††institutetext: Instituut-Lorentz, Leiden University, Niels Bohrweg 2, 2333
CA Leiden, The Netherlands

# Probing leptogenesis at the future muon collider

Oleksii Mikulenko  Mariia Marinichenko<EMAIL_ADDRESS>
###### Abstract

The future muon collider has the potential to discover feebly interacting
particles in a wide range of masses above the electroweak scale. It is
particularly suitable to search for heavy neutral leptons (HNLs), as their
production cross section $\sigma\sim m_{W}^{-2}$ is not suppressed by the new
physics scale. We demonstrate that the muon collider, with the capacity to
observe up to $10^{5}$ events in the previously unexplored TeV mass range,
provides the means to measure the fraction of lepton number violating (LNV)
processes with precision at the level of a percent. This capability enables
elucidating the nature of HNLs, allowing us to differentiate between Majorana,
Dirac, and non-minimal scenarios featuring multiple degenerate HNLs. We link
the observed fraction of LNV processes to the parameters of the model with
three degenerate HNLs, which could be responsible for generating baryon
asymmetry in the Universe. Additionally, we present a simple estimate for the
number of signal events, as well as analyze the feasibility of vector boson
fusion processes in searches for HNLs.

##  1 Introduction

A few puzzles remained to be addressed by the Standard Model (SM) of particle
physics, such as the origin of neutrino masses and the origin of matter-
antimatter asymmetry in the Universe. A notable solution is to complete the SM
with the right-handed counterparts of the left-handed neutrinos - heavy
neutral leptons (HNL). With such a little adjustment, it is possible to
explain the current neutrino oscillation data through the seesaw mechanism
ParticleDataGroup:2020ssz ; Minkowski:1977sc ; Schechter:1980gr ;
Schechter:1981cv ; Gell-Mann:1979vob ; Yanagida:1979as ; Mohapatra:1979ia ,
while simultaneously providing the HNLs with a neutrino-like weak interaction,
suppressed by a small mixing angle $U_{\alpha}^{2}\ll 1$, $\alpha=e$, $\mu$,
$\tau$. At the same time, the new particles may produce the observed amount of
baryon asymmetry in the Universe Boyarsky:2009ix ; Asaka:2005an ;
Canetti:2012vf ; Shuve:2014zua ; Abada:2015rta ; Hernandez:2015wna ;
Drewes:2016gmt ; Hernandez:2016kel ; Hambye:2017elz ; Abada:2017ieq ;
Antusch:2017pkq ; Klaric:2020phc ; Klaric:2021cpi ; Drewes:2016jae ;
Canetti:2012kh . Both these goals may be achieved with GeV-TeV scale HNLs that
can be probed by the current and proposed future experiments Alekhin:2015byh ;
SHiP:2015vad ; Aberle:2839677 ; Baldini:2021hfw ; Alviggi:2839484 ;
CortinaGil:2839661 ; DUNE:2020ypp ; DUNE:2020fgq ; Batell:2020vqn ;
MATHUSLA:2019qpy ; Cerci:2021nlb ; FASER:2018bac ; SHiP:2020sos ;
SNDLHC:2022ihg ; Bauer:2019vqk ; Aielli:2019ivi ; Dercks:2018wum ;
Boyarsky:2022epg ; Chrzaszcz:2020emg ; Antusch:2016ejd .

The muon collider Delahaye:2019omf ; AlAli:2021let  is a possible way forward
to explore physics at the TeV scale, alternative to the proposed future $pp$
and $ee$ colliders: Future Circular Collider (FCC) Agapov:2022bhm ;
Benedikt:2022kan , Circular Electron-Positron collider (CEPC)
CEPCPhysicsStudyGroup:2022uwl , Compact Linear Collider (CLIC) Linssen:2012hp
, and International Linear Collider (ILC) Behnke:2013xla . Muons can be
accelerated to high energies up to $1-10\,\mathrm{TeV}$, equivalent to a
$10-100\,\mathrm{TeV}$ proton collider Delahaye:2019omf ; Costantini:2020stv .
At the same time, muons are fundamental particles and provide a clean
environment, allowing for a simple description in high-energy scatterings.
Finally, new physics in the lepton sector may be probed more efficiently in
processes with muons, as compared to hadron colliders.

We consider the two standard setups with the energies $\sqrt{s}=3$,
$10\,\mathrm{TeV}$ and integrated luminosities $\mathcal{L}=1$,
$10\,\mathrm{ab^{-1}}$, respectively. Throughout the paper, the muon beams are
assumed to be fully polarized, with muons having negative and antimuons having
positive helicity correspondingly. The detector apparatus is designed to cover
angles above $\theta_{\text{det}}=10^{\circ}$ relative to the beam axis.

In these setups, the muon collider may serve as a powerful instrument in
searching for feebly interacting HNLs in the region above the electroweak
scale. The reason behind it relies on the following considerations. The cross
section of new physics processes at the scale $M$ has the form

$\sigma_{M}\sim\frac{g^{4}}{16\pi M^{2}}=10^{2}\,\mathrm{fb}\times
g^{4}\left(\frac{10\,\mathrm{TeV}}{M}\right)^{2},$ (1)

where $g$ is the coupling constant of the considered interaction. The
luminosity for different runs is assumed to scale as $\mathcal{L}\propto s$ in
order to account for the suppression of the cross section at larger energies.
The corresponding number of events is

$\mathcal{L}\sigma\sim 10^{6}g^{4}\sim 10^{4}\,\mathrm{events}$ (2)

for $g^{2}\sim 0.1$ (electroweak coupling). However, this is not the case for
TeV-scale HNLs, which are produced mainly in the $t$ channel process, mediated
by the $W$-boson Antusch:2016ejd . The propagator that may carry a light-like
transferred momentum $q^{2}\to 0$ introduces a divergence in the differential
cross section, which is regularized by the mediator mass. Therefore, the cross
section avoids the suppression by the new physics scale and is enhanced to

$\sigma\sim
U_{\mu}^{2}\sigma_{\text{weak}},\qquad\sigma_{\text{weak}}\sim\frac{g^{4}}{16\pi
m_{W}^{2}}\sim 200\,\mathrm{pb}\gg\sigma_{m_{N}}$ (3)

with the expected number of produced HNLs being

$N_{\text{ev}}\sim
U_{\mu}^{2}\mathcal{L}\sigma_{\text{weak}}\sim\frac{U_{\mu}^{2}}{10^{-8}}\times\frac{\mathcal{L}}{10\,\mathrm{ab^{-1}}}\,\text{events}.$
(4)

The studies of the sensitivity to HNLs of an experiment at the muon collider
have recently been performed in Kwok:2023dck ; Li:2023tbx ; Mekala:2023diu ,
with the obtained sensitivity down to $U^{2}_{\mu}\sim 10^{-6}$. The potential
to explore HNLs above the electroweak scale with a high number of events
provides a unique opportunity to study the properties of the new particles.
The possibility to infer the underlying physics model of the HNLs, if
detected, and link it to the origin of neutrino masses and baryon asymmetry
has attracted significant attention Antusch:2017pkq ; Tastet:2019nqj ;
Tastet:2021vwp ; Tastet:2021ygq ; Drewes:2018gkc ; Bondarenko:2021cpc . In the
minimal scenario of leptogenesis with two quasi-degenerate HNLs, the range of
the mixing angles is beyond the expected sensitivity reach. However, the
existence of the third HNL may significantly alter the dynamics of the
production of matter-antimatter asymmetry, yielding a much larger parameter
space Abada:2018oly ; Drewes:2021nqr , which may be probed by an experiment
hosted at the muon collider, see Fig. 1.

