- #1
Siupa
- 30
- 5
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
$$0= \Lambda \frac {\text{d} \lambda}{\text{d} \Lambda} = \Lambda \frac {\partial \lambda}{\partial \Lambda} + \Lambda \frac{\partial \lambda_0}{\partial \Lambda} \frac{\partial \lambda}{\partial \lambda_0} \implies 0 = - \mu \frac{\partial \lambda}{\partial \mu} + \beta^{(\text{B})}(\lambda) \frac{\partial \lambda}{\partial \lambda_0}$$
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