Exercises — Chapter 14: Beta Decay: The Weak Interaction in the Nucleus

Section A: Energetics and Decay Modes (14.1)

Exercise 14.1 — Q-Values for the Three Modes

Using atomic masses from AME2020, calculate the Q-values for all energetically allowed beta decay modes of the following nuclides. State which modes are allowed.

(a) $^{32}$P ($M = 31.973908\,\text{u}$); daughter $^{32}$S ($M = 31.972071\,\text{u}$)

(b) $^{22}$Na ($M = 21.994437\,\text{u}$); daughter for $\beta^+$/EC: $^{22}$Ne ($M = 21.991386\,\text{u}$)

(c) $^{40}$K ($M = 39.963998\,\text{u}$); daughters: $^{40}$Ca ($M = 39.962591\,\text{u}$), $^{40}$Ar ($M = 39.962383\,\text{u}$)

(d) $^{64}$Cu ($M = 63.929764\,\text{u}$); daughters: $^{64}$Zn ($M = 63.929142\,\text{u}$), $^{64}$Ni ($M = 63.927966\,\text{u}$). This is one of the rare nuclides that can undergo all three modes. Verify this.

Hint: $1\,\text{u} = 931.494\,\text{MeV}/c^2$; $m_e = 0.000549\,\text{u} = 0.511\,\text{MeV}/c^2$.

Exercise 14.2 — Electron Capture vs. $\beta^+$ Competition

(a) Show that whenever $\beta^+$ decay is energetically allowed, electron capture is also allowed (neglecting the atomic electron binding energy).

(b) $^{7}$Be has $Q_\text{EC} = 0.862\,\text{MeV}$. Explain why it cannot undergo $\beta^+$ decay.

(c) For what range of atomic mass differences $\Delta M = M_\text{parent} - M_\text{daughter}$ (in u) is EC allowed but $\beta^+$ forbidden?

(d) In the Sun's core, $^7$Be captures a free electron (not bound). Does this change the energetics? Explain.

Exercise 14.3 — Recoil Energy

In $\beta^-$ decay of $^{14}$C ($Q = 156.5\,\text{keV}$), the daughter $^{14}$N recoils.

(a) If the electron is emitted with its maximum kinetic energy and the neutrino is massless, what is the kinetic energy of the recoiling $^{14}$N nucleus? Express in eV.

(b) Justify the approximation that the nuclear recoil energy is negligible compared to $Q$.

(c) For what decay would the nuclear recoil energy be non-negligible? Consider the free neutron decay ($Q = 0.782\,\text{MeV}$, $M_p = 938.3\,\text{MeV}/c^2$).

Exercise 14.4 — Isobaric Parabolas

For the $A = 135$ isobar chain, the following atomic masses are known (AME2020):

where $M = A \times 1\,\text{u} + \Delta / 931.494$.

(a) Plot the mass excess as a function of $Z$ and identify the most stable isobar.

(b) For each pair of adjacent isobars, determine whether $\beta^-$ or $\beta^+$/EC is energetically allowed.

(c) Explain why $^{135}$Cs is stable even though $^{135}$Ba has a lower mass excess. (Hint: both are odd-$A$; there is only one parabola.)

Wait — recheck the data: if $^{135}$Ba has $\Delta = -84.63$ and $^{135}$Cs has $\Delta = -84.93$, then $^{135}$Cs is more tightly bound. Is $^{135}$Cs stable? Verify by checking whether any decay mode is energetically allowed for $^{135}$Cs.

Section B: Fermi's Theory and the Beta Spectrum (14.3)

Exercise 14.5 — Phase Space Derivation

Starting from Fermi's golden rule, derive the expression for the density of final states in beta decay.

(a) Show that the number of electron states in a box of volume $V$ with momentum between $p$ and $p + dp$ is $dn = V/(2\pi\hbar)^3 \cdot 4\pi p^2 dp$.

(b) Using energy conservation $Q = T_e + E_\nu$ (with $m_\nu = 0$), eliminate the neutrino momentum and show that the spectrum shape is $N(T_e) \propto p_e E_e (Q - T_e)^2$ (before the Coulomb correction).

(c) Verify that this spectrum vanishes at both $T_e = 0$ and $T_e = Q$, and find the energy at which the spectrum peaks for the case $Q \gg m_e c^2$.

Exercise 14.6 — Fermi Function Calculation

(a) Calculate the non-relativistic Fermi function $F(Z, T_e) = 2\pi\eta / (1 - e^{-2\pi\eta})$ for $\beta^-$ decay of $^{60}$Co ($Z' = 28$) at electron kinetic energies of $T_e = 10, 50, 100, 200, 300\,\text{keV}$.

