Past exam of the quantum field theory course of the University of Oxford / 2017 / 4 / a / Solution

At fixed external momenta, fixed regulator and fixed bare coupling, define the total scale derivativeThe bare vertex has no independent dependence on the arbitrary renormalization scale. Differentiating ${\Gamma }^{(n)}=Z_{\varphi }^{n/2}{\Gamma }_0^{(n)}$ therefore givesDefining the two scale-response functions as in the source yieldsThis is the Callan-Symanzik equation. Its minus sign and factor $n/2$ follow from the specified $Z_{\varphi }^{n/2}$ multiplying the bare vertex. In particular, the source's $\gamma =\mu d\log Z_{\varphi }/d\mu$ is twice the field anomalous dimension defined using $\log Z_{\varphi }^{1/2}$. A convention change must be made consistently in both definitions and equation.