Unstable particles in One Loop Calculations

TL;DR

This paper develops a gauge-invariant, pole-based framework for calculating one-loop corrections in processes with unstable particles, addressing divergences that arise from resonant propagators by isolating the pole position $M^2$ and residue $w$ from an off-shell amplitude ${\cal A} = \frac{w}{p^2-M^2} + n(p^2)$. It generalizes M. Veltman’s method to multiple charged resonances, separating factorizable and non-factorizable resonant diagrams and providing explicit prescriptions to compute the necessary quantities from perturbative expansions, while carefully treating threshold regions, mixing, and infrared issues via a complex-mass approach and soft Bremsstrahlung. The approach yields a gauge-invariant, practical recipe for tree-level and one-loop calculations, handling both neutral and charged unstable particles and addressing the non-trivial resonant structure that appears when multiple unstable states participate. The framework is particularly applicable to $W$-pair production at LEP II and offers possible reductions in computational effort by isolating the gauge-invariant components and systematically incorporating width effects without compromising gauge invariance.

Abstract

We present a gauge invariant way to compute one loop corrections to processes involving the production and decay of unstable particles.

Figures (7)

• Figure 1: The Bjorken process at $\sqrt{s}\approx m_Z$ (left) and $\sqrt{s}\stackrel{\hbox{$>$}} {\hbox{$\sim$}} m_Z+m_H$ (left). $Z^*$ denotes an off-shell $Z$ boson; the thick line indicates a propagator which needs to be resummed.
• Figure 2: The structure of the factorizable resonant diagrams
• Figure 3: The structure of the non-factorizable resonant diagrams
• Figure 4: The two on-shell divergent scalar three point functions with physical examples
• Figure 5: Example of a diagram which is not analytic at $p^2=m^2$: the one loop photonic contribution to the self energy of a $W$ boson.