                  \overrightarrow{\partial_{\beta}}}g(x)$$

The NCQED action, using the above line of reasoning, is $$S_{NCQED} = \int d^Dx \Bigg( -\frac{1}{4g^2}F^{\mu\nu}(x)\ast F_{\mu\nu}(x)
               + \iota\bar{\psi}(x)\gamma^{\mu}\ast D_{\mu}\psi(x)
               - m\bar{\psi}(x)\ast\psi(x)\Bigg)$$ where $g$ is the coupling and $$F_{\mu\nu} = \partial_{\mu}A_{\nu}(x) - \partial_{\nu}A_{\mu}(x)
              + \iota g[A_{\mu}(x),A_{\nu}(x)]_{\ast}$$ The covariant derivative is given by $$D_{\mu}\psi(x) = \partial_{\mu}\psi(x) + \iota gA_{\mu}(x)\ast\psi(x)$$ The action is invariant under the noncommutative $U(1)$ transformations obtained by replacing all the products in the standard transformations by the corresponding star products.

The noncommutativity is encoded in the star product and from the above expressions it is quite evident that the field strength, even in the case of $U(1)$, theory is nonlinear in gauge field and it is precisely this nonlinearity that gives rise to additional vertices for the gauge field. It is now a straight forward task to derive the Feynman rules from the above action [@sh1], Arfaei et.al [@pheno]. It is found that apart from generating the three and four point vertices for the gauge field self interaction, each interaction vertex picks up a momentum dependent phase factor, whose argument typically has the structure $\frac{\iota}{2} p \wedge k$. The $\wedge$ product, in general, is defined as $$p \wedge k = p_{\mu} \Theta^{\mu\nu} k_{\nu}$$ In the case of theories with only space-like noncommutativities, only the space-space elements contribute and using Eq.(2) it simply reduces to the usual vector cross-product of the two three momenta i.e. $$p \wedge k = \vec{p}\times \vec{k}$$ The process, $\gamma\gamma \longrightarrow \gamma\gamma$ takes place at the one loop level in standard QED as well as SM and thus is quite suppressed. But the presence of Yang-Mills type coupling for the photon field in NCQED enables the process to take place at the tree level. This makes the a