⚛️ nuclear theory

Muon-to-positron conversion from local dimension-nine operators: pion-pole-improved nuclear multipoles

This paper initiates a systematic study of neutrinoless muon-to-positron conversion by constructing pion-pole-improved nuclear multipoles for local dimension-nine operators, deriving exact two-body nuclear tensors that account for unequal momentum transfers and form-factor arguments, and specializing the results for $0^+\to0^+$ transitions to facilitate direct contraction with charge-changing two-body transition densities.

Original authors: Yi Liao, Hao-Lin Wang

Published 2026-09-30

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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 vast, invisible landscape of the subatomic world, particles called muons and electrons are cousins, both belonging to a family known as leptons. Under the standard rules of physics, these particles are incredibly stable in their identities; a muon will not simply turn into an electron, nor will it vanish into a positron, the antimatter twin of the electron. However, a deeper theory suggests that these rules might be broken. If nature allows a muon to transform directly into a positron, it would mean that a fundamental quantity called lepton number is not conserved. This violation is a crucial clue, potentially explaining why the universe is made of matter rather than being a balanced mix of matter and antimatter. Scientists have long searched for this rare transformation, hoping to catch a glimpse of physics beyond what we currently understand.

The specific process researchers are hunting for involves a muon, usually captured by an atom's nucleus, turning into a positron while the nucleus itself changes its identity, losing two protons in the process. This event is so rare that if it happens at all, it is buried under a mountain of background noise and requires incredibly precise calculations to predict what the signal should look like. The challenge lies in the fact that the muon is not just a point particle floating in empty space; it is trapped inside an atom, interacting with a dense cluster of protons and neutrons. To find the signal, physicists must understand exactly how the muon interacts with this nuclear crowd, accounting for the complex forces that bind the nucleus together and the way the emitted positron moves away.

A team of researchers has now taken a major step forward in this effort by constructing a new, highly detailed map of how this transformation occurs. They focused on a specific type of theoretical interaction where the muon and the nucleus exchange energy through very short-range forces, described by a set of mathematical rules known as dimension-nine operators. Rather than relying on rough approximations that treat the nucleus as a simple, uniform blob, the team built a sophisticated model that respects the individual nature of the protons and neutrons inside. They carefully tracked how momentum flows between the particles, ensuring that the calculation accounts for the fact that the two nucleons involved in the reaction do not necessarily receive the same "kick" from the interaction.

A key innovation in their work is how they handle the movement of the outgoing positron. In previous studies, scientists often used a simplified correction to account for the electric pull of the nucleus, which works well for estimating the total rate of events but fails to capture the subtle details of the positron's path. The new model uses a more refined approach that preserves the specific spatial phase of the positron wave as it escapes the nucleus. This is vital because the positron's momentum is comparable to the size of the nucleus itself, meaning the wave-like nature of the particle interacts with the nuclear structure in a complex way that simple corrections miss. By keeping these details intact, the researchers created a more accurate description of the physical reality inside the atom.

The team also refined how they describe the forces acting between the protons and neutrons. They included all the known ways these particles can interact, such as their magnetic properties and their spin, while carefully filtering out effects that are theoretically expected to be negligible. They introduced a specific improvement regarding the role of pions, which are particles that mediate the strong force holding the nucleus together. Instead of discarding the long-range influence of these pions, they kept them in the calculation, ensuring that the model captures the full strength of the interaction. This "pion-pole-improved" method allows them to retain the most important physical effects without getting bogged down in unnecessary complexity.

The result of this work is a precise formula for the nuclear operator that governs the muon-to-positron conversion. This formula is designed to be used directly with modern computer simulations of atomic nuclei, allowing experimentalists to compare their data with theory more accurately than ever before. The researchers demonstrated that their method works perfectly for a specific, common type of nuclear transition where the nucleus starts and ends in a state with zero spin. In this case, many of the complicated mathematical terms that usually clutter the equations cancel out, leaving a clean, usable expression that connects the fundamental particle physics to the observable nuclear behavior.

This study does not claim to have discovered the muon-to-positron conversion; that discovery remains the goal of future experiments. Instead, the paper provides the essential theoretical toolkit needed to interpret those future results. By establishing a rigorous framework that avoids common shortcuts and preserves the intricate details of the nuclear environment, the authors have cleared a path for more reliable predictions. Their work ensures that when the next generation of detectors, such as the Mu2e and COMET experiments, finally look for this rare event, they will be looking with the sharpest possible theoretical eyes, ready to distinguish a genuine signal from the background noise of the universe.

