A Novel On-Shell Recursive Relation

We present a novel framework for deriving on-shell recursion relations, with a specific focus on biadjoint and pure Yang-Mills theories. Starting from the double-cover CHY factorization formulae, we identify a suitable set of independent kinematic variables that enables the reconstruction of amputated currents from amplitudes. As a byproduct, this new recursive structure recasts the BCJ numerators into an explicitly on-shell factorized form.

💡 Research Summary

The paper introduces a new on‑shell recursive framework for tree‑level scattering amplitudes, derived from the double‑cover formulation of the Cachazo‑He‑Yuan (CHY) representation. By extending the usual SL(2,ℂ) gauge fixing to GL(2,ℂ), the authors fix four punctures on the double‑cover Riemann sphere, which turns one of the scattering equations into a propagator‑like factor. This modification yields off‑shell objects—amputated currents J₍ϕ³₎ⁿ—that carry one off‑shell leg while the remaining legs stay on shell.

A central observation is that these currents depend only on a reduced set of Mandelstam invariants. For an n‑point amplitude the total number of independent kinematic invariants is n(n‑3)/2, but the invariants associated with the effective mass of the off‑shell leg (e.g. s₁₂, s₂… ) never appear in the currents. The authors therefore define a “common set” ˜Kₙ of independent variables that excludes the off‑shell mass invariants. For n≥5 this set is simply ˜Kₙ, while for n=4 they augment it with a single invariant s₁₃ to obtain K₄. The full common set Kₙ = ˜Kₙ ∪ {s₁₃} contains n(n‑3)/2 − 1 independent invariants.

Because the currents are independent of the off‑shell mass invariant, one can consistently set that invariant to zero—effectively projecting the off‑shell leg onto the massless shell. This projection allows any amputated current to be reconstructed directly from the corresponding on‑shell amplitude. The authors demonstrate the procedure explicitly for four‑, five‑, six‑, seven‑ and eight‑point examples. At four points the factorization is trivial: the amplitude splits into two channels, 1/s₃₄ + 1/s₂₃, reflecting the product of two three‑point currents (which are unity). For higher points the factorization formula derived from the double‑cover CHY integral (equation 14) is applied recursively, yielding a compact on‑shell recursion relation (equation 30): \