###### Abstract

Lepton flavor violation (LFV) has been discovered in the neutrino sector by
neutrino oscillation experiments. The minimal extension of the Standard Model
(SM) to include neutrino masses allows LFV in the charged sector (CLFV) at the
loop-level, but at rates that are too small to be experimentally observed.
Lepton number violation (LNV) is explicitly forbidden even in the minimally
extended SM, so the observation of a LNV process would be unambiguous evidence
of physics beyond the SM. The search for the LNV and CLFV process (referred to
as $\MuToEp$ conversion) complements decay searches, and is sensitive to
potential flavor effects in the neutrino mass generation mechanism. A
theoretical motivation for $\MuToEp$ conversion is presented along with a
review of the status of past $\MuToEp$ conversion experiments and future
prospects. Special attention is paid to an uncertain and potentially dominant
background for these searches, namely, radiative muon capture (RMC). The RMC
high energy photon spectrum is theoretically understudied and existing
measurements insufficiently constrain this portion of the spectrum, leading to
potentially significant impacts on current and future $\MuToEp$ searches.

###### keywords:

Muon; muon conversion; charged lepton flavor violation; CLFV; lepton number
violation; LNV; radiative muon capture

1 1 0 https://doi.org/ Muon to positron conversion Muon to positron conversion
MyeongJae Lee 1,† and Michael MacKenzie 2,† Correspondence:
<EMAIL_ADDRESS>These authors contributed equally to this
work.

##  1 Introduction

The incoherent conversion of a negative muon into a positron in a muonic atom,
(referred to as $\mu^{-}\to e^{+}$ hereafter), is an exotic process that is
both lepton flavor violating (LFV) and lepton number violating (LNV) with a
change in lepton number by two units ($\Delta L=2$). The conservation of
charged lepton number and flavor has been very well experimentally
established. The symmetry corresponding to the conservation of lepton flavor
is broken in the Standard Model (SM) by the introduction of neutrino masses,
and the symmetry corresponding to the conservation of lepton number can
additionally be broken by the introduction of new interactions and particles.
In the last few decades, LFV decays of the muon have been studied
experimentally through three processes, each forbidden before the introduction
of neutrino masses: $\mu^{+}\to e^{+}\gamma$, $\mu^{+}\to e^{+}e^{-}e^{+}$,
and $\mu^{-}+N\to e^{-}+N$ (referred to as $\mu^{-}\to e^{-}$). Experiments
searching for $\mu^{-}\to e^{-}$ can also typically search for $\mu^{-}\to
e^{+}$, and so are able to investigate both LFV and LNV in the muon sector.
The most recent result on muon LFV processes comes from the MEG experiment
searching for $\mu^{+}\to e^{+}\gamma$, reporting $B(\mu^{+}\to
e^{+}\gamma)<4.2\times 10^{-13}$ at 90% confidence level (CL) Baldini, A. M.;
et al. (2016) (MEG collaboration). The upgrade of the MEG experiment to
improve the experimental sensitivity by an order of magnitude is in progress
Baldini, A. M.; et al. (2021) (MEG collaboration). The new Mu3e Arndt K.; et
al. (2021) (Mu3e collaboration) at PSI will search for $\mu^{+}\to
e^{+}e^{-}e^{+}$, improving the sensitivity by three orders of magnitude in
Phase I. New results for $\mu^{-}\to e^{-}$ with 100 - 10,000 times better
sensitivity are expected by the end of the decade from the COMET Abramishvili,
R.; et al. (2020) (COMET collaboration) and Mu2e Bartoszek, L.; et al. (2015)
(Mu2e collaboration) experiments.

The theoretical branching ratio expectations for muon LFV decays in the SM
minimally extended to include neutrino masses are extremely small, below
$10^{-50}$ for all three searched for processes. The rate is suppressed by
$((\Delta m_{\nu}^{2})/M_{W}^{2})^{2}$ where $m_{\nu}$ is the neutrino mass
and $M_{W}$ is the W-boson mass Marciano et al. (2008), and also due to the
GIM (Glashow–Iliopoulos–Maiani) mechanism. Therefore, observation of any
charged LFV (CLFV) process is direct evidence of physics beyond the SM.

In the LNV case, such as $\mu^{-}\to e^{+}$, the process is not allowed as a
perturbative process. The minimal extension of the SM with a Majorana neutrino
allows for the famous LNV process, neutrinoless double-beta decay (), to
occur. In fact, the Feynman diagrams for $\mu^{-}\to e^{+}$ and mediated by a
Majorana neutrino are very similar, as depicted in Figure 1, except that
$\mu^{-}\to e^{+}$ also involves a change of lepton flavor. The $\mu^{-}\to
e^{+}$ and processes are complimentary in the sense that involves same-flavor
(or “flavor-diagonal”) transitions while $\mu^{-}\to e^{+}$ involves a
different flavor (of “flavor-off-diagonal”) transition. In some theories for
physics beyond the SM, flavor-diagonal transitions are suppressed and
observable signals of new physics are associated with different-flavor
processes.

Benefiting from progress in detector technology and the feasibility of
obtaining a large amount of target material, models involving LNV in general
are more easily observable at experiments, such as KamLAND-Zen Gando (2020),
than in $\mu^{-}\to e^{+}$ searches. This is true for the case that a light
Majorana neutrino mediates the LNV, where would be more easily measurable than
$\mu^{-}\to e^{+}$. The effect on the and $\mu^{-}\to e^{+}$ processes in a
specific new physics model, for example, a heavy sterile neutrino model, is
not very well studied. The “Black box theorem” Schechter and Valle (1982);
Nieves (1984); Takasugi (1984); Hirsch et al. (2006) relates the Majorana
neutrino mass to the amplitude of and any $\Delta L=2$ processes even in the
presence of new physics, showing that the occurrence of any $\Delta L=2$
process, including $\mu^{-}\to e^{+}$, implies a nonzero Majorana neutrino
mass. An enhancement or suppression of the rate of $\mu^{-}\to e^{+}$ over was
suggested by some new physics models Berryman et al. (2017); Geib et al.
(2017); Geib and Merle (2017); Chen et al. (2007); King et al. (2014);
Pritimita et al. (2016); Cirigliano et al. (2004). In addition, some theories
explain the LNV of without a Majorana neutrino exchange, but instead using a
new mediator particle from a supersymmetric theory or a Majoron: see the
discussion by Engel and Menéndez Engel and Menéndez (2017). These models
discussed by Engel and Menéndez can also be used to describe $\mu^{-}\to
e^{+}$. Therefore, from one perspective, $\mu^{-}\to e^{+}$ is a LNV search
mediated by a Majorana neutrino similar to experiments, and from another
perspective, it is a new physics search regarding the flavor effect on the
neutrino mixing in the framework of the theory beyond the SM. While there are
other LNV searches such as $\tau^{-}\to e^{+}\pi^{-}\pi^{-}$ Miyazaki, Y.; et
al. (2013) (Belle collaboration) or $K^{\pm}\to\pi^{\mp}\mu^{\pm}\mu^{\pm}$
Cortina Gil, E.; et al. (2019) (NA62 collaboration), those are not similar
processes to or $\mu^{-}\to e^{+}$, so their direct comparison with
$\mu^{-}\to e^{+}$ or is very difficult. A few very recent studies have
suggested comparing the results with the collider experimental results for the
$t$-channel process of same-sign W-boson scattering Fuks et al. (2021a, b);
Cirigliano et al. (2021).

