# Muon-induced baryon number violation

Patrick J. Fox<EMAIL_ADDRESS>Theory Division, Fermilab, Batavia, IL 60510,
USA  Matheus Hostert<EMAIL_ADDRESS>Department of Physics &
Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge,
MA 02138, USA  Tony Menzo<EMAIL_ADDRESS>Department of Physics,
University of Cincinnati, Cincinnati, Ohio 45221, USA  Maxim Pospelov
<EMAIL_ADDRESS>School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455, USA  William I. Fine Theoretical Physics Institute,
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN
55455, USA  Jure Zupan<EMAIL_ADDRESS>Department of Physics,
University of Cincinnati, Cincinnati, Ohio 45221, USA

###### Abstract

The search for charged-lepton flavor violation in muon capture on nuclei is a
powerful probe of heavy new physics. A smoking gun signal for $\mu\to e$
conversion is a monochromatic electron with energy almost equal to the muon
mass. We show that light new physics can mimic this signature and that it can
also lead to electrons above the $\mu\to e$ signal peak. A concrete example of
such light new physics is $\mu^{-}$-nucleon annihilation into a light dark
sector, which can produce an energetic $e^{-}$ as well as $e^{+}e^{-}$
byproducts. Due to the size of the muon mass, the exotic muon capture process
can be kinematically allowed, while the otherwise stringent constraints, e.g.,
from proton decay, are kinematically forbidden. We also discuss other relevant
constraints, including those from the stability of nuclei and muon capture in
the interior of neutron stars.

††preprint: FERMILAB-PUB-24-0314-T

##  I Introduction

In the Standard Model (SM) of particle physics, taus, muons, and electrons do
not change flavor. With the addition of neutrino masses, the rate for
conversion is non-zero but remains experimentally unobservable. Therefore,
charged-lepton flavor violation (cLFV) provides a smoking gun signature for
new physics. The most sensitive probes of cLFV include flavor-changing muon
decays, such as $\mu\to e\gamma$ and $\mu\to eee$ Feinberg:1958zzb , and muon-
to-electron conversion on nuclei, $\mu^{-}A\to e^{-}A^{*}$ Marciano:1977cj .
Due to significantly improved beam intensities and background reductions,
next-generation experimental facilities are expected to achieve major leaps in
sensitivity to these rare processes. The Mu2e experiment at Fermilab
Mu2e:2012eea ; Mu2e:2014fns ; Mu2e-II:2022blh ; CGroup:2022tli  and COMET
Kuno:2013mha ; COMET:2018auw  experiment at J-PARC will reach $\mu\to e$
conversion rates over two orders of magnitude below current bounds by the end
of the decade, while the DeeMe experiment at J-PARC Teshima:2019orf  will
improve bounds by an order of magnitude on a shorter timescale. In terms of
new-physics reach, these experiments will probe cLFV new physics at scales as
high as $\Lambda\sim 10^{8}$ GeV (see Ref. Fernandez-Martinez:2024bxg  for a
recent global analysis of cLFV in effective theories).

It is also conceivable that light new particles can manifest in the ultra-rare
rates probed by the cLFV experiments. Some examples already studied in the
literature include the production of an invisible axion-like-particle $X$ in
muon decays such as the two body $\mu\to eX$ decays GarciaiTormo:2011cyr ;
Uesaka:2020okd ; Calibbi:2020jvd ; Hill:2023dym ; Knapen:2023zgi , virtual
exchange of axion-like-particles Fuyuto:2023ogq , as well as other signatures
from muon decay induced production of light new particles that then further
decay to visible SM particles — electrons, positrons and/or photons
Echenard:2014lma ; Heeck:2017xmg ; Hostert:2023gpk ; Knapen:2023iwg . Note
that in all these cases the kinematic reach for the production of new
particles is set by the available energy, the mass difference $m_{\mu}-m_{e}$.

In this manuscript, we explore a distinct new possibility: a class of models
where the production of new particles is a result of $\mu^{-}$ annihilation
with a proton or neutron bound inside a nucleus. The energy available for the
production of new particles is now much larger, $\sim m_{\mu}+m_{p}$. However,
the process also violates the baryon number and faces stringent constraints.

These scenarios constitute an example of new physics models that can appear
only in muon conversion experiments and would not be probed directly by other
cLFV probes. As we will show below, the sample models we will discuss can be
probed with the primary running mode and design parameters of Mu2e and COMET,
complementing searches for light new physics that would otherwise rely on
dedicated runs or on decays of muons in orbit GarciaiTormo:2011cyr ;
Uesaka:2020okd ; Xing:2022rob ; Hill:2023dym .

In choosing the new physics benchmarks, we have two questions in mind that we
want to explore quantitatively in specific examples:

  1. 1.

Can light new physics ‘fake’ a conventional $\mu\to e$ conversion signal?

  2. 2.

Can light new physics lead to electrons or positrons with energies greater
than $m_{\mu}$?

We will show that the answer to both questions is yes. With the goal of
providing proof of existence, we study a dark sector model where baryon number
violation is induced by muonic reactions. Constraints from proton decay and
nuclear stability can be avoided via a judicious choice of new particle
spectra and flavor structures. Ultimately, our study demonstrates that there
are other new physics signatures beyond the mono-energetic conversion
electrons that the $\mu\to e$ experiments can look for.

This article is organized as follows. In Section II we describe muon-proton
annihilation induced muon capture on nuclei and the resulting possible
production channels of new particles. We study quantitatively an example of
such a transition in more detail in Section III, deriving predictions for the
Mu2e and COMET experiments and addressing constraints from other rare
processes. Section IV discusses the necessary ingredients for the UV
completion of these models, while Section V contains our conclusions.

##  II Particle production in muon capture

Figure 1: A schematic of the different particle production mechanisms in muon
capture. Each quadrant corresponds to a different rule for baryon and lepton
flavor number conservation or (apparent) violation.

We are interested in models of light new physics that can generate the
following types of transitions at the nucleon level,

$\displaystyle\mu^{-}+p\to e^{-}+\\{X\\}^{+},\text{~{}\,or~{}~{}}\mu^{-}+n\to
e^{-}+\\{X\\}^{0},$ (1)

where $\\{X\\}$ denotes dark and/or visible sector final state particles, out
of which an odd number are fermions. We are specifically interested in
transitions with $e^{-}$ in the final state since these can result in a
similar signal to the usual one in the $\mu\to e$ conversion searches but are
still distinct from it when examined in detail. How much the process in Eq. 1
will differ from the usual channel, $\mu^{-}+A\to e^{-}+A$, will depend on
which particles are part of the $\\{X\\}^{+}$ and $\\{X\\}^{0}$ final states,
as well as on the nature of the new interactions. For instance, the electron
may be a by-product of the decays in the dark sector, in which case the
kinematics of these processes no longer ensure that the electron is mono-
energetic. As is well appreciated in the literature, the fragmentation of on-
shell “dark” states $\\{X\\}$ opens up vast possibilities for the multiple
production of light SM states, such as photons, electrons, and positrons.

In $\mu\to e$ conversion, the small recoil energy of the nucleus can be
neglected to a very good approximation. That is, the electron takes away
essentially all of the available energy,

$E_{e}^{\rm conv}\simeq m_{\mu}-E_{b},$ (2)

where $E_{b}\sim(Z\alpha)^{2}m_{\mu}/2\sim 450$ keV is the binding energy of
the muonic aluminum atom, and thus $E_{e}^{\rm conv}\simeq 104.98$ MeV
Haxton:2022piv . The distinct signature of $\mu\to e$ conversion is,
therefore, mono-energetic electrons, with energy slightly below the muon mass
and thus above the SM backgrounds. The kinematics of $\mu\to e$ transitions
that involve the production of light new physics, Eq. (1), is different.
Depending on the details of the model, i.e., which particles are present in
the final state and what their masses are, the energy of the final state
electron may be well below or even well above the muon mass. If the outgoing
electron is significantly less energetic than $E_{e}^{\rm conv}$, the
backgrounds from the decays of muons in orbit, as well as from radiative
decays followed by either Dalitz or Bethe-Heitler conversion, are expected to
be prohibitively large. Furthermore, due to the small acceptance of the
tracker for low-transverse momentum electrons ($p_{T}\gtrsim 90$ MeV at Mu2e
Mu2e:2014fns ), such less energetic electrons and other visible particles may
be experimentally unobservable. Therefore, we will be particularly interested
in new physics scenarios in which $E_{e}>E_{e}^{\rm conv}$.

In what follows, we start with the general discussion of the minimal scenarios
for the light particle production in muon capture, Eq. (1), and then focus on
a particular concrete example, the model discussed in Section III. In general,
we can divide the models into apparent baryon number conserving (aBNC) and
apparent baryon number violating (aBNV) scenarios, and also into apparent
lepton flavor conserving (aLFC) and apparent lepton flavor violating (aLFV)
scenarios, giving four distinct classes of models. By “apparent conservation
or violation”, we refer to the conservation of baryon number or lepton flavors
within visible sector particles, including neutrinos. These two quantum
numbers need not necessarily be violated since the dark sector particles can
also carry baryon or lepton numbers. Our discussion focuses on processes
involving muonic aluminum atoms ($\mu\,\isotope[27\\!]{Al}$), since
$\isotope[27\\!]{Al}$ is the currently planned target material in the Mu2e
experiment. Should other targets for muon capture be used, our models can be
easily generalized to those target atoms as well. Notably, the choice of a
different target only changes $E_{e}^{\rm conv}$ by at most $\approx 5\%$, and
thus has negligible impact on the kinematics of the final state particles for
the models we consider Haxton:2022piv .

aLFC and aBNC scenarios. These involve charged-current (CC) reactions, such
as,

$\displaystyle\mu^{-}+\isotope[27]{Al}$
$\displaystyle\to(X_{0}\to\nu_{\mu}e^{+}e^{-})+\isotope[27]{Mg}^{(*)},$ (3)
$\displaystyle\mu^{-}+\isotope[27]{Al}$ $\displaystyle\to\nu_{\mu}+(X_{0}\to
e^{+}e^{-})+\isotope[27]{Mg}^{(*)},$ (4)

where $X_{0}$ is an electromagnetically neutral dark sector particle. For
example, $X_{0}$ in Eq. 3 could be a heavy neutral lepton, $X_{0}=N$, produced
through electroweak charge-current interaction, $\mu^{-}p\to Nn$, that is
induced via $N$ mixing with the SM neutrinos, followed by the
$N\to\nu_{\mu}e^{+}e^{-}$ decay. The sterile-active neutrino mixing is
strongly constrained for light sterile neutrinos, $m_{N}\lesssim m_{\mu}$.
However, it could still induce the muon-to-$N$ capture rate as large as $R\sim
10^{-12}-10^{-10}$ (see Eq. (26) below for the definition of $R$). The sterile
neutrino $N$ can decay within the target, for instance, if the decay is
mediated through a new dark force, such as $N\to\nu(A^{\prime}\to
e^{+}e^{-})$, where $A^{\prime}$ is a dark photon. Related NP transitions have
been considered in the literature; for instance, the muon capture producing a
sterile neutrino, $\mu^{-}p\to(N\to\nu\gamma)n$ was discussed in Ref.
McKeen:2010rx . The dark photon $A^{\prime}$ can also be the dark state in the
transition in Eq. 4. In the notation of Eq. 4, $X_{0}=A^{\prime}$, so that the
dark photon gets produced in the $\mu^{-}p\to\nu_{\mu}nA^{\prime}$ transition
and then decays to an electron-positron pair, $A^{\prime}\to e^{+}e^{-}$. If
the $A^{\prime}$ boson is emitted directly, as in Eq. 4, it can be emitted
from any of the lines in the diagram through couplings to muon, neutrinos, or
nucleons.

