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

Understand

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 $\beta$-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.