 Doubt while computing the animalous dimension of $\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi\bar{\psi}\gamma^{\nu}\partial_{\nu}\psi$

+ 3 like - 0 dislike
109 views

I am following the conventions of http://isites.harvard.edu/fs/docs/icb.topic1146665.files/III-9-RenormalizationGroup.pdf. Consider the QED Lagrangian

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

where $\mathcal{O}_j=Z_j\partial^n\gamma^mA_{\mu}\ldots{}A_{\nu}\bar{\psi}\ldots\psi$ are operators with all fields evaluated at the same point that have any number of photons, fermions, gamma matrices, factors of the metric... and analytic dependence on derivatives. Everything is written using renormalized fields. 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 get the anomalous dimension of this operator at one loop. I know that in order to do that I need to get $Z$ but I am clueless of how to proceed. I have the feeling that I have to consider a correlation function but I don't know which.  Any indication wouldbe greatly appreciated.

asked Oct 29, 2015

 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$\hbar$ys$\varnothing$csOverflowThen 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.