$U(N)$ Yang-Mills in non-commutative space time

We present an approach to $U_\star(N)$ Yang-Mills theory in non-commutative space based upon a novel phase-space analysis of the dynamical fields with additional auxiliary variables that generate Lorentz structure and colour degrees of freedom. To illustrate this formalism we compute the quadratic terms in the effective action focusing on the planar divergences so as to extract the $β$-function for the Yang-Mills coupling constant. Nonetheless the method presented is general and can be applied to calculate the effective action at arbitrary order of expansion in the coupling constant, including both planar and non-planar contributions, and is well suited to the computation of low energy one-loop scattering amplitudes.