In both types of transitions, Eqs. (3) and (4), the energy of the outgoing
electron is significantly smaller than $m_{\mu}$ (as is the energy of the
outgoing positron). The kinematic end-point is given by the back-to-back decay
configuration of electron recoiling against all the other particles, resulting
in $E_{e}\lesssim m_{\mu}/2$. This kinematic limitation is a result of both
the fact that the above CC reactions conserve baryon number and thus the
masses of final and initial nuclei are very close to each other,
$M({\isotope[27]{Mg}})-M({\isotope[27]{Al}})\sim 3$ MeV, as well as due to the
fact that the remaining final state particles, $e^{+}$ and $\nu_{\mu}$, are
very light. Therefore, while the current constraints on the couplings of new
particles do not preclude several thousands of such transitions from occurring
at Mu2e or COMET, identifying the new signal events would be challenging for
the current experimental designs.

aLFV and aBNC scenarios. An example is

$\mu^{-}+\isotope[27]{Al}\to e^{-}+X_{0}+\isotope[27]{Al},$ (5)

which could be a result of a $\mu^{-}\to e^{-}X_{0}$ decay, with nucleus not
playing any significant role. The $X_{0}$ can be stable on collider time-
scales, resulting in $\mu^{-}+\isotope[27]{Al}\to
e^{-}+\isotope[27]{Al}+\text{inv}$ signature. A concrete realization of this
scenario is an axion-like-particle, $X_{0}=a$, coupled to the
$\overline{\mu}\gamma^{5}e$ current GarciaiTormo:2011cyr ; Uesaka:2020okd ;
Calibbi:2020jvd ; Knapen:2023zgi ; Hill:2023dym ; Calibbi:2016hwq . However,
searching for $\mu^{-}\to e^{-}a$ decays is intrinsically more challenging
than searching for the signal of $\mu\to e$ conversion, since in the former
the electron is necessarily much softer, $E_{e}\lesssim m_{\mu}/2$. Similar
conclusions apply also in the case the decays of $X_{0}$ are visible, such as
$X_{0}\to e^{+}e^{-}$.

aLFC and aBNV scenarios. The simplest transitions of this type are

$\displaystyle\mu^{-}+\isotope[27]{Al}$
$\displaystyle\to\nu_{\mu}+\left(X_{1}\to
e^{+}e^{-}\right)+\isotope[26]{Mg}^{(*)},$ (6)
$\displaystyle\mu^{-}+\isotope[27]{Al}$
$\displaystyle\to\nu_{\mu}+\left(X_{2}\to
X_{1}e^{+}e^{-}\right)+\isotope[26]{Mg}^{(*)},$ (7)

where $X_{1,2}$ are dark sector states. Compared to the previous two types of
models, the main new ingredient is that the number of SM baryons changes. In
principle, this releases an additional $m_{N}\sim 1\,$GeV of energy and thus
one can have $E_{e}>E_{e}^{\rm conv}$, as long as $m_{1}\lesssim m_{n},m_{p}$.

Note that the transitions in Eqs. 6 and 7 need not be BNV since $X_{1,2}$ can
carry nonzero baryon number. Nevertheless, in general, the constraints on BNV
processes, i.e., the bounds on the (in)stability of the proton and the bound
neutrons, do place stringent bounds on aBNV processes. That is, the
transitions in Eqs. 6 and 7 above imply nucleon decays either through tree-
level diagrams or via loop diagrams with off-shell muons, which then naively
precludes any signatures at the $\mu\to e$ conversion experiments. However, as
we will show below, through a combination of judicious charge assignments and
kinematic suppressions in the form of mass hierarchies, these contributions
can be sufficiently suppressed either due to off-shellness, higher loop
orders, or phase space suppression, ensuring phenomenologically-viable nucleon
stability.

In the two examples above, the nucleon decay transitions are
$p\to\mu^{+*}\nu_{\mu}X_{1,2}$. These are kinematically forbidden if the mass
spectra satisfy $m_{X_{1,2}}>m_{p}-m_{e}$, neglecting neutrino masses and
using the fact that the off-shell $\mu^{+*}$ decays leptonically, $\mu^{+}\to
e^{+}\overline{\nu}_{\mu}\nu_{e}$. Since $X_{1}$ and $X_{2}$ are allowed to
decay, one also needs to consider the following full decay chains,

$\displaystyle p$ $\displaystyle\to\nu_{\mu}+(X_{1}^{*}\to
e^{+}e^{-})+(\mu^{+*}\to e^{+}\nu_{e}\bar{\nu}_{\mu}),$ (8) $\displaystyle p$
$\displaystyle\to\nu_{\mu}+(X_{2}^{*}\to X_{1}e^{+}e^{-})+(\mu^{+*}\to
e^{+}\nu_{e}\bar{\nu}_{\mu}),$ (9)

as well as their variants. As we will show below, these decays can be
sufficiently suppressed in the region of parameter space where
$E_{e^{+}/e^{-}}>E_{e}^{\rm conv}$. We will illustrate this within the
scenario in Eq. 7, identifying $X_{1,2}$ with dark sector fermions,
$X_{1,2}=\chi_{1,2}$, and also generalizing it by replacing $\nu_{\mu}$ with
some generic light particle $\chi_{0}$. The muon capture and the associated
nucleon decay channel schematics for this case are shown in Fig. 2.

Figure 2: The scenario for muon-induced (apparent) baryon number violation
considered in this work (Eq. 14). On the left, we show the muon capture on
protons, and on the right, the tree-level proton decay channel. In both cases,
the dark photon $A^{\prime}$ can be either on or off shell.

aLFV and aBNV scenarios. Transitions of this type are

$\displaystyle\mu^{-}+\isotope[27]{Al}$ $\displaystyle\to
e^{-}+X_{1}+\isotope[26]{Al},$ (10) $\displaystyle\mu^{-}+\isotope[27]{Al}$
$\displaystyle\to e^{-}+(X_{2}\to X_{1}e^{+}e^{-})+\isotope[26]{Al},$ (11)

and are induced by the $\mu^{-}n\to e^{-}X$ nucleon level transitions.
Similarly to other cases, these transitions can also result in energetic
electrons above the $\mu\to e$ endpoint, $E_{e^{-}}\gtrsim E_{e}^{\rm conv}$.
In (10), this occurs for $m_{X_{1}}\lesssim m_{n}$, and in Eq. 11, if either
$m_{X_{2}}\lesssim m_{n}$ and/or if $m_{X_{2}}\gg m_{X_{1}}$.

As in the previous classes of models, also here the most stringent constraints
are due to the bounds on BNV nucleon decays. For instance, if the decays of
bound neutrons, $n\to\mu^{+(*)}e^{-}X_{0}$, are kinematically forbidden, this
also means that the transition Eq. 10 will not result in energetic electrons
in Mu2e. The decay chain in Eq. 11 is of greater phenomenological interest.
Choosing $m_{X_{1}}>m_{n}$, the decay $n\to\mu^{+(*)}e^{-}(X_{1}^{*}\to
X_{0}e^{+}e^{-})$ comes with a second virtual particle in the final state,
which, combined with the phase-space suppression caused by the smallness of
$\Delta E=m_{n}-m_{X_{0}}-3m_{e}$, can make the decays of neutrons bound
inside nuclei sufficiently rare, at least at tree-level. In the transition in
Eq. 11, the electron from the primary LFV vertex is soft, $E_{e^{-}}<m_{\mu}$.
However, the positron and the electron from the $X_{2}\to X_{1}e^{+}e^{-}$
decay can be energetic enough to be observed in the experiment.

##  III Muon-induced baryon number violation

Figure 3:  The energy spectrum for electrons and positrons produced in exotic
muon capture for a few choices of dark particle masses. Couplings are fixed as
indicated in the left panel of each row. On the left, we show the individual
electron/positron energies, and on the right, their combined energy. The usual
$\mu\to e$ conversion electron energy is shown as a narrow green band around
$E_{e^{-}}^{\rm cap}\simeq 104.98$ MeV, and the region where muon decay-in-
orbit dominates is shaded in gray Czarnecki:2011mx .

Figure 4:  Same as Fig. 3 but for $\alpha_{D}\ll 1$ and $d_{\rm tr}=10^{-3}$
GeV-1. The $\chi_{2}\to\chi_{1}e^{+}e^{-}$ transition is induced by the tensor
interaction of Eq. 15 and leads to a preference for more energy-asymmetric
$e^{+}e^{-}$.

Next, we discuss in detail a concrete model of muon capture on a nucleus that
results in multiple particle production. We specialize to the aLFC/aBNV
scenario in Eq.(7), but with three dark sector particles, $\chi_{0,1,2}$,

$\mu^{-}+\isotope[27]{Al}\to\chi_{0}+\left(\chi_{2}\to\chi_{1}e^{+}e^{-}\right)+\isotope[26]{Mg}^{(*)}.$
(12)

The transition conserves lepton flavor number if $\chi_{0}$ carries a muon
lepton number (for instance, $\chi_{0}$ can be $\nu_{\mu}$ in which case this
is exactly the scenario in Eq. (7)). Similarly, the transition conserves
baryon number if either $\chi_{0}$ or $\chi_{2,1}$ carry baryon number. We are
particularly interested in the part of the parameter space that results in
$e^{+}e^{-}$ pairs with high enough energy to be observable at Mu2e and COMET.

We start by realizing the scenario within a U$(1)_{\rm D}$ extension of the
SM, to which we add three dark fermions, $\chi_{0,1,2}$ with masses
$m_{0,1,2}$. The U$(1)_{\rm D}$ gauge symmetry is spontaneously broken, giving
rise to a dark photon $A^{\prime}$ with mass $m_{A^{\prime}}$. Kinetic mixing
gives rise to the interaction term

$\mathscr{L}\supset e\varepsilon A^{\prime}_{\mu}J^{\mu}_{\rm EM},$ (13)

where $J^{\mu}_{\rm EM}$ is the SM electromagnetic current.

The flavor-violating interaction at low energies is assumed to be given by the
following effective dimension 6 operator,

$\mathscr{L}\supset\frac{G_{\mu
p}}{\sqrt{2}}\left(\bar{p}\chi_{2}\right)\left(\bar{\mu}\chi_{0}\right)+\text{h.c.}$
(14)

One can also contemplate other chiral and Lorentz structures than the one
above; we discuss some variations and their impact on our conclusions in
Section IV. At the quark level, the above interaction is generated from
operators of dimension 9 or higher. The underlying new-physics scale
associated with the nucleon-level four-fermion operator is, therefore, more
accurately estimated as $\Lambda\sim\left(\Lambda_{\rm QCD}^{3}/G_{\mu
p}\right)^{1/5}\sim 100$ GeV, which points to new color-charged particles at
the electroweak scale (see Section IV for a more detailed discussion of UV
completions).

Note that if $\chi_{0}$ carries baryon number, $B(\chi_{0})=1$, then baryon
number is conserved by the interaction in Eq. 14. Since we are interested in
scenarios where $m_{0}<m_{p}$, this means that $\chi_{0}$ would be the
lightest baryon and remain stable. If we can consistently assign
$L(\chi_{0})=1$, then lepton number is also conserved. The other possibility
is that $B(\chi_{2})=B(\chi_{1})=1$, and $L(\chi_{2})=L(\chi_{1})=1$, which
also keeps the baryon and lepton numbers conserved. In this case, the lightest
baryon is $\chi_{1}$. Which of the dark sector particles carries baryon and
lepton numbers has no bearing on the phenomenology that we are interested in,
though. The important point for us is that there is a muon-induced exothermic
transition between nucleons and dark sector particles that reduces the number
of nucleons by one unit.

