Self-Assessment Quiz — Chapter 32

Test your understanding of the core concepts before moving on. Try to answer each question before checking the solutions at the end.

Q1. (Multiple Choice) What is the primary reason nuclei are useful as precision laboratories for fundamental physics?

(a) Nuclei are large enough to be seen with microscopes (b) Coherent enhancement amplifies tiny effects by factors of $A$ or $A^2$ (c) Nuclear experiments are cheaper than particle accelerators (d) The strong force is easier to calculate than the electromagnetic force

Q2. (True/False) The Wu experiment demonstrated parity violation in the electromagnetic interaction.

Q3. (Multiple Choice) In parity-violating electron scattering, the asymmetry $A_{\text{PV}}$ arises from the interference between:

(a) Strong and electromagnetic amplitudes (b) Electromagnetic ($\gamma$ exchange) and weak ($Z^0$ exchange) amplitudes (c) Two different electromagnetic multipoles (d) Nuclear absorption and scattering amplitudes

Q4. (Short Answer) Why does parity-violating electron scattering on a heavy nucleus primarily probe the neutron distribution rather than the proton distribution?

Q5. (True/False) A permanent electric dipole moment of the neutron would violate parity (P) but not time-reversal (T) symmetry.

Q6. (Multiple Choice) The current upper limit on the neutron EDM is approximately:

(a) $10^{-18}\,e\cdot\text{cm}$ (b) $10^{-22}\,e\cdot\text{cm}$ (c) $10^{-26}\,e\cdot\text{cm}$ (d) $10^{-32}\,e\cdot\text{cm}$

Q7. (Short Answer) State Schiff's theorem and explain why it does not completely shield atomic EDMs from nuclear EDMs.

Q8. (Multiple Choice) The nuclear Schiff moment is particularly enhanced in nuclei with:

(a) Spherical shapes and closed shells (b) Large quadrupole deformation (c) Octupole deformation (pear shapes) (d) High spin isomeric states

Q9. (Short Answer) What quantity do superallowed $0^+ \to 0^+$ Fermi transitions determine, and why are they uniquely suited for this measurement?

Q10. (Multiple Choice) The "Cabibbo angle anomaly" refers to:

(a) The discovery that the Cabibbo angle is larger than expected (b) A $\sim 3\sigma$ deficit in first-row CKM unitarity ($|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 < 1$) (c) A disagreement between nuclear and particle physics measurements of the weak mixing angle (d) The unexplained mass of the strange quark

Q11. (True/False) The triple correlation coefficient $D$ in nuclear beta decay ($\vec{J} \cdot (\vec{p}_e \times \vec{p}_\nu)$) is a test of time-reversal symmetry.

Q12. (Multiple Choice) In coherent elastic neutrino-nucleus scattering (CE$\nu$NS), the cross section is proportional to:

(a) $Z^2$ (b) $A^2$ (c) $N^2$ (approximately) (d) $A^{1/3}$

Q13. (Short Answer) The COHERENT experiment at Oak Ridge was the first to observe CE$\nu$NS. What neutrino source did it use, and why was the pulsed timing structure important?

Q14. (Multiple Choice) The experimental signature of neutrinoless double beta decay ($0\nu\beta\beta$) is:

(a) A continuous electron energy spectrum (b) A sharp peak in the two-electron sum energy at the $Q$-value (c) The emission of two gamma rays (d) A proton recoil with no visible electrons

Q15. (True/False) Observation of neutrinoless double beta decay would prove that neutrinos have nonzero mass and are Majorana particles.

Q16. (Short Answer) Explain in one or two sentences why the nuclear matrix element (NME) problem is the dominant theoretical uncertainty in $0\nu\beta\beta$ experiments.

Q17. (Multiple Choice) Which of the following next-generation experiments does NOT search for $0\nu\beta\beta$?

(a) LEGEND-1000 (b) nEXO (c) LZ (d) CUPID

Q18. (Multiple Choice) In WIMP dark matter direct detection, the spin-independent scattering cross section is enhanced for heavy nuclei because:

(a) Heavy nuclei have more electrons (b) The WIMP couples coherently to all nucleons, giving $\sigma \propto A^2$ (c) Heavy nuclei are more radioactive (d) The Helm form factor increases with $A$

Q19. (Short Answer) What is the "neutrino floor" (or "neutrino fog") in dark matter direct detection, and why is it significant for next-generation experiments?

Q20. (True/False) Nuclear form factors are unimportant for dark matter direct detection because WIMPs are point-like particles.

Solutions

Q1. (b) Coherent enhancement amplifies tiny effects by factors of $A$ or $A^2$. When a probe interacts with all nucleons simultaneously, the amplitude adds coherently, producing measurable signals from otherwise undetectable effects.

Q2. False. The Wu experiment demonstrated parity violation in the weak interaction (beta decay of ${}^{60}\text{Co}$), not the electromagnetic interaction.

Q3. (b) The parity-violating asymmetry arises from interference between the parity-conserving electromagnetic amplitude ($\gamma$ exchange) and the parity-violating weak neutral current amplitude ($Z^0$ exchange).

