My question is about the loopwise expansion of the effective action $\Gamma(\varphi)$ up to 1-loop contributions. I've understood well the results for both $Z[J]$ and $W[J]$ functionals loopwise expansions. But then something is missing when I follow the path towards the expansion of the effective action. Following this excerpt from Zinn-Justin's "Quantum field theory and critical phenomena":
I don't understand the statement: "Therefore a correction of order $\hbar$ to the relation between $J(x)$ and $\varphi(x)$ will produce a change of order $\hbar^2$ to the r.h.s. of equation (6.47)"
Could someone please write down some more details?
My undestanding at this stage is that the relation between $J(x)$ and $\varphi$ is fixed by the stationarity of the (6.47) (as for any Legendre transformation). But then, how to proceed?