Figure 1: The tree level diagram of $\mu^{-}\to e^{+}$ (left) and (right) in a
Majorana neutrino model.

From the experimental point of view, $\mu^{-}\to e^{+}$ is attracting more
interest as the future $\mu^{-}\to e^{-}$ experiments progress. The COMET
Abramishvili, R.; et al. (2020) (COMET collaboration) and Mu2e Bartoszek, L.;
et al. (2015) (Mu2e collaboration) experiments will search for $\mu^{-}\to
e^{-}$ with an experimental sensitivity down to $10^{-17}$. Although the
experimental designs were not optimized for $\mu^{-}\to e^{+}$, these
experiments can significantly improve the sensitivity reach of $\mu^{-}\to
e^{+}$ owing to the considerable increase in the number of muons. The current
best experimental limits on $\mu^{-}\to e^{+}$ were obtained by the SINDRUM II
experiment with titanium nuclei: $B(\mu^{-}+{~{}\rm Ti}\to e^{+}+{~{}\rm
Ca})<1.7\times 10^{-12}$ at 90% CL to the ground state of calcium Kaulard et
al. (1998).

The purpose of this article is to emphasize the importance of the search for
$\mu^{-}\to e^{+}$ and assess its feasibility in future $\mu^{-}\to e^{-}$
experiments. Section 2 reviews the theoretical estimation of the $\mu^{-}\to
e^{+}$ rate in connection with the search. It also describes the past
experimental results for $\mu^{-}\to e^{+}$. Section 3 introduces the future
$\mu^{-}\to e^{-}$ experiments where $\mu^{-}\to e^{+}$ can be studied. A
major background component is analyzed and a mitigation strategy for
suppressing the background in order to improve the signal sensitivity of
$\mu^{-}\to e^{+}$ searches is also described.

##  2 Theories and past results

###  2.1 Estimation of the $\mu^{-}\to e^{+}$ rate in the extended SM with a
Majorana neutrino

A generic model of neutrino masses includes at least three light, left-handed
neutrinos, with neutrino flavor oscillations described by the
Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix Pontecorvo (1957a, b); Maki et
al. (1962). A right-handed neutrino, or any other hypothetical neutrino, may
have significantly larger mass than the light neutrinos, as is expected, for
example, from the Seesaw Mechanism Minkowski (1977); Gell-Mann et al. (1979);
Yanagida (1980); Glashow (1980); Mohapatra and Senjanović (1980); Schechter
and Valle (1980). In the minimal extension of the SM with a Majorana neutrino,
the $\mu^{-}\to e^{+}$ process is allowed through a light or heavy Majorana
neutrino exchange, as depicted in Figure 1. But, as described before, it is
also possible that other new particles beyond the SM or new interactions
between quarks and leptons may mediate the interaction.

Assuming the light or heavy Majorana neutrino exchange interaction, the
leading order $\mu^{-}\to e^{+}$ matrix element when the initial and final
states of nuclei are both ground states can be written as Domin et al. (2004):

$\displaystyle\mathscr{M}_{fi}$ $\displaystyle=$
$\displaystyle-i\left(\frac{G_{F}}{\sqrt{2}}\right)^{2}\frac{1}{\left(2\pi\right)^{3/2}}\frac{1}{\sqrt{4E_{\mu^{-}}E_{e^{+}}}}\bar{v}(k_{e^{+}})(1+\gamma_{5})u(k_{\mu^{-}})$
(1) $\displaystyle\times\frac{m_{e}g_{A}^{2}}{2\pi R}\left[\frac{\langle
m_{\nu}\rangle_{\mu e}}{m_{e}}\mathcal{M}_{\nu}+\langle M_{N}^{-1}\rangle_{\mu
e}m_{p}\mathcal{M}_{N}\right]\times
2\pi\delta\left(E_{\mu^{-}}+E_{i}-E_{f}-E_{e^{+}}\right)$
$\displaystyle\propto$ $\displaystyle\left\\{\begin{array}[]{ll}\langle
m_{\nu}\rangle_{\mu e}\mathcal{M}_{\nu}&{\rm(light~{}neutrino)}\\\ \langle
M_{N}^{-1}\rangle_{\mu
e}\mathcal{M}_{N}&{\rm(heavy~{}neutrino)}\end{array}\right.$ (4)

In this equation, $G_{F}$ is the Fermi constant, $m_{e}$ is the electron mass,
$g_{A}$ is the weak axial coupling constant, and $R$ is the nuclear radius.
$E_{(\mu^{-},e^{+},f,i)}$ represent the energy of the muon, positron, final
nuclear ground state, and initial nuclear ground state, respectively. The
effective neutrino masses are defined as:

$\langle m\rangle_{\alpha\beta}=\sum_{k}U_{\alpha k}U_{\beta k}m_{k}~{},$ (5)

for the light neutrino, and

$\langle M^{-1}_{N}\rangle_{\alpha\beta}=\sum_{k}\frac{U_{\alpha k}U_{\beta
l}}{M_{k}}~{},$ (6)

for the heavy neutrino, where $\alpha$ and $\beta$ are flavor indexes, and
$m_{k}$ and $M_{k}$ represent the light neutrino mass and the heavy neutrino
mass, respectively. $U$ is the neutrino mixing matrix, or PMNS matrix for the
light neutrino case. The effective neutrino mass matrix can be calculated from
the current estimation of the neutrino masses and mixing angles, in the case
of the light neutrino exchange model. For example, from a cosmological
observation, the sum of three light neutrino masses is less than 0.42 eV,
which translates to $\langle m_{\nu}\rangle_{l_{1}l_{2}}<0.14\,{\rm eV}$ Atre
et al. (2005).

