# Conditions for renormalization of effective potential

Currently I am reading Das' Finite Temperature Field Theory. After computing the first order correction to the potential with a cut-off (eq. 6.77) Das states that, "we add counterterms and renormalize our theory by demanding that the renormalized mass and the coupling constant of the theory up to one loop are given by"

$$\begin{equation} \left. \frac{d^2 V(\phi_c)}{d\phi_c^2}\right|_{\phi_c = 0} = m^2 \end{equation}$$

$$\begin{equation} \left. \frac{d^4 V(\phi_c)}{d\phi_c^4}\right|_{\phi_c = 0} = \lambda \end{equation}$$ where $$V$$ is the classical potential, $$\phi_c$$ is the classical field and $$V^0 = (m^2/2)\phi^2 + (\lambda/4!)\phi^4$$.

Why is this the case? Are these condition equivalent to the 'renormalization conditions' on pg. 325 of Peskin and Schroeder? Is this standard procedure for renormalization?

• I don't have access to a copy of Das, but if $V$ is the renormalized potential and $\phi_c$ is the extrema of the effective potential, then these statements are equivalent to the usual conditions because these derivatives of the effective potential are equal to the 2-pt and 4-pt amplitudes at zero momenta (hence the kinetic terms don't contribute). – Richard Myers Nov 6 '20 at 20:40