#
Flavino Dark Matter
in a Non-Abelian Discrete Flavor Model

Takaaki Nomura<EMAIL_ADDRESS>Yusuke Shimizu
<EMAIL_ADDRESS>and Towa Takahashi
<EMAIL_ADDRESS>

1College of Physics, Sichuan University, Chengdu 610065, China  2Graduate
School of Science and Technology, Niigata University,
Niigata, 950-2181, Japan  3Department of Information, Kaishi Professional
University,
Niigata 950-0916, Japan

( Abstract
We study a relic density of the “flavino” dark matter in modified Altarelli
and Feruglio $A_{4}$ model which is respecting the $SU(2)_{L}\times
A_{4}\times Z_{3}\times U(1)_{R}$ symmetry. We calculate the Lagrangian from
the superpotential in the model. In estimating the relic density, we consider
the relevant interactions from the Lagrangian that realize the vacuum
expectation value alignments and charged lepton masses where we assume that
the supersymmetry breaking effects are small for “flavon” sector. As a result,
we find the degenerate masses between the lightest “flavon” and “flavino”, and
only two parameters in the potential determines the relic density. Then the
allowed parameter space of these parameters are estimated from the relic
density calculation taking a constraint from lepton flavor violation into
account. We also briefly discuss other dark matter physics such as the direct
detection, indirect detection, and collider search. )

##  1 Introduction

The standard model (SM) is one of the successful ones with discovery of the
Higgs boson. However, there are some mysterious problems in the SM, e.g. the
origin of the generations for particle physics. In the SM, one cannot explain
the differences of the mixing angles and masses for the quarks and leptons.
Yukawa couplings are completely free parameters so that the mixing angles and
masses cannot be predicted in the SM. Furthermore, the neutrinos are massless
for the renormalizable operators. One of the attractive phenomena to solve the
problems is the neutrino oscillation which is the evidence of the beyond the
SM. Actually, the neutrino oscillation experiments provide us the important
informations which are the two neutrino mass squared differences and two large
mixing angles. The reactor neutrino experiments reported the last mixing angle
which is non-zero value [1, 2]. Furthermore, the neutrino oscillation
experiments go to new stage to measure the Dirac CP phase in the lepton
sector. The neutrinos are thus windows to the new physics beyond the SM.

The flavor symmetry can be applied to the three generations of the SM
fermions. The Froggatt-Nielsen mechanism [3] which was proposed by C. D.
Froggatt and H. B. Nielsen applies a global $U(1)_{FN}$ symmetry. Thanks to
the symmetry, one can naturally explain the fermion mass hierarchies. The non-
Abelian discrete symmetry (See for the review [4]-[8].) can naturally explain
the lepton mixing angles which are tri-bimaximal mixing (TBM) [9, 10] before
the reactor experiments reported the non-zero reactor angle $\theta_{13}$ [1,
2]. After measurements of the $\theta_{13}$, many authors have studied the
breaking or deviation from the TBM [11]-[33] or other patterns of the lepton
mixing angles, e.g. tri-bimaximal-Cabibbo mixing [34, 35]. One of the
successful flavor models was proposed by G. Altarelli and F. Feruglio [36,
37]. The Altarelli and Feruglio (AF) model realizes the TBM by using the non-
Abelian discrete symmetry $A_{4}$. They mainly introduced two SM gauge singlet
scalar fields so-called “flavons”. Taking the vacuum expectation value (VEV)
alignments of the $A_{4}$ triplets as $(1,0,0)$ and $(1,1,1)$, one can
naturally explain that the charged lepton mass matrix is diagonal and neutrino
mixing is TBM, respectively. However, the correct VEV alignments cannot be
driven from the potential analysis. Then, they adopted the supersymmetry
(SUSY) and introduced the so-called “driving” fields [36, 37].

Introduction of the SUSY can solve another mysterious problem that is
existence of dark matter (DM) in the universe since the lightest
supersymmetric particle (LSP) is stable due to $R$-parity and it can be DM if
it is neutral one [38, 39]. In the minimal supersimmetric SM (MSSM), the
lightest neutralino is a good DM candidate and its properties are well
discussed such as parameter space explaining observed relic density [40]. Also
the neutralino DM is searched for in direct/indirect detection and collider
experiments [41, 42, 43, 44, 45, 46]. However we have not seen any clear
evidence of a DM in these experiments yet and nature of DM is an open
question. Interestingly, in SUSY models with flavons, we have new candidate of
DM that is the superpartner of the flavon which we call the ”flavino” [47].
The lightest flavino can be DM if it is the LSP since flavinos are neutral
particles. Thus it is worth exploring the possibility of flavino DM.

In this paper, we discuss a relic density of the flavino DM in modified AF
$A_{4}$ model [27],[48]-[51] which is respecting the $SU(2)_{L}\times
A_{4}\times Z_{3}\times U(1)_{R}$ symmetry. In estimating the relic density,
we consider relevant interactions from superpotential that realize the VEV
alignments and charged lepton masses where we assume SUSY breaking effects are
small for flavon sector. As a result, we find degenerate mass between the
lightest flavon and flavino, and only two parameters in the potential
determines the relic density. Then allowed parameter space of these parameters
are estimated from relic density calculation taking a constraint from lepton
flavor violation (LFV) into account. We also briefly discuss other DM physics
such as the direct detection, indirect detection and collider search.

