beta_shape_pnpi/docs/gA-extraction/02-ssnme.md

title, date, tags, status, aliases, related_notes

title	date	tags	status	aliases	related_notes
sNME — Small Relativistic Vector NME	2026-05-14	gA/extraction/ssnme beta-spectrum/forbidden nuclear-structure/NME	active	small relativistic vector nuclear matrix element sNME CVC vector current	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