Widths of unstable particles
In nonrelativistic quantum mechanics the instability of a particle is described by a complex potential or by a function \(\psi(t) \sim e^{-iEt - \Gamma t/2}\). In quantum field theory the decay width arises from the imaginary part of the self-energy.
In this project you will trace how the width arises from diagrams and how it is related to observable quantities.
Theory
Let the scalar field \(\phi\) interact with a light field \(\chi\) through \[\mathcal{L}_{\text{int}} = -g \phi \chi^2.\] Draw the second-order (in \(g\)) self-energy diagram of \(\phi\). Compute the imaginary part of the loop with the help of the optical theorem.
Show that the field \(\phi\) has the modified propagator \[\frac{i}{p^2 - m^2 + i m\Gamma}.\] Explain the physical meaning of \(\Gamma\) by comparing with the exponential decay law.
Show that the observed cross section near the resonance has the Breit–Wigner form \[\sigma(E) \propto \frac{1}{(E - m)^2 + (\Gamma/2)^2}.\] Derive this expression from the full amplitude with the field \(\phi\) in the intermediate state.
Explain why one cannot speak of a fixed mass of an unstable particle. What is the meaning of the diagram with the decay \(A \to B\phi \to B\chi\chi\)?
Discuss how the decay width is related to mass measurements: why is the mass of the \(Z\) boson determined from the position of the peak of the cross section \(e^+e^-\to f\bar f\), and the width from its shape?
A numerical example
Compute the decay width of a hypothetical scalar field, \(\phi \to \chi\chi\), for \(m_\phi = 1\) GeV, \(m_\chi = 0.2\) GeV, \(g = 1\) GeV.
Plot \(\sigma(E)\) near the resonance and show how the shape of the peak changes as \(\Gamma\) grows. How does the width depend on the mass and the coupling?
Find the widths of the real resonances \(\rho\), \(K^*\), \(\Delta\) in the PDG tables. Compare with the Breit–Wigner formula. Why does it work poorly in some cases?