The heaviest dark fermions are required to interact with dark photon via the
off-diagonal terms, giving rise to $\chi_{2}\to\chi_{1}+A^{\prime}$ decays. We
consider two ways of generating such interactions, either through dimension 4
and dimension 5 operators,

$\mathscr{L}\supset
g_{D}\big{(}\bar{\chi}_{2}\gamma^{\mu}\chi_{1}\big{)}A_{\mu}^{\prime}+d_{\rm
tr}\big{(}\bar{\chi}_{2}\sigma^{\mu\nu}\chi_{1}\big{)}F_{\mu\nu}^{\prime}+\text{h.c.}.$
(15)

The first term arises if $\chi_{1}$ and $\chi_{2}$ are maximally mixed
Majorana fermions (a well-known example is, e.g., inelastic dark matter
Tucker-Smith:2001myb ). The second term is a non-renormalizable transition
magnetic moment between the two (Dirac or Majorana) fermions through the dark
photon, where
$F_{\mu\nu}^{\prime}=\partial_{\mu}A^{\prime}_{\nu}-\partial_{\nu}A^{\prime}_{\mu}$.
In the remainder of the paper, we will refer to the following benchmark for
Mu2e signatures,

$\begin{split}&\text{Benchmark (I)}\\\ &m_{2}=1030\text{
MeV},\,m_{1}=900\text{ MeV},\\\ &m_{0}=0,\,m_{A^{\prime}}=20\text{ MeV},\\\
&G_{\mu p}=(300\text{
TeV})^{-2},\,\varepsilon=10^{-4},\,\alpha_{D}=10^{-3}.\end{split}$ (16)

Similarly, in the case of a transition magnetic moment,

$\begin{split}&\text{Benchmark (II)}\\\ &m_{2}=1030\text{
MeV},\,m_{1}=900\text{ MeV},\\\ &m_{0}=0,\,m_{A^{\prime}}=20\text{ MeV},\\\
&G_{\mu p}=(300\text{ TeV})^{-2},\,\varepsilon=10^{-4},\,d_{\rm
tr}=10^{-3}\text{ GeV}^{-1}.\end{split}$ (17)

This choice of spectrum ensures that the energy release in $\mu\to e$
transitions can be large $E_{e^{+}/e^{-}}>E_{e}^{\rm conv}$ while guaranteeing
that at least two particles must be off-shell in nucleon decay.

Given Eq. (13) and our benchmarks, the partial width for the leptonic decay of
the dark photon is given by

$\Gamma_{A^{\prime}\rightarrow
e^{+}e^{-}}=\frac{1}{3}\alpha\varepsilon^{2}m_{A^{\prime}}\sqrt{1-4r_{e}^{2}}(1+2r_{e}^{2})~{},$
(18)

where $r_{e}\equiv m_{e}/m_{A^{\prime}}$. For our benchmarks, this gives a
lifetime of $\tau_{A^{\prime}}^{0}\approx 1.4$ ps. The $\chi_{2}$ lifetime
depends on which interaction term in Eq. 15 is considered. Assuming the
channel is open, the partial width for $\chi_{2}$ to decay to $\chi_{1}$ and
$A^{\prime}$ through the vector interaction is given by

$\Gamma^{V}_{\chi_{2}\rightarrow\chi_{1}A^{\prime}}=\frac{1}{4}\alpha_{D}\frac{m_{2}^{3}}{m_{A^{\prime}}^{2}}F\left(r_{1},r_{A^{\prime}}\right)~{},$
(19)

where

$\begin{split}F(r_{1},r_{A^{\prime}})=&\left(r_{1}^{4}-2r_{A^{\prime}}^{4}+r_{1}^{2}(r_{A^{\prime}}^{2}-2)+r_{A^{\prime}}^{2}(1-6r_{1})+1\right)\\\
&\times\sqrt{r_{1}^{4}-2r_{1}^{2}(r_{A^{\prime}}^{2}+1)+(1-r_{A^{\prime}}^{2})^{2}}~{},\end{split}$
(20)

and $r_{1}\equiv m_{1}/m_{2}$ and $r_{A^{\prime}}=m_{A^{\prime}}/m_{2}$.
Giving a benchmark lifetime $\tau_{\chi_{2}}^{0}\approx 6\times 10^{-14}\text{
ns}$. In the limit that $r_{A^{\prime}}\rightarrow 0$, the mass splitting
between $\chi_{1}$ and $\chi_{2}$ also vanishes ($r_{1}\rightarrow 0)$ and the
width is well defined. The tensor interaction gives the partial rate

$\Gamma^{T}_{\chi_{2}\rightarrow\chi_{1}A^{\prime}}=\frac{1}{4\pi}d_{\text{tr}}^{2}m_{2}^{3}\tilde{F}(r_{1},r_{A^{\prime}})~{},$
(21)

where

$\begin{split}\tilde{F}(r_{1},r_{A^{\prime}})=&\left(2(1-r_{1}^{2})^{2}-r_{A^{\prime}}^{2}(r_{A^{\prime}}^{2}+r_{1}^{2}+6r_{1}+1)\right)\\\
&\times\sqrt{r_{1}^{4}-2r_{1}^{2}(r_{A^{\prime}}^{2}+1)+(1-r_{A^{\prime}}^{2})^{2}},\end{split}$
(22)

giving a benchmark lifetime of $\tau_{\chi_{2}}^{0}\approx 3\times
10^{-7}\text{ ns}$. These decays are all well below the
$\mathcal{O}(1\,\mu{\rm s})$ signal time window at Mu2e and COMET. This is no
longer true when the dark photon becomes off-shell in
$\chi_{2}\to\chi_{1}e^{+}e^{-}$ decays. For instance, when
$m_{A^{\prime}}=200$ MeV, the $\chi_{2}$ lifetime is of the order of $\mu$s
and $s$ for the couplings in LABEL:eq:benchmarkI and LABEL:eq:benchmarkII,
respectively. In comparison with the timescales of the beam and muon capture,
these events will constitute a constant rate in time and may be harder to
discriminate from beam-induced backgrounds.

No. | Decay channel | Mass range | Decay rate scaling
---|---|---|---
1) | $p\rightarrow\mu\chi_{2}\chi_{0}$ | $m_{p}>m_{\mu}+m_{2}+m_{0}$ |  $\Gamma\propto\dfrac{G_{\mu p}^{2}Q^{5}}{8\pi\times 16\pi^{2}}$
2) | $p\rightarrow\mu(\chi_{2}^{*}\to\chi_{1}A^{\prime})\chi_{0}\qquad\qquad\,\biggr{\\{}$ | $m_{p}>m_{\mu}+m_{A^{\prime}}+m_{1}+m_{0},$ |  $\Gamma\propto\dfrac{G_{\mu p}^{2}\alpha_{D}Q^{7}}{(16\pi^{2})^{2}m_{2}^{2}}$
$m_{1}+m_{A^{\prime}}>m_{2}$ or $m_{2}>m_{p}-m_{\mu}-m_{0}$
3) | $p\rightarrow\mu(\chi_{2}^{*}\to\chi_{1}(A^{\prime}{}^{*}\to ee))\chi_{0}\,\,\,\Biggr{\\{}$ | $m_{p}>m_{\mu}+2m_{e}+m_{0}$ |  $\Gamma\propto\dfrac{G_{\mu p}^{2}\alpha_{D}(e\varepsilon)^{2}Q^{11}}{(16\pi^{2})^{3}m_{2}^{2}m_{A^{\prime}}^{4}}$
$m_{1}+2m_{e}>m_{2}$ or $m_{2}>m_{p}-m_{\mu}-m_{0}$
$2m_{e}>m_{A^{\prime}}$ or $m_{A^{\prime}}>m_{p}-m_{\mu}-m_{0}-m_{1}$ |
4) | $p\rightarrow(\mu^{*}\to e\nu\nu)\chi_{2}\chi_{0}$ |  $m_{p}>m_{2}+m_{0}+m_{e}$ and $m_{\mu}>m_{p}-m_{2}-m_{0}$ |  $\Gamma\propto\dfrac{G_{\mu p}^{2}G_{F}^{2}Q^{11}}{8\pi\times(16\pi^{2})^{3}m_{\mu}^{2}}$
5) | $p\rightarrow(\mu^{*}\to e\nu\nu)(\chi_{2}^{*}\to\chi_{1}A^{\prime})\chi_{0}\,\Biggr{\\{}$ | $m_{p}>m_{1}+m_{0}+m_{e}+m_{A^{\prime}}$ |  $\Gamma\propto\dfrac{G_{\mu p}^{2}G_{F}^{2}\alpha_{D}Q^{13}}{(16\pi^{2})^{4}m_{\mu}^{2}m_{2}^{2}}$
$m_{\mu}>m_{p}-m_{1}-m_{0}-m_{A^{\prime}}$
$m_{2}<m_{1}+m_{A^{\prime}}$ or $m_{p}-m_{0}-m_{e}<m_{2}$
Table 1: Proton decay channels and the corresponding constraints on the dark
sector mass spectrum that we impose so as to forbid them at tree level. The
naive scaling of each decay channel, assuming all final state particles are
relativistic, is shown in the right-most column. Generalization to non-
relativistic final state particles is straightforward – see Sec. III.2 for
more details.

###  III.1 Muon capture signature

While the prediction for the aBNV transition
$\mu^{-}+\isotope[27]{Al}\to\chi_{0}+\left(\chi_{2}\to\chi_{1}e^{+}e^{-}\right)+\isotope[26]{Mg}^{(*)}$
requires a calculation of inelastic nuclear response function, we can make a
naive estimate for its rate by using the muon capture rates in the SM. That
is, an ${\mathcal{O}}(1)$ fraction of muon capture on 27Al involves an
emission of a neutron,
$\mu^{-}+\isotope[27]{Al}\to\nu_{\mu}+n+\isotope[26]{Mg}^{(*)}$, which has
similar $1\to 3$ decay kinematics and involves the same types of nuclear wave-
functions in the initial and final state. We assume that ignoring the phase
space corrections, the ratio of aBNV and SM muon captures remains roughly the
same, irrespective of the initial nucleus. This implies, for instance

$\begin{split}\frac{|\mathcal{M}(\mu\,\isotope[27]{Al}\to\chi_{0}\chi_{2}\,\isotope[26]{Mg})|^{2}}{|\mathcal{M}(\mu\,\isotope[27]{Al}\to\nu_{\mu}n\,\isotope[26]{Mg})|^{2}}\simeq&\frac{|\mathcal{M}(\mu
p\to\chi_{0}\chi_{2})|^{2}}{|\mathcal{M}(\mu
p\to\nu_{\mu}n)|^{2}}~{}.\end{split}$ (23)

Including the phase space correction, we thus have

$\Gamma(\mu\,\isotope[27]{Al}\to\chi_{0}\chi_{2}\,\isotope[26]{Mg})\simeq
r_{\rm p.s.}\frac{G_{\mu
p}^{2}}{G_{F}^{2}}\Gamma(\mu\,\isotope[27]{Al}\to\nu_{\mu}n\,\isotope[26]{Mg}),$
(24)

where we estimate the effect of phase space with the ratio of two Dalitz plot
areas

$\begin{split}r_{\rm p.s.}\simeq&\frac{A_{\rm
Dalitz}(\mu\,\isotope[27]{Al}\to\chi_{0}\chi_{2}\,\isotope[26]{Mg})}{A_{\rm
Dalitz}(\mu\,\isotope[27]{Al}\to\nu_{\mu}n\,\isotope[26]{Mg})}\\\
\simeq&\frac{\big{(}M-m_{{\rm Mg}}-m_{0}-m_{2}\big{)}^{2}\big{(}M-m_{\rm
Mg}+m_{0}+m_{2}\big{)}}{\big{(}M-m_{{\rm Mg}}-m_{n}\big{)}^{2}\big{(}M-m_{\rm
Mg}+m_{n}\big{)}}.\end{split}$ (25)

In the last line, we neglected the neutrino mass. Above, $M$ is the mass of
the muonic atom, $M=m_{\mu}+m_{\rm Al}-E_{b}$, with $E_{b}\approx 0.463$ MeV
the binding energy, $m_{\rm Al}$ the mass of the $\isotope[27]{Al}$ and
$m_{\rm Mg}$ the mass of the $\isotope[26]{Mg}$ atom. Note that for
$m_{0}+m_{2}=m_{n}$ the above ratio is $r_{\rm p.s.}=1$. This is exact only if
$m_{0}$ is massless, showing the limitation of our approximations. For our
benchmarks, LABEL:eq:benchmarkI and LABEL:eq:benchmarkII, we find $r_{\rm
p.s.}\simeq 4\times 10^{-3}$.

