Problem Set

Problem 1. Using the Gell-Mann—Nishijima formula, determine the electric charge of the $\Sigma^*$ baryon with quark content $sss$.

Problem 2. A particle $X$ decays via the strong interaction: $X \to p + K^+$. Determine The quark content, baryon number, strangeness, and charge of $X$. Classify $X$ as a meson or Baryon.

Problem 3. Show that the decay $\Lambda^0 \to n + \pi^0$ is allowed, while $\Lambda^0 \to p + \pi^-$ is forbidden by strangeness conservation. Through which Interaction does the $\Lambda^0$ actually decay?

Problem 4. Compute the branching ratio upper bound for $\pi^0 \to e^+e^-$ relative to $\pi^0 \to \gamma\gamma$ using dimensional analysis and helicity suppression.

Problem 5. In the decay $\mu^- \to e^- + \bar{\nu}_e + \nu_\mu$Verify that Electric charge, baryon number, and all three lepton family numbers are conserved.

Problem 6. Draw the Feynman diagram for $e^+e^- \to e^+e^-$ (Bhabha scattering) at tree Level. Identify both the $s$-channel and $t$-channel diagrams.

Problem 7. For Compton scattering $e^-\gamma \to e^-\gamma$ in the low-energy limit, Derive the Thomson cross section $\sigma_T = 8\pi\alpha^2/(3m_e^2)$ from the Klein—Nishina Formula.

Problem 8. Calculate the QED running coupling $\alpha$ at the $Z$ boson mass scale ($m_Z = 91.2$ GeV), given $\alpha(m_e) \approx 1/137$.

Problem 9. For the Higgs potential $V = \mu^2\phi^\dagger\phi + \lambda(\phi^\dagger\phi)^2$ With $\mu^2 \lt 0$Show that expanding the field around the VEV $\phi = (0, v)^T/\sqrt{2}$ Produces a physical scalar with mass $m_H = \sqrt{2\lambda}\,v$. Identify the three Goldstone Modes.

Problem 10. The top quark has mass $m_t = 173$ GeV/$c^2$ and the electron has mass $m_e = 0.511$ MeV/$c^2$. Compute the ratio of their Yukawa couplings. What does this imply About the origin of the fermion mass hierarchy?

Problem 11. Verify that the SU(3) structure constant $f^{123} = 1$ by explicitly computing $[\lambda^1, \lambda^2]$ and showing it equals $2i\lambda^3$.

Problem 12. Use the Gell-Mann—Okubo mass formula for the baryon octet, $\frac{1}{2}(m_N + m_\Xi) = \frac{1}{4}(3m_\Lambda + m_\Sigma)$And the experimental masses $m_p = 938.3$ MeV, $m_n = 939.6$ MeV, $m_{\Xi^0} = 1314.9$ MeV, $m_{\Xi^-} = 1321.7$ MeV, $m_\Lambda = 1115.7$ MeV, $m_{\Sigma^+} = 1189.4$ MeV, $m_{\Sigma^0} = 1192.6$ MeV, $m_{\Sigma^-} = 1197.4$ MeV. Compute both sides and find the percentage discrepancy.

Problem 13. Compute the QCD beta function coefficient $b_0 = 11 - 2n_f/3$ for $n_f = 4$, $5$And $6$ active flavours. At what number of flavours does asymptotic freedom Break down?

Problem 14. For a flat, matter-dominated universe with $H_0 = 70$ km/s/Mpc, compute: (a) the critical density $\rho_c$ in kg/m$^3$(b) the age of the universe $t_0 = 2/(3H_0)$ And (c) the Hubble distance $d_H = c/H_0$.

Problem 15. A supernova at redshift $z = 0.5$ is observed to be fainter than predicted By the matter-dominated Friedmann model. Using the deceleration parameter $q_0$Show that $q_0 \lt 0$ is required and that this implies $\Omega_\Lambda \gt \Omega_m/2$.

Problem 16. Solar neutrinos are produced by $p + p \to d + e^+ + \nu_e$ with energy $E_\nu \leq 0.42$ MeV. Using the two-flavour oscillation formula with $\Delta m^2_{21} = 7.5 \times 10^{-5}$ eV$^2$ and $\sin^2(2\theta_{12}) = 0.84$Calculate The oscillation probability $P(\nu_e \to \nu_\mu)$ at the distance $L = 1.5 \times 10^{11}$ m (Earth—Sun distance). Take $E_\nu = 0.3$ MeV. (Express $L$ and $E$ in natural units.)

Problem 17. In the seesaw mechanism, if the Dirac mass is $m_D = m_{\mathrm{top} = 173}$ GeV And the heavy Majorana mass is $M = 10^{14}$ GeV, calculate the resulting light neutrino mass. Compare this with the cosmological bound $\sum m_\nu \lt 0.12$ eV.

Problem 18. The $\Omega^-$ baryon ($sss$) was predicted by Gell-Mann in 1962 using the Decuplet equal-spacing rule. Given the masses $m_\Delta = 1232$ MeV, $m_{\Sigma^*} = 1385$ MeV, And $m_{\Xi^*} = 1533$ MeV, predict $m_{\Omega^-}$. Compare with the measured value of $1672.5$ MeV and comment on the agreement.