Probing ALP Lepton Flavour Violation at $\mu$TRISTAN

Lorenzo Calibbi<EMAIL_ADDRESS>Tong Li<EMAIL_ADDRESS>Lopamudra Mukherjee<EMAIL_ADDRESS>Yiming Yang
<EMAIL_ADDRESS>
School of Physics, Nankai University, Tianjin 300071, China

Axion-like particles (ALPs) with lepton flavour violating (LFV) interactions
are predicted within a wide range of flavoured ALP models. The proposed
$\mu$TRISTAN high-energy $e^{-}\mu^{+}$ and $\mu^{+}\mu^{+}$ collider will
provide a good opportunity to explore flavour physics in the charged lepton
sector. In this work, based on a model-independent effective Lagrangian
describing the ALP leptonic interactions, we investigate the potential of
$\mu$TRISTAN to probe ALP LFV couplings. We analyse the testability of
selected ALP production channels with potential sensitivity at $\mu$TRISTAN,
considering different beams and collision energies, including $e^{-}\mu^{+}\to
a\gamma$, $e^{-}\mu^{+}\to e^{-}\tau^{+}a$,
$\mu^{+}\mu^{+}\to\mu^{+}\tau^{+}a$, and $e^{-}\mu^{+}\to\tau^{-}\mu^{+}a$.
The produced ALP $a$ is either long-lived or can promptly decay to flavour
violating or conserving charged lepton final states. In particular, combining
the above LFV ALP production modes with a suitable LFV decay mode, one can
identify signatures that are virtually free of Standard Model background. We
show the resulting sensitivity of $\mu$TRISTAN to LFV ALP couplings and
compare it with multiple low-energy leptonic constraints and the future
improvements thereof. We find that $\mu$TRISTAN can be generally complementary
to searches for low-energy LFV processes and measurements of the leptonic
magnetic dipole moments and has the capability to explore unconstrained
parameter space for ALP masses in the $\mathcal{O}(1)$ to $\mathcal{O}(100)$
GeV range. In the light ALP regime, however, the parameter space that
$\mu$TRISTAN is sensitive to, has been already excluded by low-energy searches
for LFV decays.

###### Contents

  1. 1 Introduction
  2. 2 ALP-lepton interactions
  3. 3 LFV processes at $\mu$TRISTAN
    1. 3.1 ALP-mediated Lepton Flavour Violation
    2. 3.2 ALP production through LFV interactions
  4. 4 $\mu$TRISTAN sensitivity to LFV ALPs
    1. 4.1 ALP production through $e^{-}\mu^{+}\to a\gamma$
    2. 4.2 ALP production through LFV $\tau$ interactions
  5. 5 Summary and conclusions
  6. A ALP decay into photons
  7. B Low-energy leptonic processes
    1. B.1 Muonium antimuonium oscillations
    2. B.2 LFV muon and tau decays
    3. B.3 Anomalous magnetic moments
    4. B.4 Loop functions

##  1 Introduction

Axions and axion-like particles (ALPs) are predicted within a wide class of
new physics (NP) models. Specifically, these fields are pseudo Nambu-Goldstone
bosons (PNGB), that is, light pseudoscalars arising from the spontaneous
breaking of a global $\text{U}(1)$ symmetry. Examples are provided by the
Peccei-Quinn symmetry [1, 2, 3], the lepton number [4, 5, 6], or a flavour
symmetry [7, 8, 9, 10, 11, 12, 13, 14, 15], in the context of well-motivated
NP frameworks. These models can address some open questions of the Standard
Model (SM) of particle physics, such as the strong CP problem, the origin of
neutrino masses, the hierarchical structure of fermion masses and mixing.

Searching for light ALPs is therefore an effective way to test new fundamental
dynamics at energy scales possibly beyond the direct reach of high-energy
colliders. This appealing connection — besides a robust dedicated axion
experimental programme and a large number of studies on possible cosmological
and astrophysical probes of these light fields, for reviews cf. [16, 17, 18] —
recently motivated a renewed interest also in the potential of axion/ALP
searches at flavour [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44] and collider experiments [45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63].

In this article, we study ALPs in the context of the recently proposed
$\mu$TRISTAN project [64]. The idea behind $\mu$TRISTAN is to employ in a
high-energy collider the mature methods for cooling and focusing positive muon
beams that have been developed by the J-PARC muon $g-2$ experiment [65]. The
$\mu^{+}$ beams could be accelerated up to 1 TeV and made to collide with a 30
GeV $e^{-}$ beam provided by the TRISTAN ring at KEK, resulting in a
$\mu^{+}e^{-}$ “Higgs factory” with centre of mass (c.m.) energy
$\sqrt{s}\simeq 346$ GeV and an expected integrated luminosity
$\mathcal{L}=1~{}\text{ab}^{-1}$. In addition, the machine could be also
operated as a $\mu^{+}\mu^{+}$ collider with $\sqrt{s}=2$ TeV, thus
constituting a high-energy leptonic “discovery machine” already feasible with
present-day technologies. In the original proposal [64], it was estimated that
the $\mu^{+}\mu^{+}$ mode could attain a luminosity
$\mathcal{L}=100~{}\rm{fb}^{-1}$. Both setups can be used to study Higgs
phyisics [64], to perform precision measurements [66, 67], as well as to probe
new physics up to the TeV scale [68, 69, 70, 71, 72, 73, 74, 75, 76, 77].

In the following pages, we explore the capability of the $\mu$TRISTAN proposal
to search for ALPs with lepton flavour violating (LFV) interactions. Lepton
flavour violation is a generic prediction of a wide range of ALP models [30]
and, as we are going to show, it would give rise to distinctive signatures
virtually affected by no (irreducible) SM background (BG). Furthermore, while
LFV ALP couplings are strongly constrained by low-energy LFV experiments if
muons and taus can decay _into_ an on-shell ALP $a$ — both for ALPs decaying
inside the detector [27, 36] and in the case of invisible ALPs [30] — the LFV
interactions of heavier ALPs, for which the decays $\mu\to e\,a$ and
$\tau\to\ell\,a$ are kinematically forbidden, are comparatively poorly
constrained. The aim of this work is to systematically study the role that
$\mu$TRISTAN could play in this context by assessing its sensitivity to LFV
ALP interactions and comparing it with present and future limits set by low-
energy LFV and $g-2$ experiments.

The rest of the paper is organised as follows. In Section 2, we review the
general ALP-lepton interaction Lagrangian and discuss the ALP decay length. In
Section 3, we list and analyse the possible processes sensitive to LFV ALP
interactions in $e^{-}\mu^{+}$ and $\mu^{+}\mu^{+}$ collisions. By computing
the cross sections of these processes, we single out a number of ALP
production modes that have the potential to be tested at $\mu$TRISTAN:
$e^{-}\mu^{+}\to a\gamma$, $e^{-}\mu^{+}\to e^{-}\tau^{+}a$,
$\mu^{+}\mu^{+}\to\mu^{+}\tau^{+}a$, and $e^{-}\mu^{+}\to\tau^{-}\mu^{+}a$. In
Section 4, we present the results of our simulations of these processes and
compute the resulting sensitivity to the LFV ALP couplings that could be
achieved at $\mu$TRISTAN. Furthermore, we compare these results with the
present and future constraints from the relevant low-energy leptonic
processes, in particular LFV $\mu$ and $\tau$ decays. We summarise our
findings and draw our conclusions in Section 5. Useful formulae and
experimental information are collected in the Appendices.

##  2 ALP-lepton interactions

Instead of adopting a specific ALP model, we will work within the model-
independent framework specified by the following effective (dimension-5)
Lagrangian — see e.g. [78, 30] — that describes the interactions of an ALP $a$
with vector and axial-vector leptonic currents in the basis where the leptons
are mass eigenstates:

$\displaystyle\mathcal{L}_{a\ell\ell}\supset\sum_{\ell_{i}}\frac{\partial_{\mu}a}{2f_{a}}\,\bar{\ell_{i}}C_{\ell_{i}\ell_{i}}^{A}\gamma^{\mu}\gamma_{5}\ell_{i}+\sum_{i\neq
j}\frac{\partial_{\mu}a}{2f_{a}}\,\bar{\ell_{i}}\gamma^{\mu}\left(C_{\ell_{i}\ell_{j}}^{V}+C_{\ell_{i}\ell_{j}}^{A}\gamma_{5}\right)\ell_{j}\,,$
(1)

where $i,j=1,2,3$ are flavour indices with $\ell_{i}=(e,\mu,\tau)$, $f_{a}$ is
the ALP decay constant, a cut-off scale related to the breaking scale of the
$\text{U}(1)$ symmetry, and the dimensionless coefficients
$C_{\ell_{i}\ell_{j}}^{A,V}$ form hermitian matrices in flavour space,
$C_{\ell_{i}\ell_{j}}^{A,V\,\ast}=C_{\ell_{j}\ell_{i}}^{A,V}$. Within a
specific model, the coefficients $C_{\ell_{i}\ell_{j}}^{A,V}$ are calculable
in terms of the lepton $\text{U}(1)$ charges and the lepton fields’ rotations
to the mass eigenbasis. In the following, we do not commit to a specific
scenario and just take $C_{\ell_{i}\ell_{j}}^{A,V}$ as free parameters.

Upon integrating by parts and employing the equations of motion of the lepton
fields, one can show that the above Lagrangian is equivalent, up to a shift of
the model-dependent anomalous ALP coupling to photons, to the following
dimension-4 scalar/pseudoscalar interactions:

$\displaystyle\mathcal{L}_{a\ell\ell}\supset-\mathrm{i}\sum_{i,j}\left[\frac{C_{\ell_{i}\ell_{j}}^{V}(m_{\ell_{j}}-m_{\ell_{i}})}{{2}f_{a}}\,a\bar{\ell_{i}}\ell_{j}+\frac{C_{\ell_{i}\ell_{j}}^{A}(m_{\ell_{i}}+m_{\ell_{j}})}{{2}f_{a}}\,a\bar{\ell_{i}}\gamma_{5}\ell_{j}\right]\,.$
(2)

From this expression, we can define dimensionless scalar/pseudoscalar
couplings:

$\displaystyle
g_{ij}^{V}\equiv\frac{C_{\ell_{i}\ell_{j}}^{V}(m_{\ell_{j}}-m_{\ell_{i}})}{2f_{a}},\quad
g_{ij}^{A}\equiv\frac{C_{\ell_{i}\ell_{j}}^{A}(m_{\ell_{j}}+m_{\ell_{i}})}{2f_{a}},$
(3)

which, due to the mass hierarchy $m_{e}\ll m_{\mu}\ll m_{\tau}$, read to a
good approximation:

$\displaystyle
g_{ii}^{A}=\frac{C_{\ell_{i}\ell_{i}}^{A}m_{\ell_{i}}}{f_{a}},\quad
g_{ij}^{A,V}\simeq\frac{C_{\ell_{i}\ell_{j}}^{A,V}m_{\ell_{j}}}{2f_{a}}~{}~{}~{}(i<j)\,.$
(4)

From Eq. (2), one can see that the strength of the interactions is
approximately proportional to the mass of the heaviest lepton involved. This
expression also shows why flavour-conserving vector currents are unphysical
and were consequently not introduced in Eq. (1).

Throughout the paper, we do not consider ALP fields that are genuine massless
Nambu-Goldstone boson. Instead, we introduce a mass term
$\frac{1}{2}m_{a}a^{2}$, assuming that it stems from explicit $\text{U}(1)$
breaking, and we treat the ALP mass $m_{a}$ as a free parameter. Following
from the above interactions, it is then clear that, if kinematics allows, the
ALP will decay into lepton pairs. The resulting width reads [30]:

$\displaystyle\Gamma(a\to\bar{\ell_{i}}\ell_{j})=~{}$
$\displaystyle\frac{m_{a}}{8\pi}\sqrt{\left(1-\frac{(m_{\ell_{i}}+m_{\ell_{j}})^{2}}{m_{a}^{2}}\right)\left(1-\frac{(m_{\ell_{i}}-m_{\ell_{j}})^{2}}{m_{a}^{2}}\right)}\,\times$
$\displaystyle\left[|g^{V}_{ij}|^{2}\left(1-\frac{(m_{\ell_{i}}+m_{\ell_{j}})^{2}}{m_{a}^{2}}\right)+|g^{A}_{ij}|^{2}\left(1-\frac{(m_{\ell_{i}}-m_{\ell_{j}})^{2}}{m_{a}^{2}}\right)\right]\,.$
(5)

For $m_{a}\gg m_{\ell_{i}}+m_{\ell_{j}}$, this decay width corresponds to the
following ALP proper decay length:

$\displaystyle c\tau(a\to\ell_{i}\ell_{j})\approx$ $\displaystyle{16\pi\over
m_{a}}{1\over|g^{V}_{ij}|^{2}+|g^{A}_{ij}|^{2}}\approx 10^{-6}~{}{\rm
m}~{}\left({1~{}{\rm GeV}\over
m_{a}}\right)\left({10^{-8}\over|g^{V}_{ij}|^{2}+|g^{A}_{ij}|^{2}}\right)\;,$
(6)

where we have summed over the final state charges. As we will see in the
following sections, we expect $\mu$TRISTAN to be able to probe LFV ALP
couplings as small as $g^{A,V}_{ij}\sim 10^{-3}-10^{-4}$. Hence, Eq. (6)
implies that ALPs whose decays into charged leptons are allowed will always
decay promptly inside the detector for the range of couplings we are
interested in. Therefore, in this case, ALPs can be searched through
distinctive nearly-background-free LFV signals.

