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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?

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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.

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