Relationship between bare and renormalized beta functions

[Siupa]

I'm looking at a proof of beta function universality in $\phi^4$ theory, and at one point they do the following step: after imposing that the renormalized coupling $\lambda$ is independent of the cutoff $\Lambda$, we have
$$0= \Lambda \frac {\text{d} \lambda}{\text{d} \Lambda} = \Lambda \frac {\partial \lambda}{\partial \Lambda} + \Lambda \frac{\partial \lambda_0}{\partial \Lambda} \frac{\partial \lambda}{\partial \lambda_0} \implies 0 = - \mu \frac{\partial \lambda}{\partial \mu} + \beta^{(\text{B})}(\lambda) \frac{\partial \lambda}{\partial \lambda_0}$$
This seems to imply
$$\Lambda \frac{\partial \lambda}{\partial \Lambda} = -\mu \frac{\partial \lambda}{\partial \mu}$$
Why is this true? Where does it come from? $\lambda_0$ is the bare coupling and $\mu$ the arbitrary mass scale

 

[PeterDonis]

@Siupa inline LaTeX needs to be enclosed in double dollar signs, not single ones. I have used magic moderator powers to edit your OP to fix this.

 

[vanhees71]

Can you give the reference, where you got this from? What's the renormalization scheme used?

I have some RG stuff applied to $\phi^4$ theory in my QFT notes:

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