Muon-to-Positron Conversion

• Muon-to-positron conversion is a nuclear process where a bound muon converts into a positron with a nuclear charge reduction, violating lepton number and flavor by two units.
• The process involves detailed energy kinematics and nuclear matrix element evaluations, serving as a complementary probe to neutrinoless double beta decay.
• Experimental designs focus on distinguishing the narrow monoenergetic positron signal from backgrounds like radiative muon capture through optimal target selection.

Searching arXiv for the cited muon-to-positron conversion papers to ground the article in current arXiv records. First, I’ll look up the 2021 review on muon-to-positron conversion. Searching arXiv for (Lee et al., 2021) Muon-to-positron conversion denotes the nuclear process

$\mu^- + N(A,Z)\to e^+ + N'(A,Z-2),$

in which a negative muon bound in a muonic atom is converted into a positron while the nuclear charge decreases by two units. It is a charged-lepton process that violates both lepton flavor and total lepton number by two units, and is therefore conceptually distinct from ordinary $\mu^- \to e^-$ conversion, ordinary muon capture, and other positron-producing reactions in stopped-muon experiments. In the contemporary literature it is treated as a complementary probe to neutrinoless double beta decay, particularly because it accesses mixed $e\mu$ flavor structure rather than the $ee$ sector alone (Lee et al., 2021, Geib et al., 2016).

1. Definition, quantum numbers, and conceptual status

The defining reaction changes both flavor and lepton number. In the notation used in recent studies, the process has

$\Delta L_\mu=-1,\qquad \Delta L_e=+1,$

so the net lepton number changes by two units. It is therefore simultaneously lepton-flavor violating and lepton-number violating. This distinguishes it from ordinary muon capture,

$\mu^- + (A,Z)\to \nu_\mu + (A,Z-1)^*,$

which conserves lepton number, and from ordinary $\mu^- \to e^-$ conversion,

$\mu^- + (A,Z)\to e^- + (A,Z),$

which violates flavor but conserves total lepton number (Sahu et al., 9 Jul 2025, Mackenzie et al., 2020).

The process occupies a special position in beyond-the-Standard-Model searches. A 2021 review emphasizes that charged-lepton flavor violation in the minimally extended Standard Model with neutrino masses appears only through loop effects and is suppressed to branching ratios below $10^{-50}$, while lepton number violation remains forbidden there. In that sense, observation of $\mu^- \to e^+$ conversion would constitute unambiguous evidence for physics beyond the Standard Model (Lee et al., 2021).

Its closest conceptual analogue is neutrinoless double beta decay. Both are $\mu^- \to e^-$0 observables and both can be mediated by Majorana neutrino exchange, but they probe different flavor combinations. In $\mu^- \to e^-$1, the relevant quantity is flavor-diagonal, whereas muon-to-positron conversion probes mixed-flavor structure, often written in terms of the off-diagonal effective Majorana mass

$\mu^- \to e^-$2

This flavor complementarity is one of the main reasons the process remains theoretically important even though its minimal light-neutrino rate is extremely small (Berryman et al., 2016, Sahu et al., 9 Jul 2025).

A terminological complication persists in the literature. One review describes the general process as “incoherent” because the nucleus usually changes state, while the short-range EFT treatment developed for ground-state transitions explicitly focuses on coherent conversion. This suggests that “muon-to-positron conversion” functions as an umbrella term covering both the general nuclear transition and the experimentally most favorable ground-state channel (Lee et al., 2021, Geib et al., 2016).

2. Signal kinematics and nuclear structure

For a ground-state transition, the emitted positron is monoenergetic. The signal energy is written as

$\mu^- \to e^-$3

This is the central kinematic relation in experimental searches because it determines the endpoint region in which a narrow positron line would appear (Lee et al., 2021).

