations. This matrix elements describes the amplitude to keep a $t$-channel colour flow $\sigma$.
• Figure 4: Same as figure \ref{['figures:diagonal']} for an off-diagonal matrix element. The matrix element considered describes the transition from a $u$-channel colour flow $\tau$ to a $t$-channel one, $\sigma$.
• Figure 5: Comparison of the prime resummation prescription compared to the native one for the same parameters as used in figure \ref{['figures:diagonal']}. Typically, a N$^2$LC' summation reaches a similar accuracy as a N$^3$LC one, both providing sub-permille agreement with the exact result.