††thanks: ORCID: 0000-0003-4655-2866.††thanks: ORCID:
0000-0003-2653-5962.††thanks: ORCID: 0000-0003-4471-2336.

# Neutrino mass models at $\mu$TRISTAN

P. S. Bhupal Dev<EMAIL_ADDRESS>Department of Physics and McDonnell Center
for the Space Sciences, Washington University, St. Louis, Missouri 63130, USA
Julian Heeck<EMAIL_ADDRESS>Department of Physics, University of
Virginia, Charlottesville, Virginia 22904-4714, USA  Anil Thapa
<EMAIL_ADDRESS>Department of Physics, University of Virginia,
Charlottesville, Virginia 22904-4714, USA

###### Abstract

We study the prospects of probing neutrino mass models at the newly proposed
antimuon collider $\mu$TRISTAN, involving $\mu^{+}e^{-}$ scattering at
$\sqrt{s}=346\,\mathrm{GeV}$ and $\mu^{+}\mu^{+}$ scattering at
$\sqrt{s}=2\,\mathrm{TeV}$. We show that $\mu$TRISTAN is uniquely sensitive to
leptophilic neutral and doubly-charged scalars naturally occurring in various
neutrino mass models, such as Zee, Zee–Babu, cocktail, and type-II seesaw
models, over a wide range of mass and coupling values, well beyond the current
experimental constraints. It also allows for the possibility to correlate the
collider signals with neutrino mixing parameters and charged lepton flavor
violating observables.

##  I Introduction

The origin of neutrino mass and mixing remains one of the important open
questions in fundamental physics [1, 2]. It clearly requires the introduction
of new particles beyond the particle content of the Standard Model (SM).
Qualitatively, we can expect these new particles to induce novel experimental
signatures, such as lepton number violation (LNV) and charged lepton flavor
violation (LFV), which are either forbidden or highly suppressed in the SM.
Arguably, the cleanest method to identify the new particle(s) would be via
their direct production at a high-energy collider. By studying the subsequent
decays of these new particles to SM particles, preferably involving LNV and/or
LFV to reduce SM background, one might be able to pinpoint the underlying
neutrino mass model. A summary of existing collider constraints on various
neutrino mass models can be found in Refs. [3, 4]. Similarly, a summary of the
LFV constraints can be found in Refs. [5, 6].

All past and current high-energy colliders constructed so far [7] involve
electron or proton beams and are therefore particularly sensitive to new
particles that couple to electrons or quarks. An entirely new class of
couplings could be probed using muon colliders, originally proposed long ago
[8]. The main advantage is that leptons provide a much cleaner collision
environment than hadrons, and muon beams suffer less synchrotron radiation
loss than electron beams, thus making muon colliders capable of reaching
higher center-of-mass energies with a reasonable-size circular ring design [9,
10]. They have gained considerable attention in recent years [11, 12, 13, 14,
15], as novel muon cooling techniques are now available [16], and other
technical difficulties related to the muon lifetime and radiation seem
solvable [15], making muon colliders an increasingly realistic and desirable
option. Most work has been done in the context of future $\mu^{+}\mu^{-}$
colliders [17], which would mimic LEP [18] and could reach a center of mass
energy of 10 TeV or more.

Here, we will focus on a different experimental setup, $\mu$TRISTAN [19],
which is a proposed high-energy lepton collider using the ultra-cold antimuon
technology developed at J-PARC [20]. It can run in the $\mu^{+}e^{-}$ mode
with $\sqrt{s}=346$ GeV, and later, in the $\mu^{+}\mu^{+}$ mode [21] with
$\sqrt{s}=2$ TeV or higher. It can serve as a Higgs factory and do precision
physics [22]. Other new physics studies for the $\mu^{+}e^{-}$ and
$\mu^{+}\mu^{+}$ collider options can be found in Refs. [23, 24, 25] and [26,
27, 25, 28], respectively. As we will show in this article, the unique initial
states of $\mu$TRISTAN make it especially sensitive to neutrino mass models
involving leptophilic neutral and/or doubly-charged scalars, allowing for
direct production and study of these new scalars in regions of parameter space
otherwise untestable. We take examples from both tree- and loop-level neutrino
mass models. Specifically, we use the Zee model [29], Zee–Babu model [30, 31],
cocktail model [32], and type-II seesaw model [33, 34, 35, 36, 37] as concrete
examples, and we consider the cleanest final states (with the least SM
background), i.e., the LFV channels
$\mu^{+}e^{-}\to\ell_{\alpha}^{+}\ell_{\beta}^{-}$ and
$\mu^{+}\mu^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{+}$ mediated by the scalars,
as well as the associated production of scalars with a photon or $Z$
boson.111All models under consideration also generate LNV signatures, such as
$\mu^{+}\ell_{\alpha}^{\pm}\to W^{+}W^{\pm}$, but since these are typically
suppressed by a product of many couplings or even the neutrino mass, we will
focus on LFV processes. We show that $\mu$TRISTAN can provide unprecedented
sensitivity well beyond existing constraints and complementary to future low-
energy LFV searches.

The rest of this article is organized as follows: in Sec. II we briefly
describe the details of the $\mu$TRISTAN collider. In Sec. III we go through
several neutrino mass models (both radiative and tree-level), derive
$\mu$TRISTAN’s sensitivity and compare to other LFV observables, notably
lepton flavor violation. We conclude in Sec. IV.

##  II $\mu$TRISTAN

The ultra-cold antimuon technology developed for the muon anomalous magnetic
moment and electric dipole moment experiment at J-PARC [20] uses laser
ionization of muonium atoms to provide a low-emittance $\mu^{+}$ beam, which
can be re-accelerated to high energies [38]. Allowing a 1 TeV $\mu^{+}$ beam
to collide with a high-intensity $e^{-}$ beam at the TRISTAN (Transposable
Ring Intersecting Storage Accelerators in Nippon [39]) energy of 30 GeV in a
storage ring of the same size as TRISTAN (3 km circumference), one can realize
the $\mu^{+}e^{-}$ mode of $\mu$TRISTAN with a center-of-mass energy
$\sqrt{s}=346\,\mathrm{GeV}$.222A larger storage ring allows for higher-energy
collisions. One can reach $\sqrt{s}=775\,\mathrm{GeV}$ with 50 GeV electrons
and 3 TeV muons. Taking into account muon decay, the deliverable instantaneous
luminosity for a single detector at any collision point in the storage ring is
estimated as $4.6\times 10^{33}\ {\rm cm}^{-2}\ {\rm s}^{-1}$ [22], which
translates to an integrated luminosity of $100\ {\rm fb}^{-1}\ {\rm
year}^{-1}$.

Using the same 3 km storage ring and 1 TeV $\mu^{+}$ beams, one can also
consider a $\mu^{+}\mu^{+}$ collider [21] with $\sqrt{s}=2\,\mathrm{TeV}$ (or
6 TeV for the larger ring option). The beam intensity will be lower than in
the $\mu^{+}e^{-}$ mode due to both muons decaying in the storage ring. The
instantaneous luminosity is estimated as $5.7\times 10^{32}\ {\rm cm}^{-2}\
{\rm s}^{-1}$ [22], which translates to an integrated luminosity of $12~{}{\rm
fb}^{-1}~{}{\rm year}^{-1}$.

The precise luminosity numbers depend on various efficiencies for the muon
production, as well as the detailed designs of the muon accelerator and
storage ring. For instance, a higher luminosity is, in principle, achievable
with better focusing of the $e^{-}$ beam (compared to the $\mu^{+}$ beam
[20]), following the SuperKEKB design [40]. We will use the numbers given
above from Ref. [22] as realistic but conservative order-of-magnitude
estimates to work with. Assuming negligible SM background for the LFV signals
we study below, the above-mentioned luminosities correspond to a minimum
signal cross section of 0.09 (0.75) fb in the $\mu^{+}e^{-}$
($\mu^{+}\mu^{+}$) mode in order to achieve $3\sigma$ sensitivity with 1 year
runtime. To be conservative, we will use a signal cross section of 0.1 (1) fb
in the $\mu^{+}e^{-}$ ($\mu^{+}\mu^{+}$) mode to derive our sensitivity
limits. These limits can be easily scaled for a longer runtime. For instance,
10 years of runtime with 1 ab-1 integrated luminosity can achieve the same
level of sensitivity with a signal cross section ten times smaller, thus being
capable of probing a larger model parameter space than what is shown here.

