Abstract
The recursive method of Berends and Giele to compute tree-level gluon amplitudes is revisited using the framework of ten-dimensional super Yang-Mills. First we prove that the pure spinor formula to compute SYM tree amplitudes derived in 2010 reduces to the standard Berends-Giele formula from the 80s when restricted to gluon amplitudes and additionally determine the fermionic completion. Second, using BRST cohomology manipulations in superspace, alternative representations of the component amplitudes are explored and the Bern-Carrasco-Johansson relations among partial tree amplitudes are derived in a novel way. Finally, it is shown how the supersymmetric components of manifestly local BCJ-satisfying tree-level numerators can be computed in a recursive fashion.
Highlights
Problem [3, 4], and the BRST operator in the associated pure spinor superspace powerfully embodies gauge invariance and supersymmetry [5]
We prove that the pure spinor formula to compute SYM tree amplitudes derived in 2010 reduces to the standard Berends-Giele formula from the 80s when restricted to gluon amplitudes and determine the fermionic completion
Using BRST cohomology manipulations in superspace, alternative representations of the component amplitudes are explored and the Bern-Carrasco-Johansson relations among partial tree amplitudes are derived in a novel way
Summary
The theta-expansions of ten-dimensional multiparticle superfields have recently [15] been simplified using supersymmetric Berends-Giele currents which generalize the gluonic currents defined by Berends and Giele [16] Using these simplified expansions, the pure spinor superspace formula to compute ten-dimensional color-ordered SYM amplitudes at tree level [6], ASYM(1, 2, . + (X12γmX5)em34 + (X34γmX12)em5 + (X5γmX34)em12 + cyclic(12345) , streamlining the earlier approach in [17] based on the above J1m2...p. Using the generating series of supersymmetric Berends-Giele currents discussed in [15, 18], it will be shown that the generating series of ten-dimensional SYM tree-level amplitudes takes a very simple form, Tr. Note that the left-hand side of (1.5) matches the ten-dimensional SYM Lagrangian evaluated on the generating series Fmn(x, θ = 0) and Wα(x, θ = 0) defined below
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