This paper shows how to build the ingredients of scattering calculations in planar Yang–Mills theory by stitching together simpler off‑shell building blocks. The authors apply an off‑shell recursion method that starts from the classical field equation solved by the perturbiner method. “Off‑shell” here means intermediate fields that are not constrained to satisfy the particle energy–momentum relation. Using those multi‑particle off‑shell currents, they construct loop integrands in an organized, recursive way.

A main new point is a matrix formalism for the pure‑gluon part of the planar integrands. The authors package contributions from three‑point and four‑point gluon interactions into two kinds of matrices (called B and C in the paper). Loop ingredients then appear as chains of these matrices, with extra matrices for contact terms. This matrix picture separates the role of different vertices and of different external legs. For the one‑loop case they write a compact “bare kernel” in that matrix language and explain how symmetry factors convert it to the usual kernel used in integrand construction.

The paper also treats the ghost fields that appear in the gauge‑fixed Yang–Mills Lagrangian. Ghosts are anti‑commuting fields (usually called b and c) whose propagators carry specific minus signs. The authors write the corresponding ghost multi‑particle currents and include all ghost sewing cases in the recursion. They note technical points such as which legs must occupy the first position in an ordered sewing and when factors of 1/2 are required. The ancillary files reportedly contain explicit one‑loop kernels including ghost contributions, and the authors checked their one‑loop results against FeynCalc.

Why this matters: the usual way to build higher‑loop integrands is to sum many Feynman diagrams. That quickly becomes cluttered and hides structure. This off‑shell recursive approach generalizes the Berends–Giele recursion (a standard tree‑level tool) to loop level. Writing the gluon sector in matrix form makes the sewing step local in the sense that only the matrices near the sewn legs must be handled. The authors argue this could help uncover relations among amplitudes at higher loop order and make the off‑shell structure clearer.