The literature gives several benchmark ground-state signal energies:

Target nucleus	$\mu^- \to e^-$4
$\mu^- \to e^-$5	$\mu^- \to e^-$6 MeV
$\mu^- \to e^-$7	$\mu^- \to e^-$8 MeV
$\mu^- \to e^-$9	$e\mu$0 MeV
$e\mu$1	$e\mu$2 MeV

These values matter because the principal experimental background, radiative muon capture, has its own endpoint, and the relative placement of the signal and background endpoints depends strongly on nuclear masses. The 2021 review therefore formulates a practical target-selection criterion,

$e\mu$3

so that the $e\mu$4 signal lies above the radiative-muon-capture endpoint and is kinematically protected. Titanium, sulfur, and calcium are highlighted as favorable in this respect, whereas aluminum is described as less favorable because the signal lies close to the radiative-muon-capture spectrum (Lee et al., 2021).

The gold-target case illustrates the role of atomic binding and recoil corrections in detail. For $e\mu$5, the appendix of the SINDRUM-II reanalysis writes

$e\mu$6

with

$e\mu$7

yielding

$e\mu$8

The same analysis notes that the reconstructed signal position is shifted lower by energy loss before tracking, and that on gold the expected $e\mu$9 peak is about $ee$0 below the $ee$1 conversion peak (Mackenzie et al., 2020).

Nuclear structure enters not only through masses but also through transition matrix elements. The 2021 review writes the nuclear matrix element schematically as

$ee$2

with Gamow–Teller, tensor, and Fermi components, and notes that the Gamow–Teller term is typically dominant. It also stresses that the scale of nuclear-matrix-element uncertainty is important, by analogy with $ee$3, because different nuclear models can vary by up to a factor of three (Lee et al., 2021).

3. Effective descriptions and microscopic mechanisms

In the light-Majorana-neutrino picture, the conversion rate normalized to ordinary muon capture is written as

$ee$4

For titanium, the same review quotes

$ee$5

for light neutrino exchange with normal hierarchy,

$ee$6

for inverted hierarchy, and

$ee$7

for heavy-neutrino exchange. Even the heavy-neutrino value remains far below foreseeable experimental sensitivity if only the minimal mechanism is present (Lee et al., 2021).

A complementary formulation uses short-range effective operators of dimension nine. The general effective Lagrangian is written as

$ee$8

In the coherent non-relativistic nuclear limit, the $ee$9, $\Delta L_\mu=-1,\qquad \Delta L_e=+1,$0, and $\Delta L_\mu=-1,\qquad \Delta L_e=+1,$1 structures do not contribute. The detailed 2016 treatment therefore focuses on $\Delta L_\mu=-1,\qquad \Delta L_e=+1,$2, the single short-range operator for which the corresponding nuclear matrix element is presently known (Geib et al., 2016).

For that benchmark operator,

$\Delta L_\mu=-1,\qquad \Delta L_e=+1,$3

the factorized amplitude takes the form

$\Delta L_\mu=-1,\qquad \Delta L_e=+1,$4

and the corresponding rate is

$\Delta L_\mu=-1,\qquad \Delta L_e=+1,$5

This formalism makes explicit how UV models are mapped onto a small set of Wilson coefficients and then onto nuclear matrix elements (Geib et al., 2016).

The broader operator survey reaches a complementary conclusion. For dimension-five, -seven, and -nine lepton-number-violating operators regarded as the source of neutrino masses, the predicted $\Delta L_\mu=-1,\qquad \Delta L_e=+1,$6 rates are generally extremely small, typically

$\Delta L_\mu=-1,\qquad \Delta L_e=+1,$7

That survey also emphasizes that $\Delta L_\mu=-1,\qquad \Delta L_e=+1,$8 probes the $\Delta L_\mu=-1,\qquad \Delta L_e=+1,$9 flavor component while $\mu^- + (A,Z)\to \nu_\mu + (A,Z-1)^*,$0 probes $\mu^- + (A,Z)\to \nu_\mu + (A,Z-1)^*,$1, so the latter can remain informative even when $\mu^- + (A,Z)\to \nu_\mu + (A,Z-1)^*,$2 is structurally suppressed by flavor (Berryman et al., 2016).

