Gauge-fixing independence of IR divergences in non-commutative U(1), perturbative tachyonic instabilities and supersymmetry

It is argued that the quadratic and linear non-commutative IR divergences that occur in U(1) theory on non-commutative Minkowski spacetime for small non-commutativity matrices $\theta^{\mu\nu}$ are gauge-fixing independent. This implies in particular that the perturbative tachyonic instability produced by the quadratic divergences of this type in the vacuum polarization tensor is not a gauge-fixing artifact. Supersymmetry can be introduced to remove from the renormalized Green functions at one loop, not only the non-logarithmic non-commutative IR divergences, but also all terms proportional to $\theta^{\mu\nu}p_\nu