beta_shape_pnpi/docs/14-gA-shape-factor-parameterization.md

---

title: "14 – gA-Driven Shape Factor Parameterization" date: 2026-05-13 tags:

• beta-spectrum/shape-factor
• beta-spectrum/gA
• beta-spectrum/forbidden status: active aliases: [Spectrum Shape Method, SSM, gA quenching, shape factor parameterization] related_components: 10-nuclear-structure, 07-recoil-effects

---

gA-Driven Shape Factor Parameterization — The Spectrum Shape Method (SSM)

Overview

For forbidden nonunique β decays, the shape factor C(Z,W) depends on the weak axial-vector coupling constant g_A in a nontrivial, energy-dependent way. This creates a powerful observable: by fitting the theoretical spectrum shape to high-precision experimental data, one can extract the effective value of $g_A$ (and g_V) for a given nuclear transition. This technique is called the Spectrum Shape Method (SSM), introduced in Kostensalo et al. (2017).

[!tip] Why this matters for 99Tc The ^{99}\text{Tc} β spectrum is a second-forbidden nonunique transition (9/2_1^+ \to 5/2_1^+). Its shape factor is strongly sensitive to g_A — making it an excellent laboratory for determining the quenched effective coupling constants. Paulsen et al. (2024) measured the spectrum with MMCs down to <1 keV and extracted g_V^{\text{eff}} = 0.376(5), g_A^{\text{eff}} = 0.574(36).

1. Theoretical Foundation

1.1 Shape Factor Decomposition

The shape factor C(w_e) decomposes into vector, axial-vector, and mixed parts:

C(w_e) = g_V^2\, C_V(w_e) + g_A^2\, C_A(w_e) + g_V g_A\, C_{VA}(w_e) \tag{1}

where w_e = W is the total electron energy in units of m_e c^2. The three components C_V, C_A, C_{VA} are purely nuclear-structure dependent — they are computed from nuclear matrix elements (NMEs) and are independent of the coupling constants.

After integration over the spectrum:

\tilde{C} = g_V^2\, \tilde{C}_V + g_A^2\, \tilde{C}_A + g_V g_A\, \tilde{C}_{VA} \tag{2}

The integrated components \tilde{C}_i are energy-independent constants.

1.2 The β Spectrum Equation

The full differential spectrum (Paulsen et al., 2024, Eq. 5):

N(W)\,dW = \frac{G_F^2 \cos^2\theta_C}{2\pi^3}\, F(Z,W)\, p W (W_0 - W)^2\, X(W)\, C(W)\, \rho(Z,W)\, dW

where:

• F(Z,W) — Fermi function (Coulomb correction)
• X(W) — screening and exchange corrections
• C(W) — shape factor (nuclear structure)
• \rho(Z,W) — atomic overlap correction
• G_F \cos\theta_C — effective Fermi constant

1.3 Traditional Parametrization (Ad-Hoc)

Historically, the shape factor for ^{99}\text{Tc} has been approximated as first-forbidden unique (Reich & Schüpferling, 1974; standard reference [15]):

C_{\text{adhoc}}(W) = q^2 + \lambda p^2 = (W_0 - W)^2 + \lambda (W^2 - 1) \tag{3}

with \lambda as a free fitting parameter. This is not derived from nuclear structure — it is an empirical ansatz. Paulsen et al. (2024) extracted \lambda \approx 0.65 from fits, but this parametrization:

• Cannot distinguish g_V from g_A effects
• Provides no insight into nuclear structure
• Fails below $\sim$100 keV when compared to high-precision data

1.4 SSM Parametrization (Theory-Driven)

The SSM replaces Eq. (3) with the microscopically computed shape factor:

C_{\text{SSM}}(W; g_A) = g_V^2\, C_V(W) + g_A^2\, C_A(W) + g_V g_A\, C_{VA}(W) \tag{4}

where C_V(W), C_A(W), C_{VA}(W) are computed from shell-model (or MQPM) one-body transition densities (OBTDs).