Another important term in Equation 1 is $\mathcal{M}_{\nu/N}$, representing
the nuclear matrix element (NME) for the light and heavy neutrino,
respectively. This is a transition probability matrix of the nucleon from the
initial state to the final state:

$\mathcal{M}_{i}\propto\int dq\sum_{n}\langle f|J^{\mu}(x)|n\rangle\langle
n|J^{\nu}(y)|i\rangle=\mathcal{M}_{i}^{(GT)}+\mathcal{M}_{i}^{(T)}+\mathcal{M}_{i}^{(F)}~{},$
(7)

where $n$ is an intermediate nuclear state and $q$ and $J$ represent the
momentum transfer and hadronic current, respectively. Theoretically, the
matrix element is the sum of three components: the axial vector
($\mathcal{M}_{i}^{(GT)}$, Gamow-Teller term), tensor
($\mathcal{M}_{i}^{(T)}$), and vector ($\mathcal{M}_{i}^{(F)}$, Fermi term)
terms, where the axial vector NME is the dominant one Engel and Menéndez
(2017); Ejiri (2019). Because of uncertainties in the nucleus models, and
difficulty of calculating many body dynamics, the NME calculation usually
needs to be approximated. Approximation methods of NME calculations used for
experiments, such as the interacting shell model (ISM), quasiparticle random-
phase approximation (QRPA), interacting boson model (IBM), projected Hartree-
Fock-Bogoliubov model (PHFB), and energy density functional method (EDF), are
reviewed by Vergados et al. Vergados et al. (2012) and Engel and Menéndez
Engel and Menéndez (2017). Engel and Menéndez Engel and Menéndez (2017) also
show the NME calculations for in different materials and with different
approximation methods differ by up to a factor of three. This shows the
importance of understanding the NME for the process, and therefore for the
similar $\mu^{-}\to e^{+}$ process, in the minimal extension of the SM with a
Majorana neutrino.

Using this formulation, the theoretical estimation of the $\mu^{-}\to e^{+}$
rate was obtained by Domin et al. Domin et al. (2004):

$\displaystyle\mathcal{R}^{\mu^{-}e^{+}}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma(\mu^{-}+N(A,Z)\to
e^{+}+N^{\prime}(A,Z-2))}{\Gamma(\mu^{-}+N(A,Z)\to{\rm(All~{}muon~{}captures)})}$
(8) $\displaystyle=$ $\displaystyle 2.6\times
10^{-22}\times\left\\{\begin{array}[]{ll}\left|\langle m_{\nu}\rangle_{\mu
e}/m_{e}\right|^{2}\left|\mathcal{M}_{\nu}\right|^{2}&{\rm(light~{}neutrino)}\\\
\left|\langle M_{N}^{-1}\rangle_{\mu
e}m_{p}\right|^{2}\left|\mathcal{M}_{N}\right|^{2}&{\rm(heavy~{}neutrino){}}\end{array}\right\\}$
(11)

Applying the effective neutrino mass obtained from the experimental data on
the neutrinos masses and their mixing, and the NME for titanium from the QRPA
method, $\mathcal{R}^{\mu^{-}e^{+}}({\rm Ti})$ is:

$\begin{array}[]{rl}(0.008-1.7)\times 10^{-41}&{\rm
for~{}a~{}light~{}neutrino,~{}normal~{}neutrino~{}mass~{}hierarchy,}\\\
(0.05-6.7)\times 10^{-40}&{\rm
for~{}a~{}light~{}neutrino,~{}inverted~{}neutrino~{}mass~{}hierarchy,~{}and}\\\
\leq 3.8\times 10^{-24}&{\rm for~{}a~{}heavy~{}neutrino.}\end{array}$

While the estimated $\mu^{-}\to e^{+}$ rate is much higher in the heavy
Majorana neutrino case than the light neutrino one, it is far smaller than the
feasible experimental reach.

Any $\Delta L=2$ interaction other than the tree-level interaction depicted in
Figure 1 may lead to $\mu^{-}\to e^{+}$. This was studied by Berryman et al.
Berryman et al. (2017) by using an effective operator description and
normalizing the estimation of the Majorana neutrino exchange case (including
one- and two-loop corrections) with the above tree-level calculation. The
conversion rate was estimated according to the new physics energy scale
($\Lambda$), shown in Figure 2. From the expected sensitivity of Mu2e, the new
physics energy scale accessible from a $\mu^{-}\to e^{+}$ measurement is
comparably low, around 40 GeV, compared to the new physics scale accessible
from a $\mu^{-}\to e^{-}$ measurement at Mu2e of $\mathcal{O}(10^{4})$ TeV.
For all possible $\Delta L=2$ interactions leading to $\mu^{-}\to e^{+}$ and ,
the new physics scale reach of $\mu^{-}\to e^{+}$ searches is a few orders of
magnitude smaller than experiments. According to Berryman et al. Berryman et
al. (2017), the observation of $\mu^{-}\to e^{+}$ would imply that: first, the
neutrino is a Majorana fermion; second, flavor effects suppress while
enhancing $\mu^{-}\to e^{+}$; and third, a combination of complex interactions
other than the tree-level interaction is responsible for the physics of
nonzero neutrino mass.

Figure 2: The estimation of the $\mu^{-}\to e^{+}$ rate, as a function of the
new physics scale parameter $\Lambda$, taken from Ref. Berryman et al. (2017).

###  2.2 Past $\mu^{-}\to e^{+}$ experiments

The results from past $\mu^{-}\to e^{+}$ experiments are listed in Table 2.2
Particle Data Group (2020). The experimental techniques were similar for most
all of these searches: a muon beam interacted with a nuclear target, where
muons quickly fell to a 1s orbit in a target atom. The outgoing positron
spectrum from the nuclear target was measured in a tracking detector,
searching for the $\mu^{-}\to e^{+}$ signal. The experiment using
${}^{127}{\rm I}$ as the nuclear target Abela et al. (1980) used a
radiochemical method: a muon beam interacted with a NaI target, and then the
target was chemically treated to extract ${}^{127}{\rm Sb}$ or ${}^{127}{\rm
Te}$, where the decay rate of ${}^{127}{\rm Te}$ was measured to detect an
excess of ${}^{127}{\rm Sb}$.