In this paper, we discuss the possible implications of the muon collider to
reveal the underlying properties of the HNLs. Namely, we analyze the
possibility of probing the scenario of leptogenesis with 3 HNLs. In Sec. 2, we
complement the previous simulation-based result with simple analytical
estimates for the signal yield for HNLs. Additionally, we discuss caveats and
limitations associated with probing HNL couplings to $e$, $\tau$ lepton
flavors via vector boson fusion (VBF). In Sec. 3, we obtain the sensitivity to
observations of lepton number violation and the implications of such
measurement to a model of 3 HNLs. We discuss our findings in Sec. 4.

Figure 1: Constraints on heavy neutral leptons from Abdullahi:2022jlv , see
reference therein. The shaded areas are excluded regions by direct detection
and electroweak precision measurements (above) or seesaw limit (below). The
parameter space above $m_{N}>100\,\mathrm{GeV}$ is consistent with 3 HNL
leptogenesis Drewes:2021nqr  and is a target for searches at multi-TeV
colliders.

##  2 Analytic estimate of the sensitivity reach

In the framework of the type-I seesaw mechanism, the Lagrangian with $n$
right-handed neutrinos has the following form Abdullahi:2022jlv :

$\mathcal{L}=\mathcal{L}_{SM}+i\overline{N}_{I}\gamma^{\mu}\partial_{\mu}N_{I}-F_{\alpha
I}\overline{L}_{\alpha}N_{I}\tilde{H}-\frac{1}{2}M_{IJ}\overline{N}^{C}_{I}N_{J}+h.c.,$
(5)

where $N_{I}$, $I=\overline{1,n}$ are HNL states with Majorana mass matrix
$M_{IJ}$; $F_{\alpha I}$ are Yukawa couplings to active lepton flavors
$\alpha=e,\mu,\tau$; $L_{\alpha}=(\nu_{\alpha},l_{\alpha})^{T}$ is the lepton
left doublet; and $H$ is the Higgs doublet.

Due to the mixing with heavy sterile neutrinos, the active neutrino flavor
eigenstates become

$\nu_{\alpha}=U^{\text{PMNS}}_{\alpha j}\nu_{j}+\theta_{\alpha I}N_{I}^{C},$
(6)

where $\nu_{j}$ and $N_{I}$ are mass eigenstates, and $U^{\text{PMNS}}$ is the
Pontecorvo–Maki–Nakagawa–
Sakata matrix Pontecorvo:1957qd . The small parameters $\theta_{\alpha I}$
control the interaction strength of the HNLs with the $W$, $Z$, and $H$
bosons. Their explicit form is given by

$\theta_{\alpha I}=\frac{v}{\sqrt{2}}\sum_{I}F_{\alpha J}(M^{-1})_{IJ}$ (7)

with $v=246\,\mathrm{GeV}$ being the Higgs vacuum expectation value.

To characterize HNL interactions, it is convenient to define the mixing
angles:

$U_{\alpha I}^{2}=|\theta_{\alpha I}|^{2},\quad\quad
U^{2}_{\alpha}=\sum_{I}U_{\alpha I}^{2},\quad\quad
U^{2}_{I}=\sum_{\alpha}U_{\alpha I}^{2},\quad\quad U^{2}=\sum_{\alpha
I}U_{\alpha I}^{2}.$ (8)

At high-energy colliders, the detection of an HNL occurs by reconstructing a
fully visible decay $N\to lW\to lqq$, without a neutrino in the final state.
The total number of events can be estimated by

$N_{\text{ev}}=2\mathcal{L}\times\sum_{\alpha}U^{2}_{\alpha}\sigma_{N_{\alpha}}\times\text{Br}(N\to
W(qq)l)\times\epsilon_{\text{eff}}.$ (9)

Here, the factor $2$ takes into account the Majorana nature of the HNLs, and
$\sigma_{N_{\alpha}}$ represents the production cross section of a Dirac HNL-
particle (i.e., excluding charge-conjugated channels) with unit mixing angle
$U^{2}_{\beta}=\delta_{\alpha\beta}$. The detection efficiency
$\epsilon_{\text{eff}}$ is the probability of observing an HNL decay when it
occurs. The branching ratio of the HNL decay is

$\text{Br}(N\to lW\to lqq)=\text{Br}(N\to lW)\cdot\underbrace{\text{Br}(W\to
qq)}_{=0.676}.$ (10)

The expressions for the decay widths of right-handed neutrinos are presented
in Appendix A. In the TeV mass range, when $M_{N}$ significantly exceeds the
Higgs boson mass ($M_{N}\gg m_{H}$), the branching ratio of HNL decay into a
$W$ boson approaches $\text{Br}(N\to lW)\approx\frac{1}{2}$.

###  2.1 $\mu$-mixing

Figure 2:  Feynman diagram for $\mu^{-}\mu^{+}\to\nu N$ with $W$-boson
exchange in the $t$-channel.

$\sigma_{N_{\mu}}\approx\frac{g_{W}^{4}}{16\pi
m_{W}^{2}}\left(1-\frac{m_{N}^{2}}{s}\right)=\underbrace{\sigma_{\text{weak}}}_{\sim
200\,\mathrm{pb}}\left(1-\frac{m_{N}^{2}}{s}\right).$ (11)

The production in the collinear process comes at the cost that the produced
particles move at small angles with respect to the axis, which may suppress
the detection efficiency. Suppose we assume that the detector system covers
angles above $\theta_{\text{det}}$. In that case, the detection efficiency can
be estimated by the fraction of HNL whose decay products can deviate by angles
$\theta>\theta_{\text{det}}$ from the initial beam axis. We choose the
reference value $\theta_{\text{det}}=10^{\circ}$ (corresponding to the pseudo-
rapidity cut $\eta=2.44$). There are two limiting cases for the calculation of
efficiency:

  1. 1.

If HNLs are not too light, they are produced with small boosts, and their
decay products are not focused at the beam line. The typical opening angle
between the decay products for a boosted particle may be estimated as
$\theta_{\text{dec.}}\sim 1/\gamma_{N}$, where $\gamma_{N}$ is the gamma
factor of the HNL. The detection efficiency, i.e. the probability of emitting
decay products in the detector coverage is simply:

$\epsilon_{\text{eff}}\sim
1-\left(\frac{\theta_{\text{det}}}{\theta_{\text{dec.}}}\right)^{2}\sim
1-\theta_{\text{det}}^{2}\frac{4sm_{N}^{2}}{(s+m_{N}^{2})^{2}}.$ (12)

  2. 2.

