Renormalization Group Equations (RGEs)
When regularizing a Green’s function $\Gamma$ in dimensional regularization you get
$$\Gamma(p_i, \alpha, m, \epsilon)= Z_\Gamms(\epsilon, \mu)\Gamma_R(p_i, \alpha_R, m_R, \mu).$$
- $p_i$ are the momenta, $R$-subscript denotes the renormalized quantaties
- $Z_\mu$ denotes the renormalization constant
$\Gamma$ is independent of $\mu$:
$$\mu\frac{\text{d}}{\text{d}\mu}\Gamma = \mu\frac{\text{d}}{\text{d}\mu}(Z_\Gamma\Gamma_R) = 0$$
Applying product rule
$$(\mu \frac{\text{d}{\text{d}\mu} + \gamma_\Gamma)\Gamma_R = 0$$ with $\gamma_\Gamma = \frac{\text{d}}{\text{d}\log\mu}\log Z_\Gamma$
In a mass independet renormalization scheme $\gamma_\Gamma$ depends on $\alpha_R$
The solution can be obtained by integrating from $\mu_0$ to $\mu$ $$\Gamma_R(p_i, d_R(\mu), m_R(\mu), \mu) = \Gamma(p_i, d_R(\mu_0), m_R(\mu_0), \mu_0) e^{-\int^{\alpha_R(\mu)}{\alpha_R(\mu_0)}\frac{\text{d}\alpha}{\alpha} \frac{\gamma\Gamma(\alpha)}{\beta(\alpha)}}$$ where the running of $\alpha(\mu)$ was used for substitution in the integral.
Describes the evolution from $\mu_0$ to $\mu$
If we scale all momenta by $\xi$: $p_i \to \xi p_i$, it follows from $\Gamma_R(\xi p_i) = \xi^{d_\Gamma} \Gamma_R(p_i)$ that in the RG equation $\gamma_\Gamma \to d_\Gamma + \gamma_\Gamma$. Since $d_\Gamma$ is the mass dimension of $\Gamma$, $\gamma_\Gamma$ can be interpreted as anomalous dimension of $\Gamma