beta_shape_pnpi/.llm-wiki/raw/sources/SRC-2026-05-27-001/extracted.md

title, date, tags, status, aliases, related_notes

title	date	tags	status	aliases	related_notes
SSM — Spectrum Shape Method for gA Extraction	2026-05-14	gA/extraction/algorithm gA/ssm beta-spectrum/forbidden	active	Spectrum Shape Method gA fitting shape factor fit	gA-extraction/02-ssnme gA-extraction/03-tc99-case-study gA-extraction/04-implementation-todo nuclear-structure/01-shape-factors corrections/05-radiative

Spectrum Shape Method (SSM) — Step-by-Step Algorithm for gA Extraction

1. Physical Basis

For forbidden nonunique β decays, the shape factor C(w_e) decomposes into three energy-dependent components with distinct coupling-constant dependence:

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 is the total electron energy in units of m_e c^2. The components C_V, C_A, C_{VA} are computed from nuclear many-body theory and are independent of the coupling constants. The spectrum shape is sensitive to g_A through the interference between vector, axial-vector, and mixed terms — a purely nuclear-structure effect unique to nonunique forbidden decays.

[!note] Why nonunique decays? Unique forbidden decays have a single dominant NME → universal spectrum shape independent of nuclear structure. Nonunique decays involve multiple NMEs → the relative weight of vector vs. axial contributions depends on g_A, creating observable spectral shape differences.

2. The Full Spectrum Model

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

N(W) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A, g_V) \cdot \rho(Z,W)

where:

• \mathcal{N} — overall normalization (absorbs G_F^2 \cos^2\theta_C / 2\pi^3)
• F(Z,W) — Fermi function (Coulomb correction) — corrections/01-fermi-function
• p W (W_0 - W)^2 — phase space — 02-phase-space
• X(W) — atomic exchange correction — corrections/04-exchange
• S(Z,W) — atomic screening — corrections/03-screening
• R(W) — outer radiative correction — corrections/05-radiative
• C(W; g_A, g_V) — shape factor — nuclear-structure/01-shape-factors
• \rho(Z,W) — atomic overlap correction — corrections/08-atomic-overlap

3. The Algorithm

Step 1: Compute shape factor components from nuclear model

Input: Nuclear many-body model (NSM, MQPM, etc.) with appropriate valence space and Hamiltonian for the transition of interest.

Output: Three energy-dependent lookup tables: C_V(W), C_A(W), C_{VA}(W) for W \in [1, W_0].

Method:

1. Run shell model (or MQPM) to obtain one-body transition densities (OBTDs)
2. For each OBTD, compute the contribution to each shape factor component using the Behrens-Bühring formalism (truncated lepton current approximation)
3. Sum over all contributing multipole orders K and lepton quantum numbers k_e, k_\nu

CVC constraint for vector current: The relativistic vector current includes the small relativistic NME (sNME), which is related to the large vector NME (l-NME) via the CVC hypothesis:

\text{sNME} = \frac{1}{2} \left[ W_0 - (m_n - m_p) + E_C \right] \cdot \text{l-NME}

where E_C is the Coulomb displacement energy. The sNME is critical for small-Q decays like ^{99}\text{Tc} ([W_0 - (m_n - m_p)] = -0.487 MeV).

Practical note: For ^{99}\text{Tc}, the dominant transition is 2d_{5/2} \to 1g_{9/2} with OBTD \approx 0.478. The sub-dominant 1g_{7/2} \to 1g_{9/2} transition contributes OBTD \approx 0.010. See Kostensalo & Suhonen (2017), Table VII.

Step 2: Build theoretical spectrum for trial g_A

For each trial value of g_A (with g_V = 1.0 fixed by CVC):

1. Compute shape factor:

   C(W; g_A) = C_V(W) + g_A^2\, C_A(W) + g_A\, C_{VA}(W)

   (using g_V = 1.0)

2. Compute full spectrum:

   N_{\text{theo}}(W; g_A) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A) \cdot \rho(Z,W)

3. Normalize: Choose \mathcal{N} so that \int_1^{W_0} N_{\text{theo}}(W; g_A)\, dW = 1 (or match the total counts in the experimental spectrum)

Step 3: Compare with experimental spectrum

Input: Experimental spectrum data (W_i, N_i, \sigma_i) for i = 1, \dots, N_{\text{bins}}.

χ² function:

\chi^2(g_A) = \sum_{i \in \text{fit range}} \frac{\left[N_i - N_{\text{theo}}(W_i; g_A)\right]^2}{\sigma_i^2}

Fit range selection (Paulsen et al., 2024):

• Lower bound: ~20 keV — below this, detector threshold effects and atomic exchange modeling dominate
• Upper bound: ~275 keV for ^{99}\text{Tc} — above this, endpoint region with poor statistics and resolution distortion
• Rationale: the $g_A$-driven shape differences are most pronounced in the mid-energy range

Step 4: Minimize χ² and extract g_A

Grid search: Evaluate \chi^2(g_A) on a grid (e.g., g_A \in [0.4, 1.8] with step 0.01).