If HNLs are sufficiently light, such that $1/\gamma_{N}<\theta_{\text{det}}$,
and the decay products do not deviate sufficiently from the initial direction
of the HNL motion. Therefore, it becomes necessary to account for the nonzero
angle of the HNLs itself. It can be easily computed from the angular
dependence of the cross section $\frac{d\sigma}{dt}\propto(m_{W}^{2}-t)^{-2}$
with the requirement of deflection by an angle larger than
$\theta_{\text{det}}$.

$\epsilon_{\text{eff}}\sim\frac{\sigma_{N\nu}[\theta>\theta_{\text{det}}]}{\sigma_{N\nu}}=\left[1+\frac{s}{m_{W}^{2}}\left(1-\frac{m_{N}^{2}}{s}\right)\sin^{2}\left(\frac{\theta_{\text{det}}}{2}\right)\right]^{-1}.$
(13)

It should be noted that this effect becomes relevant in the regime when
$m_{N}^{2}\ll s$, therefore, the detection efficiency approaches a constant
value independent of HNL mass. This approximation might be inaccurate for very
light HNLs, where the separation of the decay products and reconstruction of
the kinematics becomes more challenging. Hence, we restrict ourselves to
$m_{N}>200\,\mathrm{GeV}$.

In our discussion, we omit direct cuts on the $p_{T}$ of the decay products,
because in both scenarios it is limited by
$p_{T}\gtrsim\frac{\sqrt{s}}{2}\theta_{\text{det}}>100\,\mathrm{GeV}$.
Therefore, we approximate the detection efficiency with the two simple
formulas (12), (13), keeping the maximal of the two.

$\epsilon_{\text{eff}}(m_{N}|s,\theta_{\text{det}})=\text{max}\left[1-\theta^{2}_{\text{det}}\frac{4sm_{N}^{2}}{(s+m_{N}^{2})^{2}},\left(1+\theta^{2}_{\text{det}}\frac{s}{4m_{W}^{2}}\right)^{-1}\right].$
(14)

To verify our estimates, we made a toy Monte-Carlo generator of the decay of
an HNL into two massless particles. The HNL is produced at various angles
$\theta$ with the weights proportional to $\frac{d\sigma}{d\cos\theta}$. The
decay products are created in HNL’s reference frame and are boosted to the lab
frame. The fraction of decays where both decay products have
$\theta>\theta_{\text{det}}$ is shown in Fig. 3, together with the estimate
(14).

Figure 3: Comparison of the detection efficiency, computed with a simple
formula (14), and the results of a toy Monte-Carlo generator. The generator
simulates HNL production and decay into two massless particles, and estimates
the fraction of decay with both daughter particles moving into the detector.

To estimate the sensitivity of an experiment at the muon collider, it is
necessary to know the number of expected SM background events
$N_{\text{bkg}}$. Then, the exclusion region (at $1\sigma$ confidence level)
corresponds to the expected number of HNL events

$N_{\text{ev}}=\sqrt{N_{\text{bkg}}}.$ (15)

The number of background events lacks a simple analytical estimate, therefore
we extract it from the simulations performed in the previous studies. For this
purpose, we used the results of the previous simulation-based studies
Kwok:2023dck ; Li:2023tbx ; Mekala:2023diu . By fitting the background
spectrum from Kwok:2023dck  and choosing 100 GeV bins, equal to the size of a
peak for the reconstructed mass, we found that for a 3 TeV muon collider
$N_{\text{bkg}}$ can be approximated as

$N_{\text{bkg}}=4.0\cdot
10^{4}\left(\exp\bigg{[}-1.6\cdot\frac{m_{N}}{1\,\mathrm{TeV}}\bigg{]}+0.03\right).$
(16)

To validate our estimates, we compared (16) with the results from Li:2023tbx
and Kwok:2023dck . Our calculations indicate that there are roughly $\sim
10^{4}$ events in the 100 GeV bin.

For the 10 TeV collider, the background is dominated by
$\mu^{+}\mu^{-}\rightarrow qqlll\nu$ for $m_{N}\;\lesssim\;5$ TeV and
$\mu^{+}\mu^{-}\rightarrow qql\nu$ for $m_{N}\;\gtrsim\;5$ TeV. Additionally,
there is a contribution from $\mu^{+}\mu^{-}\rightarrow qqll$, uniform in the
invariant mass. To determine the background in this scenario, we used the
ratio between cross sections of those three processes (0.40 pb, 0.21 pb, and
0.46 pb correspondingly) to the total cross section from Li:2023tbx  assuming
a linear $N_{\text{bkg}}(m_{N})$ dependence. Our derived equation is

$N_{\text{bkg}}=1.5\cdot
10^{5}\>\left(1-0.07\cdot\frac{m_{N}}{1\,\mathrm{TeV}}\right).$ (17)

In addition, we performed a check with Kwok:2023dck  and Mekala:2023diu .
Here, we noticed a disagreement between the latter and previously mentioned
papers, where the background is 10 times larger. Unfortunately, we are unable
to explain this discrepancy.

Figure 4: The $N_{\text{ev}}$ map in the $(m_{N},U^{2})$ parameter space. The
black band represents the sensitivity reach with the use of the background
fits (16), (17), allowing them to vary by a factor of two. The gray area at
the level $\sim 10^{-3}$ is excluded by constraints from the electroweak
precision observables delAguila:2008pw ; Chrzaszcz:2019inj . The FCC
sensitivity is taken from Antusch:2016ejd . The current charge lepton flavor
violation constraints (cLFV) Alonso:2012ji ; Abada:2015oba ; deGouvea:2015euy
; Fernandez-Martinez:2016lgt ; Abada:2018nio ; Urquia-Calderon:2022ufc  may in
principle exclude mixing angles up to $10^{-4}$. However, they rely on the
simultaneous mixing of a HNL with electron and muon neutrinos. While not
directly probing $U^{2}_{\mu}$, these constraints are nevertheless relevant
for realistic HNL scenarios.

The sensitivity to HNLs obtained by the presented simple estimates is shown in
Fig. 4. Given the somewhat arbitrary estimates for the background, we plot the
sensitivity in the form of a band, allowing for variation in the background by
factor two. We note that the consistency of our estimates with the previous
results is at the same level as the (dis)agreement between them.

###  2.2 Feasibility of vector boson fusion for $e$, $\tau$-mixings

A natural process that may experience a collinear enhancement of the cross
section and produce HNL through mixing with $\nu_{e}$/$\nu_{\tau}$ is vector
boson fusion (VBF), see Fig. 5. This type of interaction has been identified
as a promising process for the muon collider to probe new physics
Costantini:2020stv ; Ruiz:2021tdt ; Li:2023lkl . In this section, we
demonstrate that this process does not provide an advantage compared to the
$s$-channel annihilation process into $N\nu$. Moreover, we highlight the
caveats of the effective $V$ approximation when considering HNL production.

Figure 5: Feynman diagrams of HNL production through vector boson fusion.

