Charge dependence of operators in QED renormalization

[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?

Thanks in advance!

 

[eljose79]

Thank you for your question. Determining the dependence of 1-loop corrections on the bare charge $e$ can be a challenging task, but there are a few approaches you can take.

One method is to use dimensional regularization, where the bare charge $e$ is treated as a dimensionful parameter. This allows you to easily track the dependence on $e$ through the renormalization process. In this approach, the bare charge will appear in the counterterms for the electron and photon propagators, as well as in the vertex corrections.

Another method is to use the background field method, where the gauge field is split into a classical background field and a quantum fluctuation. This allows you to separate the bare charge from the quantum corrections, making it easier to track the dependence on $e$.

In terms of constructing the 1-loop Feynman diagrams for the operators you mentioned, it would be helpful to first write them in terms of the fields and their derivatives. For example, the first operator can be written as $\bar{\psi}\partial^2 A_{\mu} \partial_{\nu} A^{\nu}\psi$. From here, you can use Feynman rules to construct the diagrams, keeping track of the vertices and propagators. The number of vertices will determine the dependence on $e$.

I hope this helps. Let me know if you have any further questions.