ones which have a higher mass dimension than the original starting Lagrangian. For example, for gravitational corrections to Quantum Electrodynamics (QED) the divergences generated at one loop have the forms $${\cal L}_{ct} = Gc_1 F^{\mu\nu} \partial^2F_{\mu\nu} + Gc_2\bar{\psi}\gamma_\mu \psi \partial_\nu F^{\mu\nu} + Gc_3 \bar{\psi}\gamma_\mu D_\nu \psi F^{\mu\nu}\,,$$ when regularized dimensionally. (We will discuss regularization with a dimensionful cutoff in the next section.) Equivalently field redefinitions and/or equations of motion can be used to rewrite some of these operators (and the accompanying divergences) as contact interactions such as $${\cal L'}_{ct} =  Gc_4\bar{\psi}\gamma_\mu \psi \bar{\psi}\gamma^\mu \psi\,.$$ The divergences can be absorbed into renormalized parameters of these higher dimensional operators, as in standard effective field theory practice. The scale dependence that arises in dimensional regularization will then be associated with the coefficients of the higher order operators, i.e. the set $c_i$, rather than the original coupling constant, which in the QED case would be the electric charge. Finite remainders will also carry this higher order momentum dependence, i.e. they will be of order $Gq^2$ or $Gq^2\ln q^2$ higher than the original coupling.

Because loops generate these higher order effects in the momentum expansion, attempts to have the loops contribute to the running of 