A beam of with momentum strikes protons at rest in a bubble chamber

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

Particle Physics, Ch Resonances, widths and the Breit-Wigner line shape

Problem

A beam of

\pi^+

with momentum

300\,\mathrm{MeV}/c

strikes protons at rest in a bubble chamber. The

\pi^+p

cross section shows a bump whose full width at half maximum, corrected for the negligible momentum spread of the beam, is

\Gamma = 117\,\mathrm{MeV}

. Take

m_{\pi^+}c^2 = 139.57\,\mathrm{MeV}

m_pc^2 = 938.3\,\mathrm{MeV}

and

\hbar = 6.582\times10^{-22}\,\mathrm{MeV\,s}

. ((a)) Reconstruct the invariant mass

Mc^2

of the system made in this collision. ((b)) Convert the quoted width

\Gamma

into the mean lifetime

\tau

of the resonance. ((c)) The

\pi^+p

initial state has isospin

I_3 = +3/2

(

I_3=+1

for the

\pi^+

I_3=+1/2

for the proton). Explain why this fact alone, together with the rule for adding two isospins, guarantees that the resonance seen here is a pure

I=3/2

state, without needing to compare it against the

\pi^-p

channels.

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