A spin-zero parent of mass moves along with speed and Lorentz factor

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

Symmetry and Flavour, Ch Four-vectors at work: boosts, invariants and the shape of a final state

Problem

A spin-zero parent of mass

M

moves along

z

with speed

\beta

and Lorentz factor

\gamma

, and decays isotropically in its rest frame to particle

1

(rest-frame energy

E_1^*

, momentum

p_1^*

) and particle

2

. ((a)) Derive the laboratory energy spectrum

\mathrm dN/\mathrm dE_1

of particle

1

, including its end points, and show that it integrates to one. The chapter states this result for massive products without proof. ((b)) A

K^+

(

m_K=493.68\,\mathrm{MeV}

) with laboratory momentum

75.0\,\mathrm{GeV}

decays to

\mu^+\nu_\mu

(

m_\mu=105.66\,\mathrm{MeV}

, neutrino massless). Using the two-body momentum at rest, find the lowest and highest muon energy in the laboratory and the height of the flat spectrum in

\mathrm{GeV^{-1}}

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