Exercises — Chapter 32

Parity Violation

Problem 32.1 ⭐ In the Wu experiment, ${}^{60}\text{Co}$ nuclei ($J^{\pi} = 5^+$) are cooled to $\sim 10\,\text{mK}$ in a magnetic field to achieve polarization.

(a) Explain why cooling is necessary. Estimate the temperature at which the thermal energy $k_B T$ equals the nuclear magnetic splitting $\mu B$ for a typical laboratory field $B = 0.05\,\text{T}$ and nuclear magnetic moment $\mu \sim 3\mu_N$ ($\mu_N = 3.15 \times 10^{-8}\,\text{eV/T}$).

(b) The beta asymmetry parameter for the ${}^{60}\text{Co}$ transition ($5^+ \to 4^+$, pure Gamow-Teller) is $A = -1$. Show that this means the electrons are emitted preferentially antiparallel to the nuclear spin direction.

(c) Explain why a scalar interaction ($V - A$ replaced by $S$) would produce $A = 0$ for this transition. What does the measured $A \neq 0$ tell us about the Lorentz structure of the weak interaction?

Problem 32.2 ⭐ The parity-violating asymmetry in electron-proton scattering at low momentum transfer is:

$$A_{\text{PV}} \approx -\frac{G_F Q^2}{4\sqrt{2}\pi\alpha} Q_W^p$$

where $Q_W^p = 1 - 4\sin^2\theta_W$.

(a) Using $G_F = 1.166 \times 10^{-5}\,\text{GeV}^{-2}$, $\alpha = 1/137$, and $\sin^2\theta_W = 0.238$, calculate $A_{\text{PV}}$ at $Q^2 = 0.025\,\text{GeV}^2$.

(b) If you measure the cross section with $10^{13}$ electrons (roughly the Q-weak statistics), what is the statistical uncertainty on the asymmetry? (Hint: $\delta A \sim 1/\sqrt{N}$.)

(c) The Q-weak experiment achieved $\delta Q_W^p / Q_W^p \approx 6\%$. How does the fractional uncertainty on $A_{\text{PV}}$ translate to a fractional uncertainty on $\sin^2\theta_W$?

Problem 32.3 ⭐⭐ The weak charge of a nucleus with $Z$ protons and $N$ neutrons is:

$$Q_W = -N + Z(1 - 4\sin^2\theta_W)$$

(a) Calculate $Q_W$ for ${}^{208}\text{Pb}$ ($Z = 82$, $N = 126$) and for ${}^{133}\text{Cs}$ ($Z = 55$, $N = 78$).

(b) Show that $Q_W \approx -N$ for heavy nuclei. Why does this make PVES a probe of the neutron distribution?

(c) The PREX-II experiment measured a neutron skin thickness $\Delta r_{np} = 0.283 \pm 0.071\,\text{fm}$ in ${}^{208}\text{Pb}$. If the proton rms radius is $r_p = 5.503\,\text{fm}$, what is the neutron rms radius $r_n$?

Problem 32.3a ⭐⭐ The PREX-II experiment measured the parity-violating asymmetry on ${}^{208}\text{Pb}$ to be $A_{\text{PV}} = 550 \pm 16\,\text{ppb}$ at $Q^2 = 0.00616\,\text{GeV}^2$.

(a) The asymmetry is related to the neutron skin through the nuclear weak form factor. At leading order: $A_{\text{PV}} \propto Q_W F_W(Q^2)$, where $F_W$ is the weak (neutron) form factor. Using $Q_W^n/Q_W^p \approx -14$, explain why the asymmetry is dominated by the neutron distribution.

(b) The extracted neutron skin thickness is $\Delta r_{np} = 0.283 \pm 0.071\,\text{fm}$. Using the empirical correlation $L \approx (5.5\,\text{fm}^{-1}) \times \Delta r_{np}({}^{208}\text{Pb}) \times 50\,\text{MeV}$, estimate the slope of the nuclear symmetry energy $L$ and its uncertainty.