(b) Repeat for $\beta^+$ decay of $^{18}$F ($Z' = 8$) at the same energies.

(c) At what electron energy does the Fermi function for $^{60}$Co $\beta^-$ exceed $F = 2$? What does this imply for the shape of the low-energy part of the spectrum?

(d) Explain physically why $F \to 1$ as $T_e \to \infty$ for both $\beta^-$ and $\beta^+$.

Exercise 14.7 — Spectrum Peak Energy

For an allowed $\beta^-$ transition with $F \approx 1$ (neglecting the Coulomb correction):

(a) Show that $N(T_e) \propto \sqrt{T_e(T_e + 2m_ec^2)}(T_e + m_ec^2)(Q - T_e)^2$.

(b) For the non-relativistic limit ($T_e \ll m_ec^2$, relevant for tritium): show that the spectrum simplifies to $N(T_e) \propto \sqrt{T_e}(Q - T_e)^2$ and find the peak energy $T_\text{peak}$.

(c) For the ultra-relativistic limit ($T_e \gg m_ec^2$, relevant for high-$Q$ decays): show that the spectrum becomes $N(T_e) \propto T_e^2(Q - T_e)^2$ and find $T_\text{peak}$.

(d) Calculate $T_\text{peak}$ for tritium ($Q = 18.6\,\text{keV}$) using part (b) and for $^{32}$P ($Q = 1.711\,\text{MeV}$) using part (c). Compare to the full numerical result.

Exercise 14.8 — Total Decay Rate Integration

The total decay rate for an allowed transition is $\lambda = C \cdot f(Z', Q)$ where $f$ is the Fermi integral.

(a) For a very simple estimate, set $F = 1$ and work in the ultra-relativistic limit. Show that the Fermi integral becomes:

$$f \approx \frac{1}{30}\left(\frac{Q}{m_ec^2}\right)^5$$

(b) Use this to estimate the half-life of $^{32}$P ($Q = 1.711\,\text{MeV}$, superallowed) given $G_F/(\hbar c)^3 = 1.166 \times 10^{-5}\,\text{GeV}^{-2}$ and $|M_{fi}|^2 \approx 3$ (Gamow-Teller dominant).

(c) Compare your estimate to the measured half-life of 14.3 days. Comment on the quality of the approximation.

Exercise 14.8a — Average Electron Energy

(a) For the non-relativistic limit ($Q \ll m_ec^2$), show that the mean kinetic energy of the emitted electron (with $F = 1$) is:

$$\langle T_e \rangle = \frac{\int_0^Q T_e \sqrt{T_e} (Q - T_e)^2 dT_e}{\int_0^Q \sqrt{T_e} (Q - T_e)^2 dT_e}$$

Evaluate this integral (use the substitution $x = T_e/Q$) and show that $\langle T_e \rangle \approx Q/3$.

(b) For tritium ($Q = 18.6\,\text{keV}$), compute $\langle T_e \rangle$ numerically, including the Fermi function ($Z' = 2$). How much does the Coulomb correction change the average energy?

(c) The "missing energy" per decay, $Q - \langle T_e \rangle$, is carried away by the neutrino (on average). For $^{210}$Bi ($Q = 1.161\,\text{MeV}$), estimate the average neutrino energy and comment on how this compares to the antineutrino energy from a nuclear reactor ($\sim 3\,\text{MeV}$ average from fission product decays).

Section C: Selection Rules and ft-Values (14.4–14.5)

Exercise 14.9 — Classifying Transitions

For each of the following beta decays, identify the type of transition (Fermi, Gamow-Teller, or mixed), the degree of forbiddenness (allowed, first forbidden, etc.), and predict the approximate $\log ft$ range.

Exercise 14.10 — ft-Value Calculation

The decay $^{14}$O $\to$ $^{14}$N* ($0^+ \to 0^+$) has $Q = 2.831\,\text{MeV}$, $t_{1/2} = 70.62\,\text{s}$, and branching ratio $\text{BR} = 99.4\%$.

(a) Compute the partial half-life $t$.

(b) The Fermi integral for this decay has been calculated to be $f = 43.53$. Calculate the $ft$-value and $\log ft$.

(c) This is a superallowed $0^+ \to 0^+$ transition with $|M_F|^2 = 2$. Using the relation $ft = K/|M_{fi}|^2$ with $K = 6144\,\text{s}$, check whether your $ft$-value is consistent.

