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

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.

Read the line styles and fermion-number arrows in the original PDF: a wavy line is a photon, and a coiled line is a gluon. The valid fundamental interaction vertices areBoth (a) and (f) have one electron-line arrow entering and one leaving a photon vertex; changing the drawing's orientation does not change the quantum electrodynamics interaction. The Quantum chromodynamics vertices (b) and (g) are the cubic and quartic gluon self-interactions.

The remaining cases fail for different reasons. Diagram (c) couples an isolated electron line to gluons and is not an allowed fermion vertex. Diagram (d) changes an Electron to a muon at a photon vertex, violating the flavour-diagonal electromagnetic coupling. Diagram (e) couples a colourless Electron to a gluon. Diagram (h) changes an up quark into a down quark at a gluon vertex, whereas Quantum chromodynamics preserves quark flavour; that assignment also fails electric charge conservation.

One measurement of the strong coupling constant uses the fraction of resolved three-jet events in $e^{+}{e}^{-}\to q\bar{q}g$: with a fixed jet-resolution definition, its leading ratio to two-jet production is proportional to ${\alpha }_s$, with calculable higher-order corrections.

The intermediate current is ${\gamma }^{\ast }$ or $Z^{\ast }$, and the extra gluon can be emitted from either the quark or the antiquark. Both Feynman diagrams contribute coherently. The observed jets arise from hadronization; a resolved-jet definition is needed because arbitrarily soft or collinear radiation is not a separate measurable third jet.

The original chart has $Z$ vertically and $N$ horizontally. Thus its region (a) is proton-rich relative to the beta-stable band, whereas (b) is neutron-rich.

Electron capture, $e^{-}+(Z,N)\to (Z-1,N+1)+{\nu }_e$, is the other proton-to-neutron weak process; it remains possible for some neutral atoms whose beta-plus decay lacks the $2m_e$ atomic-mass threshold.The emitted alpha particle contains two protons and two neutrons. Very heavy nuclei can also compete through nuclear fission; the chart alone does not determine each nuclide's branching fractions.

The quark transitions are $u\to d+W^{+}$ followed by $W^{+}\to e^{+}{\nu }_e$ in (a), and $d\to u+W^{-}$ followed by $W^{-}\to e^{-}{\bar{\nu }}_e$ in (b). The additional panel shows electron capture as $u+e^{-}\to d+{\nu }_e$ through W boson exchange. The remaining quarks in each nucleon are spectators.

Regions (c) and (d) lie beyond the proton and neutron binding limits, respectively. In (c), unbound protons can be emitted, $(Z,N)\to (Z-1,N)+p$; in (d), unbound neutrons can be emitted, $(Z,N)\to (Z,N-1)+n$. These are the proton drip line and neutron drip line limits, with one- or several-nucleon emission possible according to the nuclear separation energies. They are not merely more distant points on a beta-decay trajectory.