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Muon-to-positron conversion from light Majorana neutrino exchange: nuclear responses and pion-pole-improved multipoles

This paper derives the nuclear responses and pion-pole-improved multipole formulas for muon-to-positron conversion mediated by light Majorana neutrino exchange, providing explicit expressions for the conversion rate and closure operators to facilitate future nuclear matrix element calculations without offering specific isotope predictions.

Original authors: Yi Liao, Hao-Lin Wang

Published 2026-10-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yi Liao, Hao-Lin Wang

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

In the subatomic world, there is a fundamental rule that nature seems to follow: the total number of particles known as leptons remains constant. Electrons and muons are two types of these particles, and in standard physics, they can change into one another, but only if a matching neutrino is involved to balance the books. However, a long-standing mystery in physics asks whether this rule is absolute. Some theories suggest that neutrinos are their own antiparticles, a property that would allow them to break the lepton-number rule. If this is true, it would explain why the universe is made of matter rather than antimatter. Scientists have been searching for evidence of this violation for decades, primarily by watching for a rare event called neutrinoless double-beta decay, where a nucleus emits two electrons without any neutrinos.

There is, however, a second way to look for this same violation, one that involves a different pair of particles: a muon and a positron. A muon is a heavy cousin of the electron, while a positron is the antimatter twin of the electron. In a process called muon-to-positron conversion, a muon captured by an atomic nucleus could transform directly into a positron, changing the nucleus in the process. This event would violate the lepton-number rule by two units, just like the double-beta decay, but it probes a different combination of neutrino properties. While experiments have set limits on how often this might happen, the theoretical tools needed to predict exactly what to look for in different atomic nuclei have been incomplete. Understanding the precise mechanics of this conversion is essential for designing future experiments that could finally catch this elusive signal.

A team of researchers has now provided a comprehensive new framework for calculating how this conversion happens inside an atomic nucleus. Their work focuses on the "mass mechanism," a specific way the process could occur if neutrinos have mass and are their own antiparticles. The scientists started from the basic laws of particle physics and built a detailed map of how the interaction unfolds within the complex environment of a nucleus. They treated the nucleus not as a single point, but as a collection of protons and neutrons, each moving and interacting with the others. By carefully tracking the flow of energy and momentum between the incoming muon, the outgoing positron, and the invisible neutrino that mediates the change, they derived a complete set of instructions for calculating the probability of this event.

One of the key achievements of this study is the refinement of how the forces inside the nucleus are described. The researchers incorporated several subtle effects that previous calculations often ignored or simplified too much. They included the influence of a particle called the pion, which acts as a carrier of the strong nuclear force, and they accounted for the motion of the protons and neutrons as they react to the conversion. They also developed a method to handle the fact that the positron, as it flies out of the nucleus, is repelled by the positive charge of the remaining protons. By using an improved approximation for this repulsion, they ensured that the predicted rates for the conversion are more accurate than before. The team also clarified how the process depends on the specific arrangement of the protons and neutrons, showing that the calculation must account for the spins and positions of individual nucleons rather than just the average properties of the nucleus.

The researchers applied their new formulas to a specific case where the starting and ending nuclei have zero spin, a situation that simplifies the math but still captures the core physics. They found that the conversion rate depends on two main components: a symmetric part, which behaves like a simple scalar quantity, and an antisymmetric part, which involves the vector nature of the interaction. Crucially, they demonstrated that both parts contribute to the final result and cannot be ignored. This finding challenges older assumptions that suggested the antisymmetric part might vanish in certain cases. The team provided explicit mathematical recipes, free of the complex intermediate steps, that allow other scientists to plug in data for specific elements like titanium or aluminum and get a precise prediction for the conversion rate.

Despite these advances, the authors are careful to note that their work is a theoretical foundation rather than a final answer. They have not yet calculated the specific rates for any particular element, leaving that task to nuclear physicists who specialize in modeling the complex structure of atomic nuclei. Their formulas are designed to be used with different nuclear models, allowing for a comparison of results to understand the uncertainties involved. They also acknowledge that their current treatment uses some approximations, such as assuming the nucleus does not recoil significantly during the process, and that future work will need to include more complex interactions between pairs of nucleons. Nevertheless, this study provides the necessary bridge between the abstract theory of neutrino physics and the concrete calculations required to guide experimental searches.

The paper serves as a vital update to the toolkit available to scientists hunting for new physics. By laying out the nuclear responses in a clear and systematic way, the researchers have removed a significant barrier to progress. Their work ensures that when future experiments, such as those planned for the Mu2e and COMET projects, report their findings, the theoretical predictions used to interpret the data will be based on a more complete and rigorous understanding of the nuclear environment. This precision is essential, as the signal they are looking for is incredibly faint, and distinguishing it from background noise requires the highest possible accuracy in the theoretical models. The study does not claim to have discovered the conversion process, but it has successfully mapped the terrain where that discovery might one day be made.

