Getting more flavour out of one-flavour QCD

TL;DR

This work shows that tree-level massless QCD amplitudes can be computed without invoking flavour, by constructing a minimal basis of one-flavour primitives organized by Dyck-tree topologies and KK relations. A flavour recursion expresses $k$-flavour primitives in terms of one-flavour primitives, with the recursion terminating at a highest-maturity Dyck configuration; consequently, all massless QCD tree amplitudes can be obtained from the closed-form tree-level solution of $ ext{N}=4$ SYM. By exploiting planarity and group-theory relations, the one-flavour sector effectively embodies $ ext{N}=1$ supersymmetry, enabling a projection from $ ext{N}=4$ SYM to QCD primitives and allowing loop amplitudes to be built via unitarity from tree-level inputs. This framework reduces flavour to combinatorics and offers potential computational benefits for high-multiplicity jet processes and fixed-order QCD predictions at colliders, while suggesting connections to BCJ relations and alternative recursion formalisms.

Abstract

We argue that no notion of flavour is necessary when performing amplitude calculations in perturbative QCD with massless quarks. We show this explicitly at tree-level, using a flavour recursion relation to obtain multi-flavoured QCD from one-flavour QCD. The method relies on performing a colour decomposition, under which the one-flavour primitive amplitudes have a structure which is restricted by planarity and cyclic ordering. An understanding of SU(3)_c group theory relations between QCD primitive amplitudes and their organisation around the concept of a Dyck tree is also necessary. The one-flavour primitive amplitudes are effectively N=1 supersymmetric, and a simple consequence is that all of tree-level massless QCD can be obtained from Drummond and Henn's closed form solution to tree-level N=4 super Yang-Mills theory.

Figures (6)

• Figure 1: Constructing quark line graphs based around the Dyck words $XYXY$ and $XXYY$. The first column shows the two Dyck topologies, and the second column shows the two possible flavour pair allocations for each topology. Following the direction of the arrows, each row then depicts the four possible choices of signature for each flavour pair allocation. Four permutations corresponding to one graph out of each of these rows constitute the Dyck permutations which are then used to construct a basis of QCD primitives.
• Figure 2: The quark line graph for the above Dyck word is shown on the left. The quark line directions have been chosen in accordance with an all-positive signature. On the right is the associated rooted oriented Dyck tree, which can be seen as a dual graph to the quark line graph once the line $(1\to2)$ is removed and the circle representing the edge of the plane is identified as a node. The levels $v_i$ of the tree are given on the far right.
• Figure 3: The Dyck word and quark line graph of a primitive with $k$ quark lines, for which the rooted oriented Dyck tree has maximum height, or maturity.
• Figure 4: Path back to the lowest node below two different nodes of the tree, one at level $v_i$, associated with quark line $\bar{q}_1\to q_2$, and the other at level $v_j$, associated with the quark line $\bar{q}_3\to q_4$.
• Figure 5: The termination of a scalar $\phi_{AB}$ results in two external gluinos $\tilde{g}_A$ and $\tilde{g}_B$ (to be identified as quarks) of different flavours, since the coupling requires $A\ne B$.