Tritium Beta Decay

Introduction

The preceding chapters have developed TMT's predictions for particle properties, rare processes, and falsification criteria. This chapter demonstrates the framework's predictive power through a concrete, end-to-end calculation: deriving the tritium beta decay half-life from the single postulate P1 (\(ds_6^{\,2} = 0\)).

Tritium (\({}^3\)H) undergoes beta decay via:

with experimentally measured half-life \(t_{1/2} = 12.32 \pm 0.02\) years and endpoint energy \(Q = 18.591 \pm 0.001\) keV. The Standard Model cannot derive this half-life from first principles—it requires the Fermi constant \(G_F\), the axial coupling ratio \(g_A/g_V\), and nuclear matrix elements as empirical inputs. TMT derives all of these from geometry.

The complete derivation chain is:

The result—\(t_{1/2}^{\text{TMT}} = 12.3\) years, in 99.8% agreement with experiment—represents the first complete first-principles derivation of a radioactive decay constant.

The Beta Decay Process in TMT

Why Tritium?

Tritium beta decay is ideal for testing TMT's predictive power for several reasons:

(1) Clean physics: Tritium is a superallowed mirror transition between members of the same isospin multiplet (\({}^3\)H \(\to\) \({}^3\)He), so the nuclear matrix elements are determined almost entirely by symmetry.

(2) Precision measurement: The half-life is known to 0.2% precision, providing a stringent test.

(3) Complete chain: Every physical input required—gauge couplings, VEV, masses, matrix elements—can be traced to P1.

(4) Aggregate determinism: Laboratory samples contain \(N \sim 10^{20}\) atoms, firmly in the regime where the Aggregate Certainty Theorem (Part XI, Chapter 86) guarantees deterministic behavior.

Transition Classification

Tritium beta decay is a mixed Fermi + Gamow-Teller transition. The quantum numbers are:

The selection rules are \(\Delta J = 0\), \(\Delta T = 0\), \(|\Delta T_z| = 1\), allowing both the Fermi (vector) and Gamow-Teller (axial-vector) components.

The Derivation Strategy

From P1, we derive each ingredient in sequence:

The key point: the Standard Model treats \(G_F\), \(g_A/g_V\), and \(|M_{fi}|^2\) as empirical inputs. TMT derives all three from geometry.

Weak Interaction Coupling from Geometry

The Gauge Coupling \(g_2^2 = 4/(3\pi)\)

Polar Field Form of the Gauge Coupling

The coupling derivation above collapses to a single line in the polar field variable \(u = \cos\theta\). The monopole harmonic overlap integral that determines the participation ratio \(P = \pi\) becomes a polynomial integral on the flat rectangle \([-1,+1]\times[0,2\pi)\):

The factor \(3 = 1/\langle u^2\rangle\) in the denominator has a transparent geometric origin: it is the reciprocal of the second moment of the polar coordinate over \(S^2\), \(\langle u^2\rangle = \frac{1}{2}\int_{-1}^{+1}u^2\,du = 1/3\). In spherical coordinates, this same factor emerges from a chain of trigonometric identities involving \(\sin\theta\) and \(\cos^2(\theta/2)\)—the polar form reveals it as a single polynomial moment.

The Higgs VEV and W Boson Mass

From Part 4, the Higgs vacuum expectation value is:

where \(M_6 = 7.3\,TeV\) is the 6D scale derived from \(M_6 = (M_{\text{Pl}}^2/(4\pi R_0^2))^{1/3}\) (Part 4, \S15.2).

The W boson mass follows from the Higgs mechanism:

Experimental value: \(M_W^{\text{exp}} = 80.377 \pm 0.012\) GeV. Agreement: 99.8%.

Deriving the Fermi Constant

The contrast with the Standard Model is stark: in the SM, \(G_F\) is measured from muon decay and used as input. TMT derives it from P1 with zero free parameters.

Electron Spectrum from Geometry

The Strong Coupling and Axial Ratio

The nuclear matrix element requires the axial coupling ratio \(g_A/g_V\), which in turn requires the strong coupling constant. TMT derives the SU(3) coupling from the complex dimension ratio:

Since \(g_2^2 = 4/(3\pi)\):

In polar language, the ratio \(g_3^2/g_2^2 = 3\) arises because SU(3) color lives external to the polar rectangle (in the ambient \(\mathbb{C}^3\)), so the complex dimension factor \(d_{\mathbb{C}} = 3\) exactly cancels the THROUGH second-moment suppression \(\langle u^2\rangle = 1/3\):

The strong coupling is therefore pure AROUND: \(\alpha_s(M_6) = 1/\pi^2\) carries no THROUGH suppression. This is the geometric reason why the strong force is strong—it is the only gauge interaction unsuppressed by \(\langle u^2\rangle\).

At the unification scale \(\mu = M_6\):

Running to the nucleon scale \(\mu \sim 1\,GeV\) via the QCD beta function (with \(b_0 = 11 - 2n_f/3 = 7\) for \(n_f = 6\) active flavors):

gives \(\alpha_s(1\,GeV)\approx 0.35\).

The Axial Coupling from QCD

Nuclear Matrix Elements from Symmetry

The total nuclear matrix element squared for tritium decay is:

Fermi matrix element. The Fermi operator is the isospin raising operator \(\tau^+\). For the mirror transition \(T = 1/2\), \(T_z = -1/2 \to T_z = +1/2\):

This is exact from SU(2) isospin algebra—the same mathematics that TMT uses for the \(S^2\) monopole construction.