(c) The neutron star radius for a $1.4 M_\odot$ star correlates with $L$ as approximately $R_{1.4} \approx 10 + 0.05L\,\text{km}$ (with $L$ in MeV). Using your answer from (b), estimate $R_{1.4}$ and compare to the NICER measurements ($R_{1.4} \approx 12.4 \pm 0.5\,\text{km}$).

Electric Dipole Moments

Problem 32.4 ⭐ A permanent electric dipole moment $\vec{d} = d\,\vec{S}/S$ of a spin-$\frac{1}{2}$ particle placed in parallel electric ($\vec{E}$) and magnetic ($\vec{B}$) fields experiences an energy shift:

$$\Delta E = -2dE \cdot m_s$$

where $m_s = \pm 1/2$ is the spin projection.

(a) Show that the Larmor precession frequency shifts by $\Delta\nu = 4dE/h$ when the electric field is reversed.

(b) For the current neutron EDM limit $|d_n| < 1.8 \times 10^{-26}\,e\cdot\text{cm}$ and a typical experimental field $E = 12\,\text{kV/cm}$, calculate $\Delta\nu$ in Hz. Compare to the magnetic precession frequency for $B = 1\,\mu\text{T}$.

(c) Explain why this measurement is so sensitive to systematic effects from the $\vec{v} \times \vec{E}$ motional magnetic field.

Problem 32.5 ⭐⭐ Schiff's theorem states that a system of point charges held together by electrostatic forces has zero net EDM, even if the constituents have EDMs.

(a) Give a physical argument for why this must be so. (Hint: think about what happens to the center of charge in an external electric field.)

(b) List the three physical effects that provide loopholes to Schiff's theorem and explain which is most important for: (i) the neutron, (ii) a diamagnetic atom like ${}^{199}\text{Hg}$, (iii) a paramagnetic atom like Tl.

(c) The nuclear Schiff moment enhancement scales roughly as $Z^2 A^{2/3} \beta_2$ for deformed nuclei. Estimate the ratio of Schiff moment sensitivities for ${}^{225}\text{Ra}$ ($Z = 88$, $A = 225$, $\beta_3 \approx 0.1$, with $10^3 \times$ octupole enhancement) versus ${}^{199}\text{Hg}$ ($Z = 80$, $A = 199$, $\beta_2 \approx 0$).

Problem 32.6 ⭐⭐⭐ The Standard Model predicts a neutron EDM from the CKM CP-violating phase at $d_n^{\text{SM}} \sim 10^{-32}\,e\cdot\text{cm}$. Supersymmetric models with CP-violating phases of order unity predict $d_n^{\text{SUSY}} \sim 10^{-25}\,e\cdot\text{cm}$ for superpartner masses at the TeV scale.

(a) If the next-generation n2EDM experiment at PSI achieves a sensitivity of $d_n \sim 10^{-27}\,e\cdot\text{cm}$ and finds a null result, what lower bound does this place on the SUSY CP-violating mass scale? (Hint: $d_n \propto \sin\phi_{\text{CP}}/M_{\text{SUSY}}^2$.)

(b) An alternative BSM scenario involves a QCD vacuum angle $\bar{\theta}$, which produces $d_n \sim 3.6 \times 10^{-16}\,\bar{\theta}\,e\cdot\text{cm}$. What is the current upper limit on $\bar{\theta}$ from the neutron EDM measurement?

(c) Explain the "strong CP problem" — why is $\bar{\theta}$ so small? — and briefly describe the axion solution.

CKM Unitarity and Beta-Decay Correlations

Problem 32.7 ⭐ The corrected $\mathcal{F}t$ value for superallowed $0^+ \to 0^+$ Fermi transitions is $\overline{\mathcal{F}t} = 3072.27 \pm 0.72\,\text{s}$.

(a) Using $\mathcal{F}t = K / [2G_V^2(1 + \Delta_R^V)]$ with $K = 8120.2776 \times 10^{-10}\,\text{GeV}^{-4}\text{s}$ and $\Delta_R^V = 0.02467$, extract $G_V$.

(b) From $G_V = G_F |V_{ud}|$ with $G_F = 1.1663788 \times 10^{-5}\,\text{GeV}^{-2}$, determine $|V_{ud}|$ and its uncertainty.