The relevant reactions for HNL production are $W^{-}W^{+}\to\nu N$,
$ZW^{\pm}\to l^{\pm}N$, $\gamma W^{\pm}\to l^{\pm}N$, and $ZZ\to\nu N$, shown
in Fig 5. Their cross sections can be estimated with the use of the effective
$V$ approximation. The two necessary ingredients of the computations are the
probability distribution $f_{V}(\xi,Q^{2})$ to emit a vector boson with energy
fraction $\xi\equiv\frac{E_{V}}{E_{\mu}}$ and virtuality $q^{2}=-Q^{2}<0$, and
the fusion cross section $\sigma(\hat{s},Q_{1}^{2},Q_{2}^{2})$, with the
center of mass energy $\hat{s}=\xi_{1}\xi_{2}s$. A detailed treatment of the
emission of $V$ may be found elsewhere, see e.g. Costantini:2020stv ;
Borel:2012by ; Kuss:1995yv ; Dawson:1984gx ; Kane:1984bb .

For our purposes, we capture the general behavior of the emission probability
by approximating

$f_{V}(\xi,Q^{2})\sim\frac{g_{V}^{2}}{16\pi^{2}}\frac{1}{\xi}\frac{1}{Q^{2}+m_{V}^{2}},$
(18)

which results in

$\sigma_{\text{VBF}}\sim\frac{g_{V}^{4}}{256\pi^{4}}\int\frac{d\xi_{1}}{\xi_{1}}\frac{d\xi_{2}}{\xi_{2}}\frac{dQ_{1}^{2}}{Q_{1}^{2}+m_{V}^{2}}\frac{dQ_{2}^{2}}{Q_{2}^{2}+m_{V}^{2}}\times\theta(\hat{s}-m_{N}^{2})\sigma(\hat{s},Q_{1}^{2},Q_{2}^{2})$
(19)

with $\theta$ being the Heaviside step-function.

The next step of the effective $V$ approximation is to replace the off-shell
cross section by its value for on-shell vector bosons
$\sigma(\hat{s},-m_{V}^{2},-m_{V}^{2})$. However, this step becomes invalid
for the specific process of HNL production. For instance, consider a process
$W^{-}W^{+}\to\nu N$. The total cross section for real $W$ bosons is divergent
because of the $t$-channel singularity. Namely, the range of variable $t$ in
the limit $m_{W}^{2}\ll m_{N}^{2},s$ is

$m_{W}^{2}-s\left(1-\frac{m_{N}^{2}}{s}\right)\leq t\leq
m_{W}^{2}\left[1-\frac{1}{4}\left(1-\frac{m_{N}^{2}}{s}\right)\right].$ (20)

Therefore, this variable crosses the zero value, i.e. contains an on-shell
lepton in the $t$-channel propagator, for HNL masses

$m^{2}_{N}<s-m_{W}^{2},$ (21)

i.e. the singularity appears for almost all HNL masses except for a narrow
region near the kinematic threshold $s\approx m_{N}^{2}$.

This type of singularity has been first noticed in the context of hadron
collisions peierls1961possible ; Li:1997zb ; Li:1997bi , and has received
attention for applications in cosmology Grzadkowski:2021kgi ; Iglicki:2022jjf
and in the context of the processes at a muon collider ginzburg1995effect ;
ginzburg1996initial ; melnikov1997processes ; dams2003singular . It had been
suggested to regularize the singularity by accounting for the decay width of
the incoming particles or the transversal size of the beam, in the case of the
muon collider. Both suggestions rely on the fact that the initial state
particles cannot be treated as well-defined plane waves. This violates the
energy-momentum conservation and prevents the particle in the propagator from
being exactly on-shell.

It is important to recognize that in the context of HNL production, the
singularity cannot be resolved by implying kinematic cuts on the final
particles. As discussed in Sec. 2.1, sufficiently heavy HNLs may be produced
with small boosts, and their decay products can be detected with $\approx 1$
efficiency. The key parameter is the total number of HNLs created in the
process.

Therefore, the cross section $\sigma(\hat{s},Q_{1}^{2},Q_{2}^{2})$ should be
evaluated strictly for off-shell vector bosons with negative invariant mass
$m_{W}^{2}\to-Q^{2}$. In this case, there is no zero crossing in the $t$ range
(20). Using the unitarity constraints that forbid $\sigma$ from growing with
energy, we can estimate the upper bounds on the cross section by:

$\sigma(s,Q_{1}^{2},Q_{2}^{2})\lesssim\frac{g^{4}}{16\pi}\begin{cases}\frac{1}{m_{W}\Gamma_{W}},&Q^{2}\lesssim
m_{W}\Gamma_{W},\\\ \frac{1}{Q^{2}},&m_{W}\Gamma_{W}\lesssim Q^{2}\lesssim
m_{W}^{2},\\\ \frac{1}{m_{W}^{2}},&m_{W}^{2}\lesssim Q^{2},\end{cases}$ (22)

where $Q^{2}$ represents any of the quantities $Q_{1}^{2}$, $Q_{2}^{2}$. The
estimate for the case $Q^{2}\lesssim m_{W}\Gamma_{W}$ accounts for the
instability of the incoming particles, essentially employing the same argument
that was used for the $t$-channel singularity ginzburg1996initial .

The integration over $Q^{2}_{1,2}$ in Eq. (19) can be split into three ranges,
listed in Eq. (22). This results in the following upper bound

$\hat{\sigma}(\hat{s})\equiv\int\frac{dQ^{2}_{1}}{Q_{1}^{2}+m_{W}^{2}}\frac{dQ^{2}_{2}}{Q_{2}^{2}+m_{W}^{2}}\sigma(\hat{s},Q_{1}^{2},Q_{2}^{2})\lesssim\\\
\lesssim\frac{g^{4}}{16\pi}\left(\pi\frac{(m_{W}\Gamma_{W}/m_{W}^{2})^{2}}{m_{W}\Gamma_{W}}+\pi\frac{1-\Gamma_{W}/m_{W}}{m_{W}^{2}}+\frac{1}{m_{W}^{2}}\left(\ln\frac{\hat{s}}{m_{W}^{2}}\right)^{2}\right),$
(23)

where the first two terms are negligible compared to the third term, which is
the usual $\sigma_{\text{weak}}$ cross section enhanced by the collinear
emission of the vector bosons. Therefore, the final VBF cross section for HNL
production (19) is bounded by

$\sigma_{\text{VBF}}\lesssim\sigma_{\text{weak}}\times\frac{g^{4}}{256\pi^{4}}\times\left(\ln\frac{s}{m_{W}^{2}}\right)^{2}\times\ln\frac{s}{m_{N}^{2}}\approx
10^{-3}\times\sigma_{\text{weak}}.$ (24)

For comparison, the cross section of $s$-channel production is

$\sigma_{s}\sim\frac{m_{W}^{2}}{s}\sigma_{\text{weak}}=(10^{-4}\,\mbox{---}\,10^{-3})\times\sigma_{\text{weak}}$
(25)

for the center-of-mass energies $\sqrt{s}=3$ – $10\,\mathrm{TeV}$.