(a) LFV ($e$-$\mu$) couplings only (b) LFV and LFC couplings

Figure 1:  Contours in the plane
$m_{a}-g_{ij}^{A,V}~{}(=m_{\ell_{j}}\,C_{\ell_{i}\ell_{j}}^{A,V}/f_{a})$ of
the proper decay length $c\tau_{a}$ of an ALP whose only non-vanishing
couplings are $C_{e\mu}^{V}=C_{e\mu}^{A}$ (left panel) and an ALP with all
couplings $C_{\ell_{i}\ell_{j}}^{A,V}$ ($i\neq j$) and
$C_{\ell_{i}\ell_{i}}^{A}$ equal (right panel).

For $m_{a}<2\,m_{e}$ or, if the model does not admit flavour-conserving
interactions, $m_{a}<m_{e}+m_{\mu}$, the ALP can still decay into neutrinos —
either because of a direct coupling or through $W$ exchange at one loop — but
such process is extremely suppressed by small neutrino masses and irrelevant
in our context. In addition, even if an UV interaction with photons is not
present, lepton flavour-conserving (LFC) couplings necessarily induce
$a\to\gamma\gamma$. The decay width for this mode is reported in Appendix A.

We now illustrate the above discussion showing the ALP proper decay length for
different scenarios. In the left panel of Figure 1, the ALP has only LFV
interactions to electrons and muons with $C_{e\mu}^{V}=C_{e\mu}^{A}$. In this
case, the ALP is always long-lived below the $a\to e\mu$ kinematic threshold,
always short-lived above it unless the coupling $g^{A,V}_{e\mu}$ is smaller
than $10^{-5}-10^{-6}$, values beyond the $\mu$TRISTAN reach, as we will see.

In the right plot of Figure 1, we consider an ALP with all the (LFC and LFV)
coefficients $C_{\ell_{i}\ell_{j}}^{A,V}$ set to be equal. As we can see,
below the $m_{\mu}+m_{e}$ threshold, the ALP tends to decay into $e^{+}e^{-}$
inside the detector in the large coupling regime, while an ALP lighter than
$2\,m_{e}$ always escapes from the detector appearing as missing energy even
if the flavour-conserving couplings inducing $a\to\gamma\gamma$ are
present.111Here we set the model-dependent UV ALP-photon coupling
$E_{\text{UV}}$ to zero, see Appendix A. Although this would give an
interesting signature at colliders, the $\mu$TRISTAN sensitivity to such light
ALPs compares unfavourably with the stringent constraints from searches for
LFV $\mu$ and $\tau$ decays, as we will show in the following sections.

##  3 LFV processes at $\mu$TRISTAN

In this section, we examine several processes involving LFV ALPs that could be
probed at $\mu$TRISTAN either in the $e^{-}\mu^{+}$ or the $\mu^{+}\mu^{+}$
mode. We first present ALP-mediated LFV processes and then processes with a
final-state ALP produced through its LFV interactions. After implementing the
LFV ALP Lagrangian introduced in the previous section in FeynRules [79, 80],
we compute the cross sections of the processes we are interested in by means
of Madgraph [81, 82]. All numbers reported in this section refer to the
leading-order cross sections after the standard generator-level cuts on the
transverse momentum ($|p_{T}|>10~{}\text{GeV}$) and pseudo-rapidity
($|\eta|<2.5$) of all visible final state particles are applied.

###  3.1 ALP-mediated Lepton Flavour Violation

Process | ALP couplings | Cross section (fb)
---|---|---
$e^{-}\mu^{+}\to e^{+}\mu^{-}$ |  $C^{A}_{e\mu}$ or $C^{V}_{e\mu}$ | $9.7\times 10^{-9}$
$C^{A}_{e\mu}=C^{V}_{e\mu}$ | $5.1\times 10^{-8}$
$e^{-}\mu^{+}\to\gamma\gamma$ |  $C^{A}_{e\mu}$ or $C^{V}_{e\mu}$, $C_{\gamma}$ | $4.3\times 10^{-7}$
$C^{A}_{e\mu}=C^{V}_{e\mu}$, $C_{\gamma}$ | $8.5\times 10^{-7}$
$e^{-}\mu^{+}\to\mu^{-}\tau^{+}$ |  $C^{A}_{\mu\tau}=C^{A}_{e\mu}$ or $C^{V}_{\mu\tau}=C^{V}_{e\mu}$ | $3.1\times 10^{-6}$
$C^{A}_{\mu\tau}=C^{V}_{\mu\tau}=C^{A}_{e\mu}=C^{V}_{e\mu}$ | $7.3\times 10^{-6}$
$e^{-}\mu^{+}\to\tau^{-}\tau^{+}$ |  $C^{A}_{\mu\tau}=C^{A}_{e\tau}$ or $C^{V}_{\mu\tau}=C^{V}_{e\tau}$ | $2.9\times 10^{-4}$
$C^{A}_{\mu\tau}=C^{V}_{\mu\tau}=C^{A}_{e\tau}=C^{V}_{e\tau}$ | $1.0\times 10^{-3}$
$\mu^{+}\mu^{+}\to e^{+}e^{+}$ |  $C^{A}_{e\mu}$ or $C^{V}_{e\mu}$ | $2.3\times 10^{-10}$
$C^{A}_{e\mu}=C^{V}_{e\mu}$ | $1.2\times 10^{-9}$
$\mu^{+}\mu^{+}\to\tau^{+}\tau^{+}$ |  $C^{A}_{\tau\mu}$ or $C^{V}_{\tau\mu}$ | $2.3\times 10^{-5}$
$C^{A}_{\tau\mu}=C^{V}_{\tau\mu}$ | $9.4\times 10^{-5}$
Table 1:  Cross sections of some ALP-mediated LFV processes in $e^{-}\mu^{+}$
collisions (with beam energies $E_{e}=30$ GeV, $E_{\mu}=1$ TeV) and
$\mu^{+}\mu^{+}$ collisions (with $E_{\mu}=1$ TeV for both beams). For the ALP
mass and decay constant we take $m_{a}=1$ GeV, $f_{a}=100$ GeV. The only non-
vanishing couplings are indicated in the second column and are set equal to
one, $C^{A,V}_{\ell_{i}\ell_{j}}=1$, with the exception of the coupling to
photons, $C_{\gamma}\frac{a}{f_{a}}F\widetilde{F}$, that is set to
$C_{\gamma}=\alpha/4\pi$ — that corresponds to $E_{\text{UV}}=1$ according to
the notation in Appendix A. All cross sections scale as
$(100~{}\text{GeV}/f_{a})^{4}$.

In Table 1, we display a number of LFV processes that can be induced in
$e^{-}\mu^{+}$ and $\mu^{+}\mu^{+}$ collisions by an ALP exchange in the $t$
channel or — only in the case of $e^{-}\mu^{+}\to\gamma\gamma$ — in the $s$
channel. The couplings are set as indicated in the table and, for the sake of
illustration, we take $m_{a}=1$ GeV. We checked that the $t$-channel processes
have a weak dependence on the ALP mass. These LFV processes could be in
principle sensitive to a wide range of ALP masses, even with $m_{a}$ larger
than the c.m. energy $\sqrt{s}$ at which $\mu$TRISTAN is operated, and are
virtually free of any irreducible SM background. However, all processes
require two ALP vertices and thus suffer a $\sim 1/f_{a}^{4}$ suppression.
Hence, as we can see, the resulting cross sections are in general too small to
be realistically probed at $\mu$TRISTAN, even assuming that an integrated
luminosity as large as $\mathcal{L}=10~{}\text{ab}^{-1}$ can be achieved.

Notice that we set $f_{a}=100$ GeV and the relevant couplings
$C^{A,V}_{\ell_{i}\ell_{j}}=1$, which corresponds to $g^{A,V}_{e\mu}=5.3\times
10^{-4}$ and $g^{A,V}_{\ell\tau}=8.9\times 10^{-3}$, as one can see from Eq.
(4). As we mentioned, the scale $f_{a}$ is related to the vacuum expectation
value (vev) of the scalar field that breaks the underlying $\text{U}(1)$
symmetry. As a consequence, we expect that viable UV completions of the
effective Lagrangian in Eq. (1) will require $f_{a}\gtrsim 100$ GeV, hence
$g^{A,V}_{e\mu}\lesssim 10^{-3}$ and $g^{A,V}_{\ell\tau}\lesssim 10^{-2}$.
Hence, the cross sections in Table 1 are arguably the largest possible within
realistic ALP models and we can conclude that $\mu$TRISTAN would not be
capable to test LFV ALPs through these processes.

###  3.2 ALP production through LFV interactions

Process |  ALP couplings |  $m_{a}$ (GeV) |  Cross section (fb)
---|---|---|---
$e^{-}\mu^{+}\to a\gamma$ | $C^{A}_{e\mu}=C^{V}_{e\mu}$ | 1 | $3.0\times 10^{-2}$
10 | $3.0\times 10^{-2}$
100 | $3.3\times 10^{-2}$
$e^{-}\mu^{+}\to aZ$ | $C^{A}_{e\mu}=C^{V}_{e\mu}$ | 1 | $5.9\times 10^{-2}$
10 | $4.8\times 10^{-2}$
100 | $3.1\times 10^{-2}$
$e^{-}\mu^{+}\to\mu^{-}\mu^{+}a$ | $C^{A}_{e\mu}=C^{V}_{e\mu}$ | 1 | $1.4\times 10^{-3}$
10 | $1.1\times 10^{-3}$
100 | $7.2\times 10^{-4}$
$e^{-}\mu^{+}\to e^{-}e^{+}a$ | $C^{A}_{e\mu}=C^{V}_{e\mu}$ | 1 | $8.0\times 10^{-3}$
10 | $4.2\times 10^{-3}$
100 | $9.8\times 10^{-4}$
$e^{-}\mu^{+}\to e^{-}\tau^{+}a$ | $C^{A}_{\mu\tau}=C^{V}_{\mu\tau}$ | 1 | 1.4
10 | 0.93
100 | $8.6\times 10^{-2}$
$e^{-}\mu^{+}\to\tau^{-}\mu^{+}a$ | $C^{A}_{e\tau}=C^{V}_{e\tau}$ | 1 | $3.3\times 10^{-2}$
10 | $1.7\times 10^{-2}$
100 | $2.3\times 10^{-3}$
$\mu^{+}\mu^{+}\to\mu^{+}e^{+}a$ | $C^{A}_{e\mu}=C^{V}_{e\mu}$ | 1 | $8.7\times 10^{-4}$
10 | $5.1\times 10^{-4}$
100 | $1.8\times 10^{-4}$
$\mu^{+}\mu^{+}\to\mu^{+}\tau^{+}a$ | $C^{A}_{\mu\tau}=C^{V}_{\mu\tau}$ | 1 | $0.23$
10 | $0.14$
100 | $0.05$
Table 2:  Same as Table 1 for ALP production processes that only involve one
LFV ALP vertex, hence the cross sections scale as
$(100~{}\text{GeV}/f_{a})^{2}$. The resulting cross sections are the same if
the ALP couples to leptons with
$C^{A}_{\ell_{i}\ell_{j}}=-C^{V}_{\ell_{i}\ell_{j}}$, while they are a factor
of two smaller if either $C^{A}_{\ell_{i}\ell_{j}}=0$ or
$C^{V}_{\ell_{i}\ell_{j}}=0$. The most promising processes (highlighted in
blue) are discussed in detail in the next sections.