(c) Verify the Cabibbo angle anomaly: using $|V_{us}| = 0.2243 \pm 0.0005$ and $|V_{ub}| = 0.00382 \pm 0.00020$, calculate $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2$ and the deviation from unity in units of standard deviations.

Problem 32.8 ⭐⭐ For a mixed Fermi + Gamow-Teller transition with mixing ratio $\rho = g_A M_{GT}/(g_V M_F)$, the beta asymmetry parameter is:

$$A = \frac{-2|\rho|^2 + 2\text{Re}(\rho)}{1 + 3|\rho|^2}$$

(a) Show that for a pure Fermi transition ($\rho = 0$), $A = 0$.

(b) Show that for a pure Gamow-Teller transition ($\rho \to \infty$), $A \to -2/3$.

(c) Neutron decay has $\rho = g_A/g_V \approx -1.276$. Calculate $A$ for the neutron and compare to the measured value $A = -0.11985 \pm 0.00017$.

(d) Why is the neutron an important complement to superallowed decays for determining $V_{ud}$?

Problem 32.9 ⭐⭐ The isospin symmetry-breaking correction $\delta_C$ for the superallowed decay of ${}^{14}\text{O}$ ($T = 1$, $T_z = -1$) to ${}^{14}\text{N}^*$ ($T = 1$, $T_z = 0$) is approximately $\delta_C \approx 0.3\%$.

(a) Explain physically why $\delta_C > 0$. What interactions break the isospin symmetry that makes the Fermi matrix element exactly $\sqrt{2}$?

(b) If $\delta_C$ were systematically underestimated by $0.1\%$ across all 15 superallowed transitions, by how much would $|V_{ud}|$ shift? Would this resolve the Cabibbo angle anomaly?

(c) Explain how mirror transitions ($T = 1/2$, $T_z = \pm 1/2$) provide an independent cross-check on the $V_{ud}$ extraction.

CE$\nu$NS

Problem 32.10 ⭐ The maximum nuclear recoil energy in CE$\nu$NS is:

$$T_R^{\max} = \frac{2E_\nu^2}{M + 2E_\nu}$$

(a) For ${}^{133}\text{Cs}$ ($M \approx 124\,\text{GeV}$) and $E_\nu = 30\,\text{MeV}$ (from stopped-pion neutrinos), calculate $T_R^{\max}$ in keV.

(b) Repeat for ${}^{40}\text{Ar}$ ($M \approx 37.2\,\text{GeV}$) at the same neutrino energy. Which target gives larger recoils?

(c) Why does a lighter target produce larger recoils but a smaller coherent cross section? Discuss the experimental trade-off.

Problem 32.11 ⭐⭐ The CE$\nu$NS cross section for a spin-zero nucleus at zero momentum transfer is:

$$\sigma_0 = \frac{G_F^2}{4\pi} Q_W^2 E_\nu^2$$

(a) Calculate $\sigma_0$ for ${}^{133}\text{Cs}$ ($N = 78$) at $E_\nu = 30\,\text{MeV}$. Use $Q_W \approx -N + Z(1 - 4\sin^2\theta_W)$ with $\sin^2\theta_W = 0.238$.

(b) Compare this to the inverse beta decay cross section at the same energy: $\sigma_{\text{IBD}} \approx 9.6 \times 10^{-44}(E_\nu/\text{MeV})^2\,\text{cm}^2$.

(c) The coherence condition requires $qR \ll 1$. For CsI ($R \approx 6.0\,\text{fm}$), at what recoil energy $T_R$ does $qR = 1$? (Use $q = \sqrt{2MT_R}$.)

Problem 32.12 ⭐⭐ The COHERENT experiment at Oak Ridge observed CE$\nu$NS using a pulsed neutrino beam from pion decay at rest ($\pi^+ \to \mu^+ \nu_\mu$, then $\mu^+ \to e^+ \nu_e \bar{\nu}_\mu$).

(a) What are the energies of the $\nu_\mu$ from pion decay at rest? What is the energy spectrum of the $\nu_e$ and $\bar{\nu}_\mu$ from muon decay? (Maximum energies?)

