The baryon, mass , total width and spin , is produced as an elastic resonance in…

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

\Lambda(1520)

baryon, mass

M = 1.5195\,\mathrm{GeV}

, total width

\Gamma = 15.6\,\mathrm{MeV}

and spin

J = 3/2

, is produced as an elastic resonance in

K^-p

scattering (

K^-

: spin

0

, mass

0.494\,\mathrm{GeV}

; proton: spin

1/2

, mass

0.938\,\mathrm{GeV}

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

k

of the

K^-p

system at

\sqrt{s} = M

. ((b)) The

\Lambda(1520)

decays to

\bar KN

(

K^-p

\bar K^0n

) 45 per cent of the time, shared equally between the two charge states, so the elastic branching ratio is

\mathcal{B}_{\rm el} = \Gamma_{K^-p}/\Gamma = 0.225

. Using the spin-counting factor for

J=3/2

formed from a spin-

0

and a spin-

\frac12

particle, compute the unitarity limit for this channel and predict the resonant peak elastic cross section

\sigma_{K^-p\to\Lambda(1520)\to K^-p}

. What fraction of the limit does it reach? ((c)) A separate, coarser-resolution exposure (beam energy spread

\sim 40\,\mathrm{MeV}

\sqrt s

, too broad to resolve

\Gamma = 15.6\,\mathrm{MeV}

) can measure only the integrated area under the elastic peak. Treat the line as

\sigma_{\rm peak}(\Gamma^2/4)/[(\sqrt s-M)^2+\Gamma^2/4]

with

k

held at its peak value, and integrate across it to show

\int\sigma_{\rm el}\,\mathrm{d}\sqrt s = 2\pi^2\frac{2J+1}{(2s_a+1)(2s_b+1)}\frac{\Gamma_{\rm el}^2}{k^2\Gamma}.

Explain why the

e^+e^-

area formula has

M^2

where this has

4k^2

, evaluate the expected area in

\mathrm{mb\,MeV}

, and say how precisely a

10

per cent measurement of that area fixes

\Gamma_{\rm el}