Figure 3 shows the most recent $\mu^{-}\to e^{+}$ experimental search Kaulard
et al. (1998) and the comparison to the most recent $\mu^{-}\to e^{-}$ search
Bertl et al. (2006), both from SINDRUM II. For $\mu^{-}\to e^{+}$, the nucleus
in the final state can be either in the ground state or an excited state. In
the case of the transition to the ground state, the signal positron is mono-
energetic with an energy ($E_{\mu^{-}e^{+}}$) given by:

$E_{\mu^{-}e^{+}}=m_{\mu}+M(A,Z)-M(A,Z-2)-B_{\mu}-E_{\rm{recoil}}\,,$ (12)

where $m_{\mu}$ is the muon mass, $M(A,Z)$ is the mass of the nucleus
$N(A,Z)$, $B_{\mu}$ is the binding energy of the muonic atom, and $E_{\rm
recoil}$ is the recoil energy of the outgoing nucleus. Similarly to photo-
nuclear reactions, the $\mu^{-}\to e^{+}$ process may leave the target nuclei
in an excited state, a Giant Dipole Resonance (GDR), which is a collective
vibration of protons against neutrons with a dipole spatial pattern Snover
(1986). For example, a GDR with a 20 MeV width was assumed in Ref. Kaulard et
al. (1998), resulting in a much wider energy distribution for the signal
positron, as can be seen in Figure 3. The past experimental results in Table
2.2 are separately reported for the final ground state and GDR state of the
nucleus. Table A lists the $E_{\mu^{-}e^{+}}$ of the ground state transition
for some selected nuclei used by or considered for $\mu^{-}\to e^{+}$
experiments.

[bt]  The past $\mu^{-}\to e^{+}$ experimental results Particle Data Group
(2020). The limits are quoted at 90 % CL, and $\rm N_{capture}$ is the number
of muon captures in the experiments. GS and GDR represent ground state and
giant dipole resonance excitation of the final nucleus status. While the
natural abundance of ${}^{32}{\rm S}$ and ${}^{127}{\rm I}$ among their
isotopes are almost 100 %, Ti and Cu are not, which is the reason for not
specifying the mass of Ti and Cu.  Nuclei Upper limit $\mathbf{N_{capture}}$
Year Experiment Detector GS/GDR Reference Ti $3.6\times 10^{-11}$ } $2.5\times
10^{13}$ 1998 SINDRUM II Drift chamber GDR Kaulard et al. (1998); Kaulard
(1997) $1.7\times 10^{-12}$ GS $8.9\times 10^{-11}$ } $4.9\times 10^{12}$ 1993
SINDRUM II Drift chamber GDR Dohmen et al. (1993) $4.3\times 10^{-12}$ GS
$1.7\times 10^{-10}$ $9\times 10^{12}$ 1988 (TRIUMF) TPC GDR Ahmad et al.
(1988) ${}^{32}{\rm S}$ $9\times 10^{-10}$ $6.7\times 10^{11}$ 1980 SIN
Streamer chamber GDR Badertscher, A.; et al. (1980) $1.5\times 10^{-9}$
$1.2\times 10^{11}$ 1978 SIN Streamer chamber GDR Badertscher et al. (1979,
1978) Cu $2.6\times 10^{-8}$ 1972 Spark chamber GS/GDR Bryman et al. (1972)
$2.2\times 10^{-7}$ $2.2\times 10^{9}$ 1962 (CERN) Spark chamber Conforto et
al. (1962) ${}^{127}{\rm I}$ $3\times 10^{-10}$ $2.1\times 10^{12}$ 1980
Radiochemical GS Abela et al. (1980) 2

Figure 3: The energy spectrum of the signal estimation and backgrounds of
$\mu^{-}\to e^{+}$ (left) and $\mu^{-}\to e^{-}$ (right) from the SINDRUM II
experiment Kaulard et al. (1998); Bertl et al. (2006). For $\mu^{-}\to e^{+}$,
the and lower plots are the data when the muon beam is on and off,
respectively. For $\mu^{-}\to e^{-}$, the and lower plots are the data with
the radiative pion background candidates removed and enhanced, respectively.

2

##  3 Future $\MuToEp$ experimental searches

###  3.1 Upcoming experimental prospects

The COMET Phase-I Abramishvili, R.; et al. (2020) (COMET collaboration) and
Mu2e Bartoszek, L.; et al. (2015) (Mu2e collaboration) experiments will have
unprecedented sensitivity to $\MuToEp$ using aluminum as their nuclear
targets, with single event sensitivities (SES) on the order of $10^{-15}$ and
$10^{-17}$ respectively in the $\mu^{-}\mbox{$\rightarrow$}~{}e^{-}$
conversion search. These experiments are expected to have similar
sensitivities in the $\MuToEp$ channel. After COMET Phase-I, the detector will
be upgraded for COMET Phase-II which will have an SES on the order of
$10^{-17}$ Abramishvili, R.; et al. (2020) (COMET collaboration). COMET Phase-
II will not be able to search for positron and electron signals
simultaneously, so a $\mu^{-}\to e^{+}$ search would be a special run of the
$\mu^{-}\to e^{-}$ experiment and not necessarily reach the same muon
statistics as estimated for the electron channel. There has also been an
expression of interest in upgrading the Mu2e detector, called Mu2e-II, with an
SES on the order of $10^{-18}$ Abusalma, F.; et al. (2018).

Both COMET Phase-I and Mu2e use similar principles to search for the
$\mu^{-}\mbox{$\rightarrow$}~{}e^{-}$ and $\MuToEp$ processes: (1) a pulsed
proton beam is used to produce pions and muons in a production target within a
superconducting solenoid; (2) the low momentum pions and muons are guided by
the superconducting production solenoid field to a curved transport
superconducting solenoid, which has an acceptance designed to eliminate high
momentum particles; (3) an off-axis collimator selects the desired charge of
the beam, utilizing the sign-dependent drift of the beam along the curved
beamline; (4) the muons from production and pion decays along the beamline are
stopped in a nuclear target within a superconducting solenoid containing the
cylindrical detector elements; (5) non-stopped beam particles continue through
the solenoid, passing through a central axis hole in the detector elements;
(6) after a time sufficient for nearly all pions to either decay or be
captured on the nuclear target, signal candidates are reconstructed in the
detectors (where the central holes blind them to the high intensity, low
momentum muon decay backgrounds). COMET Phase-II is similar to Phase-I, except
after the stopping target there is a second curved transport solenoid followed
by the detectors, where the second transport solenoid provides the momentum
filtering, eliminating the need for a central hole in the detectors. This
second transport solenoid is also what prevents the simultaneous search for
$\mu^{-}\mbox{$\rightarrow$}~{}e^{-}$ and $\MuToEp$. The pulsed proton beam
significantly reduces beam related backgrounds during the signal measurement
period by delaying the signal measurement period until after the beam products
arrive. This time delay method requires that the lifetime of the muon in the
nuclear target (see Table A) is large enough that there are a sufficient
number of muon captures/decays in the target during the signal search period.
Due to this restriction, nuclear target materials with $Z\gtrsim 40$ cannot be
easily studied at experiments like COMET and Mu2e.

