Width and Partial Widths of Unstable Particles in the Light of the Nielsen Identities

TL;DR

Using Nielsen identities, the paper shows that conventional on-shell mass and width definitions become gauge dependent at NNLO, while the pole position is gauge invariant to all orders. It establishes that pole residues are gauge independent to all orders and develops both residue-based and NNLO additive definitions for partial widths, with a refined all-orders construction via a hat-vertex in the $\Gamma_2\to 0$ limit. It further demonstrates that box diagrams integrate into a unique, gauge-invariant full amplitude and clarifies the analytic structure of NI functions through dispersion relations and explicit one-loop examples in the $Z$–$\gamma$ sector. The results provide a theoretically consistent, gauge-invariant framework for unstable-particle properties and their physical observables.

Abstract

Fundamental properties of unstable particles, including mass, width, and partial widths, are examined on the basis of the Nielsen identities (NI) that describe the gauge dependence of Green functions. In particular, we prove that the pole residues and associated definitions of branching ratios and partial widths are gauge independent to all orders. A simpler, previously discussed definition of branching ratios and partial widths is found to be gauge independent through next-to-next-to-leading order. It is then explained how it may be modified in order to extend the gauge independence to all orders. We also show that the physical scattering amplitude is the most general combination of self-energy, vertex, and box contributions that is gauge independent for arbitrary s, discuss the analytical properties of the NI functions, and exhibit explicitly their one-loop expressions in the Z-gamma sector of the Standard Model.