Technical Summary: Muon-to-positron conversion from light Majorana neutrino exchange: nuclear responses and pion-pole-improved multipoles

Problem Statement
This paper investigates the muon-to-positron conversion process in nuclei, $\mu^- + (A, Z) \to e^+ + (A, Z-2)$ , mediated by the exchange of light Majorana neutrinos. This process violates lepton number by two units and probes the electron-muon sector, offering a flavor-complementary probe to neutrinoless double-beta decay ( $0\nu\beta\beta$ ). While the particle-physics parameter for this process is the effective mass $m_{\mu e} = \sum_j U_{\mu j} U_{ej} m_j$ , calculating the nuclear matrix elements (NMEs) required to connect this parameter to observable rates is complex. Unlike local interactions, light-neutrino exchange involves physical intermediate nuclear states and distinct momentum transfers at the two hadronic vertices. Previous studies have addressed aspects of this, but a systematic derivation of nuclear responses using modern nucleon form factors and a complete angular reduction for the mass mechanism has been lacking.

Methodology
The authors derive the conversion amplitude starting from the second-order effective four-Fermi interaction Hamiltonian. The methodology proceeds through several rigorous steps:

Effective Interaction and S-Matrix: The process is treated as a second-order perturbation involving two weak currents. The S-matrix element is constructed by integrating over spacetime coordinates, leading to a neutrino propagator and time-ordered products of nuclear currents.
Nucleon Currents (PI Prescription): The authors employ a "pion-pole improved" (PI) prescription for the single-nucleon currents within the one-body impulse approximation. This includes convection, axial charge, weak magnetism, and the longitudinal pion-pole current. They explicitly retain terms proportional to $m_N^{-2}$ and avoid a complete Foldy-Wouthuysen expansion, instead using a specified current truncation. A static limit benchmark is also defined for comparison.
Intermediate States and Time Orderings: The nuclear response is derived by summing over physical intermediate nuclear states. The authors explicitly handle the two physical time orderings (muon vertex first vs. positron vertex first) and the distinct momentum transfers associated with each. This leads to resolvent operators that depend on the energy differences between intermediate and initial/final states.
Angular Reduction and Closure: The formalism performs a complete angular reduction of the nuclear response.
- Spectral Response: The neutrino direction is integrated while retaining intermediate states, expressing the response in terms of one-current multipoles and radial integrals.
- Closure Approximation: When explicit intermediate states are unavailable, the authors derive a closure operator. This replaces excitation energies with average values, converting the response into an instantaneous two-body operator. This operator includes spin-gradient terms and is constructed to be compatible with nuclear structure calculations.
Lepton Contraction and Rate: The lepton tensor is contracted with the nuclear response tensor. The authors distinguish between symmetric (scalar) and antisymmetric (vector) responses, showing that both contribute to the spin-averaged, angle-integrated rate. The effective momentum approximation (EMA) is applied to account for Coulomb distortion of the emitted positron.

Key Contributions

Derivation of Nuclear Responses: The paper provides a first-principles derivation of the nuclear response functions for the light Majorana neutrino mass mechanism, extending previous work on local dimension-nine operators.
Pion-Pole Improved Formalism: The authors retain a comprehensive set of nucleon form factors (convection, axial charge, weak magnetism, and pion pole) within a specified prescription, rather than relying on simplified static limits.
Complete Angular Reduction: The work derives the closure operator and its two-body multipoles, including spin-gradient terms, which are necessary for accurate nuclear structure calculations.
Symmetric and Antisymmetric Responses: The paper demonstrates that both symmetric (scalar) and antisymmetric (vector) lepton-hadron Lorentz contractions contribute to the rate. It explicitly shows that the antisymmetric response does not vanish even for $0^+ \to 0^+$ transitions, as the positron direction supplies the necessary vector structure.
Explicit Formulas for $0^+ \to 0^+$ : For transitions between spin-zero ground states (e.g., $^{48}\text{Ti} \to ^{48}\text{Ca}$ ), the authors provide explicit formulas in terms of two coherent radial amplitudes, $A_0(P)$ and $B_0(P)$ , without restricting intermediate nuclei to spin zero.

Results
The paper presents the theoretical framework and analytical formulas required to compute the conversion rate.

The conversion rate is expressed in terms of nuclear matrix elements of operators constructed from two-body currents and closure resolvents.
For $0^+ \to 0^+$ transitions, the rate depends on the coherent sum of scalar and vector radial amplitudes: $\Gamma \propto |A_0(P) + B_0(P)|^2$ .
The formalism distinguishes between the spectral calculation (requiring intermediate state energies and transition matrix elements) and the closure calculation (requiring two-body transition densities).
The authors provide a guide for nuclear calculations, including the necessary inputs (masses, binding energies, form factors) and the structure of the operators in both spectral and closure limits.

Significance and Scope
The authors position this work as the second paper in a series, following their study of local dimension-nine interactions. Its primary significance lies in providing a rigorous "interface" between particle physics parameters and nuclear structure calculations. By deriving the nuclear responses and closure operators explicitly, the paper enables nuclear physicists to calculate the required NMEs using various many-body methods (e.g., QRPA, shell model).

The paper is modest in its claims regarding immediate phenomenological predictions. It explicitly states that no isotope-specific numerical predictions are attempted in this work. The treatment uses approximate lepton wavefunctions and recoil-free nuclear propagation. The authors note that two-body weak currents and systematic chiral effective-field-theory matching are left for future work. The immediate goal is to establish the formalism and assumptions (such as the PI prescription and closure averages) so that they can be tested and refined in subsequent isotope-specific calculations, particularly for targets like $^{48}\text{Ti}$ and $^{27}\text{Al}$ .

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