We now turn to the more promising class of processes featuring an on-shell ALP
in the final state. Since the dominant diagrams require a single ALP
interaction, such processes are affected by a milder suppression, $\sim
1/f_{a}^{2}$, than those considered in the previous subsection. Table 2 lists
on-shell ALP production processes involving electroweak $Z/\gamma$ boson
interactions and the LFV ALP couplings. The collision of $e^{-}$ and $\mu^{+}$
beams can yield the $2\to 2$ processes $e^{-}\mu^{+}\to a\gamma$ and
$e^{-}\mu^{+}\to aZ$ through the $C^{A,V}_{e\mu}$ interaction and a photon/$Z$
boson radiation. The others are all $2\to 3$ processes from $t$-channel
$Z/\gamma$ diagrams with an ALP emitted from an initial- or final-state
lepton. Notice that those with a $\tau$ lepton in the final state are induced
by the $C^{A,V}_{\ell\tau}$ couplings and their cross sections are enhanced by
the $\tau$ mass, that is, by a factor $\sim(m_{\tau}/m_{\mu})^{2}$ — see Eq.
(4) — compared to the processes that only involve electrons and muons.
Furthermore, processes with the ALP emitted from the muon have a substantially
larger cross section than those with the ALP emitted from the electron, due to
the large asymmetry of the beam energies in $e^{-}\mu^{+}$ collisions. The
highlighted processes have large enough cross sections to be a realistic
target at $\mu$TRISTAN.222We do not study $e^{-}\mu^{+}\to aZ$, although its
cross section can be even larger than $e^{-}\mu^{+}\to a\gamma$. In fact, the
number of signal events would decrease significantly in exclusive searches
targeting specific $Z$ decay modes. Furthermore, as we will see, the final-
state particles produced in asymmetric $e^{-}\mu^{+}$ collisions tend to have
a large pseudorapidity. This substantially reduces the reconstruction
efficiency of $Z$ bosons, as a large fraction of the decay products is outside
the geometric acceptance of the detector. Hence, in the following analysis, we
will focus on the following ALP production modes as a way to probe the three
LFV ALP couplings:

  * •

$C^{A,V}_{e\mu}$: $e^{-}\mu^{+}\to a\gamma$ ;

  * •

$C^{A,V}_{\mu\tau}$: $e^{-}\mu^{+}\to e^{-}\tau^{+}a$ and
$\mu^{+}\mu^{+}\to\mu^{+}\tau^{+}a$ ;

  * •

$C^{A,V}_{e\tau}$: $e^{-}\mu^{+}\to\tau^{-}\mu^{+}a$ .

##  4 $\mu$TRISTAN sensitivity to LFV ALPs

In the following, we assess the $\mu$TRISTAN potential to test LFV ALP
production through the processes outlined at the end of the previous section.
Signal and background events are generated by means of Madgraph [81, 82],
Pythia [83], and Madspin [84], while the detector response is simulated by
Delphes [85]. The resulting events are analysed using Root [86]. For both
signal and background events, we apply the default basic generator-level cuts
on the final-state photons or charged leptons in the Madgraph package, with
the exception of their maximum pseudorapidity, for which we consider different
options, as discussed in this section.

We employ the following definition of statistical significance

$\mathcal{S}=\frac{N_{\rm S}}{\sqrt{N_{\rm S}+N_{\rm B}}}\;,~{}~{}~{}N_{\rm
S(B)}=\sigma_{\rm S(B)}\times\varepsilon_{\rm S(B)}\times\mathcal{L}\;,$ (7)

where $N_{\rm S}$ ($\varepsilon_{\rm S}$) and $N_{\rm B}$ ($\varepsilon_{\rm
B}$) are the event numbers (efficiencies) of signal and backgrounds,
respectively. The production cross section $\sigma_{\rm S}$ ($\sigma_{\rm B}$)
is multiplied by the branching fraction of the ALP (SM gauge bosons) decay
processes relevant to the signal selection. For later convenience, we define
the signal efficiency as $\varepsilon_{\text{S}}\equiv\varepsilon_{\rm
sel}\times\varepsilon_{\rm cut}$, that is, the product of the selection
efficiency and the signal acceptance after cuts. The selection efficiency,
accounting for the geometric acceptance of the detector and the particle
identification probabilities, can be estimated counting the number of signal
events remaining after the fast detector simulation performed by Delphes. As
we will discuss, several searches we consider are affected by a reduced
$\varepsilon_{\rm sel}$, because the decay products of the ALP have a sizeable
probability to be outside the geometric acceptance of the detector due to the
large asymmetry of the electron and muon beams.

In the following, we are showing the expected 95% confidence level (CL)
exclusion potential of $\mu$TRISTAN, corresponding to $\mathcal{S}=2$ in Eq.
(7), assuming that an integrated luminosity of
$\mathcal{L}=10~{}\text{ab}^{-1}$ can be achieved. However, the most
constraining searches being practically free of SM background, one can easily
rescale the limits on the couplings we present as
$\sim\left(10~{}\text{ab}^{-1}/\mathcal{L}\right)^{1/2}$.

###  4.1 ALP production through $e^{-}\mu^{+}\to a\gamma$

We first analyse the process $e^{-}\mu^{+}\to a\gamma$ that is sensitive to
LFV couplings of the ALP with electrons and muons. The analytical expression
of the differential cross section is

$\displaystyle\frac{\differential\sigma}{\differential t}(e^{-}\mu^{+}\to
a\gamma)\simeq\frac{\alpha}{16\pi}\frac{m_{\mu}^{2}}{f_{a}^{2}}\frac{|C^{A}_{e\mu}|^{2}+|C^{V}_{e\mu}|^{2}}{t(m_{a}^{2}-t-s)}\simeq\frac{\alpha}{4\pi}\frac{|g^{A}_{e\mu}|^{2}+|g^{V}_{e\mu}|^{2}}{t(m_{a}^{2}-t-s)}\,,$
(8)

where $\alpha$ is the fine-structure and constant and
$t=(p_{\mu}-p_{\gamma})^{2}$ in the limit $m_{e},m_{\mu}\ll\sqrt{s}$. After
integrating the above expression, we obtain the total cross section that, for
$m_{a}\ll\sqrt{s}$, reads

$\displaystyle\sigma(e^{-}\mu^{+}\to
a\gamma)\simeq\alpha\frac{m_{\mu}^{2}}{f_{a}^{2}}\frac{|C^{A}_{e\mu}|^{2}+|C^{V}_{e\mu}|^{2}}{4s}|\eta|_{\text{max}}\simeq\frac{\alpha}{s}\left(|g^{A}_{e\mu}|^{2}+|g^{V}_{e\mu}|^{2}\right)|\eta|_{\text{max}}\,,$
(9)

where $|\eta|_{\text{max}}$ is the maximum pseudorapidity within the
geometrical acceptance of the detector. For $|\eta|_{\text{max}}=2.5$, the
above formula agrees with the Madgraph result reported in Table 2 within 10%.
As one can see, a larger cross section can be achieved by decreasing the
energy of the muon beam $E_{\mu}$ and thus $\sqrt{s}$. However, the
$\mu$TRISTAN luminosity is expected to decrease for lower values of $\sqrt{s}$
[64]. Here, we consider the beam energy as in the $\mu$TRISTAN proposal, i.e.
$E_{e}=30$ GeV and $E_{\mu}=1$ TeV, while we comment on the impact on the
sensitivity of different values of $\sqrt{s}$ in the concluding Section 5.

The experimental signatures that could be used to search for this production
process depend on the possible ALP decay modes, hence on its mass and
couplings. Consequently, there could be different SM backgrounds which may or
may not be trivially reducible. We consider several possible signals and
backgrounds, distinguishing a “heavy ALP” scenario (with
$m_{a}>m_{e}+m_{\mu}$), for which the ALP can decay as $a\to e\mu$ through the
same LFV interaction as the production process, and a “light ALP” case (with
$m_{a}<m_{e}+m_{\mu}$), for which such process is kinematically closed. We
start discussing the latter case.

Figure 2:  Distribution of the photon transverse momentum, $p_{T}^{\gamma}$,
for $e^{-}\mu^{+}\to\gamma+$ invisible events before cuts. The blue line
denotes the signal from production of a light invisible ALP, with
$C^{A}_{e\mu}/f_{a}=C^{V}_{e\mu}/f_{a}=0.01~{}\text{GeV}^{-1}$, corresponding
to $g^{A}_{e\mu}\simeq g^{V}_{e\mu}\simeq 5.3\times 10^{-4}$. The red line is
the SM background $e^{-}\mu^{+}\to\gamma\nu_{e}\bar{\nu}_{\mu}$. The number of
events is calculated including both production cross sections and assuming for
the integrated luminosity $\mathcal{L}=10~{}\text{ab}^{-1}$.

#### Light ALP.

As discussed in Section 2, if $m_{a}<m_{e}+m_{\mu}$ and the LFC couplings are
absent or suppressed (in particular, if $C_{ee}^{A}=0$) the ALP is long-lived
and the signal events will feature a single photon and missing energy. The SM
background is then given by the process
$e^{-}\mu^{+}\to\gamma\nu_{e}\bar{\nu}_{\mu}$ that stems from exchange of a
$W$ boson in the $t$ channel and a photon from initial state radiation or $W$
bosons’ annihilation. The cross-section is rather large, 840 fb, once the
standard Madgraph generator-level cuts are applied.

In Figure 2, we show the signal and BG distributions of the photon transverse
momentum $p_{T}^{\gamma}$, which is the same as the transverse missing energy
(MET). The distribution for the signal exhibits a peak around
$p_{T}^{\gamma}\simeq\sqrt{s}/2\simeq 170$ GeV from the Lorentz boost factor
in the two-body final state. Instead, the $p_{T}^{\gamma}$ distribution of the
background decreases exponentially with increasing energy. Therefore, we
choose the kinematic cut $p_{T}^{\gamma}>170$ GeV, in order to distinguish the
signal from the background.

If $C_{ee}^{A}\neq 0$ and the decay $a\to e^{+}e^{-}$ is kinematically
allowed, this process occurs with nearly $100\%$ branching ratio, since the
decay into photons is loop-suppressed (see Appendix A). The signature then
depends on the ALP decay length shown in Figure 1. Long-lived enough ALPs can
be still searched for in events with a single photon and MET but we expect a
reduction in the sensitivity due to the fraction of ALPs decaying inside the
detector. ALPs with a shorter lifetime can be sought in $e^{-}\mu^{+}\to\gamma
e^{+}e^{-}$ events. In this case, the SM background is given by the process
$e^{-}\mu^{+}\to\gamma e^{-}\bar{\nu}_{\mu}W^{+}(\to e^{+}\nu_{e})$, whose
cross section is $0.34$ fb. Because of the presence of neutrinos in the final
state, a cut on the MET of the events can reduce this background
significantly. In this case, we impose MET$<10$ GeV.

#### Heavy ALP.

If $m_{a}>m_{e}+m_{\mu}$, the ALP can decay as $a\to e^{\pm}\mu^{\mp}$ through
the very same interaction involved in its production. In order to avoid any
major SM background, one can select events with negative muons and positive
electrons in the final state: $e^{-}\mu^{+}\to\gamma a\to\gamma\mu^{-}e^{+}$.
In the presence of LFV couplings only, one has
$\text{BR}(a\to\mu^{-}e^{+})=50\%$. The process
$e^{-}\mu^{+}\to\gamma\nu_{e}\bar{\nu}_{\mu}W^{-}(\to\mu^{-}\bar{\nu}_{\mu})W^{+}(\to
e^{+}\nu_{e})$ can in principle produce a reducible background. However, it
can be completely neglected because of its tiny cross section,
$\sigma_{\text{B}}\simeq 2\times 10^{-5}$ fb.333 Strictly speaking, box
diagrams with neutrino and $W$ propagators could induce
$e^{-}\mu^{+}\to\mu^{-}e^{+}(\gamma)$ in the SM. However, such process is
suppressed to negligible levels by the tiny neutrino masses, as it is always
the case for LFV processes within the SM. In this case the suppression is of
the order $\left({m_{\nu}}/{m_{W}}\right)^{8}\approx 10^{-98}$!

Additionally, if LFC ALP couplings exist, one can also search for
$e^{-}\mu^{+}\to a~{}\gamma$ followed by $a\to\mu^{+}\mu^{-}$. If
$C_{\mu\mu}^{A}=C_{e\mu}^{A}=C_{e\mu}^{V}$, the ALP decay branching fractions
are ${\rm BR}(a\to\mu^{+}\mu^{-})\simeq 50\%$ and ${\rm BR}(a\to
e^{+}\mu^{-})\simeq 25\%$ for $m_{a}\gg 2\,m_{\mu}$. The main SM background in
this case is given by the process
$e^{-}\mu^{+}\to\gamma\nu_{e}\bar{\nu}_{\mu}Z/\gamma^{\ast}(\to\mu^{+}\mu^{-})$,
whose cross section is 0.12 fb. We find that a missing energy cut, MET$<$15
GeV, is sufficient to eliminate this SM background.