###  3.2 Background consideration

A potentially dominant physics background to $\mu^{-}\to e^{+}$ comes from
radiative muon capture (RMC),

$\mu^{-}+N(A,Z)\to\nu_{\mu}+N(A,Z-1)+\gamma~{},$

followed by photon pair production, $\gamma\to e^{+}e^{-}$, where the $e^{+}$
in the pair is misidentified as a signal. The photon energy spectrum of RMC,
in particular in the endpoint energy region, is poorly known experimentally
Bergbusch et al. (1999). However, the maximum allowed endpoint energy
($E^{\rm{end}}_{\rm{RMC}}$) can be kinematically determined:

$E^{\rm{end}}_{\rm{RMC}}=m_{\mu}+M(A,Z)-M(A,Z-1)-B_{\mu}-E_{\rm{recoil}}\,.$
(13)

The endpoint energies of some relevant nuclei are shown in Table A. The
nucleus in the final state may not be $N(A,Z-1)$ once nucleon emission occurs.
In this case, the RMC endpoint energy is smaller than that of Equation 13. It
is also possible that the RMC endpoint energy could be much smaller depending
on the spin state of the $N(A,Z)$ and $N(A,Z-1)$ nuclei. In order to account
for the RMC background, it is necessary to measure the RMC photon spectrum,
either at the $\mu^{-}\to e^{-}$ experiment or in a dedicated experiment, such
as the AlCap experiment Edmonds (2018). In addition, the kinematic separation
of the RMC background and the $\mu^{-}\to e^{+}$ positron can be improved in
future experiments such as COMET Phase-II or Mu2e-II by choosing a target
material where the endpoint energy of RMC is smaller than the $\mu^{-}\to
e^{+}$ positron energy, $E^{\rm{end}}_{\rm{RMC}}<E_{\mu^{-}e^{+}}$ Yeo et al.
(2017), therefore:

$M(A,Z-2)<M(A,Z-1)\,.$ (14)

Radiative pion capture (RPC),

$\displaystyle\pi^{-}+N(A,Z)$ $\displaystyle\to$ $\displaystyle
N(A,Z-1)+\gamma~{}~{}{\rm and}$ $\displaystyle\pi^{+}+N(A,Z)$
$\displaystyle\to$ $\displaystyle N(A,Z+1)+\gamma~{},$

followed by $\gamma\to e^{+}e^{-}$ is another background source, when the
converted positron is misidentified as a signal. The RPC background can be
controlled in $\mu^{-}\to e^{+}$ and $\mu^{-}\to e^{-}$ experiments by
suppressing the pion contamination in the muon beam. The pion contamination
can be suppressed by using a beamline sufficiently long where the pions decay
to muons before reaching the nuclear target. This was true for the previous
$\mu^{-}\to e^{-}$ measurements, where the $\pi/\mu$ ratio was about $10^{-7}$
for the SINDRUM II experiment for example, allowing the use of a continuous
muon beam with continuous data acquisition Kaulard et al. (1998). The COMET
and Mu2e $\mu^{-}\to e^{-}$ experiments are targeting a factor of 100 to
10,000 improvement on the upper limits for the $\mu^{-}\to e^{-}$ process in
the absence of a signal. To suppress the effects of pions and other non-muon
beam particles contaminating the beam, they are adopting a pulsed proton beam
with a high inter-beam pulse primary particle suppression and a delayed data
acquisition timing window technique Bartoszek, L.; et al. (2015) (Mu2e
collaboration); Abramishvili, R.; et al. (2020) (COMET collaboration).

Antiprotons contaminating the muon beam can annihilate in the muon stopping
target and create high energy background $e^{\pm}$ tracks, as well as
introduce delayed pion background due to interactions along the muon beamline.
Along with the pulsed beam technique, antiproton absorbers can be placed in
the muon beamline to suppress the antiproton background.

Another large background arises from cosmic-ray induced events. In general,
the cosmic-ray induced background is similar for the $\mu^{-}\to e^{-}$ and
$\mu^{-}\to e^{+}$ searches. Therefore, the consideration of the detector
design to suppress the cosmic-ray background for the $\mu^{-}\to e^{-}$ search
is also applicable for the $\mu^{-}\to e^{+}$ search. The COMET and Mu2e
experiments are developing cosmic-ray veto detector systems to detect charged
cosmic-ray particles entering the detectors and veto signal candidates
coincident with these detected cosmic-rays. Cosmic-rays can be reconstructed
in the trackers at COMET Phase-I and Mu2e to constrain the cosmic-ray veto
detector systems’ efficiencies, and in the case of COMET Phase-I where the
tracking detector covers the muon stopping target, cosmic-rays that interact
with the stopping target and produce background tracks can be fully
reconstructed Wong (2020).

Muon decay-in-orbit (DIO), which is one of the major background sources for
$\mu^{-}\to e^{-}$, is also a background source in the $\mu^{-}\to e^{+}$
search due to photons generated in DIO electron interactions pair-creating a
positron in the detector material. It is not an important background at the
SES of the planned COMET and Mu2e experiments as long as the charge
identification is sufficient to suppress DIO electrons being reconstructed as
positron events. However, it should be noted that the charge identification
will never be perfect, especially in the case that an electron or positron is
generated downstream of the detector system and propagates back to the muon
stopping target, where only the tracker hit timing and, for Mu2e and COMET
Phase-II, the calorimeter information can distinguish the particle trajectory.

###  3.3 RMC status

Both COMET and Mu2e will have far greater sensitivity to the high momentum
positron spectrum from RMC than previous muon conversion and RMC measurement
experiments. The background to the $\mu^{-}\to e^{+}$ search arises from
highly asymmetric RMC photon conversions, either from on-shell photons
converting in the detector material or off-shell photons internally
converting, and the background strongly depends on the nuclear target. The
internal conversion spectrum for RMC has never been measured, though the
internal conversion spectrum approximation from Kroll and Wada Kroll and Wada
(1955) (with corrections published by Joseph Joseph (1960)) for general
nuclear captures with RPC in mind should similarly apply for RMC, as shown by
Plestid and Hill Plestid and Hill (2020). This approximation makes simplifying
assumptions about the $e^{+}e^{-}$ matrix element by using the energy spectrum
of the on-shell photon and assumes that the virtuality of the $e^{+}e^{-}$
pair is small. Following this, the internal conversion spectrum can be
estimated directly from the on-shell photon spectrum, requiring only an on-
shell photon spectrum to predict the total on- and off-shell photon induced
background. Plestid and Hill show that this approximation is most reliable in
the high energy region of the positron spectrum, with the next order
uncertainty decreasing as the positron energy approaches the endpoint Plestid
and Hill (2020).

