Two-neutrino $ββ$ decay to excited states at next-to-leading order

This paper calculates next-to-leading order nuclear matrix elements for two-neutrino double-beta decay to excited states in key isotopes using the nuclear shell model, finding that while NLO corrections are typically small, they can become significant due to leading-order cancellations, and that nuclear deformation and seniority structure critically influence the predicted half-lives.

The Gist

The Big Picture: A Rare Double-Double Event

Imagine a nucleus (the heart of an atom) as a busy dance floor. Usually, when a dancer (a neutron) wants to leave the floor, they swap places with a partner (a proton) and leave the room with a single exit ticket (an electron) and a ghostly whisper (an antineutrino). This is normal radioactive decay.

But sometimes, two dancers decide to leave at the exact same time. They swap places, and two electrons and two antineutrinos fly out together. This is called Double-Beta Decay.

It is an incredibly rare event. If you had a pile of atoms the size of a mountain, you might only see this happen once every few billion years. Because it's so slow, scientists call it the "slowest decay ever measured."

The Mystery: The "Excited" Dance Floor

Most scientists study what happens when the dance floor starts calm and ends calm (Ground State to Ground State). But this paper asks a different question: What happens if the dance floor ends up in a state of high energy or "excitement"?

The researchers looked at specific atoms (like Germanium, Selenium, and Xenon) where the final dance floor is wobbly and excited (an excited $0^+_2$ state). They wanted to calculate exactly how long it takes for this rare "double exit" to happen.

The Tools: The "Rulebook" and the "Calculator"

To predict how long this takes, you need two things:

1. A Rulebook (Hamiltonians): This describes how the dancers (protons and neutrons) interact with each other. The authors used several different rulebooks because, in nuclear physics, we aren't 100% sure which one is perfect.
2. A Calculator (The Shell Model): This is a super-computer simulation that tracks every single dancer's move to see if the double exit is possible.

The New Twist: "Next-to-Leading Order"

In the past, scientists used a "rough draft" version of the physics rules (Leading Order). This paper introduces a "revised draft" (Next-to-Leading Order or NLO).

Think of it like baking a cake:

• Leading Order: You mix flour, sugar, and eggs. You get a cake.
• Next-to-Leading Order: You realize you forgot the vanilla extract and a pinch of salt. You add them in.

Usually, adding the vanilla (the NLO corrections) only changes the taste by a tiny bit (less than 5%). However, the paper found a funny exception: In some specific cases, the main ingredients (the "Leading Order" part) canceled each other out perfectly, leaving a flat, tasteless batter. In those rare cases, adding the vanilla (NLO) became the most important part of the recipe, changing the result dramatically.

The Shape of the Dancers: Deformation

The researchers also looked at the shape of the atoms. Some atoms are perfectly round (spherical), while others are squashed like a rugby ball or even twisted like a pretzel (triaxial).

They discovered a simple rule: If the starting atom and the ending atom look very different (one is round, the other is squashed), the double-beta decay is very hard to do. It's like trying to do a complex dance move when your partner is wearing completely different shoes. The "shape difference" acts as a barrier, making the decay take even longer.

The Results: How Long Does It Take?

The team calculated the "half-life" (the time it takes for half the atoms to decay) for these excited states.

• The Good News: Their calculations are consistent with the very latest, tiny hints from experiments in Selenium-82.
• The Bad News: For most of the atoms they studied (like Germanium-76), the predicted time is still too long to be seen easily with current machines. The predicted time is often 100 times longer than the current experimental limits.
• The Uncertainty: Because we have different "Rulebooks" (Hamiltonians), the predictions vary wildly. For some atoms, the time could be 100 years or 10,000 years (in scientific terms, that's a huge difference!). This uncertainty comes mostly from not knowing exactly which "Rulebook" is the correct one.

Why Should We Care?

You might ask, "Why bother calculating something that takes a trillion years to happen?"

1. Neutrinoless Double-Beta Decay: There is a hypothetical, even rarer version of this decay where no antineutrinos are emitted. Finding this would prove that neutrinos are their own antiparticles and could explain why the universe is made of matter instead of anti-matter.
2. The Connection: To find the "neutrinoless" version, we first need to perfectly understand the "two-neutrino" version. This paper is like calibrating the microscope. If we get the math right for the common (but rare) version, we can trust our predictions for the ultra-rare version.

