Minimal Basis in Four Dimensions and Scalar Blocks

TL;DR

The paper identifies a genuine four-dimensional minimal basis for tree-level color-ordered gauge-theory amplitudes in the N^{k-2}MHV sector, with dimension given by the Eulerian number $\eulerian{n-3}{k-2}$. It introduces scalar blocks $m_{n,k}(\alpha,\beta)$, defined via CHY and Witten–RSV formalisms, which decompose YM amplitudes as $A^{YM}_{n,k}(\gamma)=\sum_{\beta\in\mathcal{B}_k} F(\gamma,\beta) A^{YM}_{n,k}(\beta)$ with $F$ built from $m_{n,k}$ and its inverse; a parity-invariant scalar-block combination $m^{\text{scalar}}_{n,k}$ yields purely Mandelstam-coefficient expansions, at the cost of doubling the basis. The work also constructs an intrinsically four-dimensional KLT relation for gravity and extends the framework to maximally supersymmetric theories via Witten–RSV, introducing supersymmetric scalar blocks and their use in analogous expansions. Unphysical poles arising from the scalar blocks cancel across neighboring $k$-sectors, and explicit MHV/NMHV examples illustrate the machinery, including a six-point NMHV case. The results illuminate four-dimensional relations beyond the generic BCJ/KLT picture and hint at deep connections to the four-dimensional double-copy and potential string-theoretic interpretations.

Abstract

We find a construction that expresses any tree-level $n$-particle ${\rm N^{k-2}MHV}$ 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 "${\rm N^{k-2}MHV}$ 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.