The on-shell RMC photon spectrum was measured by the TRIUMF RMC Spectrometer
group on several nuclear targets, including aluminum Armstrong et al. (1992);
Bergbusch (1995); Gorringe et al. (1998); Bergbusch et al. (1999). The
experiment was interested in studying the pseudoscalar coupling constant in
weak interactions, $g_{p}$, through the ratio $g_{p}/g_{a}$, where $g_{a}$ is
the axial coupling constant. As such, the focus of the measurements was on the
total rate of RMC, not on a precise model of the high energy tail.

The closure approximation is typically used to describe the RMC photon energy
spectrum, where one assumes the sum of the nuclear final states can be
approximated with a single nuclear transition using the mean excitation
energy. This nuclear excitation energy manifests as the closure approximation
endpoint, and is typically considered a free parameter that is fit to data.
The closure approximation photon energy spectrum is shown in Equation 15,
where $x=E_{\gamma}/k_{\rm max}$ and $k_{\rm max}$ is the spectrum endpoint
Christillin et al. (1980):

$\frac{dn}{dx}=\frac{e^{2}}{\pi}\frac{k_{\rm
max}^{2}}{m_{\mu}^{2}}(1-\frac{N-Z}{A})(1-2x+2x^{2})x(1-x)^{2}~{}.$ (15)

The TRIUMF RMC Spectrometer group measured a closure approximation endpoint of
90.1 $\pm$ 1.8 MeV and a branching fraction of $(1.40\pm 0.11)\times 10^{-5}$
above 57 MeV with respect to ordinary muon capture (OMC) using aluminum as
their nuclear target Bergbusch et al. (1999). This endpoint is significantly
lower than the kinematic endpoint on aluminum, $\sim$101.9 MeV, as shown in
Appendix A. This was the case for all of the nuclear targets – the fit closure
approximation endpoint was $\sim$10 MeV below the target’s kinematic endpoint.
As nothing forbids photon energies up to the kinematic endpoint, there is no
reason to expect the spectrum to be 0 between the measured endpoint and the
kinematic endpoint, though it may be suppressed. Predictions using a Fermi gas
model by Fearing et al. Fearing and Walker (1989); Fearing and Welsh (1992)
show the photon energy spectrum falling up to the kinematic endpoint, with no
tuned endpoint as found in the closure approximation. A few example spectra
from these calculations are shown in Figure 4.

Figure 4: RMC spectra vs $k_{\rm Fermi}$ Fearing and Walker (1989) (left) and
example RMC spectra Fearing and Welsh (1992) (right) as calculated using a
Fermi gas model.

On aluminum, the nuclear target for the currently planned COMET Phase-I and
Mu2e searches, the ground state transition energy for the positron signal is
92.3 MeV, far below the RMC kinematic endpoint of $\sim$101.9 MeV. Unlike the
$\mu^{-}\mbox{$\rightarrow$}~{}e^{-}$ searches, DIO backgrounds are not a
significant background in the positron channel at the sensitivity level of the
current and future experiments, so the dominant backgrounds are RMC, RPC,
antiprotons, and cosmic-ray events. The non-RMC backgrounds are expected to
contribute similarly in the positron and electron channel, where the
expectation is less than 1 event per experiment for both searches
Abramishvili, R.; et al. (2020) (COMET collaboration); Bartoszek, L.; et al.
(2015) (Mu2e collaboration). Assuming the closure approximation with the
measured endpoint, the true positron spectrum would end at $89.6\pm 1.8$ MeV/c
with a rapidly falling spectrum, so only resolution and energy loss effects
would lead to overlaps with the signals. As the resolution is approximately
200 keV/c for both experiments Abramishvili, R.; et al. (2020) (COMET
collaboration); Bartoszek, L.; et al. (2015) (Mu2e collaboration) and the two
processes are separated by nearly 3 MeV/c, both experiments would be able to
maintain a background expectation of below 1 event per experiment for the
measured endpoint.

The dataset on aluminum for the 1999 RMC measurement from TRIUMF only had
3,051 photons above 57 MeV, so this measurement was not sensitive to photon
rates above 90 MeV that are about 3,000 times smaller than the total rate
above 57 MeV ($R(E_{\gamma}>57~{}{\rm
MeV})=\frac{\Gamma_{RMC}(E_{\gamma}>57~{}{\rm MeV})}{\Gamma_{OMC}}$). COMET
Phase-I and Mu2e will see about $10^{16}$ and $10^{18}$ muon captures
respectively Abramishvili, R.; et al. (2020) (COMET collaboration); Bartoszek,
L.; et al. (2015) (Mu2e collaboration), and so will have $\sim 10^{11}$ and
$10^{13}$ RMC photons above 57 MeV. To test their sensitivity to RMC beyond
the fit endpoint on aluminum, we assume a flat photon energy spectrum tail
above 90 MeV up to the kinematic endpoint with a rate around the sensitivity
limit of the 1999 measurement, shown in Equation 16:

$R(E_{\gamma}>90~{}{\rm MeV})=\frac{1}{3,000}\times R(E_{\gamma}>57~{}{\rm
MeV})=4.7\times 10^{-9}~{}~{}.$ (16)

We also make the following simplifying assumptions: all $e^{\pm}$ energies
from a photon conversion are equally likely, such that the energy sharing
distribution between the $e^{+}e^{-}$ pair is flat, there is a 0.1% chance of
a photon conversion in the nuclear target, and the tracking efficiency for
$\sim$90 MeV/c positrons is 10%. This leads to a positron background rate of
$\mathcal{O}(500)$ and $\mathcal{O}(50,000)$ events per MeV/c at 90 MeV/c for
COMET Phase-I and Mu2e respectively, before considering internal photon
conversions, as shown in Figure 5. In this simple model, which is consistent
with the existing data on aluminum, the RMC background estimate would change
from below 1 background event to tens of thousands of background events near
the signal for $10^{18}$ muon captures, significantly lessening the discovery
potential of $\MuToEp$ signal searches.

Figure 5: Assumed on-shell RMC photon spectrum high energy tail on aluminum
for $10^{18}$ muon captures (left) and the corresponding positron spectrum
using the simplifying assumptions about the conversion spectrum and track
reconstruction (right). The signal process is also shown using the same
tracking efficiency for a sample $\mathcal{R}^{\mu^{-}e^{+}}$ value of
$10^{-16}$, which would likely be discoverable in the absence of an RMC
background.

