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1. Phi 4 Theory

   Linked via "momentum cutoff"

   Perturbative quantum field theory calculations using Feynman diagrams show that $\phi^4$ theory in four spacetime dimensions ($D=4$) is non-renormalizable, meaning that the coupling constant $\lambda$ must be redefined at every order of perturbation theory to absorb ultraviolet divergences.
   The running of the coupling constant $\lambda(\mu)$ is gove…

2. Renormalization Group Flow

   Linked via "momentum cutoff"

   Theoretical Formalism
   The RG transformation is fundamentally a change of variables that integrates out high-momentum (short-distance) degrees of freedom, yielding a lower-energy description. This iterative procedure maps an initial theory, parameterized by a set of bare couplings $\mathbf{g}_0$, onto a sequence of effective theories characterized by running couplings $\mathbf{g}(\Lambda)$, where $\Lambda$ is the momentum cutoff or observation scale.
   The dynamics of this flow are governed by the [Callan-Symanzi…