ALP couplings | $m_{a}$ (GeV) | Signature (BR) | SM background | Cut | $\sigma_{\text{B}}(\rm{fb})\times\varepsilon_{\rm B}$ | $\varepsilon_{\text{S}}=\varepsilon_{\rm sel}\times\varepsilon_{\rm cut}$ |  $2\sigma$ limit on $g_{e\mu}^{V,A}$
---|---|---|---|---|---|---|---
($\mathcal{L}=10~{}\text{ab}^{-1}$)
$C_{e\mu}^{V}=C_{e\mu}^{A}$ | 0.1 | invisible (100%) | $e^{-}\mu^{+}\to\gamma\nu_{e}\bar{\nu}_{\mu}$ | $p_{T}^{\gamma}>170$ GeV | $840\times 0.004\%$ | $95\%\times 8.3\%$ | $1.2\times 10^{-3}$
10 | $a\to e^{+}\mu^{-}$ (50%) | negligible | – | – | $14\%\times 100\%$ | $2.2\times 10^{-4}$
100 | $14\%\times 100\%$ | $2.1\times 10^{-4}$
$C_{e\mu}^{V,A}=C^{A}_{\ell_{i}\ell_{i}}$ | 0.1 |  $a\to e^{+}e^{-}$ (100%)$\,{}^{\dagger}$ | $e^{-}\mu^{+}\to\gamma e^{-}\bar{\nu}_{\mu}W^{+}(\to e^{+}\nu_{e})$ | MET $<$ 10 GeV | $0.34\times 6\%$ | $14\%\times 73\%$ | $8.4\times 10^{-4}$
10 |  $a\to e^{+}\mu^{-}$ (25%) | negligible | – | – | $14\%\times 100\%$ | $3.1\times 10^{-4}$
$a\to\mu^{+}\mu^{-}$ (50%) | $e^{-}\mu^{+}\to\gamma\nu_{e}\bar{\nu}_{\mu}Z/\gamma^{\ast}(\to\mu^{+}\mu^{-})$ | MET $<$ 15 GeV | $0.12\times 5\%$ | $39\%\times 42\%$ | $5.9\times 10^{-4}$

Table 3:  Summary of the possible searches for ALPs produced in
$e^{-}\mu^{+}\to a\gamma$ for different ALP masses and couplings. The upper
block shows the results for exclusively LFV ALPs, the lower block for ALPs
also with LFC couplings. The third column reports the branching ratio (BR) of
the ALP decay under consideration — the process denoted by † has
$\text{BR}\simeq 100\%$ but only about 40% of the decays occur in the inner
detector for the chosen value of $m_{a}$. The applied cuts (if any), the BG
cross section $\sigma_{\text{B}}$ and efficiency $\varepsilon_{\text{B}}$, the
signal efficiency
$\varepsilon_{\text{S}}\equiv\varepsilon_{\text{sel}}\times\varepsilon_{\text{cut}}$
(see the main text for details) are shown in the fifth, sixth and seventh
columns, respectively. In the last column, we report the resulting $2\sigma$
limit on $g_{e\mu}^{V,A}\simeq m_{\mu}C_{e\mu}^{V,A}/2f_{a}$ (with
$g_{e\mu}^{V}=g_{e\mu}^{A}$) for $\mathcal{L}=10~{}\text{ab}^{-1}$. For the
BG-free searches these limits scale as
$\sim(10~{}\text{ab}^{-1}/\mathcal{L})^{1/2}$.

####  $\mu$TRISTAN sensitivity.

In Table 3, we summarise signals and backgrounds for the different ALP masses
and coupling scenarios discussed above. We also show the kinematic cuts that
effectively reduce the backgrounds in case they are present, and then compute
the background and signal cut efficiencies. In order to estimate the signal
significance, we additionally include the effect of the particle
identification capabaility and geometric acceptance of the detector,
$\varepsilon_{\rm sel}$, assuming as a requirement for the pseudorapidity of
photons and leptons $|\eta|<2.5$. We finally obtain a $2\sigma$ lower bound on
the flavour violating coupling $g_{e\mu}^{V,A}$ for an integrated luminosity
of 10 ab-1 using Poisson statistics as in Eq. (7). We find that $\mu$TRISTAN
has the capability to constrain LFV ALP couplings as low as
$g_{e\mu}^{V,A}\sim 10^{-3}-10^{-4}$.

Figure 3: Geometric efficiency of electrons and muons from ALP decays as a
function of the ALP mass, for two different designs of the detector:
$|\eta|_{\text{max}}=2.5$ (solid lines) and $|\eta|_{\text{max}}=3.5$ (dashed
lines).

As mentioned before, because of the planned asymmetric beam configuration of
$\mu$TRISTAN, the ALP decay products tend to fly along the muon beam
direction, with a very large pseudorapidity $|\eta|$.444This partly occurs
also for Higgs decays [64], which are the main target of the $\mu$TRISTAN
proposal. In order to study the impact on the sensitivity of this effect, we
consider two possible detectors with different values of
$|\eta|_{\text{max}}$: one with a standard geometric acceptance $|\eta|<2.5$,
and one more suitable for the $\mu$TRISTAN design with $|\eta|<3.5$. In Figure
3, we show the probability — that we call “geometric efficiency”
$\varepsilon_{\text{geo}}$ — for muons and electrons from ALP decays to be
within the geometric acceptance of the detector, ignoring particle
identification. We plot $\varepsilon_{\text{geo}}$ as a function of the ALP
mass, showing how it varies in the case of a detector with
$|\eta|_{\text{max}}=3.5$ (dashed lines) instead of $|\eta|_{\text{max}}=2.5$
(solid lines). As expected, $\varepsilon_{\rm geo}$ increases if a larger
lepton pseudorapidity range is allowed. However, even in such a case, a
substantial proportion of signal events is lost, especially for light ALPs
($50\%-60\%$).555We checked that the presence of a forward muon detector
covering large pseudorapidities (up to $|\eta|_{\text{max}}=8$) does not
substantially improve the reach of the most sensitive search we consider — the
one targeting the LFV ALP decay $a\to e^{+}\mu^{-}$ — which is still limited
by the reduced probability of detecting the positron. We also checked that,
instead, the angular separation between positron and photon does not give any
further limitation on the detection of our signal. For instance, imposing
$\theta_{e\gamma}>0.2$ has no appreciable impact on our estimated
sensitivities.

In Figure 4, we plot the resulting sensitivity of $\mu$TRISTAN in the
$m_{a}-|g^{V,A}_{e\mu}|$ plane for a purely LFV ALP with
$C_{e\mu}^{V}=C_{e\mu}^{A}$ (left panel) and an ALP with all LFC and LFV
couplings $C_{\ell_{i}\ell_{j}}^{A,V}$ equal (right panel). The solid (dashed)
curves show the lower limits on the couplings that can be obtained by a
detector with $|\eta|_{\text{max}}=2.5$ ($|\eta|_{\text{max}}=3.5$) searching
for the different signatures discussed in this section, assuming an integrated
luminosity of $\mathcal{L}=10~{}\text{ab}^{-1}$. As mentioned, these limits
scale as $\sim\left(10~{}\text{ab}^{-1}/\mathcal{L}\right)^{1/2}$ in the
situations where there are no background events.

(a) LFV ($e$-$\mu$) couplings only (b) LFV ($e$-$\mu$) and LFC couplings

Figure 4: Sensitivity of $\mu$TRISTAN to ALPs from $e^{-}\mu^{+}\to a\gamma$.
The expected 95% CL exclusion limits on $|g^{V,A}_{e\mu}|$ as a function of
$m_{a}$ are shown for a purely LFV ALP with $C^{A}_{e\mu}=C^{V}_{e\mu}$ (left
panel) and an ALP with LFC and LFV couplings and
$C^{A}_{ee}=C^{A}_{\mu\mu}=C^{A,V}_{e\mu}$ (right panel). Solid (dashed) lines
denote sensitivities of a detector with $|\eta|_{\text{max}}=2.5$
($|\eta|_{\text{max}}=3.5$). The limits scale as
$\sim(10~{}\text{ab}^{-1}/\mathcal{L})^{1/2}$. The two vertical dashed lines
indicate the $m_{\mu}+m_{e}$ threshold and the centre of mass energy of the
$e^{-}\mu^{+}$ collisions respectively. The coloured regions are excluded by
searches for low-energy LFV processes: muonium-antimuonium oscillations
(blue), two-body muon decay into an electron and an invisible boson (brown),
$\mu\to eee$ (pink), $\mu\to e\gamma$ (yellow). See main text for details.

For a light ALP with LFC couplings (right plot), let us note the interplay
between the search for an invisible ALP, which loses sensitivity if the ALP
lifetime is reduced (for larger values of $m_{a}$ and larger couplings), and
the search for $a\to e^{+}e^{-}$ which is more sensitive to the latter
situation, cf. Figure 1.

As we can see from both plots, the best sensitivity on the LFV couplings can
be achieved by the background-free search for $a\to e^{+}\mu^{-}$ if that is
kinematically allowed. This search can test the parameter space substantially
beyond the limits from low-energy LFV processes (shown as coloured regions)
especially for a purely LFV ALP (left plot). For an ALP with also LFC
couplings (right plot), $\mu$TRISTAN could test some unconstrained region only
if $m_{a}\gtrsim 50$ GeV. In either scenario, low-energy constraints on light
LFV ALPs already exclude the complete range of couplings that could be tested
at $\mu$TRISTAN. In the following, we discuss in detail the current and future
limits from low-energy LFV processes.

#### Present and future LFV and $g-2$ constraints.

In Figure 5, we summarise present and expected future constraints from
searches for low-energy LFV processes, denoting them as coloured regions and
dot-dashed lines, respectively. The ALP contributions to these processes are
computed by means of the expressions reported in the Appendix B.

The most model-independent constraint on an ALP with LFV interactions with
electrons and muons stems from mixing between muonium Mu (that is, a
$\mu^{+}e^{-}$ bound state) and antimuonium $\overline{\text{Mu}}$
($\mu^{-}e^{+}$), since the ALP contribution to this process solely relies on
the $C^{A,V}_{e\mu}$ couplings. The blue regions in Figures 4 and 5 are
excluded by the MACS experiment limit on the $\text{Mu}-\overline{\text{Mu}}$
oscillation probability, $P_{\text{Mu}\,\overline{\text{Mu}}}<8.3\times
10^{-11}$ at 90% CL [87]. As we can see, this is the only extant constraint on
a heavy ALP with the LFV couplings $C^{A,V}_{e\mu}$ only. In such a case,
according to our estimate, $\mu$TRISTAN can test unconstrained regions of the
parameter space if $m_{a}\gtrsim 1$ GeV. The proposed MACE experiment is
expected to improve the limit on muonium oscillations by three orders of
magnitude, down to $P_{\text{Mu}\,\overline{\text{Mu}}}<7\times 10^{-14}$ [88,
89]. The MACE impact on our parameter space is shown as a blue-dashed line in
Figure 5. Let us note the nice complementarity of MACE and $\mu$TRISTAN, with
the former providing a better sensitivity for $m_{a}\lesssim 3-4$ GeV.

(a) LFV ($e$-$\mu$) couplings only (b) LFV ($e$-$\mu$) and LFC couplings

Figure 5: Overview of the present and future constraints on LFV ALPs in the
$m_{a}-|g^{V,A}_{e\mu}|$ plane and comparison with the $\mu$TRISTAN
sensitivity as estimated in this work. The model’s parameters and the colour
coding are as in Figure 4. The dot-dashed lines represent the expected
sensitivities of running or future LFV experiments. See main text for details.

If the heavy ALP also enjoys substantial LFC interactions with electrons or
muons, the LFV decay $\mu\to e\gamma$ is induced by an ALP-lepton loop. As
shown in the right plots of Figures 4 and 5, the current 90% CL limit set by
the MEG experiment, $\text{BR}(\mu\to e\gamma)<4.2\times 10^{-13}$ [90], gives
a more stringent constraint (yellow region) on our parameter space than
$\text{Mu}-\overline{\text{Mu}}$ oscillations, if
$C^{A}_{ee}=C^{A}_{\mu\mu}=C^{A,V}_{e\mu}$. As a consequence, $\mu$TRISTAN
could be only sensitive to this kind of ALP models for $m_{a}$ in the
$\mathcal{O}(100)$ GeV mass range. The expected limit, $\text{BR}(\mu\to
e\gamma)<6\times 10^{-14}$, of the currently running MEG II experiment [91] is
shown as a yellow dot-dashed line in Figure 5.

In the light ALP regime, LFV muon decays into an on-shell ALP provide by far
the strongest constraints. If the ALP is long-lived, the limit on
$\text{BR}(\mu\to ea)$ with an invisible $a$ depends on the ALP mass and the
chirality of its couplings [30], ranging from $5.8\times 10^{-5}$ [92] for a
(practically) massless boson coupling mainly to left-handed leptons (thus to a
$V-A$ current) to $2.5\times 10^{-6}$ [93, 30] if the couplings to right-
handed leptons (hence to a $V+A$ current) dominate, such as in our benchmark
scenario. For ALPs in the mass range $10~{}\text{MeV}\lesssim m_{a}\lesssim
90~{}\text{MeV}$, the dependence on $m_{a}$ is mild in the $V+A$ case and the
average upper bound is $6\times 10^{-6}$ [92], the limit that we employ here.
The region excluded by searches for an invisible ALP in muon decays is shown
in brown in Figure 5, while the dot-dashed brown line is the expected future
limit of the Mu3e experiment, $\text{BR}(\mu\to ea)\lesssim 10^{-8}$ [22]. In
the presence of couplings to electrons, searches for $\mu\to ea(\to
e^{+}e^{-})$ can be also sensitive to our parameter space. The limit from
$\mu\to e\gamma$ is stronger if the ALP is off-shell, while for on-shell
short-lived ALPs the current limit $\text{BR}(\mu\to eee)<1.0\times 10^{-12}$
[94] excludes the pink region of our figures. The Mu3e expected future bound,
$\text{BR}(\mu\to eee)\lesssim 10^{-16}$ [95, 96], is shown as a dot-dashed
pink line. In order to assess the relative importance of searches for
invisible and visible ALPs, we used information on the ALP lifetime —
illustrated in Figure 1 — as explained in Appendix B.