4. Search history, target choice, and projected reach

The best published direct limit summarized in the 2021 review is

$\mu^- + (A,Z)\to \nu_\mu + (A,Z-1)^*,$3

at $\mu^- + (A,Z)\to \nu_\mu + (A,Z-1)^*,$4 CL from SINDRUM II. Earlier searches also used sulfur, copper, iodine, and gold targets, with reported limits ranging from $\mu^- + (A,Z)\to \nu_\mu + (A,Z-1)^*,$5 to $\mu^- + (A,Z)\to \nu_\mu + (A,Z-1)^*,$6 depending on target and era. Experimental methods were generally direct positron searches after stopping muons in a target, although one older iodine search used a radiochemical method (Lee et al., 2021).

The SINDRUM-II gold-target positron spectrum became a focal case study because it contained a high-end excess. A 2020 reanalysis estimates $\mu^- + (A,Z)\to \nu_\mu + (A,Z-1)^*,$7 events in the interval

$\mu^- + (A,Z)\to \nu_\mu + (A,Z-1)^*,$8

with an expected background of about $\mu^- + (A,Z)\to \nu_\mu + (A,Z-1)^*,$9–$\mu^- \to e^-$0 event. The excess has a width consistent with the SINDRUM-II resolution and visually resembles a conversion peak, but its center is about $\mu^- \to e^-$1 lower than the expected $\mu^- \to e^-$2 signal. A fit in the region $\mu^- \to e^-$3 MeV/c gives a $\mu^- \to e^-$4-value of $\mu^- \to e^-$5, which strongly disfavors the exotic interpretation. The same work argues that an exclusive dipole radiative muon capture transition,

$\mu^- \to e^-$6

with branching fraction of order

$\mu^- \to e^-$7

could instead account for the bump (Mackenzie et al., 2020).

Future prospects are tied to facilities built primarily for ordinary $\mu^- \to e^-$8 conversion. The 2021 review identifies COMET Phase I, Mu2e, COMET Phase II, and a possible Mu2e-II as the main platforms, and summarizes approximate single-event sensitivities for the conversion programs as around $\mu^- \to e^-$9 for COMET Phase I, around $\mu^- + (A,Z)\to e^- + (A,Z),$0 for Mu2e and COMET Phase II, and around $\mu^- + (A,Z)\to e^- + (A,Z),$1 for Mu2e-II (Lee et al., 2021). A 2016 operator survey, making crude $\mu^- + (A,Z)\to e^- + (A,Z),$2 extrapolations for aluminum, gives

$\mu^- + (A,Z)\to e^- + (A,Z),$3

and emphasizes that experiments reconstructing the sign of the outgoing charged particle can in principle distinguish positrons from electrons, whereas transport designs that sweep positrons away make the dedicated $\mu^- + (A,Z)\to e^- + (A,Z),$4 search much harder (Berryman et al., 2016).

These considerations make target choice a first-order design variable rather than a secondary optimization. Aluminum is optimal for ordinary $\mu^- + (A,Z)\to e^- + (A,Z),$5 conversion in current first-generation programs, but the $\mu^- + (A,Z)\to e^- + (A,Z),$6 literature repeatedly stresses that the best target is one for which the signal sits above the radiative-muon-capture endpoint (Lee et al., 2021).

5. Radiative muon capture and other positron-producing backgrounds

Radiative muon capture is the dominant background issue in the modern literature. The basic process is

$\mu^- + (A,Z)\to e^- + (A,Z),$7

followed by

$\mu^- + (A,Z)\to e^- + (A,Z),$8

If the resulting positron is sufficiently energetic, it can populate the signal region. The 2021 review emphasizes that the high-energy tail of the radiative-muon-capture spectrum is poorly constrained experimentally, especially for the tiny endpoint region relevant to conversion searches (Lee et al., 2021).

The same review writes the radiative-muon-capture endpoint as

$\mu^- + (A,Z)\to e^- + (A,Z),$9

For aluminum it gives a photon endpoint of about $10^{-50}$0 MeV, corresponding to a positron endpoint about one electron mass lower. It also notes that the commonly used closure approximation fitted TRIUMF aluminum data with an endpoint parameter of about

$10^{-50}$1

roughly $10^{-50}$2 MeV below the true kinematic endpoint, and argues that nothing forbids photons in the interval between the fitted and true endpoints (Lee et al., 2021).

