# Charge dependence of operators in QED renormalization

Science Advisor
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?

Thanks in advance!

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## Answers and Replies

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?