LFV ALP interactions also contribute to both the electron and the muon
anomalous magnetic moments through ALP-lepton loops. In the presence of LFC
interactions, there are important additional contributions, especially due to
the induced ALP-photon coupling [97, 27]. However, we find that limits on non-
standard contributions to the electron and muon $g-2$ barely affect the region
of the parameter space that can be tested by $\mu$TRISTAN beyond the limits
from LFV observables, if no LFV coupling involving $\tau$ leptons is present.
For this reason and in order to avoid an excess of information in the figures
of this section, we do not show such limits here. More details and a
discussion of the currently uncertain SM predictions of both observables are
presented in Appendix B.3.

###  4.2 ALP production through LFV $\tau$ interactions

We now proceed to discuss the other promising search channels, which can
target ALP production with an associated tau lepton in the final state as
highlighted in Table 2. The cross sections of these processes as a function of
the ALP mass for beams with energies as in the $\mu$TRISTAN proposal,
$E_{e}=30$ GeV and $E_{\mu}=1$ TeV, are shown in the left plot of Figure 6.
The solid lines correspond to a generator-level cut on the pseudorapidity of
the final-state leptons (including the $\tau$) of $|\eta|<2.5$, the dashed
lines to $|\eta|<3.5$.

Figure 6:  Left panel: total cross section of ALP production processes through
LFV ALP-$\tau$ interactions at $\mu$TRISTAN for $|\eta|_{\text{max}}=2.5$
(solid lines) and $|\eta|_{\text{max}}=3.5$ (dashed lines), for
$C^{A}_{\ell\tau}/f_{a}=C^{V}_{\ell\tau}/f_{a}=0.01~{}\text{GeV}^{-1}$
($\ell=e\,\text{or}\,\mu$), corresponding to $g^{A}_{\ell\tau}\simeq
g^{V}_{\ell\tau}\simeq 8.9\times 10^{-3}$. Right panel: signal efficiency of
searches for the processes of the left plot with hadronic taus in the final
state.

For what concerns $\tau$ detection, one can use either the hadronic $\tau$
decays or the leptonic ones. In the former case, the resulting tau jets posses
distinctive features. They are very collimated, exhibit a low particle
multiplicity, mostly one or three prongs, and are associated to a relevant
fraction of electromagnetic energy deposition mainly due to the decay into
photons of neutral pions. Modern detectors are expected to yield an
identification efficiency for hadronic taus as large as 90%, see e.g. Ref.
[98].

In the following, we examine several signatures that can be probed at
$\mu$TRISTAN depending on the ALP mass and decay channels, as well as on the
selection of hadronic or leptonic $\tau$ decays.

ALP couplings | Signal | SM background |  $\sigma_{\text{B}}$ (fb)
---|---|---|---
$C^{V,A}_{\mu\tau}$ | $e^{-}\mu^{+}\to e^{-}\tau^{+}a_{\text{inv}}$ | $e^{-}\mu^{+}\to e^{-}\tau^{+}\bar{\nu}_{\mu}\nu_{\tau}$ | 24
$\mu^{+}\mu^{+}\to\mu^{+}\tau^{+}a_{\text{inv}}$ | $\mu^{+}\mu^{+}\to\mu^{+}\tau^{+}\bar{\nu}_{\mu}\nu_{\tau}$ | 180
$C^{V,A}_{e\tau}$ | $e^{-}\mu^{+}\to\tau^{-}\mu^{+}a_{\text{inv}}$ | $e^{-}\mu^{+}\to\tau^{-}\mu^{+}\bar{\nu}_{\tau}\nu_{e}$ | 4.8
Table 4: SM background processes for a light ALP, $m_{a}<m_{\tau}+m_{\ell}$,
appearing as missing energy ($a_{\text{inv}}$). The displayed
$\sigma_{\text{B}}$ are computed requiring that all visible final-state
particles have $|\eta|<2.5$.

ALP couplings | Signal | SM background |  $\sigma_{\text{B}}$ (fb)
---|---|---|---
$C^{V,A}_{\mu\tau}=C^{A}_{\mu\mu}$ | $e^{-}\mu^{+}\to e^{-}\tau^{+}a(\to\mu^{+}\mu^{-})$ | $e^{-}\mu^{+}\to e^{-}\bar{\nu}_{\mu}W^{+}(\to\tau^{+}\nu_{\tau})Z/\gamma^{*}(\to\mu^{+}\mu^{-})$ | $5.6\times 10^{-3}$
$\mu^{+}\mu^{+}\to\mu^{+}\tau^{+}a(\to\mu^{+}\mu^{-})$ | $\mu^{+}\mu^{+}\to\mu^{+}\bar{\nu}_{\mu}W^{+}(\to\tau^{+}\nu_{\tau})Z/\gamma^{*}(\to\mu^{+}\mu^{-})$ | 0.44
$C^{V,A}_{e\tau}=C^{A}_{\mu\mu}$ | $e^{-}\mu^{+}\to\tau^{-}\mu^{+}a(\to\mu^{+}\mu^{-})$ | $e^{-}\mu^{+}\to\nu_{e}\mu^{+}W^{-}(\to\tau^{-}\bar{\nu}_{\tau})Z/\gamma^{*}(\to\mu^{+}\mu^{-})$ | $7.6\times 10^{-5}$

Table 5: SM background processes for a light ALP with
$2m_{\mu}<m_{a}<m_{\tau}+m_{\ell}$ and LFC couplings to muons, such that the
decay $a\to\mu^{+}\mu^{-}$ occurs inside the detector. The displayed
$\sigma_{\text{B}}$ are computed requiring that all visible final-state
particles have $|\eta|<2.5$. The illustrative production mode
$Z/\gamma^{\ast}\to\mu^{+}\mu^{-}$ here and below actually includes all
possible production processes of $\mu^{+}\mu^{-}$ in our background event
generation.

#### Light ALP.

We first consider ALPs whose decays into $\tau$ leptons are kinematically
forbidden, that is, with $m_{a}<m_{\tau}+m_{\ell}$ (where $\ell=e$ or $\mu$).
If the LFC interactions are absent, the ALP is long-lived and the processes in
Table 2 yield signatures involving missing energy. In such a case, the SM
processes with neutrinos in the final state listed in Table 4 look exactly
like our ALP signals. As we can see, the BG cross sections are much larger
than the ALP production cross sections shown in Figure 6. In addition, we
found no kinematical distribution giving a handle to reduce the background.
Hence, the ALP emission signal induced by the LFV $C^{V,A}_{\ell\tau}$
couplings is completely overshadowed by the SM background and it is not
possible to set a limit on such couplings from $\mu$TRISTAN in this case.

For ALPs coupling with muons, one could instead search for the decay
$a\to\mu^{+}\mu^{-}$, if it is kinematically allowed. The SM background
processes for these modes are shown in Table 5. As one can see, the associated
$\sigma_{\text{B}}$ are not always negligible compared to our signals.
Furthermore, as discussed in the following, hadronic $\tau$ tagging is not
effective in the light ALP regime, such that one should rely on leptonic
$\tau$ decays, further reducing the signal cross section and introducing more
reducible SM background modes. Finally, similarly to the $\mu$-$e$ LFV case
studied above, the light ALP regime is already strongly constrained by LFV
processes. For these reasons, we refrain from a more quantitative study of
this case and we move to discuss the heavy ALP regime.

#### Heavy ALP.

In the rest of the section, we study the case $m_{a}>m_{\tau}+m_{\ell}$, where
the same LFV interactions involved in the ALP production processes are
responsible for it to decay promptly as $a\to\ell\tau$. As in the $\mu$-$e$
case, we select as signal the combination of charges of the final-state
leptons that differs from the initial-state one: $a\to\mu^{-}\tau^{+}$ for the
searches targeting ALP couplings to $\mu-\tau$, $a\to e^{+}\tau^{-}$ for the
$e-\tau$ ones. For simplicity, we do not consider the situation where
$C^{A,V}_{\mu\tau}$ and $C^{A,V}_{e\tau}$ are simultaneously non-vanishing.
Notice that in the presence of LFC couplings with muons or taus, if
$C^{A,V}_{\ell\tau}=C^{A}_{\mu\mu}=C^{A}_{\tau\tau}$
($\ell=e\,\text{or}\,\mu$),
$\text{BR}(a\to\tau^{+}\tau^{-})\approx\text{BR}(a\to\ell^{\pm}\tau^{\mp})$
and
$\text{BR}(a\to\mu^{+}\mu^{-})\approx(m_{\mu}/m_{\tau})^{2}\text{BR}(a\to\ell^{\pm}\tau^{\mp})$,
as one can see from Eq. (5). Hence, the decays into muons are irrelevant while
those into $\tau$ leptons yield double signal events than the LFV decays,
given that only one charge combination is selected for the latter ones.
However, the $a\to\tau\tau$ signal is penalised by a reduced selection
efficiency of hadronic taus — as we discuss below — or by the $\tau$ leptonic
branching fractions. Hence, we only focus on LFV ALP decays in the following.

ALP couplings | Signal | SM background |  $\sigma_{\text{B}}$ (fb)
---|---|---|---
$C^{V,A}_{\mu\tau}$ | $e^{-}\mu^{+}\to e^{-}\tau_{h}^{+}a(\to\tau_{h}^{+}\mu^{-})$ | $e^{-}\mu^{+}\to e^{-}\bar{\nu}_{\mu}W^{+}(\to\tau_{h}^{+}\nu_{\tau})W^{+}(\to\tau_{h}^{+}\nu_{\tau})W^{-}(\to\mu^{-}\bar{\nu}_{\mu})$ | $2.1\times 10^{-8}$
$\mu^{+}\mu^{+}\to\mu^{+}\tau_{h}^{+}a(\to\tau_{h}^{+}\mu^{-})$ | $\mu^{+}\mu^{+}\to\bar{\nu}_{\mu}\bar{\nu}_{\mu}W^{+}(\to\tau_{h}^{+}\nu_{\tau})W^{+}(\to\tau_{h}^{+}\nu_{\tau})Z/\gamma^{*}(\to\mu^{+}\mu^{-})$ | $1.4\times 10^{-3}$
$C^{V,A}_{e\tau}$ | $e^{-}\mu^{+}\to\tau_{h}^{-}\mu^{+}a(\to\tau_{h}^{-}e^{+})$ | $e^{-}\mu^{+}\to\nu_{e}\mu^{+}W^{+}(\to e^{+}\nu_{e})W^{-}(\to\tau_{h}^{-}\bar{\nu}_{\tau})W^{-}(\to\tau_{h}^{-}\bar{\nu}_{\tau})$ | $1.1\times 10^{-8}$

Table 6:  SM background processes for a heavy ALP, $m_{a}>m_{\tau}+m_{\ell}$,
with hadronically decaying $\tau$ leptons (denoted as $\tau_{h}$) in the final
state. The displayed $\sigma_{\text{B}}$ are computed requiring that all
visible final-state particles have $|\eta|<2.5$.

ALP couplings | Signal | SM background |  $\sigma_{\text{B}}$ (fb)
---|---|---|---
$C^{V,A}_{\mu\tau}=C^{A}_{\mu\mu}$ | $e^{-}\mu^{+}\to e^{-}\tau_{e}^{+}a(\to\tau_{e}^{+}\mu^{-})$ | $e^{-}\mu^{+}\to\nu_{e}\bar{\nu}_{\mu}W^{-}(\to\mu^{-}\bar{\nu}_{\mu})W^{+}(\to e^{+}\nu_{e})Z/\gamma^{*}(\to e^{+}e^{-})$ | $2.3\times 10^{-7}$
$\mu^{+}\mu^{+}\to\mu^{+}\tau_{e}^{+}a(\to\tau_{e}^{+}\mu^{-})$ | $\mu^{+}\mu^{+}\to\bar{\nu}_{\mu}\bar{\nu}_{\mu}W^{+}(\to e^{+}\nu_{e})W^{+}(\to e^{+}\nu_{e})Z/\gamma^{*}(\to\mu^{+}\mu^{-})$ | $3.3\times 10^{-3}$
$C^{V,A}_{e\tau}=C^{A}_{\mu\mu}$ | $e^{-}\mu^{+}\to\tau_{\mu}^{-}\mu^{+}a(\to\tau_{e}^{+}\mu^{-})$ | $e^{-}\mu^{+}\to\nu_{e}\bar{\nu}_{\mu}W^{-}(\to\mu^{-}\bar{\nu}_{\mu})W^{+}(\to e^{+}\nu_{e})Z/\gamma^{*}(\to\mu^{+}\mu^{-})$ | $2.3\times 10^{-7}$

Table 7: Same as in Table 6 for signal selection involving leptonic $\tau$
decays. $\tau_{\ell}$ denotes $\tau\to\ell\nu\bar{\nu}$ ($\ell=e,\mu$).