(b) COHERENT observed $134 \pm 22$ events. The Standard Model prediction was $173 \pm 48$. Is this consistent within uncertainties? Calculate the ratio and propagate the uncertainty.

(c) If the cross section scales as $N^2$, what would be the event rate ratio between CsI ($N_{\text{Cs}} = 78$, $N_{\text{I}} = 74$) and liquid argon ($N_{\text{Ar}} = 22$), assuming equal detector masses and the same neutrino flux? (Hint: account for the different atomic masses.)

Problem 32.13 ⭐⭐⭐ Nonstandard neutrino interactions (NSI) can be parameterized as modifications to the SM neutral-current coupling:

$$Q_W^{\text{NSI}} = Q_W^{\text{SM}} + Z(2\varepsilon_{ee}^{uV} + \varepsilon_{ee}^{dV}) + N(2\varepsilon_{ee}^{dV} + \varepsilon_{ee}^{uV})$$

where $\varepsilon$ are dimensionless BSM coupling constants.

(a) If $\varepsilon_{ee}^{uV} = \varepsilon_{ee}^{dV} \equiv \varepsilon$, show that $Q_W^{\text{NSI}} = Q_W^{\text{SM}} + 3A\varepsilon$.

(b) The COHERENT measurement constrains the CE$\nu$NS cross section to within $\pm 30\%$ of the SM value. What is the resulting bound on $|\varepsilon|$ for CsI ($A \approx 133$)?

(c) How would measuring CE$\nu$NS on multiple targets (CsI, Ar, Ge) help disentangle different NSI couplings?

Problem 32.13a ⭐⭐ The neutrino floor from CE$\nu$NS limits the ultimate sensitivity of dark matter detectors.

(a) The ${}^{8}\text{B}$ solar neutrino flux is $\Phi = 5.16 \times 10^6\,\text{cm}^{-2}\text{s}^{-1}$. For a xenon detector ($N = 77$, $A = 131$, $M = 122\,\text{GeV}$), estimate the CE$\nu$NS cross section at $E_\nu = 8\,\text{MeV}$ (the average ${}^8\text{B}$ neutrino energy).

(b) Calculate the expected CE$\nu$NS event rate per tonne-year in the recoil energy window $1\,\text{keV} < T_R < 10\,\text{keV}$. (Approximate by assuming the integrated cross section above $T_R = 1\,\text{keV}$ is $\sim 50\%$ of the total.)

(c) A 6-GeV WIMP with cross section $\sigma_p = 10^{-48}\,\text{cm}^2$ would produce a similar recoil spectrum. Using $\langle v \rangle = 220\,\text{km/s}$ and $\rho_\chi = 0.3\,\text{GeV/cm}^3$, estimate the WIMP event rate per tonne-year and compare to your CE$\nu$NS estimate. This is the origin of the "neutrino floor."

Neutrinoless Double Beta Decay

Problem 32.14 ⭐ The $Q$-value for double beta decay of ${}^{76}\text{Ge}$ is $Q = 2039.061 \pm 0.007\,\text{keV}$.

(a) In the $0\nu\beta\beta$ mode, all the decay energy goes to the two electrons (the nuclear recoil is negligible). What is the expected two-electron sum energy?

(b) In the $2\nu\beta\beta$ mode, the energy is shared among 4 leptons. Sketch the expected sum electron energy spectrum. Where does it peak relative to $Q$?

(c) Explain why energy resolution is crucial for distinguishing a $0\nu\beta\beta$ signal from the $2\nu\beta\beta$ background.

Problem 32.15 ⭐ The half-life for $2\nu\beta\beta$ of ${}^{76}\text{Ge}$ is $T_{1/2}^{2\nu} = 1.926 \times 10^{21}\,\text{yr}$.

(a) How many ${}^{76}\text{Ge}$ atoms are in 1 kg of germanium enriched to 87% in ${}^{76}\text{Ge}$?

(b) Calculate the $2\nu\beta\beta$ decay rate per kg per year.

