  We introduce and study a new class of power-counting non-renormalisable gauge
theories in four space-time dimensions. The Lagrangian is an arbitrary function
of the self-dual part of the field strength. The resulting perturbation theory
has the property that whenever two derivatives act on an internal line
propagator, the result is a delta-function and the line collapses to a point.
This means that there remains at most one derivative on each internal line,
which gives improved ulta-violet behaviour. For many purposes, this class of
theories behaves just like ordinary Yang-Mills theory. In particular, they all
share the Yang-Mills theory MHV amplitudes. Moreover, these theories remain
constructible in the sense that higher-point tree level scattering amplitudes
can be obtained from the lower-point amplitudes using the BCFW recursion
relations. Also, the square of these gauge-theory amplitudes gives the
scattering amplitudes of "deformations" of General Relativity, at least for the
low particle numbers that we checked. We compute the one-loop beta-function of
the first new coupling constant, and find it to be positive, which signals the
associated non-renormalisable interaction becoming important in the
ulta-violet.
