⚛️ phenomenology

Radiative corrections to double- $\beta$ decay and their impact on experimental analyses

This paper presents a comprehensive calculation of $\mathcal{O}(\alpha)$ radiative corrections to two-neutrino and neutrinoless double- $\beta$ decay using heavy-nucleus effective field theory, revealing that neglecting these effects—particularly real-photon emission and a universal double-weak Sirlin function—significantly biases experimental analyses and increases the neutrinoless decay rate by approximately 5%.

Original authors: Jordy de Vries, Emanuele Mereghetti, Saad el Morabit, Stefan Sandner

Published 2026-10-22

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CC BY 4.0

Original authors: Jordy de Vries, Emanuele Mereghetti, Saad el Morabit, Stefan Sandner

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, atomic nuclei occasionally undergo a rare transformation known as double-beta decay. In this process, a nucleus converts two of its neutrons into protons, ejecting two electrons and two invisible particles called neutrinos in the bargain. While this event is a standard occurrence in the universe, it is so infrequent that it took decades of observation to confirm it in a laboratory. Today, massive experiments buried deep underground are hunting for an even rarer variation of this decay, one where no neutrinos are emitted at all. Finding this "neutrinoless" version would prove that neutrinos are their own antiparticles and could help explain why the universe is made of matter rather than antimatter. To find this needle in a haystack, scientists must first understand the ordinary, two-neutrino version with extreme precision. They treat the two-neutrino decay not just as background noise, but as a high-precision tool to test the fundamental laws of physics and to map the complex internal structure of the atomic nucleus.

However, measuring this process with the required accuracy has hit a subtle wall. For years, scientists have tried to predict exactly how the energy of the emitted electrons should be distributed, but their calculations have been missing a crucial piece of the puzzle: the effect of light itself. Even though these decays happen inside the nucleus, the two speeding electrons interact with the electromagnetic force, meaning they can emit tiny flashes of light, or photons, as they fly away. These flashes carry away energy, subtly altering the total energy measured by detectors. Until now, most theoretical models treated these two electrons as if they were independent travelers, each emitting light on their own, ignoring the fact that they are part of a single, coordinated event. A new, comprehensive study has corrected this oversight, revealing that the light emitted by these electrons is far more complex and interconnected than previously thought, and that ignoring it leads to significant errors in interpreting experimental data.

The researchers, working with a mathematical framework designed to handle heavy atomic nuclei, performed a complete calculation of these light-emission effects, known as radiative corrections. They mapped out every possible way a photon could be created, whether it was a virtual particle popping in and out of existence or a real photon flying out of the detector. They discovered that the two electrons do not act independently; instead, they influence each other through the exchange of light in a way that cannot be simply added together. The study derived a new, universal function that describes this double-weak interaction, showing that the old method of simply adding two separate corrections together was wrong. In fact, the old method overestimated the size of the correction by nearly a factor of two and got the shape of the energy spectrum completely wrong.

The impact of this discovery depends heavily on how the experiment is set up. In some detectors, the energy of the emitted photon is captured along with the electrons, effectively adding it back into the total energy count. In these cases, the distortion caused by the light emission largely cancels out, and the measured energy looks very close to the simple prediction. But in other scenarios, or if the photon escapes detection, the energy spectrum is significantly warped. The researchers found that if scientists ignore these new corrections, they will extract the wrong values for the nuclear properties they are trying to measure. Specifically, the study showed that neglecting these effects could bias the results by a large margin, potentially leading researchers to believe they have found new physics or a different nuclear structure when they have only missed a subtle detail of light emission.

For the search for the elusive neutrinoless decay, the findings are equally important. The study calculated that the total rate of this rare event increases by about five percent when these light-emission effects are properly included. This increase is a fixed amount that does not depend on the specific type of nucleus or the mass of the neutrino, making it a reliable correction that can be applied immediately. Furthermore, the emission of real photons means that some events that should appear at the very top of the energy scale will instead appear slightly lower, potentially hiding the signal scientists are looking for. The team provided detailed maps of how these photons are emitted, including their energy and direction, allowing experimentalists to simulate exactly how their detectors will respond.

