  We propose a new \emph{CPT}-even and Lorentz-violating nonminimal coupling
between fermions and Abelian gauge fields involving the CPT-even tensor
$(K_{F})_{\mu\nu\alpha\beta}$ of the standard model extension. We thus
investigate its effects on the cross section of the electron-positron
scattering by analyzing the process $e^{+}+e^{-}\rightarrow\mu^{+}+\mu^{-}$.
Such a study was performed for the parity-odd and parity-even nonbirefringent
components of the Lorentz-violating $(K_{F})_{\mu\nu\alpha\beta}$ tensor.
Finally, by using experimental data available in the literature, we have
imposed upper bounds as tight as $10^{-12}(eV)^{-1}$ on the magnitude of the
CPT-even and Lorentz-violating parameters while nonminimally coupled.
