Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 7.9

Feynman rules — Schwartz M.D. z-lib.org

Physics — QFT · unstable-particles-breit-wigner

Problem

Unstable particles. Unstable particles pick up imaginary parts that generate a width Γ\Gamma in their resonance line shape. This problem will develop an understanding of what is meant by the terms width and pick up. (a) What would the cross section be for ss-channel scattering if the intermediate propagator were ip2−m2+imΓ\frac{i}{p^2 - m^2 + im\Gamma}, where Γ>0\Gamma > 0? This is called the Breit--Wigner distribution. (b) Sketch the cross section as a function of x=sm2x = \frac{s}{m^2} for Γm\frac{\Gamma}{m} small and for Γm\frac{\Gamma}{m} large. (c) Show that a propagator only has an imaginary part if it goes on-shell. Explicitly, show that Im(M)=−πδ(p2−m2)\mathrm{Im}(\mathcal{M}) = -\pi\delta(p^2 - m^2), when iM=ip2−m2+iεi\mathcal{M} = \frac{i}{p^2 - m^2 + i\varepsilon}. (d) Loops of particles can produce effective interactions that have imaginary parts. Suppose we have another particle ψ\psi and an interaction ϕψψ\phi\psi\psi in the Lagrangian. Loops of ψ\psi will have imaginary parts if and only if ψ\psi is lighter than half of ϕ\phi, that is, if ϕ→ψψ\phi \to \psi\psi is allowed kinematically. Draw a series of loop corrections to the ϕ\phi propagator. Show that, if these give an imaginary number, you can sum the graphs to reproduce the propagator in part (a). (e) What is the connection between parts (c) and (d)? Can you see why the width is related to the decay rate?
Intuitive explanation

When an unstable particle can decay, its propagator picks up an imaginary part imΓim\Gamma that broadens its resonance peak—this is the Breit–Wigner form. The key insight is that the imaginary part only appears when the particle is kinematically allowed to decay (goes on-shell), and the width Γ\Gamma is directly connected to the decay rate through the optical theorem. Understanding this requires seeing how loop corrections generate the imaginary piece and how it reshapes the cross section from a sharp delta function into a characteristic Lorentzian bump.

Intuitive explanation

When an unstable particle can decay, its propagator picks up an imaginary part imΓim\Gamma that broadens its resonance peak—this is the Breit–Wigner form. The key insight is that the imaginary part only appears when the particle is kinematically allowed to decay (goes on-shell), and the width Γ\Gamma is directly connected to the decay rate through the optical theorem. Understanding this requires seeing how loop corrections generate the imaginary piece and how it reshapes the cross section from a sharp delta function into a characteristic Lorentzian bump.

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