Quantum Field Theory renormalization dictates that the physical pole mass and residue of a propagator remain invariant by defining counter-terms to exactly cancel loop-induced shifts in these parameters. This mechanism relies on summing infinite series of one-particle irreducible (1PI) self-energy diagrams, represented formally as $\Sigma(p^2)$, where renormalization conditions enforce specific values for the mass counter-term ($\delta m$) and field strength counter-term ($\delta Z$). Consequently, the observable spectrum is determined by requiring that divergent loop corrections are absorbed into redefined physical parameters, rendering scattering amplitudes finite without altering the underlying pole structure of the S-matrix.
Quantum Field Theory renormalization dictates that the physical pole mass and residue of a propagator remain invariant by defining counter-terms to exactly cancel loop-induced shifts in these paramet…