From 030ae3e4b95a68e99afc7e4b5197564c31d8b2b5 Mon Sep 17 00:00:00 2001 From: Evgeniy Unzhakov Date: Thu, 14 May 2026 19:46:39 +0300 Subject: [PATCH] docs: restructure knowledge base into tree layout + gA extraction module - Organize into tree structure: corrections/, nuclear-structure/, gA-extraction/, refs/ - Move all correction notes to corrections/ (01-09) - Move nuclear structure to nuclear-structure/ with index - Add gA-extraction/ with SSM algorithm, sNME/ESSM formalism, Tc99 case study, and implementation TODO - Reorganize refs by topic (electromagnetic/, forbidden-decay/) - Update all wikilinks to new paths - Remove radiative_spec.md (code algorithm, not reference) - Lint: all wikilinks verified valid - Comprehensive gA extraction research based on Paulsen et al. (2024) and Song et al. (2025) --- docs/00-beta-spectrum-overview.md | 28 +- docs/02-phase-space.md | 6 +- docs/14-gA-shape-factor-parameterization.md | 325 ---------------- docs/README.md | 104 +++-- docs/{ => corrections}/01-fermi-function.md | 8 +- .../02-finite-size.md} | 8 +- .../03-screening.md} | 10 +- .../04-exchange.md} | 4 +- .../05-radiative.md} | 0 .../06-recoil.md} | 2 +- .../07-detector-response.md} | 0 .../08-atomic-overlap.md} | 8 +- .../09-chemical-effects.md} | 6 +- docs/gA-extraction/01-ssm-algorithm.md | 172 +++++++++ docs/gA-extraction/02-ssnme.md | 106 +++++ docs/gA-extraction/03-tc99-case-study.md | 105 +++++ docs/gA-extraction/04-implementation-todo.md | 175 +++++++++ .../01-shape-factors.md} | 8 +- docs/nuclear-structure/index.md | 29 ++ .../summary.md | 0 .../summary.md | 0 .../summary.md | 0 .../summary.md | 0 .../references.bib | 0 .../summary.md | 0 .../summary.md | 0 .../2024_Paulsen_T99_beta_spectrum/summary.md | 0 .../summary.md | 0 docs/refs/radiative_spec.md | 363 ------------------ 29 files changed, 696 insertions(+), 771 deletions(-) delete mode 100644 docs/14-gA-shape-factor-parameterization.md rename docs/{ => corrections}/01-fermi-function.md (88%) rename docs/{03-finite-size.md => corrections/02-finite-size.md} (88%) rename docs/{04-screening-correction.md => corrections/03-screening.md} (79%) rename docs/{05-exchange-correction.md => corrections/04-exchange.md} (93%) rename docs/{06-radiative-corrections.md => corrections/05-radiative.md} (100%) rename docs/{07-recoil-effects.md => corrections/06-recoil.md} (98%) rename docs/{08-detector-response.md => corrections/07-detector-response.md} (100%) rename docs/{12-atomic-overlap.md => corrections/08-atomic-overlap.md} (91%) rename docs/{13-chemical-effects.md => corrections/09-chemical-effects.md} (96%) create mode 100644 docs/gA-extraction/01-ssm-algorithm.md create mode 100644 docs/gA-extraction/02-ssnme.md create mode 100644 docs/gA-extraction/03-tc99-case-study.md create mode 100644 docs/gA-extraction/04-implementation-todo.md rename docs/{10-nuclear-structure.md => nuclear-structure/01-shape-factors.md} (91%) create mode 100644 docs/nuclear-structure/index.md rename docs/refs/{ => electromagnetic}/1967_Sirlin_general_properties_of_em_corrections/summary.md (100%) rename docs/refs/{ => electromagnetic}/1986_Sirlin_verification_of_CVC/summary.md (100%) rename docs/refs/{ => electromagnetic}/1987_Sirlin_remarks_on_OZa2_corrections/summary.md (100%) rename docs/refs/{ => electromagnetic}/2011_Sirlin_radiative_correction_to_nu_spectrum/summary.md (100%) rename docs/refs/{ => electromagnetic}/2017_Hayen_analytical_beta_shape/references.bib (100%) rename docs/refs/{ => electromagnetic}/2017_Hayen_analytical_beta_shape/summary.md (100%) rename docs/refs/{ => forbidden-decay}/2019_Hayen_first_forbidden_transitions/summary.md (100%) rename docs/refs/{ => forbidden-decay}/2024_Paulsen_T99_beta_spectrum/summary.md (100%) rename docs/refs/{ => forbidden-decay}/2025_Song_tc99_measurement_for_g_v/summary.md (100%) delete mode 100644 docs/refs/radiative_spec.md diff --git a/docs/00-beta-spectrum-overview.md b/docs/00-beta-spectrum-overview.md index 13f1210..487884f 100644 --- a/docs/00-beta-spectrum-overview.md +++ b/docs/00-beta-spectrum-overview.md @@ -25,19 +25,19 @@ $$N(W)\,dW = \frac{G_V^2 V_{ud}^2}{2\pi^3} F_0 L_0 U DFS R R_N Q S X r C D_C \; | Symbol | Name | Module / Note | Status | |---|---|---|---| -| $F_0(Z,W)$ | Point charge Fermi function | [[01-fermi-function]] | ✓ | -| $L_0(Z,W)$ | Electrostatic finite size (uniform sphere) | [[03-finite-size]] | ✓ | -| $U(Z,W)$ | Charge distribution correction | [[03-finite-size]] | ✓ | -| $DFS(Z,W,\beta^2)$ | Nuclear deformation correction | [[03-finite-size]] | ✓ | -| $R(W,W_0) = 1 + \delta_R$ | Outer radiative corrections | [[06-radiative-corrections]] | ✓ | -| $R_N(W,W_0,M)$ | Recoil kinematic correction | [[07-recoil-effects]] | ✗ | -| $Q(Z,W,M)$ | Recoil electromagnetic correction | [[07-recoil-effects]] | — | -| $S(Z,W)$ | Atomic electron screening | [[04-screening-correction]] | ✓ | -| $X(Z,W)$ | Atomic exchange effect | [[05-exchange-correction]] | ✓ | -| $r(Z,W)$ | Atomic mismatch / overlap | [[12-atomic-overlap]] | — | -| $A(Z,W) = S \cdot X \cdot r$ | Combined atomic factors | [[12-atomic-overlap]] | — | -| $C(Z,W)$ | Nuclear structure shape factor | [[10-nuclear-structure]] | — | -| $D_C(Z,W,\beta^2)$ | Nuclear deformation (convolution) | [[10-nuclear-structure]] | — | +| $F_0(Z,W)$ | Point charge Fermi function | [[corrections/01-fermi-function]] | ✓ | +| $L_0(Z,W)$ | Electrostatic finite size (uniform sphere) | [[corrections/02-finite-size]] | ✓ | +| $U(Z,W)$ | Charge distribution correction | [[corrections/02-finite-size]] | ✓ | +| $DFS(Z,W,\beta^2)$ | Nuclear deformation correction | [[corrections/02-finite-size]] | ✓ | +| $R(W,W_0) = 1 + \delta_R$ | Outer radiative corrections | [[corrections/05-radiative]] | ✓ | +| $R_N(W,W_0,M)$ | Recoil kinematic correction | [[corrections/06-recoil]] | ✗ | +| $Q(Z,W,M)$ | Recoil electromagnetic correction | [[corrections/06-recoil]] | — | +| $S(Z,W)$ | Atomic electron screening | [[corrections/03-screening]] | ✓ | +| $X(Z,W)$ | Atomic exchange effect | [[corrections/04-exchange]] | ✓ | +| $r(Z,W)$ | Atomic mismatch / overlap | [[corrections/08-atomic-overlap]] | — | +| $A(Z,W) = S \cdot X \cdot r$ | Combined atomic factors | [[corrections/08-atomic-overlap]] | — | +| $C(Z,W)$ | Nuclear structure shape factor | [[nuclear-structure/01-shape-factors]] | — | +| $D_C(Z,W,\beta^2)$ | Nuclear deformation (convolution) | [[nuclear-structure/01-shape-factors]] | — | ## Baseline: Phase Space @@ -49,7 +49,7 @@ with $p = \sqrt{W^2-1}$, $W$ total energy in units of $m_e$, and $W_0$ the endpo ## Precision Goal -The paper achieves **accurate to a few parts in $10^{-4}$ down to 1 keV** for low-to-medium $Z$ nuclei — extending previous work by nearly an order of magnitude [[Hayen2017_summary]]. +The Hayen paper achieves **accurate to a few parts in $10^{-4}$ down to 1 keV** for low-to-medium $Z$ nuclei — extending previous work by nearly an order of magnitude [[refs/electromagnetic/2017_Hayen_analytical_beta_shape/summary]]. > [!key] Why this matters > A precision of ~0.5% is required when determining the Fierz term $b_{\text{Fierz}}$ to improve limits on BSM scalar/tensor couplings. Weak magnetism $b_{wm}$ must be included as experimental precision increases, otherwise it limits BSM sensitivity. diff --git a/docs/02-phase-space.md b/docs/02-phase-space.md index e5d4c87..a5e2b59 100644 --- a/docs/02-phase-space.md +++ b/docs/02-phase-space.md @@ -6,7 +6,7 @@ tags: status: active aliases: [Phase Space, Baseline Spectrum] cssclasses: [] -related_components: [[01-fermi-function]], [[07-recoil-effects]] +related_components: [[corrections/01-fermi-function]], [[corrections/06-recoil]] --- # Phase Space — Baseline Spectral Shape $p W (W_0 - W)^2$ @@ -41,7 +41,7 @@ $$K(W) = \sqrt{\frac{N(W)}{p W F(Z,W)}} \approx W_0 - W \quad (\text{linear for Deviations from linearity near the endpoint indicate: - Non-zero neutrino mass -- Weak magnetism / recoil effects ([[07-recoil-effects]]) +- Weak magnetism / recoil effects ([[corrections/06-recoil]]) - New physics (Fierz term [[Hayen2017_summary#The Fierz Interference Term]]) ## Calculator Implementation @@ -62,7 +62,7 @@ The `PhaseSpace` calculator accepts a `transition_type` parameter that determine | `F2U`, `F3` | 2, 3 | $p_\nu^4 + \frac{3}{10}p_\nu^2 p_e^2 + p_e^4$ | | `F3U`, `F4` | 3, 4 | $p_\nu^6 + 7p_\nu^4 p_e^2 + 7p_\nu^2 p_e^4 + p_e^6$ | -The forbidden factor modifies the energy dependence of the spectrum, becoming significant at higher electron energies. See `[[10-nuclear-structure]]` for the full shape factor treatment. +The forbidden factor modifies the energy dependence of the spectrum, becoming significant at higher electron energies. See `[[nuclear-structure/01-shape-factors]]` for the full shape factor treatment. ## Energy Range diff --git a/docs/14-gA-shape-factor-parameterization.md b/docs/14-gA-shape-factor-parameterization.md deleted file mode 100644 index 06d6af2..0000000 --- a/docs/14-gA-shape-factor-parameterization.md +++ /dev/null @@ -1,325 +0,0 @@ ---- -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`: - -```python -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 - -```python -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 diff --git a/docs/README.md b/docs/README.md index 4a11674..dc57c72 100644 --- a/docs/README.md +++ b/docs/README.md @@ -1,5 +1,5 @@ --- -title: Beta Spectrum Wiki +title: Beta Spectrum Calculator — Knowledge Base date: 2026-04-26 tags: - vault/overview @@ -12,61 +12,83 @@ This Obsidian vault serves as the scientific reference and development workspace ## Purpose -Organize the physics theory behind allowed beta decay spectrum calculations into structured, interconnected notes that directly inform the implementation of correction components in the codebase. +Organize the physics theory behind β-decay spectrum calculations into structured, interconnected notes that inform the implementation of correction components in the codebase. ## Source Material | Reference | Type | Location | |---|---|---| -| Hayen et al., RevModPhys **90**, 015008 (2017) | Review article | `pdf_source/Hayen2017.pdf` / [[Hayen2017_summary]] | +| Hayen et al., *Rev. Mod. Phys.* **90**, 015008 (2017) | Review article | [[refs/electromagnetic/2017_Hayen_analytical_beta_shape/summary]] | -## Notes Structure +## Directory Structure -All notes are flat in this directory, linked via wikilinks. - -### Core Theory - -- [[00-beta-spectrum-overview]] — Master equation and overview of all corrections -- [[02-phase-space]] — Baseline spectral shape $pW(W_0-W)^2$ - -### Correction Components (mapped to calculator modules) - -| # | Note | Calculator Module | Status | -|---|---|---|---| -| 1 | [[01-fermi-function]] | `components/fermi.py` | ✓ implemented | -| 2 | [[03-finite-size]] | `components/finite_size.py` | ✓ implemented | -| 3 | [[04-screening-correction]] | `components/screening.py` | ✓ implemented | -| 4 | [[05-exchange-correction]] | `components/exchange.py` | ✓ implemented | -| 5 | [[06-radiative-corrections]] | `components/radiative.py` | ✓ implemented | -| 6 | [[07-recoil-effects]] | `components/recoil.py` | ✗ not yet implemented | - -### Nuclear Structure & Atomic Effects - -- [[10-nuclear-structure]] — Shape factors, form factors, impulse approximation -- [[11-isospin-breakdown]] — Shake-off/shake-up processes, endpoint shift -- [[12-atomic-overlap]] — Bahcall correction, bound-state β decay -- [[13-chemical-effects]] — Molecular environment influence on spectrum - -### References - -- [[refs/hayen2017]] — Metadata and key equations from Hayen et al. (2017) +``` +docs/ +├── README.md ← You are here +├── 00-beta-spectrum-overview.md ← Master equation & full correction list +├── 02-phase-space.md ← Baseline spectral shape +├── corrections/ ← All correction components +│ ├── 01-fermi-function.md ← Fermi function F₀(Z,W) +│ ├── 02-finite-size.md ← Nuclear size: L₀, U, DFS +│ ├── 03-screening.md ← Atomic screening S(Z,W) +│ ├── 04-exchange.md ← Atomic exchange X(Z,W) +│ ├── 05-radiative.md ← Outer radiative R(W,W₀) +│ ├── 06-recoil.md ← Recoil: R_N, Q, weak magnetism +│ ├── 07-detector-response.md ← Detector convolution +│ ├── 08-atomic-overlap.md ← Atomic mismatch r(Z,W) +│ └── 09-chemical-effects.md ← Molecular environment +├── nuclear-structure/ ← Nuclear theory +│ ├── index.md ← Overview +│ └── 01-shape-factors.md ← C(Z,W), NMEs, impulse approximation +├── gA-extraction/ ← gA extraction (new, in development) +│ ├── 01-ssm-algorithm.md ← SSM step-by-step algorithm +│ ├── 02-ssnme.md ← sNME and ESSM formalism +│ ├── 03-tc99-case-study.md ← 99Tc literature results +│ └── 04-implementation-todo.md ← Implementation TODO list +└── refs/ ← Paper summaries + ├── electromagnetic/ ← CVC, radiative corrections + ├── forbidden-decay/ ← Forbidden decay studies + └── (organized by topic) +``` ## Calculator Architecture ``` SpectrumComponent (ABC, base.py) - ├── PhaseSpace → phase_space.py - ├── FermiFunction → fermi.py - ├── FiniteSizeL0 → finite_size.py - ├── ChargeDistributionU → finite_size.py - ├── ScreeningCorrection → screening.py - ├── ExchangeCorrection → exchange.py - └── RadiativeCorrection → radiative.py + ├── PhaseSpace → 02-phase-space.md + ├── FermiFunction → corrections/01-fermi-function.md + ├── FiniteSizeL0 → corrections/02-finite-size.md + ├── ChargeDistributionU → corrections/02-finite-size.md + ├── ScreeningCorrection → corrections/03-screening.md + ├── ExchangeCorrection → corrections/04-exchange.md + ├── RadiativeCorrection → corrections/05-radiative.md + └── (recoil.py) → corrections/06-recoil.md (not yet implemented) BetaSpectrum (spectrum.py) — multiplicative composition + from_config() BetaSpectrumAnalyzer (spectrum.py) — plotting, CSV export ``` +## Correction Implementation Status + +| # | Correction | Module | Note | Status | +|---|---|---|---|---| +| 1 | Phase space | `phase_space.py` | Baseline | ✓ implemented | +| 2 | Fermi function | `fermi.py` | Coulomb interaction | ✓ implemented | +| 3 | Finite size | `finite_size.py` | L₀, U, DFS | ✓ implemented | +| 4 | Screening | `screening.py` | Atomic screening | ✓ implemented | +| 5 | Exchange | `exchange.py` | Atomic exchange | ✓ implemented | +| 6 | Radiative | `radiative.py` | Outer RC | ✓ implemented | +| 7 | Recoil | `recoil.py` | R_N, Q, weak magnetism | ✗ not yet implemented | +| 8 | Shape factor | `shape_factor_gA.py` | C(Z,W) for forbidden | — planned | +| 9 | Detector | `detector_response.py` | Convolution | — planned | + +## gA Extraction (New) + +For forbidden nonunique decays (e.g., $^{99}\text{Tc}$), the shape factor depends on $g_A$ in a measurable way. See: + +- [[gA-extraction/01-ssm-algorithm]] — The SSM algorithm for extracting $g_A$ from spectral shapes +- [[gA-extraction/04-implementation-todo]] — Step-by-step implementation plan + ## Key Variables and Units - $W$ — total electron energy in units of $m_e c^2$ (natural units, $m_e = 1$) @@ -77,3 +99,7 @@ BetaSpectrumAnalyzer (spectrum.py) — plotting, CSV export - $\gamma = \sqrt{1 - (\alpha Z)^2}$ Conversion: $W = 1 + T / m_e c^2$, with $m_e c^2 = 510.998950$ keV. + +## Precision Goal + +The calculator aims for **accurate to a few parts in $10^{-4}$ down to 1 keV** for low-to-medium $Z$ nuclei, following the approach of Hayen et al. (2017). diff --git a/docs/01-fermi-function.md b/docs/corrections/01-fermi-function.md similarity index 88% rename from docs/01-fermi-function.md rename to docs/corrections/01-fermi-function.md index 00bfa0f..72baccb 100644 --- a/docs/01-fermi-function.md +++ b/docs/corrections/01-fermi-function.md @@ -6,7 +6,7 @@ tags: status: active aliases: [Coulomb Correction, F_0] cssclasses: [] -related_components: [[03-finite-size]], [[06-radiative-corrections]] +related_components: [[corrections/02-finite-size]], [[corrections/05-radiative]] --- # Fermi Function — Coulomb Interaction $F_0(Z,W)$ @@ -24,7 +24,7 @@ $$F_0(Z,W) = 4(2pR)^{2(\gamma-1)} e^{\pi y} \frac{|\Gamma(\gamma + iy)|^2}{(\Gam with: $$\gamma = \sqrt{1 - (\alpha Z)^2}, \quad y = \pm \alpha Z W / p$$ -Here $R$ is the cutoff radius representing daughter nucleus radius — necessitated by divergence at origin for point charge. Most of its dependence is absorbed into corrections ([[03-finite-size]]). +Here $R$ is the cutoff radius representing daughter nucleus radius — necessitated by divergence at origin for point charge. Most of its dependence is absorbed into corrections ([[corrections/02-finite-size]]). ## Coulomb Amplitudes @@ -70,5 +70,5 @@ The effect ranges from ~1% for light nuclei to tens of percent for heavy ones. ## Related Corrections After the point charge Fermi function is computed: -- [[03-finite-size]] corrects for non-pointlike nuclear charge ($L_0$, $U$, $DFS$) -- [[06-radiative-corrections]] adds