# Renormalisation group equation

1. Feb 23, 2016

### CAF123

1. The problem statement, all variables and given/known data
The renormalization group equations for the n-point Green’s function $\Gamma(n) = \langle \psi_{x_1} \dots \psi_{x_n}\rangle$ in a four-dimensional massless field theory is $$\mu \frac{d}{d \mu} \tilde{\Gamma}(n) (g) = 0$$ where the coupling g is defined at mass scale $\mu$.

Show that this is equivalent to $$(\beta \frac{\partial}{\partial g} + n )\tilde{\Gamma}(n) = 0,$$ where $\beta(g) = \mu \frac{d g}{d \mu}.$The field $\psi$ has mass dimension one and the Green’s function is a homogeneous function of degree n in the field.

2. Relevant equations

function of homogenous degree n is one in which the exponents of each term all add up to n.

Renormalisation of fields

3. The attempt at a solution
In renormalisation, $\psi \rightarrow Z_{\psi} \psi$ and given that the Green's function is a homogenous function of degree n, in the renormalised Green's function, we now have a factor of $(Z_{\psi})^n$ in each term. So, $$\frac{d}{d \mu} \tilde \Gamma = \frac{\partial \tilde \Gamma}{\partial \mu} + \frac{\partial \tilde \Gamma}{\partial Z_{\psi}} \frac{\partial Z_{\psi}}{\partial \mu}$$ I would say that $$\frac{\partial \tilde \Gamma}{\partial Z_{\psi}} = n (Z_{\psi})^{n-1}\tilde \Gamma$$ but this does not seem to give me correct result.

Did I assume something incorrect? Thanks!

2. Feb 28, 2016

### nrqed

You are missing a $\gamma$ in that equation, next to the factor of n , did you realize this?