= Solution
Write $\mathcal D_0=\mu\partial_\mu+\beta_0(g_0)\partial_{g_0}$. Allow a general bare coefficient $\gamma_0$ temporarily, so the starting equation is $(\mathcal D_0-n\gamma_0/2)\widehat\Gamma^{(n)}=0$. Substitute $\widehat\Gamma^{(n)}=Z_\phi(g_0)^{-n/2}\Gamma^{(n)}(p,g(g_0),\mu)$ and apply the product and chain rules:
$$
0=Z_\phi^{-n/2}\left[
\mu\partial_\mu+\beta_0\frac{dg}{dg_0}\partial_g
-\frac n2\left(\gamma_0+\beta_0\frac{d\log Z_\phi}{dg_0}\right)\right]\Gamma^{(n)}.
$$
Multiplying by $Z_\phi^{n/2}$ identifies
$$
\boxed{\beta(g(g_0))=\beta_0(g_0)\frac{dg}{dg_0},\qquad
\gamma(g(g_0))=\gamma_0(g_0)+\frac{\beta_0(g_0)}{Z_\phi(g_0)}\frac{dZ_\phi}{dg_0}.}
$$
Thus the <renormalization-group beta function> transforms by the <chain rule> under a <coupling> reparametrization, while the <field> coefficient also records the scale dependence of the finite or divergent <field> normalization. For the actual bare parametrization here, substitute $\beta_0=-\epsilon g_0$ and $\gamma_0=0$.
Back to article page