This work provides a new, more accurate foundation for the next generation of experiments. By showing that the two electrons are inextricably linked through the light they emit, the researchers have corrected a long-standing approximation that was limiting the precision of double-beta decay studies. They have demonstrated that to truly understand the nucleus and to find the rarest events in the universe, scientists must account for the full complexity of how light interacts with matter at the subatomic level. The study concludes that future experiments must use these new, detailed templates to analyze their data, ensuring that the search for the secrets of the universe is not derailed by the subtle glow of a single photon.

Technical Summary: Radiative Corrections to Double-β Decay and Their Impact on Experimental Analyses

Problem Statement
Two-neutrino double-β decay ( $2\nu\beta\beta$ ) serves as a precision observable for testing the Standard Model (SM) in the electroweak sector and constraining nuclear matrix elements (NMEs) relevant to neutrinoless double-β decay ( $0\nu\beta\beta$ ) searches. Current and next-generation experiments (e.g., LEGEND, nEXO, CUPID) aim to extract nuclear structure parameters, specifically the ratio of NMEs $\xi_{31} = M^{(-3)}_{GT}/M^{(-1)}_{GT}$ , from the electron energy and angular spectra. However, the theoretical description of the $2\nu\beta\beta$ spectral shape requires corrections at the sub-percent level.

A critical gap exists in the treatment of electromagnetic radiative corrections ( $O(\alpha)$ ). Previous analyses approximated these corrections as the sum of two independent single- $\beta$ Sirlin functions. This approximation fails to account for correlations between the two outgoing electrons, the spectrum of intermediate nuclear states, and the specific kinematics of the double-weak process. Consequently, neglecting the correct radiative structure introduces significant biases in the extraction of $\xi_{31}$ and potentially obscures signals of physics beyond the Standard Model (BSM). Furthermore, the role of real-photon emission ( $2\nu\beta\beta + \gamma$ ) and its impact on detector response had not been comprehensively addressed in the context of $2\nu\beta\beta$ .

Methodology
The authors employ a Heavy-Nucleus Effective Field Theory (EFT) to systematically compute $O(\alpha)$ radiative corrections for both $2\nu\beta\beta$ and $0\nu\beta\beta$ decays.

EFT Framework: The calculation utilizes a Lagrangian containing scalar fields for the $0^+$ initial and final nuclei and vector fields for $1^+$ intermediate nuclear states. The theory is organized as an expansion in the ratio of lepton energies to nuclear excitation energies ( $E_e/\omega_n$ ) and includes minimal electromagnetic couplings. The vector coupling $g_V$ is treated with its full $O(\alpha)$ radiative corrections, including the $\gamma W$ box diagram.
Diagrammatic Calculation: The authors compute all virtual and real-emission diagrams at $O(\alpha)$ $O (α)$ . This includes:
- Electron-nucleus triangle diagrams (coupling to initial, intermediate, and final nuclei).
- Electron-electron box diagrams (unique to the double-weak process).
- Nuclear self-energies and vertex corrections.
- Real photon emission from all charged lines.
Loop Integrals: A key technical achievement is the derivation of new closed-form expressions for scalar one-loop integrals involving heavy-nucleus propagators. These are obtained using an auxiliary-mass representation that maps the integrals onto standard relativistic ones, handling infrared (IR) divergences via dimensional regularization.
Large- $\omega_n$ Limit: The authors perform a consistency check by integrating out intermediate nuclear states to derive a "large- $\omega_n$ " EFT containing only ground-state nuclei. They demonstrate that in the limit where excitation energies are large compared to lepton energies, the corrections factorize into the tree-level rate multiplied by a universal function.
Phenomenological Analysis: The theoretical results are applied to specific isotopes ( $^{76}\text{Ge}$ , $^{100}\text{Mo}$ , $^{130}\text{Te}$ , $^{136}\text{Xe}$ ) using NME ratios from Nuclear Shell Model (NSM) and Quasiparticle Random Phase Approximation (QRPA) calculations. The impact on spectral shapes, angular distributions, and the extraction of $\xi_{31}$ is quantified under various detector scenarios (photon absorbed vs. photon escaping).