2. Algorithm for Parameterization

2.1 Input Requirements

Quantity	Source	Notes
C_V(W), C_A(W), C_{VA}(W)	Nuclear shell model (NuShellX@MSU) or MQPM	Computed for the specific transition
W_0 (endpoint)	Independent measurement or fit	Paulsen et al.: W_0 = 295.82(16) keV
Experimental spectrum N_{\text{exp}}(W)	Calorimetry / semiconductor / MMC	Corrected for detector response, background, energy losses
F(Z,W), X(W), \rho(Z,W)	Calculator modules	Already implemented in project
g_V	Fixed (CVC: g_V = 1.0, or treated as free)	Paulsen et al. used free g_V fit

2.2 Computation of Shape Factor Components

The shape factor in the Behrens-Bühring formalism (truncated lepton current approximation, Paulsen et al. Eq. 10):

C(W) = \sum_{K,k_e,k_\nu} \left[ e_k^2 W k_e\, m_K(k_e,k_\nu) - 2\mu e_k W k_e\, \gamma\, m_K(k_e,k_\nu) \right]

where the sum runs over lepton quantum numbers k_e, k_\nu and nuclear multipole order K. The key point: each component C_i(W) (i = V, A, VA) is computed separately from the NMEs.

Practical computation via shell model:

1. OBTD calculation (NuShellX):

   • Compute one-body transition densities for the transition
   • For ^{99}\text{Tc}: dominant transition 2d_{5/2} \to 1g_{9/2} with OBTD \approx 0.478
   • Sub-dominant: 1g_{7/2} \to 1g_{9/2} with OBTD \approx 0.010

2. Coulomb displacement energy E_C (for relativistic vector current):

   • Leading order: E_C^{(3)} = \frac{\int_0^\infty g_f V(r) g_i (r/R)^K r^2 dr}{\int_0^\infty g_f g_i (r/R)^K r^2 dr}
   • Energy-dependent: E_C^{(4)} includes full lepton current dependence
   • For ^{99}\text{Tc}: E_C^{(4)} \approx 10.5 MeV (GLEKPN valence space)

3. Vector current from CVC:

   V_F^{221} = -\sqrt{\frac{R}{10}}\left[W_0 - (m_n - m_p) + E_C\right] V_F^{220}

   For ^{99}\text{Tc}, [W_0 - (m_n - m_p)] = -0.487 MeV — small Q-value makes E_C critical.

2.3 Fitting Procedure

Step 1: Precompute shape factor components C_i(W) from nuclear model
        → Store as lookup tables: C_V(W), C_A(W), C_VA(W) for W ∈ [1, W0]

Step 2: For trial (gA, gV) pair:
        a. Compute C(W; gA, gV) = gV²·C_V(W) + gA²·C_A(W) + gV·gA·C_VA(W)
        b. Compute full spectrum:
           N_theo(W) ∝ p·W·(W0−W)²·F(Z,W)·X(W)·C(W; gA, gV)·ρ(Z,W)
        c. Normalize: ∫ N_theo(W) dW = 1

Step 3: Compare with experimental spectrum:
        χ²(gA, gV) = Σ [N_exp(W_i) − N_theo(W_i; gA, gV)]² / σ_i²

Step 4: Minimize χ² over (gA, gV) grid
        → Extract best-fit values and confidence intervals

2.4 Energy Range Selection

Paulsen et al. (2024) recommend fitting in the range 20–275 keV to avoid:

• Below ~20 keV: detector threshold effects, atomic exchange modeling uncertainties
• Above ~275 keV: endpoint region with poor statistics, resolution distortion

The χ² landscape is approximately quadratic, enabling simple uncertainty estimation at χ² = χ²_min + 1.

3. Results from Literature

3.1 ^{99}\text{Tc} — Paulsen et al. (2024)

From the MMC measurement (PTB/LNHB), the extracted values are:

Quantity	Value	Method
Q_\beta	295.82(16) keV	Fit to spectrum shape
g_V^{\text{eff}}	0.376(5)	From half-life normalization
g_A^{\text{eff}}	0.574(36)	From spectrum shape fit
\bar{E}_\beta	98.51(23) keV	From theoretical spectrum
\log f	−0.476 60(22)	From phase space integration
\log ft	12.3478(23)	Combined

Key finding: The shape-fit alone gives g_A^{\text{eff}} = 1.526(92) (enhanced, not quenched), but when the half-life is used to normalize the overall rate, the quenched values g_V^{\text{eff}} = 0.376, g_A^{\text{eff}} = 0.574 emerge — consistent with first-forbidden nonunique transitions (Suhonen review: g_A^{\text{eff}} \approx 0.9, $g_V^{\text{eff}} \approx 0.3$–0.7).

