Dyck Words and Multi-Quark Primitive Amplitudes

TL;DR

The paper addresses how KK relations, flavour, and quark-number conservation constrain purely multi-quark primitive amplitudes at tree level. It develops a Dyck-word–based construction to identify a minimal independent basis, showing that the number of independent primitives with distinct flavours is $(n-2)!/(n/2)!$ and providing an explicit $A(1,2,\text{Dyck})$-style basis. The approach leverages non-crossing quark-line graphs and orientational constraints to systematically reduce the primitive set, with potential implications for all-n colour decompositions and higher-loop extensions. This framework clarifies the combinatorial structure of multi-quark amplitudes and offers a constructive path toward more efficient amplitude computations in collider physics.

Abstract

I study group theory (Kleiss-Kuijf) relations between purely multi-quark primitive amplitudes at tree level, and prove that they reduce the number of independent primitives to (n-2)!/(n/2)!, where n is the number of quarks plus antiquarks, in the case where quark lines have different flavours. I give an explicit example of an independent basis of primitives for any n which is of the form A(1,2,sigma), where sigma is a permutation based on a Dyck word.

Figures (4)

• Figure 1: The planar graphs contributing to a four particle primitive amplitude with cyclic ordering 1234.
• Figure 2: (a) Quark line graph for the permutation $\sigma=(12345678)$; (b) an example of a permutation giving rise to crossed quark lines -- the corresponding primitive amplitude is zero; (c) a relation between primitive amplitudes.
• Figure 3: Dyck words for $r=3$ (top row) and the quark line graph topologies they describe.
• Figure 4: A quark line graph for a primitive multi-quark amplitude, showing the general structure of a zone with a boundary of the quark line with ends labelled $x$ and $y$, and with sub-zones $\{\alpha_1\}\ldots\{\alpha_{s}\}$. The orientation of the boundary $(i\to j)$ of sub-zone $\{\alpha_m\}$ is explicitly shown to be in the wrong direction, and the substructure of $\{\alpha_m\}$ is also explicitly shown and labelled $\{\beta\}$.