Simplify the general sum-over-trees formula for single-minus graviton amplitudes

Develop a simplified, compact expression for the stripped n-graviton single-minus amplitude M_{1…n} at general half-collinear kinematics by reducing the explicit non-recursive sum over unordered Feynman trees given in the formula labeled Eq. (Full), which was derived from the Berends–Giele recursion. The goal is to obtain a more tractable closed form beyond the decay-region product formula.

Background

The paper derives an explicit on-shell recursion and a non-recursive closed form (Eq. (Full)) for the stripped single-minus graviton amplitude M_{1…n} using a Berends–Giele approach adapted to the half-collinear regime. This expression involves a sum over trees whose number grows exponentially with n, making practical use and structural understanding difficult.

In a restricted kinematic decay region, the authors obtain a remarkably simple product formula. However, outside this chamber the general expression remains unwieldy. The authors explicitly state that simplifying the general solution is deferred to future work.