As already mentioned, modern detectors are expected to yield a large
$\tau$-jet tagging efficiency. For definiteness, we set such a parameter in
Delphes to be 90% for taus with $p_{T}>10$ GeV. In fact, the identification
efficiency is expected to degrade substantially in case of soft jets.
Furthermore, the geometric acceptance of the detector can severely reduce the
selection efficiency of the signal, especially in asymmetric $e^{-}\mu^{+}$
collisions, as discussed above in the case of searches for $e^{-}\mu^{+}\to
a\gamma$. As shown in the right plot of Figure 6, the combination of these
effects limit the signal efficiency to $\varepsilon_{\text{S}}\lesssim
10\%\,(25\%)$ for $e^{-}\mu^{+}$ ($\mu^{+}\mu^{+}$) collisions. In addition,
no sensitivity is expected for ALP masses smaller than $10-20$ GeV, because
this results in hadronic taus that are too soft to be detected, as mentioned
above.

Due to the limited sensitivity to light ALPs, we also consider signal
selection based on leptonic $\tau$ decays, for which $\sigma_{\text{S}}$ is
reduced by a factor
$\left[\text{BR}(\tau\to\ell\nu\bar{\nu})\right]^{2}\approx 3\%$ (times the
branching fraction of the selected $a\to\ell\tau$ mode) relative to the
production cross section displayed in the left panel of Figure 6.

The cross sections of the BG processes relevant to ALP searches involving
hadronic and leptonic taus are displayed, respectively, in Table 6 and Table
7. As we can see, searches for $\tau$ LFV ALPs at $\mu$TRISTAN operating as a
$e^{-}\mu^{+}$ collider are expected to be background-free for
$\mathcal{L}=10~{}\text{fb}^{-1}$. In contrast, the SM background for the
searches based on $\mu^{+}\mu^{+}$ is non-negligible, but only gives
$\mathcal{O}(10)$ events. Following from Eq. (7), this mildly limits the
resulting sensitivity on the $\mu$-$\tau$ ALP couplings.

(a) LFV ($\mu$-$\tau$) couplings only (b) LFV ($\mu$-$\tau$) and LFC couplings
(c) LFV ($e$-$\tau$) couplings only (d) LFV ($e$-$\tau$) and LFC couplings

Figure 7: Sensitivity of $\mu$TRISTAN to ALP production through LFV
interactions with $\tau$ leptons. In the first row, the expected 95% CL
exclusion limits on $|g^{V,A}_{\mu\tau}|$ as a function of $m_{a}$ are shown
for a purely LFV ALP with $C^{A}_{\mu\tau}=C^{V}_{\mu\tau}$ (left panel) and
an ALP with LFC and LFV couplings and $C^{A}_{\ell\ell}=C^{A,V}_{\mu\tau}$
(right panel). Solid (dashed) lines denote sensitivities of a detector with
$|\eta|_{\text{max}}=2.5$ ($|\eta|_{\text{max}}=3.5$). Analogously, the plots
in the second row show limits on $|g^{V,A}_{e\tau}|$ for the case
$C^{A}_{e\tau}=C^{V}_{e\tau}$ (left panel) and
$C^{A}_{\ell\ell}=C^{A,V}_{e\tau}$ (right panel). The coloured regions are
excluded by low-energy leptonic processes: two-body tau decay into muon or
electron and an invisible boson (brown), $\tau\to\ell\ell\ell^{(\prime)}$
decays (pink), $\tau\to e\gamma$ (yellow), the ALP contribution to muon or
electron $g-2$, respectively, $\Delta a_{\mu}$ and $\Delta a_{e}$ (purple).
See main text for details.

(a) LFV ($\mu$-$\tau$) and LFC couplings (b) LFV ($\mu$-$\tau$) and LFC
couplings (c) LFV ($e$-$\tau$) and LFC couplings (d) LFV ($e$-$\tau$) and LFC
couplings

Figure 8: Overview of the present and future constraints on LFV ALPs in the
$m_{a}-|g^{V,A}_{\mu\tau}|$ plane (first row) and $m_{a}-|g^{V,A}_{e\tau}|$
plane (second row) compared with the $\mu$TRISTAN sensitivity as estimated in
this work. The model’s parameters and the colour coding are as in Figure 7.
The dot-dashed lines represent the expected Belle II sensitivities on LFV
$\tau$ decays with $\mathcal{L}=50~{}\text{ab}^{-1}$. See main text for
details.

####  $\mu$TRISTAN sensitivity.

The expected sensitivity of the $\mu$TRISTAN searches discussed in this
section is summarised in Figure 7. As before, the solid (dashed) lines denote
the lower limits on the relevant LFV couplings that can be obtained by a
detector with geometric acceptance $|\eta|_{\text{max}}=2.5$
($|\eta|_{\text{max}}=3.5$) after collecting $\mathcal{L}=10~{}\text{ab}^{-1}$
of data. Searches based on leptonic $\tau$ decays can provide some sensitivity
to the LFV couplings for $m_{a}\lesssim 10$ GeV. For heavier ALPs, however,
searches for signatures with $\tau$ jets have a better sensitivity and can
constrain couplings at the $10^{-3}-10^{-2}$ level, especially for
$m_{a}=\mathcal{O}(100)$ GeV, as expected from the signal efficiencies plotted
in Figure 6 (right). Comparing the $\mu$TRISTAN sensitivity with the bounds
from low-energy observables (LFV decays and leptonic $g-2$), one can see that
$\mu$TRISTAN can test portions of the parameter space that are currently
unconstrained, especially if the ALP exclusively (or dominantly) couples with
leptons of different flavours. Our results show that, in addition, some
sensitivity to $\mu$-$\tau$ interactions beyond the present low-energy limits
is possible even if LFC couplings are present. Nevertheless, to this end, it
is important to design a detector with a geometric acceptance of light leptons
and hadronic taus up to $|\eta|_{\text{max}}=3.5$ or higher. In the $e$-$\tau$
case, our results show that only a marginal improvement beyond the bounds on
LFV $\tau$ decays can be expected. In the following, we briefly discuss the
limits set by LFV decays and muon/electron $g-2$. For further information, we
refer to Appendix B where the expressions used to compute them are collected
together with the current and forecast experimental limits.

#### Present and future LFV and $g-2$ constraints.

As shown in Figure 7, if the ALP does not couple to same-flavour leptons and
is too heavy for the $\tau\to\ell\,a$ decays to be allowed, the only low-
energy constraints that affect our parameter space stem from the anomalous
magnetic moments of the leptons. Indeed, in the presence of $\mu$-$\tau$
($e$-$\tau$) ALP interactions, an ALP-tau loop provides a non-standard
contribution $\Delta a_{\mu}$ ($\Delta a_{e}$) to the muon (electron) $g-2$.
Although the theoretical predictions within the SM are currently subject to
debate for either observables, a model giving too large values of $|\Delta
a_{\mu}|$ or $|\Delta a_{e}|$ is certainly excluded. Here, we conservatively
show the $2\sigma$ exclusion (denoted as a purple region) obtained adopting in
each case the SM prediction that is in better agreement with the experimental
measurement. These are, for the muon $g-2$, the prediction based on the most
precise determination of the hadronic vacuum polarization (HVP) from lattice
QCD [99] and, for the electron $g-2$, the prediction based on the value of the
fine-structure constant measured by means of matter-wave interferometry of
rubidium atoms [100] — see Appendix B.3 for details. In Figure 8, we also show
as a green band the values of the parameters that could explain (at the
$2\sigma$ level) the recent measurement of the Muon g-2 Experiment [101] if
instead the data-driven approach to the HVP using dispersion relations is
employed [102]. As we can see, that region is partly within the reach of our
estimated $\mu$TRISTAN sensitivity.

If $\tau$ decays into ALPs are kinematically open and/or in the presence of
LFC couplings, a number of LFV $\tau$ decays set important constraints on our
parameter space. This is better depicted in Figure 8, where the expected
future limits from the Belle II experiment [103, 104] are also shown as dot-
dashed lines. In particular, we show as brown regions the current best limits
(in the $10^{-4}-10^{-3}$ range) obtained by Belle II on the branching ratios
of $\tau\to\ell\,a$, with an invisible $a$ [41], alongside with the future
sensitivity $\text{BR}(\tau\to\ell\,a)\simeq 10^{-5}$ that can be achieved by
the same experiment with the expected full data set of $50~{}\text{ab}^{-1}$
[30]. As we can see from the figures, LFC couplings of the ALP to electrons
and muons can also mediate a number of LFV decays of the kind
$\tau\to\ell\ell\ell^{(\prime)}$ ($\ell,\ell^{\prime}=e,\mu$) that, in the
light ALP regime can be more or less constraining than the searches for an
invisible ALP depending on the lifetime of the latter. Interestingly, the
future Belle II limits $\text{BR}(\tau\to\mu\mu\mu)<3.6\times 10^{-10}$ and
$\text{BR}(\tau\to e\mu\mu)<4.5\times 10^{-10}$ [104] can set more stringent
bounds than, respectively, $\tau\to\mu\gamma$ and $\tau\to e\gamma$ also if
the ALP is off-shell, as long as $m_{a}\lesssim 20$ GeV, and show a nice
complementarity with the sensitivity that can be achieved at $\mu$TRISTAN.

##  5 Summary and conclusions

Axions and ALPs are intriguing candidates for NP beyond the SM. If these
particles couple to the lepton sector of the SM, there is no fundamental
reason why their interactions should be flavour conserving. In fact, LFV ALP
interactions are predicted within a wide range of explicit UV models [30]. The
recently proposed $\mu$TRISTAN $e^{-}\mu^{+}$ (and $\mu^{+}\mu^{+}$) collider
[64], drawing from the mature $\mu^{+}$ beam technology developed for the
J-PARC muon $g-2$ experiment [65], plays as an ideal machine to explore
possible NP in the charged lepton sector. In this work, we studied the search
potential for LFV ALPs at $\mu$TRISTAN with different beams and energies. We
assessed the sensitivity to the LFV ALP couplings that can be achieved
employing a number of ALP production channels and compared it with the
existing low-energy leptonic constraints as well as future improvements. Our
main results can be summarised as follows.

  * •

The most promising ALP production modes at $\mu$TRISTAN include
$e^{-}\mu^{+}\to a\gamma$, $e^{-}\mu^{+}\to e^{-}\tau^{+}a$,
$\mu^{+}\mu^{+}\to\mu^{+}\tau^{+}a$, and $e^{-}\mu^{+}\to\tau^{-}\mu^{+}a$.
They are induced by the LFV ALP couplings $C_{e\mu}^{A,V}$,
$C_{\mu\tau}^{A,V}$, and $C_{e\tau}^{A,V}$, respectively. We examined the
$\mu$TRISTAN sensitivity to these couplings by means of a cut-based collider
analysis.

  * •

For each signal channel, the analysis is divided into two mass regimes for the
ALP and limits are set on dimensionless coupling $g_{ij}^{V,A}\simeq
m_{j}C_{ij}^{V,A}/2f_{a}$ at an integrated luminosity of $\mathcal{L}=10$
ab-1. The limits scale as
$\sim\left(10~{}\text{ab}^{-1}/\mathcal{L}\right)^{1/2}$ for background free
signals (that are usually the most constraining ones).

  * •

We find that for improved signal sensitivity, a wider geometric acceptance of
the detector is desirable in the case of the asymmetric $e^{-}\mu^{+}$
collisions, as the ALP decay products tend to be collinear to the muon beam
direction with large pseudorapidity. In our analysis, we compare limits on the
dimensionless couplings for $|\eta|<2.5$ and $|\eta|<3.5$ respectively.

  * •

Light ALP. When $m_{a}<m_{\ell}+m_{\ell^{\prime}}$ with
$m_{\ell}<m_{\ell^{\prime}}$, the ALP is either long-lived or may promptly
decay to $\ell^{+}\ell^{-}$ depending if the LFC coupling $C^{A}_{\ell\ell}$
is allowed. For the $e^{-}\mu^{+}\to a\gamma$ channel, we find that the
parameter space that $\mu$TRISTAN can probe has already been excluded by low-
energy LFV processes such as Mu–$\overline{\text{Mu}}$ oscillation, $\mu\to
e+{\rm inv.}$ and $\mu\to eee$. For the ALP productions with $\tau$ lepton, we
are not able to place a limit on the LFV $\tau$ couplings because of the non-
negligible SM background. However, analogous LFV $\tau$ decays already
strongly constrain this regime.

  * •

Heavy ALP. In the heavier ALP regime, i.e. when
$m_{a}>m_{\ell}+m_{\ell^{\prime}}$, under different assumptions of leptonic
couplings, the ALP can decay to either flavour changing or flavour conserving
lepton final states.

    * –

The search for heavy ALP through $e^{-}\mu^{+}\to a\gamma$ can test the
parameter space beyond the limits from Mu–$\overline{\text{Mu}}$ oscillation
and $\mu\to e\gamma$ and has the potential to test the $g^{V,A}_{e\mu}$
couplings to values as low as $\sim\mathcal{O}(10^{-4})$.