Equating $\Gamma(\mu\,\isotope[27]{Al}\to\nu_{\mu}n\,\isotope[26]{Mg})$ with
the full SM muon capture rate,
$\Gamma_{\mu\text{Al}}=0.7\times\mu\mathrm{s}^{-1}=4.6\times 10^{-19}$ GeV
Suzuki:1987jf , which is in line with our other approximations, we then get
for the aBNV conversion rate

$R\equiv\frac{\Gamma_{\text{exotic}}}{\Gamma_{\mu\text{Al}}}\simeq r_{\rm
p.s.}\frac{G_{\mu p}^{2}}{G_{F}^{2}}.$ (26)

The benchmarks in LABEL:eq:benchmarkI and LABEL:eq:benchmarkII result in a
rate $R\sim 3\times 10^{-15}$, well within Mu2e and COMET’s sensitivities and,
as we will see, safe from nucleon decay bounds.

Figure 5: Estimates of the lifetime of a free proton (left) and neutron
(right) as a function of the mass of the lightest dark particle $\chi_{1}$ for
benchmark (I). The bottom panel illustrates the scaling of the lifetime with
respect to the energy release $Q$, which is proportional to the nucleon mass
minus the sum of the masses of all final state particles that are on shell,
see Eqs. 27 and 28. The nucleon lifetime $\tau_{N}^{\rm min}$ and its
respective energy release $Q_{\rm min}$ are calculated for the smallest value
of $\chi_{1}$ mass shown in the figure. We do not include loop-level proton
decay channels which may dominate at the largest masses.

Once produced, $\chi_{2}$ eventually decays to an $e^{+}e^{-}$ pairs through
the cascade $\chi_{2}\to\chi_{1}(A^{\prime}\to e^{+}e^{-})$. The resulting
energy spectra of the $e^{+}e^{-}$ pairs produced are shown for a few
representative points in Figs. 3 and 4. Note that the individual electron or
positron energies can be greater than $m_{\mu}$ due to the additional energy
released by the destruction of the proton mass. This is also manifested in the
sum of the electron and positron energies, which peaks at
$E_{e^{+}}+E_{e^{-}}\simeq\left(m_{2}^{2}-m_{1}^{2}+m_{A^{\prime}}^{2}\right)/2m_{2}$,
given that $\chi_{2}$ only has a small boost and thus the lab frame kinematics
almost coincide with the rest frame $\chi_{2}\to\chi_{1}A^{\prime}$ two body
decay kinematics ($E_{e^{+}}+E_{e^{-}}=E_{A^{\prime}}$ for $A^{\prime}$ on-
shell).

Several features can help distinguish the aBNV processes from $\mu^{-}\to
e^{-}$ conversion, most notably the differing electron spectra and the
presence of positrons in the aBNV case. A signal of equal magnitude with equal
endpoints in the electron and positron channels that go beyond the $\mu^{-}\to
e^{-}$ endpoint would be a strong indication for aBNV processes of the type
considered here. The positron signal is subject to even smaller backgrounds
than the search for $\mu^{-}\to e^{-}$ signal is; the DIO background produces
only electrons, and charge misidentification is expected to be rare. Most
background events are expected to arise from radiative muon capture (RMC) with
either internal or external photon conversion. RMC, however, has an even lower
endpoint than DIO, $E_{\rm RMC}^{\rm Al}\lesssim 101.9$ MeV Yeo:2017fej . The
positron search could be done already with the Al target, while one could also
explore other targets (for a different BSM signal producing positrons, the
lepton-number-violating $\mu^{-}\to e^{+}$ conversion, see Refs. Yeo:2017fej ;
Lee:2021hnx ). Finally, we note that the $Z$ dependence of the aBNV rate will
differ from the one for $\mu^{-}\to e^{-}$ or $\mu^{-}\to e^{+}$ conversion
(see, for instance, Ref. Borrel:2024ylg ). The planned upgrades of Mu2e using,
e.g., Au as the target Mu2e-II:2022blh , could thus help distinguish between
these scenarios.

One of the crucial aspects of the Mu2e experiment is the requirement of a very
high degree of initial proton and pion beam extinction within the signal time
window, as well as a very reliable rejection of cosmic-ray-induced events. If
a certain amount of pion “contamination” is, for example, present within the
signal-taking time interval, it may result in charged leptons with energy well
above $E_{e}^{\rm conv}$. The $\pi^{-}$ capture on a nucleus can thus result
in a $\sim 135$ MeV photon, which can Compton-scatter and produce electrons
well above $E_{e}^{\rm conv}$. The $\mu^{-}p$ annihilation electrons could,
therefore, easily be misinterpreted as being due to a pion background. The
above benchmarks for the $\mu^{-}p$ annihilation models, which can create
$E_{e}\sim m_{\pi}$, thus show that extra care needs to be taken. In the event
of a signal, a detailed investigation would be warranted to determine
definitively whether such electrons do indeed stem from the pion decays and
not from exotic new physics signatures, such as those investigated in this
section.

###  III.2 Proton decay

Figure 6:  The normalized true total visible energy spectrum (energy of the
$e^{+}e^{+}e^{-}$ system) in $p\to eee\nu\nu\chi_{1}\chi_{0}$ decays for two
choices of dark photon masses. The distributions look similar for the same
benchmarks when using transition moments instead ($\alpha_{D}\to d_{\rm tr}$),
although the total lifetimes increase.

The effective interaction in Eq. 14 violates proton number. What is an
apparent phenomenological disaster can be readily reconciled with the most
stringent constraints on proton decay by a judicious choice of mass
hierarchies. By such a choice, we can ensure that some of
$\chi_{2},\chi_{0},\mu$ must be produced off their mass shell, resulting in a
higher particle multiplicity of final states and extra suppression by small
coupling constants. Furthermore, the amount of open final state phase space
will be small, suppressing the proton decay rate. In Table 1, we list the
possible decay channels, the necessary constraints on the mass hierarchy of
the model to allow this channel, and an estimate for the proton decay width.
Before discussing how these constraints affect the allowed region of parameter
space, we first discuss how they are derived. Later on, we will calculate the
rate fully numerically using MadGraph5_v3.5.3.

##### Scaling with energy release:

The energy released in proton decay is

$\Delta E=(m_{p}-\sum_{i}^{N_{f}}m_{i}),$ (27)

where $N_{f}$ is the number of final state particles in the decay. Each of the
final particles therefore carries a typical momentum

$Q\simeq\Delta E/N_{f}~{},$ (28)

with $N_{f}\in[3,7]$ depending on which decay is being considered; we leave
the discussion of loop decay for later since it is UV model dependent. The
typical value of $Q$ in proton decay is $\sim\mathcal{O}(5)$ MeV, see Fig. 7.
That is, at the upper range of $m_{1}$, the electrons will start becoming non-
relativistic; we will discuss the effect of this below. In some of the decay
channels, there are off-shell particles whose propagators we estimate to scale
as inverse powers of the particle mass ($m^{-1}$ for fermions and $m^{-2}$ for
bosons). For instance, the muon has 3 approximately massless decay products
and scales as

$\frac{\not{p}_{\mu}+m_{\mu}}{p_{\mu}^{2}-m_{\mu}^{2}}\sim\frac{3\not{Q}+m_{\mu}}{(3Q)^{2}-m_{\mu}^{2}}\sim-\frac{1}{m_{\mu}},$
(29)

whereas for $\chi_{2}$ the decay products may be non-relativistic and it
scales as

$\frac{\not{p}_{\chi_{2}}+m_{2}}{p_{\chi_{2}}^{2}-m_{2}^{2}}\sim\frac{\not{p}_{\chi_{2}}+m_{2}}{(m_{\chi_{1}}+m_{A^{\prime}})^{2}-m_{2}^{2}}\sim-\frac{1}{m_{2}},$
(30)

where we used the fact that $\chi_{1}$ and $A^{\prime}$ are non-relativistic
(a better approximation for $\chi_{1}$ than it is for $A^{\prime}$). The
matrix element squared acquires scalings with $Q$ when carrying out
spin/polarization sums. The final state fermions contribute to
$|\mathcal{M}|^{2}$ parametrically as $\sim Q$ $(\sim m)$ if they are
relativistic (non-relativistic), while the final state vector bosons as $\sim
Q^{0}$ $(\sim Q^{2}/m_{A^{\prime}}^{2})$. Furthermore, although we focus on
the contact interaction, if the coupling is through the dipole interaction
$d_{\rm tr}$ then $|\mathcal{M}|^{2}$ picks up another factor of $Q^{2}$. The
remaining $Q$ scaling comes from the phase space integrals, where for each
final state particle

$\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{i}}\sim\left\\{\begin{matrix}\dfrac{1}{16\pi^{2}}Q^{2},&\text{relativistic},\\\\[7.22743pt]
\dfrac{1}{16\pi^{2}}\dfrac{Q^{3}}{m_{i}},&\text{non-rel.},\end{matrix}\right.$
(31)

The scaling of the matrix element and phase space for final state fermions is
such that each one contributes $Q^{3}$ in both the relativistic and non-
relativistic limits. We present the results of these approximations in Table
1, where we have assumed the final state $A^{\prime}$ is non-relativistic,
which will be the case for the parameter space we are interested in. In Table
1, we also list the scaling for a tree level $p\to\mu ee\chi_{1}\chi_{0}$
decay with $\chi_{1}$ heavy enough for $A^{\prime}$ to be required to be off-
shell, which gives an additional $Q^{4}/m_{A^{\prime}}^{4}$ suppression for
the decay width.

From these scalings, it is clear that channels 1 and 2 will lead to too rapid
proton decay unless $\alpha_{D}$ is taken very small, which removes the prompt
signal at Mu2e. Channel 3 requires considerable tuning in the mass spectrum,
so we ignore it from now on. This leaves channels 4 and 5, both of which can
be small enough to avoid constraints. The benchmarks described in
LABEL:eq:benchmarkI and LABEL:eq:benchmarkII have a decay through channel (5)
with an estimated lifetime of $\tau_{p}\sim 5\times 10^{34}$ years and
$\tau_{p}\sim 2\times 10^{41}$ years, respectively.

##### MadGraph estimates:

To calculate the rate more accurately and study the resulting kinematics, we
implement the model in MadGraph5_v3.5.3 and calculate the proton lifetime by
generating $p\to(\mu^{*}\to
e\nu_{e}\overline{\nu}_{\mu})(\chi_{2}^{*}\to\chi_{1}(A^{\prime}\to
e\bar{e}))\chi_{0}$ decays. This includes contributions from real as well as
virtual $A^{\prime}$. We calculate the decay width of the new particles for
each choice of couplings. For values of $\alpha_{D}$ larger than about
$\alpha_{D}=10^{-2}$, the $\chi_{2}$ width can lead to important corrections
to the decay rate and kinematics, but we do not explore this further here. For
the small values of kinetic mixing we are interested in, the width of the dark
photon is always negligible. Our results are shown for $10^{5}$ generated
events.

Figure 7: The momentum of the individual final state particles in $p\to
eee\nu\nu\chi_{1}\chi_{0}$ decays for two choices of dark photon masses. Here
the final state positrons $e^{+}_{1}$ and $e^{+}_{2}$ are distinguished at the
event level by the magnitude of the three-momenta, with $e^{+}_{2}$ always
being assigned to the larger momenta.

The left panel of Fig. 5 shows the resulting proton lifetime in the region of
interest for the benchmark point in LABEL:eq:benchmarkI as a function of the
$\chi_{1}$ mass. The bottom row of the left panel shows the accuracy of the
$Q^{13}$ scaling. This lifetime should be compared to experimental
constraints. The strongest constraints on the proton lifetime were obtained in
underground neutrino detectors, where, for instance, $\tau(p\to
e^{+}\pi^{0})>2.4\times 10^{34}$ years Super-Kamiokande:2020wjk . Other
similar channels have been searched for, including $\tau(p\to
e^{+}e^{+}e^{-})>3.4\times 10^{34}$ years Super-Kamiokande:2020tor . While
these channels have similar topologies to $p\to eee\nu\nu\chi_{1}\chi_{0}$,
namely three electromagnetic showers, the kinematics differs significantly
from our signal. In particular, the energy of the three showers must
reconstruct the proton, while in our case, the energy release is, in fact,
much smaller. Figure 6 shows the total visible energy of the exotic proton
decay mode, showing that the energy released into electrons and positrons is
significantly smaller than the proton mass. In this low-energy region,
experimental backgrounds, such as from neutrino interactions at underground
detectors, become larger, typically by factors of $\mathcal{O}(100)$ Super-
Kamiokande:2020wjk . Therefore, constraints on proton decays induced by the
muon-induced baryon number violation are expected to be significantly weaker
than they are for the golden $p\to e^{+}\pi^{0}$ and $p\to e^{+}e^{+}e^{-}$
decay channels. While a detailed study of the resulting limit is beyond the
scope of this paper, it appears that one can safely conclude that the current
searches do not exclude lifetimes as small as $10^{32}$ years.

