Solving renormalization group equation in QFT

[qft-El]

TL;DR Summary

I've run into this books claiming to solve the RG equation via the method of characteristics. There is a passage I don't quite understand and the method looks different from the usual method of characteristics.

I'm learning about the RG equation and Callan-Symanzik equation. In ref.1 they claim to solve the RG equation via the method of characteristics for PDE. Here's a picture of the relevant part:

Spoiler: Picture

First, the part I don't understand - the one underlined in red. What does "compatible" mean here? Upon explicit differentiation one gets
$$Z_{\text{eff}}^{-n/2}(\lambda)\left[\Lambda\frac{\partial}{\partial\Lambda}+\lambda\frac{d}{d\lambda}g_\text{eff}(\lambda)\frac{\partial}{\partial g_\text{eff}}-n\cdot\frac{1}{2}\lambda\frac{d}{d\lambda}Z_\text{eff}(\lambda)\right]\Gamma_0(p_i;g_\text{eff}(\lambda),\lambda\Lambda)=0\tag{9.97a}.$$
Even if "compatible" just meant that they don't contradict each other, I don't see why $(9.98-9.99)$ specifically should hold. If anything, the choice doesn't seem unique, does it?
Secondly, I'm unsure about the method of characteristics here: it looks very different from the standard method I know (see this question on Physics.SE). I have never seen this dilatation parameter anywhere else, except other books having ref.1 as a reference, is there any source you can advise?
By the way, the independent variables should be $g$ and $\Lambda$ in eq. $(9.94)$, so I guess that once one writes it down, we disregard that $g=g(\Lambda)$ - used to derive the equation and regard it as an independent variable, right?

See this question on Physics.SE for more context.

References

1. Quantum Field Theory and Critical Phenomena, Jean Zinn-Justin, 2021, fifth edition. Section 9.11.2-9.12, eqs. $(9.91-9.101)$.

 

[vanhees71]

It's a bit enigmatically written. You have to consider of course the $n$-point vertex functions, $\Gamma^{(n)}$ for different $n$ to get the full set of RG equations for the normalization factors, $Z$, the running coupling, etc.

Perhaps my own try to explain it (in various renormalization schemes) helps:

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf (Sect. 5.11)

 

[qft-El]

It's a bit enigmatically written. You have to consider of course the $n$-point vertex functions, $\Gamma^{(n)}$ for different $n$ to get the full set of RG equations for the normalization factors, $Z$, the running coupling, etc.

Perhaps my own try to explain it (in various renormalization schemes) helps:

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf (Sect. 5.11)

Hello, vanshees71. I'm glad to receive your feedback.
I happened to check your notes a couple of days ago, by the way. I think it was some other doubt related to renormalization. Regarding the section that you mention, I see that you solve the homogenous RGE there and it also appears to be more general than the one I have. In any case, at a first glance the path you follow seems a different (apparently, if nothing else) one, although the objective is the same. For the moment, my priority is not only to understand this matter, but specifically what is happening in the referenced source, i.e. why what they do and the order they follow is ok, since as you mentioned it's quite enigmatic.

More specifically:

1. I find the "compatibility" argument quite mystifying, so it would help to have someone flesh this step out.
2. Not only that, but also their ad hoc introduction of $Z(\lambda)$ as "running" field strength renormalization, which is not the same $Z$ appearing in the equation relating the bare and the renormalized field, which with e.g. a cutoff regularization depends on the cutoff scale (below the definition of the same reference):
   Spoiler: field-strenght renormalization

   

In other words, this specific book is my concern for the moment, a one-track mind.

 

[vanhees71]

I'm a bit lost with this book since I know it. It's a standard source by a very famous author, but I don't think it's very good to learn from it, if you don't know the material already very well.

 

[qft-El]

I know it may not be relevant to the question, but here's a summary of how I've found this book.
I was reading a brief section about the RG equation and Callan-Symanzik equation on an introductory QFT book: A Modern Introduction to Quantum Field Theory by Michele Maggiore and I found this enigmatic terminology. After hours of relentess search, I finally found out another book using this method and these terms ("dilatation parameter") that I couldn't find anywhere else in the literature. After cross-checking the bibliographies of both, I realized they both references this famous book and a comparison shows that they both were heavily inspired by it (the spitting image excluding notation, basically) in the respective sections.

