Past exam of the particle and nuclear physics course of the University of Cambridge / 2021 / Section A / 1 / Solution

Choose the positive $z$ direction along the incident Electron. With the neglected lepton masses and the paper's natural units, the initial four-momenta areThe squared centre-of-mass energy is the Lorentz invariantProducing the Upsilon meson as the sole final resonance requires $s=M_{{\rm Y}}^2$. The threshold positron energy is thereforeAt that beam setting, four-momentum conservation gives $E_{{\rm Y}}=E_{-}+E_{+}=12.10934\mathrm{GeV}$ and $p_{{\rm Y},z}=E_{-}-E_{+}=5.89066\mathrm{GeV}$. HenceThe resonance moves in the incident-Electron direction, with velocity approximately $0.486c$. The laboratory sum of beam energies exceeds the resonance rest energy because the asymmetric beams also give it laboratory kinetic energy.