Therefore, we conclude that the account for the VBF processes does not provide
a significant improvement in sensitivity compared to the $s$-channel.
Moreover, in the most optimistic scenario, the sensitivity contour can reach
$U^{2}_{\alpha}\sim 10^{-3}$, while the current constraints on $e$ and $\tau$
mixing are $U^{2}_{e}\lesssim 10^{-3}$ and $U_{\tau}\lesssim 10^{-2}$
Chrzaszcz:2019inj .

At the same time, vector boson fusion may become the dominant process for the
production of $\nu_{e}$\- and $\nu_{\tau}$-mixing HNLs at muon colliders of
much higher energy above $10\,\mathrm{TeV}$. An accurate analysis of the
effective boson approximation would be required in this case.

##  3 Probing lepton number violation and leptogenesis

The muon collider is a perfect instrument to measure the Majorana vs. Dirac
nature of the HNL. Similar studies at various collider and beam-extracted
experiments have been performed in Tastet:2019nqj ; Antusch:2016ejd ;
Drewes:2022rsk ; Antusch:2023jsa ; Deppisch:2013jxa . The specific advantage
of the muon collider stems from the fact that the HNL momentum tends to point
in the same forward direction $\theta<\pi/2$ as the muon it originated from.

Each reconstructed signal event can be classified by the initial lepton
number, which is labeled according to the corresponding hemisphere containing
the total momentum of the decay products. This lepton charge is then compared
with the charge of the produced muon in the $N\to\mu qq$ decay to eventually
label the process as lepton number violating (LNV) or conserving (LNC).

###  3.1 Sensitivity to lepton number violation

To quantify the measure of lepton number violation, we define the parameter
$A$, being twice the probability that HNL, produced in the mixing with a
lepton, decays into the opposite-charge lepton:

$A=2\times\text{Br}(N_{l}\to\bar{l}\dots).$

With this definition, a Dirac HNL corresponds to $A=0$, while for a purely
Majorana HNL $A=1$.

Given the total number of detectable HNL decays $N_{\text{ev}}$, the counts of
observed LNV-like and LNC-like events contain three contributions: from
correctly and incorrectly classified signal events, and from background
events:

$\displaystyle
N_{\text{LNV}}=\frac{A}{2}\cdot(1-f)N_{\text{ev}}+\left(1-\frac{A}{2}\right)\cdot
fN_{\text{ev}}+N_{\text{LNV, bkg}},$ $\displaystyle
N_{\text{LNC}}=\left(1-\frac{A}{2}\right)\cdot(1-f)N_{\text{ev}}+\frac{A}{2}\cdot
fN_{\text{ev}}+N_{\text{LNC, bkg}},$

where $f$ is the probability of incorrect classification of the reconstructed
HNL decay, $N_{\text{LNV/C, bkg}}$ are contributions of the SM background to
each event class.

Incorrect classification of the signal may happen due to backward production
of an HNL at large angles $\theta>\pi/2$, and because of poor reconstruction
of the total momentum. The probability $f_{B}$ of backward production
($\theta>\pi/2$) can be approximated numerically as

$f_{B}\approx
0.07\frac{m_{W}^{2}}{s}\times\frac{(1+m_{N}^{2}/s)^{2}}{1-m_{N}^{2}/s}\approx
5\cdot 10^{-5}\,\frac{3\,\mathrm{TeV}^{2}}{s},\quad s-m_{N}^{2}\gg m_{W}^{2},$
(26)

where the prefactor is computed by accounting for both $W$ and $Z$-boson-
mediated processes. The total momentum of the HNL is
$p_{N}\sim\frac{1}{2}\sqrt{s}(1-m_{N}^{2}/s)$. Assuming that the uncertainty
of momentum reconstruction is not too large, i.e. $\lesssim m_{W}$, both
effects are negligible in the large range of HNL masses up to the production
threshold $s-m_{N}^{2}\lesssim m_{W}^{2}$. For the sake of simplicity, we
ignore this specific case and set $f=0$.

We assume that the contribution of the SM background is symmetric with respect
to LNV/LNV-like events: $N_{\text{LNV, bkg}}=N_{\text{LNC,
bkg}}=N_{\text{bkg}}/2$. This assumption needs to be verified by background
simulations, given that the SM processes with the initial muons may be more
likely to produce the same sign lepton in the forward direction. In this case,
the background for the LNV-like events becomes lower, improving the overall
sensitivity. Therefore, we consider our assumption conservative.

The sensitivity to $A$ is estimated using the $\chi^{2}$ test. For a model
that is parametrized $(N_{\text{ev}},A)$, we may test an alternative model
$(\tilde{N}_{\text{ev}},\tilde{A})$ by defining the quantity

$\chi^{2}(\tilde{N}_{\text{ev}},\tilde{A}|N_{\text{ev}},A,N_{\text{bkg}})=\frac{(\tilde{N}_{\text{LNV}}-N_{\text{LNV}})^{2}}{\tilde{N}_{\text{LNV}}}+\frac{(\tilde{N}_{\text{LNC}}-N_{\text{LNC}})^{2}}{\tilde{N}_{\text{LNC}}}.$
(27)

We define the precision of the $A$ determination as the interval $\Delta A$ of
variation of $\tilde{A}$, in which $\chi^{2}(\tilde{A})$ does not exceed the
$1\sigma$ quantile of the chi-squared distribution with 2 degrees of freedom
$\chi^{2}<2.30$, after marginalization over $\tilde{N}_{\text{ev}}$. Using the
estimates for the number of signal and background events, we show this
sensitivity in Fig. 6.

The numerical results for the precision $\Delta A$ of the determination of $A$
may be approximated by a simple relation

$\Delta A\approx\frac{\sqrt{N_{\text{ev}}\cdot
A+N_{\text{bkg}}}}{N_{\text{ev}}}.$ (28)

If uncertainty is dominated by the background fluctuations, the sensitivity
scales as $\Delta A\sim 1/U^{2}$. This always holds for the Dirac case $A=0$.
Conversely, once $N_{\text{ev}}A>N_{\text{bkg}}$, the sensitivity starts to
grow with the mixing angle at a smaller pace $\propto 1/\sqrt{U^{2}}$.

Figure 6: Constraints on lepton number violation $\Delta A$ deviations for the
pure Dirac $A=0$ (left) and Majorana $A=1$ (right) cases for the
$\sqrt{s}=3\,\mathrm{TeV}$ (top) and $\sqrt{s}=10\,\mathrm{TeV}$ (bottom) muon
collider. The gray area is the excluded region. The green shaded area
represents the parameter space where successful leptogenesis with neither
thermal nor vanishing initial conditions is possible Drewes:2021nqr . The
green dashed line represents the scale above which the 3 HNLs in the Majorana-
like case would produce large loop corrections to active neutrino masses
Drewes:2019byd .

