⚛️ nuclear theory

Long-range scalar interactions in the effective field theory approach to $0\nu\beta\beta$

This paper presents a compact, model-independent effective field theory formulation for neutrinoless double-beta decay that explicitly derives the dimensionally explicit half-life expression and a positivity bound for the coherent interference between two long-range scalar mechanisms, accounting for their complex coupling phases and specific nuclear matrix element ratios.

Original authors: Fahim Ahmed

Published 2026-08-25

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CC BY 4.0

Original authors: Fahim Ahmed

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 certain heavy atoms, a rare and mysterious event can occur: two neutrons transform into two protons, and in the process, two electrons are ejected. In the standard story of particle physics, this transformation must also release two invisible particles called neutrinos to balance the books of energy and momentum. But there is a second, more exotic possibility that physicists have been hunting for decades. In this version, the neutrinos are never released at all. Instead, the two neutrons swap their identities with the two protons by exchanging a single neutrino that acts as its own antiparticle. If this "neutrinoless double-beta decay" were ever observed, it would prove that neutrinos are their own antiparticles, a property known as being Majorana particles. This discovery would rewrite our understanding of the universe's fundamental building blocks and explain why the cosmos is made of matter rather than antimatter.

The challenge in finding this decay is that it is incredibly rare, and many different theories beyond our current standard model of physics could cause it to happen. To make sense of the data, scientists use a flexible mathematical framework called effective field theory. Think of this framework as a universal translator that allows researchers to describe many different possible causes of the decay using a single, common language of interactions. Within this language, scientists can calculate how different causes might interfere with one another, much like how two sound waves can combine to make a louder noise or cancel each other out completely. The goal is to disentangle these overlapping signals to see which specific mechanism is at work.

In this work, the researchers focused on a specific set of interactions within this framework known as scalar terms. These are particular ways that the particles can interact that involve a "scalar" nature, distinct from the more familiar vector interactions. The team set out to calculate exactly how two different types of these scalar interactions would behave if they occurred simultaneously in the same atom. Previous studies had looked at these interactions one at a time, but the real world might involve both happening together. The researchers derived a new, compact formula that describes the decay rate when both scalar mechanisms are present at once. This formula accounts for how the two mechanisms interfere with each other, a process controlled by the relative phase, or timing, of their underlying forces.

The calculation required a detailed accounting of the complex environment inside the atomic nucleus. The team had to track the motion of the electrons as they escape the nucleus, including the subtle distortions caused by the electric charge of the nucleus itself. They also had to account for the recoil of the protons and neutrons as they shift positions during the decay. By using a method called the closure approximation, which treats the many possible intermediate states of the nucleus as a single average, they were able to simplify the problem without losing the essential physics. Their result is a precise mathematical expression that links the speed of the decay to the strength of the two scalar interactions and their relative phase.

A key finding of the study is that the interference between these two scalar mechanisms is a coherent contribution that must be included to correctly interpret experimental data. The researchers showed that if you ignore the fact that these two interactions can happen together, you might misjudge the strength of the forces involved. They also established a strict mathematical rule, known as a positivity bound, which ensures that the calculated decay rate remains physically possible. This rule acts as a consistency check, guaranteeing that the combination of these interactions cannot produce a negative or impossible probability.

The paper provides a solid foundation for future experiments. By giving a clear, unified formula that handles both scalar interactions at once, the author has provided the tools needed to analyze data from detectors around the world. If a signal for neutrinoless double-beta decay is found, this new framework will allow scientists to determine whether it comes from the standard mass mechanism or from these more exotic scalar interactions, and how the different possibilities might be mixing together. The work does not claim to have found the decay, nor does it prove that these scalar interactions exist. Instead, it offers a rigorous and complete map for navigating the search, ensuring that when the signal is finally detected, the scientific community will be ready to understand exactly what it means.

Technical Summary: Long-range scalar interactions in the effective field theory approach to $0\nu\beta\beta$

Problem Statement
Neutrinoless double-beta decay ( $0\nu\beta\beta$ ) serves as a critical low-energy probe for physics beyond the Standard Model (BSM), specifically testing the Majorana nature of neutrinos. While the standard mass mechanism (light Majorana neutrino exchange) is well-studied, the effective field theory (EFT) framework allows for a model-independent description involving various Lorentz structures, including scalar ( $S$ ) and pseudoscalar ( $P$ ) currents. Previous analyses have often treated scalar contributions in isolation or focused on single-coupling limits. A comprehensive understanding requires addressing the simultaneous presence of two distinct scalar interactions involving a common $(S+P)$ leptonic current coupled to either an $(S-P)$ or an $(S+P)$ hadronic current. The challenge lies in accurately calculating the coherent interference between these two scalar mechanisms and their separate interference with the standard mass mechanism, while properly accounting for nuclear recoil, electron partial waves ( $s_{1/2}$ and $p_{1/2}$ ), and the complex phases of the couplings.