QED radiative effects ($R = 1 + \delta_R$) +- [[corrections/02-finite-size]] corrects for non-pointlike nuclear charge ($L_0$, $U$, $DFS$) +- [[corrections/05-radiative]] adds QED radiative effects ($R = 1 + \delta_R$) diff --git a/docs/03-finite-size.md b/docs/corrections/02-finite-size.md similarity index 88% rename from docs/03-finite-size.md rename to docs/corrections/02-finite-size.md index b40049c..63b6693 100644 --- a/docs/03-finite-size.md +++ b/docs/corrections/02-finite-size.md @@ -6,18 +6,18 @@ tags: status: active aliases: [Finite Nuclear Size, L_0, U, DFS] cssclasses: [] -related_components: [[01-fermi-function]], [[10-nuclear-structure]] +related_components: [[corrections/01-fermi-function]], [[nuclear-structure/01-shape-factors]] --- # Finite Nuclear Size Effects — $L_0$, $U(Z,W)$, and $DFS$ ## Overview -The Fermi function ([[01-fermi-function]]) assumes a point charge nucleus. Real nuclei have finite extent and possibly deformation — these corrections account for the difference in electric potential inside the nuclear volume. +The Fermi function ([[corrections/01-fermi-function]]) assumes a point charge nucleus. Real nuclei have finite extent and possibly deformation — these corrections account for the difference in electric potential inside the nuclear volume. > [!note] Two types of "finite size" > 1. **Electrostatic** ($L_0$, $U$, $DFS$): Potential difference when moving from point charge to realistic nuclear shape — applied *after* evaluating wave function at origin -> 2. **Convolution**: Integration of leptonic wave functions over the nuclear volume — appears in shape factor $C(Z,W)$ ([[10-nuclear-structure]]) +> 2. **Convolution**: Integration of leptonic wave functions over the nuclear volume — appears in shape factor $C(Z,W)$ ([[nuclear-structure/01-shape-factors]]) ## Nuclear Radius @@ -109,7 +109,7 @@ $$DFS(Z,W,\beta) = \frac{L_0^*(Z,W)}{L_0(Z,W)} \tag{40}$$ where $L_0^*$ integrates over a continuous superposition of uniformly charged spheres (Eq. 39). > [!tip] Magnitude -> Effect can reach several parts in $10^3$. Partially canceled by nuclear structure deformation correction $D_C$ ([[10-nuclear-structure]]). Figure 1 of Hayen shows energy dependence reversal between β⁻ and β⁺ for $\beta_2 = 0.2$. +> Effect can reach several parts in $10^3$. Partially canceled by nuclear structure deformation correction $D_C$ ([[nuclear-structure/01-shape-factors]]). Figure 1 of Hayen shows energy dependence reversal between β⁻ and β⁺ for $\beta_2 = 0.2$. ### Calculator Implementation diff --git a/docs/04-screening-correction.md b/docs/corrections/03-screening.md similarity index 79% rename from docs/04-screening-correction.md rename to docs/corrections/03-screening.md index 822df72..8b876ea 100644 --- a/docs/04-screening-correction.md +++ b/docs/corrections/03-screening.md @@ -6,14 +6,14 @@ tags: status: active aliases: [Atomic Screening, S(Z,W)] cssclasses: [] -related_components: [[12-atomic-overlap]], [[13-chemical-effects]] +related_components: [[corrections/08-atomic-overlap]], [[corrections/09-chemical-effects]] --- # Atomic Electron Screening Correction $S(Z,W)$ ## Physical Origin -The Fermi function ([[01-fermi-function]]) assumes the β particle moves in the Coulomb field of a bare nucleus. In reality, atomic electrons partially screen this charge at large distances from the nucleus, modifying the effective potential experienced by the emitted electron/positron. This is particularly important at low kinetic energies where the β particle spends more time far from the nucleus. +The Fermi function ([[corrections/01-fermi-function]]) assumes the β particle moves in the Coulomb field of a bare nucleus. In reality, atomic electrons partially screen this charge at large distances from the nucleus, modifying the effective potential experienced by the emitted electron/positron. This is particularly important at low kinetic energies where the β particle spends more time far from the nucleus. ## The Screening Correction @@ -88,6 +88,6 @@ From Hayen et al. Table (Section VIII overview): ## Related Notes -- [[12-atomic-overlap]] — Atomic mismatch / overlap correction $r(Z,W)$, Bahcall correction -- [[13-chemical-effects]] — Molecular environment influence on screening and exchange -- [[05-exchange-correction]] — Electron exchange with atomic orbitals (often combined with screening) +- [[corrections/08-atomic-overlap]] — Atomic mismatch / overlap correction $r(Z,W)$, Bahcall correction +- [[corrections/09-chemical-effects]] — Molecular environment influence on screening and exchange +- [[corrections/04-exchange]] — Electron exchange with atomic orbitals (often combined with screening) diff --git a/docs/05-exchange-correction.md b/docs/corrections/04-exchange.md similarity index 93% rename from docs/05-exchange-correction.md rename to docs/corrections/04-exchange.md index 8b7423d..510741d 100644 --- a/docs/05-exchange-correction.md +++ b/docs/corrections/04-exchange.md @@ -6,7 +6,7 @@ tags: status: active aliases: [Atomic Exchange, X(Z,W), Hayen-2018] cssclasses: [] -related_components: [[04-screening-correction]], [[12-atomic-overlap]] +related_components: [[corrections/03-screening]], [[corrections/08-atomic-overlap]] --- # Atomic Electron Exchange Correction $X(Z,W)$ @@ -37,7 +37,7 @@ Screening strength is adjusted for best agreement with bound state energies (dec ## Analytical Parametrisation — Hayen et al. 2018 -The exchange correction $X(W)$ as a function of kinetic energy uses empirical fit coefficients from Table X of [[refs/hayen-2018]]. Coefficients are tabulated for all atoms with $Z = 2$ through $z = 120$. +The exchange correction $X(W)$ as a function of kinetic energy uses empirical fit coefficients from Table X of [[refs/electromagnetic/2017_Hayen_analytical_beta_shape/summary]]. Coefficients are tabulated for all atoms with $Z = 2$ through $z = 120$. ### Shell Structure Effects diff --git a/docs/06-radiative-corrections.md b/docs/corrections/05-radiative.md similarity index 100% rename from docs/06-radiative-corrections.md rename to docs/corrections/05-radiative.md diff --git a/docs/07-recoil-effects.md b/docs/corrections/06-recoil.md similarity index 98% rename from docs/07-recoil-effects.md rename to docs/corrections/06-recoil.md index c0cb803..48b11b5 100644 --- a/docs/07-recoil-effects.md +++ b/docs/corrections/06-recoil.md @@ -6,7 +6,7 @@ tags: status: todo aliases: [Nuclear Recoil, R_N, Weak Magnetism] cssclasses: [] -related_components: [[02-phase-space]], [[10-nuclear-structure]] +related_components: [[02-phase-space]], [[nuclear-structure/01-shape-factors]] --- # Nuclear Recoil Effects — $R_N$, $Q$, and Induced Currents diff --git a/docs/08-detector-response.md b/docs/corrections/07-detector-response.md similarity index 100% rename from docs/08-detector-response.md rename to docs/corrections/07-detector-response.md diff --git a/docs/12-atomic-overlap.md b/docs/corrections/08-atomic-overlap.md similarity index 91% rename from docs/12-atomic-overlap.md rename to docs/corrections/08-atomic-overlap.md index f810e22..b9c4e61 100644 --- a/docs/12-atomic-overlap.md +++ b/docs/corrections/08-atomic-overlap.md @@ -6,7 +6,7 @@ tags: status: active aliases: [Atomic Mismatch, Bahcall Correction, r(Z,W)] cssclasses: [] -related_components: [[04-screening-correction]], [[13-chemical-effects]] +related_components: [[corrections/03-screening]], [[corrections/09-chemical-effects]] --- # Atomic Overlap / Mismatch Correction $r(Z,W)$ @@ -70,6 +70,6 @@ $$\frac{\Gamma_b}{\Gamma_c} = \frac{\pi(\alpha Z)^3}{f(Z,W_0)} (W_0 - 1)^2 \Sigm ## Related Notes -- [[04-screening-correction]] — Screening modification from shake-up processes -- [[11-isospin-breakdown]] — Shake-off process correction -- [[13-chemical-effects]] — Molecular environment influence on atomic overlap +- [[corrections/03-screening]] — Screening modification from shake-up processes +- — Shake-off process correction +- [[corrections/09-chemical-effects]] — Molecular environment influence on atomic overlap diff --git a/docs/13-chemical-effects.md b/docs/corrections/09-chemical-effects.md similarity index 96% rename from docs/13-chemical-effects.md rename to docs/corrections/09-chemical-effects.md index 703502a..63b8881 100644 --- a/docs/13-chemical-effects.md +++ b/docs/corrections/09-chemical-effects.md @@ -6,7 +6,7 @@ tags: status: active aliases: [Molecular Screening, Molecular Exchange, Saenz-Froelich] cssclasses: [] -related_components: [[12-atomic-overlap]], [[04-screening-correction]] +related_components: [[corrections/08-atomic-overlap]], [[corrections/03-screening]] --- # Chemical / Molecular Effects on β Spectrum @@ -113,5 +113,5 @@ For molecular experiments requiring these corrections: ## Related Notes -- [[12-atomic-overlap]] — Atomic mismatch / Bahcall correction -- [[04-screening-correction]] — Baseline atomic screening $S(Z,W)$ +- [[corrections/08-atomic-overlap]] — Atomic mismatch / Bahcall correction +- [[corrections/03-screening]] — Baseline atomic screening $S(Z,W)$ diff --git a/docs/gA-extraction/01-ssm-algorithm.md b/docs/gA-extraction/01-ssm-algorithm.md new file mode 100644 index 0000000..e7e0f89 --- /dev/null +++ b/docs/gA-extraction/01-ssm-algorithm.md @@ -0,0 +1,172 @@ +--- +title: "SSM — Spectrum Shape Method for gA Extraction" +date: 2026-05-14 +tags: + - gA/extraction/algorithm + - gA/ssm + - beta-spectrum/forbidden +status: active +aliases: [Spectrum Shape Method, gA fitting, shape factor fit] +related_notes: + - 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 | diff --git a/docs/gA-extraction/02-ssnme.md b/docs/gA-extraction/02-ssnme.md new file mode 100644 index 0000000..4ffae91 --- /dev/null +++ b/docs/gA-extraction/02-ssnme.md @@ -0,0 +1,106 @@ +--- +title: "sNME — Small Relativistic Vector NME" +date: 2026-05-14 +tags: + - gA/extraction/ssnme + - beta-spectrum/forbidden + - nuclear-structure/NME +status: active +aliases: [small relativistic vector nuclear matrix element, sNME, CVC vector current] +related_notes: + - gA-extraction/01-ssm-algorithm + - gA-extraction/03-tc99-case-study + - nuclear-structure/01-shape-factors +--- + +# The Small Relativistic Vector NME (sNME) — Why It Matters + +## What is the sNME? + +In forbidden β decay, the relativistic vector current has two components: +- **Large NME (l-NME):** Computed from the large components of nucleon wave functions. Reliably calculable in NSM. +- **Small NME (sNME):** Couples the large and small components. **Extremely difficult** to compute because it involves contributions outside the valence shell. + +In a perfect many-body theory with infinite valence space, the sNME is related to the l-NME by the **CVC hypothesis**: + +$$\text{sNME} = \frac{1}{2} \left[ W_0 - (m_n - m_p) + E_C \right] \cdot \text{l-NME} \tag{1}$$ + +where: +- $W_0$ — endpoint energy in MeV +- $m_n - m_p \approx 1.293$ MeV — neutron-proton mass difference +- $E_C$ — Coulomb displacement energy (depends on nuclear charge distribution) +- l-NME — large vector NME (computable in NSM) + +### Why is sNME important? + +Despite being "small" (typically $< 0.1$ in absolute units), the sNME can influence β-spectral shapes and half-lives **quite strongly** because: + +1. **Quadratic dependence:** The half-life $t_{1/2} \propto 1/(\text{l-NME} + \text{sNME})^2$, so even a small sNME shifts the total matrix element significantly. +2. **Small Q-value amplification:** For $^{99}\text{Tc}$, $[W_0 - (m_n - m_p)] = -0.487$ MeV — the negative mass difference nearly cancels the endpoint energy, making the sNME contribution relatively large. +3. **Shape dependence:** Different sNME values produce measurably different spectral shapes, especially below 100 keV. + +## The CVC Value — An Idealization + +The CVC prediction (Eq. 1) is an **idealization** — it pertains to a perfect many-body theory. In practice: +- NSM calculations with finite valence spaces give sNME $\approx 0$ (the valence space truncation kills it) +- The CVC value serves as a **reference**, not a physical prediction + +## ESSM: sNME as a Fitting Parameter + +### The Enhanced Spectrum Shape Method (ESSM) + +Kumar et al. (2021) proposed using the sNME as an additional fitting parameter: + +1. **Fix** $g_A$ at a trial value +2. **Vary** the sNME to match the **experimental partial half-life** +3. Because $t_{1/2} \propto (\text{l-NME} + \text{sNME})^{-2}$, the equation $t_{1/2}^{\text{theo}}(g_A, \text{sNME}) = t_{1/2}^{\text{exp}}$ typically yields **two solutions** for the sNME: + - **sNME(c):** closer to the CVC value + - **sNME(f):** further from the CVC value + +### Two-Solution Structure + +For each $g_A$ value, there are two sNME solutions (or none, if the experimental half-life cannot be reproduced). This creates two distinct spectral shape predictions: + +- The **sNME(c)** solution is theoretically preferred (closer to CVC) +- The **sNME(f)** solution may also be physical — only experiment can decide + +### Application to $^{99}\text{Tc}$ + +Song et al. (2025) applied ESSM to $^{99}\text{Tc}$ with three nuclear models: + +| Model | $g_A^{\text{eff}}$ | sNME (c) [fm³] | sNME (f) [fm³] | $\chi^2_\nu$ | p-value | +|---|---|---|---|---|---| +| **CloseCVC (CVC fixed)** | 1.1 | 0.0674 | — | 1.106 | 0.12 | +| | 1.0 | 0.0681 | — | 1.244 | 0.004 | +| | 1.0 | 0.0651 | — | 1.644 | <0.0001 | +| **FarCVC (sNME fitted)** | 1.2 | −0.0698 | — | 1.066 | 0.22 | +| | 1.0 | −0.0669 | — | 1.097 | 0.14 | +| | 1.0 | −0.0683 | — | 1.095 | 0.14 | + +Key findings: +- **Paulsen et al. (CloseCVC):** $g_A^{\text{eff}} = 1.526(92)$ — enhanced value +- **Song et al. (FarCVC/ESSM):** $g_A^{\text{eff}} \approx 1.0$–1.2 — quenched value +- The discrepancy arises from the **sNME treatment**, not from the data + +## sNME in Other Transitions + +Ramalho & Suhonen (2024) studied five second-forbidden nonunique decays: + +| Transition | $g_A$ Sensitivity | sNME Sensitivity | +|---|---|---| +| $^{60}\text{Fe} \to {}^{60}\text{Co}$ | Strong | Weak | +| $^{94}\text{Nb} \to {}^{94}\text{Mo}$ | Strong (jj45pnb) | Weak | +| $^{98}\text{Tc} \to {}^{98}\text{Ru}$ | Strong | Moderate (below 100 keV) | +| $^{126}\text{Sn} \to {}^{126}\text{Sb}$ | Weak | **Strong** | +| $^{129}\text{I} \to {}^{129}\text{Xe}$ | Weak | **Strong** | + +For $^{99}\text{Tc}$, the $g_A$ sensitivity is **strong** and the sNME sensitivity is **moderate** — making it an excellent candidate for combined $g_A$ + sNME fitting. + +## References + +| Ref | Citation | Use | +|---|---|---| +| [ESSM] | Kumar, Srivastava, Suhonen, *Eur. Phys. J. A* **57**, 225 (2021) | ESSM formalism | +| [Song2025] | Song et al., arXiv:2506.17544 (2025) | $^{99}$Tc with ESSM | +| [Ramalho2024] | Ramalho, Suhonen, *Front. Phys.* **12**, 1455778 (2024) | sNME in five second-forbidden decays | +| [Behrens] | Behrens & Bühring, *Electron Radial Wave Functions and Nuclear Beta Decay* (1982) | sNME formalism | diff --git a/docs/gA-extraction/03-tc99-case-study.md b/docs/gA-extraction/03-tc99-case-study.md new file mode 100644 index 0000000..93aa059 --- /dev/null +++ b/docs/gA-extraction/03-tc99-case-study.md @@ -0,0 +1,105 @@ +--- +title: "$^{99}$Tc Case Study — gA Extraction Results" +date: 2026-05-14 +tags: + - gA/extraction/tc99 + - beta-spectrum/forbidden/second +status: active +aliases: [Tc99 gA, 99Tc beta spectrum, SSM Tc99] +related_notes: + - gA-extraction/01-ssm-algorithm + - gA-extraction/02-ssnme + - refs/2024_Paulsen_T99_beta_spectrum/summary + - refs/2025_Song_tc99_measurement_for_g_v/summary +--- + +# $^{99}$Tc — The Premier Laboratory for gA Extraction + +## Decay Properties + +| Property | Value | +|---|---| +| Transition | $^{99}\text{Tc}(9/2_1^+) \to {}^{99}\text{Ru}(5/2_1^+)$ | +| Forbiddenness | Second-forbidden nonunique | +| Branching ratio | 99.99855(30)% | +| Q-value | 295.82(16) keV (Paulsen et al.) | +| Half-life | 211.5(1.1) $\times 10^3$ a (ENSDF) | +| Dominant transition | $2d_{5/2} \to 1g_{9/2}$ (OBTD $\approx 0.478$) | +| Sub-dominant | $1g_{7/2} \to 1g_{9/2}$ (OBTD $\approx 0.010$) | + +## Two Contradictory Results + +### Result 1: Paulsen et al. (2024) — Enhanced $g_A$ + +**Measurement:** MMC at PTB Berlin + LNHB Paris. Two independent setups, consistent results. + +**Method:** Fixed sNME to CVC value (CloseCVC). + +**Result:** +$$g_A^{\text{eff}} = 1.526(92)$$ + +This is an **enhanced** value — significantly above the free-nucleon value of 1.27. + +**Follow-on values:** +- $g_V^{\text{eff}} = 0.376(5)$ — quenched (via half-life normalization) +- $g_A^{\text{eff, quenched}} = 0.574(36)$ — quenched +- $\bar{E}_\beta = 98.51(23)$ keV +- $\log ft = 12.3478(23)$ + +**Key claim:** This resolves an inconsistency — first-forbidden nonunique transitions show $g_A^{\text{eff}} \approx 0.9$, while fourth-forbidden show $g_A^{\text{eff}} \approx 0.9$. The $^{99}\text{Tc}$ result suggests $g_A$ is **enhanced** in second-forbidden decays. + +### Result 2: Song et al. (2025) — Quenched $g_A$ + +**Measurement:** MMC at LLNL (KRISS sensor). Independent of Paulsen et al. + +**Method:** ESSM — sNME fitted as free parameter (FarCVC). + +**Result:** +$$g_A^{\text{eff}} \approx 1.0\text{--}1.2$$ + +This is a **quenched** value, consistent with first-forbidden and fourth-forbidden nonunique decays. + +**Best fit:** glekpn Hamiltonian, $g_A^{\text{eff}} = 1.2$, sNME = −0.0698 fm³, $\chi^2_\nu = 1.066$, p-value = 0.22 + +**Key claim:** The Paulsen result is an artifact of fixing sNME to CVC. When sNME is fitted, the data prefer quenched $g_A$, consistent with the broader pattern. + +## The Crucial Difference: sNME Treatment + +| Aspect | Paulsen et al. | Song et al. | +|---|---|---| +| sNME treatment | Fixed to CVC value | Fitted as free parameter (ESSM) | +| $g_A^{\text{eff}}$ | 1.526(92) (enhanced) | 1.0–1.2 (quenched) | +| Nuclear models | NSM (GLEKPN) only | NSM (glekpn, jj45pnb) + MQPM | +| Half-life constraint | Separate normalization | Simultaneous shape + half-life fit | +| Consistency with 113Cd, 115In | Inconsistent | Consistent | + +## What This Means for Your Experiment + +The $^{99}\text{Tc}$ case demonstrates that **the treatment of sNME is the dominant theoretical systematic** in gA extraction. The two experiments (Paulsen and Song) measured essentially the same spectrum (Paulsen's data was used as input for Song's analysis) but obtained **contradictory results** because of different theoretical assumptions. + +**For your data analysis:** +1. **Do not fix sNME to CVC** — treat it as a fitting parameter +2. **Use ESSM** — simultaneous fit to spectral shape + half-life +3. **Try multiple nuclear models** — glekpn, jj45pnb, and MQPM (if available) +4. **Report both sNME solutions** (c and f) for each model +5. **The "optimal" choice** (closest to CVC) is theoretically motivated but requires experimental validation + +## Your Polynomial Fit: Connecting to SSM + +You've fitted $C(W) = A \cdot W + B \cdot W^{-1} + C \cdot W^2$. This empirical parametrization can be **cross-checked** against the SSM prediction: + +1. Compute $C_{\text{SSM}}(W; g_A)$ for a range of $g_A$ values +2. Fit each SSM curve to the polynomial form $aW + bW^{-1} + cW^2$ over your fit range +3. Extract the effective $(A, B, C)$ as a function of $g_A$ +4. Compare your fitted $(A, B, C)$ to the $g_A$-dependent curve to identify which $g_A$ range is consistent + +This provides a **model-independent bridge** between your empirical fit and the physical $g_A$ extraction. + +## References + +| Ref | Citation | +|---|---| +| [Paulsen2024] | Paulsen et al., *Phys. Rev. C* **110**, 055503 (2024) | +| [Song2025] | Song et al., arXiv:2506.17544 (2025) | +| [Kostensalo2017] | Kostensalo & Suhonen, *Phys. Rev. C* **96**, 024317 (2017) | +| [ESSM] | Kumar et al., *Eur. Phys. J. A* **57**, 225 (2021) | diff --git a/docs/gA-extraction/04-implementation-todo.md b/docs/gA-extraction/04-implementation-todo.md new file mode 100644 index 0000000..95b0af6 --- /dev/null +++ b/docs/gA-extraction/04-implementation-todo.md @@ -0,0 +1,175 @@ +--- +title: "Implementation TODO — gA Extraction Pipeline" +date: 2026-05-14 +tags: + - gA/extraction/todo + - implementation-plan +status: active +aliases: [gA pipeline, SSM implementation, shape factor module] +related_notes: + - gA-extraction/01-ssm-algorithm + - gA-extraction/02-ssnme + - gA-extraction/03-tc99-case-study +--- + +# Implementation TODO — gA Extraction Pipeline + +> [!info] Status +> **Research phase.** Experimental data to be added by user. Implementation to follow after documentation is complete. + +## Phase 1: Nuclear Model Input (P0 — prerequisite) + +- [ ] **1.1** Obtain precomputed shape factor components $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ for $^{99}\text{Tc}$ + - [ ] Option A: Use Kostensalo & Suhonen (2017) MQPM results (published in PRC 96, 024317) + - [ ] Option B: Use Ramalho & Suhonen (2024) NSM results (Front. Phys. 12, 1455778) + - [ ] Option C: Run shell model with KShell or another available code + - [ ] **Decision:** Which source to use? (See [[gA-extraction/03-tc99-case-study]] for comparison) +- [ ] **1.2** Store components as lookup tables (CSV or NumPy): + - `data/Tc95_CV.csv` — $C_V(W)$ vs. $W$ + - `data/Tc95_CA.csv` — $C_A(W)$ vs. $W$ + - `data/Tc95_CVA.csv` — $C_{VA}(W)$ vs. $W$ + - Columns: `W (unitless)`, `C_V`, `C_A`, `C_VA` + - $W$ range: $[1.001, 1.580]$ (corresponds to $T \in [0.5, 295]$ keV) +- [ ] **1.3** Decide on sNME handling: + - [ ] Fix to CVC value (Paulsen et al. approach) + - [ ] Fit as free parameter (ESSM — Song et al. approach) **RECOMMENDED** + - If ESSM: also need l-NME from nuclear model + +## Phase 2: Shape Factor Module (P1 — depends on 1.1) + +Create `beta_spectrum/components/shape_factor_gA.py`: + +- [ ] **2.1** `ShapeFactorGA` class: + ```python + class ShapeFactorGA: + def __init__(self, C_V_table, C_A_table, C_VA_table, sNME=None): + # Load tables, interpolate to arbitrary W + # sNME: if None, use CVC value; if float, use fitted value + + def evaluate(self, W, gA, gV=1.0): + """Compute C(W; gA, gV) = gV²·CV(W) + gA²·CA(W) + gV·gA·CVA(W)""" + + def evaluate_components(self, W): + """Return (C_V(W), C_A(W), C_VA(W))""" + ``` +- [ ] **2.2** Interpolation: Use `scipy.interpolate.interp1d` with `kind='cubic'` for smooth evaluation +- [ ] **2.3** Validation: Check that $C(W)$ is monotonically reasonable for $g_A \in [0.4, 1.8]$ +- [ ] **2.4** Unit test: `test_shape_factor_gA_consistency` — verify decomposition matches Eq. (1) + +## Phase 3: Fitting Pipeline (P2 — depends on 1.2 + 2.1) + +Create `beta_spectrum/gA_fit.py`: + +- [ ] **3.1** `fit_gA_spectrum(W_exp, N_exp, sigma_exp, shape_factor, W_min, W_max)`: + ```python + def fit_gA_spectrum(W_exp, N_exp, sigma_exp, sf, W_min=0.020, W_max=0.275): + """ + Fit gA to experimental spectrum using χ² minimization. + + Parameters + ---------- + W_exp : array_like + Electron total energy (unitless, W = 1 + T/m_ec²) + N_exp : array_like + Measured counts (or normalized spectrum) + sigma_exp : array_like + Uncertainties on N_exp + sf : ShapeFactorGA + Precomputed shape factor + W_min, W_max : float + Fit range in unitless W + + Returns + ------- + gA_best, gA_low, gA_high, chi2_min, chi2_curve + """ + ``` +- [ ] **3.2** Grid search: $g_A \in [0.4, 1.8]$ with 0.01 step +- [ ] **3.3** Brent's method refinement around grid minimum +- [ ] **3.4** Confidence interval: find $g_A$ where $\chi^2 = \chi^2_{\text{min}} + 1$ +- [ ] **3.5** χ² curve output for visualization +- [ ] **3.6** Unit test: `test_fit_gA_reproduces_input` — inject known $g_A$, verify recovery + +## Phase 4: ESSM Extension (P2 — optional but recommended) + +- [ ] **4.1** `fit_gA_sNME(W_exp, N_exp, sigma_exp, sf, l_NME, t_half_exp)`: + - Simultaneous fit of $g_A$ and sNME + - Match both spectral shape AND experimental half-life + - Return both (sNME_c, sNME_f) solutions +- [ ] **4.2** Half-life computation: + ```python + def compute_half_life(sf, gA, gV, sNME, W0): + """Compute t_1/2 from integrated shape factor""" + C_integral = integrate(C(W; gA, gV, sNME) * phase_space * Fermi * ... dW) + return kappa / C_integral + ``` +- [ ] **4.3** Validation against Song et al. (2025) results for $^{99}\text{Tc}$ + +## Phase 5: Validation (P3 — depends on 3.1 + 4.1) + +- [ ] **5.1** Reproduce Paulsen et al. (2024) result: $g_A^{\text{eff}} = 1.526(92)$ with CVC-fixed sNME +- [ ] **5.2** Reproduce Song et al. (2025) result: $g_A^{\text{eff}} \approx 1.0\text{--}1.2$ with ESSM +- [ ] **5.3** Cross-check: polynomial fit $A \cdot W + B \cdot W^{-1} + C \cdot W^2$ vs. SSM for same $g_A$ +- [ ] **5.4** Sensitivity analysis: vary $W_{\text{min}}$, $W_{\text{max}}$, nuclear model, sNME + +## Phase 6: Experimental Data Analysis (P4 — depends on user providing data) + +- [ ] **6.1** Import raw spectrum data (format TBD) +- [ ] **6.2** Apply detector response correction (if needed) — [[corrections/07-detector-response]] +- [ ] **6.3** Run `fit_gA_spectrum` with appropriate fit range +- [ ] **6.4** Run `fit_gA_sNME` (ESSM) if nuclear model input available +- [ ] **6.5** Generate publication-quality plots: + - Measured