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Positron-Emitting and Electron-Capturing Double-Beta Processes in the Standard Model and Beyond

Motivated by the proposed NuDoubt++ experiment, this paper analyzes positron-emitting and electron-capturing double-beta decay in candidate isotopes ${}^{78}$ Kr, ${}^{106}$ Cd, and ${}^{124}$ Xe by providing nuclear matrix elements and phase-space factors for both standard and neutrinoless modes, demonstrating that these processes can probe lepton-number-violating new physics at scales of 1–100 TeV and resolve operator degeneracies when combined with conventional double-beta decay searches.

Original authors: Lukáš Gráf, Jenni Kotila, Oliver Scholer

Published 2026-06-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lukáš Gráf, Jenni Kotila, Oliver Scholer

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

Imagine the atomic nucleus as a tiny, crowded dance floor. Usually, particles on this floor are stable, but sometimes, they need to get rid of extra energy to find balance. In the world of physics, there's a rare dance move called "double beta decay," where two particles change their identity at the same time.

Most scientists have been watching a specific version of this dance where the nucleus spits out two electrons (negatively charged particles). This is the "standard" dance everyone knows.

However, this paper focuses on a much rarer, more exotic version of the dance where the nucleus tries to emit positrons (the electron's positive twin) or swallow electrons from its own orbit. The authors are studying a proposed experiment called NuDoubt++, which aims to catch these rare moves in action.

Here is a breakdown of their findings using simple analogies:

1. The Three Types of "Positron" Dances

The paper looks at three different ways the nucleus can perform this double-beta dance involving positrons:

The "Double Positron" Dance ( $2\nu\beta^+\beta^+$ ): The nucleus tries to throw out two positrons at once.
- The Problem: It's like trying to push two heavy boulders up a very steep hill at the same time. The "hill" (physics rules) is so steep that this move is incredibly difficult. The paper says detecting this is extremely hard and would require a massive amount of time and material (like waiting for a very long time in a giant, quiet room).
The "Mixed" Dance ( $2\nu EC\beta^+$ ): The nucleus swallows one electron from its own orbit and throws out one positron.
- The Good News: This is much easier to spot. It's like pushing one boulder up a gentle slope. The authors believe the NuDoubt++ experiment could catch this happening relatively soon.
The "Double Swallow" Dance ( $2\nu ECEC$ ): The nucleus swallows two of its own electrons.
- The Good News: This is also a promising target, similar in difficulty to the "Mixed" dance. However, detecting it is tricky because it doesn't throw out big, energetic particles; instead, it leaves behind a quiet "sigh" of low-energy X-rays.

2. The "Ghost" Dance (Neutrinoless Decay)

So far, we've talked about dances where two invisible "ghosts" (neutrinos) are also released. But the paper also looks for a "Ghost-Free" version ( $0\nu\beta\beta$ ), where no neutrinos escape.

Why it matters: If we see this ghost-free dance, it proves that neutrinos are their own anti-particles (like a mirror image that is identical to the original). It would also reveal "New Physics"—rules of the universe that we don't know yet.
The Search: The authors calculated how sensitive the NuDoubt++ experiment would be to this. Even though the "positron" dance is harder to do than the standard "electron" dance, the experiment could still detect signs of new physics at energy scales of 1 to 100 TeV. Think of this as being able to see a whisper from a very distant mountain range.

3. The Power of Multiple "Isotopes" (Different Dance Floors)

The experiment plans to test three different types of atoms (Krypton-78, Cadmium-106, and Xenon-124). The authors argue this is crucial.

The Analogy: Imagine trying to figure out who is dancing by listening to the music. If you only listen to one song, you might get confused if two different dancers are moving in a way that cancels each other out (a "degeneracy").
The Solution: By listening to the music on three different dance floors (isotopes), you can tell the difference. If the dancers cancel each other out on one floor but not the others, you can figure out exactly who is doing what. This helps scientists avoid getting tricked by "false negatives" where new physics is hiding.

