Retrieving "Infrared Fixed Point" from the archives

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1. Continuum Limit

   Linked via "Infrared (IR) Fixed Point"

   | Gaussian Fixed Point (GFP) | $\beta(a) \sim \beta^*$ (constant) | Free theory; couplings vanish or remain fixed; continuum limit is typically trivial. |
   | Non-Gaussian Fixed Point (NGFP) | $\beta(a) \sim \beta^* + C a^{\gamma}$ ($\gamma > 0$) | Interacting continuum theory; relevant for critical phenomena and asymptotic freedom. |
   | Infrared (IR) Fixed Point | $\beta(a) \to \infty$ as $a \to 0$ | Leads to confinement or str…

2. Renormalization Group Equation

   Linked via "IR"

   Ultraviolet (UV) Fixed Points: These occur at $\mu \to \infty$ (or $Q^2 \to \infty$). A UV fixed point corresponds to a theory that is asymptotically free (like QCD) or conformal (if the mass terms vanish).
   Infrared (IR) Fixed Points: These occur at $\mu \to 0$ (or $Q^2 \to 0$). Fixed points in the IR often signal the presence of a non-trivial phase transition or [critical behavior](/entries/critical-phenomena/…

3. Renormalization Group Flow

   Linked via "infrared fixed points"

   \left( \Lambda \frac{\partial}{\partial \Lambda} + \beta(\mathbf{g}, \Lambda) \frac{\partial}{\partial \mathbf{g}} + \gamma(\mathbf{g}, \Lambda) \right) L(\phi, \mathbf{g}_0, \Lambda) = 0
   $$
   Here, $L$ is the Lagrangian density, $\beta$ is the $\beta$-function describing the flow of the couplings, and $\gamma$ is the anomalous dimension associated with the field $\phi$. For scale-invariant theories (the infrared fixed points), the $\beta$-f…

4. Renormalization Group Flow

   Linked via "infrared fixed point"

   Flow Trajectories and Universality Classes
   The set of trajectories generated by the RG flow defines the renormalization group flows [3]. These trajectories partition the space of all possible microscopic theories into equivalence classes, known as universality classes. Two microscopic models belong to the same class if their RG trajectories converge to the same infrared fixed point [1].
   The Criterion of $\mathcal{C}$-Theorems