The is a bound state of formed in collisions

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

The

\Upsilon(1S)

is a

J=1

bound state of

b\bar b

formed in

e^+e^-

collisions, with mass

M = 9.460\,\mathrm{GeV}

, total width

\Gamma = 54.02\,\mathrm{keV}

(PDG 2024), and hadronic branching ratio

\mathcal{B}_{\rm had} = 92.5\,\%

. Because

\Gamma

is far smaller than the typical beam-energy spread of an

e^+e^-

collider (

\sim 4\,\mathrm{MeV}

), only the area under the resonance peak can be measured, not its shape. A scan of

e^+e^- \to \Upsilon(1S) \to \mathrm{hadrons}

finds an integrated area

\int \sigma_{\rm had}\,\mathrm{d}E = 0.320\,\mathrm{nb\,GeV}

. ((a)) Using the relativistic two-body momentum formula from Chapter 2 (the Kallen function), find the centre-of-mass momentum

k

of each beam at

\sqrt{s} = M

, noting that the electron mass is negligible compared with

M

. ((b)) Using the narrow-width area formula

\int\sigma_{\rm had}\,\mathrm{d}E = 6\pi^2\Gamma_{ee}\mathcal{B}_{\rm had}/M^2

, extract

\Gamma_{ee}

. ((c)) Using

\Gamma_{ee}

from part (b), compute the peak hadronic cross section

\sigma^0_{\rm had} = 12\pi\Gamma_{ee}\Gamma_{\rm had}/(M^2\Gamma^2)

, and compare it with the unitarity limit for a

J=1

resonance formed in

e^+e^-

\sigma_{\rm max} = (4\pi/k^2)(3/4)

, using

k

from part (a). State whether the bound is respected.