3.2 ^{87}\text{Rb} — Kostensalo et al. (2017)

Third-forbidden nonunique (3/2^- \to 9/2^+):

• Strong g_A dependence in spectrum shape
• MQPM and NSM agree on shape evolution
• g_A \approx 0.9 gives best match to available data
• Best candidate for g_A determination in third-forbidden decays

3.3 ^{113}\text{Cd} and ^{115}\text{In} — Kostensalo et al. (2017, 2017a)

Fourth-forbidden nonunique:

• Most dramatic g_A dependence: bell-shaped spectrum at g_A \approx g_V, monotonic otherwise
• Consistent g_A \approx 0.9 extracted across three nuclear models (MQPM, NSM, IBFM-2)
• Demonstrates robustness of SSM against nuclear model details

3.4 Summary of g_A Sensitivity by Forbiddenness

Transition	Forbiddenness	g_A Sensitivity	SSM Viability
^{99}\text{Tc}	2nd nonunique	Strong	✅ Excellent
^{94}\text{Nb}, ^{98}\text{Tc}	2nd nonunique	Strong	✅ Good
^{87}\text{Rb}	3rd nonunique	Strong	✅ Good
^{113}\text{Cd}, ^{115}\text{In}	4th nonunique	Very strong	✅ Excellent
^{137}\text{Cs} (to 11/2^-)	1st unique	Weak (NLO only)	⚠️ Limited
^{137}\text{Cs} (to 3/2^+)	2nd nonunique	None (cancellation)	❌ Not applicable
First-forbidden nonunique	1st nonunique	Weak	❌ Not useful

Systematic pattern: Even-forbidden decays show strong g_A dependence; odd-forbidden decays are mostly insensitive.

4. Implementation Plan

4.1 New Module: components/shape_factor_gA.py

ShapeFactorGA
├── __init__: load C_V(W), C_A(W), C_VA(W) from precomputed tables or nuclear model
├── evaluate(W, gA, gV=1.0): compute C(W; gA, gV)
├── evaluate_components(W): return (C_V, C_A, C_VA) separately
├── chi2_spectrum(gA, gV, W_grid, N_exp, sigma): χ² for spectrum fit
├── fit_gA(W_grid, N_exp, sigma, W_min, W_max): minimize χ², return (gA_best, gA_err)
├── fit_gA_gV(W_grid, N_exp, sigma, W_min, W_max): fit both, return (gA, gV, cov)
└── mean_energy(gA, gV, W0): compute ⟨E⟩ from theoretical spectrum

4.2 Integration with BetaSpectrum

The shape factor becomes a new SpectrumComponent:

class ShapeFactorGA(SpectrumComponent):
    """gA-dependent shape factor for forbidden nonunique decays."""
    
    def __init__(self, C_V_table, C_A_table, C_VA_table, gA=1.27, gV=1.0):
        # Load precomputed shape factor components
        # C_i tables: W array → C_i(W) values
    
    def evaluate(self, W):
        return gV**2 * self.C_V(W) + gA**2 * self.C_A(W) + gV*gA * self.C_VA(W)
    
    def integrate(self, W0):
        """Compute integrated shape factor for half-life prediction."""
        return integrate(C(W) * p * W * (W0-W)**2 * F(Z,W) dW, 1, W0)

4.3 Precomputed Tables from Nuclear Model

For ^{99}\text{Tc}, the shape factor components should be precomputed from NuShellX@MSU (GLEKPN valence space, Mach interaction):

W (keV)	C_V(W)	C_A(W)	C_{VA}(W)
1.001	...	...	...
...	...	...	...
1.581	...	...	...

(Endpoint: W_0 = 1 + 295.82/510.998950 \approx 1.579)

4.4 Fitting Pipeline

def fit_gA_for_Tc95(data_file, W_min=20*keV, W_max=275*keV):
    """Fit gA to 99Tc beta spectrum data."""
    W, N_exp, sigma = load_spectrum(data_file)
    
    # Load precomputed shape factor components
    sf = ShapeFactorGA("Tc95_CV_CA_CVA_tables.npy")
    
    # Grid search + refinement
    best_chi2 = inf
    for gA in np.linspace(0.4, 1.8, 141):
        chi2 = sf.chi2_spectrum(gA=gA, gV=1.0, W=W, N_exp=N_exp, sigma=sigma,
                                W_min=W_min, W_max=W_max)
        if chi2 < best_chi2:
            best_chi2 = chi2
            best_gA = gA
    
