Radiative and exchange corrections for two-neutrino double-beta decay

The Gist

Imagine the atomic nucleus as a busy, crowded dance floor. Sometimes, two dancers (neutrons) decide to leave the floor at the exact same time, transforming into two new dancers (protons) and tossing out two pairs of shoes (electrons) and two invisible balloons (neutrinos). This rare event is called two-neutrino double-beta decay. It happens so slowly that it takes longer than the age of the universe for a single atom to do it, but scientists are very interested in watching it because it helps us understand the fundamental rules of the universe.

This paper is like a team of physicists putting on high-definition glasses to watch this dance more closely. They realized that previous calculations were missing two subtle "background effects" that change how the dance looks. Here is what they found, explained simply:

1. The "Ghost" Swap (Atomic Exchange Correction)

Imagine you are throwing a ball (an electron) out of a crowded room. Usually, you just throw your own ball. But in this atomic dance, there's a weird rule: the ball you throw might actually swap places with a ball that was already sitting in a chair (a bound electron) in the room. The chair-ball then jumps out the window, and your ball takes its seat.

• What the paper found: The authors calculated this "swap" effect very carefully. They found that this swapping happens much more often in this specific double-beta decay than in regular single-beta decay because the nucleus changes its electric charge by two steps instead of one.
• The result: This swap causes a huge spike in the number of low-energy electrons (the "slow" dancers) being detected. It's like noticing that suddenly, many more people are leaving the dance floor slowly rather than quickly.

2. The "Flash" Effect (Radiative Correction)

Imagine that while the dancers are throwing their shoes, they also briefly flash a camera light (a photon) to say "hello." This light doesn't change the dance steps, but it adds a little bit of extra energy to the whole event.

• What the paper found: This "flash" doesn't change the shape of the dance much, but it makes the whole event happen about 5% faster than we thought before. It's a small but important boost to the total speed of the decay.

3. The Shift in the Peak

When you combine the "Ghost Swap" and the "Flash," something interesting happens to the overall pattern of the dance.

• The Result: The "peak" of the dance (the most common energy level where the electrons are found) shifts slightly to the left. The authors calculated this shift to be about 10 keV (a tiny unit of energy).
• Why it matters: Scientists use the shape of this dance to look for "new physics" (rules beyond our current understanding). If the dance naturally shifts by 10 keV due to these background effects, scientists might mistake it for a sign of new physics if they don't account for it. It's like trying to hear a whisper in a room; if you don't account for the hum of the air conditioner, you might think the hum is a secret message.

4. The "Recipe" for the Dance (The Hypotheses)

To understand the dance, the authors used a mathematical "recipe" (Taylor expansion) that breaks the event down into different layers of complexity. They tested three different ways to imagine how the dance happens:

1. The "Single Star" Hypothesis (SSD): The dance is driven by one specific, famous dancer (the first excited state).
2. The "Crowd" Hypothesis (HSD): The dance is driven by the collective energy of many dancers higher up in the energy levels.
3. The "Real Data" Hypothesis: Using actual measurements from recent experiments.

They found that while the "Single Star" idea is a good approximation, the "Crowd" idea and real data give slightly different shapes to the dance, especially at low energies.

The Bottom Line

The authors have updated the "map" for how this rare atomic decay happens. By adding these two corrections (the swap and the flash), they have made the map more accurate.

• The total speed of the decay is about 5% faster.
• The shape of the energy distribution shifts slightly (by 10 keV).
• The low-energy part of the spectrum has a steep rise due to the swapping effect.

These refined numbers are now ready to be used by experimentalists who are building detectors to watch this decay. If they use the old, less accurate map, they might misinterpret their data. With this new, precise map, they can better distinguish between the standard dance of the universe and any potential "new physics" that might be hiding in the shadows.

Technical Summary

Technical Summary: Radiative and Exchange Corrections for Two-Neutrino Double-Beta Decay

Problem Statement
The precision of theoretical predictions for two-neutrino double-beta ($2\nu\beta\beta$) decay observables is critical for distinguishing Standard Model (SM) processes from potential New Physics scenarios and for reducing background uncertainties in experiments searching for neutrinoless double-beta decay ($0\nu\beta\beta$) and dark matter. While nuclear matrix elements (NMEs) are a primary source of uncertainty, this study addresses the often-overlooked impact of atomic and radiative corrections on the emitted electrons. Specifically, the authors investigate how atomic exchange effects and radiative corrections modify the decay rate and energy spectra of $^{100}\text{Mo}$, a key isotope for current and future experiments.

