# How to Calculate Anomalous Dimensions in (Effective) QED

+ 2 like - 0 dislike
673 views

I am following the conventions here. Consider the (effective) QED Lagrangian

$$\mathcal{L}=-\frac{1}{4}Z_3F_{\mu\nu}^2+Z_2\bar{\psi}i\gamma^{\mu}\partial_{\mu}\psi-Z_2Z_mm\bar{\psi}\psi+Z_eZ_2\sqrt{Z_3}e\bar{\psi}\gamma^{\mu}A_{\mu}\psi+\sum_j C_j\mathcal{O}_j$$

where $\mathcal{O}_j$ are local operators involving any number of $A$ fields and $\psi$ fields (and of course, derivatives). Consider in particular the operator

$$\mathcal{O}=Z\ \bar{\psi}\gamma^{\mu}\partial_{\mu}\psi\ \bar{\psi}\gamma^{\nu}\partial_{\nu}\psi$$

I want to calculate the anomalous dimension of this operator at one loop. I know that this is indicated by $Z$ but I am clueless about how to proceed.

Could anyone give me a hint or a reference which might help me perform the calculation?

This post imported from StackExchange Physics at 2016-05-31 07:24 (UTC), posted by SE-user Anarchist Birds Worship Fungus
retagged May 31, 2016

I think Andrey Grozin's http://arxiv.org/pdf/hep-ph/0508242.pdf works quite well enough if you are looking for a general strategy to calculate the anomalous dimension of an operator. You need to somehow define $Z$, i.e. you need to develop a scheme.
Now let's say you have defined your scheme or you have simply tried one of the conventional ones. The rest is easy, you need to find a place for this renormalization constant $Z$ in your theory. If you cannot find a place for your $Z$, either you are mixing things in your Lagrangian or you have not appropriately defined this counter-term.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysics$\varnothing$verflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.