Paper 4, Section II, A

Dynamics and Relativity | Part IA, 2018

Define the 4-momentum PP of a particle of rest mass mm and velocity u\mathbf{u}. Calculate the power series expansion of the component P0P^{0} for small u/c|\mathbf{u}| / c (where cc is the speed of light in vacuo) up to and including terms of order u4|\mathbf{u}|^{4}, and interpret the first two terms.

(a) At CERN, anti-protons are made by colliding a moving proton with another proton at rest in a fixed target. The collision in question produces three protons and an anti-proton. Assume that the rest mass of a proton is identical to the rest mass of an anti-proton. What is the smallest possible speed of the incoming proton (measured in the laboratory frame)?

(b) A moving particle of rest mass MM decays into NN particles with 4 -momenta QiQ_{i}, and rest masses mim_{i}, where i=1,2,,Ni=1,2, \ldots, N. Show that

M=1c(i=1NQi)(j=1NQj)M=\frac{1}{c} \sqrt{\left(\sum_{i=1}^{N} Q_{i}\right) \cdot\left(\sum_{j=1}^{N} Q_{j}\right)}

Thus, show that

Mi=1NmiM \geqslant \sum_{i=1}^{N} m_{i}

(c) A particle AA decays into particle BB and a massless particle 1 . Particle BB subsequently decays into particle CC and a massless particle 2 . Show that

0(Q1+Q2)(Q1+Q2)(mA2mB2)(mB2mC2)c2mB20 \leqslant\left(Q_{1}+Q_{2}\right) \cdot\left(Q_{1}+Q_{2}\right) \leqslant \frac{\left(m_{A}^{2}-m_{B}^{2}\right)\left(m_{B}^{2}-m_{C}^{2}\right) c^{2}}{m_{B}^{2}}

where Q1Q_{1} and Q2Q_{2} are the 4-momenta of particles 1 and 2 respectively and mA,mB,mCm_{A}, m_{B}, m_{C} are the masses of particles A,BA, B and CC respectively.

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