A vector 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 vector resonance decays in its own rest frame with amplitude

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

for

t \ge 0

, zero for

t<0

, where

M

is its mass and

\Gamma

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

\psi(t)

to find

A(E)

, locate its pole at

E = M - i\Gamma/2

, and use

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

near the peak to obtain the relativistic propagator

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

. ((b)) Show that

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

exactly, and state the extra condition, of order

\Gamma/M

, needed for a cross section built from this propagator to factorise as

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

. ((c)) The

\Upsilon(1S)

has

M = 9460.30\,\mathrm{MeV}

and

\Gamma = 54.02\,\mathrm{keV}

(PDG 2024). Evaluate

\pi/(M\Gamma)

\mathrm{GeV}^{-2}

and the fractional error, of order

\Gamma/M

, in the narrow-width factorisation for this state.