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Proof of a new colour decomposition for QCD amplitudes

Tom Melia

TL;DR

This paper proves Johansson–Ochirov's conjectured new colour decomposition for QCD tree amplitudes by a diagrammatic approach based on Mario World diagrams, which expose the Dyck-word–structured primitive basis. It develops a recursion for colour factors, constructs a linear system relating primitive colour factors to Feynman-diagram colour factors, and uses MW diagrams to realize a one-to-one mapping between basis permutations and colour factors. The proof proceeds inductively on the total nesting level $l_{\text{tot}}$, showing that the colour factors obey the proposed recursion $C_{\ldots a\,\overline{a}\,\overline{b}\ldots} = C_{\ldots \overline{ab}\ldots} + C_{\ldots \overline{b} \, a\, \overline{a} \ldots}$ and thus establishes the JO colour decomposition $\\mathcal{A}_{n,k}=\sum_{\sigma\in\text{Dyck}_{k-1}} C_{1\,\sigma\,\overline{1}} A(1,\sigma,\overline{1})$. The results imply a gauge-group and matter-representation independent structure for QCD primitives and suggest usefulness for broader diagrammatic analyses and potential loop extensions. The work also reinforces connections to KK/BCJ-type relations and Dyck-word based formalisms in multi-quark amplitudes.

Abstract

Recently, Johansson and Ochirov conjectured the form of a new colour decomposition for QCD tree-level amplitudes. This note provides a proof of that conjecture. The proof is based on "Mario World" Feynman diagrams, which exhibit the hierarchical Dyck structure previously found to be very useful when dealing with multi-quark amplitudes.

Proof of a new colour decomposition for QCD amplitudes

TL;DR

This paper proves Johansson–Ochirov's conjectured new colour decomposition for QCD tree amplitudes by a diagrammatic approach based on Mario World diagrams, which expose the Dyck-word–structured primitive basis. It develops a recursion for colour factors, constructs a linear system relating primitive colour factors to Feynman-diagram colour factors, and uses MW diagrams to realize a one-to-one mapping between basis permutations and colour factors. The proof proceeds inductively on the total nesting level , showing that the colour factors obey the proposed recursion and thus establishes the JO colour decomposition . The results imply a gauge-group and matter-representation independent structure for QCD primitives and suggest usefulness for broader diagrammatic analyses and potential loop extensions. The work also reinforces connections to KK/BCJ-type relations and Dyck-word based formalisms in multi-quark amplitudes.

Abstract

Recently, Johansson and Ochirov conjectured the form of a new colour decomposition for QCD tree-level amplitudes. This note provides a proof of that conjecture. The proof is based on "Mario World" Feynman diagrams, which exhibit the hierarchical Dyck structure previously found to be very useful when dealing with multi-quark amplitudes.

Paper Structure

This paper contains 7 sections, 20 equations, 4 figures.

Table of Contents

  1. Introduction and discussion
  2. Proof of the new colour decomposition for QCD trees
  3. The JO conjecture
  4. Recursion relation
  5. Feynman diagram system
  6. 'Mario World' diagrams
  7. Recovering eq. \ref{['eq:rec3']} with MW diagrams

Figures (4)

  • Figure 1: Johansson and Ochirov's diagrammatic notation for primitive colour factors. The colour factor on the left consists of four terms upon expanding out the $\Xi$ operators (see text for explicit expressions). The one on the right consists of twelve such terms.
  • Figure 2: Diagrammatic representation of the recursion relation, eq. \ref{['eq:rec1']}, for primitive colour factors.
  • Figure 3: Example 'Mario World' diagrams.
  • Figure :