A major refinement is the distinction between external and internal pair conversion. In radiative muon capture, a real photon can pair-produce later in surrounding material, or a virtual photon can generate the $10^{-50}$3 pair internally at the amplitude level. The latter is the more serious irreducible background for $10^{-50}$4 searches. Near the positron endpoint, the internal spectrum can be related to the measured real-photon spectrum from the same nucleus through

$10^{-50}$5

so that once $10^{-50}$6 is known, the endpoint internal-positron background can be predicted with a calculable conversion kernel (Plestid et al., 2020).

The gold-target excess discussed above sharpened this concern. In that case the closure-approximation radiative-muon-capture model predicts only about $10^{-50}$7 events above $10^{-50}$8 MeV/c, while the observed high-end tail is much larger. The reanalysis therefore argues that exclusive low-lying final states may significantly modify the endpoint spectrum and that high-resolution photon measurements are required to exploit the full physics potential of Mu2e and COMET (Mackenzie et al., 2020).

Stopped-muon facilities also produce positrons through mechanisms that are not $10^{-50}$9 conversion at all. In COMET phase-II, electron antineutrinos from decay in orbit can undergo inverse beta decay on protons in the aluminum target,

$\mu^- \to e^+$0

yielding about

$\mu^- \to e^+$1

positron events per $\mu^- \to e^+$2 stopped muons. This is a secondary neutrino-induced positron source associated with stopped muons, not a direct charged-lepton-flavor-violating or lepton-number-violating conversion signal (Mulmule, 2021).

6. Interpretive extensions and boundaries of the subject

Although the minimal neutrino-mass contribution is negligible, recent work has explored mechanisms that amplify the effective off-diagonal Majorana parameter probed by the process. One example introduces an ultralight scalar dark matter background that couples to neutrinos and induces

$\mu^- \to e^+$3

After time averaging in the fast-oscillation regime,

$\mu^- \to e^+$4

The associated paper states that the mechanism can enhance $\mu^- \to e^+$5 conversion to experimentally observable levels, can operate even when $\mu^- \to e^+$6, and may allow current and upcoming searches to constrain flavor-off-diagonal neutrino–ULSDM couplings down to about

$\mu^- \to e^+$7

for sufficiently light scalar masses, with the most relevant window roughly

$\mu^- \to e^+$8

This suggests that an observed signal would not, by itself, isolate a unique microscopic origin of lepton-number violation (Sahu et al., 9 Jul 2025).

The process is also bounded conceptually by several adjacent but distinct topics. Threshold positron annihilation,

$\mu^- \to e^+$9

as studied in LEMMA, concerns positron-driven muon-pair production rather than nuclear $\mu^- \to e^-$00 conversion. Muonium-to-antimuonium conversion,

$\mu^- \to e^-$01

is a $\mu^- \to e^-$02 atomic transition with different experimental signatures and detector systems. Dark-matter models that produce positrons through

$\mu^- \to e^-$03

likewise generate positrons only through subsequent muon decay, not through direct muon-to-positron conversion. These distinctions matter because the phrase “muon-to-positron conversion” is sometimes used loosely in surrounding discussions, whereas the nuclear reaction

$\mu^- \to e^-$04

has a sharply defined meaning in the lepton-number-violation literature (Amapane et al., 2019, Alesini et al., 2019, Lu et al., 11 Aug 2025, Tomar et al., 2014).

Within that literature, the long-term significance of the process lies in its flavor structure. It tests $\mu^- \to e^-$05 dynamics in the $\mu^- \to e^-$06 sector, where the underlying operator content need not track the $\mu^- \to e^-$07 sector probed by $\mu^- \to e^-$08. A plausible implication is that future interpretation will depend at least as much on nuclear and radiative-muon-capture control as on raw stopped-muon statistics: the experimental frontier is defined not only by sensitivity, but by whether the endpoint positron spectrum can be understood with sufficient fidelity to distinguish genuine lepton-number violation from nuclear Standard-Model backgrounds (Lee et al., 2021, Berryman et al., 2016).