Since the details of the $\mu$TRISTAN detector design and acceptance
efficiencies are currently unknown, we will only impose basic trigger-level
cuts on the transverse momenta and pseudorapidity of the outgoing leptons and
photons, i.e., the default MadGraph5 cuts $p_{T}^{\ell,\gamma}>10$ GeV and
$|\eta^{\ell,\gamma}|<2.5$ [41] while calculating the cross sections in the
$\mu^{+}\mu^{+}$ option. For the asymmetric beams in the $\mu^{+}e^{-}$
option, we only keep the trigger-level $p_{T}$ cuts and remove the $\eta$ cuts
because the final state particles are boosted in the $\mu^{+}$ direction; the
detector should be designed to cover the small-angle region from the beam
direction.

We will use unpolarized beams for both $\mu^{+}e^{-}$ and $\mu^{+}\mu^{+}$
modes to derive our sensitivity limits. Although the surface antimuons
produced by the $\pi^{+}$ decay are 100% polarized due to the $V-A$ nature of
the weak interaction, the final polarization of the antimuon beam depends on a
detailed understanding of the beam emittance under the applied magnetic field,
which in some cases can reduce the polarization down to 25% [22]. Similarly,
the beam polarization option for the $e^{-}$ beam is still under discussion
for the SuperKEKB upgrade [42]. Including realistic beam polarization effects
could modify our cross sections by a factor of few due to the chiral nature of
the scalar couplings.

##  III Neutrino mass models with leptophilic scalars

The leptonic initial states and clean environment at $\mu$TRISTAN provide an
unprecedented opportunity to directly probe heavy leptophilic particles with
possible LFV interactions. We will mainly focus on the leptophilic neutral and
doubly-charged scalars that arise in well-known neutrino mass models, both
tree-level and radiative, such as the Zee model [29], Zee–Babu model [30, 31],
cocktail model [32], and type-II seesaw model [33, 34, 35, 36, 37]. If
kinematically allowed, a neutral scalar $H$ with sizable LFV coupling $e\mu$
can be resonantly produced in $\mu^{+}e^{-}$ collisions either by itself or in
association with a photon or $Z$ boson, as shown in Fig. 1(a) and (b)
respectively, thus providing unparalleled sensitivity to the LFV scalar
sector. Even for $m_{H}>\sqrt{s}$, the dilepton channels
$\mu^{+}e^{-}\to\ell_{\alpha}^{+}\ell_{\beta}^{-}$ and
$\mu^{+}\mu^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{+}$, shown in Fig. 1(c) and
(d), respectively, are sensitive to the LFV couplings of $H$ and give rise to
a contact-interaction-type bound on the scalar parameter space. Similarly, a
doubly-charged scalar can be resonantly produced at a $\mu^{+}\mu^{+}$
collider, either by itself or in association with a photon or $Z$ boson (see
Fig. 3). The higher center-of-mass energy of the $\mu^{+}\mu^{+}$ option at
$\mu$TRISTAN allows us to probe doubly-charged scalars beyond the current LHC
constraints [43]. We only focus on the LFV final states, as they are free from
the SM background (modulo lepton misidentification, whose rate is negligible
at lepton colliders [44, 45]). Also, we do not consider processes involving
singly-charged scalars, as they necessarily involve neutrinos in the final
state, making it harder to separate our signal from the SM background.

###  III.1 Zee model

In the Zee model [29], the SM scalar sector with one Higgs doublet $H_{1}$ is
extended by adding a second Higgs doublet $H_{2}$ and an $SU(2)_{L}$-singlet
charged scalar $\eta^{+}$. The relevant Lagrangian terms are given by

$\displaystyle\mathcal{L}\supset\mu
H_{1}H_{2}\eta^{-}-f\bar{L}^{c}L\eta^{+}-\tilde{Y}\bar{\ell}L\tilde{H}_{1}-Y\bar{\ell}L\tilde{H}_{2}+\text{H.c.}\,,$
(1)

where the superscript $c$ stands for charge conjugate and $\tilde{H}_{a}\equiv
i\sigma_{2}H_{a}^{\star}$ ($a=1,2$, $\sigma_{2}$ is the second Pauli matrix).
We have suppressed the flavor and $SU(2)_{L}$ indices. Note that the Yukawa
coupling matrix $f$ is anti-symmetric in flavor space, while $Y$ is an
arbitrary complex coupling matrix. We go to the Higgs basis [46, 47], where
only $H_{1}$ acquires a vacuum expectation value, $\langle H_{1}\rangle\equiv
v/\sqrt{2}\simeq 174\,\mathrm{GeV}$, and the charged leptons obtain a diagonal
mass matrix $M_{\ell}=\tilde{Y}v/\sqrt{2}$. We work in the alignment limit
[48], as preferred by the LHC Higgs data [49], where the neutral scalars of
$H_{2}$ (the CP-even $H$ and the CP-odd $A$) do not mix with the neutral Higgs
contained in $H_{1}$ that can be identified as the SM Higgs boson. The $\mu$
term in the Lagrangian (1) will induce a mixing of $\eta^{+}$ with the charged
scalar contained in $H_{2}$ upon electroweak symmetry breaking; we denote the
mixing angle by $\phi$ and the two mass eigenstates by $h^{+}$ and $H^{+}$,
see Refs. [50, 51] for details.

The simultaneous presence of $f$, $Y$, and $\mu$ breaks lepton number by two
units and leads to a one-loop Majorana neutrino mass matrix

$\displaystyle M^{\nu}=\kappa\left(fM_{\ell}Y+Y^{T}M_{\ell}f^{T}\right),$ (2)

with prefactor $\kappa\equiv(16\pi^{2})^{-1}\sin
2\phi\log(m_{h^{+}}^{2}/m_{H^{+}}^{2})$. This matrix is manifestly symmetric
and can be diagonalized as usual via

$\displaystyle M^{\nu}=U\,\text{diag}(m_{1},m_{2},m_{3})\,U^{T}\,,$ (3)

where $U$ is the unitary Pontecorvo–Maki–Nakagawa–Sakata matrix and $m_{j}$
the neutrino masses. Through neutrino oscillations we have obtained
information about the mass splittings and the three mixing angles in $U$. The
overall neutrino mass scale, ordering, and CP phases are unknown, although
their ranges are partially restricted [52].

With the parametrization of Refs. [53, 54] we can express $Y$ in terms of
$M^{\nu}$ and $f$. The $\mu^{+}e^{-}$ run of $\mu$TRISTAN will be uniquely
sensitive to $Y_{e\mu}$ and $Y_{\mu e}$, see Fig. 1(a)-(c), so we investigate
$Y$ textures where one of these entries is non-vanishing, which is hardly a
restriction. The simultaneous presence of $Y_{e\mu}$ and $Y_{ee}$ (or
$Y_{\mu\mu}$) however would induce large LFV amplitudes, e.g. $\mu\to e\gamma$
and $\mu\to 3e$ [55, 56, 57, 58, 59], leaving little parameter space for
$\mu$TRISTAN to probe. To evade LFV constraints and simplify our analysis, we
will set as many $Y$ entries to zero as possible, leading to the four
benchmark textures

$\displaystyle Y_{A_{1}}$ $\displaystyle\propto\begin{pmatrix}0&1&0\\\
0&0&-\frac{2m_{e}}{m_{\mu}}\frac{M^{\nu}_{e\tau}}{M^{\nu}_{\mu\mu}}\\\
0&0&0\end{pmatrix}\sim\begin{pmatrix}0&1&0\\\ 0&0&0.0035\\\
0&0&0\end{pmatrix},$ (4) $\displaystyle Y_{B_{2}}$
$\displaystyle\propto\begin{pmatrix}0&1&0\\\
-\frac{m_{e}}{m_{\mu}}\frac{M^{\nu}_{ee}}{M^{\nu}_{\mu\mu}}&0&0\\\
0&0&0\end{pmatrix}\sim\begin{pmatrix}0&1&0\\\ 0.013&0&0\\\
0&0&0\end{pmatrix},$ (5) $\displaystyle Y_{B_{3}}$
$\displaystyle\propto\begin{pmatrix}0&0&1\\\
-\frac{m_{e}}{2m_{\mu}}\frac{M^{\nu}_{ee}}{M^{\nu}_{\mu\tau}}&0&0\\\
0&0&0\end{pmatrix}\sim\begin{pmatrix}0&0&1\\\ 0.0023&0&0\\\
0&0&0\end{pmatrix},$ (6) $\displaystyle Y_{B_{4}}$
$\displaystyle\propto\begin{pmatrix}0&1&0\\\ 0&0&0\\\
-\frac{m_{e}}{2m_{\tau}}\frac{M^{\nu}_{ee}}{M^{\nu}_{\mu\tau}}&0&0\end{pmatrix}\sim\begin{pmatrix}0&1&0\\\
0&0&0\\\ 0.00013&0&0\end{pmatrix}.$ (7)

