When one writes down RG-Flow equations for any theory, at some point one encounters statements like
"It is useful to properly rescale the above exact flow equations and rewrite them in dimensionless form."
or
"The rescaled form of the RG-Flow equations is also most convenient to discuss fixed points."
or
"The advantage of working with the rescaled flow equations is that we can directly read off the canonical dimensions of the couplings and thus classify all couplings according to their relevance w.r.t. a given fixed point."
(These sentences are more or less literally from the Kopietz-book.)
I don't really understand this logic; I don't really understand why one has to go to dimensionless flow-equations; Can't I just calculate the critical exponents (by diagonalizing the stability matrix and reading off it's eigenvalues, assuming that this is possible) for the RG-flow equations of the dimensional couplings?
An explanation that I could deal with is, that it is just an standard-fact that has been verified experimentally that the only "interesting" critical exponents are those, that come with the flow-equations of the dimensionless couplings.
Can anybody elaborate a good explanation please?
EDIT:
An example for dimensional flow equations:
$$ \left\{ \, \begin{aligned} \frac{ \mathrm{d} }{ \mathrm{d} \Lambda } \gamma_{1} \quad & \approx \quad 0 \\ \frac{ \mathrm{d} }{ \mathrm{d} \Lambda } \gamma_{2} \quad & \approx \quad 0 \\ \frac{ \mathrm{d} }{ \mathrm{d} \Lambda } \gamma_{3} \quad & \approx \quad + {\gamma}_{1} \cdot \left( 4 \, D_{ \Lambda } \right) \cdot {\gamma}_{2} \\ \frac{ \mathrm{d} }{ \mathrm{d} \Lambda } \gamma_{4} \quad & \approx \quad + {\gamma}_{2} \cdot \left( -16 \, D_{ \Lambda } \right) \cdot {\gamma}_{4} \end{aligned} \right. $$
An example for dimensionless flow equations:
$$ \left\{ \, \begin{aligned} \beta_{1} ( \tilde{\gamma}_{1} , \tilde{\gamma}_{2} , \tilde{\gamma}_{3} , \tilde{\gamma}_{4}) \quad & \approx \quad - 2 \tilde{\gamma}_{1} / \Lambda \\ \beta_{2} ( \tilde{\gamma}_{1} , \tilde{\gamma}_{2} , \tilde{\gamma}_{3} , \tilde{\gamma}_{4}) \quad & \approx \quad - 2 \tilde{\gamma}_{2} / \Lambda \\ \beta_{3} ( \tilde{\gamma}_{1} , \tilde{\gamma}_{2} , \tilde{\gamma}_{3} , \tilde{\gamma}_{4}) \quad & \approx \quad - 2 \tilde{\gamma}_{3} / \Lambda + \tilde{\gamma}_{1} \cdot \left( 4 \, D_{ \Lambda } \, \Lambda^2 \right) \cdot \tilde{\gamma}_{2} \\ \beta_{4} ( \tilde{\gamma}_{1} , \tilde{\gamma}_{2} , \tilde{\gamma}_{3} , \tilde{\gamma}_{4}) \quad & \approx \quad - 2 \tilde{\gamma}_{4} / \Lambda + \tilde{\gamma}_{2} \cdot \left( -16 \, D_{ \Lambda } \, \Lambda^2 \right) \cdot \tilde{\gamma}_{4} \end{aligned} \right. $$
where $D_{ \Lambda }$ is the dominant contribution for the flow of the couplings; Mode-elimination (in the FRG, analog to Wilson's RG) corresponds to decreasing $\Lambda$ from $\Lambda = \infty$ to $\Lambda = 0$ and $\tilde{\gamma }_{ \bullet } = \gamma_{ \bullet } / \Lambda^2$