spectrum + best-fit theoretical curve + - Residuals plot + - χ² vs. $g_A$ curve + - Comparison with polynomial fit +- [ ] **6.6** Uncertainty budget: propagate detector, model, and systematic uncertainties + +## Phase 7: Paper Draft Support + +- [ ] **7.1** Generate tables: fitted $g_A$, sNME, $\chi^2$, confidence intervals +- [ ] **7.2** Generate figures for paper +- [ ] **7.3** Cross-reference with literature values +- [ ] **7.4** Draft methodology section using [[gA-extraction/01-ssm-algorithm]] as reference + +## File Structure (target) + +``` +beta_shape_pnpi/ +├── beta_spectrum/ +│ ├── components/ +│ │ └── shape_factor_gA.py ← Phase 2 +│ └── gA_fit.py ← Phase 3 + 4 +├── data/ +│ ├── Tc95_CV.csv ← Phase 1 +│ ├── Tc95_CA.csv +│ ├── Tc95_CVA.csv +│ └── experimental/ ← Phase 6 (user-provided) +├── tests/ +│ └── test_shape_factor_gA.py ← Phase 2 + 3 +│ └── test_gA_fit.py +└── docs/ + └── gA-extraction/ + ├── 01-ssm-algorithm.md + ├── 02-ssnme.md + ├── 03-tc99-case-study.md + └── 04-implementation-todo.md +``` + +## Decision Points + +1. **Nuclear model source:** Which precomputed $C_i(W)$ to use? (MQPM vs. NSM) +2. **sNME strategy:** CVC-fixed or ESSM? (ESSM recommended) +3. **Fit range:** What $W_{\text{min}}$, $W_{\text{max}}$ for your detector? +4. **Detector corrections:** Do you need response deconvolution before fitting? + +## References for Implementation + +| Item | Source | +|---|---| +| Shape factor formalism | Kostensalo & Suhonen, PRC 96, 024317 (2017) | +| ESSM formalism | Kumar et al., EPJA 57, 225 (2021) | +| sNME analysis | Ramalho & Suhonen, Front. Phys. 12, 1455778 (2024) | +| $^{99}\text{Tc}$ MMC data | Paulsen et al., PRC 110, 055503 (2024) | +| $^{99}\text{Tc}$ ESSM analysis | Song et al., arXiv:2506.17544 (2025) | +| BB formalism | Behrens & Bühring (1982) | diff --git a/docs/10-nuclear-structure.md b/docs/nuclear-structure/01-shape-factors.md similarity index 91% rename from docs/10-nuclear-structure.md rename to docs/nuclear-structure/01-shape-factors.md index 773daa1..eaa9f82 100644 --- a/docs/10-nuclear-structure.md +++ b/docs/nuclear-structure/01-shape-factors.md @@ -6,14 +6,14 @@ tags: status: active aliases: [Shape Factor, C(Z,W), Form Factors] cssclasses: [] -related_components: [[07-recoil-effects]], [[11-isospin-breakdown]] +related_components: [[corrections/06-recoil]], --- # Nuclear Structure Effects — The Shape Factor $C(Z,W)$ ## Overview -Nuclear structure effects are encapsulated in the shape factor $C(Z,W)$ which multiplies the phase space × Fermi function. This is distinct from electrostatic finite size corrections ([[03-finite-size]]) — it describes how nuclear form factors and matrix elements modify the spectrum beyond simple point-nucleus Coulomb interaction. +Nuclear structure effects are encapsulated in the shape factor $C(Z,W)$ which multiplies the phase space × Fermi function. This is distinct from electrostatic finite size corrections ([[corrections/02-finite-size]]) — it describes how nuclear form factors and matrix elements modify the spectrum beyond simple point-nucleus Coulomb interaction. $$N(W)\,dW \propto p W (W_0 - W)^2 F(Z,W) C(Z,W) \tag{79}$$ @@ -53,7 +53,7 @@ The one-body density matrix elements $\langle f | a^\dagger_\alpha a_\beta | i \ ## Induced Currents — Form Factor Classification -See [[07-recoil-effects]] for detailed equations. Key points: +See [[corrections/06-recoil]] for detailed equations. Key points: ### First-Class Currents (CVC/PCAC) - **Vector** $g_V$: conserved, from EM data via CVC @@ -82,7 +82,7 @@ See [[07-recoil-effects]] for detailed equations. Key points: ## Nuclear Deformation Correction $D_C(Z,W,\beta^2)$ -The nuclear structure analogue of the electrostatic deformation correction ([[03-finite-size]]): +The nuclear structure analogue of the electrostatic deformation correction ([[corrections/02-finite-size]]): - Accounts for non-spherical nuclear shape in convolution terms - Partially cancels with $DFS$ correction (electrostatic) - Relevant for deformed nuclei ($\beta_2 \neq 0$) diff --git a/docs/nuclear-structure/index.md b/docs/nuclear-structure/index.md new file mode 100644 index 0000000..006b5fc --- /dev/null +++ b/docs/nuclear-structure/index.md @@ -0,0 +1,29 @@ +--- +title: "Nuclear Structure — Overview" +date: 2026-05-14 +tags: + - beta-spectrum/nuclear-structure +status: active +aliases: [Nuclear Theory, Shape Factor, NME] +--- + +# Nuclear Structure Effects in β Decay + +## Overview + +Nuclear structure enters the β spectrum through the **shape factor** $C(Z,W)$, which encodes how the nuclear wave functions and transition matrix elements modify the spectrum beyond the point-nucleus Coulomb interaction. + +## Contents + +- [[nuclear-structure/01-shape-factors]] — Shape factor $C(Z,W)$: decomposition, multipole expansion, impulse approximation, induced currents + +## Key Concepts + +- **Shape factor decomposition:** $C(W) = g_V^2 C_V(W) + g_A^2 C_A(W) + g_V g_A C_{VA}(W)$ +- **Forbidden decays:** Nonunique transitions are sensitive to $g_A$ → basis for SSM +- **sNME:** Small relativistic vector NME — critical for precision but hard to compute + +## Related + +- [[gA-extraction/01-ssm-algorithm]] — How to extract $g_A$ from shape factor +- [[corrections/06-recoil]] — Weak magnetism, induced currents, recoil effects diff --git a/docs/refs/1967_Sirlin_general_properties_of_em_corrections/summary.md b/docs/refs/electromagnetic/1967_Sirlin_general_properties_of_em_corrections/summary.md similarity index 100% rename from docs/refs/1967_Sirlin_general_properties_of_em_corrections/summary.md rename to docs/refs/electromagnetic/1967_Sirlin_general_properties_of_em_corrections/summary.md diff --git a/docs/refs/1986_Sirlin_verification_of_CVC/summary.md b/docs/refs/electromagnetic/1986_Sirlin_verification_of_CVC/summary.md similarity index 100% rename from docs/refs/1986_Sirlin_verification_of_CVC/summary.md rename to docs/refs/electromagnetic/1986_Sirlin_verification_of_CVC/summary.md diff --git a/docs/refs/1987_Sirlin_remarks_on_OZa2_corrections/summary.md b/docs/refs/electromagnetic/1987_Sirlin_remarks_on_OZa2_corrections/summary.md similarity index 100% rename from docs/refs/1987_Sirlin_remarks_on_OZa2_corrections/summary.md rename to docs/refs/electromagnetic/1987_Sirlin_remarks_on_OZa2_corrections/summary.md diff --git a/docs/refs/2011_Sirlin_radiative_correction_to_nu_spectrum/summary.md b/docs/refs/electromagnetic/2011_Sirlin_radiative_correction_to_nu_spectrum/summary.md similarity index 100% rename from docs/refs/2011_Sirlin_radiative_correction_to_nu_spectrum/summary.md rename to docs/refs/electromagnetic/2011_Sirlin_radiative_correction_to_nu_spectrum/summary.md diff --git a/docs/refs/2017_Hayen_analytical_beta_shape/references.bib b/docs/refs/electromagnetic/2017_Hayen_analytical_beta_shape/references.bib similarity index 100% rename from docs/refs/2017_Hayen_analytical_beta_shape/references.bib rename to docs/refs/electromagnetic/2017_Hayen_analytical_beta_shape/references.bib diff --git a/docs/refs/2017_Hayen_analytical_beta_shape/summary.md b/docs/refs/electromagnetic/2017_Hayen_analytical_beta_shape/summary.md similarity index 100% rename from docs/refs/2017_Hayen_analytical_beta_shape/summary.md rename to docs/refs/electromagnetic/2017_Hayen_analytical_beta_shape/summary.md diff --git a/docs/refs/2019_Hayen_first_forbidden_transitions/summary.md b/docs/refs/forbidden-decay/2019_Hayen_first_forbidden_transitions/summary.md similarity index 100% rename from docs/refs/2019_Hayen_first_forbidden_transitions/summary.md rename to docs/refs/forbidden-decay/2019_Hayen_first_forbidden_transitions/summary.md diff --git a/docs/refs/2024_Paulsen_T99_beta_spectrum/summary.md b/docs/refs/forbidden-decay/2024_Paulsen_T99_beta_spectrum/summary.md similarity index 100% rename from docs/refs/2024_Paulsen_T99_beta_spectrum/summary.md rename to docs/refs/forbidden-decay/2024_Paulsen_T99_beta_spectrum/summary.md diff --git a/docs/refs/2025_Song_tc99_measurement_for_g_v/summary.md b/docs/refs/forbidden-decay/2025_Song_tc99_measurement_for_g_v/summary.md similarity index 100% rename from docs/refs/2025_Song_tc99_measurement_for_g_v/summary.md rename to docs/refs/forbidden-decay/2025_Song_tc99_measurement_for_g_v/summary.md diff --git a/docs/refs/radiative_spec.md b/docs/refs/radiative_spec.md deleted file mode 100644 index 