Decomposition of one-loop QCD amplitudes into primitive amplitudes based on shuffle relations

TL;DR

This work tackles the problem of decomposing QCD partial amplitudes into primitive amplitudes at both tree and one-loop levels for arbitrary quark–gluon content. It introduces a purely combinatorial method based on generalized shuffle relations, avoiding Feynman diagrams and the inversion of large linear systems. The core contributions are the definitions of tree-level and loop-level shuffle operations (U, C, CU), a loop-closure mechanism for $U(1)$-gluons, and a constructive algorithm that expresses partial amplitudes as linear combinations of primitive amplitudes across all quark flavors, including multi-quark and $U(1)$-gluon effects. The approach generalizes known all-gluon and qq̄+gluon results, is amenable to automation, and is particularly suited for high-multiplicity QCD computations where traditional diagrammatic methods become unwieldy.

Abstract

We present the decomposition of QCD partial amplitudes into primitive amplitudes at one-loop level and tree level for arbitrary numbers of quarks and gluons. Our method is based on shuffle relations. This method is purely combinatorial and does not require the inversion of a system of linear equations.

Figures (6)

• Figure 1: Illustration of planar and non-planar cyclic orderings: The cyclic ordering $(\bar{q}_1 q_1 \bar{q}_2 q_2)$ can be drawn in a planar way on a disc (left diagram), while the cyclic ordering $(\bar{q}_1 \bar{q}_2 q_1 q_2)$ cannot be drawn in a planar way on a disc (second-to-left diagram). The cyclic ordering $(\bar{q}_1^L q_1^L \bar{q}_2^L q_2^L)$ with left/right assignments can be drawn in a planar way on an annulus (second-to-right diagram), while the cyclic ordering $(\bar{q}_1^R q_1^R \bar{q}_2^R q_2^R)$ with left/right assignments cannot be drawn in a planar way on an annulus (right diagram).
• Figure 2: Illustration of the Kleiss-Kuijf relation: Only currents consisting of particles from $\alpha_1$, ..., $\alpha_j$ or $\beta_1$, ..., $\beta_{n-2-j}$ couple to the line from $1$ to $n$. No mixed currents couple to the line from $1$ to $n$.
• Figure 3: Examples of colour-flow diagrams for the three cases discussed in the text. The first diagram shows an example with a closed string corresponding to case 1. The second diagram shows an example without a closed string. Here, the particles attached to the outer ring are colour-disconnected from particles attached to the inner ring (case 2). The third diagram shows also an example without a closed string. Here, the particles attached to the outer ring are colour-connected to the particles attached to the inner ring (case 3).
• Figure 4: Attaching a $U(1)$-gluon in all possible between the quark lines $i$ and $j$ of the primitive $(q_i\bar{q}_iq_j\bar{q}_j)$ amplitude.
• Figure 5: The $RL$-contribution can be re-drawn in the way shown on the right-hand side. The minus sign is due to the fact, that the $U(1)$-gluon is emitted now to the right side of the quark line $j$.