  The scattering equations provide a powerful framework for the study of
scattering amplitudes in a variety of theories. Their derivation from
ambitwistor string theory led to proposals for formulae at one loop on a torus
for 10 dimensional supergravity, and we recently showed how these can be
reduced to the Riemann sphere and checked in simple cases. We also proposed
analogous formulae for other theories including maximal super-Yang-Mills theory
and supergravity in other dimensions at one loop. We give further details of
these results and extend them in two directions. Firstly, we propose new
formulae for the one-loop integrands of Yang-Mills theory and gravity in the
absence of supersymmetry. These follow from the identification of the states
running in the loop as expressed in the ambitwistor-string correlator.
Secondly, we give a systematic proof of the non-supersymmetric formulae using
the worldsheet factorisation properties of the nodal Riemann sphere underlying
the scattering equations at one loop. Our formulae have the same decomposition
under the recently introduced Q-cuts as one-loop integrands and hence give the
correct amplitudes.
