Colour Decompositions of Multi-quark One-loop QCD Amplitudes

TL;DR

The paper develops a systematic colour decomposition of one-loop QCD amplitudes with up to seven coloured external states by expressing amplitudes in terms of colour-ordered primitive amplitudes. It introduces an automated algorithm to derive both leading and subleading colour contributions and to uncover relations among multi-quark primitives, driven by fermion-line routing and vertex antisymmetry. The authors provide explicit six- and seven-parton decompositions and demonstrate the approach on W+4 jets, showing subleading colour effects are at the few-percent level and hence often negligible in practice. They also explain how to incorporate colourless states (leptons, vector bosons) into the framework and supply extensive data files with explicit partial amplitudes to facilitate NLO computations at high multiplicity.

Abstract

We describe the decomposition of one-loop QCD amplitudes in terms of colour-ordered building blocks. We give new expressions for the coefficients of QCD colour structures in terms of ordered objects called primitive amplitudes, for processes with up to seven partons. These results are needed in computations of high-multiplicity scattering cross sections in next-to-leading-order (NLO) QCD. We explain the origin of new relations between multi-quark primitive amplitudes which can be used to optimise efficiency of NLO computations. As a first application we compute the full-colour virtual contribution to the cross section for W+4-jet production at the Large Hadron Collider, and verify that it is very well approximated by keeping only the leading terms in an expansion around the formal limit of a large number of colours.

Figures (9)

• Figure 1: Parent diagrams with a distinct routing of quark lines: (a) a parent with a 'left-turner' fermion labelled by '$L$', (b) a 'right-turner' fermion, labelled by '$R$', (c) '$n_f$-left-turner' ($n_fL$) fermions, (d) '$n_f$-right-turner' ($n_fR$) fermions. Two routing labels are associated to the distinct quark flavours $Q$ and ${Q_{2}}$: (e) 'left-left-turner' ($LL$) fermions, (f) 'left-right-turner' ($LR$) fermions. Note that in (f) the first fermion does not in fact enter the loop. In general as many $L/R$-labels appear as there are distinct external quark flavours. Quarks are associated with capital letter $Q$'s, as in primitive amplitudes they arise from adjoint representation colour assignments in our algorithm.
• Figure 2: Four quark partial amplitude decomposed into primitive amplitudes, in the notation of the attached file.
• Figure 3: Parent diagram representation of relations between multi-quark primitive amplitudes: Diagrams (a) show a vanishing sum of primitive amplitudes associated to parent diagrams with a closed fermion loop, as in eq. (\ref{['eq:relation_nf']}). Diagrams (b) show a vanishing sum of primitive amplitudes with at least one gluon in the loop, the pictorial form of eq. (\ref{['eq:relation']}). The relations originate from exchanging the ordering of the fermion pair $\{4_{ {Q_{2}}},5_{ {\overline{Q}_{2}}}\}$,which gives a relative sign. Parent diagrams represent classes of colour-ordered Feynman diagrams. The contributions of all colour-ordered Feynman diagrams cancel once the asymmetry of the fermion-fermion-gluon vertices is taken into account.
• Figure 4: Feynman diagrams $d_i$ contributing to the four-quark amplitude. For simplicity we consider only box and triangle diagrams. These are sufficient to obtain the colour decomposition of the four-quark partial amplitudes.
• Figure 5: A six-quark partial amplitude given as a decomposition into primitive amplitudes. The shown partial amplitude is an extract from the file ${\rm partials\_6\,q.dat}$ in the ancillary material partialsdata.