All these $Y$ textures lead to viable two-zero textures in $M^{\nu}$ [60],
indicated by their common name as a subscript, following the nomenclature of
Ref. [61]. The $M^{\nu}$ two-zero textures predict the unknown parameters in
the neutrino sector, i.e., the lightest neutrino mass and the three phases. We
show in Tab. 1 the predictions for the sum of neutrinos masses $\sum_{j}m_{j}$
(testable via cosmology [62]), the effective mass parameter for neutrinoless
double beta decay $\langle m_{\beta\beta}\rangle=\sum_{i}U^{2}_{ei}m_{i}$
(testable in the next-generation experiments [63]), and the Dirac CP phase
(testable in neutrino oscillation experiments [64, 65]). Notice that the $\sum
m_{\nu}$ predictions of the $B$ textures are already in tension [66] with
limits from cosmology, $\sum m_{\nu}<0.12\,\mathrm{eV}$ [67],333 Even stronger
limits have been obtained in Refs. [68, 69], while mild indications of a
nonzero sum of neutrino masses (in tension with the stringent Planck limits)
was suggested in Ref. [70]. but perfectly in line with laboratory constraints
[71].

name | texture zeros | $\sum_{j}m_{j}/\mathrm{eV}$ | $\langle m_{\beta\beta}\rangle/\mathrm{eV}$ | $\delta_{\rm CP}/{}^{\circ}$
---|---|---|---|---
$A_{1}$ |  $M_{ee}$, $M_{\mu e}$ |  $0.062$–$0.071$ | $0$ |  $44$–$341$
$B_{2}$ |  $M_{\tau\tau}$, $M_{\mu e}$ | $>0.13$ | $>0.036$ |  $85$-$90$ $\wedge$ $270$-$275$
$B_{3}$ |  $M_{\mu\mu}$, $M_{\mu e}$ | $>0.16$ | $>0.047$ |  $87$-$90$ $\wedge$ $270$-$273$
$B_{4}$ |  $M_{e\tau}$, $M_{\tau\tau}$ | $>0.14$ | $>0.039$ |  $90$-$94$ $\wedge$ $266$-$270$
Table 1: Predictions for the sum of neutrino masses $\sum_{j}m_{j}$, the
effective $0\nu\beta\beta$ Majorana neutrino mass $\langle
m_{\beta\beta}\rangle$, and the Dirac CP phase $\delta_{\rm CP}$ from the
texture zeros employed in the Zee model, using the $3\sigma$ normal-ordering
ranges for the oscillation parameters from NuFit 5.2 [52].

The many zeros in these four $Y$ benchmarks ensure highly suppressed LFV.
Indeed, neither of them give rise to the most stringent LFV modes, $\mu\to
e\gamma$ and $\mu\to 3e$, despite the non-zero $e\mu$ entry in $Y$. However,
all cases induce muonium–antimuonium oscillation [72, 73, 74, 75] through
those $e\mu$ entries, which will turn out to be an important constraint. In
addition, all textures except for $Y_{B_{2}}$ also give rise to LFV tauon
decays. Furthermore, all textures contribute to $(g-2)_{\mu}$, although the
$2\sigma$-preferred region turns out to be already excluded by the muonium
constraint.

The overall scale of $Y$ is degenerate with $f$ and $\kappa$ from Eq. (2) and
can effectively be adjusted at will. The $e\mu$ entry of $Y$ is then a free
parameter, subject only to perturbative unitarity constraints. The second non-
zero entry of $Y$ is not free, however, but rather predicted by lepton masses
and neutrino mass matrix entries. The latter are essentially predicted due to
the two-zero textures in $M^{\nu}$, allowing us to predict the $Y$ entries, as
already shown above. For $A_{1}$, $B_{2}$, and $B_{4}$, we find a large $e\mu$
entry in $Y$ that drives the $H$ production at $\mu$TRISTAN, plus a suppressed
second $Y$ entry that induces LFV. For $B_{3}$, the $e\tau$ entry dominates
and $\mu$TRISTAN’s reach is severely limited by tau LFV. Notice that we are
focusing on such extreme textures just for the sake of illustration to
emphasize $\mu$TRISTAN’s complementarity to other experimental probes.

Figure 1: Relevant Feynman diagrams for the processes involving the neutral
scalar $H$ in the Zee model at $\mu$TRISTAN. Figure 2: $\mu$TRISTAN
sensitivity to the Zee model parameter space for various channels as shown in
Fig. 1. The shaded regions are excluded: Purple (pink) shaded from LEP (LHC)
dilepton data, green shaded from $(g-2)_{\mu}$, and gray shaded from muonium
oscillation. The future muonium (ILC) sensitivity is shown by the black
(purple) dashed line (curve). The solid black lines show the $\tau$ LFV
constraints for different $Y$ textures ($A_{1},B_{3},B_{4}$).

Assuming $H$ to be the lightest scalar, the textures $Y_{A_{1}}$, $Y_{B_{3}}$,
and $Y_{B_{4}}$ lead to $\tau^{-}\to\mu^{-}\mu^{\pm}e^{\mp}$, $\tau^{-}\to
e^{-}\mu^{\pm}e^{\mp}$, and $\tau^{-}\to e^{-}e^{\pm}\mu^{\mp}$, respectively,
which give limits of order
$|Y_{\tau\alpha}Y_{\beta\delta}|<(m_{H}/5\,\mathrm{TeV})^{2}$, as shown by the
solid black lines in Fig. 2. For all textures except $B_{3}$ these are very
suppressed by the small $Y_{\tau\alpha}$ entry. For those textures, as well as
for the $Y_{B_{2}}$ texture which does not give rise to tau (or muon) LFV
decay, the most important LFV process is the $|\Delta L_{\mu}|=|\Delta
L_{e}|=2$ conversion of muonium ($M=e^{-}\mu^{+}$) to antimuonium
($\bar{M}=e^{+}\mu^{-}$) [72, 73, 74, 75], which only requires the $Y_{e\mu}$
entry we are interested in for $\mu$TRISTAN. The conversion probability is
currently limited to $P(M\leftrightarrow\bar{M})<8.3\times 10^{-11}$ at $90\%$
C.L. by the MACS experiment at PSI [76], while a sensitivity at the level of
$\mathcal{O}(10^{-14})$ is expected in the future by the proposed MACE
experiment [77]. The current MACS limit sets stringent constraints on the
Yukawa couplings $Y_{e\mu}$ and $Y_{\mu e}$:

$\displaystyle|Y_{e\mu,\mu e}|<\frac{m_{H}}{0.85\,\mathrm{TeV}}\,.$ (8)

This is the most important limit for $\mu$TRISTAN, as shown in Fig. 2 by the
gray-shaded region (current) and black dotted line (future).

The muonium limit can be significantly weakened due to destructive
interference in the $M-\bar{M}$ amplitude [78] if we choose $m_{A}\simeq
m_{H}$, which renders even the future MACE projection insensitive to our
parameter space of interest. However, for $m_{H}\simeq m_{A}\ll m_{H^{+}}$, we
would generate large oblique parameters due to custodial symmetry breaking
[79, 80]; this puts an upper limit on the mass splitting between the neutral
and charged scalars in the Zee model [54, 78]. On the other hand, the
leptophilic charged scalars in this model are constrained from slepton
searches at the LHC because the slepton decay
$\tilde{\ell}^{+}\to\ell^{+}\tilde{\chi}^{0}$ mimicks a charged scalar decay
$H^{+}\to\ell^{+}\nu$ in the massless neutralino limit. The current LHC bound
is $m_{H^{+}}>425$ GeV at 90% CL [81] for ${\rm
BR}(H^{+}\to\mu^{+}\nu_{e})=1$. To evade the muonium bound while satisfying
the global electroweak precision constraint [82, 83], we then require
$m_{H}\simeq m_{A}\gtrsim 320\,\mathrm{GeV}$, making direct $H$ production in
$\mu$TRISTAN’s $\mu^{+}e^{-}$ mode difficult. To extend our analysis to
lighter $H$, we therefore assume the scalar hierarchy $m_{H}\ll m_{A}\simeq
m_{H^{+}}$, subject to the muonium constraint from Eq. (8).444Note that our
results are symmetric under $m_{H}\leftrightarrow m_{A}$; we simply choose $H$
to be the lighter one for concreteness. Moreover, to set the scale of neutrino
masses, we choose the $f$ couplings to be much smaller than $Y$ and can hence
neglect the $\eta^{\pm}$-mediated processes at $\mu$TRISTAN entirely.

