Renormalization scale dependence in scalar field theory ($\phi^4$) establishes that defining a quantum effective field theory necessitates introducing a physical energy scale, such as a cutoff or renormalization mass $\mu$, to parameterize the variation of coupling constants. The core mechanism involves satisfying physical renormalization conditions—specifically fixing pole masses and residue normalizations for propagators—to absorb divergences into counterterms ($B$ and $C$), resulting in finite scattering amplitudes independent of the regulator choice. This process generates beta functions via the Renormalization Group Equations, which govern how coupling strengths flow with scale changes to preserve low-energy physical predictions despite alterations at high energies or different cutoff values.
Renormalization scale dependence in scalar field theory ($\phi^4$) establishes that defining a quantum effective field theory necessitates introducing a physical energy scale, such as a cutoff or ren…