    * –

For the associated productions of ALP–$\tau$–$\ell$ with
$m_{a}>m_{\tau}+m_{\ell}$, the leptonic (hadronic) $\tau$ decay mode is
sensitive to relatively low (high) ALP mass values. Signatures with hadronic
$\tau$ decays are able to probe values of the $g_{\ell\tau}^{V,A}$ couplings
currently allowed by $\tau$ LFV decay and leptonic $g-2$ and reach a
sensitivity of $g_{\mu\tau}^{V,A}\sim 10^{-3}-10^{-2}$ and
$g^{V,A}_{e\tau}\sim 10^{-2}$ for $m_{a}=\mathcal{O}(100)$ GeV.

(a) LFV ($e$-$\mu$) couplings only (b) LFV ($e$-$\mu$) and LFC couplings

Figure 9: Comparison of the limits on the ALP couplings to $e$-$\mu$ that
could be achieved at $\mu$TRISTAN with different $\mu^{+}$ beam energy
configurations, while keeping $E_{e}=30$ GeV: $E_{\mu}=1000$ GeV
($\sqrt{s}\simeq 346$ GeV) and $E_{\mu}=100$ GeV ($\sqrt{s}\simeq 110$ GeV).
Note that the background from $e^{-}\mu^{+}\to\gamma\mu^{+}\mu^{-}+$ MET is
negligible for $E_{\mu}=100$ GeV ($\sigma_{\text{B}}=4.8\times 10^{-4}$ fb),
so that we can combine the signal from the LFV ALP decay with the LFC one. The
model’s parameters and colour coding are as in Figure 4.

As a future outlook, one can envisage different muon beam energy
configurations for the $\mu$TRISTAN project. We have seen that the ALP
production cross-section for the $e^{-}\mu^{+}$ collisions increases for lower
c.m. energies, cf. Eq. (9). Furthermore, if the asymmetry in the beam energies
is lowered, the detecting efficiency of the signal also increases, as the ALP
decay products are less collinear to the muon beam. We find that the optimal
choice for the beam energy configuration is
$(E_{e},E_{\mu})=(30~{}\rm{GeV},100~{}\rm{GeV})$,666Such a configuration
corresponds to $\sqrt{s}\simeq 110$ GeV, hence it is clearly beyond the main
purpose of $\mu$TRISTAN to work as a Higgs factory. However, if evidence of
light NP in the $\sqrt{s}\simeq 346$ GeV run is observed, performing precision
studies of it at lower $\sqrt{s}$ is a reasonable option. and study how this
influences the collider bound on $g_{e\mu}^{V,A}$ that can be obtained in the
search for $e^{-}\mu^{+}\to\gamma a(\to e^{+}\mu^{-}/\mu^{+}\mu^{-})$.
Although in such a case the luminosity is expected to be lower due to
relativistic effects — for more details see [64] — we show in Figure 9 the
improvement on the sensitivity to $g_{e\mu}^{V,A}$ (by approximately one order
of magnitude) assuming that an integrated luminosity of $10~{}\rm{ab}^{-1}$
can still be reached in the future with advanced beam intensity and focusing
technologies. For the ALP production modes with an associated $\tau$ lepton in
the final state, we observe that $\mu$TRISTAN with the above energy
configuration does not exhibit a better performance compared to the results
obtained in this paper, although the cross sections also increase, since the
$\tau$-jet identification efficiency rapidly deteriorates for softer jets.

### Acknowledgements

We would like to thank organisers and participants — especially, Chih-Ting Lu
and Zuowei Liu — of the 2nd TOPAC meeting (Southeastern University, Nanjing,
May 30th - June 2nd 2024), where LM presented our preliminary results, for
useful feedback and valuable discussions. LC and TL acknowledge financial
support from the National Natural Science Foundation of China (NSFC) under the
grant No. 12035008. LC is additionally supported by the NSFC under the grant
No. 12211530479. TL is also supported by the NSFC (Grants No. 12375096,
11975129) and “the Fundamental Research Funds for the Central Universities”,
Nankai University (Grant No. 63196013).

##  Appendix A ALP decay into photons

The $a\to\gamma\gamma$ decay width can be written as [46, 47]

$\displaystyle\Gamma(a\to\gamma\gamma)=\frac{m_{a}^{3}\alpha^{2}}{64\pi^{3}f_{a}^{2}}\left[E_{\text{UV}}+\sum_{i}C_{\ell_{i}\ell_{i}}^{A}B\left(\frac{4m_{\ell_{i}}^{2}}{m_{a}^{2}}\right)\right]^{2},$
(10)

where the loop function $B$ reads

$\displaystyle
B(x)=1-xf(x)^{2},~{}~{}f(x)=\begin{cases}\arcsin\left(\frac{1}{\sqrt{x}}\right)&x\geq
1\,,\\\
\frac{\pi}{2}+\frac{i}{2}\log\left(\frac{\sqrt{1-x}+1}{\sqrt{1-x}+1}\right)&x<1\,.\end{cases}$
(11)

In the above expression, $E_{\text{UV}}$ is a model-dependent UV ALP-photon
coupling (related to the electromagnetic anomaly of the global $\text{U}(1)$
symmetry) and the second term is due to lepton loops that are unavoidable if
LFC couplings exist. Unless otherwise noted, we only consider the latter
model-independent contribution, that is, we set $E_{\text{UV}}=0$ throughout
the paper.

##  Appendix B Low-energy leptonic processes

###  B.1 Muonium antimuonium oscillations

The ALP mediated $\text{Mu}-\overline{\text{Mu}}$ transition probability is
given by [29]

$\displaystyle
P_{\text{Mu}\,\overline{\text{Mu}}}=\frac{m_{\mu}^{4}}{2\pi^{2}a_{B}^{6}\Gamma_{\mu}^{2}m_{a}^{4}f_{a}^{4}}$
$\displaystyle\left[|c_{0,0}|^{2}\left|(C^{V}_{e\mu})^{2}-\left(1+\frac{1}{\sqrt{1+X^{2}}}\right)(C^{A}_{e\mu})^{2}\right|^{2}+\right.$
$\displaystyle\left.|c_{1,0}|^{2}\left|(C^{V}_{e\mu})^{2}-\left(1-\frac{1}{\sqrt{1+X^{2}}}\right)(C^{A}_{e\mu})^{2}\right|^{2}\right],$
(12)

where $\Gamma_{\mu}\simeq 3.00\times 10^{-19}$ GeV is the muon decay rate, and
$a_{B}\simeq 2.69\times 10^{5}~{}\text{GeV}^{-1}$ is the muonium Bohr radius,
while $X$ is a parameter related to the magnetic field $B$ employed in the
experimental apparatus, defined as $X=6.31~{}(B/1\,\text{T})$. The field also
affects the probability of populating the initial state with angular momentum
$(J,m_{J})$, which is expressed by the quantity $|c_{J,m_{J}}|^{2}$.

The most precise search for muonium oscillations to date was performed by the
MACS experiment, whcih used $B=0.1$ T, for which $|c_{0,0}|^{2}=0.32$ and
$|c_{1,0}|^{2}=0.18$. The 90% CL upper limit on the oscillation probability
obtained by MACS is $P_{\text{Mu}\,\overline{\text{Mu}}}<8.3\times 10^{-11}$
[87]. Concerning the future prospects, the proposed MACE experiment is
expected to reach a single event sensitivity of $3\times 10^{-14}$,
corresponding to the 90% CL upper limit
$P_{\text{Mu}\,\overline{\text{Mu}}}<7\times 10^{-14}$ [88, 89]. MACE plans to
employ the same field as MACS, $B=0.1$ T, hence the quantities $X$ and
$c_{J,m_{J}}$, required in Eq. (12) to interpret the experimental results in
terms of the ALP parameters, are also the same.

###  B.2 LFV muon and tau decays

Process | Current limit | Future limit
---|---|---
$\mu\to e\gamma$ |  $4.2\times 10^{-13}$ MEG [90] |  $6\times 10^{-14}$ MEG II [91]
$\mu\to ee\bar{e}$ |  $1.0\times 10^{-12}$ SINDRUM [94] |  $10^{-16}$ Mu3e [95]
$\tau\to\mu\gamma$ |  $4.2\times 10^{-8}$ Belle [105] |  $6.9\times 10^{-9}$ Belle II [103, 104]
$\tau\to\mu\mu\bar{\mu}$ |  $1.9\times 10^{-8}$ Belle II [106] |  $3.6\times 10^{-10}$ Belle II [103, 104]
$\tau\to\mu e\bar{e}$ |  $1.8\times 10^{-8}$ Belle [107] |  $2.9\times 10^{-10}$ Belle II [103, 104]
$\tau\to e\gamma$ |  $3.3\times 10^{-8}$ BaBar [108] |  $9.0\times 10^{-9}$ Belle II [103, 104]
$\tau\to ee\bar{e}$ |  $2.7\times 10^{-8}$ Belle [107] |  $4.7\times 10^{-10}$ Belle II [103, 104]
$\tau\to e\mu\bar{\mu}$ |  $2.7\times 10^{-8}$ Belle [107] |  $4.5\times 10^{-10}$ Belle II [103, 104]
Table 8: Present and expected future 90% CL limits on branching ratios of LFV
decays. Notice that if $\ell=\ell^{-}$ then $\bar{\ell}=\ell^{+}$ and vice
versa. For the modes involving invisible ALPs, we refer to discussions and
references in Section 4.

The LFV decays giving the most stringent constraints on the paramter space
that we are interested in include $\ell_{i}\to\ell_{j}\,a$,
$\ell_{i}\to\ell_{j}\gamma$, $\ell_{i}\to\ell_{j}\ell_{k}\ell_{k}$. The
present and future experimental limits for these processes are collected in
Table 8.

If both LFV and LFC ALP interactions are present, they can induce the
radiative LFV decays $\ell_{i}\to\ell_{j}\gamma$ at one loop. The resulting
decay width reads [27]:

$\displaystyle\Gamma(\ell_{i}\to\ell_{j}\gamma)=\frac{\alpha\,m_{\ell_{i}}}{2}\bigg{(}|F_{2}(x_{i})|^{2}+|G_{2}(x_{i})|^{2}\bigg{)},$
(13)

where $x_{i}\equiv m_{a}^{2}/m_{\ell_{i}}^{2}$ and

$\displaystyle
F_{2}(x_{i})=-\frac{m_{\ell_{i}}C_{\ell_{i}\ell_{j}}^{A}}{64\pi^{2}f_{a}^{2}}\bigg{[}m_{\ell_{i}}C_{\ell_{i}\ell_{i}}^{A}g_{1}(x_{i})+m_{\ell_{j}}C_{\ell_{j}\ell_{j}}^{A}g_{2}(x_{i})\bigg{]},$
(14) $\displaystyle
G_{2}(x_{i})=-\frac{m_{\ell_{i}}C_{\ell_{i}\ell_{j}}^{V}}{64\pi^{2}f_{a}^{2}}\bigg{[}m_{\ell_{i}}C_{\ell_{i}\ell_{i}}^{A}g_{1}(x_{i})-m_{\ell_{j}}C_{\ell_{j}\ell_{j}}^{A}g_{2}(x_{i})\bigg{]}.$
(15)

The loop functions $g_{1,2}$ are listed in Appendix B.4.

If kinematically allowed, LFV ALP couplings alone induce lepton decays into
ALPs, whose decay width reads in the limit $m_{\ell_{j}}\ll m_{\ell_{i}}$ [30,
27]:

$\displaystyle\Gamma(\ell_{i}\to\ell_{j}a)=\frac{1}{64\pi}\frac{m_{\ell_{i}}^{3}}{f_{a}^{2}}\left(|C_{\ell_{i}\ell_{j}}^{V}|^{2}+|C_{\ell_{i}\ell_{j}}^{A}|^{2}\right)\left(1-\frac{m_{a}^{2}}{m_{\ell_{i}}^{2}}\right)^{2}.$
(16)

If LFC couplings exist, the 3-body decays
$\ell_{i}\to\ell_{j}\ell_{k}\ell_{k}$ are also induced. When
$2\,m_{\ell_{k}}<m_{a}<m_{\ell_{i}}-m_{\ell_{j}}$, on-shell ALP decays give
the dominant contribution to these processes and one gets in the narrow width
approximation:

$\displaystyle\Gamma(\ell_{i}\to\ell_{j}\ell_{k}\ell_{k})=\Gamma(\ell_{i}\to\ell_{j}a)\,\text{BR}(a\to\ell_{k}\ell_{k}),$
(17)

where the width process $a\to\ell_{k}\ell_{k}$ is given in Eq. (5). In order
to estimate the fraction of ALPs decaying inside the detector, thus giving
rise to the 3-body signal, we follow Ref. [27] and multiply the above
expression by a factor $\bigg{(}1-\exp(-\frac{L}{l_{a}})\bigg{)}$, where
$L\sim 1$ m is the typical radius of the detector in LFV experiments and
$l_{a}$ is the mean flight length of the ALP, which is given by

$\displaystyle l_{a}=\frac{p_{a}}{m_{a}\Gamma_{a}}\hbar c\,,\quad\quad
p_{a}=\frac{\lambda^{\frac{1}{2}}(m_{\ell_{i}}^{2},m_{\ell_{j}}^{2},m_{a}^{2})}{2m_{\ell_{i}}},$
(18)

with $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+yz+zx)$.

