  The scattering equations give striking formulae for massless scattering
amplitudes at tree level and, as shown recently, at one loop. The progress at
loop level was based on ambitwistor string theory, which naturally yields the
scattering equations. We proposed that, for ambitwistor strings, the standard
loop expansion in terms of the genus of the worldsheet is equivalent to an
expansion in terms of nodes of a Riemann sphere, with the nodes carrying the
loop momenta. In this paper, we show how to obtain two-loop scattering
equations with the correct factorization properties. We adapt genus-two
integrands from the ambitwistor string to the nodal Riemann sphere and show
that these yield correct answers, by matching standard results for the
four-point two-loop amplitudes of maximal supergravity and super-Yang-Mills
theory. In the Yang-Mills case, this requires the loop analogue of the
Parke-Taylor factor carrying the colour dependence, which includes non-planar
contributions.