Having established our benchmark scenarios and relevant LFV signatures, we can
study this region of the Zee-model parameter space at $\mu$TRISTAN. The
relevant Feynman diagrams and processes are shown in Fig. 1. Away from the
$s$-channel resonance at $\sqrt{s}\sim m_{H}$, the dilepton cross section
takes on the simple form

$\displaystyle\sigma(\mu^{+}e^{-}\to\mu^{-}e^{+})\simeq\frac{|Y_{e\mu}|^{4}}{64\pi
s}\begin{cases}1\,,&m_{H}\ll\sqrt{s}\,,\\\
\frac{s^{2}}{12m_{H}^{4}}\,,&m_{H}\gg\sqrt{s}\,.\end{cases}$ (9)

This was numerically verified in MadGraph5_aMC@NLO [41] using the general 2HDM
FeynRules model file [84]. The exact analytic expression for the cross section
is not very illuminating, and therefore, we do not show it here. We demand
this cross section to be of order $0.1\,\mathrm{fb}$ (after applying the cuts
specified in Sec. II) for a discovery, since this flavor-violating channel is
background-free. The textures $A_{1}$, $B_{2}$, and $B_{4}$ dominantly induce
this channel.555The $B_{3}$ texture dominantly gives $\mu^{+}e^{-}\to\tau e$.
We show the $\mu$TRISTAN reach of this process $\mu^{+}e^{-}\to\mu^{-}e^{+}$
in Fig. 2 (solid red curve), after applying the basic trigger cuts. We find
that the $\mu$TRISTAN sensitivity surpasses the current limit from muonium
conversion for $m_{H}>50\,\mathrm{GeV}$. The $B_{3}$ texture is the only one
that is already too constrained by tau LFV to give large
$\sigma(\mu^{+}e^{-}\to\ell^{+}_{\alpha}\ell^{-}_{\beta})$. Future muonium
data can cover almost the entire relevant parameter space for $\mu$TRISTAN’s
dilepton mode in the Zee model, offering confirmation potential in case of a
discovery.

In Fig. 2, we also show the existing collider constraints from LEP
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ data (purple shaded) [85, 86] and from LHC
$pp\to e\mu$ data (pink shaded) [87, 88].666As noted in Ref. [78], the
$3.8\sigma$ CMS excess in the $e\mu$ channel [88] can be explained by $H$
using lepton PDF, but only for $m_{H}\simeq m_{A}$ to avoid the muonium limit.
The future ILC sensitivity from $e^{+}e^{-}\to\mu^{+}\mu^{-}H$ is also shown
by the pink dashed curve [89, 90, 51] for comparison with the $\mu$TRISTAN
sensitivity. The green-shaded region is excluded by demanding the $H$
contribution to $(g-2)_{\mu}$ not to exceed $5\sigma$ deviation between the
world average of the SM prediction [91] and the experimental value
[92].777Taking the BMW result [93] instead of the world average [91] for the
SM prediction does not make much difference to our allowed parameter space,
which is dominated by the muonium limit.

For the associated production of $H$ with a photon or a $Z$ boson (cf. Fig.
1(b)), the cross sections for small $m_{H}\ll\sqrt{s}$ take the form

$\displaystyle\sigma(\mu^{+}e^{-}\to
H\gamma)\simeq\frac{\alpha_{\text{EM}}|Y_{e\mu}|^{2}}{8s}\,\log\left(\frac{s}{m_{e}m_{\mu}}\right),$
(10) $\displaystyle\sigma(\mu^{+}e^{-}\to
HZ)\simeq\frac{\alpha_{\text{EM}}|Y_{e\mu}|^{2}\,(s-m_{Z}^{2})}{32s_{w}^{2}c_{w}^{2}s^{2}}\left[\frac{s}{4m_{Z}^{2}}\right.$
(11)
$\displaystyle\left.-(1-2s_{w}^{2}+4s_{w}^{4})-(1-4s_{w}^{2}+8s_{w}^{4})\log\left(\frac{m_{H}m_{Z}}{s-m_{Z}^{2}}\right)\right],$

where $\alpha_{\rm EM}$ is the electromagnetic fine-structure constant, and
$s_{w}\equiv\sin\theta_{w}$ ($c_{w}\equiv\cos\theta_{w}$) is the (co)sine of
the weak mixing angle. These cross sections are typically larger than the
dilepton channel but are open only for $m_{H}\lesssim\sqrt{s}$ for the photon
case (or $\sqrt{s}-m_{Z}$ for the $Z$ case). The photon cross section exhibits
an infrared divergence for $\sqrt{s}\to m_{H}$ that is regulated by the cut
$p_{T}^{\gamma}>10\,\mathrm{GeV}$, reducing the total cross section compared
to the analytical expression above. The $Z$ cross section is well behaved near
the kinematic threshold but diverges for $m_{H}\to 0$, not of any concern for
us. As can be seen in Fig. 2, both modes are important for $\mu$TRISTAN and
cover parameter space that cannot be probed with other colliders or LFV.888The
only exception is texture $B_{3}$, for which only a tiny region survives the
tau LFV bound. The $H$ scalars subsequently decay promptly into
$\mu^{\pm}e^{\mp}$, half of which being background free even without any
momentum reconstruction.

The Zee model also makes predictions for $\mu$TRISTAN’s $\mu^{+}\mu^{+}$ mode,
as there are $t$-channel diagrams for
$\mu^{+}\mu^{+}\to\ell^{+}\ell^{\prime+}$ (cf. Fig. 1(d)). All textures except
$B_{3}$ induce the background free $\mu^{+}\mu^{+}\to e^{+}e^{+}$, with
testable allowed cross sections for $m_{H}>300\,\mathrm{GeV}$, as shown in
Fig. 2 by the brown curve. We find that the $H$ sensitivity in this channel is
worse than or comparable to the dilepton channel in the $\mu^{+}e^{-}$ mode,
so it can only be used as a secondary channel for verifying any signal found
in $\mu$TRISTAN’s first run.

Before we move on to other neutrino mass models, let us briefly comment on the
discrepancy in the muon magnetic moment [92]. While the status of the SM
prediction is currently unclear, it is worthwhile to entertain the possibility
that the discrepancy is real and a sign for new physics. The benchmark values
taken above are incapable of explaining $(g-2)_{\mu}$ due to LFV constraints.
A recent study [54] has shown that the Zee model is in principle able to
explain $(g-2)_{\mu}$, but this requires one of the following textures:

$\displaystyle Y=\begin{pmatrix}0&0&0\\\ 0&\times&\times\\\
0&\times&\times\end{pmatrix}\text{ or }\begin{pmatrix}\times&0&\times\\\
0&\times&0\\\ \times&0&\times\end{pmatrix}.$ (12)

The first (second) requires $M^{\nu}_{ee}=0$ ($M^{\nu}_{\mu\mu}=0$) and
effectively conserves electron (muon) number, which makes it obvious that muon
LFV is evaded, including muonium conversion. The first texture could only show
up in $\mu$TRISTAN’s $\mu^{+}\mu^{+}$ run via
$\mu^{+}\mu^{+}\to\mu^{+}\tau^{+}$ or $\tau^{+}\tau^{+}$; the second texture
can give $\mu^{+}e^{-}\to\mu^{+}\tau^{-}$ in $\mu$TRISTAN’s first run. A
dedicated study of this scenario will be postponed until the $(g-2)_{\mu}$
anomaly is clarified.

Overall, we see that $\mu$TRISTAN could probe the Zee model in regions of
parameter space that are inaccessible by other means. A exhaustive study of
the Zee model at $\mu$TRISTAN goes beyond the scope of this work but the
benchmarks discussed here indicate a very promising situation.

###  III.2 Zee–Babu model

In the Zee–Babu model [30, 31], we extend the SM by two $SU(2)_{L}$-singlet
scalars $h^{+}$ and $k^{++}$ with hypercharge $1$ and $2$, respectively, which
have the following couplings relevant for neutrino masses:

$\displaystyle-\mathcal{L}\supset
f\bar{L}^{c}Lh^{+}+g\bar{\ell}^{c}\ell\,k^{++}+\mu
h^{-}h^{-}k^{++}+\text{H.c.}$ (13)

The matrix $g$ ($f$) is symmetric (antisymmetric) in flavor space. Taken
together, these couplings break lepton number and generate a Majorana neutrino
mass matrix

$\displaystyle M^{\nu}\simeq
16\mu\,I(m_{h},m_{k})\,fM_{\ell}g^{*}M_{\ell}f\,,$ (14)

where $I(m_{h},m_{k})$ is a two-loop function [94, 95]. The antisymmetry of
$f$ leads to $\det M^{\nu}=0$ and thus predicts one massless neutrino.

Figure 3: Relevant Feynman diagrams for the doubly-charged scalars in the
Zee–Babu, cocktail, and triplet seesaw models.

