A recursive method for SYM n-point tree amplitudes

TL;DR

This work develops a recursive, purely supersymmetric framework for color-ordered $n$-point tree amplitudes in ten-dimensional ${ m N}=1$ SYM using pure spinor BRST cohomology. Central to the approach are BRST-covariant building blocks $T_{12oldots p}$ and Berends-Giele currents $M_{i_1oldots i_p}$, organized so that the amplitude ${ m A}_n$ can be written as ${ m A}_n=igra E_{i_1oldsymbol{ angle i_{n-1}}}V_nig rangle$, with a recursion $QM_{12oldots p}= extstylerac{}{} extstyleigl( extstyle M_{12oldots j}M_{j+1oldots p}igr)=E_{12oldots p}$ and a maximal rank limited to $[n/2]$. The paper provides explicit, manifestly cyclic expressions for $n ext{--point}$ amplitudes up to $n=10$ and shows how BRST integration by parts yields compact, cyclic forms. It also connects to the full tree-level superstring amplitude via a string-inspired construction, recovering the field-theory results in the $oldsymbol{ abla' o 0}$ (or $oldsymbol{rac{}{} oldsymbol{ ext{alpha}'}}$) limit and clarifying how the pure spinor framework unifies field theory and string theory viewpoints.

Abstract

We present a recursive method for super Yang-Mills color-ordered n-point tree amplitudes based on the cohomology of pure spinor superspace in ten space-time dimensions. The amplitudes are organized into BRST covariant building blocks with diagrammatic interpretation. Manifestly cyclic expressions (no longer than one line each) are explicitly given up to n=10 and higher leg generalizations are straightforward.