$\Lambda$ Scattering Equations

Understand

The CHY representation of scattering amplitudes is based on integrals over the moduli space of a punctured sphere.

• We replace the punctured sphere by a double-cover version.
• The resulting scattering equations depend on a parameter $\Lambda$ controlling the opening of a branch cut.
• The new representation of scattering amplitudes possesses an enhanced redundancy which can be used to fix, modulo branches, the location of four punctures while promoting $\Lambda$ to a variable.