Renormalization

Definition

Consider a family of quantum field theories with a regulator $\Lambda$, a cutoff that makes every Feynman diagram finite. The theory has bare parameters $g_0 = (g_{0,1}, \dots, g_{0,k})$ and a field $\phi_0$, with correlation functions $G_n(x_1, \dots, x_n; g_0, \Lambda)$. These are formal power series in the bare couplings. A renormalization of the theory is a choice of the following data.

1. Renormalized parameters $g$, and bare parameters $g_0 = g_0(g, \Lambda)$ given as formal power series in $g$.
2. A field strength factor $Z = Z(g, \Lambda)$, also a formal power series, with renormalized field $\phi = Z^{-1/2}\phi_0$.
3. The condition: for every $n$, the renormalized correlation function $Z^{-n/2} G_n(x; g_0(g, \Lambda), \Lambda)$ has a finite limit as $\Lambda \to \infty$, coefficient by coefficient in $g$.

The differences $g_0 - g$ and $Z - 1$ are the counterterms. The theory is renormalizable if this is possible with counterterms of the same form as the terms already in the Lagrangian.