(d) The corrected $\mathcal{F}t$ value for this transition (after radiative and isospin-breaking corrections) is $3072.4 \pm 1.0\,\text{s}$. What is the percentage correction from $ft$ to $\mathcal{F}t$?

Exercise 14.11 — Free Neutron Decay

The free neutron has $\tau = 878.4\,\text{s}$ (mean lifetime), $Q = 0.7824\,\text{MeV}$.

(a) Convert to half-life $t_{1/2}$.

(b) The Fermi integral for neutron decay is $f = 1.6887$. Calculate $ft$ and $\log ft$. Verify that it is superallowed.

(c) This is a mixed Fermi+GT transition. The matrix elements are $|M_F|^2 = 1$ and $|M_{GT}|^2 = 3$. Using $|M_{fi}|^2 = g_V^2|M_F|^2 + g_A^2|M_{GT}|^2$ and $K/ft = |M_{fi}|^2$, extract $|g_A/g_V|$. Compare to the PDG value of $1.2754 \pm 0.0013$.

Exercise 14.12 — Forbidden Decay of $^{40}$K

$^{40}$K ($J^\pi = 4^-$) decays to $^{40}$Ca ($J^\pi = 0^+$) via $\beta^-$ emission with $t_{1/2} = 1.248 \times 10^9$ years (total for the $\beta^-$ branch, BR = 89.3%).

(a) What is the degree of forbiddenness of this transition? Justify using the selection rules.

(b) The Fermi integral is $f = 278.2$. Calculate $\log ft$ for this transition.

(c) Is your $\log ft$ consistent with the expected range for this degree of forbiddenness?

(d) This decay is used in K-Ar dating of rocks. Explain why the long half-life makes it suitable for geological dating.

Section D: Kurie Plots and Spectrum Analysis (14.5)

Exercise 14.13 — Constructing a Kurie Plot

The following (simulated) data are from a measurement of the $^{63}$Ni $\beta^-$ spectrum ($Q = 66.98\,\text{keV}$, $Z' = 29$):

(a) Calculate $F(Z', T_e)$ at each energy using the non-relativistic approximation.

(b) Compute the Kurie function $K(T_e) = \sqrt{N/(F \cdot p_e \cdot E_e)}$ at each point (use appropriate units or work dimensionlessly).

(c) Plot $K$ vs. $T_e$ and perform a linear fit. Extract $Q$ from the x-intercept.

(d) Does the plot appear linear? What would a concave-upward deviation indicate? A concave-downward deviation?

Exercise 14.14 — Neutrino Mass from the Endpoint

Near the endpoint of the tritium beta spectrum, the Kurie function for a neutrino of mass $m_\nu$ is:

$$K(T_e) \propto (Q - T_e) \left[1 - \frac{m_\nu^2 c^4}{(Q - T_e)^2}\right]^{1/4}$$

(a) Show that this reduces to $K \propto (Q - T_e)$ for $m_\nu = 0$.

(b) At what electron energy does the Kurie plot deviate significantly (say, by 10%) from the $m_\nu = 0$ line for $m_\nu = 1\,\text{eV}$?

(c) KATRIN measures the tritium endpoint with an energy resolution of $\sim 1\,\text{eV}$. Explain why the fraction of total decays in the last 1 eV of the spectrum is only $\sim 2 \times 10^{-13}$, making this an extraordinarily challenging measurement.

Section E: Parity Violation (14.6)

Exercise 14.15 — The Wu Experiment: Quantitative Analysis

In the Wu experiment, $^{60}$Co ($J^\pi = 5^+$) decays to $^{60}$Ni* ($J^\pi = 4^+$) with polarized nuclei.

(a) Show that $\mathbf{J} \cdot \mathbf{p}_e$ is a pseudoscalar (changes sign under parity). Explain why a nonzero expectation value of $\mathbf{J} \cdot \mathbf{p}_e$ implies parity violation.

(b) The angular distribution of electrons from polarized nuclei is $W(\theta) = 1 + \alpha(v/c)\cos\theta$ where $\alpha$ is the asymmetry parameter and $v/c$ is the electron speed. For maximum parity violation in a pure GT transition, theory predicts $\alpha = -1$. What does this mean physically about the direction of electron emission?

(c) At the operating temperature of 0.01 K and a magnetic field of 0.05 T, estimate the degree of nuclear polarization for $^{60}$Co ($\mu = 3.799\,\mu_N$) using the Boltzmann factor. (Hint: $\mu_N = 5.051 \times 10^{-27}\,\text{J/T}$, $k_B = 1.381 \times 10^{-23}\,\text{J/K}$.)

