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Charge exchange reactions with central and tensor interactions and nuclear matrix elements associated with double-beta-decay neutrinos and astro neutrinos

This paper proposes and demonstrates a novel experimental method to extract Gamow-Teller and tensor nuclear matrix elements for neutrinoless double-beta decays and inverse-beta decays by analyzing the momentum-transfer dependence of charge exchange reaction cross-sections while explicitly accounting for both central and tensor interactions.

Original authors: Hiroyasu Ejiri, Yoritaka Iwata

Published 2026-10-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hiroyasu Ejiri, Yoritaka Iwata

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the heart of matter, where protons and neutrons bind together to form atomic nuclei, a subtle and complex struggle plays out. This is the domain of nuclear physics, a field dedicated to understanding how these tiny building blocks interact and transform. One of the most intriguing questions in this realm concerns the behavior of neutrinos, ghostly particles that rarely interact with anything at all. Scientists are trying to determine if these particles are their own antiparticles and to measure their mass, properties that could rewrite our understanding of the universe. To do this, they study specific nuclear reactions, such as double beta decay, where a nucleus spontaneously changes its identity by emitting two electrons. The speed and likelihood of this transformation depend entirely on a hidden value called a nuclear matrix element. Think of this element as a measure of how easily the nucleus can rearrange itself during the reaction; if scientists cannot calculate this value accurately, they cannot interpret the neutrino data correctly. The challenge is that these nuclei are crowded, chaotic systems where the forces between particles are incredibly difficult to model with pure theory alone.

For decades, researchers have relied on a technique called charge exchange reactions to peek inside these nuclei. In a typical experiment, a beam of particles is fired at a target nucleus, causing it to swap a neutron for a proton or vice versa. By measuring how the beam scatters, scientists can infer the internal structure of the nucleus. However, a long-standing assumption in this field has created a blind spot. Scientists have traditionally assumed that the force driving this swap is purely "central," meaning it acts like a simple push or pull between particles. They believed a more complex force, known as the tensor interaction, was too weak to matter in these measurements. This tensor force is a bit like a directional grip that depends on the orientation of the particles, rather than just their distance. The prevailing view was that ignoring this grip would not significantly alter the results, allowing researchers to extract the necessary nuclear values with confidence.

A new study challenges this assumption, revealing that the tensor force is not a negligible background noise but a significant player that distorts previous measurements. The researchers, working with data from experiments at the Research Center for Nuclear Physics in Osaka, Japan, developed a method to separate the effects of the central force from the tensor force. They realized that while the central force changes slowly as the momentum of the collision increases, the tensor force grows rapidly. By carefully analyzing how the scattering patterns changed as the angle of the collision shifted, they could isolate the contribution of the tensor interaction. This approach allowed them to extract two distinct values for the first time: the strength of the standard central interaction and the strength of the tensor interaction for specific nuclear states.

The team applied this refined method to several nuclei that are critical for neutrino research, including Gallium-71, which is used in solar neutrino studies, and Molybdenum-100 and Tellurium-128, which are central to the search for neutrinoless double beta decay. In the case of Gallium-71, the results were particularly revealing. When the researchers corrected for the tensor force, they found that the nuclear matrix elements for certain excited states were significantly smaller than previously thought. This correction is vital because it directly impacts the interpretation of the "solar neutrino anomaly," a long-standing puzzle where the number of detected neutrinos from the sun did not match theoretical predictions. The study shows that the discrepancy is not caused by a misunderstanding of how the Gallium nucleus responds to neutrinos. Instead, the nuclear response is consistent with standard physics, suggesting that the anomaly likely points to new physics, such as the existence of sterile neutrinos, rather than a flaw in the nuclear model.

The findings also shed light on the structure of the nuclei themselves. The study confirmed that the tensor force is most influential in transitions where the internal arrangement of protons and neutrons is complex and "forbidden" by simple rules. In these cases, the tensor force can be just as strong as the central force, fundamentally altering the outcome of the reaction. For other nuclei, like Molybdenum-100, the tensor contribution was found to be much smaller, confirming that the old method was reasonably accurate for those specific cases. This distinction is crucial because it means scientists can no longer apply a single, simplified rule to all nuclear reactions. They must now account for the specific shape and orientation of the forces at play in each unique nucleus.

