= Solution
The dimensionless bare parameter is $g_0=\lambda_0\mu^{-\epsilon}$, so direct differentiation at fixed $\lambda_0$ gives
$$
\boxed{\beta_0(g_0)=\mu\frac{dg_0}{d\mu}\bigg|_{\lambda_0}=-\epsilon g_0.}
$$
The dimensionful bare vertex itself is scale independent. Reexpressing it as $\widehat\Gamma^{(n)}(p,g_0,\mu)=\Gamma_0^{(n)}(p,g_0\mu^\epsilon)$ introduces only the combination $g_0\mu^\epsilon$. Hence
$$
\left(\mu\partial_\mu-\epsilon g_0\partial_{g_0}\right)\widehat\Gamma^{(n)}=0.
$$
There is no multiplicative <field> rescaling in this bare parametrization, so
$$
\boxed{\gamma_0(g_0)=0.}
$$
This does not assert that the interacting renormalized <field> has zero <anomalous dimension>. It says that the unrescaled bare <field> has no additional normalization factor under this change of the arbitrary scale.
Back to article page