Figure 7 shows the momentum distribution of the final state particles in the
exotic proton decay $p\to eee\nu\nu\chi_{1}\chi_{0}$. As expected, the
available energy $\Delta E$ is split unevenly between the final states thanks
to the large imbalance of masses of the particles involved. In fact, the final
states produced in the chain $\chi_{2}\to\chi_{1}+A^{\prime}$ carry more
energy thanks to the large mass of $\chi_{1}$ and $\chi_{2}$. Similarly,
positrons emitted from the $\mu^{+}\to e^{+}\nu\nu$ chain carry less energy
than those from the decay of the on-shell dark photon.

Finally, an important consideration is that the tree-level six-body proton
decay channel, for which we estimated the rates above, is not the only
possible proton decay mode that can be generated via $\mu p$ annihilation
interactions, once radiative corrections are included. In fact, at the loop
level, many new decay channels open up, with a smaller number of final state
particles, which relaxes the steep suppression of the decay rate by high
powers of $Q$. An example is a loop correction obtained by attaching the dark
photon line to any of the charged particles, which then leads to
$p\to(\chi_{1}\chi_{0})(e^{+}\nu_{e}\overline{\nu_{\mu}})$ decays. In similar
ways, corresponding decay modes for bound neutrons can be found. Since these
contributions are UV-sensitive, we return to them in Section IV, where we will
show that this reduction in $Q$ dependence is compensated by the necessary
inclusion of small couplings/masses.

###  III.3 Neutron stars

Neutron star properties such as mass-radius relation and their cooling
function are well understood and can be used to probe many aspects of new
physics Yakovlev:2004iq ; Lattimer:2006xb . Among others, neutron stars can
also provide an indirect probe of the new $\mu^{-}$-nucleon interactions in
Eq. 1, due to a large abundance of muons, not only during the initial stage of
the core-collapse supernova explosion but also at much later stages in the
form of the quasi-degenerate Fermi fluid of $\mu^{-}$. A related topic has
been addressed in the literature of “dark neutron” models Fornal:2018eol ,
from where we borrow several results in our discussion (see, in particular,
Refs. McKeen:2018xwc ; Motta:2018rxp ; Baym:2018ljz ; Cline:2018ami ;
Strumia:2021ybk ; Darini:2022mcy ).

The zero-temperature nuclear matter inside neutron stars is in chemical
equilibrium due to weak interaction transitions, such as $n\leftrightarrow
p\ell^{-}\overline{\nu}_{\ell}$ and $p\,\ell^{-}\leftrightarrow n\nu_{\ell}$,
where $\ell=\\{e,\mu\\}$. These ensure that the chemical potentials $\mu_{i}$
for $i=n,p,e,\mu,$ satisfy $\mu_{n}-\mu_{p}=\mu_{e}=\mu_{\mu}$. Thanks to the
large value of $\mu_{e}$ it is possible for beta-decay type reactions to
create muons inside the neutron star despite their larger mass. The large
value of $\mu_{e}$ also makes $\mu^{-}$ stable inside the neutron star. The
muon fraction inside the core of the neutron star is determined by nuclear
density and equation of state, and, therefore, depends on the modeling of
nuclear matter.111In some models of nuclear matter it is hypothesized that at
extremely large densities there is a stable population of heavier baryons and
hyperons such as $\Delta(1232)^{-}$ and $\Sigma^{-}$, which dominate the
number density of negatively charged particle species, and de-leptonize the
core. This can significantly reduce the muon density and, therefore, the rate
of $\mu^{-}p$ or $\mu^{-}n$ reactions. Similar arguments apply to the higher-
density models where the transition to quark matter reduces the muon chemical
potential $\mu_{\mu}$. In Ref. Zhang:2020wov , it was estimated that the
fraction of the muon to baryon number densities could be as high as
$n_{\mu}/n_{B}\sim 20\%$, to be compared with the maximum proton fraction of
about $n_{p}/n_{B}\sim 40\%$.

The muon-induced aBNV transitions of the form $\mu^{-}n\to e^{-}\\{X\\}^{0}$
or $\mu^{-}p\to e^{-}\\{X\\}^{+}$, cf. Eq. 1, can change the above
equilibrium, since now neutrons and/or protons can be converted into dark
sector states inside the neutron star. That is, in the concrete model we are
considering in detail, the process $\mu^{-}p\to\chi_{0}\chi_{2}$ (and the
$\nu_{\mu}n\to\chi_{0}\chi_{2}$ process, generated at 1-loop) will become
energetically favorable and deplete the number of nucleons. The capture rate
inside the neutron star is given by

$\begin{split}\Gamma^{\rm
NS}_{\mu^{-}p\to\chi_{0}\chi_{2}}&=\langle\sigma_{\mu p}v\rangle n_{p}\simeq
G_{\mu p}^{2}m_{\mu}m_{p}n_{p},\end{split}$ (32)

where $\sigma_{\mu p}$ is the cross section for scattering of muons on
protons, $v$ is their relative velocity, and $\langle\dots\rangle$ indicates
that we should average over the phase space of the particles involved. For the
benchmark value of $G_{\mu p}$, LABEL:eq:benchmarkI and LABEL:eq:benchmarkII,
this gives

$\Gamma^{\rm
NS}_{\mu^{-}p\to\chi_{0}\chi_{2}}\simeq\frac{1}{100\,s}\biggr{(}\frac{G_{\mu
p}}{(300\,\text{TeV})^{2}}\biggr{)}^{2}.$ (33)

Interestingly, for dynamics on such intermediate time-scales
$\sim\mathcal{O}({\rm min}-{\rm months})$ there are no stringent constraints
from the properties of neutron stars, as long as $\chi_{0}$ and $\chi_{2}$ do
not escape from the stellar interior. The constraints on properties of the
proto-neutron star come from the measurements of the neutrino flux that was
generated by the supernova SN-1987a explosion. This probes dynamics on the
timescales $\lesssim 10$s of seconds. The constraints on properties of neutron
stars, on the other hand, apply only to much longer timescales. Typically,
there are no direct observations of young neutron stars following the
supernova explosion since these are obscured by the remnants of the outer
envelope of a progenitor.

The muon-induced aBNV processes in Eq. (1) would heat up the star, emitting
visible energy in the form of $e^{+}$ and $e^{-}$. The $\mu^{-}p$ and/or
$\mu^{-}n$ annihilations would create holes in the nucleon and $\mu^{-}$ Fermi
seas, which then get refilled, resulting in additional energy release. If such
processes are rather slow, i.e., occurring on time-scales of millions of
years, the late-time observables of neutron stars, such as the temperature
McKeen:2020oyr ; McKeen:2021jbh  of the coldest neutron stars Guillot:2019ugf
or the periods of pulsars Goldman:2019dbq , will be sensitive to the muon-
induced aBNV.

However, for the capture rates that Mu2e and COMET are sensitive to, the
nucleon depletion is fast and will lead to a new equilibrium inside the star
with a lower pressure and a softer equation of state (EoS) due to the
redistribution of energy over a larger number of degrees of freedom. The dark
sector dynamics will determine whether or not the new EoS will support stars
with radii and masses consistent with observation. These considerations are
similar to what happens in mirror neutron models McKeen:2018xwc ;
Motta:2018rxp ; Baym:2018ljz ; Cline:2018ami , with the added requirement that
muons participate in the conversion reactions. It is clear that the models
with stable massive fermions in the dark sector, similar to our benchmark
cases, are vastly favored over the models in which the dark sector states
decay to very light fermions and/or bosons. In the latter case, the newly
produced dark states are not retained by the neutron stars and/or do not
contribute to the Fermi pressure.

A new dark force can help restore the pressure lost to the increased number of
degrees of freedom. To illustrate this, consider a scenario where
${\mathcal{O}}(50\%)$ of nucleons get converted into a population of new
fermions, $\chi$. This may be $\chi_{1}$ or $\chi_{0}$, depending on the
details of the model. If $k_{F,0}$ is the original Fermi momentum of neutrons
in a neutron star without dark states, then by particle number conservation,
$k_{F,0}^{3}=k_{F,n}^{3}+k_{F,\chi}^{3}$. If half of the existing nucleons go
to $\chi$, then $k_{F,n}\simeq k_{F,\chi}<k_{F,0}$, thereby reducing the
pressure (which is a highly nonlinear function of $k_{F}$).

A repulsive self-interaction between the dark sector states contributes to the
pressure and could compensate for the reduction of the Fermi momentum. The
change in pressure due to a single new dark particle $\chi$ subject to a new
long-range force mediated by $A^{\prime}$ is

$\Delta
P=\frac{1}{15\pi^{2}}\left(\frac{k_{F,n}^{5}-k_{F,0}^{5}}{m_{n}}+\frac{k_{F,\chi}^{5}}{m_{\chi}}\right)+\frac{g_{\chi}^{2}k_{F,_{\chi}}^{6}}{18\pi^{4}m_{A^{\prime}}^{2}}.$
(34)

Therefore, when
$(g_{\chi}^{2}/\pi^{2}m_{A^{\prime}}^{2})\times(k_{F,0}m_{n})\gg 1$, this
leads to a positive change in pressure compared to the one for a neutron star
in the SM, indicating that it is indeed possible to counteract the loss of
pressure through a repulsive force. In our scenario, the required size of
$g_{\chi}^{2}/m_{A^{\prime}}^{2}$ will depend on the dynamics of the dark
sector, the nature of $\chi_{0}$, and the origin of the new force (the
mediator of this dark force may be the dark photon of Section III, but in
general this may not be the case and could be due to a different light
mediator).

An alternative possibility for adjusting the neutron star properties was
pointed out in Refs. Strumia:2021ybk ; Darini:2022mcy , where the large
multiplicity of the dark fermions in the final state was used to correct for
the loss of Fermi pressure in the nucleon fluid. In our framework this can be
implemented by, e.g., further decays along the “dark decay chain”, such as
$\chi_{1}\to 3\chi_{3}$ with $m_{\chi_{3}}$ not too far from $m_{\chi_{1}}/3$.
This model would then not require any additional pressure from the
$A^{\prime}$ exchanges.

In conclusion, dark sector models offer sufficient flexibility to compensate
for the loss of nucleon Fermi pressure. Having outlined a potential
reconciliation between observable exotic muon capture and the physics of
neutron stars, we leave an investigation of the broader impact of muon-induced
baryon number violation on neutron stars to future literature.

##  IV UV completions

At first glance, the expected sensitivity of the Mu2e experiment at the level
of ${\rm R}(\mu\to e)\sim\mathcal{O}(10^{-16})$ translates to a very
impressive NP reach, on the order of $G_{\mu p}\propto 10^{-8}G_{F}$, or in
terms of the effective NP scale, $G_{\mu p}^{-1/2}\sim 10^{3}$ TeV. However,
$G_{\mu p}^{-1/2}$ is not a fundamental UV scale. Proton is composed of
quarks, and thus we can expect

$G_{\mu p}\sim\frac{\Lambda_{\rm QCD}^{3}}{\Lambda_{\rm col}^{d_{\rm
col}}\Lambda_{\rm sin}^{5-d_{\rm col}}},$ (35)

where the QCD scale is $\Lambda_{\rm QCD}\sim 0.3$ GeV, while the value of the
exponent, $d_{\rm col}$, depends on the details of the model. In Eq. 35 we
already anticipated that the phenomenologically favored situation is when the
mediators charged under QCD are much heavier than the color singlet mediators,
and we have thus split the two scales, $\Lambda_{\rm col}\gg\Lambda_{\rm sin}$
(for simplicity, we still assume that each of the two types of mediators have
a common mass).

