  We present new and explicit formulae for the one-loop integrands of
scattering amplitudes in non-supersymmetric gauge theory and gravity, valid for
any number of particles. The results exhibit the colour-kinematics duality in
gauge theory and the double-copy relation to gravity, in a form that was
recently observed in supersymmetric theories. The new formulae are expressed in
a particular representation of the loop integrand, with only one quadratic
propagator, which arises naturally from the framework of the loop-level
scattering equations. The starting point in our work are the expressions based
on the scattering equations that were recently derived from ambitwistor string
theory. We turn these expressions into explicit formulae depending only on the
loop momentum, the external momenta and the external polarisations. These
formulae are valid in any number of spacetime dimensions for pure Yang-Mills
theory (gluon) and its natural double copy, NS-NS gravity (graviton, dilaton,
B-field), and we also present formulae in four spacetime dimensions for pure
gravity (graviton). We perform several tests of our results, such as checking
gauge invariance and directly matching our four-particle formulae to previously
known expressions. While these tests would be elaborate in a Feynman-type
representation of the loop integrand, they become straightforward in the
representation we use.
