Renormalization group flow

Elementary theory

For a certain transformation of the state variables ${\displaystyle \{s_{i}\}\to \{{\tilde {s}}_{i}\}}$, the number of ${\displaystyle {\tilde {s}}_{i}}$is less than the number of ${\displaystyle s_{i}}$. If the ${\displaystyle Z}$ function can be rewritten only in terms of the ${\displaystyle {\tilde {s}}_{i}}$ by a certain change in the parameters, ${\displaystyle \{J_{k}\}\to \{{\tilde {J}}_{k}\}}$, then the theory is said to be renormalizable. The change in the parameters is implemented by a certain beta function: ${\displaystyle \{{\tilde {J}}_{k}\}=\beta (\{J_{k}\})}$, which is said to induce a renormalization group flow (or RG flow) on the ${\displaystyle J}$-space. The values of ${\displaystyle J}$ under the flow are called running couplings.