Cornwall-Jackiw-Tomboulis effective potential for canonical noncommutative field theories

We apply the Cornwall-Jackiw-Tomboulis (CJT) formalism to the scalar $\lambda \phi^{4}$ theory in canonical-noncommutative spacetime. We construct the CJT effective potential and the gap equation for general values of the noncommutative parameter $\theta_{\mu\nu}$. We observe that under the hypothesis of translational invariance, which is assumed in the effective potential construction, differently from the commutative case ($\theta_{\mu\nu}= 0$), the renormalizability of the gap equation is incompatible with the renormalizability of the effective potential. We argue that our result, is consistent with previous studies suggesting that a uniform ordered phase would be inconsistent with the infrared structure of canonical noncommutative theories.