Summary in a Nutshell

The authors used super-computers to simulate a very rare atomic dance where two neutrons leave at once, leaving the atom in an "excited" state. They found that:

• New, more precise physics rules (NLO) usually don't change the result much, unless the main rules cancel each other out.
• If the starting and ending atoms have very different shapes, the dance is harder to perform.
• Their predictions are mostly consistent with new experimental hints but suggest that detecting this specific event will be very difficult in the near future.
• Getting these numbers right is crucial for the bigger hunt: finding the "neutrinoless" decay that could unlock the secrets of the universe.

Technical Summary

1. Problem Statement

The paper addresses the theoretical calculation of two-neutrino double-beta decay ($2\nu\beta\beta$) specifically for transitions from the ground state to the first-excited $0^+_2$ state ($0^+_{gs} \to 0^+_2$). While $2\nu\beta\beta$ decays to the ground state ($0^+_{gs} \to 0^+_{gs}$) have been measured in several nuclei, transitions to excited states are less understood theoretically and experimentally.

Key challenges identified include:

• Theoretical Overestimation: Standard many-body methods (like the Nuclear Shell Model, NSM) often overestimate Nuclear Matrix Elements (NMEs) for Gamow-Teller (GT) transitions, requiring phenomenological "quenching."
• Missing Physics: Traditional calculations often neglect higher-order weak currents and specific nuclear structure effects (like deformation and triaxiality) that significantly influence decay rates.
• Experimental Context: Recent experimental indications for $82\text{Se}$ and limits for other isotopes ($^{76}\text{Ge}$, $^{136}\text{Xe}$, etc.) require precise theoretical predictions to interpret data and constrain neutrinoless double-beta decay ($0\nu\beta\beta$) models, as the NMEs for both modes are correlated.

2. Methodology

The authors employ the Nuclear Shell Model (NSM) to calculate NMEs for $^{48}\text{Ca}$, $^{76}\text{Ge}$, $^{82}\text{Se}$, $^{124}\text{Sn}$, $^{130}\text{Te}$, and $^{136}\text{Xe}$.

A. Hamiltonians and Interactions

Multiple shell-model Hamiltonians were used for each mass region to assess theoretical uncertainty:

• $pf$ shell ($^{48}\text{Ca}$): KB3G, GXPF1A.
• $sdg$ shell ($^{76}\text{Ge}, ^{82}\text{Se}$): JJ4BB, GCN2850, JUN45, RG.
• $sdg$ shell ($^{124}\text{Sn}, ^{130}\text{Te}, ^{136}\text{Xe}$): GCN5082, QX.

B. Operator Formalism

The study improves upon leading-order (LO) calculations in two main ways:

1. Lepton Energy Expansion: Following Ref. [52], the lepton energy denominator is Taylor-expanded up to $m \le 2$. This introduces subleading NMEs ($M^{(-3)}_{GT}, M^{(-5)}_{GT}$) coupled with new phase-space factors (PSFs).
2. Next-to-Leading Order (NLO) $\chi$EFT: For the first time in these specific decays, NLO terms derived from Chiral Effective Field Theory ($\chi$EFT) are included. These terms account for:
   • Weak magnetism.
   • One-pion-exchange (OPE) diagrams.
   • The expansion includes terms proportional to $\Delta_0$ and $\Delta_2$, which combine LO and NLO OPE NMEs. Short-range NLO terms (dependent on unknown couplings) are omitted.

C. Quenching and Operators

Two GT operators were tested:

• Bare GT Operator ($\sigma_j$): Requires phenomenological quenching ($q_{\beta\beta}$ or $q_\beta$) to reproduce experimental ground-state half-lives.
• Effective GT Operator ($GT_{eff}$): Renormalized within the valence space (Ref. [41]), which implicitly captures some many-body correlations and two-nucleon currents.

D. Nuclear Structure Analysis

The authors analyzed shape invariants ($Q_n$) to quantify deformation parameters ($\beta_2$) and triaxiality ($\gamma$). They calculated the deformation difference ($\delta_{def}$) between initial and final states to correlate structural changes with NME magnitudes.

