# Charge dependence of operators in QED renormalization

1. Dec 13, 2014

### WannabeNewton

Hi all. Consider a UV cutoff regulator $\Lambda$ with an effective QED lagrangian $\mathcal{L}_{\Lambda} = \bar{\psi}_{\Lambda}(i\not \partial - m_{\Lambda})\psi_{\Lambda} - \frac{1}{4}(F^{\mu\nu}_{\Lambda})^2 - e_{\Lambda}\bar{\psi}_{\Lambda}\not A_{\Lambda}\psi_{\Lambda}$. One can of course add more local operators to $\mathcal{L}_{\Lambda}$ by considering corrections from various 1-loop diagrams of the full theory. The $\Lambda$ dependence of these 1-loop corrections is easy to determine from the dimension of the operator.

But how does one systematically go about determining the dependence on the bare charge $e$ of the 1-loop corrections given the operators? For example, if I have the operator $\bar{\psi}(\partial^2 F^{\mu\nu})\sigma_{\mu\nu}\psi$ or the operator $\bar{\psi}\partial_{\mu}F^{\mu\nu}\gamma_{\nu}\psi$ or even $\bar{\psi}\not D^3 \psi$ (which are allowed by the theory since they respect gauge invariance, Lorentz invariance, and parity) how can I tell how the associated 1-loop correction will depend on $e$?

One obvious thing to do would be to try and construct the 1-loop Feynman diagrams that these operators are generated by in the full theory and get the $e$ dependence by looking at the number of vertices but I do not have a good enough intuition to do this. For example what Feynman diagrams would generate the three operators above?