Methodology
The authors build upon a previously developed formalism that utilizes a Taylor expansion of lepton energy parameters within the denominators of the NMEs. This approach decomposes the decay rate into partial components governed by the ratios $\xi_{31}$ and $\xi_{51}$, allowing for a flexible description that connects to both the Single State Dominance (SSD) and Higher State Dominance (HSD) hypotheses.

To incorporate the corrections, the study employs the following specific techniques:

• Radiative Corrections: The authors account for the exchange of a virtual photon and the emission of a real photon during the decay process, utilizing the leading-order radiative correction function $R(E_e, E_{max}^e)$.
• Atomic Exchange Corrections: The study models the possibility that an emitted electron exchanges with a bound atomic electron, which simultaneously transitions to a continuum state. Crucially, the authors employ a modified Dirac-Hartree-Fock-Slater (DHFS) self-consistent framework. This modification ensures strict orthogonality between the continuum and bound electron wave functions in the final atomic system, a requirement identified as essential for accurate results.
• Hypothesis Testing: The corrections are applied under three distinct scenarios for the NME ratios: the HSD hypothesis ($\xi_{31} = \xi_{51} = 0$), the SSD hypothesis (fixed theoretical values), and values derived from recent experimental measurements by the CUPID-Mo collaboration.

Key Contributions and Results
The paper presents a refined theoretical description of $2\nu\beta\beta$ decay for $^{100}\text{Mo}$, yielding the following specific findings:

1. Decay Rate Increase: The combined radiative and exchange corrections result in an approximate 5% increase in the total decay rate of $^{100}\text{Mo}$. The authors identify the radiative correction as the dominant factor driving this increase, while the exchange correction contributes less to the total rate but significantly alters the spectral shape.
2. Spectral Shape Modifications:
   • Single-Electron Spectrum: The atomic exchange correction induces a steep increase in the number of events in the low-energy region (0–100 keV). This behavior aligns with previous studies on single $\beta$-decay. In contrast, the radiative correction has a negligible effect on the shape of the single-electron spectrum.
   • Summed-Electron Spectrum: The combination of both corrections causes a leftward shift of approximately 10 keV in the maximum of the summed electron spectrum. The authors note that these effects are constructive, meaning they do not cancel each other out.
3. Orthogonality Importance: The study demonstrates that maintaining orthogonality between continuum and bound states in the DHFS framework is critical. Non-orthogonal states were shown to significantly impact the behavior of the exchange correction.
4. Comparison of Hypotheses: The paper validates the Taylor expansion formalism by comparing it against the "exact" SSD formalism, showing that including terms up to the next-to-next-to-leading order provides high precision. Furthermore, the study shows that the single-electron spectrum is highly sensitive to the values of $\xi_{31}$ and $\xi_{51}$, particularly in the low-energy region, whereas the summed spectrum shows sensitivity primarily around the spectral maximum (~1.1 MeV) and specific regions around 0.3 MeV and 2 MeV.

Significance
The authors assert that these refined theoretical predictions serve as precise inputs for double-beta decay experiments. The primary significance lies in the potential impact on the interpretation of experimental data:

• New Physics Constraints: Since many New Physics (BSM) scenarios predict a shift in the maximum of the summed electron distribution, the calculated 10 keV shift caused by standard radiative and exchange corrections must be accounted for to avoid misinterpreting standard effects as new physics or vice versa.
• Parameter Extraction: The corrections may influence future experimental measurements and constraints on the strength parameters $\xi_{31}$ and $\xi_{51}$.
• Background Modeling: The improved precision is relevant for experiments using liquid Xenon and other detectors where $2\nu\beta\beta$ decay represents an inevitable background source for Weakly Interacting Massive Particles (WIMPs) and Coherent Elastic Neutrino-Nucleus Scattering (CE$\nu$NS) searches.

The paper concludes that while the SSD hypothesis remains a useful approximation for testing the truncation order of the Taylor series, the inclusion of these corrections is necessary for high-precision comparisons with current and future experimental statistics.