To facilitate a simultaneous treatment of an arbitrary number of colors in representation theory-based descriptions of QCD color structure, we derive an $\textit{N}$-independent reduction of SU($\textit{N}$) tensor products. To this end, we label each irreducible representation by a pair of Young diagrams, with parts acting on quarks and antiquarks. By combining this with a column-wise multiplication of Young diagrams, we generalize the Littlewood-Richardson rule for the product of two Young diagrams to the product of two Young diagram pairs, achieving a general-$\textit{N}$ decomposition.

Germany

Stefan Keppeler

Sweden

Malin Sjodahl

Austria

Denmark

Bernanda Telalovic

An $\textit{N}$-independent tensor decomposition for SU($\textit{N}$)

The efficient computation of color-summed QCD amplitudes at high parton multiplicities remains a central challenge for precision collider predictions. Existing approaches using trace, color-flow, or adjoint bases suffer from non-orthogonality, which complicates the color algebra and scales poorly with multiplicity. In this work, we present an off-shell recursive framework for computing all-gluon tree-level amplitudes directly in orthogonal multiplet bases. Utilizing Wigner 6$\textit{j}$ coefficients, we construct an algorithm that builds multiplet-projected off-shell currents from lower-point currents. By optimizing the recursion through partial summation and caching, we find that the computational complexity of calculating $\textit{n}$-gluon color-summed squared amplitudes scales as $$ \mathcal{O}\left({17}^n\right) $$ . This demonstrates the potential competitiveness of multiplet bases for high-multiplicity processes.

Oskar Bolinder

Rikkert Frederix

All-gluon amplitudes with off-shell recursion in multiplet bases

A complication in the treatment of any strongly charged particle is the SU(3) color structure. For the standard model quarks, antiquarks and gluons there are various well-known strategies for dealing with the color structure, including orthogonal multiplet bases. For exotic representations, beyond triplets, antitriplets and octets, non-orthogonal bases have been systematically worked out only recently. In this letter we demonstrate how to construct orthogonal multiplet bases also for processes including exotic representations.

Orthogonal color bases for exotic representations

We construct a set of Wigner 6𝑗 symbols with gluon lines (adjoint representations) in closed form, expressed in terms of similar 6𝑗 symbols with quark lines (fundamental representations). Together with these Wigner 6𝑗 symbols with quark lines, this gives a set of 6𝑗 symbols sufficient for treating QCD color structure for any number of external particles, in or beyond perturbation theory. This facilitates a complete treatment of QCD color structure in terms of orthogonal multiplet bases, without the need of ever explicitly constructing the corresponding bases. We thereby open up for a completely representation theory based treatment of SU(N) color structure, with the potential of significantly speeding up the color structure treatment.

Simon Plätzer

Wigner 6j symbols with gluon lines: completing the set of 6j symbols required for color decomposition

The chirality-flow formalism, combined with good choices of gauge reference vectors, simplifies tree-level calculations to the extent that it is often possible to write down amplitudes corresponding to Feynman diagrams immediately. It has also proven to give a very sizable speedup in a proof of concept implementation of massless tree-level QED in MadGraph5_aMC@NLO. In the present paper we extend this analysis to QCD, including massive quarks. We define helicity-dependent versions of the gluon vertices, derive constraints on the spinor structure of propagating gluons, and explore the Schouten identity to simplify the four-gluon vertex further. For massive quarks, the chirality-flow formalism sheds light on how to exploit the freedom to measure spin along any direction to shorten the calculations. Overall, this results in a clear speedup for treating the Lorentz structure at high multiplicities.

Emil Boman

Andrew Lifson

Adam Warnerbring

CERN

Zenny Wettersten

Simplifying QCD event generation with chirality flow, reference vectors and spin directions

In a recent paper we introduced the chirality-flow formalism, a method for simple and transparent calculations of Feynman diagrams based on the left- and right-chiral  $$\mathfrak {sl}(2,\mathbb {C})$$  <math> <mrow> <mi>sl</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mi>C</mi> <mo>)</mo> </mrow> </math>   nature of spacetime. While our previous work focused on massless QED and QCD at tree-level, we here extend the chirality-flow formalism to the full (tree-level) Standard Model, including massive particles and electroweak interactions – for which the W-interaction simplifies elegantly due to its chiral nature. We illustrate how values of Feynman diagrams can be immediately written down with some representative examples.

Joakim Alnefjord

Christian Reuschle

The chirality-flow formalism for the standard model

We take a fresh look at Feynman diagrams in the spinor-helicity formalism. Focusing on tree-level massless QED and QCD, we develop a new and conceptually simple graphical method for their calculation. In this pictorial method, which we dub the chirality-flow formalism, Feynman diagrams are directly represented in terms of chirality-flow lines corresponding to spinor inner products, without the need to resort to intermediate algebraic manipulations.

The chirality-flow formalism

We develop an algorithm for recursively constructing orthogonal multiplet bases for the color space of QCD, for any order of partons and any N  c . This recipe is then applied for explicitly constructing some of these bases. Using the bases, a corresponding set of Wigner 6j coefficients are calculated. The Wigner coefficients offer a method of using multiplet bases without resorting to the explicit expressions of the basis vectors, which lead to a significant speed-up compared to other methods of treating full color structure.

Johan Thorén

QCD multiplet bases with arbitrary parton ordering

We investigate the effects of keeping the full color structure for parton emissions in parton showers for both LEP and LHC. This is done within the Herwig 7 dipole shower, and includes gluon emission, gluon splitting, initial state branching processes, as well as hadronization. The subleading N  c  terms are included as color matrix element corrections to the splitting kernels by evolving an amplitude-level density operator and correcting the radiation pattern for each parton multiplicity, up to a fixed number of full color emissions, after which a standard leading color shower takes over. Our results are compared to data for a wide range of LEP and LHC observables and show that the subleading N  c  corrections tend to be small for most observables probing hard, perturbative dynamics, for both LEP and LHC. However, for some of these observables they exceed 10%. On soft physics we find signs of significantly larger effects.

Color matrix element corrections for parton showers

We illustrate how QCD color structure elegantly can be decomposed into orthogonal multiplet bases corresponding to irreducible representations of SU( N c ) with the aid of Wigner 3 j and 6 j coefficients. We also show how to calculate the relevant 3 j and 6 j coefficients using multiplet bases and birdtrack techniques and argue that only a relatively small number of Wigner 3 j and 6 j coefficients are required. For up to six gluons plus quark-antiquark pairs we explicitly calculate all 6 j coefficients required for up to NLO calculations.