Scalar diagrammatic rules for Born amplitudes in QCD

TL;DR

The authors present a scalar diagrammatic framework for Born-level QCD amplitudes, showing that all amplitudes can be built from scalar propagators and a finite set of three- and four-valent primitive vertices, applicable to any number of massive or massless quark pairs. They introduce an off-shell continuation using a fixed null vector $q$, define off-shell amplitudes $O_n$ with degree $d = n_- - 1$, and prove that the diagram degree equals the sum of vertex degrees, enabling recursive constructions that mirror MHV-like structures in a purely field-theoretic setting. The pure-gluon sector uses primitive vertices $V_3$ and $V_4$ with helicity constraints that restrict to $d=0$ or $d=1$, while the massive-quark extension adds quark-specific $V_3$ and $V_4$ vertices and spin-flip terms, all absorbed into a unified scalar-propagator calculus. The framework provides a practical, index-free approach suitable for fast computer implementation and offers a clear connection to Cachazo–Svrček–Witten prescriptions for Born amplitudes.

Abstract

We show that all Born amplitudes in QCD can be calculated from scalar propagators and a set of three- and four-valent vertices. In particular, our approach includes amplitudes with any number of quark pairs. The quarks may be massless or massive. The proof of the formalism is given entirely within quantum field theory.

Figures (3)

• Figure 1: Diagrams contributing to the four-valent gluon vertex. In the last two diagrams only the $\varepsilon^0$ polarisation is exchanged.
• Figure 2: Diagrams contributing to the four-valent quark-gluon vertex. For the internal propagator only "0"-polarisations are exchanged.
• Figure 3: Diagrams contributing to the four-valent quark-quark vertex. For the internal propagator only "0"-polarisations are exchanged.