 

[qft-El]

I've put some thoughts and here's what I'll flesh out what I think is going on. I would appreciate to have feedback if you think this is flawed or not.
To avoid confusion with the notation, I'll switch the notation from $\lambda\to u$ for the parameter. Furthermore, I'll use the $_R$ subscript for renormalized quantities and $_0$ bare quantities.
I'll start from the very beginning, i.e. how the RGE is derived in my case. After a $\Lambda$ cutoff regularization and field-renormalization, the bare $n-$point function is related to the renormalized (with momenta $p_i$) as follows
$$\Gamma_R(p_i; g_R, \mu)=Z^{-n/2}\left(g_0(\Lambda), \frac{\Lambda}{\mu}\right)\Gamma_0(p_i; g_0(\Lambda), \Lambda)\tag{1}\label{1}$$
where $\mu$ is a renormalization scale, such as $\mu=m_R$ and the bare coupling has cutoff dependence.
Then, using the fact that $\Gamma_R$ is independent of $\Lambda$,
$$\Lambda\frac{d\Gamma_R}{d\Lambda}=0\overset{(1)}{\implies}\left[\Lambda\frac{d}{d\Lambda}+\Lambda\frac{dg_0}{d\Lambda}\frac{\partial}{\partial g_0}-\frac{n}{2}\Lambda \frac{d}{d\Lambda}\log Z\right]\Gamma_0(p_i; g_0(\Lambda), \Lambda)=0\tag{2}\label{2}$$
Now let $$\beta(g_0)=\Lambda\frac{dg_0}{d\Lambda} \qquad \eta(g_0)=\frac{1}{2}\Lambda \frac{d}{d\Lambda}\log Z\tag {2a}\label{2a}$$
so the remaining RGE is
$$\left[\Lambda\frac{\partial}{\partial\Lambda}+\beta(g_0)\frac{\partial}{\partial g_0}-n\eta(g_0)\right]\Gamma_0(p_i; g_0, \Lambda)=0.\tag{3}\label{3}$$
Now that we have derived the equation, we may regard $g_0$ as an independent variable and study the solution of the PDE. Either using \eqref{3} (where $g_0$ and thus $\beta$ and $\eta$ are independent of $\Lambda$) or \eqref{2} , it's evident that upon rescaling $\Lambda\to u\Lambda$ in the argument of $\Gamma_0$, equation{3} is unchanged and we find that for $u\in\mathbb{R}$
$$\left[\Lambda\frac{\partial}{\partial\Lambda}+\beta(g_0)\frac{\partial}{\partial g_0}-n\eta(g_0)\right]\Gamma_0(p_i; g_0, u\Lambda)=0.\tag{3a}\label{3a}$$
Ok, now I have what I need. To find a solution of \eqref{3}, we define two functions $g(u)$ and $Z(u)$ such that (for the sake of clarity, I'm still calling the derivative of $\Gamma_0$ wrt the first and the second argument respectively as $\partial\Gamma_0/\partial \Lambda$ and $\partial\Gamma_0/\partial g_0$)
$$u\frac{d}{du}\left[Z^{-n/2}(u)\Gamma_0(p_i;g(u), u\Lambda)\right]:=0\iff Z^{-n/2}(u)\left[\Lambda\frac{\partial}{\partial\Lambda}+u\frac{d}{du}g(u)\frac{\partial}{\partial g_0}-n\cdot\frac{1}{2}u\frac{d}{du}Z(u)\right]\Gamma_0(p_i;g(u),u\Lambda):=0\tag{4}\label{4}.$$
where the $:=$ symbol is a reminder that $g(u)$ and $Z(u)$ are to be determined such that the equality is satisfied. Comparing \eqref{4} to \eqref{3} we immediately see that if
$$\begin{cases}u\frac{d}{du}g(u)=\beta(g(u)) \\
\frac{1}{2}u\frac{d}{du}\log Z(u)=\eta(g(u)) \end{cases}\tag{5}\label{5}$$
then \eqref{4} is reduced to \eqref{3a} (which is true being out starting point), with $g_0\to g(u)$, so \eqref{5} is indeed a definition (not unique I guess) that grants \eqref{4} to be true as we wanted.
Finally, observe that our condition \eqref{4} amounts to $u$-independence, i.e.
$$Z^{-n/2}(u)\Gamma_0(p_i;g(u), u\Lambda)=Z^{-n/2}(1)\Gamma_0(p_i;g(1), \Lambda)\overset{(\star)}{=}\Gamma_0(p_i;g_0, \Lambda)\tag{6}\label{6}$$
from which we see that sensible initial conditions for \eqref{5} are given by $g(1)=g_0$ and $^{(\star)}Z(1)=1$.
At this point we may stop treating $g_0$ as an independent variable and restore its cutoff dependence in \eqref{6}.