Appendix D. Problems and selected solutions

Problem 1 — Electron versus proton mass #

Using $m_pc^2=938.27208943\,\mathrm{MeV}$ and $m_ec^2=0.51099895069\,\mathrm{MeV}$, calculate $m_p/m_e$ and compare with the CODATA ratio.

Solution.

consistent with the CODATA value within the rounding used here. CODATA recommended values of the fundamental physical constants: 2022 (2024)

Problem 2 — Photon energy #

Find the energy of a $500\,\mathrm{nm}$ photon.

Solution.

Problem 3 — Weak length scale #

Estimate $\hbar c/M_Wc^2$ using $M_W\approx80.4\,\mathrm{GeV}$.

Solution.

This is only an order-of-magnitude range estimate; weak amplitudes are computed from propagators, not from a literal finite flight path of a virtual $W$ boson.

Problem 4 — Higgs Yukawa coupling of the electron #

Estimate the electron Yukawa coupling using $m_e=y_ev/\sqrt2$ and $v=246.22\,\mathrm{GeV}$.

Solution.

The smallness of this dimensionless number is part of the unexplained flavor hierarchy.

Problem 5 — QCD crossover temperature in kelvin #

Use $k_B=8.617333262\ldots\times10^{-5}\,\mathrm{eV/K}$ to convert $156.5\,\mathrm{MeV}$ to kelvin.

Solution.

CODATA recommended values of the fundamental physical constants: 2022 (2024) · Chiral crossover in QCD at zero and non-zero chemical potentials (2019)

Problem 6 — Why a photon has no rest frame #

Show that the Lorentz factor diverges as $v\to c$.

Solution. Since $1-v^2/c^2\to0^+$,

A finite Lorentz boost cannot transform an inertial observer into a lightlike rest frame.

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