Methodology
The paper employs an EFT approach to the long-range part of the $0\nu\beta\beta$ amplitude, utilizing a dimension-six Hamiltonian that includes light neutrino exchange. The calculation proceeds through the following steps:

Formalism and Amplitudes: The author derives the S-matrix elements for the decay, considering the interference between the standard $V-A$ mass mechanism and the scalar terms. They retain two scalar amplitudes simultaneously, defined by amplitude-level couplings $\epsilon_-$ (coupling to $S-P$ hadronic current) and $\epsilon_+$ (coupling to $S+P$ hadronic current).
Leptonic and Hadronic Reduction:
- Leptonic: The calculation includes the $s_{1/2}$ and $p_{1/2}$ Coulomb-distorted electron waves. The chiral projectors distinguish between mass-dependent terms (standard mechanism) and momentum-dependent terms (scalar mechanisms).
- Hadronic: The quark currents are matched to nucleon currents using form factors ( $F_S, F_{PS}, g_V, g_A, g_M, g_P$ ) and expanded non-relativistically. The calculation retains terms up to first order in nucleon recoil (single-nucleon impulse approximation).
- Nuclear Structure: The closure approximation is applied to sum over intermediate nuclear states, replacing excitation energies with an average closure energy.
Operator Reduction: The scalar amplitudes are reduced to a set of nuclear matrix elements (NMEs). The author identifies three key dimensionless NME ratios: $\chi'_{SP,F}$ (Fermi-type), $\chi'_{SP,B}$ (pseudoscalar recoil), and $\chi'_{SP,D}$ (vector recoil).
Decay Rate Calculation: The spin-summed square of the total amplitude is computed. The differential decay rate is expressed in terms of isotropic and angular-correlation functions, which are then integrated over the electron phase space to derive the inverse half-life.

Key Contributions and Results
The primary contribution of this work is the derivation of a compact, dimensionally explicit half-life formula that simultaneously accounts for both scalar interactions and their mutual interference.

Interference Terms: The derived inverse half-life expression includes:
- The standard mass mechanism term ( $|\eta_\nu|^2 C_1$ ).
- Linear interference terms between the mass mechanism and each scalar coupling ( $|\eta_\nu||\epsilon_\pm| C_{SP}^{2, 2+} \cos \psi_{3,4}$ ).
- Quadratic scalar terms ( $|\epsilon_\pm|^2 C_{SP}^{3, 3+}$ ).
- Crucially, a new term representing the mutual interference between the two scalar mechanisms: $8|\epsilon_-||\epsilon_+| C_{SP}^{+-} \cos \psi_{+-}$ , where $\psi_{+-}$ is the relative complex phase of the couplings.
Explicit Coefficients: The paper provides explicit expressions for the interference coefficients ( $C_{SP}^{2}, C_{SP}^{2+}, C_{SP}^{3}, C_{SP}^{3+}, C_{SP}^{+-}$ ) in terms of the NME ratios ( $\chi'_{SP,F}, \chi'_{SP,B}, \chi'_{SP,D}$ ) and integrated phase-space factors ( $G_{SP}^{0k}$ ).
Positivity Bound: By formulating the pure-scalar rate as a quadratic form involving a matrix of phase-space factors, the author derives a positivity bound on the scalar-scalar interference coefficient: $|C_{SP}^{+-}|^2 \leq C_{SP}^{3} C_{SP}^{3+}$ . This serves as an internal consistency check for the formalism.
Consistency Checks: The derived formula successfully reproduces the corrected single-coupling scalar $P$ -wave results found in previous literature (Refs. [7, 8]) when either $\epsilon_+$ or $\epsilon_-$ is set to zero.

Significance and Claims
The paper claims to provide a consistent basis for the correlated analysis of the two scalar interactions in $0\nu\beta\beta$ . By retaining both scalar mechanisms coherently and explicitly deriving the mutual interference term, the work addresses a gap in previous treatments that often considered these terms separately.

The author emphasizes that their formulation:

Corrects and Extends: It cross-checks and extends previous scalar $P$ -wave results by including the simultaneous presence of both scalar couplings and their relative phase dependence.
Enables Correlated Constraints: The explicit dependence on the relative phase and the derived positivity bound allow for more rigorous constraints on the effective couplings when analyzing experimental data, preventing unphysical parameter combinations.
Foundation for Future Work: The dimensionally explicit formulation is intended to serve as a foundation for numerical implementations (such as the $\nu$ DoBe package) and future analyses that may incorporate pion-exchange and short-distance operators from chiral EFT.

The paper remains modest in its scope, focusing strictly on the analytical derivation of the long-range scalar amplitudes and their interference within the EFT framework, without proposing new experimental setups or claiming immediate resolution of the $0\nu\beta\beta$ mechanism.

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