(d) Why was it essential to perform the experiment at millikelvin temperatures? What would happen at room temperature?

Exercise 14.16 — Helicity and the V$-$A Theory

(a) Define helicity $h = \hat{\boldsymbol{\sigma}} \cdot \hat{\mathbf{p}}$. What are its eigenvalues?

(b) In the V$-$A theory, only left-handed particles and right-handed antiparticles participate in charged-current weak interactions. For a massless neutrino, explain why this means only $\nu_L$ and $\bar{\nu}_R$ interact.

(c) For a massive particle, helicity is not Lorentz invariant. Explain why a small neutrino mass allows a tiny $\nu_R$ component, and why this is relevant for neutrinoless double beta decay.

(d) The Goldhaber experiment (1958) measured the helicity of the neutrino emitted in EC of $^{152}$Eu. The result was $h_\nu = -1$ (left-handed). Explain qualitatively how resonant scattering of the subsequent gamma ray was used to determine the neutrino helicity.

Exercise 14.16a — Beta-Neutrino Angular Correlation

In beta decay from unpolarized nuclei, the electron-neutrino angular correlation is:

$$W(\theta_{e\nu}) \propto 1 + a \frac{\mathbf{p}_e \cdot \mathbf{p}_\nu}{E_e E_\nu}$$

where $a$ is the angular correlation coefficient.

(a) For a pure Fermi transition, theory predicts $a = +1$. Explain physically why the electron and neutrino tend to be emitted in the same direction (for a Fermi transition, the lepton pair has total spin $S = 0$).

(b) For a pure Gamow-Teller transition, $a = -1/3$. Show that this means there is a slight preference for back-to-back emission. How does this relate to the $S = 1$ spin coupling of the lepton pair?

(c) For the neutron ($|g_A/g_V| = 1.276$), the mixing ratio is $\rho = g_A M_{GT} / (g_V M_F)$ with $M_F = 1$ and $M_{GT} = \sqrt{3}$. Calculate $a = (1 - \rho^2/3) / (1 + \rho^2)$.

(d) The aSPECT and aCORN experiments measure $a$ for neutron decay. Current result: $a = -0.10430 \pm 0.00084$. Compare to your calculation. What does the sign tell you about the relative strength of V and A contributions?

Section F: Double Beta Decay (14.7)

Exercise 14.17 — Double Beta Decay Energetics

(a) For the $A = 76$ isobar chain: $M(^{76}\text{Ge}) - M(^{76}\text{Se}) = 0.00221\,\text{u}$. Calculate the $Q$-value for $^{76}$Ge double beta decay in MeV.

(b) Verify that single beta decay of $^{76}$Ge to $^{76}$As is energetically forbidden. (Mass excess of $^{76}$As: $-68.86\,\text{MeV}$; $^{76}$Ge: $-73.21\,\text{MeV}$.)

(c) Draw a schematic of the $A = 76$ mass parabola showing $^{76}$Ge, $^{76}$As, $^{76}$Se. Explain why even-even nuclei can undergo double beta decay even though single beta decay is forbidden.

Exercise 14.18 — $0\nu\beta\beta$ Sensitivity

The half-life for $0\nu\beta\beta$ is related to the effective Majorana mass by:

$$\left(T_{1/2}^{0\nu}\right)^{-1} = G_{0\nu} |M_{0\nu}|^2 \langle m_{\beta\beta} \rangle^2$$

where $G_{0\nu}$ is a phase-space factor and $M_{0\nu}$ is the nuclear matrix element.

(a) For $^{76}$Ge: $G_{0\nu} = 2.36 \times 10^{-15}\,\text{yr}^{-1}\text{eV}^{-2}$ and $|M_{0\nu}| \approx 3.5$ (shell model). Calculate the half-life corresponding to $\langle m_{\beta\beta} \rangle = 50\,\text{meV}$.

(b) LEGEND-1000 aims for a half-life sensitivity of $10^{28}$ years. What effective Majorana mass does this correspond to (using the same nuclear parameters)?

(c) The inverted mass ordering predicts $\langle m_{\beta\beta} \rangle \gtrsim 15\,\text{meV}$. Will LEGEND-1000 be able to test this?

(d) Explain why the factor-of-2-3 uncertainty in $|M_{0\nu}|$ is the dominant source of uncertainty in extracting $\langle m_{\beta\beta} \rangle$ from a measured half-life.