###  3.4 RMC considerations at future $\mu^{-}\to e^{+}$ experiments

The discovery potential of future experiments searching for $\MuToEp$ is
entwined with their understanding of the RMC background, where any claim of
new physics must also be able to accurately subtract the SM background. As the
RMC spectrum is not well constrained by existing measurements, muon conversion
experiments should plan to perform RMC measurements on the relevant nuclear
targets. COMET and Mu2e should investigate how to measure the RMC spectrum
using the electron and positron spectra, paying special attention to how to
avoid unblinding the $\MuToEp$ signal region of the positron spectrum. If both
the electron and positron track can be reconstructed for an RMC photon
conversion, this will allow one to measure the photon energy while also likely
avoiding the $\MuToEp$ signal unblinding. A calorimeter measurement, using
reconstructed photon clusters, would allow a direct measurement of the photon
energy spectrum, where the positron spectrum due to off-shell conversions can
be described using the results from Plestid and Hill given a measurement of
the on-shell photon spectrum Plestid and Hill (2020). A dedicated experiment
to measure the RMC spectrum on the intended nuclear target should also be
investigated.

Future muon conversion experiments can also consider choosing a nuclear target
whose $\MuToEp$ ground state transition energy is above the kinematic endpoint
of the RMC spectrum, kinematically suppressing this background. This
translates into a requirement that the nuclear mass of N(A,Z-2) is smaller
than the nuclear mass of N(A,Z-1). Several potential target choices were
identified by Yeo et al. Yeo et al. (2017), where all of the nuclear targets
considered had $\MuToEp$ ground state transition energies greater than the RMC
positron kinematic endpoints. Example targets well suited for $\MuToEp$
measurements are titanium (a common candidate for
$\mu^{-}\mbox{$\rightarrow$}~{}e^{-}$ experiments), sulfur, and calcium, which
satisfy this mass difference requirement while they also have long enough muon
lifetimes to be useful at experiments with delayed data acquisition to
suppress pion backgrounds. For each nuclear target considered, the authors
assumed a closure approximation with the endpoint set to be the kinematic
endpoint and the branching fraction above 57 MeV to be the measured value by
the TRIUMF RMC Spectrometer group. For an example $10^{18}$ muon stops, they
found COMET Phase-II would be able to achieve limits at 90% CL of
$\mathcal{O}(10^{-15})$ for the ground state transition, a three order of
magnitude improvement upon the current limit from Ref. Kaulard et al. (1998).
This is compared to the potential one order of magnitude gain by using
aluminum as the nuclear target Yeo et al. (2017), showing how critical the
nuclear target choice can be for the $\MuToEp$ search.

These potential upper limits do not include an assessment of the impact on the
limit due to the systematic uncertainty on the RMC background modeling. The
closure approximation does not take into account exclusive transitions to low
lying states, which could lead to a more complicated background spectrum with
kinks or knees in the photon energy spectrum. These are likely to be smoothed
by the (internal) pair conversion spectrum, but without understanding the RMC
spectrum it will be difficult to have confidence that an observation of 5-10
events is definitive evidence of $\MuToEp$ conversion, and not instead due to
an RMC transition. An observation of this size is sufficient to claim a
discovery at the upcoming $\mu^{-}\mbox{$\rightarrow$}~{}e^{-}$ searches. As
the goal of these searches is to discover new physics, the experiments must be
prepared for the potential discovery of the LFV and LNV process, $\mu^{-}\to
e^{+}$, which requires the confident rejection of RMC as an alternate
hypothesis.

##  4 Concluding remarks: Towards future $\mu^{-}\to e^{+}$ measurements

CLFV is a long sought-after signal of physics beyond the SM. Many experiments
have searched for $\mu^{-}\to e^{-}$ and $\mu^{-}\to e^{+}$ processes, where
the results are complementary to the best-measured CLFV channel, $\mu\to
e\gamma$. The $\mu^{-}\to e^{+}$ process is also an LNV channel, which may
give insight into the Majorana property of the neutrino. Although it is not as
strong a channel as is in general, its discovery would provide insight on
potential flavor effects in the neutrino mass generation. Current and future
experiments will have unprecedented sensitivity to both $\mu^{-}\to e^{-}$ and
$\mu^{-}\to e^{+}$ processes. However, the search for $\mu^{-}\to e^{+}$
requires more careful considerations in terms of the nuclear target selection
in order to reach the full discovery potential of these experiments.

RMC is one of the least understood backgrounds at $\mu^{-}\to e^{-}$
experiments, with the potential to be the largest background in the
$\mu^{-}\to e^{+}$ search. RMC is theoretically understudied, and previous RMC
measurements do not have the necessary sensitivity in the high energy photon
region to sufficiently constrain this background at the currently planned
COMET Phase-I and Mu2e experiments. These experiments will need to measure the
RMC photon energy spectrum in order to confidently reject the SM background in
the case of a $\mu^{-}\to e^{+}$ signal.

An important consideration for the currently planned COMET and Mu2e
experiments using aluminum as their nuclear target is what the next steps will
be if they discover the process of $\mu^{-}\to e^{+}$. A common idea is to
attempt to determine the nature of the new physics by testing the nuclear
target dependence of the conversion rate Abusalma, F.; et al. (2018). For
$\mu^{-}\to e^{-}$, Ref. Kitano et al. (2002) and Cirigliano et al. (2009)
show for example models this rate may only vary by $\sim$10-30% between the
models for nuclear targets feasible for COMET Phase-II and Mu2e-II. The total
number of muon captures will likely only be known to $\sim$10% Bartoszek, L.;
et al. (2015) (Mu2e collaboration), uncorrelated between the targets, making a
comparison with less than 15% uncertainty in the difference between the rates
difficult. It would also require a high statistics discovery to have a precise
measurement of the branching fraction, far beyond the threshold of $\sim$5-10
events needed to claim a discovery for the $\mu^{-}\to e^{-}$ (and potentially
$\mu^{-}\to e^{+}$) searches.

In the $\mu^{-}\to e^{+}$ case, there is more than the ground state transition
to consider. Experiments can search for the GDR transition in addition to the
ground-state transition, and measure the relative rate between the ground
state and the GDR transition. As this is a ratio, independent of the number of
muon captures, it is possible this can be better used to test the rate
dependence on the nuclear target, helping to understand the underlying
mechanism for the LNV process. The nuclear dependence of the $\mu^{-}\to
e^{+}$ process needs to be further studied to determine the potential next
steps experiments should take in the case of a $\mu^{-}\to e^{+}$ discovery.

