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

TL;DR

This paper applies the CJT formalism to scalar noncommutative field theories, revealing that translational invariance assumptions lead to incompatibility between the renormalizability of the gap equation and the effective potential.

Contribution

It constructs the CJT effective potential and gap equation for noncommutative scalar field theory, highlighting issues with renormalizability under translational invariance.

Findings

Renormalizability of the gap equation is incompatible with that of the effective potential.

Translational invariance assumptions may be inconsistent with the infrared structure of noncommutative theories.

Uniform ordered phases are likely incompatible with the infrared behavior of canonical noncommutative models.

Abstract

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.