A Lie bracket for the momentum kernel

Abstract

We develop new mathematical tools for the study of the double copy and colour-kinematics duality for tree-level scattering amplitudes using the properties of Lie polynomials. We show that the $S$-map that was defined to simplify super-Yang--Mills multiparticle superfields is in fact a new Lie bracket on the dual space of Lie polynomials. We introduce {\it Lie polynomial currents} based on Berends-Giele recursion for biadjoint scalar tree amplitudes that take values in Lie polynomials. Field theory amplitudes are obtained from the Lie polynomial amplitudes by numerators characterized as homomorphisms from the free Lie algebra to kinematic data. Examples are presented for the biadjoint scalar, Yang--Mills theory and the nonlinear sigma model. That these theories satisfy the Bern-Carrasco-Johansson amplitude relations follows from the identities we prove for the Lie polynomial amplitudes and the existence of BCJ numerators. A KLT map from Lie polynomials to their dual is obtained by nesting the S-map Lie bracket; the matrix elements of this map yield a recently proposed `generalized KLT matrix', and this reduces to the usual KLT matrix when its entries are restricted to a basis. Using this, we give an algebraic proof for the cancellation of double poles in the KLT formula for gravity amplitudes. We finish with some remarks on numerators and colour-kinematics duality from this perspective.

48 pp

References in corpus (14)

• New Relations for Gauge-Theory Amplitudes
• Minimal Basis for Gauge Theory Amplitudes
• The Momentum Kernel of Gauge and Gravity Theories
• Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program
• ICTP Lectures on Covariant Quantization of the Superstring
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• Non-abelian $Z$-theory: Berends-Giele recursion for the ${\alpha }^{\prime }$-expansion of disk integrals
• Gravity and Yang-Mills Amplitude Relations
• A Proof of the Explicit Minimal-basis Expansion of Tree Amplitudes in Gauge Field Theory
• Multiparticle SYM equations of motion and pure spinor BRST blocks
• Recursion Relations for Tree-level Amplitudes in the SU(N) Non-linear Sigma Model
• Fundamental BCJ Relation in N=4 SYM From The Connected Formulation
• Planar binary trees in scattering amplitudes
• Off-shell BCJ Relation in Nonlinear Sigma Model

Cited by in corpus (11)

• Covariant Color-Kinematics Duality
• Off-shell extended graphic rule and the expansion of Berends-Giele currents in Yang-Mills theory
• Differential Operators and Unifying Relations for 1-loop Feynman Integrands from Berends-Giele Currents
• Celestial Berends-Giele current
• Color-Kinematic Numerators for Fermion Compton Amplitudes
• Next-to-MHV Yang-Mills kinematic algebra
• The Algebraic Structure of the KLT Relations for Gauge and Gravity Tree Amplitudes
• Note on graph-based BCJ relation for Berends-Giele currents
• Algebraic Consistency and Explicit Construction of One-Loop BCJ Numerators of Yang-Mills and Related Theories
• Systematic approach to $\ell$-loop planar integrands from the classical equation of motion
• The bi-adjoint scalar $\ell$-loop planar integrand recursion and graded inverse variables