As we will see below, credible UV models of $\mu^{-}p$ annihilation to dark
states will require _both_ heavy new physics, above the electroweak scale, as
well as light, GeV scale, states. The goal of this section is to demonstrate
that such UV completions exist without providing a comprehensive study of all
the models that can lead to the effective operator in Eq. 14.

Below, we discuss in detail the triplet-triplet-singlet model that leads to a
large scale $\Lambda_{\rm col}$ suppression in Eq. 35 with $d_{\rm col}=3$. We
find this to be rather typical, though models with smaller $d_{\rm col}=2$
(and thus a higher value of $\Lambda_{\rm col}$ for which there is still an
observable signal at Mu2e and COMET) do exist (see Section IV.3). It is also
important to keep in mind that other UV realizations can lead to distinct LHC
and high-energy phenomenology, which can differ significantly from the model
we discuss in detail.

Figure 8:  The left diagram shows the UV completion of Section IV that
realizes the contact interaction $G_{\mu p}$ of Eq. 14. A heavy di-quark
$T_{1}$ and a lepto-quark $T_{2}$ scalar couple to an intermediate-mass
gateway particle $\Phi$. The diagram on the right shows how the gateway
particle $\Phi$ can be produced on-shell in muon capture with a third-body
rescattering ($Z$ represents a nucleus of atomic number $Z$).

###  IV.1 Triplet-triplet-singlet model

A UV model that we consider in detail extends the SM with two scalar color
triplets, the di-quark $T_{1}$ ($Q=-1/3$, $B=-2/3$, $L=0$) and the leptoquark
$T_{2}$ ($Q=-1/3$, $B=1/3$, $L=1$), which have the following couplings to the
SM fermions222For charge conjugate fields we use the notation from
book:Tanagina:Fukugita . In particular,
$\psi_{R}=P_{R}\psi=\tfrac{1}{2}(1+\gamma_{5})\psi$, and
$\psi_{R}^{C}=P_{L}\psi^{C}=C\bar{\psi}_{R}^{T}=[\psi_{R}(x)]^{C}$. (Note that
this differs from Dreiner:2008tw  where $\psi_{R}^{C}$ denotes
$[\psi_{L}(x)]^{C}$.)

$\mathscr{L}\supset-
y_{ud}(\overline{u_{R}^{iC}}d^{j}_{R})\epsilon_{ijk}T_{1}^{k}-y_{\mu
u}(\overline{u_{R}^{iC}}\mu_{R})(T_{2}^{*})_{i}+{\rm h.c.},$ (36)

as well as by a complex color singlet scalar $\Phi$ ($Q=0$, $B=1$, $L=1$), and
the dark sector fermions $\chi_{0}\equiv\nu_{R}$ and $\chi_{2}$

$\mathscr{L}\supset\rho
T_{1}^{k*}T_{2k}\Phi^{*}+y_{\chi}\Phi(\overline{\nu_{R}}\chi_{2R}^{C})+{\rm
h.c.}.$ (37)

All the new states are singlets under $SU(2)_{L}$. The color contractions over
$i,j,k$ indices are shown explicitly, while the Lorentz contractions are not;
in the latter, $C$ denotes the charge conjugation Dirac matrix. Note that the
coupling constants, $y_{ud},y_{\mu u},y_{\chi}$ are dimensionless, while
$\rho$ has mass dimension $1$. In what follows, we require that $\Phi$ does
not develop a vacuum expectation value. If the $\nu_{R}\chi_{2}$ bilinear
carries the same global quantum numbers as $\Phi$, the above Lagrangian has
both $B$ and $L$ conserved.

We are interested in the limit of very heavy $T_{1,2}$ so that at low
energies, these can be integrated out. The resulting effective interaction is

$\mathscr{L}_{\rm eff}^{\Phi}=\frac{1}{\Lambda_{\rm
col}^{3}}\big{(}\overline{u_{R}^{iC}}\mu_{R}\big{)}\epsilon_{ijk}\big{(}\overline{u_{R}^{jC}}d^{k}_{R}\big{)}\Phi^{*}+{\rm
h.c.},$ (38)

where

$\frac{1}{\Lambda_{\rm col}^{3}}=y_{ud}y_{\mu
u}\frac{\rho}{m_{T_{1}}^{2}m_{T_{2}}^{2}}.$ (39)

Below the QCD confinement scale, this then leads to the effective interaction
with the proton,

$\mathscr{L}_{\rm int}^{\Phi}=\lambda_{\rm
eff}\Phi^{*}\big{(}\overline{p_{R}^{C}}\mu_{R}\big{)}+{\rm h.c.},$ (40)

with

$\lambda_{\rm eff}\simeq y_{ud}y_{\mu u}\frac{\rho\,\Lambda^{3}_{\rm
QCD}}{m_{T_{1}}^{2}m_{T_{2}}^{2}},$ (41)

where for the matrix element between the tri-quark operator and the proton we
used the NDA estimate, $\langle
0|\overline{u_{R}^{iC}}(\overline{u_{R}^{jC}}d_{R}^{k})\epsilon_{ijk}|p\rangle\simeq\Lambda_{\rm
QCD}^{3}\langle 0|\overline{p^{C}_{R}}|p\rangle$, with the parenthesis
denoting a Lorentz contraction.

####  IV.1.1 $\Phi$ contributing as an off-shell state

If $\Phi$ is heavy enough that it can also be integrated out, this gives the
effective interaction

$\mathscr{L}_{\rm eff}=\frac{1}{\Lambda_{\rm col}^{3}}\frac{1}{\Lambda_{\rm
sin}^{2}}\big{(}\overline{u_{R}^{iC}}\mu_{R}\big{)}\epsilon_{ijk}\big{(}\overline{u_{R}^{jC}}d^{k}_{R}\big{)}\left(\overline{\nu_{R}}\chi_{2R}^{C}\right)+{\rm
h.c.},$ (42)

with $\Lambda_{\rm col}$ given in Eq. 39, and

$\frac{1}{\Lambda_{\rm sin}^{2}}=\frac{y_{\chi}}{m_{\Phi}^{2}},$ (43)

or in terms of interactions with the proton333The structure of the dimension
six operator is similar to the operator considered in Eq. 14, but with a
different Lorentz structure. This operator further suppresses nucleon decay
while leading to ${\mathcal{O}}(1)$ difference in the muon capture
phenomenology. These quantitative differences, however, are not important for
the more qualitative discussion we are focusing on here.

$\mathscr{L}_{\rm int}=\widetilde{G}_{\mu
p}\left(\overline{p_{R}^{C}}\mu_{R}\right)\left(\overline{\nu_{R}}\chi_{2R}^{C}\right),$
(44)

where $\widetilde{G}_{\mu p}$ is given in Eq. 35 with $d_{\rm col}=3$, and
$\Lambda_{\rm col}$ and $\Lambda_{\rm sin}$ given in Eq. 39 and Eq. 43,
respectively.

For observable $\mu^{-}p$ annihilation rate with $R\simeq 3\times 10^{-15}$,
we require, from Eq. 26, that $\widetilde{G}_{\mu p}\simeq 10^{-6}G_{F}$. In
terms of the UV completion this is,

$\begin{split}&\widetilde{G}_{\mu p}=\frac{\lambda_{\rm
eff}y_{\chi}}{m^{2}_{\Phi}}=2\cdot
10^{-6}G_{F}\,y_{\chi}\left(\frac{\lambda_{\rm
eff}}{10^{-10}}\right)\left(\frac{2{\,\rm GeV}}{m_{\Phi}}\right)^{2}\\\ &\sim
10^{-6}G_{F}\,y_{ud}y_{\mu u}y_{\chi}\left(\frac{1\text{
TeV}}{\sqrt{m_{T_{1}}m_{T_{2}}}}\right)^{4}\left(\frac{\rho}{4\text{
TeV}}\right)\left(\frac{2\text{ GeV}}{m_{\Phi}}\right)^{2}.\end{split}$ (45)

In the numerical example in the last line, we show a possible case
$m_{T_{1},T_{2}}\sim\rho\gg m_{\Phi}\gtrsim m_{p}$. A light $\Phi$ enhances
the $\mu^{-}p$ annihilation rate, and so would lighter $T_{1,2}$. However, the
masses of $T_{1},T_{2}$ are required to be at least in the TeV range in order
to evade the LHC monojet constraints. Note that the trilinear scalar coupling
$\rho$ was taken to be large. The $\Phi$ mass receives radiative corrections
of size $m_{\Phi}^{2}\sim\rho^{2}/16\pi^{2}$, making our choice of $m_{\Phi}$
in the GeV range somewhat fine-tuned.

The phenomenology of Section III is obtained if $\chi_{2}$ decays inside the
detector to a lighter dark state $\chi_{1}$ via $\chi_{2}\to
A^{\prime}\chi_{1}$, and the dark photon then decays to an electron-positron
pair, $A^{\prime}\to e^{+}e^{-}$. We assume that the $\chi_{2}\to
A^{\prime}\chi_{1}$ transition is due to a dipole operator $d_{\rm tr}$ in Eq.
15, and that the states in the SM and in the triplet-triplet-singlet model,
including $\chi_{2,1}$, are not charged under the $U(1)_{d}$. The dipole
operator is generated from other heavier states charged under $U(1)_{d}$
running in the loop. The possibility where $\chi_{2,1}$ are charged under
$U(1)_{d}$ is discussed in Section IV.3.

####  IV.1.2 On-shell $\Phi$ production

The $\mu^{-}p$ annihilation rate can be further enhanced if the $\Phi$ scalar
is light enough to be produced on-shell, cf. Fig. 8. In this case, the muonic
atom undergoes a transition $\mu^{-}(A,Z)\to\Phi(A-1,Z-1)$. The
phenomenologically favored kinematics is when the $\Phi$ scalar can be
produced on-shell in muon capture, while a proton decay would require both
$\Phi$ and $\mu^{+}$ to be off-shell, thus suppressing the proton decay rate.
This occurs for $\Phi$ mass in the range $m_{p}<m_{\Phi}<m_{\mu}+m_{p}$
(ignoring the differences in the nuclear binding energies), while the
$\chi_{0}$ and $\chi_{2}$ masses satisfy
$m_{p}<m_{\mu}+m_{\chi_{0}}+m_{\chi_{2}}$, with
$m_{\chi_{0}}+m_{\chi_{2}}<m_{\Phi}$. The observable signal in the $\mu^{-}p$
annihilation arises from the decay of $\Phi$ to the dark sector states,
$\Phi\to\chi_{2}\chi_{0}$, followed by the subsequent decay of $\chi_{2}$,
which involves visible states, such as $\chi_{2}\to\chi_{1}e^{+}e^{-}$.