4. The Main Takeaways

The "Easy" Wins: The experiment is most likely to see the "Mixed" dance (swallowing one, throwing one) and the "Double Swallow" dance. These are the most accessible targets.
The "Hard" Climb: Seeing the "Double Positron" dance (throwing two out) is very difficult and might take a huge amount of time and equipment.
The "Ghost" Hunt: Even though the positron versions are harder to produce than the standard electron versions, they are still powerful tools for finding new laws of physics. They act as a different kind of magnifying glass.
Teamwork: Using multiple types of atoms together is the secret sauce. It helps scientists untangle complex scenarios where different new physics theories might look identical if you only looked at one type of atom.

In short, this paper is a roadmap for a new experiment. It says, "If you build this detector, you probably won't catch the hardest dance move immediately, but you will likely catch the easier ones, and you'll have a powerful new way to hunt for the invisible 'ghosts' of new physics that standard experiments might miss."

Technical Summary: Positron-Emitting and Electron-Capturing Double-Beta Processes in the Standard Model and Beyond

Problem Statement
Neutrinoless double beta decay ( $0\nu\beta\beta$ ) experiments are primary probes for lepton-number-violating (LNV) physics and the Majorana nature of neutrinos. However, conventional searches focus almost exclusively on the double-electron-emitting ( $0\nu\beta^-\beta^-$ ) mode due to favorable phase space. The recent proposal of the NuDoubt++ collaboration to search for double beta decay in positron-emitting ( $\beta^+$ ) and electron-capturing (EC) isotopes necessitates a rigorous theoretical framework to interpret these complementary processes. While these modes offer independent constraints on nuclear structure and Beyond-the-Standard-Model (BSM) physics, they suffer from phase-space suppression and complex detection signatures. There is a need for a complete set of nuclear matrix elements (NMEs) and phase-space factors (PSFs) for these modes, as well as an analysis of their sensitivity to LNV operators within the Standard Model Effective Field Theory (SMEFT).

Methodology
The authors perform a dedicated Effective Field Theory (EFT) analysis of the projected sensitivity of the NuDoubt++ experiment to both Standard Model (SM) and BSM double-beta processes.

Framework: The study utilizes the Interacting Boson Model (IBM-2) with isospin restoration and the closure approximation to calculate NMEs. For the neutrinoless modes, the analysis employs a chain of EFTs: starting with dimension-7 LNV operators in the SMEFT, matching them to the low-energy EFT (LEFT), and finally to chiral perturbation theory ( $\chi$ PT) to derive nuclear operators.
Phase-Space Factors (PSFs): The authors calculate PSFs by numerically solving the radial Dirac equation for electron/positron wavefunctions in a Coulomb potential generated by a uniform nuclear charge, incorporating electron screening via the Thomas-Fermi (TF) function. They present an updated formulation for the screening potential, particularly for bound-state electrons in capture modes.
Isotopes: The analysis covers the three primary NuDoubt++ candidate isotopes ( $^{78}\text{Kr}$ , $^{106}\text{Cd}$ , $^{124}\text{Xe}$ ) as well as other naturally occurring $\beta^+\beta^+$ candidates ( $^{96}\text{Ru}$ , $^{130}\text{Ba}$ , $^{136}\text{Ce}$ ).
Sensitivity Projections: The projected sensitivity is evaluated for a hypothetical NuDoubt++ setup with a half-life limit of $T_{1/2} = 10^{24}$ years. The authors interpret these limits in terms of the new-physics scale $\Lambda_{NP}$ for dimension-7 SMEFT operators, accounting for 1-loop renormalization group evolution (RGE) and matching procedures.

