Gravitational corrections to two-loop beta function in quantum electrodynamics

In this work, we use the framework of effective field theory to couple Einstein's Gravity to Quantum Electrodynamics (QED) and determine the gravitational corrections to the two-loop beta function of the electric charge. Our results indicate that gravitational corrections do not alter the running behavior of the electric charge; on the contrary, we observe that it gives a positive contribution to the beta function, making the electric charge grow faster. The opposite occurs to the $\lambda$ beta function in the Einstein-Scalar-QED system, where at one-loop order we observe that gravity gives a negative contribution to the $\lambda$ beta function, indicating that, if the scalar particle has a mass a few orders below Planck scale, $\lambda$ can be asymptotically free.