Refinement: Use scipy.optimize.minimize_scalar (Brent's method) around the grid minimum.

Uncertainty: Find g_A values where \chi^2 = \chi^2_{\text{min}} + 1. These define the 68% confidence interval.

Output: (g_A^{\text{best}}, g_A^{\text{low}}, g_A^{\text{high}})

Step 5 (optional): Simultaneous g_A and g_V fit

If g_V is not fixed to 1.0 (e.g., testing CVC violation):

\chi^2(g_A, g_V) = \sum_i \frac{\left[N_i - N_{\text{theo}}(W_i; g_A, g_V)\right]^2}{\sigma_i^2}

Fit both parameters simultaneously. The correlation between g_A and g_V is typically strong.

Step 6 (optional): Half-life normalization — quenched g_A

The g_A extracted from shape alone can be enhanced (>1.27) because the normalization constant \mathcal{N} absorbs the overall rate. To obtain the quenched values consistent with half-life:

1. Compute the theoretical partial half-life for the best-fit (g_A, g_V):

   t_{1/2}^{\text{theo}} = \frac{\kappa}{\tilde{C}}, \quad \tilde{C} = \int_1^{W_0} C(W; g_A, g_V) \cdot p W (W_0 - W)^2 \cdot F(Z,W) \cdot X(W) \cdot S(Z,W) \cdot R(W) \, dW

2. Compare with experimental partial half-life:

   t_{1/2}^{\text{exp}} = \frac{t_{1/2}^{\text{total}}}{\text{branching ratio}}

3. Extract quenched values:

   g_i^{\text{quenched}} = g_i^{\text{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.

4. The sNME Problem and ESSM

The Problem

The sNME (small relativistic vector NME) is extremely difficult to compute in nuclear shell model because it involves contributions outside the valence shell. The CVC prediction gives a "canonical" value, but this assumes a perfect many-body theory — which finite valence spaces are not.

The Solution: Enhanced SSM (ESSM)

Kumar et al. (2021) proposed treating the sNME as a fitting parameter alongside g_A. This is the Enhanced Spectrum Shape Method (ESSM):

1. Fix g_A at a trial value
2. Vary the sNME to simultaneously fit the experimental half-life (not just the shape)
3. Because the half-life depends quadratically on sNME, there are typically two solutions for each g_A value:
   • sNME(c): closer to the CVC value
   • sNME(f): further from the CVC value
4. Compare both solutions' spectral shapes with the data

The "optimal" choice (closest to CVC) is theoretically motivated, but experimental validation is needed. See gA-extraction/02-ssnme for details.

5. Uncertainty Budget

Component	Uncertainty on g_A	Relative	Dominant for
Fit method (statistics)	0.0745	4.9%	All SSM analyses
Coulomb displacement energy	0.0425	2.8%	Small-Q decays
Nuclear model	0.0294	1.9%	Model comparison
Lepton current treatment	0.0127	0.8%	High-Z nuclei
Fit range	0.0061	0.4%	All
Endpoint energy	0.0038	0.2%	Precision $Q$-value needed
Radiative corrections	0.0033	0.2%	All
Atomic exchange	0.0029	0.2%	Low-energy region
Combined	0.0919	6.0%	—

Source: Paulsen et al. (2024), Table VIII.

6. Key References

Ref	Citation	Use
[SSM-orig]	Kostensalo, Haaranen, Suhonen, Phys. Rev. C 95, 044313 (2017)	SSM introduction; 26 decays
[SSM-Tc94-98]	Kostensalo, Suhonen, Phys. Rev. C 96, 024317 (2017)	gA-driven evolution; $^{94,98}$Tc, $^{99}$Tc
[Paulsen2024]	Paulsen et al., Phys. Rev. C 110, 055503 (2024)	$^{99}$Tc MMC measurement; g_A extraction (enhanced value)
[Song2025]	Song et al., arXiv:2506.17544 (2025)	$^{99}$Tc MMC measurement; quenched g_A via ESSM
[ESSM]	Kumar, Srivastava, Suhonen, Eur. Phys. J. A 57, 225 (2021)	Enhanced SSM formalism
[Ramalho2024]	Ramalho, Suhonen, Front. Phys. 12, 1455778 (2024)	sNME analysis; five second-forbidden decays
[Hayen2017]	Hayen et al., Rev. Mod. Phys. 90, 015008 (2017)	BB formalism; shape factor theory