Exercise 14.19 — $2\nu\beta\beta$ Spectrum Shape

In $2\nu\beta\beta$ decay, the summed kinetic energy of the two electrons $T = T_1 + T_2$ has the distribution:

$$\frac{dN}{dT} \propto T(Q - T)^5$$

(neglecting electron masses and Coulomb corrections).

(a) Show that this spectrum peaks at $T_\text{peak} = Q/6$.

(b) For $0\nu\beta\beta$, the two electrons share the full Q-value: the summed energy is $T = Q$ (a delta function). Sketch both spectra on the same axes and explain why the $0\nu\beta\beta$ signal is a peak at the endpoint of the $2\nu\beta\beta$ continuum.

(c) If the energy resolution of the detector is $\sigma_E$, the $0\nu\beta\beta$ peak is broadened to a Gaussian of width $\sigma_E$. Explain why excellent energy resolution is critical for distinguishing $0\nu\beta\beta$ from the tail of the $2\nu\beta\beta$ spectrum.

Section G: Computational Exercises

Exercise 14.20 — Beta Spectrum Plotter (Python)

Using the beta_spectrum.py code from this chapter:

(a) Add a new nucleus: $^{32}$P ($Z' = 16$, $Q = 1711\,\text{keV}$). Plot its beta spectrum with and without the Fermi function. How does the Fermi function distortion compare to $^{60}$Co? Why?

(b) Create a Kurie plot for $^{32}$P and verify that it is linear (allowed transition).

(c) Modify the code to plot the $\beta^+$ spectrum of $^{18}$F ($Z' = 8$, $Q = 634\,\text{keV}$). How does the Fermi function affect the spectrum shape differently for $\beta^+$ compared to $\beta^-$?

Exercise 14.21 — Monte Carlo Spectrum Simulation

Write a Python program that:

(a) Generates $10^5$ beta decay events for $^{14}$C using rejection sampling from the theoretical spectrum shape (with Fermi function).

(b) Histograms the electron energies and overlays the theoretical spectrum (properly normalized).

(c) Constructs a Kurie plot from the simulated data and extracts $Q$ by linear fitting. Report the extracted $Q$ with statistical uncertainty from the fit.

(d) Adds Gaussian energy smearing (resolution $\sigma = 3\,\text{keV}$) and repeats the Kurie analysis. How does finite resolution affect the extracted $Q$?

Exercise 14.22 — ft-Value Calculator

Write a Python function that:

(a) Takes as input: $Z_\text{daughter}$, $Q$ (in MeV), $t_{1/2}$ (in seconds), and branching ratio.

(b) Numerically computes the Fermi integral $f(Z', Q)$ using the non-relativistic Fermi function.

(c) Calculates $ft$ and $\log ft$.

(d) Test your function on: (i) free neutron decay ($Z'=1, Q=0.782\,\text{MeV}, t_{1/2}=609\,\text{s}$); (ii) $^{14}$O ($Z'=7, Q=2.831\,\text{MeV}, t_{1/2}=70.6\,\text{s}$, BR=99.4%); (iii) $^{60}$Co ($Z'=28, Q=0.318\,\text{MeV}, t_{1/2}=1.663 \times 10^8\,\text{s}$, BR=99.9%).

Challenge Problems

Exercise 14.23 — Coulomb Correction from First Principles

The Fermi function can be derived from the ratio of the Coulomb wavefunction to the free-particle wavefunction evaluated at the nuclear surface.

(a) For a non-relativistic electron in a Coulomb potential $V(r) = -Z'e^2/(4\pi\epsilon_0 r)$, write down the radial Schrodinger equation for $l = 0$.

(b) Show that the Sommerfeld parameter is $\eta = Z' \alpha / \beta_e$ and that it governs the ratio of the Coulomb parameter to the particle wavelength.

(c) The $l = 0$ Coulomb wavefunction at the origin is $|\psi_C(0)|^2 / |\psi_\text{free}(0)|^2 = 2\pi\eta / (e^{2\pi\eta} - 1)$ for an attractive Coulomb field. Verify that this gives the Fermi function formula used in this chapter (with the appropriate sign convention for $\eta$).

Exercise 14.24 — Sargent's Rule

Before Fermi's theory, empirical studies showed that $\log t_{1/2}$ was approximately linearly related to $\log Q$ for allowed beta decays (Sargent, 1933).

(a) Using the ultra-relativistic approximation $f \propto Q^5$ (Exercise 14.8), show that $t_{1/2} \propto Q^{-5}$ for a given nuclear matrix element. This is Sargent's rule.

