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.

Problem 19. (Dark matter) The rotational velocity of stars in the Milky Way is observed to be approximately constant at $v \approx 220$ km/s out to at least $r = 20$ kpc. Assuming a spherical dark matter halo with density profile $\rho(r) = \rho_0 / (1 + (r/r_s)^2)$ , compute the enclosed mass $M(r)$ and show that $v(r)$ is approximately flat for $r \gg r_s$ .

Problem 20. (Neutrino oscillations) A neutrino produced as $\nu_\mu$ with energy $E = 1$ GeV travels $L = 1000$ km. Given $\Delta m_{32}^2 = 2.5 \times 10^{-3}$ eV $^2$ and $\sin^2(2\theta_{23}) = 1.0$ , calculate the oscillation probability $P(\nu_\mu \to \nu_\tau)$ . Express $L$ and $E$ in natural units ( $\hbar = c = 1$ ).

Problem 21. (Cosmic microwave background) The CMB temperature today is $T_0 = 2.725$ K. (a) Compute the energy density of the CMB photon gas. (b) At what redshift $z$ did the CMB temperature equal $T = 3000$ K (recombination epoch)? (c) Estimate the baryon-to-photon ratio $\eta$ given that the baryon density today is $\Omega_b h^2 = 0.022$ .

Problem 22. (Running coupling unification) Plot (conceptually) the running of the three gauge couplings $\alpha_1$ , $\alpha_2$ , $\alpha_3$ in the Standard Model as a function of energy scale. Explain why minimal SU(5) unification fails: the couplings do not meet at a single point. How does supersymmetry (MSSM) resolve this?

Selected Solutions

Problem 9. The Higgs potential minimum occurs at $\phi^\dagger\phi = -\mu^2/(2\lambda) = v^2/2$ . Writing $\phi = (0, v + h)^T/\sqrt{2}$ and substituting gives the Higgs mass $m_H = \sqrt{2\lambda}\,v$ . The three Goldstone modes correspond to the three broken generators of SU(2) $_L$ , which become the longitudinal components of $W^\pm$ and $Z$ .

Problem 10. The Yukawa coupling is $y_f = \sqrt{2}\,m_f/v$ where $v = 246$ GeV. $y_t/y_e = m_t/m_e = 173\ \mathrm{GeV} / 0.511\ \mathrm{MeV} \approx 3.4 \times 10^5$ . This large hierarchy suggests that the fermion masses arise from a more fundamental mechanism.

Problem 14. $\rho_c = 3H_0^2/(8\pi G) \approx 1.88 \times 10^{-26}\,h^2\ \mathrm{kg/m}^3$ with $h = H_0/(100\ \mathrm{km/s/Mpc}) = 0.7$ , giving $\rho_c \approx 9.2 \times 10^{-27}\ \mathrm{kg/m}^3$ . $t_0 = 2/(3H_0) \approx 9.3 \times 10^9$ years (for a matter-dominated flat universe). $d_H = c/H_0 \approx 4280\ \mathrm{Mpc}$ .

Problem 16. In natural units $\hbar = c = 1$ : $L = 1.5 \times 10^{11}\ \mathrm{m} \approx 7.6 \times 10^{20}\ \mathrm{GeV}^{-1}$ , $E = 0.3$ MeV. The oscillation phase is $\Delta m^2 L/(4E) \approx 1.24$ , giving $P(\nu_e \to \nu_\mu) = \sin^2(2\theta_{12}) \sin^2(\Delta m^2 L/(4E)) \approx 0.84 \times 0.95 \approx 0.80$ .

Problem 17. $m_\nu = m_D^2/M \approx (173\ \mathrm{GeV})^2 / 10^{14}\ \mathrm{GeV} \approx 0.3$ eV. This is above the cosmological bound $\sum m_\nu < 0.12$ eV, suggesting either a smaller Dirac mass or a larger Majorana mass.

Problem 18. The decuplet equal-spacing rule predicts $m_{\Omega^-} = m_{\Xi^*} + (m_{\Xi^*} - m_{\Sigma^*}) = 1533 + (1533 - 1385) = 1681$ MeV. The measured value is $1672.5$ MeV, a discrepancy of $0.5\%$ , which was a spectacular confirmation of SU(3) flavour symmetry and led to the discovery of the $\Omega^-$ at Brookhaven in 1964.

Problem 19. For $\rho(r) = \rho_0/(1 + (r/r_s)^2)$ , the enclosed mass is $M(r) = 4\pi\int_0^r \rho(r') r'^2\,dr' = 4\pi\rho_0 r_s^3[r/r_s - \arctan(r/r_s)]$ . The circular velocity is $v^2(r) = GM(r)/r$ . For $r \gg r_s$ , $M(r) \propto r$ , giving $v \approx \sqrt{4\pi G\rho_0 r_s^2}$ , which is constant — explaining the flat rotation curves.

Problem 22. The gauge couplings run according to the renormalisation group equations: $\mu d\alpha_i^{-1}/d\mu = -b_i/(2\pi)$ . In the SM, $b = (41/10, -19/6, -7)$ for U(1), SU(2), SU(3) respectively. The couplings nearly meet at $M_U \sim 10^{15}$ GeV but do not intersect at a single point. In the MSSM, the beta coefficients change to $(33/5, 1, -3)$ , and the couplings unify at $M_U \sim 2\times 10^{16}$ GeV, providing indirect evidence for supersymmetry.

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