    # Refine around minimum
    gA_opt = optimize.minimize_scalar(...)
    
    # Uncertainty from χ² = χ²_min + 1
    gA_low, gA_high = find_chi2_one_boundary(gA_opt, sf, ...)
    
    return gA_opt, gA_low, gA_high

5. Uncertainty Budget (from Paulsen et al., 2024)

Component	Uncertainty on g_A^{\text{eff}}	Relative
Fit method (statistics)	0.0745	4.9%
Coulomb displacement energy	0.0425	2.8%
Nuclear model	0.0294	1.9%
Lepton current treatment	0.0127	0.8%
Fit range	0.0061	0.4%
Maximum energy	0.0038	0.2%
Radiative corrections	0.0033	0.2%
Atomic exchange	0.0029	0.2%
Combined	0.0919	6.0%

Dominant uncertainty: fit method / statistics. The nuclear model uncertainty is surprisingly small (1.9%), confirming SSM robustness.

6. Key References

Ref	Citation	Relevance
[SSM-1]	Kostensalo, Haaranen, Suhonen, Phys. Rev. C 95, 044313 (2017)	Original SSM paper; 26 decays studied
[SSM-2]	Kostensalo, Suhonen, Phys. Rev. C 96, 024317 (2017)	gA-driven evolution; 16 decays; Table I with integrated shape factors
[SSM-3]	Paulsen et al., Phys. Rev. C 110, 055503 (2024)	^{99}\text{Tc} high-precision MMC measurement; g_A extraction
[SSM-4]	Kostensalo et al., Phys. Rev. C 95, 024327 (2017)	^{113}\text{Cd} analysis with three nuclear models; SSM robustness
[Hayen]	Hayen et al., Rev. Mod. Phys. 90, 015008 (2017)	Comprehensive review; BB formalism; Eq. 79–81
[Suhonen]	Suhonen, From Nucleons to Nucleus (Springer, 2007)	Nuclear structure background; effective coupling constants
[Behrens]	Behrens & Bühring, Electron Radial Wave Functions and Nuclear Beta Decay (1982)	BB formalism reference; tabulated I(k_e,m,n,\rho) functions

7. Connection to Existing Modules

This module integrates with:

• components/fermi.py — Fermi function F(Z,W)
• components/exchange.py — Exchange correction X(W) (extended to forbidden transitions)
• components/finite_size.py — Finite-size correction (daughter nucleus radius)
• components/screening.py — Screening correction
• spectrum.py — BetaSpectrum multiplicative composition
• docs/10-nuclear-structure.md — Theoretical background on shape factors

8. Practical Considerations

8.1 Which transitions to prioritize?

For g_A extraction via SSM, the selection criteria are:

1. Strong g_A dependence in spectrum shape (even-forbidden, nonunique)
2. High branching ratio (preferably > 90%)
3. Measurable Q-value with good energy resolution
4. Low background environment

For the ^{99}\text{Tc} project, this is the primary candidate — it satisfies all criteria.

8.2 Quenching vs. Enhancement

The extracted g_A^{\text{eff}} from shape fits alone can be enhanced (>1.27) because the shape factor normalization absorbs the overall rate. The quenched values emerge only when the half-life is used for absolute normalization:

g_i^{\text{eff, quenched}} = g_i^{\text{eff, shape}} \times \sqrt{\frac{t_{1/2}^{\text{theo}}}{t_{1/2}^{\text{exp}}}}

For ^{99}\text{Tc}: t_{1/2}^{\text{theo}} \approx 30 \times 10^3 a vs. t_{1/2}^{\text{exp}} \approx 212 \times 10^3 a → quenching factor \approx 0.38.

8.3 Nuclear Model Dependencies

While SSM is robust against nuclear model details (MQPM vs. NSM vs. IBFM-2 agree to $\sim$10%), the absolute normalization of shape factor components varies:

• For ^{99}\text{Tc}: MQPM gives $\sim$2× larger \tilde{C} than NSM
• This affects half-life predictions but not the shape evolution
• The g_A extraction from shape is thus model-independent to $\sim$2% (see uncertainty budget)

9. Future Extensions

1. First-forbidden unique decays: Include NLO terms for g_A sensitivity
2. Highly forbidden decays (4th–6th order): ^{96}\text{Zr}, ^{48}\text{Ca} — strong g_A dependence in even-forbidden channels
3. Two-body currents: Recent ab initio calculations suggest quenching may be unnecessary when 2BC are included (Carbone et al.)
4. Multi-parameter fits: Simultaneous fit of g_A, g_V, Q_\beta, and nuclear structure parameters
5. Machine learning surrogate: Train emulator on shape factor components for rapid χ² evaluation