###  3.2 Implications for leptogenesis

In the scenario with two degenerate HNLs $N_{1}$, $N_{2}$, the coupling matrix
$\theta_{\alpha I}$ and mass matrix has the form

$\theta_{\alpha I}=\frac{1}{\sqrt{2}}\begin{pmatrix}U_{e}&iU_{e}\\\
U_{\mu}&iU_{\mu}\\\ U_{\tau}&iU_{\tau}\end{pmatrix},\qquad\qquad
M_{IJ}=\begin{pmatrix}m_{N}-\frac{\Delta M}{2}&0\\\ 0&m_{N}+\frac{\Delta
M}{2}\end{pmatrix}.$ (29)

The two HNLs form a quasi-Dirac pair,
$\Psi=\frac{1}{\sqrt{2}}(|N_{1}\rangle+i|N_{2}\rangle)$,
$\bar{\Psi}=\frac{1}{\sqrt{2}}(|N_{1}\rangle-i|N_{2}\rangle)$. This pair
violates the lepton charge only due to the relative oscillations between the
HNLs arising from a slight difference in their mass $\Delta M$.

The expression for the parameter $A$ has the form:

$A=2\frac{\int_{0}^{\infty}|g_{-}(t)|^{2}\,dt}{\int_{0}^{\infty}(|g_{+}(t)|^{2}+|g_{-}(t)|^{2})dt},$
(30)

where $g_{\pm}$ are the matrix elements for LNC/LNV transitions, defined as

$g_{+}(t)=\Psi^{\dagger}\mathcal{U}(t)\Psi,\qquad
g_{-}(t)=\bar{\Psi}^{\dagger}\mathcal{U}(t)\Psi,$ (31)

where $U(t)$ is the evolution matrix. In the 2 HNL model, it has the form

$\mathcal{U}(t)=\exp\left(-itM_{IJ}-\frac{\Gamma}{2}t\right),$ (32)

and the lepton number violation parameter is given by the ratio
Anamiati:2016uxp ; Deppisch:2015qwa :

$A=\frac{\Delta M^{2}}{\Gamma^{2}+\Delta M^{2}}.$ (33)

In the 3 HNL scenario, the coupling and mass matrix can be rewritten as

$\theta_{\alpha I}=\frac{1}{\sqrt{2}}\begin{pmatrix}U_{e}&iU_{e}&0\\\
U_{\mu}&iU_{\mu}&0\\\ U_{\tau}&iU_{\tau}&0\end{pmatrix},\qquad\qquad
M_{IJ}=m_{N}\cdot\mathbbm{1}_{3\times 3}+\Gamma\begin{pmatrix}0&0&\xi_{1}\\\
0&\mu_{2}&\xi_{2}\\\ \xi_{1}&\xi_{2}&\mu_{3}\end{pmatrix},$ (34)

assuming that the couplings to the active neutrino sector explicitly respect
the lepton symmetry. All deviations in the mass matrix are defined as
normalized to the decay width. In this case, the evolution matrix has the form

$\mathcal{U}(t)=e^{-im_{N}t}\exp\left[-\Gamma
t\begin{pmatrix}\frac{1}{2}&0&i\xi_{1}\\\ 0&\frac{1}{2}+i\mu_{2}&i\xi_{2}\\\
i\xi_{1}&i\xi_{2}&i\mu_{3}\end{pmatrix}\right].$ (35)

The larger number of independent parameters adds two difficulties in relating
the LNV parameter to the underlying model. First, a simple analytic expression
of the form (33) is missing, and in general, $A$ should be evaluated
numerically. Second, the measurement of $A$ adds only one constraint to the
set of splitting parameters $\\{\mu_{i}$, $\xi_{i}\\}$, insufficient for
reconstructing all the parameters.

To quantitatively explain the way LNV occurs, we need to separate the two main
effects. If the oscillations between the interacting states $|N_{1}\rangle$
and $|N_{2}\rangle$ are rapid enough $\mu_{2}\gtrsim 1$, we return to the case
of 2 HNLs that exhibit Majorana-like behavior $A\sim 1$. If, in contrast,
these oscillations are suppressed $\mu_{2}\ll 1$, the third HNL starts to play
a role. If $|\xi|\gg|\mu_{3}|$, the massive states of sterile neutrinos become
completely misaligned with the initial interacting states, and the mass
splittings become of order $\sim\Gamma\xi$. Finally, if $|\xi|\ll|\mu_{3}|$, a
situation similar to the seesaw mechanism occurs: the interacting states
$\Psi$ acquire a small admixture of the third HNL with an amplitude of order
$\frac{\xi^{2}}{\mu_{3}}$. Then the production and decay processes with a
Dirac-like HNL with mixing angles $U^{2}$ are accompanied by the processes
with a Majorana HNL $|N_{3}\rangle$ with mixing angles suppressed to
$U^{2}\cdot\frac{\xi}{\mu_{3}}$.

If the parameters that define the scale of the probability of LNV to happen
are sufficiently small, i.e. $|\mu_{2}|$, $|\xi|\ll 1$ (but arbitrary
$\mu_{3}$), the leading term expansion may be written explicitly:

$A=\mu_{2}^{2}+\frac{2(\xi_{1}^{2}+\xi_{2}^{2})}{1+4\mu_{3}^{2}}.$ (36)

In general, this expression cannot be directly linked to the mass splittings.
To fix the convention, we define the “third” massive state with mass $M_{3}$
as the eigenstate of the mass matrix that is closest to the initial third
state in the basis of Eq. (34), and the first two as $M_{2}>M_{1}$. Then,
there are two mass splittings $\Delta M_{12}\geq 0$ and $\Delta
M_{3}=M_{3}-\frac{M_{1}+M_{2}}{2}$. To show the variation in $A$ for various
combinations of masses, we performed a scan in the parameters $\\{\mu_{i}$,
$\xi_{i}\\}$. The minimal and maximal values of $A$ that can be achieved for
given $\Delta M_{12}$, $|\Delta M_{3}|$ are shown in Fig. 7.

While $A$ is not fixed for given mass splittings, one can nevertheless
estimate the order of magnitude of the mass splittings $\Delta M\sim\Delta
M_{12},|\Delta M_{3}|$.

$A\sim\frac{\Delta M^{2}}{\Gamma^{2}}:\qquad\frac{\Delta
M}{m_{N}}\sim\frac{\Gamma(m_{N})\sqrt{A}}{m_{N}}\approx\sqrt{A}\,U^{2}\left(\frac{m_{N}}{1\,\mathrm{TeV}}\right)^{2},\quad
m_{N}\gtrsim 200\,\mathrm{GeV}.$ (37)

Figure 7: The minimal and maximal values of $A$ for given pair of mass
splittings $\Delta M_{12}$ and $\Delta M_{3}$, see text for conventions.

This relation has been confirmed by the scan, which shows that the approximate
variation range of the parameter $A$ is $[0.5\Delta
M^{2}_{12},\text{max}(\Delta M^{2}_{12},\Delta M^{2}_{3})]/\Gamma^{2}$ for the
mass splittings that differ less by an order of magnitude.