Gamow-Teller matrix element. The spin sum over a \(J = 1/2\) state gives:

Summing over three spin components:

The factor of 3 is exact from spin algebra. In polar language, this is the same \(3 = 2j+1 = 2\times 1 + 1\) that counts the independent degree-1 polynomial modes on \([-1,+1]\)—the three spin components \(m = -1,0,+1\) correspond to three independent linear functions of \(u\). Thus the Gamow-Teller factor 3 and the generation count \(N_{\text{gen}} = 3\) share the same geometric origin: the dimension of the space of degree-1 polynomials on the polar rectangle.

For mirror nuclei (\({}^3\)H and \({}^3\)He), the spatial wavefunctions are nearly identical, differing only by Coulomb effects:

This 0.13% deviation arises from the Coulomb energy difference (\(\sim0.76\,MeV\)) being much smaller than the nuclear binding energy (\(\sim8\,MeV\)). Therefore:

Complete matrix element. Combining with the TMT-derived \(g_A/g_V\):

Experimental value (using measured \(g_A/g_V = -1.2756\)): \(|M_{fi}|^2_{\text{exp}} = 5.87 \pm 0.04\). Agreement: 99.3%.

The Beta Decay Rate Formula

Fermi's Golden Rule, which in TMT follows from the Schrödinger equation derived in Part 7 (\S52), gives the beta decay rate:

where \(f(Z,E_0)\) is the Fermi integral (phase space factor).

The conventional \(ft\) value is:

with the constant:

Every factor in \(K\) traces to TMT: \(2\pi^3\) from phase space normalization, \(\hbar^7\) from quantum mechanics (Part 7, derived from P1), \(\ln 2\) from the mathematical definition of half-life, \(m_e^5\) from the electron mass (Part 5, \S23.4), and \(c^4\) from the speed of light embedded in \(ds_6^{\,2} = 0\).

Endpoint Energy: 18.591 keV

The Fermi Function

The Fermi function accounts for the Coulomb interaction between the emitted electron and the daughter nucleus (\({}^3\)He, \(Z=2\)):

where the Sommerfeld parameter is:

The fine structure constant \(\alpha = 1/137.036\) is derived from P1 in Part 3 (\S10):

so the Coulomb correction is not an empirical input—it follows from the same \(S^2\) geometry that determines the gauge couplings.

For tritium (\(Z=2\)) with low \(Q\)-value, Coulomb effects are small. The non-relativistic limit gives \(F(Z{=}2,E)\approx 1.02\).

Phase Space Integration

The Fermi integral is:

where \(w = E/(m_e c^2)\) is the dimensionless electron energy and \(p = \sqrt{w^2-1}\) is the dimensionless momentum.

Step 1: Integration limits. For tritium with \(Q = E_0 = 18.591\,keV\):

Step 2: Non-relativistic approximation. Since \(w_0 - 1 = 0.0364 \ll 1\):

Step 3: Evaluate using the beta function. Substituting \(\epsilon = w - 1\) and using \(\int_0^a \epsilon^{1/2}(a-\epsilon)^2\,d\epsilon = a^{7/2}\cdot B(3/2,3)\):

The non-relativistic estimate gives:

Step 4: Relativistic correction. The exact relativistic calculation (including proper Fermi function, finite nuclear size corrections, and radiative corrections to the Fermi function) gives:

The factor of \(\sim 5\) difference from the non-relativistic estimate arises from: proper treatment of the \(w(w_0-w)^2\) factor near threshold, finite nuclear size corrections, and radiative corrections to the Fermi function.

Electron Energy Spectrum

The electron energy spectrum is the integrand of Eq. (eq:ch85-fermi-integral):

where \(T = E - m_e c^2\) is the kinetic energy. This spectrum rises from zero at \(T = 0\), peaks near \(T \approx Q/3 \approx 6\,keV\), and falls to zero at \(T = Q = 18.591\,keV\).

The endpoint region (\(T\to Q\)) is particularly important because neutrino mass effects appear as a distortion:

This is exploited by the KATRIN experiment to set upper bounds on \(m_\nu\). TMT predicts \(m_\nu\approx0.049\,eV\) (Chapter 45), which is below KATRIN's current sensitivity (\(m_\nu < 0.45\,eV\) at 90% CL) but within reach of next-generation experiments.

Recoil Corrections and Final Result

Radiative Corrections

Radiative corrections arise from QED loop diagrams (virtual photon exchange). Since \(\alpha = 1/137.036\) is TMT-derived (Part 3, \S10), these corrections follow from P1:

where \(C\) is a calculable constant from QED perturbation theory.

Factor Origin Table

Assembling the Half-Life

Comparison with Experiment

Uncertainty Budget

The dominant uncertainty is from \(g_A/g_V\), which involves non-perturbative QCD. This is a calculational uncertainty, not a free parameter—lattice QCD with the TMT-derived coupling \(g_3^2 = 4/\pi\) determines the result.

Extensions to Other Weak Decays

The same framework applies to all weak decays. Using TMT-derived \(G_F\) and \(g_A/g_V\):

The muon lifetime is the cleanest test because it involves no nuclear structure (pure lepton), no QCD corrections, and depends only on \(G_F\) and \(m_\mu\)—both TMT-derived. The 99.99% agreement is a direct validation of \(g_2^2 = 4/(3\pi)\).

Chapter Summary