Symmetric-group decomposition of SU(N) group-theory constraints on four-, five-, and six-point color-ordered amplitudes at all loop orders

The space of tree level color structures for gluon scattering was determined recently along with its transformation properties under permutations. Here we generalize the discussion to loops, demonstrating a reduction of an arbitrary color diagram to its vacuum skeleton plus rays. For 1-loop there are no residual relations and we determine the space of color structures both diagrammatically and algebraically in terms of certain sunny diagrams. We present the generating function for the characteristic polynomials and a list of irreducible representations for 3 ≤ n ≤ 9 external legs. Finally we present a new proof for the 1-loop shuffle relations based on the cyclic shuffle and split operations.

Section: )supporting

confidence: 58%

Section: Introductionmentioning

confidence: 99%

1-loop color structures and sunny diagrams

Kol

Shir

2015

“…A very helpful consistency test of these formulae was provided by the relations among trace coefficients studied in [84][85][86], which we checked up to five points.…”

Section: Colour Trace Decompositionmentioning

confidence: 99%

Two-loop scattering amplitudes from ambitwistor strings: from genus two to the nodal Riemann sphere

Geyer

Monteiro

2018

111

We derive from ambitwistor strings new formulae for two-loop scattering amplitudes in supergravity and super-Yang-Mills theory, with any number of particles. We start by constructing a formula for the type II ambitwistor string amplitudes on a genus-two Riemann surface, and then study the localisation of the moduli space integration on a degenerate limit, where the genus-two surface turns into a Riemann sphere with two nodes. This leads to scattering amplitudes in supergravity, expressed in the formalism of the two-loop scattering equations. For super-Yang-Mills theory, we import 'half' of the supergravity result, and determine the colour dependence by considering a current algebra on the nodal Riemann sphere, thereby completely specifying the two-loop analogue of the Parke-Taylor factor, including non-planar contributions. We also present in appendices explicit expressions for the Szegő kernels and the partition functions for even spin structures, up to the relevant orders in the degeneration parameters, which may be useful for related investigations in conventional superstring theory. arXiv:1805.05344v3 [hep-th] 17 Dec 2018 6 The amplitude on the bi-nodal Riemann sphere 51 6.1 The measure on the bi-nodal Riemann sphere 52 6.2 The scattering equations on the bi-nodal Riemann sphere 53 6.3 The chiral integrand on the bi-nodal Riemann sphere 55 6.4 Absence of unphysical poles 58 7 Super-Yang-Mills amplitudes 62 7.1 Parke-Taylor factor on the bi-nodal Riemann sphere 62 7.2 Colour trace decomposition 64 8 Discussion 65 -i -A Useful identities 67 A.1 Identities involving the chiral partition function 67 A.2 Useful identities for the PCO gauge slice 68 B Modular transformations and M g,n 68C The separating degeneration q 12 = 0 and uniqueness of f (q 12 )

“…which can be expressed [34][35][36] as i ℓ i c i = 0, where ℓ i are left null vectors of the matrix M iα :…”

Section: Review Of Color-kinematic Dualitymentioning

confidence: 99%

Scattering equations and virtuous kinematic numerators and dual-trace functions

Naculich

2014

Self Cite

Inspired by recent developments on scattering equations, we present a constructive procedure for computing symmetric, amplitude-encoded, BCJ numerators for npoint gauge-theory amplitudes, thus satisfying the three virtues identified by Broedel and Carrasco. We also develop a constructive procedure for computing symmetric, amplitudeencoded dual-trace functions τ for n-point amplitudes. These can be used to obtain symmetric kinematic numerators that automatically satisfy color-kinematic duality. The S n symmetry of n-point gravity amplitudes formed from these symmetric dual-trace functions is completely manifest. Explicit expressions for four-and five-point amplitudes are presented.