Nuclear matrix element of $2\nu\beta\beta$ decay of $^{76}$Ge: roles of high-lying states and two-body currents

This paper presents a microscopic analysis of the $2\nu\beta\beta$ decay of $^{76}$Ge, revealing that the nuclear matrix element converges at excitation energies below 5 MeV due to the cancellation of fragmented high-lying states and is further reduced by approximately 10% due to two-body current effects.

The Gist

The Big Picture: A Cosmic Detective Story

Imagine physicists are trying to solve a mystery about the universe: Do neutrinos act like their own antiparticles? To find out, they are looking for a very rare event called "neutrinoless double-beta decay." It's like looking for a needle in a haystack that hasn't been found yet.

To find this needle, they need to understand the "haystack" perfectly. The paper focuses on a specific type of atom, Germanium-76 (76Ge), which is one of the best candidates for this experiment. The scientists are trying to calculate a number called the Nuclear Matrix Element (NME). Think of the NME as the "difficulty score" of the decay. If you know the difficulty score, you can predict how long you have to wait to see the event happen.

The Problem: Too Many Paths to Count

When an atom decays, it doesn't just jump from start to finish. It passes through a "middle ground" (an intermediate nucleus, in this case, Arsenic-76).

In the past, scientists thought they had to add up the contributions of every single possible path the atom could take through this middle ground.

• The Analogy: Imagine trying to calculate the total noise in a stadium. You know there are thousands of fans. If you try to add up the voice of every single fan, it's a nightmare.
• The Reality: As the energy of these "middle ground" states gets higher, the number of possible paths explodes. There are thousands of them packed into every tiny slice of energy.

The Discovery 1: The "Noise Cancellation" Effect

The authors used a powerful computer method (the Projected Shell Model) to look at these thousands of paths. They found something surprising:

• The Analogy: Imagine a choir where some singers are singing a note slightly sharp, and others are singing the same note slightly flat. If you add them all up, the sharp ones cancel out the flat ones, and the total sound becomes very quiet.
• The Finding: The scientists found that at high energy levels, the "signs" (positive or negative values) of these thousands of paths become random. When you add them all together, they cancel each other out.
• The Result: You don't need to count the thousands of high-energy paths. They effectively disappear. The calculation "saturates" (stops changing) once you include states up to about 5 MeV (a specific energy level). Anything higher than that adds nothing to the final answer. This is a huge relief because it means we don't need to model the impossible "thousands of states" to get an accurate answer.

The Discovery 2: The "Teamwork" of Particles

For a long time, scientists assumed that when a particle decays, it acts alone (like a soloist). This is called the "one-body current." However, this paper looked at what happens when two particles inside the nucleus interact and work together (a "two-body current").

• The Analogy: Imagine you are trying to push a heavy car.
  • One-body current: You push alone.
  • Two-body current: You and a friend push together, but your friend is pushing slightly against you or in a weird angle.
• The Finding: The paper found that this "teamwork" (two-body currents) does happen, but it doesn't change the outcome drastically. It acts like a slight "brake" or "quenching" on the process.
• The Result: Including this teamwork reduces the calculated "difficulty score" (NME) by about 10%. Because the decay is slightly harder to calculate, it means the atom will live slightly longer before decaying. Specifically, the predicted time for the atom to decay increases by about 30%.

Why This Matters

1. Simplifying the Math: The paper proves that for heavy atoms like Germanium-76, we can ignore the chaotic, high-energy "noise" because it cancels itself out. This makes future calculations much more reliable.
2. Refining the Prediction: By including the "teamwork" of particles (two-body currents), the scientists refined the prediction of how long the Germanium atom lives. This helps experimentalists (like those running the LEGEND experiment) know exactly what to look for and how long they might have to wait.

Summary

The paper is like a guide for a treasure hunt. It tells the hunters:

1. Don't look everywhere: You only need to look at the low-energy paths; the high-energy ones cancel out and don't matter.
2. Adjust your map: When you account for particles working together, the "treasure" (the decay event) is slightly harder to find, meaning you might have to wait a bit longer than previously thought.

This helps ensure that when we finally find (or don't find) the mysterious neutrinoless decay, our calculations are as solid as possible.

Technical Summary

Technical Summary: Nuclear Matrix Element of 2νββ Decay of 76Ge: Roles of High-Lying States and Two-Body Currents

Problem Statement
The nuclear matrix element (NME) is the critical theoretical component required to interpret searches for neutrinoless double-beta ($0\nu\beta\beta$) decay, a process that would confirm the Majorana nature of neutrinos. However, calculating the NME for $0\nu\beta\beta$ decay is challenging due to the lack of direct observables and the complexity of nuclear structure. The two-neutrino double-beta ($2\nu\beta\beta$) decay serves as a Standard Model counterpart with identical initial and final nuclear states, offering a pathway to constrain nuclear models used for $0\nu\beta\beta$ calculations.