Key Contributions

NMEs and PSFs: The paper provides a complete set of NMEs calculated in the IBM-2 model and updated PSFs for all relevant SM ( $2\nu\beta^+\beta^+$ , $2\nu\text{EC}\beta^+$ , $2\nu\text{ECEC}$ ) and neutrinoless ( $0\nu\beta^+\beta^+$ , $0\nu\text{EC}\beta^+$ , $0\nu\text{ECEC}$ ) modes for the candidate isotopes.
SM Mode Analysis: The authors estimate the exposure required to observe SM modes. They find that $2\nu\text{ECEC}$ and $2\nu\text{EC}\beta^+$ are the most experimentally accessible, whereas the fully positron-emitting $2\nu\beta^+\beta^+$ mode is strongly phase-space suppressed.
BSM Sensitivity: The study interprets the projected NuDoubt++ sensitivity in terms of dimension-7 SMEFT operators. It demonstrates that despite the phase-space suppression relative to $\beta^-\beta^-$ , the experiment can probe new-physics scales of order $1\text{--}100$ TeV.
Degeneracy Resolution: The authors show that measurements across multiple isotopes can resolve degeneracies in multi-operator scenarios, specifically lifting constraints caused by destructive interference between operators.
Spectra and Correlations: The paper analyzes leptonic phase-space observables, including single and summed positron spectra and angular correlations, identifying that specific dimension-7 operators (e.g., $\mathcal{O}^{(7)}_{LHDe}$ and $\mathcal{O}^{(7)}_{LeudH}$ ) yield distinct signatures compared to the standard light-neutrino exchange mechanism.

Results

Standard Model Modes: For the NuDoubt++ candidate $^{78}\text{Kr}$ , the detection of the $2\nu\text{EC}\beta^+$ mode is expected with an exposure of $7\text{--}100$ tonne $\times$ days. In contrast, the $2\nu\beta^+\beta^+$ mode requires a significantly higher exposure, estimated at $10^2\text{--}10^4$ tonne $\times$ days (depending on the true half-life and background assumptions), placing it beyond the reach of the initial NuDoubt++ proposal. The $2\nu\text{ECEC}$ mode faces challenges due to low-energy backgrounds (X-rays and Auger electrons) but requires similar exposure to $2\nu\text{EC}\beta^+$ .
Neutrinoless Modes: Interpreting a projected half-life sensitivity of $T_{1/2} = 10^{24}$ years, the NuDoubt++ experiment can probe dimension-7 SMEFT operators corresponding to new-physics scales $\Lambda_{NP} \sim \mathcal{O}(1)\text{--}\mathcal{O}(10^2)$ TeV. This sensitivity is comparable to current $0\nu\beta^-\beta^-$ limits (e.g., KamLAND-Zen) within an order of magnitude, despite the lower half-life sensitivity, due to the scaling relation $\Lambda_{NP} \propto (T_{1/2}^{0\nu})^{1/6}$ .
Multi-Operator Scenarios: In scenarios involving two active operators, destructive interference can create unconstrained directions in the Wilson coefficient parameter space. The authors demonstrate that combining data from $^{78}\text{Kr}$ , $^{106}\text{Cd}$ , and $^{124}\text{Xe}$ can break these degeneracies, as the allowed regions in parameter space are isotope-dependent.
Resonant $0\nu\text{ECEC}$ : The paper notes that while $0\nu\text{ECEC}$ is a resonant process, the large uncertainties in half-life estimates (spanning 6–7 orders of magnitude) preclude a robust detailed analysis at this time.

Significance
The paper establishes that positron-emitting and electron-capturing double-beta decays serve as a valuable complement to conventional neutrinoless double beta decay experiments. While the fully positron-emitting modes are phase-space suppressed, the electron-capture channels offer accessible Standard Model targets. More importantly, the study highlights that multi-isotope measurements in a single experimental setup (like NuDoubt++) provide a unique diagnostic tool. By constraining destructive interference directions in multi-operator scenarios, these searches can help distinguish between different LNV mechanisms, offering insights into the underlying BSM physics that half-life measurements alone cannot provide. The work provides the necessary theoretical infrastructure (NMEs, PSFs, and EFT interpretations) to support the upcoming NuDoubt++ experimental program.

1. The Three Types of "Positron" Dances

2. The "Ghost" Dance (Neutrinoless Decay)

3. The Power of Multiple "Isotopes" (Different Dance Floors)

4. The Main Takeaways

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