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030ae3e4b9
- 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)
107 lines
5.0 KiB
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107 lines
5.0 KiB
Markdown
---
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title: "sNME — Small Relativistic Vector NME"
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date: 2026-05-14
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tags:
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- gA/extraction/ssnme
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- beta-spectrum/forbidden
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- nuclear-structure/NME
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status: active
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aliases: [small relativistic vector nuclear matrix element, sNME, CVC vector current]
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related_notes:
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- gA-extraction/01-ssm-algorithm
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- gA-extraction/03-tc99-case-study
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- nuclear-structure/01-shape-factors
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---
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# The Small Relativistic Vector NME (sNME) — Why It Matters
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## What is the sNME?
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In forbidden β decay, the relativistic vector current has two components:
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- **Large NME (l-NME):** Computed from the large components of nucleon wave functions. Reliably calculable in NSM.
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- **Small NME (sNME):** Couples the large and small components. **Extremely difficult** to compute because it involves contributions outside the valence shell.
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In a perfect many-body theory with infinite valence space, the sNME is related to the l-NME by the **CVC hypothesis**:
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$$\text{sNME} = \frac{1}{2} \left[ W_0 - (m_n - m_p) + E_C \right] \cdot \text{l-NME} \tag{1}$$
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where:
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- $W_0$ — endpoint energy in MeV
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- $m_n - m_p \approx 1.293$ MeV — neutron-proton mass difference
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- $E_C$ — Coulomb displacement energy (depends on nuclear charge distribution)
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- l-NME — large vector NME (computable in NSM)
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### Why is sNME important?
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Despite being "small" (typically $< 0.1$ in absolute units), the sNME can influence β-spectral shapes and half-lives **quite strongly** because:
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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.
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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.
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3. **Shape dependence:** Different sNME values produce measurably different spectral shapes, especially below 100 keV.
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## The CVC Value — An Idealization
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The CVC prediction (Eq. 1) is an **idealization** — it pertains to a perfect many-body theory. In practice:
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- NSM calculations with finite valence spaces give sNME $\approx 0$ (the valence space truncation kills it)
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- The CVC value serves as a **reference**, not a physical prediction
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## ESSM: sNME as a Fitting Parameter
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### The Enhanced Spectrum Shape Method (ESSM)
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Kumar et al. (2021) proposed using the sNME as an additional fitting parameter:
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1. **Fix** $g_A$ at a trial value
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2. **Vary** the sNME to match the **experimental partial half-life**
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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:
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- **sNME(c):** closer to the CVC value
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- **sNME(f):** further from the CVC value
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### Two-Solution Structure
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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:
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- The **sNME(c)** solution is theoretically preferred (closer to CVC)
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- The **sNME(f)** solution may also be physical — only experiment can decide
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### Application to $^{99}\text{Tc}$
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Song et al. (2025) applied ESSM to $^{99}\text{Tc}$ with three nuclear models:
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| Model | $g_A^{\text{eff}}$ | sNME (c) [fm³] | sNME (f) [fm³] | $\chi^2_\nu$ | p-value |
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|---|---|---|---|---|---|
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| **CloseCVC (CVC fixed)** | 1.1 | 0.0674 | — | 1.106 | 0.12 |
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| | 1.0 | 0.0681 | — | 1.244 | 0.004 |
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| | 1.0 | 0.0651 | — | 1.644 | <0.0001 |
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| **FarCVC (sNME fitted)** | 1.2 | −0.0698 | — | 1.066 | 0.22 |
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| | 1.0 | −0.0669 | — | 1.097 | 0.14 |
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| | 1.0 | −0.0683 | — | 1.095 | 0.14 |
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Key findings:
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- **Paulsen et al. (CloseCVC):** $g_A^{\text{eff}} = 1.526(92)$ — enhanced value
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- **Song et al. (FarCVC/ESSM):** $g_A^{\text{eff}} \approx 1.0$–1.2 — quenched value
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- The discrepancy arises from the **sNME treatment**, not from the data
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## sNME in Other Transitions
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Ramalho & Suhonen (2024) studied five second-forbidden nonunique decays:
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| Transition | $g_A$ Sensitivity | sNME Sensitivity |
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|---|---|---|
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| $^{60}\text{Fe} \to {}^{60}\text{Co}$ | Strong | Weak |
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| $^{94}\text{Nb} \to {}^{94}\text{Mo}$ | Strong (jj45pnb) | Weak |
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| $^{98}\text{Tc} \to {}^{98}\text{Ru}$ | Strong | Moderate (below 100 keV) |
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| $^{126}\text{Sn} \to {}^{126}\text{Sb}$ | Weak | **Strong** |
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| $^{129}\text{I} \to {}^{129}\text{Xe}$ | Weak | **Strong** |
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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.
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## References
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| Ref | Citation | Use |
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|---|---|---|
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| [ESSM] | Kumar, Srivastava, Suhonen, *Eur. Phys. J. A* **57**, 225 (2021) | ESSM formalism |
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| [Song2025] | Song et al., arXiv:2506.17544 (2025) | $^{99}$Tc with ESSM |
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| [Ramalho2024] | Ramalho, Suhonen, *Front. Phys.* **12**, 1455778 (2024) | sNME in five second-forbidden decays |
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| [Behrens] | Behrens & Bühring, *Electron Radial Wave Functions and Nuclear Beta Decay* (1982) | sNME formalism |
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