3. Key Contributions

• First NLO Calculation for Excited States: This work provides the first calculation of NLO $\chi$EFT contributions specifically for $0^+_{gs} \to 0^+_2$ transitions.
• Systematic NSM Study: A comprehensive survey of six isotopes using multiple Hamiltonians to quantify the uncertainty arising from nuclear structure modeling.
• Deformation-Triaxiality Correlation: A detailed analysis linking the difference in deformation (including triaxiality) between parent and daughter nuclei to the magnitude of the NMEs.
• Seniority Structure Analysis: Investigation into how the seniority structure of nuclear states affects cancellations in the NME sum.

4. Results

A. Half-Life Predictions

• General Magnitude: Predicted half-lives for $0^+_{gs} \to 0^+_2$ are generally very long, often exceeding current experimental lower limits by orders of magnitude.
• Exceptions:
  • $^{76}\text{Ge}$: The lower range of predictions approaches current experimental limits, though uncertainties span two orders of magnitude.
  • $^{82}\text{Se}$: The predicted range is consistent with a very recent experimental indication ($1\sigma$) for this decay.
• Uncertainty Sources: The dominant source of uncertainty is the choice of the nuclear Hamiltonian, not the operator type (bare vs. effective). In some cases (e.g., $^{136}\text{Xe}$ with QX interaction), the uncertainty spans three orders of magnitude due to cancellations in the leading-order NME.

B. NLO and Lepton Expansion Contributions

• Lepton Expansion ($\epsilon_{Taylor}$): Generally contributes a few percent. However, for $^{82}\text{Se}$ and $^{136}\text{Xe}$, where leading-order NMEs undergo significant cancellation, these terms become more relevant.
• NLO $\chi$EFT ($\epsilon_{NLO}$):
  • Typically small ($\sim 2\%$) for most nuclei.
  • Enhanced in specific cases: In $^{124}\text{Sn}$ and $^{136}\text{Xe}$, where the leading-order NME is suppressed by cancellations, NLO contributions can reach 10–30% (or even dominate in $^{136}\text{Xe}$ with the effective operator).
  • The sign of the contribution depends on the specific Hamiltonian and the nature of the cancellation.

C. Nuclear Structure Insights

• Deformation Correlation: A linear relationship was found between the NME magnitude and the deformation difference ($\delta_{def}$) between initial and final states. Larger deformation differences generally lead to smaller NMEs.
• Role of Triaxiality: Triaxiality ($\gamma$) is crucial. Neglecting it breaks the linear correlation observed in some Hamiltonians (e.g., $^{76}\text{Ge}$).
• Spectroscopic Moments: The Hamiltonian that best reproduces experimental spectroscopic quadrupole moments ($Q_s$) often predicts the longest half-lives, suggesting that accurate structural description is vital.
• Seniority: In $^{82}\text{Se}$, the large NME variation between Hamiltonians (JJ4BB vs. JUN45) is attributed to the seniority structure. In the JJ4BB case, cancellations between seniority components that usually suppress the NME do not occur.

5. Significance

• Experimental Guidance: The results provide critical benchmarks for upcoming experiments (e.g., CUPID, LEGEND, nEXO) searching for $2\nu\beta\beta$ to excited states. The consistency with the $^{82}\text{Se}$ indication validates the NSM approach for these transitions.
• Neutrinoless Decay Constraints: Since $2\nu\beta\beta$ and $0\nu\beta\beta$ NMEs are correlated, refining the understanding of $2\nu\beta\beta$ (including NLO effects and structural dependencies) reduces uncertainties in predictions for the neutrinoless mode, which is the primary target for determining the absolute neutrino mass scale.
• Theoretical Refinement: The study demonstrates that while NLO terms are usually small, they become non-negligible when leading-order terms cancel out. It highlights the necessity of including triaxiality and seniority structures in nuclear models to avoid large systematic errors in decay rate predictions.

In conclusion, the paper establishes that while $2\nu\beta\beta$ to excited states is generally suppressed, specific nuclear structures (like those in $^{76}\text{Ge}$ and $^{82}\text{Se}$) make them accessible to current or near-future experiments. The inclusion of NLO $\chi$EFT terms and a rigorous treatment of nuclear deformation significantly improves the reliability of these theoretical predictions.