One-loop $\mathbf{\beta }$ function of noncommutative scalar $QE{D}_4$

In this paper, we consider the $\beta$ function at one-loop approximation for noncommutative scalar QED. The renormalization of the full theory, including the basic vertices, and the renormalization group equation are fully established. Next, the complete set of the one-loop diagrams corresponding to the first-order radiative corrections to the basic functions is considered: gauge, charged scalar and ghost fields self-energies, and three- and four-point vertex functions $<{\varphi }^{\mathrm{†}}\varphi A>$ and $\left<\phi^\dag \phi A A\right>$, respectively. We pay special attention to the noncommutative contributions to the renormalization constants. To conclude, the one-loop $\beta$ function of noncommutative scalar QED is then computed and comparison to known results is presented.