(c) The current $0\nu\beta\beta$ limit is $T_{1/2}^{0\nu} > 1.8 \times 10^{26}\,\text{yr}$. How many $0\nu\beta\beta$ events would a 200-kg detector expect in 1 year if the half-life were exactly at this limit?

Problem 32.16 ⭐⭐ The effective Majorana mass is:

$$\langle m_{\beta\beta} \rangle = |m_1 |U_{e1}|^2 e^{i\alpha_1} + m_2 |U_{e2}|^2 e^{i\alpha_2} + m_3 |U_{e3}|^2|$$

where $\alpha_1$, $\alpha_2$ are Majorana phases and the PMNS mixing parameters are approximately $|U_{e1}|^2 = 0.681$, $|U_{e2}|^2 = 0.297$, $|U_{e3}|^2 = 0.022$.

(a) For the normal ordering ($m_1 \approx 0$), express $\langle m_{\beta\beta} \rangle$ in terms of $\Delta m_{21}^2 = 7.5 \times 10^{-5}\,\text{eV}^2$ and $|\Delta m_{32}^2| = 2.5 \times 10^{-3}\,\text{eV}^2$, and the Majorana phases.

(b) Show that for normal ordering with $m_1 \to 0$, the minimum $\langle m_{\beta\beta} \rangle$ (with optimal cancellation) can be as small as $\sim 1\,\text{meV}$.

(c) For the inverted ordering ($m_3 \approx 0$, $m_1 \approx m_2 \approx \sqrt{|\Delta m_{32}^2|}$), show that $\langle m_{\beta\beta} \rangle \gtrsim 15\,\text{meV}$ regardless of Majorana phases.

Problem 32.17 ⭐⭐ The $0\nu\beta\beta$ half-life is:

$$\left[T_{1/2}^{0\nu}\right]^{-1} = G^{0\nu} |M^{0\nu}|^2 \left(\frac{\langle m_{\beta\beta} \rangle}{m_e}\right)^2$$

For ${}^{76}\text{Ge}$: $G^{0\nu} = 2.36 \times 10^{-15}\,\text{yr}^{-1}$.

(a) If $M^{0\nu} = 3.0$ (shell model value) and $\langle m_{\beta\beta} \rangle = 50\,\text{meV}$, calculate the predicted $T_{1/2}^{0\nu}$.

(b) If instead $M^{0\nu} = 5.5$ (QRPA value), recalculate $T_{1/2}^{0\nu}$ for the same $\langle m_{\beta\beta} \rangle$. By what factor do the two predictions differ?

(c) LEGEND-1000 aims for a sensitivity of $T_{1/2}^{0\nu} > 10^{28}\,\text{yr}$. What is the corresponding $\langle m_{\beta\beta} \rangle$ sensitivity for $M^{0\nu} = 3.0$? For $M^{0\nu} = 5.5$?

Problem 32.18 ⭐⭐⭐ The nuclear matrix element for $0\nu\beta\beta$ can be decomposed as:

$$M^{0\nu} = M_{GT}^{0\nu} - \frac{g_V^2}{g_A^2} M_F^{0\nu} + M_T^{0\nu}$$

where $M_{GT}$, $M_F$, and $M_T$ are the Gamow-Teller, Fermi, and tensor contributions, with $g_A/g_V \approx 1.27$.

(a) In a simplified model, $M_F^{0\nu} \approx -M_{GT}^{0\nu}/3$ and $M_T^{0\nu}$ is small. Show that $M^{0\nu} \approx M_{GT}^{0\nu}(1 + 1/(3 \times 1.27^2)) \approx 1.21 M_{GT}^{0\nu}$.

(b) The "quenching" of $g_A$ in nuclear medium is a long-standing puzzle. If the effective $g_A^{\text{eff}} = 0.9 g_A^{\text{free}}$, how does this affect $M^{0\nu}$? (Note: the GT part scales as $g_A^2$ while the Fermi part does not.)

(c) Explain qualitatively why short-range correlations between nucleons affect $M^{0\nu}$ — what is the typical distance scale probed by the neutrino propagator, and how does it compare to the nucleon hard-core repulsion range?