Q4. The weak charge of the proton is $Q_W^p = 1 - 4\sin^2\theta_W \approx 0.07$, while the weak charge of the neutron is $Q_W^n \approx -1$. Since the neutron weak charge is $\sim 14$ times larger, the $Z^0$ boson couples primarily to neutrons, making PVES sensitive to the neutron distribution.

Q5. False. A permanent EDM violates both P and T symmetry (and hence CP, via the CPT theorem). Under parity, $\vec{d} \to -\vec{d}$ but $\vec{S} \to \vec{S}$; under time reversal, $\vec{d} \to \vec{d}$ but $\vec{S} \to -\vec{S}$.

Q6. (c) $|d_n| < 1.8 \times 10^{-26}\,e\cdot\text{cm}$.

Q7. Schiff's theorem states that in a nonrelativistic system of point charges bound by electrostatic forces, the constituent EDMs are completely shielded — the system rearranges to produce zero net EDM. The three loopholes are: (1) finite nuclear size (the Schiff moment), (2) relativistic effects near the nucleus, and (3) magnetic interactions. For diamagnetic atoms, the Schiff moment (loophole 1) dominates; for paramagnetic atoms, the relativistic electron-EDM coupling (loophole 2) dominates.

Q8. (c) Octupole deformation produces near-degenerate parity doublets that are strongly mixed by P,T-violating interactions, enhancing the Schiff moment by factors of $10^2$–$10^3$ compared to spherical nuclei.

Q9. Superallowed $0^+ \to 0^+$ transitions determine $|V_{ud}|$, the up-down element of the CKM quark mixing matrix. They are uniquely suited because the nuclear matrix element is fixed by isospin symmetry ($\sqrt{2}$ for $T = 1$ transitions), eliminating the dominant source of nuclear structure uncertainty.

Q10. (b) The Cabibbo angle anomaly is the observation that $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 \approx 0.9985$, about $3\sigma$ below the unitarity prediction of 1.

Q11. True. The triple product $\vec{J} \cdot (\vec{p}_e \times \vec{p}_\nu)$ is odd under time reversal ($\vec{J}$, $\vec{p}_e$, $\vec{p}_\nu$ all reverse), so a nonzero coefficient $D$ signals T violation.

Q12. (c) $\sigma_{\text{CE}\nu\text{NS}} \propto Q_W^2 \approx N^2$, because the weak charge of the nucleus is dominated by the neutron contribution ($Q_W^n \approx -1 \gg Q_W^p \approx 0.07$).

Q13. COHERENT used neutrinos from the Spallation Neutron Source (SNS) at Oak Ridge — specifically, neutrinos from pion decay at rest ($\pi^+ \to \mu^+ \nu_\mu$) and subsequent muon decay ($\mu^+ \to e^+ \nu_e \bar{\nu}_\mu$). The pulsed timing was critical because it allowed the prompt $\nu_\mu$ signal (within $\sim 1\,\mu\text{s}$ of the beam pulse) to be distinguished from backgrounds and from the delayed muon-decay neutrinos ($\tau_\mu = 2.2\,\mu\text{s}$).

Q14. (b) In $0\nu\beta\beta$, no neutrinos carry away energy, so both electrons share the full $Q$-value. The experimental signature is a monoenergetic peak in the two-electron sum energy spectrum at $E = Q$.

Q15. True. The process requires a helicity flip of the virtual neutrino, which is only possible if the neutrino has mass. The absorption of a $\bar{\nu}_e$ as a $\nu_e$ at the second vertex requires the neutrino to be its own antiparticle (Majorana).

Q16. The NME involves the transition of a correlated neutron pair to a proton pair mediated by a virtual neutrino, and it is sensitive to short-range correlations, pairing, and the detailed structure of both initial and final nuclear states. Different theoretical methods (shell model, QRPA, IBM, EDF, ab initio) disagree by factors of 2–3, which propagates to a factor of 4–9 uncertainty in the extracted $\langle m_{\beta\beta} \rangle$.

Q17. (c) LZ (LUX-ZEPLIN) is a dark matter direct detection experiment, not a $0\nu\beta\beta$ experiment. LEGEND-1000, nEXO, and CUPID all search for $0\nu\beta\beta$.

Q18. (b) For spin-independent (scalar) WIMP-nucleon interactions, the WIMP couples to all $A$ nucleons coherently, giving an amplitude $\propto A$ and a cross section $\propto A^2$. This is the same coherent enhancement principle that governs CE$\nu$NS.

Q19. The neutrino floor is the WIMP cross section below which CE$\nu$NS events from solar, atmospheric, and supernova neutrinos become an irreducible background that mimics a WIMP signal. It is significant because next-generation detectors (XLZD) will approach this floor, requiring new strategies (directional detection, multiple targets, annual modulation analysis) to discriminate WIMPs from neutrinos.

Q20. False. Nuclear form factors are essential for dark matter detection. They describe how the WIMP-nucleus cross section decreases at higher momentum transfer (where the probe resolves the nuclear structure). For heavy targets and high recoil energies, the form factor suppression can be $30$–$70\%$, significantly affecting the interpretation of experimental limits.