Similar to the Zee model, we can make the overall scale of $g$ as large as we
want and compensate for that with a smaller $f$ matrix or $\mu$ coupling. For
simplicity we assume $h^{+}$ to be very heavy and the $f$ couplings to be
small, effectively decoupling $h^{+}$. This leaves us with the doubly charged
$k^{++}$ with coupling matrix $g$. At $\mu$TRISTAN’s $\mu^{+}\mu^{+}$ run,
this $k^{++}$ leads to dilepton and associated production signatures as long
as $g_{\mu\mu}\neq 0$, see Fig. 3(a)-(b). We show $\mu$TRISTAN’s reach and
competing constraints in Fig. 4, having computed the cross sections with
MadGraph5_aMC@NLO [41] using the model file given in Ref. [96].

Figure 4:  $\mu$TRISTAN sensitivity to the Zee–Babu and cocktail model
parameter space for various channels as shown in Fig. 3. The shaded purple
region is excluded from LHC dilepton data [43], the dashed purple line shows
the HL-LHC reach [96]. The diagonal non-solid lines indicate LFV constraints
on the coupling products $|g_{\mu\mu}g_{\alpha\beta}|$. For the Zee–Babu $g$
texture from Eq. (15), only $g_{\mu\mu}$ and $g_{\mu\tau}$ are relevant. For
the cocktail-model texture from Eq. (18), mainly $g_{\mu\mu}$ and $g_{e\tau}$
are relevant.

$\mu$TRISTAN can easily probe a large region of parameter space as long as
$g_{e\mu}$ is somewhat suppressed compared to $g_{\mu\mu}$ to evade the
$\mu\to e\gamma$ constraint. This is hardly a restriction and we can even find
$g$ textures that eliminate almost all LFV constraints, e.g.

$\displaystyle g^{*}$ $\displaystyle\propto\begin{pmatrix}0&0&0\\\
0&1&-\frac{m_{\mu}}{m_{\tau}}\,\frac{M^{\nu}_{\mu\tau}}{M^{\nu}_{\tau\tau}}\\\
0&-\frac{m_{\mu}}{m_{\tau}}\,\frac{M^{\nu}_{\mu\tau}}{M^{\nu}_{\tau\tau}}&\frac{m_{\mu}^{2}}{m_{\tau}^{2}}\,\frac{M^{\nu}_{\mu\mu}}{M^{\nu}_{\tau\tau}}\end{pmatrix}\sim\begin{pmatrix}0&0&0\\\
0&1&0.1\\\ 0&0.1&5\times 10^{-3}\end{pmatrix}.$ (15)

This structure does not lead to any $e$LFV. The only process we could worry
about is $\tau\to 3\mu$, which is however not particularly stringent and could
be further suppressed by tuning $|M^{\nu}_{\mu\tau}/M^{\nu}_{\tau\tau}|\ll 1$.
$\mu$TRISTAN has a large region of testable parameter space even without this
tuning. Notice that the dominant $g_{\mu\mu}$ entry here leads to the dominant
channels $\mu^{+}\mu^{+}\to\mu^{+}\mu^{+}$ and
$\mu^{+}\mu^{+}\to\gamma/Z\,(k^{++}\to\mu^{+}\mu^{+})$; these are not exactly
background free, even though invariant mass distributions and angular
observables can be used to isolate new-physics contributions. The subleading
channels $\mu^{+}\mu^{+}\to\mu^{+}\tau^{+}$ and
$\mu^{+}\mu^{+}\to\gamma/Z\,(k^{++}\to\mu^{+}\tau^{+})$ on the other hand are
smoking-gun observables.

The texture from Eq. (15) does not induce any interesting signatures in the
$\mu^{+}e^{-}$ run, but other textures might, see Fig. 3(c). For example, a
$\mu\mu$ and $ee$ entry in $g$ would give the very clean
$\mu^{+}e^{-}\to\mu^{-}e^{+}$ (in addition to $\mu^{+}\mu^{+}\to e^{+}e^{+}$),
allowed by current muonium-conversion constraints, as shown in Fig. 4.

We also show other relevant constraints in Fig. 4. The $(g-2)_{\mu}$ excluded
region is shown by the black shaded region on top left corner. The vertical
pink shaded region is the current LHC bound [43], and the vertical pink dashed
line is the future HL-LHC sensitivity [96]. Thus, we find that $\mu$TRISTAN
will probe a wide range of the Zee–Babu model parameter space well beyond the
HL-LHC sensitivity. Similar sensitivities are also achievable at a future
$\mu^{+}\mu^{-}$ collider [97].

###  III.3 Cocktail model

The cocktail model [32] is an SM extension by two $SU(2)_{L}$-singlet scalars
$h^{-}$ and $k^{++}$, as well as a second Higgs doublet $H_{2}$. The field
content is reminiscent of the Zee and Zee–Babu models, but here an extra
$\mathbb{Z}_{2}$ symmetry is imposed under which $h^{-}$ and $H_{2}$ are odd,
which leaves the following relevant terms in the Lagrangian:

$\displaystyle\begin{split}-\mathcal{L}&\supset
g\bar{\ell}^{c}\ell\,k^{++}+\mu
h^{-}h^{-}k^{++}+\kappa\tilde{H}_{2}^{\dagger}H_{1}h^{-}\\\
&\quad+\xi\tilde{H}_{2}^{\dagger}H_{1}h^{+}k^{--}+\frac{\lambda_{5}}{2}(H_{1}^{\dagger}H_{2})^{2}+\text{H.c.}\,,\end{split}$
(16)

where $g$ is once again a symmetric Yukawa matrix in flavor space. Lepton
number is broken explicitly if all the above couplings are non-zero. We assume
parameters in the scalar potential so that $\langle H_{2}\rangle=0$, leaving
the $\mathbb{Z}_{2}$ unbroken. In that case, Majorana neutrino masses arise at
three-loop level:

$\displaystyle
M^{\nu}\simeq\frac{F_{\text{cocktail}}}{(16\pi^{2})^{3}\,m_{k^{++}}}\,M_{\ell}gM_{\ell}\,,$
(17)

where $F_{\text{cocktail}}$ is a complicated dimensionless loop function that
depends on scalar masses and couplings [98, 99]. The three-loop suppression
factor and additional suppression by charged-lepton masses require large
entries in $g$ that are easily in the non-perturbative regime, even when all
scalar masses are close to their experimental limits and the scalar-potential
couplings as large as allowed by perturbative unitarity. To keep $g$
perturbative and evade stringent constraints from muon LFV, one is more or
less forced to consider the two-zero texture $A_{1}$ for $M^{\mu}$ [98, 99],
which then results in a $g$ matrix

$\displaystyle g$ $\displaystyle\propto\begin{pmatrix}0&0&1\\\
0&\frac{m_{e}m_{\tau}}{m_{\mu}^{2}}\,\frac{M^{\nu}_{\mu\mu}}{M^{\nu}_{e\tau}}&\frac{m_{e}}{m_{\mu}}\,\frac{M^{\nu}_{\mu\tau}}{M^{\nu}_{e\tau}}\\\
1&\frac{m_{e}}{m_{\mu}}\,\frac{M^{\nu}_{\mu\tau}}{M^{\nu}_{e\tau}}&\frac{m_{e}}{m_{\tau}}\,\frac{M^{\nu}_{\tau\tau}}{M^{\nu}_{e\tau}}\end{pmatrix}\sim\begin{pmatrix}0&0&1\\\
0&0.24&0.01\\\ 1&0.01&6\times 10^{-4}\end{pmatrix}$ (18)

and the neutrino-parameter predictions from the first row of Tab. 1. The
strongest LFV constraint mediated by $k^{++}$ then comes from $\tau^{-}\to
e^{+}\mu^{-}\mu^{-}$, requiring $|g_{e\tau}|<0.17\,m_{k^{++}}/\mathrm{TeV}$,
although, by coincidence, $\mu\to e\gamma$ gives essentially the same limit
for this texture.

The LFV constraints of this texture are severe enough that $\mu$TRISTAN in the
$\mu^{+}e^{-}$ mode would not observe the characteristic
$\mu^{+}e^{-}\to\tau^{+}\mu^{-}$, see Fig. 4. However, $\mu$TRISTAN in the
$\mu^{+}\mu^{+}$ run could potentially see $\mu^{+}\mu^{+}\to e^{+}\tau^{+}$
or $\mu^{+}\mu^{+}\to k^{++}\gamma/Z$ followed by prompt $k^{++}\to
e^{+}\tau^{+}$ decays.

Notice that the $\mathbb{Z}_{2}$ symmetry renders the lightest particle among
the $H_{2}$ and $h^{-}$ stable. We can choose scalar-potential parameters to
make this one of the neutral scalars inside $H_{2}$, which could then form
dark matter. We will not discuss this here since there is very limited
connection to $\mu$TRISTAN.

