Minimal Basis in Four Dimensions and Scalar Blocks

Abstract

We find a construction that expresses any tree-level $n$-particle ${\mathrm{N}}^{\mathrm{k}-2}\mathrm{MHV}$ color-ordered partial amplitude in gauge theory as a linear combination of a basis of dimension $\eulerian{n-3}{k-2}$. Here $\eulerian{p}{q}$ denotes the $(p,q)$ Eulerian number. The coefficients of the expansion are independent of the helicities of the particles. This basis is a four-dimensional refinement of the $(n-3)!$-element BCJ basis which is valid in any number of dimensions. The construction uses a new kind of objects which we call {\it scalar blocks}. Here we initiate the study of these objects. Scalar blocks provide an "${\mathrm{N}}^{\mathrm{k}-2}\mathrm{MHV}$ sector" decomposition of a bi-adjoint scalar amplitude in four dimensions. As byproducts of the construction, we also find an intrinsically four-dimensional version of KLT relations for gravity amplitudes.

30 pages

References in corpus (6)

• New Relations for Gauge-Theory Amplitudes
• Minimal Basis for Gauge Theory Amplitudes
• Gravity from Rational Curves
• A simple formula for gravitational MHV amplitudes
• A "Twistor String" Inspired Formula For Tree-Level Scattering Amplitudes in N=8 SUGRA
• Resultants and Gravity Amplitudes

Cited by in corpus (8)

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• Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism
• CHY formula and MHV amplitudes
• Chiral Splitting and $\mathcal{N}=4$ Einstein--Yang--Mills Tree Amplitudes in 4d
• One-loop integrand from generalised scattering equations