One-Loop Beta Function (λφ⁴ Theory)

$$\beta(\lambda) = \frac{3\lambda^2}{16\pi^2} + O(\lambda^3)$$

Status: Proven.

This is the real, standard one-loop renormalization-group beta function for a scalar field with quartic self-interaction ($\lambda\phi^4$ theory) — a foundational, textbook result in quantum field theory describing how the coupling constant $\lambda$ “runs” (changes) with energy scale. Renormalization group flow itself is one of the best-tested frameworks in theoretical physics, underlying the entire concept of how forces unify at high energy in the Standard Model.

Real, independent confirmation: this specific coefficient is a standard, independently-reproducible perturbative QFT calculation, found in any quantum field theory textbook treatment of scalar field theory.

Used in: Paper 8, quoted correctly as part of PBT’s own hierarchical renormalization scheme. The formula itself is real and correctly stated; whether PBT’s broader claim — that its own infinite-hierarchy structure renormalizes divergences the way this formula’s ordinary RG-flow context does — has been independently shown is a separate, unproven claim (see Effective Gravitational Coupling for the related hierarchical-scaling formula this connects to).