Dark Matter Direct Detection

Problem 32.19 ⭐ A WIMP with mass $m_\chi = 100\,\text{GeV}/c^2$ scatters elastically off a xenon nucleus ($A = 131$, $M = 122\,\text{GeV}/c^2$).

(a) Calculate the reduced mass $\mu$ of the WIMP-nucleus system.

(b) For a WIMP velocity $v = 220\,\text{km/s}$, calculate the maximum recoil energy $T_R^{\max} = 2\mu^2 v^2 / M$ in keV.

(c) Repeat for a germanium target ($A = 73$, $M = 67.9\,\text{GeV}/c^2$). Which target gives larger recoil energies?

Problem 32.20 ⭐⭐ The Helm nuclear form factor is:

$$F(q) = \frac{3j_1(qR_0)}{qR_0} e^{-q^2 s^2/2}$$

where $j_1(x) = \sin(x)/x^2 - \cos(x)/x$, $R_0 \approx 1.2 A^{1/3}\,\text{fm}$, and $s \approx 0.9\,\text{fm}$.

(a) For ${}^{131}\text{Xe}$, calculate $R_0$ and the momentum transfer $q = \sqrt{2MT_R}$ at $T_R = 30\,\text{keV}$.

(b) Evaluate $F^2(q)$ at this momentum transfer. By what fraction is the cross section reduced compared to the zero-momentum-transfer value?

(c) Show that $F^2 \to 1$ as $q \to 0$ (verify the normalization).

Problem 32.21 ⭐⭐ The spin-independent WIMP-nucleus cross section for isospin-invariant couplings ($f_p = f_n$) is:

$$\sigma_{\text{SI}}(A) = \frac{\mu_A^2}{\mu_p^2} A^2 \sigma_p$$

where $\sigma_p$ is the WIMP-proton cross section and $\mu_A$, $\mu_p$ are the WIMP-nucleus and WIMP-proton reduced masses.

(a) Show that for $m_\chi \gg M_A$ (heavy WIMP limit), $\sigma_{\text{SI}} \propto A^4$ (scales as the fourth power of $A$).

(b) Show that for $m_\chi \ll m_p$ (light WIMP limit), $\sigma_{\text{SI}} \propto A^2$ (scales as $A^2$).

(c) For $m_\chi = 50\,\text{GeV}$, calculate the ratio $\sigma_{\text{SI}}(\text{Xe})/\sigma_{\text{SI}}(\text{Ar})$ for the same $\sigma_p$. Use $A_{\text{Xe}} = 131$, $A_{\text{Ar}} = 40$.

Problem 32.22 ⭐⭐⭐ The "neutrino floor" is the cross section below which CE$\nu$NS events from solar neutrinos become an irreducible background in dark matter detectors.

(a) For a xenon detector, ${}^8\text{B}$ solar neutrinos ($E_\nu \sim 5$–$15\,\text{MeV}$, flux $\sim 5.2 \times 10^6\,\text{cm}^{-2}\text{s}^{-1}$) produce nuclear recoils in the same energy range as $\sim 6$-GeV WIMPs. Estimate the CE$\nu$NS event rate per tonne-year from ${}^8\text{B}$ neutrinos, given $\sigma_{\text{CE}\nu\text{NS}} \sim 10^{-39}\,\text{cm}^2$ at $E_\nu = 10\,\text{MeV}$.

(b) If a dark matter detector sees $N_\nu$ neutrino events and $N_\chi$ WIMP events, explain why distinguishing them requires either directional sensitivity, annual modulation analysis, or a different target material.

(c) How does the annual modulation of the WIMP signal (due to Earth's orbital motion through the dark matter halo) differ from the seasonal variation of the ${}^8\text{B}$ neutrino flux?

Problem 32.22a ⭐⭐ The event rate in a dark matter direct detection experiment is:

$$R = \frac{\rho_\chi}{m_\chi M} \int_{v_{\min}}^{v_{\max}} v f(v) \sigma(v, T_R)\,dv$$

where $\rho_\chi = 0.3\,\text{GeV/cm}^3$ is the local dark matter density and $f(v)$ is the WIMP velocity distribution (Maxwellian with $v_0 = 220\,\text{km/s}$, truncated at galactic escape velocity $v_{\text{esc}} = 544\,\text{km/s}$).