Technical Summary: Muon-to-Positron Conversion from Local Dimension-Nine Operators

Problem Statement
This paper initiates a systematic study of neutrinoless muon-to-positron conversion ( $\mu^- + (A, Z) \to e^+ + (A, Z-2)$ ), a process that violates lepton number by two units ( $\Delta L = 2$ ) and changes charged-lepton flavor. While neutrinoless double-beta decay ( $0\nu\beta\beta$ ) is the primary probe for $\Delta L = 2$ interactions, it predominantly accesses the electron-flavor sector. The muon-to-positron conversion channel offers a complementary probe, particularly for short-range interactions described by dimension-nine (dim-9) operators in the low-energy effective field theory (LEFT).

Existing theoretical treatments of this process often rely on approximations that limit precision. Specifically, previous studies have frequently adopted a "common momentum" approximation for nucleon form factors, neglecting the fact that the two currents in the dim-9 operator can carry distinct momentum transfers. Furthermore, treatments of the outgoing positron often utilize simple Fermi-function rescalings that fail to capture the Coulomb-modified spatial phase and partial-wave dependencies, despite the positron momentum being comparable to the inverse nuclear radius.

Methodology
The authors construct a rigorous framework for calculating the nuclear matrix elements of local dim-9 operators containing a scalar or pseudoscalar lepton bilinear multiplied by two contracted vector or axial quark currents. The methodology proceeds through the following steps:

Spacetime and Momentum Formalism: Starting from the first-order transition element, the authors perform a spatial reduction. They explicitly demonstrate that the interaction-coordinate integral enforces momentum conservation only for the sum of the two current momenta ( $q_1 + q_2 = -p_{\text{eff}}$ ), not for their individual values. This necessitates retaining two independent momentum transfers at the nucleon vertices rather than assuming a common momentum.
Lepton Wave Functions: The bound muon wave function is treated via a nuclear average of the dominant $1s$ component. For the outgoing positron, the authors employ the Effective-Momentum Approximation (EMA). This approach incorporates the Coulomb normalization and a shifted spatial momentum ( $p_{\text{eff}}$ ) into an effective plane wave, capturing the essential Coulomb phase and magnitude effects without resorting to full numerical Dirac solutions.
Nuclear Currents and Impulse Approximation: The quark currents are reduced to one-body nucleon currents using the impulse approximation. The authors utilize the complete set of six nucleon form factors (vector, weak magnetism, induced scalar, axial, induced pseudoscalar, and induced tensor). To maintain physical clarity while managing complexity, they adopt two controlled specializations:
- Second-class currents (induced scalar and induced tensor) are set to zero.
- Pion-Pole-Improved (PI) Reduction: The remaining currents are reduced to Pauli operators. Crucially, the longitudinal axial current is treated with a pion-pole-improved prescription. This retains the full spin-longitudinal pion-pole current intact before contraction, rather than expanding it in inverse nucleon mass ( $1/m_N$ ) and discarding higher-order terms. This ensures that the $1/m_N^2$ terms arising from the pion pole are preserved, as they are physically significant.
Multipole Expansion: The authors perform an exact recoupling of the two-body nuclear operators. They address the non-polynomial angular dependence arising from unequal form-factor arguments ( $Q_1^2 \neq Q_2^2$ ) by expanding the form-factor products into relative-coordinate multipoles using Racah-normalized harmonics.

Key Contributions and Results

Exact Two-Body Operator for $0^+ \to 0^+$ Transitions: The primary result is the derivation of an exact PI two-body nuclear operator specifically for $0^+ \to 0^+$ nuclear transitions. While the scalar selection rule eliminates external angular momentum and magnetic sums, the authors show that finite-momentum internal orbital and spin tensors remain non-zero.
Generalization of Multipole Structure: The work demonstrates that unequal form-factor arguments generate relative-coordinate multipoles that are absent in the common-momentum approximation. This provides a more accurate description of the nuclear response.
Direct Contractibility: The final expression is formulated to be directly contractible with charge-changing two-body transition densities (TBTDs), facilitating immediate application in many-body nuclear calculations.
Systematic Framework: The paper establishes the spacetime, lepton-wave, single-nucleon-current, and nuclear-multipole framework for local dim-9 interactions. It provides a general six-form-factor construction and arbitrary angular momentum transitions in the appendices, though the main text focuses on the physically important $0^+ \to 0^+$ case.

Significance and Claims
The paper claims to provide the necessary theoretical infrastructure for precise calculations of muon-to-positron conversion rates driven by short-range dim-9 operators. By moving beyond the common-momentum approximation and incorporating the pion-pole-improved reduction, the authors address specific deficiencies in previous nuclear calculations. The work is presented as the first in a series, establishing the foundational operator structure while leaving quantitative many-body applications and other mechanisms for subsequent studies. The authors emphasize that their approach is generalizable to other local dim-9 operators by replacing the elementary lepton and nucleon-current tensors, thereby offering a robust tool for interpreting future experimental data from facilities like Mu2e and COMET.

Technical Summary: Muon-to-Positron Conversion from Local Dimension-Nine Operators

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