By providing a way to measure these forces directly from experimental data rather than relying solely on theoretical guesses, this work offers a more reliable foundation for future neutrino experiments. The researchers demonstrated that ignoring the tensor force could lead to errors of up to twenty percent in the calculated values for certain nuclear states. This level of uncertainty is too high for the precision required in modern neutrino physics. The study does not claim to have solved the mystery of the neutrino's mass or nature, but it has removed a major source of error from the equation. It establishes that the nuclear matrix elements, the key to unlocking neutrino secrets, must be derived with a more nuanced understanding of the forces inside the nucleus. As experiments continue to probe the deepest questions of the universe, this refined view of the atomic nucleus ensures that the answers scientists find are truly about the neutrinos, and not an artifact of an incomplete model.

Technical Summary: Charge Exchange Reactions with Central and Tensor Interactions and Nuclear Matrix Elements

Problem Statement
Nuclear matrix elements (NMEs) are fundamental for determining neutrino properties, specifically the Majorana mass and potential right-handed weak interactions, via neutrinoless double beta decays (DBDs). They are also critical for understanding astrophysical neutrino (astro- $\nu$ ) production and oscillations through inverse-beta decays (IBDs). However, theoretical evaluations of NMEs for medium-heavy nuclei ( $A \approx 100$ ) are highly sensitive to nucleonic and non-nucleonic correlations, leading to significant scatter (factors of 5–10) depending on the model and parameters used.

Experimentally, charge exchange reactions (CERs), specifically $(^3\text{He}, t)$ reactions at intermediate energies, have been used to probe Gamow-Teller (GT) NMEs. A critical limitation in previous experimental analyses is the assumption that CER cross-sections are dominated solely by the central interaction ( $V_C$ ), neglecting the tensor interaction ( $V_T$ ). While $V_T$ is smaller than $V_C$ , its interference with the GT amplitude becomes appreciable unless the tensor NME ( $M(T)$ ) is negligible compared to the GT NME ( $M(GT)$ ). Furthermore, $V_T$ increases steeply with momentum transfer ( $q$ ), whereas $V_C$ decreases slowly. Consequently, standard analyses that extract NMEs solely from forward-angle ( $q \approx 0$ ) cross-sections may contain uncorrected tensor contributions, leading to inaccuracies, particularly for states where $M(GT)$ is small.

Methodology
The authors propose an experimental method to extract both GT and tensor NMEs by analyzing the momentum transfer ( $q$ ) dependence of CER cross-sections. The core of the methodology relies on the distinct $q$ -dependence of the central and tensor interactions:

Interaction Dynamics: The effective nuclear interaction is modeled as $V_{\text{eff}} = V_C + V_T$ . The tensor interaction, arising primarily from pion exchange, increases rapidly with $q$ , reaching a maximum around $q \approx 140$ MeV/c, while the central interaction decreases slowly.
Cross-Section Formulation: The differential cross-section $D(110, q)$ for GT transitions is expressed as a product of kinematic factors, a distortion factor, and a term containing the reduced width $B(110)$ . Crucially, $B(110)$ is modified by a tensor correction factor $f(q) = \epsilon \frac{M(T)}{M(C)} \frac{V_T(q)}{V_C(q)}$ .
Extraction Strategy: Instead of assuming $f(q)=0$ $f (q) = 0$ at $q \approx 0$ $q \approx 0$ , the authors measure the angular distribution (which maps to $q$ $q$ ) in the forward region ( $\theta = 0\text{--}2^\circ$ $θ = 0 - 2^{\circ}$ ). By plotting the square root of the ratio of the cross-section of a target state ( $i$ $i$ ) to a reference spin-stretched quasi-particle (SQP) state ( $r$ $r$ ) against the ratio $V_T(q)/V_C(q)$ $V_{T} (q) / V_{C} (q)$ , they can separate the contributions:
- The intercept at $V_T/V_C = 0$ yields the pure GT NME ratio $|M_i(GT)|/|M_r(GT)|$ .
- The gradient of the linear fit yields the tensor contribution ratio $\epsilon M_i(T)/M_i(GT)$ .
Reference Standard: A reference GT state with a large $M(GT)$ and negligible tensor contribution (e.g., the ground state transition in $^{71}\text{Ga}$ , $^{100}\text{Mo}$ , or $^{128}\text{Te}$ ) is used to normalize the data.