###  III.4 Type-II or triplet seesaw

In the type-II or triplet seesaw mechanism [33, 34, 35, 36, 37], we extend the
SM by an $SU(2)_{L}$-triplet with hypercharge $+2$, usually written as the
$SU(2)_{L}$ matrix

$\displaystyle\Delta=\begin{pmatrix}\Delta^{+}/\sqrt{2}&\Delta^{++}\\\
\Delta^{0}&-\Delta^{+}/\sqrt{2}\end{pmatrix}.$ (19)

This triplet couples to the left-handed lepton doublets $L_{e,\mu,\tau}$ and
the SM scalar doublet $H$, giving rise to the Lagrangian

$\displaystyle-\mathcal{L}\supset Y\bar{L}^{c}{\rm i}\sigma_{2}\Delta L+\mu
H^{\dagger}{\rm i}\sigma_{2}\Delta H^{*}+\text{H.c.}$ (20)

This Lagrangian breaks lepton number and induces a small vacuum expectation
value $\langle\Delta^{0}\rangle=v_{\Delta}/\sqrt{2}$, which in turn generates
the Majorana neutrino mass matrix $M^{\nu}=\sqrt{2}Yv_{\Delta}$. The Yukawa
couplings thus inherit the structure from the neutrino mass matrix but come
with an unknown scaling factor $v_{\Delta}$.

In the limit of $v_{\Delta}\ll v$, the mass eigenstates that dominantly come
from the triplet, $H^{++}\simeq\Delta^{++}$, $H^{+}\simeq\Delta^{+}$,
$H\simeq\sqrt{2}\text{Re}\,\Delta^{0}$, and
$A\simeq\sqrt{2}\text{Im}\,\Delta^{0}$, have mass splittings

$\displaystyle m_{H}^{2}\simeq m_{A}^{2}\simeq
m_{H^{+}}^{2}+\frac{\lambda_{4}v^{2}}{4}\simeq
m_{H^{++}}^{2}+\frac{\lambda_{4}v^{2}}{2}\,,$ (21)

specified exclusively by the coupling
$\lambda_{4}\,H^{\dagger}\Delta\Delta^{\dagger}H$ [100, 101]. For simplicity
we will assume an almost degenerate spectrum here, even though a mass
splitting could resolve [102, 103, 104] the recently observed discrepancy in
CDF’s $W$-boson mass measurement [105]. The large Yukawa couplings required to
produce $\Delta^{++}$ at $\mu$TRISTAN also lead to strong constraints from
searches at the LHC, which exclude masses below 1 TeV [43] and can be improved
at the HL-LHC [106].

Even more importantly, the triplet scalars induce LFV decays, for example
[107, 108, 109, 101]

$\displaystyle\text{BR}(\mu\to
e\gamma)\simeq\frac{\alpha_{\text{EM}}\left|(M^{\nu\,\dagger}M^{\nu})_{e\mu}\right|^{2}}{48\pi
G_{F}^{2}v_{\Delta}^{4}}\left(\frac{1}{m_{H^{+}}^{2}}+\frac{8}{m_{H^{++}}^{2}}\right)^{2},$
(22) $\displaystyle\text{BR}(\mu^{+}\to e^{+}e^{-}e^{+})\simeq
4\frac{\left|M^{\nu}_{ee}M^{\nu}_{\mu
e}\right|^{2}}{G_{F}^{2}v_{\Delta}^{4}m_{H^{++}}^{4}}\,,$ (23)

where $G_{F}$ is the Fermi coupling constant. $\mu\to e\gamma$ is particularly
important because the prefactor
$\left|(M^{\nu\,\dagger}M^{\nu})_{e\mu}\right|^{2}$ is completely specified by
the known neutrino oscillation parameters [110] and is limited from below by
$(0.016\,\mathrm{eV})^{4}$, using the $2\sigma$ range from NuFit 5.2 [52]. The
current limit $\text{BR}(\mu\to e\gamma)<4.2\times 10^{-13}$ [111] then gives
$m_{\Delta^{++}}>1.5\,\mathrm{TeV}(\mathrm{eV}/v_{\Delta})$. The $\mu\to
e\gamma$ limit can be improved by almost an order of magnitude with MEG-II
[112, 113] but will eventually be surpassed by muon-conversion in Mu2e [114,
115], which probes the same coupling in our case and effectively has a
sensitivity down to $\text{BR}(\mu\to e\gamma)<2\times 10^{-14}$. This would
improve the limit to
$m_{\Delta^{++}}>3\,\mathrm{TeV}(\mathrm{eV}/v_{\Delta})$.

Notice that the other LFV decays, notably $\mu\to 3e$ [116], could give even
stronger limits on $v_{\Delta}m_{\Delta^{++}}$, especially with the upcoming
Mu3e [117], but depend on the so-far unknown neutrino parameters such as the
lightest neutrino mass and the Majorana CP phases. These allow us, for
example, to set $M^{\nu}_{ee}=0$ and thus eliminate $\mu\to 3e$ entirely. For
simplicity we will therefore ignore these other LFV processes and only
consider the unavoidable $\mu\to e\gamma$.

Figure 5:  $\mu$TRISTAN sensitivity to the triplet/type-II seesaw model
parameter space for various channels as shown in Fig. 3. We have set
$M^{\nu}_{\mu\mu}=0.05\,\mathrm{eV}$ to fix the $\Delta^{++}\mu\mu$ coupling,
see text for details.

In Fig. 5, we show the LFV and LHC constraints together with the $\mu$TRISTAN
sensitivities in various channels. We have implemented the model file in
FeynRules [84] and computed the cross sections using MadGraph5_aMC@NLO [41].
To specify the production Yukawa coupling $Y_{\mu\mu}$ we set
$M^{\nu}_{\mu\mu}=0.05\,\mathrm{eV}$; this satisfies the cosmology bound $\sum
m_{\nu}<0.12\,\mathrm{eV}$ [67], otherwise we could go to larger
$M^{\nu}_{\mu\mu}$ values and increase the $\mu$TRISTAN cross sections without
changing the LFV bound.

The cross section $\sigma(\mu^{+}\mu^{+}\to\ell^{+}_{\alpha}\ell^{+}_{\beta})$
scales with $|M^{\nu}_{\mu\mu}|^{2}|M^{\nu}_{\alpha\beta}|^{2}$, at least away
from the resonance. The on-shell produced $\Delta^{++}$ has decay rates into
charged leptons proportional to $|M^{\nu}_{\alpha\beta}|^{2}$. Our current
lack of information about the lightest neutrino mass and the CP phases
preclude us from making definite predictions for these final states, but this
will improve with future neutrino data [118]. Generically, we expect final
states with more muons and tauons than electrons at $\mu$TRISTAN from
$\Delta^{++}$ processes for normal-ordered neutrino masses. Diboson decays
$\Delta^{++}\to W^{+}W^{+}$ are heavily suppressed by $v_{\Delta}$ in our
region of interest [119, 120]. Similarly, the cascade decays of $\Delta^{++}$
involving neutral or singly-charged scalars depend on the choice of mass
spectrum and can be ignored here.

Unlike for the doubly charged scalars in the Zee–Babu or cocktail models, the
$\Delta^{++}$ in the triplet model cannot generate clean
$\mu^{+}e^{-}\to\ell^{+}\ell^{\prime-}$ signatures in $\mu$TRISTAN’s first
run, since this region of parameter space is already excluded by $\mu\to
e\gamma$ (Fig. 5).

###  III.5 Other neutrino mass models

The $\mu^{+}\mu^{+}$ mode of $\mu$TRISTAN will also be uniquely sensitive to
the LNV/LFV signatures arising from other neutrino mass models. For instance,
the heavy neutral leptons appearing in type-I [121, 122, 123, 124, 125] and
type-III [126] seesaw models will induce a clean LNV signal $\mu^{+}\mu^{+}\to
W^{+}W^{+}\to{\rm jets}$, which is like an inverse neutrinoless double beta
decay [127] but in the muon sector [21]. This channel has been recently
analyzed in Refs. [27, 128], so we will not repeat this analysis here.
Similarly, the $\mu$TRISTAN sensitivities for the neutral and/or doubly-
charged scalars derived here can also be applied to other models, such as the
left–right symmetric model [129, 130, 131], and other radiative neutrino mass
models [58], although the connection to neutrino mass may not be as direct as
in the models studied here.