The $2\to 1$ process, $\mu^{-}p\to\Phi$, is enabled by the re-scatterings on
the other nucleons inside a nucleus. In this sense, the $\mu^{-}p\to\Phi$
annihilation is analogous to the well-known SM process of pion capture,
$\pi^{-}p\to n$, which at face value is also a $2\to 1$ process. Following
Section III.1 we compute a naive estimate for $\Phi$ production through a
ratio with the off-shell channel whose SM-normalized rate is given in Eq. 26.
The ratio of the on- and off-shell capture rate is given roughly by

$\frac{\Gamma_{\mu Z\to\Phi(Z-1)}}{\Gamma_{\mu
Z\to\chi_{0}\chi_{2}(Z-1)}}\sim\frac{m_{\Phi,{\mathrm{off}}}^{4}}{y_{\chi}^{2}E_{\chi_{2}}E_{\chi_{0}}}\frac{16\pi^{2}}{\big{(}M-m_{\mathrm{Mg}}\big{)}^{3}}\tilde{r}_{\mathrm{p.s.}},$
(46)

where the phase space factor can be estimated as

$\tilde{r}_{\mathrm{p.s.}}\simeq\frac{\big{(}1-x_{\Phi,{\mathrm{on}}}^{2}\big{)}^{1/2}}{\big{(}1-x_{02}\big{)}^{2}\big{(}1+x_{02}\big{)}}~{},$
(47)

with $x_{\Phi,{\rm on}}=m_{\Phi,{\mathrm{on}}}/(M-m_{\rm Mg})$ and
$x_{02}=(m_{\chi_{0}}+m_{\chi_{2}})/(M-m_{\rm Mg})$. Here the $m_{\Phi,{\rm
on}}$ is the mass of the $\Phi$ that is produced on-shell, while $m_{\Phi,{\rm
off}}$ is the mass of the virtual $\Phi$ leading to the three body transition.
Taking $m_{\Phi,\text{on}}=1\text{ GeV}$ and comparing to the off-shell
benchmark, Eqs. (LABEL:eq:benchmarkI), (LABEL:eq:benchmarkII),
(LABEL:eq:Gmup:estimate), the on-shell rate is significantly enhanced by
$\approx 10^{9}-10^{10}\times y^{-2}_{\chi}$ depending on the phase space
configuration of the three-body off-shell decay. Here, the additional
suppression of the three body off-shell transition, beyond the naive
dimensional analysis factor $y_{\chi}^{2}/16\pi^{2}$, comes from the
suppression of the matrix element by a factor of $Q\sim E_{\chi_{0}}$ and the
fact that the phase space is more squeezed, resulting in large $\tilde{r}_{\rm
p.s.}$. The on-shell capture rate is controlled by $\lambda_{\text{eff}}$,
where $\lambda_{\text{eff}}\lesssim 10^{-14}$ gives capture rates below
current $\mu\rightarrow e$ limits, assuming a similar signal of ‘conversion’
electrons to that of the off-shell $\Phi$, see, e.g., Figs. 3 and 4, after
$\Phi$ decays promptly. Note that because $R_{\text{on-
shell}}\propto\lambda^{2}_{\text{eff}}\propto(m_{T_{1}}m_{T_{2}})^{-4}$ the
significant enhancement of the on-shell capture rate only translates to an
increase of $m_{T_{1,2}}$ by $\approx$ an order of magnitude.

###  IV.2 Loop-induced processes

So far, we have focused on processes directly related, at tree level, to those
responsible for the exotic signals at Mu2e. Within the triplet-triplet-singlet
UV completion, Section IV.1, we can also estimate the radiatively generated
processes. For proton decay, these loop-induced contributions are potentially
important since they can lead to proton decay with a smaller number of
particles in the final state than those that we used for our estimates in
Section III.2. The radiative corrections can also give rise to new processes,
such as bound neutron decays and $n-\chi$ mixing. Below, we address the
possible constraints due to each of these processes.

##### Proton decay.

If any of the final legs in Fig. 2 (right) can be contracted to form a
loop,444Remember that we identified $\chi_{0}=\nu_{R}$. this lowers the number
of powers of $Q$ suppressing the proton decay rate, and on dimensional grounds
also removes the appropriate powers of $G_{F}$ suppression. However,
contraction of external legs requires small mass and/or coupling insertions.
For instance, loop-level proton decay $p\to(\chi_{1}A^{\prime})(\nu_{e}e^{+})$
is obtained by contracting the $\bar{\nu}_{\mu}$ and $\nu_{R}$ external lines,
which requires a neutrino mass insertion. Contracting $e^{-}$ with the $e^{+}$
line from the weak vertex gives rise to the 5-body decay $p\to
e^{+}\chi_{1}\nu_{e}\bar{\nu}_{\mu}\nu_{R}$, but requires an off-shell
$A^{\prime}$ and is thus suppressed by small kinetic mixing, due to coupling
of $A^{\prime}$ to the $e^{+}e^{-}$ pair. Similarly, the $p\to
e^{+}\chi_{1}\nu_{e}$ decay, arising from a two-loop diagram with both
$\bar{\nu}_{\mu}$ contracted with $\nu_{R}$ and $e^{-}$ contracted with the
$e^{+}$ line from the weak vertex, is suppressed by the neutrino mass and by
kinetic mixing. Because of these suppressions, the loop-induced proton decays
always have smaller partial decay widths than the tree-level processes
discussed in Sec. III.2.

##### Neutron decay.

A one loop $W$ exchange converts the $u-\mu-T_{2}$ vertex in Eq. 36 to a
$d-\nu_{\mu}-T_{2}$ one

$\displaystyle(\overline{u^{iC}_{R}}\mu_{R})(T_{2}^{*})_{i}\rightarrow\frac{y_{\mu}y_{u}}{16\pi^{2}}\times(\overline{d_{L}^{iC}}\nu_{\mu})(T_{2}^{*})_{i}~{},$
(48)

which after the $T_{1,2},\Phi$ are integrated out, and the higher dimension
operator is run down to the weak scale, gives the effective Lagrangian

$\mathscr{L}_{\rm int}=\kappa\widetilde{G}_{\mu
p}\left(\overline{n_{L}^{C}}\nu_{\mu}\right)\left(\overline{\nu_{R}}\chi_{2R}^{C}\right),$
(49)

where

$\kappa\simeq\frac{y_{\mu}y_{u}}{16\pi^{2}}\log\left(\frac{\Lambda_{\rm
UV}^{2}}{M^{2}_{W}}\right)\sim 10^{-10}~{},$ (50)

with $\Lambda_{\rm UV}\sim m_{T_{1}},m_{T_{2}}$. The muon and up-quark Yukawa
couplings arise due to mass insertions required for converting right-handed
fields to left-handed ones, which then couple to the $W$ boson.

In the same way as the $u-\mu-T_{2}$ vertex leads to the proton decay inducing
operator in Eq. 42, after $T_{1,2}$ and $\Phi$ are integrated out, the
$d-\nu_{\mu}-T_{2}$ interaction in Eq. 48 will induce the decay of the neutron
into dark sector particles,

$n\to(\chi_{2}^{*}\to\chi_{1}A^{\prime})\nu_{R}\bar{\nu}_{\mu}~{},$ (51)

however, with a highly suppressed rate proportional to $\kappa^{2}$,

$\Gamma_{n}\simeq\left(\frac{g_{D}\kappa\,G_{\mu
p}}{m_{2}}\right)^{2}\frac{Q^{7}}{\left(16\pi^{2}\right)^{2}}.$ (52)

Here, $Q\approx(m_{n}-\epsilon_{b}-m_{1}-m_{A^{\prime}}-m_{\nu_{R}})/4$ is the
typical momentum of the final state particles. In the expression for it, we
already anticipated that the phenomenologically most relevant are the decays
of neutrons bound inside nuclei, with $\epsilon_{b}$ the average binding
energy of a neutron inside a nucleus. This binding energy is large enough that
the bound neutron decay will be kinematically forbidden for the squeezed mass
spectra we consider.

The bounds from searches for exotic decays of free neutrons of the type
$n\rightarrow e^{+}e^{-}+\mathrm{inv}$ are relatively weak UCNA:2018hup ;
Klopf:2019afh . However, if these transitions occur inside a nucleus, they
would appear as decays of stable or long-lived nuclei, which then places
stringent constraints on new physics models. In particular,
12C$\to^{11}$C$+\mathrm{inv}$ and 16O$\to^{15}$O$+\mathrm{inv}$ decays are
well constrained, since the de-excitations of the daughter nucleus would leave
visible signals in the underground large-volume detectors such as Borexino
Borexino:2003igu , KamLAND KamLAND:2005pen , and SNO/SNO+ SNO:2003lol ;
SNO:2022trz . Since the daughter nuclei are meta-stable, the de-excitation
signatures can be targeted experimentally, independently of the
electron/positron pair signature that arises from the decay of $A^{\prime}$. A
lack of any positive signal places a bound $\tau_{n}\ \raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}\ 10^{29}$ years,
which should be compared with the expectations in Eq. 52, shown in the right
panel of Fig. 5. Note that we take $\kappa\sim 10^{-10}$ and have neglected
the additional penalty from the binding energy.

In summary, the operator behind neutron decay indeed lifts part of the phase-
space suppression in free proton decay but also gets suppressed in our model
by $u$-quark and muon mass insertions. Due to the binding energy of the
nucleus, it may even be kinematically forbidden in part of the parameter
space.

Figure 9: Neutron mixing with $\chi_{2}$ generated at the two-loop level. At
an additional loop, the mixing between $\chi_{1}$ and $n$ is generated by an
$A^{\prime}$ line.

#####  $n-\chi$ mixing.

Since in our specific UV completion $\chi_{0}=\nu_{R}$, the diagrams such as
the one in Fig. 9 induce mixing between $n$ and $\chi_{2}$ ($\chi_{1}$) at
two-loops (three-loops), giving rise to $n\to\chi_{1}$ transitions. The
relevant operators are

$\mathscr{L}_{\rm eff}\supset
m_{n\chi_{1}}\bar{n}\chi_{1}+m_{n\chi_{2}}\bar{n}\chi_{2}+\text{h.c}.$ (53)

The $m_{n\chi_{1}}$ parameter has greater phenomenological significance as it
leads to kinematically-allowed $n\to\chi_{1}$ transitions in nuclei. Let us
first estimate $m_{n\chi_{2}}$ in our UV completion using naive dimensional
analysis,

$\displaystyle m_{n\chi_{2}}$
$\displaystyle\sim\frac{y_{u}y_{\mu}y_{\nu}}{(16\pi^{2})^{2}}(\lambda_{\rm
eff}y_{\chi})v_{\rm EW}\log^{2}\left(\frac{\Lambda_{\rm
UV}^{2}}{m_{W}^{2}}\right)$ (54) $\displaystyle\sim y_{\chi}\times
10^{-28}\text{ MeV}\times\left(\frac{\lambda_{\rm
eff}}{10^{-10}}\right)\left(\frac{y_{\nu}}{10^{-12}}\right).$

Note that the mixing parameter is proportional to the neutrino coupling
$\kappa$ in Eq. 50 and neutrino masses. One more loop is then required to
transform $\chi_{2}$ into $\chi_{1}$ and is generated by an $A^{\prime}$ line
that either connects to $\nu_{R}$, if charged under $U(1)_{D}$, or to an
electrically charged state through kinetic mixing. We discuss the possible
$U(1)_{D}$ charge assignments in Section IV.3. For now, we focus on the latter
possibility, which leads to an estimate of $m_{n\chi_{1}}\sim
m_{n\chi_{2}}\times\epsilon^{2}\alpha_{D}/4\pi$.

The rate for neutron disappearance inside a nucleus is given by Fermi’s golden
rule,

$\Gamma_{n}=2\pi\left|\langle\psi_{f}|m_{n\chi_{1}}|\psi_{i}\rangle\right|^{2}\delta(E_{f}-E_{i})\rho(E_{f})~{},$
(55)

with $\rho(E_{f})$ the density of states of the final state and $\psi_{i}$ and
$\psi_{f}$ are the wavefunctions of the initial neutron and final state
$\chi_{1}$ state, respectively. The momentum of the final state, a free
$\chi_{1}$ particle, is determined by energy conservation with
$p_{f}^{2}/2m_{1}=m_{n}-\epsilon_{b}-m_{1}$ and $\epsilon_{b}$ is the binding
energy of the neutron. Taking into account that there are $N=A-Z$ neutrons
inside a nucleus, the rate for nuclear transition induced by the $n-\chi_{1}$
mixing is given by

$\Gamma_{\mathrm{N}}=\frac{N}{\pi}m_{n\chi_{1}}^{2}m_{1}p_{f}\xi^{2}V_{\mathrm{nucl.}}~{}.$
(56)

The overlap of the final state wavefunction with the neutron, $\psi_{i}$,
bound inside a nucleus of volume $V_{\mathrm{nucl.}}$, is encoded in the
dimensionless quantity $\xi$, where

$\xi^{2}=\left|\frac{1}{\sqrt{V_{\mathrm{nucl.}}}}\int
d^{3}r\,\psi_{i}\exp\left(i\vec{p}\cdot\vec{r}\right)\right|^{2}~{}.$ (57)

For large energy release, $p_{f}^{3}\gg V_{\mathrm{nucl.}}$, this overlap is
small. Instead, we assume $\xi\approx 1$ and arrive at an estimate for the
decay time of a carbon nucleus, where $\epsilon_{b}\sim 10{\,\rm MeV}$, due to
neutron disappearance of

$\begin{split}\tau_{\mathrm{C}}\simeq\,&10^{33}\,\mathrm{years}\times\left(\frac{850{\,\rm
MeV}}{m_{1}}\right)\left(\frac{100{\,\rm MeV}}{p_{f}}\right)\\\
&\times\left(\frac{100\,\mathrm{fm}^{3}}{V_{\mathrm{nucl.}}}\right)\left(\frac{10^{-28}{\,\rm
MeV}}{m_{n\chi_{1}}}\right)^{2}~{}.\end{split}$ (58)

This lifetime should be compared to bounds on Pauli exclusion transitions from
Borexino Borexino:2004hfc , which searched for a spontaneous formation of
holes in the closed nucleon shells of 12C. For both the Pauli excluded
transitions or a neutron conversion to $\chi_{1}$ in a carbon nucleus, the
remaining nucleons will refill shells, releasing an $\mathcal{O}(10\,{\,\rm
MeV})$ photon. Borexino’s lack of a signal places a constraint of
$\tau_{\mathrm{C}}\ \raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}\ 10^{27}$ years, which is easily satisfied in our
model.