Key Contributions and Results
The study applies this method to CERs on nuclei relevant to solar neutrino anomalies and DBD/IBD studies: $^{71}\text{Ga}$ , $^{100}\text{Mo}$ , and $^{128}\text{Te}$ .

$^{71}\text{Ga}$ (Solar Neutrinos/Ga Anomaly):
- The ground state transition ( $3/2^- \to 1/2^-$ ) was used as the reference.
- Excited states at 175 keV ( $5/2^-$ ) and 500 keV ( $3/2^-$ ) in $^{71}\text{Ge}$ showed a clear deviation from the kinematic distribution, indicating significant tensor contributions.
- The extracted GT NMEs for these excited states were found to be significantly smaller ( $M(GT) \approx 0.06$ and $0.18$) than values derived without tensor correction. The tensor NMEs were found to be comparable to the GT NMEs, attributed to $l$ -forbidden transitions ( $1f_{5/2} \to 2p_{3/2}$ ).
- Implication for Ga Anomaly: The contribution of the 1st and 2nd excited states to the solar neutrino cross-section was re-evaluated. The ratio of their contribution to the ground state was found to be $K = 1.028 \pm 0.003$ . This result is consistent with previous values and suggests that the "Ga anomaly" is not caused by an incorrect nuclear response (NME) but supports the hypothesis of oscillation to a sterile neutrino.
$^{100}\text{Mo}$ and $^{128}\text{Te}$ (DBD/IBD):
- For $^{100}\text{Mo} \to ^{100}\text{Tc}$ and $^{128}\text{Te} \to ^{128}\text{I}$ , the ground state transitions are spin-stretched SQP transitions with negligible tensor contributions.
- Excited states in $^{100}\text{Tc}$ (838 keV, 1339 keV) and $^{128}\text{I}$ (1153 keV, 1222 keV) were analyzed.
- The 1339 keV state in $^{100}\text{Tc}$ showed no appreciable tensor contribution ( $M(T) \le 0.03$ ), confirming it is primarily a GT transition.
- The 838 keV state in $^{100}\text{Tc}$ showed a small tensor contribution.
- The strongly excited 1153 keV and 1222 keV states in $^{128}\text{I}$ showed no appreciable tensor contribution, indicating they are mainly GT transitions.
Comparison with Theory:
- The experimental NMEs were compared with Interacting Shell Model (ISM) calculations using JUN45 and jj44b codes.
- Theoretical models failed to reproduce both the excitation energies and the specific GT/T NME ratios observed experimentally.
- The authors note that ISM calculations often lack explicit $\Delta$ -resonance (non-nucleonic) effects, which are known to quench axial-vector NMEs. Even with quenching factors, the models did not fully align with the experimental data for the excited states.

Significance and Claims
The paper claims to present the first experimental method to explicitly extract both GT and tensor NMEs from CER data by correcting for the tensor interaction effect using the $q$ -dependence of the cross-section.

Correction of Previous Assumptions: The authors demonstrate that assuming $f(q)=0$ at forward angles can lead to overestimation of GT NMEs for states with small $M(GT)$ , where the tensor interference is relatively large. The correction is modest for strong GT states but critical for weak transitions.
Experimental vs. Theoretical Reliability: The work emphasizes that experimental NMEs derived via this method are free from the uncertainties inherent in theoretical nuclear models (such as the choice of interaction or the treatment of non-nucleonic correlations).
Impact on Neutrino Physics: By providing accurate, experimentally derived NMEs, the study refines the interpretation of DBD and IBD experiments. Specifically, it strengthens the case that the Ga anomaly is a neutrino physics phenomenon (sterile neutrino oscillation) rather than a nuclear structure issue.
Methodological Scope: The authors note that while the method relies on a reference SQP state, it is most reliable for states near the reference in excitation energy and mass number. For other states, rigorous Distorted Wave Born Approximation (DWBA) analyses of angular distributions are required.

In conclusion, the paper establishes that CERs with intermediate-energy projectiles are a realistic probe for disentangling central and tensor nuclear interactions, providing essential experimental benchmarks for neutrino physics that theoretical models currently struggle to reproduce accurately.

Technical Summary: Charge Exchange Reactions with Central and Tensor Interactions and Nuclear Matrix Elements

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