##  IV Conclusion

Neutrino masses provide the most convincing laboratory evidence for physics
beyond the SM, making searches for the underlying new particles highly
motivated. In this article, we have shown that $\mu^{+}e^{-}$ and
$\mu^{+}\mu^{+}$ colliders in the vein of the recently proposed $\mu$TRISTAN
experiment offer a new way to search for a variety of neutrino mass models. As
exemplified by several benchmark scenarios of the popular Zee, Zee–Babu,
cocktail, and triplet seesaw models, we showed that $\mu$TRISTAN could probe
regions of parameter space that are out of reach of other experiments, be it
future hadron colliders or future low-energy LFV searches.

## Acknowledgements

The work of BD is supported in part by the U.S. Department of Energy under
grant No. DE-SC 0017987 and by a URA VSP fellowship. The work of JH and AT was
supported in part by the National Science Foundation under Grant PHY-2210428.
For facilitating portions of this research, BD and AT wish to acknowledge the
Center for Theoretical Underground Physics and Related Areas (CETUP*), The
Institute for Underground Science at Sanford Underground Research Facility
(SURF), and the South Dakota Science and Technology Authority for hospitality
and financial support, as well as for providing a stimulating environment. BD
and JH would like to thank the Fermilab Theory Group for their hospitality
during the completion of this work.

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# Evaluating Deep Learning in SystemML using Layer-wise Adaptive Rate Scaling
(LARS) Optimizer

Kanchan Chowdhury
Arizona State University
Tempe, Arizona
<EMAIL_ADDRESS>Ankita Sharma
Arizona State University
Tempe, Arizona
<EMAIL_ADDRESS>Arun Deepak Chandrasekar
Arizona State University
Tempe, Arizona
<EMAIL_ADDRESS>
###### Abstract

Increasing the batch size of a deep learning model is a challenging task.
Although it might help in utilizing full available system memory during
training phase of a model, it results in significant loss of test accuracy
most often. LARS[11] solved this issue by introducing an adaptive learning
rate for each layer of a deep learning model. However, there are doubts on how
popular distributed machine learning systems such as SystemML[1] or MLlib[7]
will perform with this optimizer. In this work, we apply LARS optimizer to a
deep learning model implemented using SystemML.We perform experiments with
various batch sizes and compare the performance of LARS optimizer with
Stochastic Gradient Descent. Our experimental results show that LARS optimizer
performs significantly better than Stochastic Gradient Descent for large batch
sizes even with the distributed machine learning framework, SystemML.

##  1 Introduction

Applications of deep learning to solve problems such as classification,
regression and prediction have seen tremendous growth in this era because of
its success in solving these problems. The availability of large-scale
datasets and high-performance computing devices have also enlarged the scope
of deep learning algorithms. Although deep learning is helping us find
automated solutions to many real-world problems, the time required for
training a large deep learning model has always been a bottleneck. It is seen
that some deep learning models even take several days to finish. The time
required for training a deep learning model depends mainly on two things:
volume of the dataset and size of the model.

Nowadays, because of the availability of various sensor devices, data volume
is increasing rapidly. It is a common and also required practice to feed a
deep learning model with a huge volume of dataset. This immense size of
training data results in long training time which is a bottleneck of deep
learning models. Besides the size of the dataset, various giant technology
companies need to train deep learning models with a lot of layers and
parameters. For example, a popular deep learning model, ResNet-50[4] has
around 25.6 million parameters. This numerous number of parameters is another
reason for slow deep learning training.

As data size increases, deep learning models need to process a large set of
data in memory. Today, because of the improvement of computing devices,
feeding large data into memory is not a problem. Google and some other
companies have computing devices that are so powerful that even the largest
deep learning models cannot make full use of these computing devices.
Considering that these super-powerful computing devices are available, is it
possible to reduce the training time of the largest deep learning models
significantly without affecting the accuracy?

It is observed that an increase in the batch size results in a decrease in
total training time. Increasing the batch size up to a certain threshold is
also good for the accuracy of deep learning. Another important fact to note is
that increasing the batch size indefinitely reduces the model accuracy
significantly which prevents deep learning practitioners to keep the batch
size within reasonable limit although they might have powerful computing
devices to feed the data into memory. Besides increasing the batch size,
another possible solution to reduce the training time is parallelizing deep
learning. Training process of a deep learning model can be parallelized in two
ways: parallelizing data and parallelizing models. Machine learning systems
such as TensorFlow allow us to parallelize deep learning training. Besides, we
also have distributed systems such as Apache Spark[12]. Some other machine
learning libraries are also available on top of Apache Spark which include but
are not limited to SystemML[1], MLlib[7], etc. All these distributed systems
are helping us parallelize deep learning training to reduce training time. Is
it possible to further reduce the training time as well as increase the batch
size while maintaining the test accuracy of the model?

As we discussed earlier, training of a deep learning model can be parallelized
through either data parallelism or model parallelism. We cannot only rely on
data parallelism because the model size (number of parameters and layers) is
too big sometimes which makes model parallelism a mandatory option. Again,
model parallelism can be done in two ways: 1) parallelize within each layer
and 2) parallelize across different layers. Each of these approaches has
problems. The problem with parallelizing within each layer is that the wide
model is not efficient. A deep model can be better than a wide model. Also,
the problem with parallelizing across different layers is that parallel
efficiency is low which is 1/P. So, the solution is combining model
parallelism with data parallelism.

Scaling a machine learning model is a difficult task. Most often, it results
in the generalization problem. The generalization problem indicates we might
end up in high training accuracy while test accuracy is very low. A promising
solution can be auto-tuning the learning rate while scaling the batch size.
Another idea is using different learning rates for different layers. This idea
of layer-wise adapting the learning rate for increased batch size was first
introduced by LARS[11] for deep learning in systems such as TensorFlow. It
showed that batch size can be increased significantly even on the CPU by the
use of a layer-wise adaptive learning rate technique.

A standard value of learning rate is effective to train a model perfectly, but
it always results in a long training time. Again, a larger value learning rate
speeds up the training process, but it might result in under-fitting the
model. Learning rate decay is a proven solution to mitigate this problem. The
idea of learning rate decay is to start with an initial learning rate and to
change the learning rate after every epoch at a constant rate. The problem is
updating the learning rate at a constant rate after every epoch does not help
much which inspires the invention of algorithms such as LARS[11] and LAMB[10].
Inspired by the success of LARS and LAMB, our proposal is to apply these
algorithms to speed up machine learning on top distributed machine learning
systems such as SystemML. For this phase of our project, we focus only on LARS
optimizer instead of trying both LARS and LAMB optimizers.

We selected SystemML in this project for two reasons. Firstly, this
distributed machine system is not yet as optimized as TensorFlow or PyTorch
for deep learning. The LARS implementation is not available for this
framework. Our second reason for selecting SystemML is that it can be used on
top of Apache Spark. Although distributed deep learning is supported by
TensorFlow and PyTorch, Apache Spark seems to be very effective for inter-
cluster communication. This is because Apache Spark is mainly developed for
distributed and large scale systems. Spark RDDs are very fast and effective
for remote memory access. The summary of our contribution is outlined below:

  * •

Applying LARS optimizer to a deep learning model implemented using SystemML.

  * •

Comparing test accuracy of LARS optimizer with that of Stochastic Gradient
Descent.

  * •

Evaluating test and training accuracy by increasing batch size.

  * •

Reducing the generalization error using LARS optimizer.

  * •

Finding the challenges in applying LARS optimization technique to SystemML.

##  2 Related Work

The use of machine learning and deep learning-based algorithms has been
increasing for a long time for various applications. Various platforms like
Tensorflow, PyTorch, and Keras provide extensive functionality in order to
support data parallelism and model parallelism. While data parallelism has its
own disadvantage of not getting fit if the model is too large even if it can
support accelerating the training process. The model parallelism has been
proposed in various applications in order to solve this issue. Recently model
parallelism has been successfully implemented for relational databases[5].
This computation on an RDBMS has been implemented by applying machine
learning-based algorithms. They proposed that different parts of the model can
be stored in a set of tables and SQL queries can be used for computation
purposes. RDBMS has been studied widely for distributed computing for a long
time. The paper has also discussed challenges in involving RDBMS with ML/DL
algorithms. They have introduced multi-dimensional array-like indices for
database tables, the improved query optimizer. It uses SimSQL which scales to
large model size and can outperform TensorFlow. It shows that model
parallelism for RDBMS based ML system can be scaled well and perform better
than even Tensorflow.

The paper LARS[11] focuses on speeding up deep neural network training. Their
approach focuses on increasing batch size using Layer-wise Adaptive Rate
Scaling(LARS) for efficient use of massive resources. They train Alexnet and
Resnet-50 using the Imagenet-1k dataset while maintaining its state of the art
accuracy. However, they successfully increase the batch size to larger than
16k, thereby, reducing the training time of 100 epoch Alexnet from hours to 11
minutes and the 90-epoch ResNet-50 from hours to 20 minutes. Their system, for
90 epoch Resnet 50 returns an accuracy of 75.4% with a batch size of 32k but
reduces to 73.2% when it is 64k. Their system clearly shows the extent to
which large scale computers are capable of accelerating the training of a DNN
by using massive resources with standard Imagenet-1k.

