Given a $n$-point bare Green function in a massless asymptotically free theory, we have that the following limit exists and is finite \begin{equation} \lim_{\Lambda\rightarrow\infty} Z^{-n/2}(g_0,\Lambda/\mu)G^0_n(x_1,\ldots,x_n,g_0,\Lambda) = G_n(x_1,\ldots,x_n,g,\mu) \end{equation} where $\Lambda$ is an UV regulator. This equation for $\Lambda\gg\mu$ can be written as \begin{equation} \tag{1} Z^{-n/2}(g_0,\Lambda/\mu)G^0_n(x_1,\ldots,x_n,g_0,\Lambda) \approx G_n(x_1,\ldots,x_n,g,\mu) \end{equation} up to terms which vanish for $\Lambda\rightarrow \infty$.

Now if I hold fixed the renormalized coupling $g$ and the renormalization scale $\mu$ and I differentiate the equation in $\log\Lambda$, since the right hand side is cutoff independent I get

\begin{equation} \frac{d}{d\log\Lambda} \left[Z^{-n/2}(g_0,\Lambda/\mu)G^0_n(x_1,\ldots,x_n,g_0,\Lambda)\right] = 0 \end{equation} which becomes \begin{align} &\left[\Lambda\frac{\partial}{\partial \Lambda}+\beta(g_0,\Lambda/\mu)\frac{\partial}{\partial g}-\frac{n}{2}\gamma(g_0,\Lambda/\mu)\right] G^0_n(x_1,\ldots,x_n,g_0,\Lambda) = 0\\ &\beta(g_0,\Lambda/\mu) = \Lambda\frac{\partial}{\partial \Lambda}\biggr\rvert_{g,\mu}g_0\\ &\gamma(g_0,\Lambda/\mu) = \Lambda\frac{\partial}{\partial \Lambda}\biggr\rvert_{g,\mu}\log Z(g_0,\Lambda/\mu) \end{align} where $\gamma$ and $\beta$ do not really depend from $\Lambda/\mu$ since they can be expressed in terms of quantities which don't depend on $\mu$. The bare coupling can also be thought as a function $g_0 = g_0(g,\Lambda/\mu)$ and, since it is constant in $\mu$, can be related to the renormalized coupling with the equation \begin{equation} \beta(g) \frac{\partial g_0}{\partial g} = \beta(g_0) \end{equation}

Now if we take as an example a $d$ dimensional, scalar two point function $G_0(x,g_0,\Lambda)$ we can write it as \begin{equation} \tag{2} G_0(x,g_0,\Lambda) = \frac{1}{x^d}\mathcal{G}^{(2)}(g_0,x\Lambda)\exp\left[{\int_{g_0(\Lambda)}^{g_0(1/x)}dg_0\frac{\gamma(g_0)}{\beta(g_0)}}\right] \end{equation} where $\mathcal{G}^{(2)} = 1+\sum_{n\geq1} \gamma_ng^n$ is a RG invariant. My questions are basically two:

How do I relate eq. (2) asymptotically to the usual perturbative renormalization, namely is it possible to show that I can write eq. (2) right hand side as a divergent renormalization factor times a finite green function (and maybe a finite remainder) such as eq. (1) would suggest, i.e. $G_0(x,g_0,\Lambda) = Z(g,\Lambda x) G(g,x)$?

Is it enough to replace $g_0 = g+\sum_{n>1} c_n(\Lambda x)g^n$ (where the $c_n$ should basically be the coupling counterterms?) in the formula and separate the terms? Is it there a less naive approach to it?

I would also consider acceptable as an answer a reference.



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