If $m_{a}>m_{\ell_{i}}-m_{\ell_{j}}$, the 3-body decays can be still mediated
by an off-shell ALP mediated. We neglect the loop-suppressed ALP-photon
coupling induced by the LFC interactions and use the following expression
[27]:

$\displaystyle\Gamma(\ell_{i}\to\ell_{j}a^{*}\to\ell_{j}\ell_{k}\ell_{k})=\frac{|C_{\ell_{k}\ell_{k}}^{A}|^{2}\left(|C_{\ell_{i}\ell_{k}}^{V}|^{2}+|C_{\ell_{i}\ell_{k}}^{A}|^{2}\right)}{512\pi^{3}}\frac{m_{\ell_{i}}^{3}m_{\ell_{k}}^{2}}{f_{a}^{4}}\phi_{0}^{jk}(x_{i}),$
(19)

where $x_{i}\equiv m_{a}^{2}/m_{\ell_{i}}^{2}$ and

$\displaystyle\phi^{jj}_{0}(x)=$
$\displaystyle-\frac{11}{4}+4x-\left(\frac{1}{2}x^{2}\log\left(\frac{2x-1}{x}\right)-1+5x-4x^{2}\right)\log(\frac{x-1}{x})$
$\displaystyle+\frac{1}{2}x^{2}\left(\text{Li}_{2}\left(\frac{x-1}{2x-1}\right)-\text{Li}_{2}\left(\frac{x}{2x-1}\right)\right)\,$
(20) $\displaystyle\phi^{j\neq k}_{0}(x)$
$\displaystyle=\left(3x^{2}-4x+1\right)\log\left(\frac{x-1}{x}\right)+3x-\frac{5}{2}\,.$
(21)

In addition, the same loops inducing $\ell_{i}\to\ell_{j}\gamma$ also
contribute to $\ell_{i}\to\ell_{j}\ell_{k}\ell_{k}$ through a virtual photon
exchange [27, 109]:

$\displaystyle\Gamma(\ell_{i}\to\ell_{j}\gamma^{*}\to\ell_{j}\ell_{k}\ell_{k})\simeq\frac{\alpha}{3\pi}\left[\log\left(\frac{m_{\ell_{i}}}{m_{\ell_{j}}}\right)^{2}-3+\frac{\delta_{jk}}{4}\right]\,\Gamma(\ell_{i}\to\ell_{j}\gamma)\,.$
(22)

###  B.3 Anomalous magnetic moments

The ALP-induced contribution $\Delta a_{\ell}$ to the anomalous magnetic
dipole moments of the leptons — defined as $a_{\ell}=(g_{\ell}-2)/2$ — can
have two parts, depending on whether the LFC couplings are present or not [36,
27]. The contribution from the LFC couplings is given by

$\displaystyle\Delta
a_{\ell_{i}}^{\text{LFC}}=-\frac{m_{\ell_{i}}^{2}}{16\pi^{2}f_{a}^{2}}\left[(C_{\ell_{i}\ell_{i}}^{A})^{2}h_{1}(x_{i})+16\pi\alpha\,C_{\ell_{i}\ell_{i}}^{A}C_{\gamma\gamma}^{\text{loop}}\left(\log(\frac{4f_{a}^{2}}{m_{\ell_{i}}^{2}})-h_{2}(x_{i})\right)\right],$
(23)

where the loop functions $h_{1,2}$ are given in Appendix B.4 and
$x_{i}={m_{a}^{2}}/{m_{\ell_{i}}^{2}}$. The LFC couplings also induce the
$a\gamma\gamma$ coupling at one loop — see Appendix A — such that
$C_{\gamma\gamma}^{\text{loop}}$ reads

$\displaystyle
C_{\gamma\gamma}^{\text{loop}}=\frac{1}{8\pi^{2}}\sum_{i}C_{\ell_{i}\ell_{i}}^{A}B\left(\frac{4}{x_{i}}-i\epsilon\right)\,,$
(24)

where the loop function $B$ is given in Eq. (11).

The LFV contribution is given by

$\displaystyle\Delta
a_{\ell_{i}}^{\text{LFV}}=\frac{m_{\ell_{i}}m_{\ell_{k}}}{32\pi^{2}f_{a}^{2}}\left\\{\left[(C_{\ell_{i}\ell_{k}}^{V})^{2}+(C_{\ell_{i}\ell_{k}}^{A})^{2}\right]\left[\frac{m_{\ell_{i}}^{3}}{m_{\ell_{k}}^{3}}\int_{0}^{1}\differential
x\frac{x(1-x)^{2}}{\Delta_{i\to
i}}+\frac{m_{\ell_{i}}}{m_{\ell_{k}}}\int_{0}^{1}\differential
x\frac{(1-x)^{2}(x-2)}{\Delta_{i\to i}}\right]\right.$
$\displaystyle\left.\left[(C_{\ell_{i}\ell_{k}}^{V})^{2}-(C_{\ell_{i}\ell_{k}}^{A})^{2}\right]\left[\int_{0}^{1}\differential
x\frac{(1-x)^{2}}{\Delta_{i\to
i}}+\frac{m_{\ell_{i}}^{2}}{m_{\ell_{k}}^{2}}\int_{0}^{1}\differential
x\frac{(1-x)^{2}(1-2x)}{\Delta_{i\to i}}\right]\right\\}\,,$ (25)

where $\Delta_{i\to
i}=xx_{k}+x(x-1){m_{\ell_{i}}^{2}}/{m_{\ell_{k}}^{2}}+1-x$. In this paper, we
focus on $V\pm A$ interactions, hence the second line vanishes. Then, in the
limit $m_{\ell_{i}}\ll m_{\ell_{k}}$,

$\displaystyle\Delta
a_{\ell_{i}}^{\text{LFV}}=\frac{m_{\ell_{i}}^{2}[(C_{\ell_{i}\ell_{k}}^{V})^{2}+(C_{\ell_{i}\ell_{k}}^{A})^{2}]}{32\pi^{2}f_{a}^{2}}\frac{6(1-2x_{k})x_{k}^{2}\log(x_{k})+(x_{k}-1)((20x_{k}-19)x_{k}+5)}{6(x_{k}-1)^{4}}\,,$
(26)

while in the limit $m_{\ell_{i}}\gg m_{\ell_{k}}$

$\displaystyle\Delta
a_{\ell_{i}}^{\text{LFV}}=\frac{m_{\ell_{i}}^{2}[(C_{\ell_{i}\ell_{k}}^{V})^{2}+(C_{\ell_{i}\ell_{k}}^{A})^{2}]}{32\pi^{2}f_{a}^{2}}\left(x_{i}^{2}\log(\frac{x_{i}}{x_{i}-1})-x_{i}-\frac{1}{2}\right)\,.$
(27)

(a) LFV ($e$-$\mu$) couplings only (b) LFV ($e$-$\mu$) and LFC couplings (c)
LFV ($e$-$\mu$) couplings only (d) LFV ($e$-$\mu$) and LFC couplings

Figure 10: Electron (first row) and muon (second row) $g-2$ constraints on the
same parameter space regions depicted in Figure 4. Purple areas correspond to
ALP contributions $\Delta a_{\ell}$ beyond the conservative $2\sigma$ ranges
discussed in the text. In the green regions, the ALP can explain the
discrepancy between the measured value of $a_{\mu}$ and the SM prediction of
the WP [102] at the $2\sigma$ level.

As mentioned in Section 4, the SM predictions of both electron and muon $g-2$
are currently under scrutiny. In order for $a_{e}$ to test NP effects,
theoretical calculations of the SM contributions must be supplemented by a
very precise measurement of the fine-structure constant $\alpha$ (that is,
with an accuracy of $\approx 0.1$ parts per billion) [110]. In recent years,
this has become possible through atomic matter-wave interferometry. However,
the two most precise measurements to date, which respectively employed cesium
(Cs) [111] and rubidium (Rb) [100] atoms, are in strong mutual tension.
Depending on which result is adopted, the comparison between SM prediction and
the experimental measurements [112, 113] yields

$\displaystyle\Delta a_{e}^{\text{Cs}}$ $\displaystyle~{}\equiv
a_{e}^{\text{Cs}}-a_{e}^{\text{exp}}=-(8.8\pm 3.6)\times 10^{-13}\,,$ (28)
$\displaystyle\Delta a_{e}^{\text{Rb}}$ $\displaystyle~{}\equiv
a_{e}^{\text{Rb}}-a_{e}^{\text{exp}}=(4.8\pm 3.0)\times 10^{-13}\,.$ (29)

As we can see, the Cs-based prediction is in mild tension with the
experimental measurement, while the Rb one is in good agreement. Therefore, we
impose throughout the paper that the ALP contributions to $a_{e}$ do not
exceed the $2\sigma$ range of Eq. (29).

As is well known, the SM prediction of $a_{\mu}$ is affected by a certain
tension between the lattice QCD calculation of the the leading hadronic
contribution performed by the BMW collaboration [99] and that based on
dispersion relations and data of hadron production in low-energy $e^{+}e^{-}$
collisions, as summarised by the 2020 white paper (WP) [102]. The two
calculations have comparable precision and, once compared with the latest
measurement of the Muon g-2 collaboration [101], give

$\displaystyle\Delta a_{\mu}^{\text{WP}}$ $\displaystyle~{}\equiv
a_{\mu}^{\text{WP}}-a_{\mu}^{\text{exp}}=(2.49\pm 0.48)\times 10^{-9}\,,$ (30)
$\displaystyle\Delta a_{\mu}^{\text{BMW}}$ $\displaystyle~{}\equiv
a_{\mu}^{\text{BMW}}-a_{\mu}^{\text{exp}}=(1.05\pm 0.61)\times 10^{-9}\,.$
(31)

The first range would exclude most of the parameter space of our ALP models,
especially the regions predicting negative values of $\Delta a_{\mu}$.
However, it does exclude the SM itself at the $5\sigma$ level. Since it is not
possible at this stage to claim that a NP contribution of this size is indeed
necessary, we again adopt a conservative approach and use Eq. (31) as a
constraint in our analysis. As one can see, this can exclude both positive and
negative ALP contributions $\Delta a_{\mu}$ if too large in absolute value.

The impact of the $g-2$ constraints on the regions of the parameter space
studied in Section 4.1 is shown in Figure 10. The purple regions are outside
the $2\sigma$ ranges of Eqs. (29) and (31), while the green stripes correspond
to values of $\Delta a_{\mu}$ _within_ the $2\sigma$ range preferred by Eq.
(30). We can see that, if the ALP only couples to $e$-$\mu$ (left plots),
neither $\Delta a_{e}$ nor $\Delta a_{\mu}$ exclude portions of our parameter
space beyond the limits from searches for $\text{Mu}-\overline{\text{Mu}}$
oscillations (blue region). Notice that here we plot the rather special case
$C^{A}_{e\mu}=C^{V}_{e\mu}$, that is, a model with ALPs only coupling with
right-handed leptons, for which the second line of Eq. (25) vanishes. However,
we checked that the above conclusion is barely affected even if left-handed
couplings are also present (e.g. setting $C^{V}_{e\mu}=0$, $C^{A}_{e\mu}\neq
0$), consistently with the results of Ref. [29].

If the ALP also interacts with LFC currents (right plots), we see that the
$g-2$ constraints become stronger and there is the possibility of changes of
sign and multiple cancellations between the contribution in Eq. (25) and the
two terms of Eq. (23). Still, this does not imply additional constraints
beyond the limit from $\mu\to e\gamma$ (yellow area) on the region that can be
tested by $\mu$TRISTAN. As we can see, this is not the case for ALPs that are
too heavy to be produced at $\mu$TRISTAN, for which the leptonic $g-2$, in
particular $\Delta a_{\mu}$, can provide the most important bounds on the ALP
couplings.

###  B.4 Loop functions

The expressions for the loop functions appearing in this section are the
following [27]:

$\displaystyle g_{1}(x)=\frac{(x-3)x^{2}\log
x}{x-1}-2x+1-2\sqrt{x-4}x^{\frac{3}{2}}\log\left(\frac{\sqrt{x-4}+\sqrt{x}}{2}\right),$
(32) $\displaystyle g_{2}(x)=1-2x+2(x-1)x\log\left(\frac{x}{x-1}\right),$ (33)
$\displaystyle h_{1}(x)=1+2x-(x-1)x\log
x+2(x-3)\sqrt{\frac{x}{x-4}}x\log\left(\frac{\sqrt{x-4}+\sqrt{x}}{2}\right),$
(34) $\displaystyle h_{2}(x)=1+\frac{1}{6}x^{2}\log
x-\frac{x}{3}-\frac{1}{3}(x+2)\sqrt{x(x-4)}\log\left(\frac{\sqrt{x-4}+\sqrt{x}}{2}\right).$
(35)

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