(a) Two identical particles of mass collide head-on, each with beam energy . Starting…

PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Particle Physics & Quantum Fields · Particles, interactions and experiment

Particle Physics, Ch Invariant mass: decays, thresholds and the collider frame

Problem

(a) Two identical particles of mass

m

collide head-on, each with beam energy

E

. Starting from

s=(m_1^2+m_2^2)c^4+2E_1E_2-2\mathbf p_1\cdot\mathbf p_2c^2

, derive an exact expression for

\sqrt{s}_{\rm collider}

. Then derive an exact expression for

\sqrt{s}_{\rm fixed}

for the same beam energy

E

striking a stationary target particle of the same mass

m

. Hence show that, in the limit

E\gg mc^2

\frac{\sqrt{s}_{\rm collider}}{\sqrt{s}_{\rm fixed}}\to\sqrt{\frac{2E}{mc^2}}

(b) A proposed muon collider delivers two beams of muons, each of energy

E=1.50\,\mathrm{TeV}

, colliding head-on; take

m_\mu c^2=105.7\,\mathrm{MeV}

. Calculate (i)

\sqrt{s}

for the collider, (ii)

\sqrt{s}

for a fixed-target experiment with a stationary muon target and the same beam energy, and (iii) the ratio of the two, and confirm the ratio agrees with the high-energy formula of part (a).

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