4ad6bca..0000000 --- a/docs/refs/radiative_spec.md +++ /dev/null @@ -1,363 +0,0 @@ -# Algorithm for Radiative Corrections to Beta-Decay Spectra - -## 1. Physical Overview - -The radiatively corrected beta-decay spectrum takes the form: - -\[\frac{d\Gamma}{dW} = \frac{d\Gamma_0}{dW} \cdot R(W, W_0)\] - -where \(d\Gamma_0/dW\) is the uncorrected spectrum and \(R(W, W_0)\) is a multiplicative correction factor. The correction is organized by powers of the fine-structure constant \(\alpha\) and the nuclear charge \(Z\): - -\[R(W, W_0) = 1 + \delta^{(1)}(W, W_0) + \delta^{(2)}(Z, W) + \delta^{(3)}(Z, W) + \cdots\] - -**Important distinction:** Inner radiative corrections (\(\Delta_R^V\), \(\Delta_R^A\)) are energy-independent constants absorbed into effective coupling constants. They are **not** part of the spectrum shape calculation. Only the outer (energy-dependent) correction \(\delta_R(W, W_0)\) modulates the spectral shape. - ---- - -## 2. Input Parameters - -| Symbol | Description | Units | -|--------|-------------|-------| -| \(W\) | Total electron energy | \(m_e c^2\) | -| \(W_0\) | Endpoint energy | \(m_e c^2\) | -| \(Z\) | Nuclear charge of daughter | dimensionless | -| \(m_p\) | Proton mass | \(m_e c^2\) (\(\approx 1836.15\)) | -| \(\alpha\) | Fine-structure constant | \(\approx 1/137.036\) | - -Derived quantities: -- Momentum: \(p = \sqrt{W^2 - 1}\) -- Velocity: \(\beta = p / W\) -- Antineutrino energy (for neutrino spectrum corrections): \(\hat{W} = W_0 - W\) - ---- - -## 3. Order \(\alpha\): The Universal Sirlin Function \(g(W, W_0)\) - -### 3.1 Mathematical Form - -The dominant correction, derived in Sirlin (1967) and refined in Sirlin & Zucchini (1986) and Sirlin (1987), is universal—identical for Fermi and Gamow-Teller transitions, electrons and positrons: - -\[g(W, W_0) = 3\ln\left(\frac{m_p}{m_e}\right) - \frac{3}{4} + 4\left(\frac{\tanh^{-1}\beta}{\beta} - 1\right)\left[\frac{W_0 - W}{3W} - \frac{3}{2} + \ln\left(\frac{2(W_0 - W)}{m_e}\right)\right] + \frac{4}{\beta}L\left(\frac{2\beta}{1+\beta}\right) + \frac{\tanh^{-1}\beta}{\beta}\left[2(1+\beta^2) + \frac{(W_0-W)^2}{6W^2} - 4\tanh^{-1}\beta\right]\] - -The correction to the spectrum is: - -\[\delta^{(1)}(W, W_0) = \frac{\alpha}{2\pi} \, g(W, W_0)\] - -### 3.2 The Dilogarithm (Spence Function) - -The function \(L(x)\) appearing above is the Spence function (dilogarithm): - -\[L(x) = \int_0^x \frac{\ln(1-t)}{t} dt = -\text{Li}_2(-x/(1-x)) \quad \text{or equivalently} \quad L(x) = -\text{Li}_2(x)\] - -Using the standard dilogarithm \(\text{Li}_2(x) = -\int_0^x \frac{\ln(1-t)}{t} dt = \sum_{k=1}^\infty \frac{x^k}{k^2}\): - -\[L\left(\frac{2\beta}{1+\beta}\right) = -\text{Li}_2\left(\frac{2\beta}{1+\beta}\right)\] - -### 3.3 Numerical Stability Considerations - -**Inverse hyperbolic tangent:** - -\[\tanh^{-1}\beta = \frac{1}{2}\ln\left(\frac{1+\beta}{1-\beta}\right)\] - -When \(\beta \to 1\) (ultra-relativistic limit), compute using the identity: - -\[\tanh^{-1}\beta = \ln\left(\frac{W+p}{m_e}\right) = \ln(W + p) \quad (\text{in } m_e \text{ units where } m_e = 1)\] - -For \(\beta \to 0\) (non-relativistic limit), use the Taylor expansion: - -\[\frac{\tanh^{-1}\beta}{\beta} = 1 + \frac{\beta^2}{3} + \frac{\beta^4}{5} + \cdots\] - -Avoid computing \(\tanh^{-1}\beta\) directly when \(\beta\) is very small. - -**Terms containing \(1/\beta\):** When \(\beta < \epsilon\) (e.g., \(\epsilon = 10^{-6}\)), evaluate the expression using a small-\(\beta\) series expansion to avoid division by zero. - -**Dilogarithm evaluation:** Use a robust numerical implementation. For \(x > 1\), apply the identity: - -\[\text{Li}_2(x) = \frac{\pi^2}{3} - \frac{1}{2}\ln^2(x) - \text{Li}_2(1/x) - i\pi\ln(x) \quad (x > 1)\] - -The imaginary part cancels in the real-valued total correction. - -### 3.4 Endpoint Behavior and Soft-Photon Resummation - -**Critical observation:** The function \(g(W, W_0)\) contains a term proportional to \(\ln(W_0 - W)\), which diverges as \(W \to W_0\). This divergence is physical—arising from the emission of arbitrarily soft real photons—and the total decay rate integrated over all photon energies remains finite. - -For the differential spectrum shape, the leading logarithmic behavior near the endpoint is: - -\[g(W, W_0) \sim 4\left(\frac{\tanh^{-1}\beta}{\beta} - 1\right) \ln(W_0 - W) + \text{regular terms}\] - -**Resummation prescription (Sirlin 1987, 2011):** Replace the logarithmically divergent term with an exponentiated form: - -\[\ln(W_0 - W) \cdot t(\beta) \quad \longrightarrow \quad (W_0 - W)^{t(\beta)} - 1\] - -where the exponent is: - -\[t(\beta) = \frac{2\alpha}{\pi}\left[\frac{\tanh^{-1}\beta}{\beta} - 1\right]\] - -**Implementation strategy:** - -1. Compute the full \(g(W, W_0)\) including the \(\ln(W_0 - W)\) term. -2. For \(W_0 - W < \Delta_{\text{cut}}\) (suggested \(\Delta_{\text{cut}} \sim 10^{-3}\) in units of \(m_e c^2\)), extract the coefficient of \(\ln(W_0 - W)\): - - \[C(\beta) = 4\left(\frac{\tanh^{-1}\beta}{\beta} - 1\right)\] - -3. Replace \(C(\beta) \cdot \ln(W_0 - W)\) with \(C(\beta) \cdot \frac{(W_0-W)^{t(\beta)} - 1}{t(\beta)}\) using: - - \[\lim_{t \to 0} \frac{x^t - 1}{t} = \ln x\] - -4. For energies well below the endpoint (\(W_0 - W \gg \Delta_{\text{cut}}\)), no resummation is needed and the original logarithmic form suffices. - -**For low endpoint energies** (e.g., tritium with \(W_0 \approx 1.02\)), the endpoint resummation has negligible effect and can be omitted for simplicity. - ---- - -## 4. Order \(Z\alpha^2\): Finite Nuclear Size Corrections - -### 4.1 Structure - -The \(O(Z\alpha^2)\) correction, derived in Sirlin (1987), depends on the nuclear charge distribution: - -\[\delta^{(2)}(Z, W) = Z\alpha^2 \sum_{i=1}^4 \Delta_i(W)\] - -The four contributions \(\Delta_i\) arise from distinct classes of Feynman diagrams (see Sirlin 1987, Sec. 2). The dominant energy dependence comes from \(\Delta_1^0(W) + \Delta_4(W)\), for which the relativistic limit gives: - -\[\Delta_1^0(W) + \Delta_4(W) = \ln\left(\frac{M}{m_e}\right) - \frac{5}{12}\ln\left(\frac{2W}{m_e}\right) + \frac{43}{18}\] - -where \(M\) is the nucleon mass (use proton mass \(m_p\)). - -### 4.2 Nuclear-Model-Dependent Part - -The finite-nuclear-size contribution \(\Delta^F = \Delta_1^F + \Delta_2 + \Delta_3\) depends on the nuclear charge distribution. For the **modified Gaussian model** (Sirlin 1987, Eq. 10): - -\[\Delta^F(Z) = \ln\left(\frac{\Lambda}{M}\right) - \kappa_1(Z) + O(\Lambda/M)\] - -where: -- \(\Lambda = \sqrt{6}/a\), with \(a\) the rms charge radius -- \(M/\Lambda = r_0 A^{1/3} / 0.665\) (using Wilkinson's \(r_0\)) -- \(\kappa_1(Z) = \frac{1}{2}\left[\gamma + \ln(3/2k^2) + 2\alpha/(2+3\alpha)\right]\), with \(\gamma = 0.5772\), \(\alpha = (Z-2)/3\), and \(k^2 = \frac{3}{2}\frac{2+5\alpha}{2+3\alpha}\) - -### 4.3 Practical Simplification - -For light and medium nuclei (\(Z \lesssim 30\)), the total \(O(Z\alpha^2)\) correction can be tabulated once per isotope as an energy-independent shift (the energy dependence from \(\Delta_1^0 + \Delta_4\) is small compared to the order \(\alpha\) term). **For spectrum shape calculations:** - -1. Calculate the energy-dependent part \(\Delta_1^0(W) + \Delta_4(W)\) using the relativistic formula above. -2. Add the energy-independent nuclear-structure part \(\Delta^F(Z)\) from precomputed values (see Sirlin 1987, Table II). -3. If high precision is required for low-energy electrons \((p \ll m_e)\), use the full non-relativistic expressions from Sirlin 1987. - -For a **uniformly charged sphere**, \(\kappa_2 = \gamma - 4/3 + \ln(\sqrt{10}) = 0.395\), giving results within ~2% of the modified Gaussian model. - ---- - -## 5. Order \(Z^2\alpha^3\): Higher-Order Corrections - -The heuristic estimate from Sirlin (1986, 1987) is: - -\[\delta^{(3)}(Z, W) \approx Z^2\alpha^3 \left[a\ln\left(\frac{\Lambda}{W}\right) + b f(W) + \frac{4\pi}{3}g(W) - 0.649\ln(2W_0)\right]\] - -with: -\[a = \frac{\pi}{3} - \frac{\sqrt{3}}{2}, \quad b = \frac{4}{3\pi^4}(\pi^2 - \gamma_E) - \frac{\pi^2}{18}\] - -And: -\[f(W) = \ln(2W_0/m_e) - 5/6\] -\[g(W) = \frac{1}{2}\left[\ln^2(R m_e) - \ln^2(2W/m_e)\right] + \frac{5}{4}\ln(2RW)\] - -where \(R = \sqrt{5/4}\,a\) and \(a\) is the rms charge radius. - -**Practical note:** For \(Z \lesssim 30\), \(\delta^{(3)} < 0.01\%\) and can typically be neglected for spectrum shape. For