Key Contributions

The Double-Weak Sirlin Function: The paper derives a universal "double-weak Sirlin function" ( $f_{2\nu}$ ) that captures the ultrasoft radiative corrections. This function depends on both electron energies and their relative angle. It is shown to differ substantially from the sum of two single- $\beta$ Sirlin functions previously used in the literature.
Closed-Form Loop Integrals: The authors provide explicit, closed-form solutions for the scalar integrals required for radiative corrections with heavy-baryon propagators, resolving previous implicit formulations and facilitating future calculations in heavy-baryon EFTs.
Real-Photon Emission Spectra: The work provides the first fully differential photon spectra (in energy and direction) for the radiative modes $2\nu\beta\beta + \gamma$ and $0\nu\beta\beta + \gamma$ . This allows for explicit simulation of photon interactions in detectors.
Factorization in the Large- $\omega_n$ Limit: It is demonstrated that for realistic nuclear spectra, the radiative corrections factorize to high accuracy (within 2%) into the tree-level spectrum multiplied by the factor $(g_V^4 + \frac{\alpha}{2\pi}f_{2\nu})$ . This simplifies the implementation of corrections in experimental analyses.
$0\nu\beta\beta$ Corrections: The study computes the $O(\alpha)$ corrections for neutrinoless decay, finding a universal increase in the inclusive rate of approximately 5.2%, independent of NMEs or the effective Majorana mass.

Results

Spectral Distortions: The double-weak Sirlin function induces spectral distortions comparable in magnitude to the leading nuclear-structure effects governed by $\xi_{31}$ . The shape of the distortion differs significantly from the single- $\beta$ approximation, which overestimates the correction by nearly a factor of two and fails to reproduce the correct spectral shape.
Bias on $\xi_{31}$ : Fitting tree-level templates to radiatively corrected spectra introduces a significant bias in the extracted value of $\xi_{31}$ $ξ_{31}$ .
- If the photon escapes detection, the bias is substantial (e.g., $\Delta\xi_{31} \approx -0.06$ to $-0.08$ for $^{76}\text{Ge}$ and $^{136}\text{Xe}$ ), becoming comparable to statistical uncertainties for event samples of $\sim 10^5$ .
- If the photon is fully absorbed (calorimetric detection), the bias is significantly reduced ( $|\Delta\xi_{31}| \lesssim 0.01$ ) for energies above 0.5 MeV, though a residual bias remains at lower energies.
- The bias depends critically on the detector's ability to resolve and reject events with hard photons.
Angular Distributions: Radiative corrections suppress the distribution of nearly collinear electrons due to Coulomb repulsion (electron-electron box diagram). This effect is distinct from single- $\beta$ decay and must be included to correctly interpret angular correlation measurements.
$0\nu\beta\beta$ Impact: The inclusive $0\nu\beta\beta$ rate increases by 5.2%. Real-photon emission creates a radiative tail below the endpoint, potentially reducing signal efficiency by a few percent depending on the energy resolution and event selection cuts (e.g., multi-site rejection).
Comparison with Other Corrections: The magnitude of radiative corrections is comparable to other subleading effects such as weak magnetism and atomic exchange, necessitating their simultaneous inclusion in precision analyses.

Significance and Claims
The paper claims that the omission of the correct double-weak radiative corrections constitutes a systematic error that is no longer negligible for next-generation experiments. The authors assert that:

Precision Requirement: To extract $\xi_{31}$ and test the SM at the sub-percent level, experimental templates must consistently include the derived double-weak Sirlin function and real-photon emission spectra.
Detector Dependence: The impact of radiative corrections is not universal but depends crucially on the detector setup (specifically, whether the photon energy is measured or absorbed). Therefore, a "one-size-fits-all" correction is insufficient; experiments should simulate the photon explicitly as a final-state particle.
Methodological Advance: The derived closed-form integrals and the factorization property (Eq. 112) provide a robust and computationally efficient framework for incorporating these corrections into spectral generators.
$0\nu\beta\beta$ Relevance: While the $0\nu\beta\beta$ signal has not yet been observed, the 5.2% shift in the inclusive rate and the modification of the background tail are essential for accurate phase-space factor calculations and signal efficiency estimates in future searches.

The authors conclude that a precise determination of nuclear structure parameters and BSM constraints requires templates that include these radiative corrections alongside weak magnetism and atomic exchange effects, moving beyond the tree-level approximations currently used in many analyses.

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