A vector meson decays in its own rest frame with amplitude for and 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 vector meson decays in its own rest frame with amplitude

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

for

t \ge 0

and

\psi(t) = 0

for

t < 0

, where

M

is its mass and

\Gamma

its total width (natural units,

\hbar = c = 1

). ((a)) Fourier transform

\psi(t)

to obtain the amplitude

A(E) = \int_0^\infty e^{iEt}\psi(t)\,\mathrm{d}t

as a function of energy

E

, and show it has a pole at

E = M - i\Gamma/2

. Using the near-peak identification

E - M \to (\sqrt{s} - M)

, obtain the relativistic propagator

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

. ((b)) Evaluate the integral

\int_{-\infty}^{\infty} \dfrac{\mathrm{d}s}{(s-M^2)^2 + M^2\Gamma^2}

exactly for any

\Gamma > 0

(substitute

u = s - M^2

), and explain why a cross section built from this propagator factorises into a production cross section times a branching ratio when

\Gamma \ll M

. ((c)) The

\omega(782)

meson has

M = 782.65\,\mathrm{MeV}

and

\Gamma = 8.68\,\mathrm{MeV}

(PDG 2024). Evaluate the integral of part (b) in

\mathrm{GeV}^{-2}

, and give the fractional size, of order

\Gamma/M

, of the error made when other slowly varying factors in the cross section are treated as constant across the resonance.