##  Appendix A RMC endpoint calculation on aluminum

The RMC kinematic endpoint on aluminum can be calculated by considering the
“decay” of the $\mu^{-}~{}N(A,Z)$ system to $\gamma~{}N(A,Z-1)\nu_{\mu}$ with
$p(\nu_{\mu})=0$, which is then a two body decay with a photon energy given by
Equation 13. Since the muon is $\sim$200 times heavier than the electron, and
therefore has a much closer orbit in the atom, the electrons do not
participate in the process. The relevant masses are then the muon mass and the
incoming and outgoing nuclei masses, not the atomic masses, where the nuclear
mass is given by $M(A,Z)=M_{A}-Z\cdot m_{e}$, where $M_{A}=A_{r}\cdot u$,
$A_{r}$ is the relative mass, u is the atomic mass unit, and $m_{e}$ is the
electron mass. The final state particles then satisfy
$p_{\gamma}=-p_{{}^{27}Al}$ and the energy of the recoiling magnesium nucleus
is given by:

$E_{{}^{27}Mg}=\frac{M^{2}+M_{N}(^{27}Mg)^{2}}{2M}=E_{\rm
recoil}+M_{N}(^{27}Mg)~{},$ (17)

where $M$ is the mass of the muonic aluminum system,
$M=M(^{27}Al)+m_{\mu}-B_{\mu}$. Table A shows the input parameters for
Equations 17 and 13. The resulting kinematic RMC endpoint on aluminum is
101.867 MeV. The corresponding positron energy endpoint that is relevant for
$\mu^{-}\to e^{+}$ experiments is one electron mass below this, which is
listed in Table A for some example nuclei.

[H] Parameters used in the RMC endpoint energy calculation on aluminum.
Parameter Value Reference $m_{\mu}$ 105. 6583745 MeV/c2 Mohr et al. (2016)
$m_{e}$ 0. 5109989461 MeV/c2 Mohr et al. (2016) $1u$ 931. 49410242 MeV/c2 Mohr
et al. (2016) $B_{\mu}$ 0. 464 MeV Czarnecki et al. (2011) $A_{r}(^{27}Al)$
26. 98153841 u Wang et al. (2017) $M_{N}(^{27}Al)$ 25126. 501 MeV/c2
$A_{r}(^{27}Mg)$ 26. 98434063 u Wang et al. (2017) $M_{N}(^{27}Mg)$ 25129. 622
MeV/c2 $E_{\rm recoil}$ 0. 206 MeV/c2

[H] The energy of the signal positron ($E_{\mu^{-}e^{+}}$) obtained from
Equation 12, the RMC positron endpoint energy ($E^{\rm{end}}_{\rm{RMC}}$) of
the ground state transition obtained from Equation 13, and the lifetime of the
muonic atom ($\tau_{\mu^{-}}$) of some nuclei considered for $\mu^{-}\to
e^{+}$ experiments. The energy of the signal electron ($E_{\mu^{-}e^{-}}$) for
$\mu^{-}\to e^{-}$ are also shown for comparison. Nuclear masses required for
these calculations are taken from AME2016 data Wang et al. (2017). The
lifetime data is from Ref. Suzuki et al. (1987).  Nuclide $E_{\mu^{-}e^{+}}$
[MeV] $E^{\rm{end}}_{\rm{RMC}}$ [MeV] $\tau_{\mu^{-}}$ [ns] $E_{\mu^{-}e^{-}}$
[MeV] ${}^{27}{\rm Al}$ 92.30 101.36 864 104.97 ${}^{32}{\rm S}$ 101.80 102.03
555 104.76 ${}^{40}{\rm Ca}$ 103.55 102.06 333 104.39 ${}^{48}{\rm Ti}$ 98.89
99.17 329 104.18

Writing—original draft preparation, M. L. and M. M.; writing—review and
editing, M.L. and M. M. All authors have read and agreed to the published
version of the manuscript.

This research of M.L. was supported by Institute for Basic Science
(IBS-R017-D1-2021-a00) of Korea. This research of M.M. was in part funded by
the “Research in the Energy, Cosmic and Intensity Frontiers at Northwestern
University” award, DE-SC0015910.

Not applicable.

Not applicable.

The data used in this study can be provided by the authors upon request.

###### Acknowledgements.

The authors M.L. and M.M. are collaborators of the COMET and Mu2e
collaborations, respectively. The author M.L. thanks KEK and J-PARC, Japan for
their support of infrastructure and the operation of COMET. This work of M.L.
is supported in part by: Japan Society for the Promotion of Science (JSPS)
KAKENHI Grant Nos. 25000004 and 18H05231; JSPS KAKENHI Grant No.JP17H06135;
Belarusian Republican Foundation for Fundamental Research Grant F18R-006;
National Natural Science Foundation of China (NSFC) under Contracts No.
11335009 and 11475208; Research program of the Institute of High Energy
Physics (IHEP) under Contract No. Y3545111U2; the State Key Laboratory of
Particle Detection and Electronics of IHEP, China, under Contract
No.H929420BTD; Supercomputer funding in Sun Yat-Sen University, China;
National Institute of Nuclear Physics and Particle Physics (IN2P3), France;
Shota Rustaveli National Science Foundation of Georgia (SRNSFG), grant No.
DI-18-293; Deutsche Forschungsgemeinschaft grant STO 876/7-1 of Germany; Joint
Institute for Nuclear Research (JINR), project COMET #1134; Institute for
Basic Science (IBS) of Republic of Korea under Project No.
IBS-R017-D1-2021-a00; Ministry of Education and Science of the Russian
Federation and by the Russian Fund for Basic Research grants: 17-02-01073,
18-52-00004; Science and Technology Facilities Council, United Kingdom; JSPS
London Short Term Predoctoral Fellowship program, Daiwa Anglo-Japanese
Foundation Small Grant; and Royal Society International Joint Projects Grant.
The author M.M. is grateful for the vital contributions of the Fermilab staff
and the technical staff of the participating institutions to Mu2e. This work
was supported by the US Department of Energy; the Istituto Nazionale di Fisica
Nucleare, Italy; the Science and Technology Facilities Council, UK; the
Ministry of Education and Science, Russian Federation; the National Science
Foundation, USA; the Thousand Talents Plan, China; the Helmholtz Association,
Germany; and the EU Horizon 2020 Research and Innovation Program under the
Marie Sklodowska-Curie Grant Agreement No. 690835, 734303, 822185, 858199.
This document was prepared in part by members of the Mu2e Collaboration using
the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S.
Department of Energy, Office of Science, HEP User Facility. Fermilab is
managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-
AC02-07CH11359. The authors declare no conflict of interest. References

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