(a) For a xenon target ($M = 122\,\text{GeV}$), $m_\chi = 50\,\text{GeV}$, and a WIMP-nucleon cross section $\sigma_p = 10^{-47}\,\text{cm}^2$, estimate the event rate per tonne-year. (Use $\sigma_{\text{SI}} \approx (\mu_A/\mu_p)^2 A^2 \sigma_p$ and an average velocity $\langle v \rangle \approx 220\,\text{km/s}$.)

(b) The LZ detector has an active mass of 7 tonnes and a target exposure of 1000 live-days. How many WIMP events would be expected at this cross section?

(c) Why do dark matter experiments typically report limits rather than discoveries? Estimate the number of background events expected in the LZ signal region and compare to your answer in (b).

Problem 32.22b ⭐⭐⭐ The annual modulation of the WIMP scattering rate has amplitude:

$$\frac{\Delta R}{R} \approx \frac{v_{\text{orb}} \cos\gamma}{v_\odot} \approx 0.07$$

where $v_{\text{orb}} = 30\,\text{km/s}$ is Earth's orbital speed and $\gamma \approx 60°$ is the ecliptic inclination.

(a) The modulation peak occurs around June 2, when Earth's velocity through the halo is maximized. Explain why this is roughly opposite to the phase of the maximum ${}^8\text{B}$ solar neutrino flux (which is essentially constant, but the Earth-Sun distance varies).

(b) If a xenon detector observes 100 events total over 5 years, estimate the expected amplitude of the annual modulation signal in events per year. Is this statistically significant?

(c) The DAMA/LIBRA experiment reports a modulation with $9.5\sigma$ significance in NaI(Tl). Other experiments exclude the standard WIMP interpretation. Describe two alternative explanations for the DAMA signal that are consistent with the null results from other experiments.

Synthesis and Integration

Problem 32.23 ⭐⭐ Coherent enhancement appears in multiple contexts in this chapter. For each of the following, state whether the enhancement scales as $A$, $N^2$, $Z^2$, or something else, and explain physically why:

(a) CE$\nu$NS cross section

(b) WIMP spin-independent cross section

(c) Nuclear Schiff moment contribution to atomic EDM

(d) Nuclear anapole moment

Problem 32.24 ⭐⭐⭐ Compare and contrast the nuclear physics challenges in three frontier experiments:

(a) Extracting $\langle m_{\beta\beta} \rangle$ from $0\nu\beta\beta$ (nuclear matrix element problem)

(b) Extracting CP-violating couplings from atomic EDM measurements (Schiff moment calculation)

(c) Extracting WIMP cross sections from dark matter direct detection (nuclear response functions)

For each, identify: (i) the nuclear structure quantity needed, (ii) the theoretical methods used to calculate it, (iii) the current level of theoretical uncertainty, and (iv) the impact of that uncertainty on the physics reach of the experiment.

Problem 32.25 ⭐⭐⭐ (Research) Read the COHERENT Collaboration's original paper (Akimov et al., Science 357, 1123, 2017).

(a) How did the experiment exploit the pulsed time structure of the SNS beam to reduce backgrounds?

(b) What was the energy threshold of the CsI detector, and how was it calibrated?

(c) The observed event rate was $77 \pm 16\%$ of the Standard Model prediction. Discuss whether this discrepancy is statistically significant and what BSM physics could explain a deficit.

Problem 32.26 ⭐⭐⭐ (Research) The Cabibbo angle anomaly depends critically on the transition-independent radiative correction $\Delta_R^V$.

(a) Research the most recent values of $\Delta_R^V$ in the literature. How has the central value changed since the 2018 Seng-Gorchtein-Ramsey-Musolf calculation?

(b) How might lattice QCD calculations of the $\gamma W$-box diagram resolve the discrepancy?

(c) Discuss how independent measurements of $V_{ud}$ from neutron beta decay could test whether the anomaly is due to nuclear structure effects (the $\delta_C$ corrections) versus electroweak radiative corrections (which are common to all systems).