###  IV.3 Alternative scenarios

Above, we discussed only one possible UV completion in detail, the triplet-
triplet-singlet model. While this model is a good representative of how the
experimental constraints can be evaded, primarily though compressed spectra,
we expect that many other SM extensions exist that can lead to observable
$\mu-p$ annihilation while satisfying current constraints. Below, we discuss
some of the possible variants of the triplet-triplet-singlet model as well as
its extensions. Some of these variants have already been touched upon above.

##### Alternative scenarios for $\Phi$ produced on-shell.

In the numerical analysis, we primarily focused on the case of a heavy singlet
$\Phi$. However, the $\Phi$ can also be lighter, even so light that it is
produced on-shell in $\mu^{-}p$ annihilation. The $\Phi$ can then decay into
the dark sector states and the $e^{+}e^{-}$ pair, which is the same final
state particle content as the one we discussed in detail in the case of heavy
$\Phi$, but with modified kinematics. It is also possible that the dominant
decay of $\Phi$ is into other visible particles (and dark states), not the
$e^{+}e^{-}$ pair. For instance, $\Phi\to 4e+$dark states or
$\Phi\to\gamma+$dark states are both possible and can have drastically
different experimental efficiencies at an experiment such as Mu2e.

##### Production of kaons.

An interesting alternative for releasing more energy is $\mu^{-}p$
annihilation to kaons. This could, for instance, be mediated via the following
low-energy interactions, $\mathscr{L}\supset\lambda_{\rm
eff}\Phi^{*}\overline{p_{R}^{C}}\mu_{R}+\lambda_{KK}\Phi(K^{0})^{2}$. For
$\Phi$ that has a mass in the range $2m_{K}<m_{\Phi}<m_{p}+m_{\mu}$, the muon
induced conversion process is then $\mu^{-}p\to\Phi+{\rm nucl\,recoil}\to
K^{0}K^{0}$. Proton decay, on the other hand, can only proceed via two virtual
particles, $p\to(\mu^{+})^{*}(\Phi^{*}\to KK^{*})$. As before, the main idea
is to use the muon mass as a means to enable baryon number violation in $\mu
p$ capture while at the same time suppressing proton decay via a judicious
choice of the mass spectrum. The novelty in this scenario is that the
byproducts of the capture reaction are fixed by the branching ratios of the
kaon, leading to the production of pions, muons, and photons in addition to
electrons and positrons. Note, however, that this scenario is quite likely
challenged by the neutron star physics. In the neutron star, the final state
$K$-mesons will get re-absorbed by nuclei, resulting in a loss of two fermions
per every $\mu^{-}p$ annihilation, with strong implications for the properties
of the neutron stars.

##### Alternative dark photon couplings.

In the main part of the text, we assumed that the
$\chi_{2}\to\chi_{1}A^{\prime}$ transition proceeds through dimension 5 dipole
operators. However, this transition can also be mediated by renormalizable
interactions. To show this, we first assume that the dark photon mass arises
from its coupling to dark higgs, $h_{d}$, after it acquires a vev. The
interactions of the dark $U(1)_{D}$ gauge boson, $A_{\mu}^{\prime}$, depend on
the assigned $U(1)_{D}$ charges of the dark states $\chi_{1,2},\nu_{R}$. To
allow for the transition $\chi_{2}\rightarrow\chi_{1}A^{\prime}$, $\chi_{1}$
and $\chi_{2}$ must have the same charge. To avoid charging SM particles under
the $U(1)_{D}$, $\nu_{R}$ must be oppositely charged to $\chi_{1,2}$, or the
second term in Eq. 37 must contain an insertion of $h_{d}$. One possibility is
that none of these carry a $U(1)_{D}$ charge, and thus the
$\chi_{2}\to\chi_{1}A^{\prime}$ decay is due to the dipole interaction in Eq.
15. If the $\chi_{i}$’s carry a $U(1)_{D}$ charge, the decay takes place
through the contact operator in Eq. 15. In either case, the coupling to the
dark photon is off-diagonal in $\chi$ flavor space.

This off-diagonal form of the coupling can be achieved if there are 3 Weyl
fermions in the dark sector, two of which make up a pseudo-Dirac pair.
Consider 3 states $\psi_{0},\psi_{1},\psi_{1}^{c}$, with a vector-like mass
$m\psi_{1}\psi_{1}^{c}$ and a small Majorana mass $\delta\psi_{1}\psi_{1}$.
This can be generated, for instance, by integrating out a massive fermion with
a small Yukawa coupling with $\psi_{1}$ and $h_{d}$. The mass eigenstates
$\chi_{1,2}$ are approximately equal admixtures of $\psi_{1}$, $\psi_{1}^{c}$.
Since Majorana fermions can neither carry a charge nor have a magnetic dipole
moment, the mass mixing implies that the couplings will be of the transition
type. If $\psi_{i}$ carry a magnetic dipole moment555These dipole moments may
be generated perturbatively through loops of $U(1)_{D}$ charged states, in
which case the typical size is $\sim g_{D}^{3}/(16\pi^{2}m_{\pm})$. Or they
may be generated non-perturbatively, as for the SM baryons, in which case we
expect the size to be $\sim g_{D}/m_{\chi}$. the mass eigenstates $\chi_{1,2}$
will acquire a magnetic dipole transition operator. If instead
$(\psi_{0},\psi_{1},\psi_{1}^{c})$ carry charge $(1,1,-1)$ respectively, with
the massive state which generates the Majorana mass having charge 2, then
$\chi_{1,2}$ will couple in an off-diagonal fashion to the dark photon.

Even if the SM fields are not charged under the dark gauge group, the coupling
of the dark photon to the SM fields can occur through kinetic mixing of
$U(1)_{D}$ with $U(1)_{Y}$. This would be generated at one loop by having
fields charged under both groups running in the loop. At low energies, this
leads to a kinetic mixing between the dark photon and the SM photon of size
$\epsilon\sim eg_{D}/(16\pi^{2})$. The dark photon couplings to charged
leptons are then of size $\epsilon e$.

##### Models that lead to $\mu^{-}n$ annihilation.

A completely separate set of models that can lead to energetic electrons at
Mu2e and COMET are models that have the $\mu^{-}n$ annihilation as the leading
effect, while $\mu^{-}p$ is induced only at the subleading loop level. An
example are models that result in neutron-to-dark-state $\chi_{n}$ transition,
induced by the mass mixing
$\mathscr{L}\supset\epsilon(\bar{n}\chi_{n}+{\bar{\chi}_{n}}n)$, as in Ref.
Fornal:2018eol . If the UV model contains $\mu\to e$ CLFV interactions, then
the $\mu^{-}n\to e^{-}\chi_{2}$ transition becomes possible.

##  V Conclusions

In this manuscript, we have constructed new physics scenarios with muon-
induced (apparent) baryon number violation. Using a specific example, we
demonstrated that these models lead to a spectrum of higher energy electrons
and positrons in the $\mu^{-}A\to e^{-}A$ conversion experiments, such as Mu2e
and COMET, while satisfying experimental bounds on proton stability. The main
idea consists of tapping into the baryonic energy reservoir of the nucleus $A$
by destroying protons or neutrons in the muon capture process. This releases
energies greater than $m_{\mu}$ into dark particles that eventually decay to
$e^{+}e^{-}$ pairs. The $e^{+}$ or $e^{-}$ can reach energies as high as
$\mathcal{O}(130)$ MeV, depending on the choice of parameters, significantly
overshooting the energy of electrons produced in decay-in-orbit as well as of
eventual monochromatic electrons from $\mu^{-}\isotope[27]{Al}\to
e^{-}\isotope[27]{Al}$ ($E_{e^{-}}^{\rm Al-conv}=104.98$ MeV).

These signatures are not excluded by limits on the proton lifetime thanks to a
coincidence between the mass of the proton and the dark particles, which
forces the protons to decay via two or more off-shell states. The stability of
nuclei (more specifically, that of bound neutrons) also imposes meaningful
constraints due to radiative corrections. These are again bypassed due to
strong phase-space suppression as well as by chiral suppression in the UV
completions we considered. The crucial component for constructing models that
enhance the $\mu^{-}-$nucleon annihilation while suppressing nucleon decays is
that the ratio of muon to proton mass is $m_{\mu}/m_{p}\sim 1/9$, which is not
a very small ratio. Consequently, changes in mass splittings in the dark
sector at the level of $\sim 20$ MeV (i.e., at a few percent level) are very
consequential for the signals discussed here and can change answers by many
orders of magnitude.

While our scenarios are specifically designed to hide nucleon decay and
highlight muon capture, they demonstrate that Mu2e and COMET can be sensitive
to electrons that come from new physics sources other than the $\mu\to e$
lepton flavor violating neutral currents. Such scenarios are particularly
attractive in the event that high-energy electrons or positrons are detected
at Mu2e. Our proposal also shows that such high-energy events may not
necessarily come only from backgrounds such as the decay of late-time pions or
cosmics.

Finally, we also emphasize that positrons may present a particularly
interesting final state. Light dark sectors can lead to positron energies
beyond those induced by radiative muon capture and by the lepton-number-
violating channel $\mu^{-}A\to e^{+}A^{\prime}{}^{*}$ Geib:2016atx ;
Berryman:2016slh . The latter includes a monochromatic positron with energy of
about $E_{e^{+}}^{\rm conv}\simeq 92.3$ MeV for $\mu^{-}\,\isotope[27]{Al}\to
e^{+}\,\isotope[27]{Na}({\rm g.s.})$ transitions. It should be noted that past
searches for $\mu^{-}\to e^{+}$ at TRIUMF Ahmad:1988ur  and PSI
SINDRUMII:1993gxf ; SINDRUMII:1998mwd ; SINDRUMII:2006dvw  on Ti and Au
targets have all reported excesses of positrons above $E_{e^{+}}\gtrsim 90$
MeV but below $E_{e^{+}}\lesssim m_{\mu}$ (see also earlier searches in
Conforto:1962zz ; Bryman:1972rf ; Abela:1980rs ). While the modeling of
radiative muon capture may be insufficient in this energy region Lee:2021hnx ,
these events represent an immediate application of our scenarios. In fact,
since the positron energies in these excess events are still less energetic
than $m_{\mu}$, an explanation through muon-induced BNV would be feasible even
for parameters that render protons and bound neutrons stable.

## Acknowledgements

We want to thank David McKeen for valuable discussions at the Muons in
Minneapolis workshop 2023, and Bertrand Echenard during the 2024 summer
program at Aspen Center for Physics. We acknowledge support from the Simons
Foundation Targeted Grant 920184 to the Fine Theoretical Physics Institute. JZ
and TM acknowledge support in part by the DOE grant de-sc0011784 and NSF
OAC-2103889. TM acknowledges support in part from the Visiting Scholars Award
Program of the Universities Research Association. MP is supported in part by
the DOE grant DE-SC0011842. The work of MH is supported by the Neutrino Theory
Network Program Grant #DE-AC02-07CHI11359 and the US DOE Award #DE-SC0020250.
PJF is supported by the DOE under contract #DE-SC0007859 and Fermilab,
operated by Fermi Research Alliance, LLC under contract #DE-AC02-07CH11359
with the United States Department of Energy. This work was performed in part
at the Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-2210452.

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