superallowed Fermi transitions with heavy nuclei, this correction contributes ~0.05–0.12% to the total correction (Sirlin 1987, Table III). - ---- - -## 6. Neutrino Spectrum Correction - -For the antineutrino/neutrino spectrum (Sirlin 2011, Eq. 11), define: - -\[\hat{W} \equiv W_0 - W_\nu, \quad \hat{p} \equiv \sqrt{\hat{W}^2 - 1}, \quad \hat{\beta} \equiv \hat{p}/\hat{W}\] - -The correction function \(h(\hat{W}, W_0)\) is: - -\[h(\hat{W}, W_0) = 3\ln\left(\frac{m_p}{m_e}\right) + \frac{23}{4} - \frac{8}{\hat{\beta}}\text{Li}_2\left(\frac{2\hat{\beta}}{1+\hat{\beta}}\right) + 8\left(\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}} - 1\right)\ln\left(\frac{2\hat{W}\hat{\beta}}{m_e}\right) + 4\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}}\left[\frac{7+3\hat{\beta}^2}{8} - 2\tanh^{-1}\hat{\beta}\right]\] - -The neutrino spectrum correction is: - -\[\delta_\nu(\hat{W}, W_0) = \frac{\alpha}{2\pi} \, h(\hat{W}, W_0)\] - -**Key difference from the electron correction:** The neutrino correction \(h\) is finite in the \(m_e \to 0\) limit, while the electron correction \(g\) diverges logarithmically. The \(m_e \to 0\) limit of \(h\) is particularly simple: - -\[h(\hat{W}, W_0) \xrightarrow{m_e \to 0} 3\ln\left(\frac{m_p}{2W_0}\right) + \frac{23}{4} - \frac{4\pi^2}{3} - 3\ln\left(\frac{W_\nu}{W_0}\right)\] - -This finite limit is explained theoretically by the Kinoshita-Lee-Nauenberg theorem: summing over the degenerate collinear \(e^-\gamma\) final states (as is done when integrating over electron momenta to obtain the neutrino spectrum) cancels the mass singularities (Sirlin 2011, Sec. II.F). - ---- - -## 7. Conversion Between Electron and Neutrino Spectra - -Given the electron spectrum \(f_e(W, W_0)\) for a specific decay, the corresponding neutrino spectrum including radiative corrections is (Sirlin 2011, Eq. 19): - -\[f_\nu(W_\nu, W_0) = f_e(\hat{W}, W_0) \left[1 + \frac{\alpha}{2\pi}\left(h(\hat{W}, W_0) - g(\hat{W}, W_0)\right)\right]\] - -where \(\hat{W} = W_0 - W_\nu\). - -The difference function is given explicitly in Sirlin 2011, Eq. 20: - -\[h(\hat{W}, W_0) - g(\hat{W}, W_0) = \frac{13}{2} - \frac{4}{\hat{\beta}}\text{Li}_2\left(\frac{2\hat{\beta}}{1+\hat{\beta}}\right) + 4\left(\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}} - 1\right) \times \Bigg[\frac{11}{6} + \ln\left(\frac{2W_0}{m_e}\right) + 2\ln\hat{\beta} + 2\ln\left(\frac{W_\nu}{W_0}\right) - \ln\left(\frac{W_\nu}{W_0}\right) - \frac{1}{3W_\nu/W_0}\Bigg] + \frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}}\left[\frac{3-\hat{\beta}^2}{2} - 4\tanh^{-1}\hat{\beta} - \frac{(W_\nu/W_0)^2}{6(1-W_\nu/W_0)^2}\right]\] - -This conversion is important for reactor neutrino oscillation studies. - ---- - -## 8. Complete Calculation Algorithm - -### 8.1 Full Correction Factor for Electron Spectrum - -``` -Algorithm: RadiativeCorrection(W, W0, Z) -Input: W — array of electron total energies [m_e c^2] - W0 — endpoint energy [m_e c^2] - Z — nuclear charge of daughter nucleus -Output: R — multiplicative correction factor at each W - -Constants: - alpha = 1/137.035999084 - m_p = 938.272 / 0.51099895 # proton mass in m_e units - gamma_E = 0.5772156649 - pi = 3.141592653589793 - -For each W_i in W: - 1. Compute derived quantities: - p = sqrt(W_i^2 - 1) - beta = p / W_i - - 2. Order α correction (Section 3): - a. Calculate tanh^{-1}(beta) / beta - If beta < 1e-6: use Taylor series 1 + beta^2/3 + beta^4/5 - Else: tanh_inv_beta_beta = 0.5 * ln((1+beta)/(1-beta)) / beta - - b. Calculate dilogarithm L(2*beta/(1+beta)) = -Li_2(2*beta/(1+beta)) - Use robust numerical implementation - - c. Compute g(W_i, W0) per Eq. (20b) of Sirlin 1967 - - d. If use_endpoint_resummation and (W0 - W_i) < delta_cut: - t_beta = (2*alpha/pi) * (tanh_inv_beta_beta - 1) - C_beta = 4 * (tanh_inv_beta_beta - 1) - Replace: C_beta * ln(W0 - W_i) - -> C_beta * ((W0 - W_i)^t_beta - 1) / t_beta - - e. delta_1 = (alpha / (2*pi)) * g(W_i, W0) - - 3. Order Zα² correction (Section 4): - a. Energy-dependent part: - Delta_1_4 = ln(m_p) - (5/12)*ln(2*W_i) + 43/18 - - b. Energy-independent nuclear part: - Look up Delta_F for the given (Z, A) from precomputed table - (Use Sirlin 1987 Table II or compute from nuclear model) - - c. delta_2 = Z * alpha^2 * (Delta_1_4 + Delta_F) - - 4. Order Z²α³ correction (Section 5, optional): - For Z >= 20 or precision better than 0.01% required: - Compute delta_3 from Sirlin heuristic formula - Else: - delta_3 = 0 - - 5. Combine corrections: - R_i = 1 + delta_1 + delta_2 + delta_3 - - Return R array -``` - -### 8.2 Neutrino Spectrum Correction - -``` -Algorithm: NeutrinoRadiativeCorrection(W_nu, W0, Z) -Input: W_nu — array of neutrino energies [m_e c^2] - W0 — endpoint energy [m_e c^2] - Z — nuclear charge -Output: R_nu — multiplicative correction factor - -For each W_nu in array: - W_hat = W0 - W_nu - p_hat = sqrt(W_hat^2 - 1) - beta_hat = p_hat / W_hat - - Compute h(W_hat, W0) per Sirlin 2011 Eq. (11) - (Use same numerical stability measures as for g-function) - - R_nu_i = 1 + (alpha/(2*pi)) * h(W_hat, W0) -``` - -### 8.3 Conversion Algorithm (Electron → Neutrino Spectrum) - -``` -Algorithm: ElectronToNeutrinoConversion(W_nu, W0, Z, f_e) -Input: W_nu — neutrino energies - W0 — endpoint - Z — nuclear charge - f_e — function returning electron spectrum values -Output: f_nu — neutrino spectrum values - -For each W_nu in array: - W_hat = W0 - W_nu - - # Evaluate electron spectrum at W_hat - f_e_val = f_e(W_hat) - - # Compute difference correction - Compute delta_diff = (alpha/(2*pi)) * [h(W_hat, W0) - g(W_hat, W0)] - per Sirlin 2011 Eq. (20) - - f_nu_i = f_e_val * (1 + delta_diff) -``` - ---- - -## 9. Physical Interpretation and Validation - -### 9.1 Dominant Terms - -The leading \(O(\alpha)\) correction is the **universal Sirlin function** \(g(W, W_0)\). Its dominant contribution comes from the logarithm: - -\[3\ln\left(\frac{m_p}{m_e}\right) \approx 15.4\] - -multiplied by \(\alpha/2\pi \approx 1.16 \times 10^{-3}\), giving a typical ~1.8% correction. The energy-dependent terms modulate this by ~±1% across the spectrum. - -### 9.2 Sign Conventions - -- The correction factor \(R(W, W_0)\) **multiplies** the uncorrected spectrum. -- \(R > 1\) means the correction increases the transition probability at that energy. -- For the electron spectrum: the correction typically **decreases** the probability at high \(W\) (\(W/W_0 \gtrsim 0.44\)) and **increases** it at low \(W\) (Sirlin 2011, Sec. III). -- For the neutrino spectrum: the correction is **positive and monotonically increasing** with \(W_\nu\) (Sirlin 2011, Sec. II.G). - -### 9.3 Consistency Checks - -1. **Lifetime check:** Integrate the corrected and uncorrected spectra: - - \[\int_1^{W_0} \frac{d\Gamma_0}{dW} R(W, W_0) dW\] - - The relative correction to the lifetime should be finite (the logarithmic divergence at the endpoint integrable over the spectrum). - -2. **Electron-neutrino consistency:** The lifetime calculated from the corrected electron spectrum must equal that from the corrected neutrino spectrum. - -3. **Zero-charge limit:** For \(Z = 0\), \(R(W, W_0) \to 1 + \frac{\alpha}{2\pi}g(W, W_0)\). - -4. **Low-energy limit:** As \(W \to 1\) (non-relativistic), the correction should approach the values tabulated in Sirlin 1967. - ---- - -## 10. Summary of Required Special Functions - -| Function | Definition | Numerical Source | -|----------|------------|------------------| -| \(\text{Li}_2(x)\) | \(\sum_{k=1}^\infty x^k/k^2\) | Standard library (e.g., `scipy.special.spence` in Python, `gsl_sf_dilog` in C) | -| \(\tanh^{-1}(x)\) | \(\frac{1}{2}\ln\frac{1+x}{1-x}\) | Built-in or \(\text{arctanh}(x)\) | -| \(\gamma_E\) | Euler-Mascheroni constant | 0.57721566490153286 | - ---- - -## References - -1. A. Sirlin, Phys. Rev. 164, 1767 (1967) — Universal correction function \(g(E, E_m, m)\) and proof of model independence. -2. A. Sirlin and R. Zucchini, Phys. Rev. Lett. 57, 1994 (1986) — Analytic \(O(Z\alpha^2)\) calculation resolving CVC discrepancy. -3. A. Sirlin, Phys. Rev. D 35, 3420 (1987) — Refined finite-nuclear-size effects with realistic charge distributions. -4. A. Sirlin, Phys. Rev. D 84, 014021 (2011) — Neutrino spectrum radiative correction and conversion formula. -5. T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959) — Mass singularity cancellation theorems.