I need an relation between the renormalization constant matrix and the anamalous dimension matrix. Now I found the following derivation \begin{equation} \begin{split} \gamma (a_\mu) &= Z^{-1}(\mu) \mu \frac{d}{d\mu} Z(\mu) \\ &= Z^{-1}(a_\mu) \mu \frac{da_\mu}{d\mu} \frac{d}{da_\mu}Z(a_\mu) \\ &= -\beta (a_\mu)Z^{-1}(a_\mu)\frac{d}{da_\mu}Z(a_\mu). \end{split} \end{equation} The first three steps are clear to me. Just add some derivatives and insert the beta function. But now my source is jumping to the final expression of the first order $\gamma^{(1)}$. I know it has something to do with the pertubative expansion of Z \begin{equation} Z = \mathbb{1} + Z^{(1)}\frac{a_\mu}{\epsilon} + \mathcal{O}(a^2_\mu), \end{equation} but do not now where the simple expression in the next line comes from \begin{equation} \gamma^{(1)}(a_\mu) = -2Z^{(1)}? \end{equation}


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.