In LAMB[10] they use a novel layerwise adaptive large batch optimization
technique called LAMB. While LARS works well for Resnet, it performs poorly
for attention models like BERT[3]. However, empirical results showed superior
results with LAMB across systems like BERT with very little hyperparameter
tuning. BERT could be trained with huge batch sizes of 32868 without
degradation in performance. Hence, the training time of BERT reduced from 3
days to 76 minutes.

##  3 Methodology

In this section, we define the architecture of our convolutional neural
network at first. After that, we discuss LARS[11] optimization technique in
details.

###  3.1 Model Architecture

Our CNN model consists of 2 convolution layers followed by 3 fully connected
layers including the classification layer. The first convolution layer has 6
filters of size 5 × 5 with zero padding. The second convolution layer has 16
filters of size 5×5 with zero padding. There are two MAX Pooling layers with
2x2 filter. Each pooling layer follows one convolution layer. The first and
second fully connected layers have dimensions 120 and 84 respectively, while
the third fully connected layer is of size 10. We apply ReLU activation
function for all layers except the last layer, which has softmax activation.
We apply cross-entropy loss function in the last layer. We don't apply any
dropout to any layer. The architectural structure of our convolutional neural
network is shown in Figure 1.

Figure 1: Architecture of our Convolutional Neural Network

###  3.2 LARS Optimizer

Standard Stochastic Gradient Descent (SGD)[9] uses same Learning Rate (LR) in
each layer. The problems are two-fold. Firstly, if the learning rate is small,
the convergence of training might take too much time. Sometimes, it might not
even converge. Secondly, if the learning rate is high, the training process
might end up with under-fitting of the model. In order to prevent under-
fitting, the initial phase of training should be highly sensitive to the
weight initialization and initial learning rate[11]. LARS[11] claims that the
L2-norm of weights and gradients varies significantly between weights and
biases, and between different layers. Training might become unstable if
learning rate is large compared to this ratio in some layers. In order to
solve this problem, there is a popular approach called learning rate warm-up.
This approach starts with a small learning rate which can be used for any
layer and slowly increases the learning rate.

Instead of maintaining a single global learning rate, LARS maintains separate
local learning rate for each layer. Equation 3.2 is used to calculate the
weight using the local and global learning rate.

$\Delta w_{t}^{l}=\gamma*\lambda^{l}*\nabla{L(w_{t})}$

In Equation 3.2, $\gamma$ is the global learning rate and $\lambda^{l}$ is the
layer-wise local learning rate. Local learning rate for each layer is defined
through a trust co-efficient which is much smaller than 1. Trust co-efficient
defines how much a layer can be trusted to change its weight during update.
Local learning rate can be calculated using Equation 3.2.

$\lambda^{l}=\eta*\frac{||w^{l}||}{||\nabla{L(w^{l})}||}$

In Equation 3.2, $\eta$ is the trust coefficient. In case of Stochastic
Gradient Descent, a term known as weight decay[8] can be used. Equation for
calculating local learning rate can also be extended to balance the learning
rate with weight decay according to Equation 3.2.

$\lambda^{l}=\eta*\frac{||w^{l}||}{||\nabla{L(w^{l})}||+\beta*||w^{l}||}$

In Equation 3.2, $\beta$ is the weight decay term. Local learning rate helps
to partially eliminate vanishing and exploding gradient problems[11].

##  4 Experimental Evaluation

In this section, we define evaluation strategy and report experimental
results. At first, we show the configuration of our machine. After that, we
state evaluation metric, evaluation dataset and parameter settings. Finally,
we graphically report experimental results.

###  4.1 System Configuration

We run our experiments in a Intel(R) Core(TM) i7-4790 CPU. The CPU frequency
is 3.6 GHz and has 8 cores. L1d, L1i, L2, and L3 caches are 32K, 32K, 256K and
8192K respectively. Operating system version is Ubuntu 18.04.3 LTS x86_64. We
use 4 parallel batches to investigate the performance in a parallel and
distributed setting.

###  4.2 Evaluation Metric and Parameter Setting

Dataset Used: MNIST Dataset[6]

Evaluation Metrics:

We used three different evaluation metrics to evaluate the performance of our
system which include Test Accuracy, Train Accuracy, and Generalization Error.

Generalization Error indicates the difference between training accuracy and
test accuracy. Generalization error is high when a model gains very high
training accuracy but low test accuracy. Normally, increasing batch size in
stochastic gradient descent results in large generalization error. We select
this evaluation metric because the goal of our system is to reduce the
generalization error.

Parameter Setting: The values of various parameters used for our CNN model
training is reported in table 4.2.

o — c — c — Parameters | Values
---|---
Initial Learning rate | 0.01
Learning rate Decay | 0.0001
Weight Decay | 0.0001
Momentum | 0.9
Trust Coefficient | 0.001
Number of Parallel Batches | 4
Table 1: Values of various parameters used during model training

###  4.3 Experimental Results

This subsection reports all experimental results in a graphical way. Figure 2
compares the test accuracy achieved by both SGD[9] and LARS[11] optimizers.
From the figure, it is clear that both approaches perform extremely good for
small batch sizes. Both optimizers maintain a test accuracy of around 99%
until batch size reaches 1024. After that, test accuracy starts decreasing
gradually although they maintain an accuracy of more than 90% until batch size
reaches 8000. The difference between performances of these two optimizers is
observable after that. The test accuracy of SGD optimizer starts dropping
significantly once batch size reaches 16000 while LARS optimizer maintains
much higher test accuracy compared to SGD. Test accuracy for SGD drops below
40% after 28000 batch size while LARS optimizer maintains more than 55% test
accuracy for similar batch size. Test accuracy for both optimizers drops
significantly once batch size reaches 32000 although LARS optimizer performs
better yet.

Figure 2: Test Accuracy of SGD and LARS Optimizers with Increasing Batch Size

Figure 3 compares the training accuracy achieved by both SGD and LARS. Similar
to test accuracy in Figure 2, train accuracy is also higher for both
approaches in case of small batch sizes. It is noticeable that train accuracy
for SGD is better than its test accuracy. Still, it cannot outperform LARS for
large batch sizes. Similar to test accuracy, train accuracy for SGD also
decreases significantly once batch size reaches 16000. LARS outperforms SGD
for both train accuracy and test accuracy if batch size is larger than or
equal to 16000.

Figure 3: Training Accuracy of SGD and LARS Optimizers with Increasing Batch
Size

Figure 3 compares the generalization error for both SGD and LARS optimizers.
As we stated earlier, generalization error is very important in our evaluation
because it indicates whether a model is consistent in train accuracy and test
accuracy. More generalization error indicates more inconsistency. That's why
lower generalization error is always desired. Figure 4 indicates that
generalization error for SGD is much higher than LARS optimizer. Although
generalization error for SGD is within a desired range for batch sizes smaller
than 8000, it increases significantly after that. It can be stated that
generalization error for LARS is much smaller than SGD for batch sizes larger
than 8000.

Figure 4: Generalization Error of SGD and LARS Optimizers with Increasing
Batch Size

##  5 Conclusion

The encouragement towards training deep learning models with growing amount of
data has been discussed thoroughly. Processing huge amount of data leads to so
many real-time issues. The inability to use large batch size to utilize full
system resources in highly configured environments is one of those issues
which we focus here. We compare test accuracy of SGD and LARS optimizers for
various batch sizes in a distributed deep learning setting using SystemML. For
the ease of our evaluation, we choose a simple convolutional neural network
model and MNIST[6] dataset for our experimental evaluation. Based on our
evaluation, we can conclude that LARS optimizer can be used to train with
large batch sizes with distributed machine learning systems such as SystemML.

##  6 Future Work and Challenges

We plan to extend this task by performing similar experiments with a large
deep learning model such ResNet[4]. So far, we used MNIST[6] dataset for our
experiments. We are also interested in evaluating our approach with a larger
dataset such as ImageNet dataset[2]. Since LAMB[10] optimizer proved to
perform excellent for even large models such as BERT[3], our another goal is
to evaluate the performance of LAMB optimizer with distributed machine
learning framework, SystemML.

We faced many challenges when finishing this task. Most of the challenges got
raised due to many bugs of our adopted machine learning framework, SystemML.
The latest release of this framework has several issues and raised errors once
we tried to utilize some of its features. Despite all of these issues with
SystemML, we hope to utilize this framework for our future experiments with
complex model and large dataset.

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