Two primary theoretical difficulties hinder accurate NME calculations for open-shell heavy deformed nuclei like $^{76}\text{Ge}$:

1. High Level Density: The intermediate nucleus ($^{76}\text{As}$) possesses thousands of virtual states within each 1 MeV excitation energy bin. Properly accounting for the contributions of these high-lying states without closure approximations is computationally demanding.
2. Two-Body Currents (2BC): Standard calculations often rely on the impulse approximation (one-body currents). However, the weak axial-vector coupling constant ($g_A$) in the nuclear medium may require quenching mechanisms arising from two-body currents in the transition operators, which are difficult to incorporate without normal ordering approximations.

Methodology
The authors employ a microscopic shell-model diagonalization method utilizing the Projected Shell Model (PSM) to calculate the $2\nu\beta\beta$-decay NME of $^{76}\text{Ge} \to {}^{76}\text{Se}$.

• Model Space and Configuration: The study utilizes a large model space (three major shells, $N=2,3,4$) and a large configuration space. The nuclear many-body wave functions are constructed from multi-quasiparticle (qp) configurations (up to 4-qp for even-even nuclei and specific 4-qp combinations for odd-odd nuclei).
• Symmetry Restoration: To handle the broken symmetries in the intrinsic frame, the authors apply angular-momentum projection techniques using the Hill-Wheeler equation to restore rotational symmetry in the laboratory frame.
• Transition Operators: The calculation explicitly includes:
  • One-Body Currents (1BC): The standard Gamow-Teller operator.
  • Two-Body Currents (2BC): The leading space piece of the axial 2BC operator in coordinate space, derived from chiral effective field theory, including terms dependent on low-energy constants ($\bar{c}_3, \bar{c}_4, \bar{d}_1, \bar{d}_2$).
• Calculation Approach: Unlike previous studies that might use closure approximations, this work explicitly sums over thousands of intermediate $1^+_n$ levels of $^{76}\text{As}$ up to an excitation energy ($E_n$) of approximately 18 MeV. The authors truncate the configuration space at a quasiparticle energy ($E_{qp}$) of 18 MeV, yielding roughly 25,000 intermediate levels.

Key Results

1. Fragmentation and Cancellation at High Energies:

   • The single-$\beta$ Gamow-Teller matrix elements connecting the initial/final states to the intermediate states exhibit distinct behaviors based on excitation energy.
   • At low $E_n$, matrix elements are sparse and larger. However, as $E_n$ increases (specifically $E_n \gtrsim 5$ MeV for $n \to f$ transitions and $E_n \gtrsim 9$ MeV for $i \to n$ transitions), the matrix elements become highly fragmented, very small in magnitude, and display seemingly random sign patterns.
   • This leads to a cancellation mechanism in the summation of the NME (Eq. 2). Consequently, the cumulative NME saturates and converges at $E_n \approx 5$ MeV.
   • Implication: The contribution of high-lying states ($E_n > 5$ MeV) to the $2\nu\beta\beta$ NME is negligible due to this cancellation. This contrasts with some previous QRPA or shell-model studies on different nuclei that suggested saturation at higher energies (e.g., 10–15 MeV).

2. Impact of Two-Body Currents (2BC):

   • The inclusion of 2BC in the transition operator results in a quenching of approximately 10% in the $2\nu\beta\beta$-decay NME of $^{76}\text{Ge}$.
   • The magnitude of this quenching is sensitive to the coupling constants ($\bar{c}_D$) used in the 2BC operator. For instance, with specific coupling constants, the NME decreases from 0.144 to 0.127 (a ~12% reduction).
   • This quenching translates to an increase in the calculated half-life from $1.0 \times 10^{21}$ yr to $1.3 \times 10^{21}$ yr.
   • The authors note that this 2BC-induced quenching is distinct from and smaller than the phenomenological quenching often achieved by simply reducing $g_A$ to 1.0, which yields a half-life of $\approx 2.5 \times 10^{21}$ yr.

3. Significance of Matrix Element Signs:

   • The study highlights that the signs of single-$\beta$ matrix elements are crucial for the NME calculation, even though they are not directly observable (unlike the absolute GT strengths measured in charge-exchange reactions).
   • A hypothetical calculation ignoring signs (summing absolute values) shows a continuous, rapid increase in NME with energy, failing to reproduce the saturation observed in the realistic calculation.

Significance and Claims
The paper claims to provide the first microscopic calculation for a typical open-shell deformed nucleus ($^{76}\text{Ge}$) that simultaneously treats the high level density of the intermediate nucleus and two-body currents without relying on closure or normal ordering approximations.

The primary findings are:

• Convergence: The $2\nu\beta\beta$ NME for $^{76}\text{Ge}$ is dominated by low-lying intermediate states ($E_n \lesssim 5$ MeV) due to the cancellation of high-energy contributions. This suggests that experimental constraints via charge-exchange reactions need not focus on the extremely high-energy region to constrain the NME effectively.
• 2BC Contribution: Two-body currents provide a non-negligible (~10%) quenching effect on the NME, which must be accounted for to accurately predict half-lives and constrain $g_A$ quenching mechanisms.
• Future Outlook: The authors state that similar work regarding the more critical $0\nu\beta\beta$-decay NME will be undertaken in the near future using this framework.

The authors conclude that reliable determination of the coupling constants in the two-body current is essential, as the current results show sensitivity to these parameters.