(b) Test Sargent's rule using the following data for superallowed decays:

Plot $\log t_{1/2}$ vs. $\log Q$ and fit a power law. Is the exponent close to $-5$?

(c) Why does Sargent's rule break down for low-$Q$ decays (like tritium)?

Exercise 14.25 — The Neutrino Mass Problem

The KATRIN experiment places an upper limit of $m_\nu < 0.45\,\text{eV}$ (2024).

(a) What fraction of tritium beta decays produce electrons within 1 eV of the endpoint? Evaluate $\int_{Q-1\,\text{eV}}^Q N(T)\,dT / \int_0^Q N(T)\,dT$ numerically.

(b) KATRIN collects data at a rate of $\sim 10^{10}$ tritium decays per day. How many decays per day produce electrons in the last 1 eV?

(c) If $m_\nu = 0.3\,\text{eV}$, calculate the fractional change in the integral $\int_{Q-5\,\text{eV}}^Q N(T)\,dT$ compared to the $m_\nu = 0$ case. This gives a sense of the statistical precision required.

(d) Project 8 aims to measure individual electron energies via cyclotron radiation emission spectroscopy (CRES). An electron with kinetic energy $T_e$ in a magnetic field $B$ emits cyclotron radiation at frequency $f_c = eB / (2\pi\gamma m_e)$, where $\gamma = (T_e + m_ec^2)/(m_ec^2)$. For $B = 1\,\text{T}$ and $T_e = 18.6\,\text{keV}$, calculate $f_c$ and the frequency shift corresponding to a 1 eV change in $T_e$.

Exercise 14.26 — Beta Decay in Stellar Environments

In the hot, dense environment of a stellar core, beta decay rates can be dramatically different from terrestrial values due to the thermal population of nuclear excited states and the high electron chemical potential.

(a) At the center of the Sun ($T \approx 1.5 \times 10^7\,\text{K}$, $k_BT \approx 1.3\,\text{keV}$), the thermal energy is much less than typical nuclear excitation energies ($\sim 1\,\text{MeV}$). Explain why terrestrial beta decay rates are a good approximation for the pp chain.

(b) In a core-collapse supernova ($T \sim 10\,\text{MeV}$, $\rho \sim 10^{14}\,\text{g/cm}^3$), the electron chemical potential can exceed 10 MeV. Explain qualitatively why electron capture rates on protons and nuclei are enormously enhanced compared to terrestrial conditions, and why this drives the core to neutron-rich compositions.

(c) In the neutron-rich ejecta of a neutron star merger ($T \sim 1\,\text{MeV}$), beta decay half-lives of very neutron-rich nuclei far from stability determine the r-process path. Many of these half-lives have never been measured. Explain why theoretical predictions of these half-lives are uncertain and why radioactive beam facilities (like FRIB) are essential.

(d) The decay of free neutrons ($t_{1/2} = 609\,\text{s}$) plays a role in Big Bang nucleosynthesis (Chapter 24). If the neutron lifetime were 10% shorter, would the primordial helium abundance increase or decrease? Explain qualitatively.

Exercise 14.27 — The CKM Matrix and Beta Decay

The CKM matrix element $V_{ud}$ connects the $u$ and $d$ quarks in the weak interaction. Its determination from nuclear beta decay is one of the precision tests of the Standard Model.

(a) The relationship between the vector coupling constant measured in nuclear beta decay ($G_V$) and the Fermi coupling constant from muon decay ($G_F$) is $G_V = G_F V_{ud}$. Using the corrected $\mathcal{F}t = 3072.24\,\text{s}$ and the relation $\mathcal{F}t = K / (2 G_V^2 |M_F|^2)$ with $K = 6144.2\,\text{s}$ and $|M_F|^2 = 2$, verify that $|V_{ud}| = 0.9737$.

(b) The first-row CKM unitarity condition is $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1$. Using $|V_{us}| = 0.2243 \pm 0.0005$ (from kaon decays) and $|V_{ub}| = 0.00382 \pm 0.00020$ (from B meson decays), test whether unitarity is satisfied with the $V_{ud}$ from part (a).

(c) A persistent tension at the $\sim 2\sigma$ level in first-row CKM unitarity has motivated searches for physics beyond the Standard Model — including right-handed currents, exotic scalar interactions, and additional generations of fermions. If unitarity were violated at the $0.1\%$ level, what new physics scenarios could explain it? (Qualitative discussion.)