Computational Exercises

Problem 32.27 ⭐⭐ (Computational) Using the provided symmetry_tests.py code as a starting point:

(a) Modify the Majorana mass plot to add a third band showing the "quasi-degenerate" regime ($m_1 \approx m_2 \approx m_3 \gg \sqrt{|\Delta m_{32}^2|}$). In this regime, $\langle m_{\beta\beta} \rangle \approx m_0 |c_{12}^2 c_{13}^2 + s_{12}^2 c_{13}^2 e^{i\alpha_{21}} + s_{13}^2 e^{i\alpha_{31}}|$ where $m_0$ is the common mass scale. Plot $\langle m_{\beta\beta} \rangle$ vs $m_0$ for $m_0 = 0.05$–$1\,\text{eV}$.

(b) Add the cosmological constraint $\sum m_i < 0.12\,\text{eV}$ (Planck 2018) as a vertical band on the Majorana mass plot. For normal ordering, what is the maximum allowed $\langle m_{\beta\beta} \rangle$ consistent with this cosmological bound?

(c) Add horizontal lines for the sensitivity of CUORE ($\langle m_{\beta\beta} \rangle \sim 90$–$300\,\text{meV}$), KamLAND-Zen 800 ($\sim 36$–$156\,\text{meV}$), and LEGEND-1000 ($\sim 9$–$21\,\text{meV}$), where the ranges reflect the NME uncertainty. Comment on which experiments probe which mass ordering scenarios.

Problem 32.28 ⭐⭐ (Computational) Implement the Helm nuclear form factor:

$$F(q) = \frac{3[\sin(qR_0) - qR_0\cos(qR_0)]}{(qR_0)^3}\,e^{-q^2 s^2/2}$$

where $R_0 = \sqrt{(1.23 A^{1/3})^2 - 5s^2}\,\text{fm}$ and $s = 0.9\,\text{fm}$.

(a) Plot $F^2(q)$ as a function of nuclear recoil energy $T_R$ (using $q = \sqrt{2MT_R}$) for xenon ($A = 131$), germanium ($A = 73$), and argon ($A = 40$) on the same axes, for $T_R$ from 0 to 100 keV.

(b) For each target, find the recoil energy at which $F^2 = 0.5$ (the "half-suppression" energy). Verify that heavier targets are more strongly suppressed.

(c) Explain qualitatively why the first minimum of $F^2$ occurs at lower $T_R$ for heavier nuclei. Connect this to the nuclear radius.

(d) Overlay on the same plot the differential WIMP scattering rate $dR/dT_R$ (which includes the exponential velocity distribution factor in addition to the form factor), and show how the form factor shapes the observed recoil spectrum for each target.

Problem 32.29 ⭐⭐⭐ (Computational/Research) Compare nuclear matrix elements for $0\nu\beta\beta$ across different candidate nuclei:

(a) Create a table of $G^{0\nu}$, $M^{0\nu}$ (from at least three theoretical methods), and the resulting $T_{1/2}^{0\nu}$ predictions for $\langle m_{\beta\beta} \rangle = 50\,\text{meV}$, for the four leading candidates: ${}^{76}\text{Ge}$, ${}^{100}\text{Mo}$, ${}^{130}\text{Te}$, and ${}^{136}\text{Xe}$. Use published NME values from the literature.

(b) For each nucleus, calculate the range of $\langle m_{\beta\beta} \rangle$ that corresponds to $T_{1/2}^{0\nu} = 10^{27}\,\text{yr}$, given the spread of NME values.

(c) Plot the "discovery potential" — the minimum $\langle m_{\beta\beta} \rangle$ detectable at $3\sigma$ — as a function of exposure (tonne-years) for germanium and xenon experiments, assuming zero background. Include the NME uncertainty as a band.

(d) Discuss: if $0\nu\beta\beta$ is observed in ${}^{76}\text{Ge}$ but not in ${}^{136}\text{Xe}$ (or vice versa) at the expected ratio, what would this tell us about the NME calculations? What if the rates are consistent — what constraint does this provide on nuclear theory?