All of this question refers to ref. 1. The equation are numbered alike. The author claims to solve a renormalization group (RG) equation using the Method of characteristics, but there is a passage which I can't reconnect to such method and I don't understand why it is necessary at all. I post it here because I suspect that it may have some physical motivation.
We start from a renormalized $n$-point function $\Gamma_R$, with $\mu$ the renormalization scale, $p_i$ the ingoing/outgoing particle momenta, $\Lambda$ the cutoff scale and $g$ the renomalized coupling. The $R$ and $0$ subscript denote respectively renormalized and bare quantities.
$$\Gamma_R=\Gamma_R(p_i; g_R, \mu) \qquad\Gamma_R(p_i; g_R, \mu)=Z^{-n/2}\left(g_0(\Lambda), \frac{\Lambda}{\mu}\right)\Gamma_0(p_i; g_0, \Lambda).\tag{5.160 - 5.161}$$ Then, using the fact that $\Gamma_R$ is independent of $\Lambda$, they write $$\Lambda\frac{d\Gamma_R}{d\Lambda}=0\overset{(5.161)}{\implies}\left[\Lambda\frac{d}{d\Lambda}+\Lambda\frac{dg_0}{d\Lambda}\frac{\partial}{\partial g_0}-\frac{n}{2}\Lambda \frac{d}{d\Lambda}\log Z\right]\Gamma_0(p_i; g_0, \Lambda)=0\tag{5.163}$$ Where in the last passage we've used equation $(5.161)$, divided by $Z^{-n/2}$ and made some arrangements. We now rewrite it as follows $$\left[\Lambda\frac{\partial}{\partial\Lambda}+\beta(g_0)\frac{\partial}{\partial g_0}-n\eta(g_0)\right]\Gamma_0(p_i; g_0, \Lambda)=0\qquad\beta(g_0)=\Lambda\frac{dg_0}{d\Lambda} \qquad \eta(g_0)=\frac{1}{2}\Lambda \frac{d}{d\Lambda}\log Z.\tag{5.163 - 5.164 - 5.165}$$ In principle $\beta$ would also depend on $\Lambda/\mu$ but it leads to subleading terms, so we ignore it. Alright, here's where they claim this equation can be solved using the method of characteristics mentioned above. Everything seems in place to do it straightforwardly with $g_0$ and $\Lambda$ as independente variables, in fact it looks like the equation is already evaluated on a curve since the bare coupling $g_0$ is a function of $\Lambda$. Instead, they introduce a "dilatation parameter" $u$ as follows $$\Gamma_0\left(p_i; g_0, \frac{\Lambda}{\mu}\right)=Z_{\text{eff}}^{-n/2}(u)\Gamma_0(p_i; g_{\text{eff}}(u), \Lambda)\tag{5.166}$$ (I think that in the $g_0$ is evaluated at $\Lambda/\mu$ but it is not mentioned) where the new functions are defined as the solutions of $$u\frac{dg_{\text{eff}}}{du}=\beta(g_{\text{eff}}(u)) \qquad \frac{1}{2}u\frac{d}{du}\log Z_{\text{eff}}=\eta(g_{\text{eff}})\tag{5.167 - 5.168}$$ and they finally use these equation to find information about the beta function. As I said, I can't find any mention to this "dilatation paramater" elsewhere, nor I see why it is needed and its connection to the method of characteristics. I suspect the variable is introduced because they later use the $u\to0$ limit as a removal of the cutoff. Then again, I don't see why this is necessary and what they're doing. I should also mention that I'm aware of the running coupling constants - I've done calculations in perturbation theory as well - and this might be related, but again I don't see what this has to do with the mentioned method. Can you please shed some light on it?
I'd also appreciate if you mentioned any sources doing things this way.
As a small addendum, the outlined procedure is also used for Callan-Symanzik equation $\mu\frac{d\Gamma_0}{d\mu}=0$ mutatis mutandis.
References
- A Modern Introduction to Quantum Field Theory. Michele Maggiore, 2005. Chapter 5, section 5.9, pag. 147-151.