Given these relations, for a specific measurement of $A$, one can potentially
reduce the parameter space of the models with 3 HNL which may explain the
baryon asymmetry of the Universe. A scan of the values of $A$ with a mapping
onto the sensitivity map of the muon collider experiments may reveal some
potential benefits of this measurement. The lower bounds of $\Delta M/m_{N}$
defined by Eq. (37) that may be probed by a muon collider (replacing $A$ by
$\Delta A$) are shown in Fig. 8.

Figure 8: The lower bounds on the scale of mass splitting $\Delta M/m_{N}$
(defined as Eq. (37)) that may be probed by the LNV measurements at a muon
collider, assuming an approximately Dirac-like scenario $A\ll 1$. The black
dashed line represents the sensitivity $\Delta A=0.1$, for which measurements
of the mass splitting via (37) is possible. Below it down to $\Delta A=0.9$
(black solid line), the precision of the $A$ measurement is insufficient, and
only a preference towards the Majorana/Dirac nature can be established. In
this region, $(\Delta M/m_{N})$ represents the transition point between the
two cases.

##  4 Conclusions

In this paper, we examined the potential of future muon colliders to measure
lepton number violation in searches for heavy neutral leptons. This probe may
address the role played by the newfound particles in explaining the matter-
antimatter imbalance. The sensitivity of the muon collider down to
$U_{\mu}^{2}\sim 10^{-6}$ provides a unique probe of the TeV-scale HNL,
superior to the competing FCC, CEPC, CLIC, and ILC projects, and enters the
parameter space where these sterile neutrinos might account for the observed
value baryon asymmetry.

We discussed the procedure for measuring lepton number violation in HNL decays
and estimated the sensitivity to such searches. With the potential to observe
up to $10^{5}$ events in the unexplored region, an experiment at the muon
collider affords a sensitivity to the ratio of lepton number violating decays
that can reach a percent level, as shown in Fig. 6. This sensitivity is
crucial for deciphering the true nature of the HNL — be it Majorana, Dirac, or
situated in a non-minimal framework involving multiple nearly degenerate heavy
neutral leptons.

In the sensitivity range of the muon collider, the minimal scenario of
leptogenesis with two HNL fails to produce the observed matter-antimatter
imbalance, and three degenerate species are needed. We relate the properties
of these HNLs to the lepton number violation parameter $A$. We showed that in
this case there is no direct link between the LNV ratio and the mass
splittings, contrary to the minimal case of two HNLs. This distinction is
attributed to a greater number of parameters responsible for breaking the
approximate lepton symmetry. Nevertheless, the overall scale of the largest
mass splitting between the HNLs is bounded by the measurements of LNV
processes. The most stringent lower bounds on the $(\Delta M/m_{N})$ ratio
that could be attained at the muon collider are shown in Fig. 8.

Lastly, we complemented the previous simulation-based estimates of the
sensitivity reach of an experiment at the muon collider to sterile neutrinos.
Moreover, we demonstrated the failure of the vector boson fusion processes to
probe couplings to electron- and tau-neutrinos beyond the excluded bounds, as
well as highlighted theoretical nuances accompanying the description of these
processes.

## Acknowledgements

We thank Juraj Klaric for the discussions on the leptogenesis in the model of
3 HNLs and for proofreading the manuscript. The evaluation of the cross
sections has been done with the use of FeynCalc Shtabovenko:2016sxi ;
Shtabovenko:2020gxv . This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (GA 694896), from the NWO Physics Vrij Programme “The
Hidden Universe of Weakly Interacting Particles”, No. 680.92.18.03, which is
partly financed by the Dutch Research Council NWO, and from the Den Adel Fund
in the form of a scholarship.

##  Appendix A HNL decay width

There are 3 decay channels for HNLs. Their partial decay widths are
gluza1997heavy ; del2006neutrino :

$\Gamma(N\rightarrow
Wl_{\alpha})=\frac{g^{2}}{64\pi}U^{2}\frac{m_{N}^{3}}{M_{W}^{2}}\bigg{(}1-\frac{M_{W}^{2}}{m_{N}^{2}}\bigg{)}^{2}\bigg{(}1+2\frac{M_{W}^{2}}{m_{N}^{2}}\bigg{)},$
(38) $\Gamma(N\rightarrow
Z\nu_{\alpha})=\frac{g^{2}}{128\pi}U^{2}\frac{m_{N}^{3}}{M_{W}^{2}}\bigg{(}1-\frac{M_{Z}^{2}}{m_{N}^{2}}\bigg{)}^{2}\bigg{(}1+2\frac{M_{Z}^{2}}{m_{N}^{2}}\bigg{)},$
(39) $\Gamma(N\rightarrow
H\nu_{\alpha})=\frac{g^{2}U^{2}_{\alpha}}{128\pi}\frac{m_{N}^{3}}{M_{W}^{2}}\bigg{(}1-\frac{M_{H}^{2}}{m_{N}^{2}}\bigg{)}^{2},$
(40)

and approach the ratio $\Gamma(N\to Wl):\Gamma(N\to Zl):\Gamma(N\to
H\nu)=2:1:1$ in the limit $m_{N}\gg M_{Z,W,H}$.

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# Efficient Data Distribution Estimation for Accelerated Federated Learning

Yuanli Wang
Boston University
<EMAIL_ADDRESS>
&Lei Huang
Boston University
<EMAIL_ADDRESS>

##  1 Introduction

Federated Learning(FL) is a privacy-preserving machine learning paradigm where
a global model is trained in-situ across a large number of distributed edge
devices. These systems are often comprised of millions of user devices and
only a subset of available devices can be used for training in each epoch.
Designing a device selection strategy is challenging, given that devices are
highly heterogeneous in both their system resources and training data. This
heterogeneity makes device selection very crucial for timely model convergence
and sufficient model accuracy [12]. To tackle the FL client heterogeneity
problem, various client selection algorithms have been developed, showing
promising performance improvement in terms of model coverage and accuracy [2,
7].

In this work, we study the overhead of client selection algorithms in a large
scale FL environment. Then we propose an efficient data distribution summary
calculation algorithm to reduce the overhead in a real-world large scale FL
environment. The evaluation shows that our proposed solution could achieve up
to 30x reduction in data summary time, and up to 360x reduction in clustering
time.

##  2 Heterogeneity-aware cluster based Federated Learning

In FL, data is not independent and identically distributed (IID) across all
edge devices, resulting in statistical heterogeneity across devices [11]. Many
existing works [13, 10, 1, 8, 7, 9] exploit client’s local data to quantify
the statistical heterogeneity, and cluster client devices based on their
discernible statistical differences. Then for every training iteration, they
select a cluster of devices based on client’s system heterogeneity and
statistical heterogeneity. As an example, Figure 1 shows the workflow of
HACCS[13].

