A resonance decays in its own rest frame with amplitude for

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Particle Physics & Quantum Fields · Particles, interactions and experiment

Symmetry and Flavour, Ch Resonances: the relativistic Breit-Wigner, unitarity and the Z line shape

Problem

A resonance decays in its own rest frame with amplitude

\psi(t)=\psi(0)e^{-iMt-\Gamma t/2}

for

t\ge0

, zero otherwise, where

M

is its mass and

\Gamma

its total width (natural units). ((a)) Fourier transform

\psi(t)

, locate the pole of

A(E)

E=M-i\Gamma/2

, and convert to the relativistic propagator

1/(s-M^2+iM\Gamma)

using

E-M\to\sqrt s - M

near the peak. ((b)) Show

\int_{-\infty}^{\infty}\mathrm{d}s/[(s-M^2)^2+M^2\Gamma^2]=\pi/(M\Gamma)

exactly, and state why the narrow-width factorisation

\sigma\to\sigma_{\rm prod}\mathcal B

additionally requires

\Gamma\ll M

. ((c)) The

\Delta(1232)

has

M=1232\,\mathrm{MeV}

and

\Gamma=117\,\mathrm{MeV}

(PDG 2024). Evaluate

\pi/(M\Gamma)

\mathrm{GeV}^{-2}

and the fractional error, of order

\Gamma/M

, in treating the

\Delta

with the narrow-width approximation.