One-loop Photon's Effective Action in the Noncommutative Scalar QED\(_{3}\)

In this paper, we consider the evaluation of the effective action for photons coupled to charged scalar fields in the framework of a \((2+1)\)-dimensional noncommutative spacetime. In order to determine the noncommutative Maxwell Lagrangian density, we follow a perturbative approach, by integrating out the charged scalar fields, to compute the respective graphs for the vev's \(\left\langle AA \right\rangle\), \(\left\langle AAA \right\rangle\) and \(\left\langle AAAA \right\rangle\). Surprisingly, it is shown that these contributions are planar and that, in the highly noncommutative limit, correspond to the Maxwell effective action and its higher-derivative corrections. It is explicitly verified that the one-loop effective action is gauge invariant, as well as under discrete symmetries: parity, time reversal and charge conjugation. Moreover, a comparison of the main results with the noncommutative QED\(_{3}\) is established. In particular, the main difference is the absence of parity violating terms in the photon's effective action coming from integrating out the charged scalar fields.