HACCS[13] proposed two types of data-sampling based _distribution summaries_
to quantify the statistical heterogeneity, motivated by techniques for
identifying IID violations. Standard distributed machine learning models
assume that the response variables $y_{i}$ and the predictor variables $X_{i}$
at each device are drawn IID from a shared joint distribution, $P(X,y)$. This
joint distribution can be factored as follows:

$P(X,y)=P(y)\;P(X\;|\;y)$ (1)

Therefore, if $P(y)$ (the marginal distribution of the response labels) or
$P(X\;|\;y)$ (the data distribution conditioned on the response) differs at
any device, we have a violation of the IID assumption. When IID violations
occur, it follows that one of these distributions must differ across one or
more devices [5]. Based on this insight, they propose $P(y)$ and $P(X\;|\;y)$
as distribution summaries. Their evaluation on a 50 clients setting with
FEMNIST and CIFAR-10 datasets shows that $P(y)$ and $P(X\mid y)$ can provide
18%-38% reduction in model training time compared to the state of the art,
without any compromise in accuracy.

Figure 1: system overview and workflow of query deployment.

###  2.1 Limitations

HACCS[13] only computes distribution summary of each client once in the first
training iteration. Thus, the overhead of computing distribution summary has
negligible impact on overall training speed. However, in the real case, as
users’ application running, the data distributions of the clients may be time-
varying and non-stationary [3, 14]. In order to do adaptive client selection,
we need to re-compute distribution summary periodically as data changes. For
slow edge computing devices with a larger scale of data samples, computing
distribution summaries is time-consuming with high memory consumption. Also,
when the number of clients becomes large, clustering a large amount of client
distribution summaries will take a long time. The overhead of these algorithms
may become large and slow down training.

In FL setting, devices also have system heterogeneity, which means devices
have different processing capacity, network bandwidth, and power. Since the
available resources of each device change rapidly, we need each device to
periodically send its resource status to the central server. We propose
efficient data distribution summary algorithm that enables devices to
periodically update their distribution summary, which could enable more robust
FL system in real world environment.

##  3 Motivation study

###  3.1 Dataset and motivation study

To motivate our solution, we first perform an empirical evaluation study to
quantify the overhead of computing distribution summaries in existing work
[13]. The results of our study motivate the design of our proposed efficient
data distribution summary algorithm.

We use two large scale real-world datasets in Table 1 to evaluate $P(X\mid
y)$and $P(y)$ distribution summary methods. The datasets are pre-processed and
partitioned to different clients by [6].

Dataset |  Application |  sample size |  number of classes |  number of clients |  number of samples per client
---|---|---|---|---|---
FEMNIST |  handwritten alphanumeric classification |  28x28x1 grayscale images |  62 |  2800 |  average: 109
max: 6709
std: 211.63
OpenImage |  image classification |  3x256x256 color images |  600 |  11325 |  average: 228
max: 465
std: 89.05
Table 1: Datasets used for evaluation

Table 2 shows the overhead of two existing distribution summary methods
$P(X\mid y)$ and $P(y)$ . We found that for larger dataset, $P(X\mid y)$needs
up to 553s to finish calculating summary, which is quite slow. Also, we
monitored that it uses more than 64GB memory for computation, which is not
acceptable for mobile devices that typically have less than 16 GB memory.
Although $P(y)$ can efficiently compute on large dataset, it fails to capture
the data heterogeneity of the feature of samples under the same label, since
it only considers the label distribution (For example, images of both cats and
dogs might be labeled as "animals", but their features could be quite
different).

We also find that since the size of summary becomes large, it will take long
time to do clustering, even up to more than 2 days. Furthermore, the cluster
algorithm (DBSCAN) is sensitive to parameter setting. When we reuse the
parameters tuned for one dataset to another setting, it can sometimes put all
devices to the same group, and can not return a meaningful clustering
solution.

##  4 Proposed Solution

###  4.1 Distribution summary calculation

We propose to use dimension reduction and coreset to allow efficient
calculation of the distribution summary on each device. In this way we can
both accelerate the computation time and reduce the size of the summary. For
each device, we construct the coreset by sampling $k$ elements from the
dataset on this device, while maintaining its original label proportions. Then
for each element in the coreset, we use encoder to do dimension reduction on
its feature. Specifically, we modified MobileNet[4] network and extract the
output of a hidden layer as the feature vector. The advantage of using encoder
rather than PCA or Johnson–Lindenstrauss Lemma is: (1). the encoding
computation could be easily accelerated by GPU. (2). MobileNet is pre-trained
for image classification tasks and can capture the spatial information of
pixels on the image (feature).

After coreset selection and dimension reduction, for each device, we construct
the distribution summary as a flat vector that includes: (1). the element-wise
mean of feature vectors of samples under each unique label. (2). the
distribution of each unique label. As the result, suppose the output of
dimension reduction is a vector with size $H$, the number of classes is $C$,
then the shape of distribution summary will be $C*H+C$. It is much smaller
than the histogram representation used in $P(X\mid y)$, but still included the
distribution information of both sample features and labels.

###  4.2 Device clustering

We use K-means for device clustering, since it fits to our simplified
distribution summary. K-means is a popular clustering algorithm that
partitions a dataset into $k$ distinct, non-overlapping clusters. The target
of the K-means clustering is to minimize the variance within each cluster as
denoted by:

$\text{minimize}\quad
J=\sum_{j=1}^{k}\sum_{i=1}^{n}\left\|x_{i}^{(j)}-c_{j}\right\|^{2}$ (2)

The objective function $J$ is the sum of distances between all data points to
their respective centroids. The procedure of K-means is as follows: (i) it
first randomly defines $k$ centroids and assigns each data point to its
closest centroid forming $k$ clusters. (ii) For each cluster, it obtains the
new centroid by calculating the mean position among all data points within the
cluster. It then repeats the step (i) to form new $k$ clusters. (iii) Repeat
the previous 2 steps until convergence or a termination condition is reached
(e.g. centroids do not change anymore).

K-means is an effective clustering method that minimizes the variance within
each cluster and guarantees explicit $k$ partitions of the data set.

##  5 Preliminary Results and Future works

We implemented our solution into the FL framework proposed in HACCS [13]. Our
proposed solution is complementary to privacy-preserving methods that could be
applied on the data summaries, such as differential privacy used in HACCS
[13]. Table 2 compares the speed of computing data distribution summary on the
two datasets. We can see that our proposed solution could achieve up to 30x
reduction in data summary time, and up to 360x reduction in clustering time.

| Time(s) calculating summary | Time(s) clustering devices
---|---|---
| OpenImage | FEMNIST | OpenImage | FEMNIST
| Avg | Max | Avg | Max
$P(y)$ | <0.01 | <0.01 | <0.01 | <0.01 | 835.69 | 24.5
$P(X\mid y)$ | 9.13 | 553.41 | 3.23 | 6.68 | more than 2 days | 1866.40
Encoder+Kmeans | 4.6 | 18.7 | 5.4 | 11.2 | 477.2 | 30
Table 2: Overhead comparison of different summary algorithms

In a large-scale FL setting, the size of the data summary impacts both the
network bandwidth usage between clients and the server, and the memory
consumption on the server during clustering tasks. For future work, we plan to
explore additional dimension reduction methods to more effectively compress
the data summary while maintaining the integrity of statistical diversity
information.

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