On the Resonance Coupling and Width in Quantum Field Theory

TL;DR

This work addresses the challenge of connecting scheme-dependent renormalized resonance parameters in quantum field theory to physically observable resonance properties. By analyzing both quadratically and logarithmically divergent self-energy diagrams, it shows how to define a scheme-independent resonance coupling $g_M$ as the residue at the pole, using a carefully subtracted real part ${\rm Re}\,B_R(s,M^2)$ of the self-energy $B(s)$. The key result is a pole-residue based parameterization of the scattering amplitude: $A(s)=\frac{g_M^2}{M^2-s- g_M^2 {\rm Re}\,B_R(s,M^2) - i g_M^2 {\rm Im}\,B(s)}$, with a finite, scheme-independent width $\Gamma = g_M^2 {\rm Im}\,B(M^2)/M$. The paper further shows that even finite Lagrangian couplings can differ from the physical residue (as in the logarithmic divergence case), emphasizing that $g_M$ provides the correct, observable coupling for resonance phenomenology, particularly for broad resonances.

Abstract

In quantum field theory, characteristics of resonances are related to self-energy diagrams, which are ultra-violet divergent and require renormalization. We demonstrate the proper way to define the resonance coupling $g_M$ such that the resonance properties calculated in quantum field theory are finite and scheme-independent quantities.

Figures (1)

• Figure 1: The diagrams describing the series for $A$ in Eq. (\ref{['a1']}).