This paper is organized as follows. In Section 2, we briefly introduce the
modified AF $A_{4}$ model which is respecting the $SU(2)_{L}\times A_{4}\times
Z_{3}\times U(1)_{R}$ symmetry. In Section 3, we discuss the relic density of
the flavino DM calculating its annihilation cross section. Section 4 is
devoted to summary. We show the relevant multiplication rule of the $A_{4}$
group in Appendix A. The LFV constraint for our model is shown in Appendix B.

##  2 Model

In this section, we briefly introduce an $SU(2)_{L}\times A_{4}\times
Z_{3}\times U(1)_{R}$ model [27],[48]-[51]. Three generations of the left-
handed lepton doublet superfields $\Phi_{\ell}=(\Phi_{\ell 1},\Phi_{\ell
2},\Phi_{\ell 3})$ are assigned to the triplet as $\bf 3$ for $A_{4}$
symmetry. The right-handed charged lepton superfields $\Phi_{e_{R}^{c}}$,
$\Phi_{\mu_{R}^{c}}$, and $\Phi_{\tau_{R}^{c}}$ are assigned to the singlets
as $\bf 1$, $\bf 1^{\prime\prime}$, and $\bf 1^{\prime}$ for $A_{4}$ symmetry,
respectively. One of the Higgs doublet superfields is defined as $\Phi_{d}$
and assigned to the singlet as $\bf 1$ for $A_{4}$ symmetry. We introduce
superfields $\Phi_{T}=(\Phi_{T1},\Phi_{T2},\Phi_{T3})$ which are so-called
“flavon superfields” and assigend to the triplet as $\bf 3$ for $A_{4}$
symmetry. In order to obtain the relevant couplings, we add the $Z_{3}$
symmetry. We also add the $U(1)_{R}$ symmetry so that we can generate the VEVs
and VEV alignments through the $F$-terms by coupling flavon superfields to so-
called “driving superfields”
$\Phi_{0}^{T}=(\Phi_{01}^{T},\Phi_{02}^{T},\Phi_{03}^{T})$ which are assigned
to triplet as $\bf 3$ for $A_{4}$ symmetry and carry the $R$ charge $+2$ under
$U(1)_{R}$ symmmetry. The charge assignments of $SU(2)_{L}$, $A_{4}$, $Z_{3}$,
and $U(1)_{R}$ are summarized in Tab. 1.

| $\Phi_{\ell}$ | $\Phi_{e_{R}^{c}}$ | $\Phi_{\mu_{R}^{c}}$ | $\Phi_{\tau_{R}^{c}}$ | $\Phi_{d}$ | $\Phi_{T}$ | $\Phi_{0}^{T}$
---|---|---|---|---|---|---|---
$SU(2)_{L}$ | $2$ | $1$ | $1$ | $1$ | $2$ | $1$ | $1$
$A_{4}$ | ${\bf 3}$ | ${\bf 1}$ | ${\bf 1^{\prime\prime}}$ | ${\bf 1^{\prime}}$ | ${\bf 1}$ | ${\bf 3}$ | ${\bf 3}$
$Z_{3}$ | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega^{2}$ | $1$ | $1$ | $1$
$U(1)_{R}$ | $1$ | $1$ | $1$ | $1$ | $0$ | $0$ | $2$
Table 1: The charge assignments of $SU(2)_{L}\times A_{4}\times Z_{3}\times
U(1)_{R}$ symmetry in our model.

In these setup, we can now write douwn the superpotential for respecting
$SU(2)_{L}\times A_{4}\times Z_{3}\times U(1)_{R}$ symmetry at the leading
order in terms of the $A_{4}$ cut-off scale $\Lambda$ as

$\displaystyle w_{\Phi_{T}}$ $\displaystyle=w_{\ell}+w_{d}^{T},$
$\displaystyle w_{\ell}$
$\displaystyle=y_{e}\Phi_{T}\Phi_{\ell}\Phi_{e_{R}^{c}}\Phi_{d}/\Lambda+y_{\mu}\Phi_{T}\Phi_{\ell}\Phi_{\mu_{R}^{c}}\Phi_{d}/\Lambda+y_{\tau}\Phi_{T}\Phi_{\ell}\Phi_{\tau_{R}^{c}}\Phi_{d}/\Lambda,$
$\displaystyle w_{d}^{T}$
$\displaystyle=-M\Phi_{0}^{T}\Phi_{T}+g\Phi_{0}^{T}\Phi_{T}\Phi_{T},$ (1)

where $\Phi_{i}$’s are chiral superfields
$\Phi_{i}=\phi_{i}+\sqrt{2}\theta\psi_{i}+\theta\theta F_{i}$, $y_{j}$’s are
Yukawa couplings, $M$ is mass paremeter for the flavon superfields $\Phi_{T}$
and $\Phi_{0}^{T}$ 444Hereafter we also call deriving field $\Phi_{0}^{T}$ as
flavon field for simplicity., and $g$ is trilinear coupling for flavon
superfields. The superpotential in Eq. (1) is rewritten as

$\displaystyle w_{\ell}$
$\displaystyle=y_{e}(\Phi_{T1}\Phi_{\ell_{1}}+\Phi_{T2}\Phi_{\ell_{3}}+\Phi_{T3}\Phi_{\ell_{2}})\Phi_{e_{R}^{c}}\Phi_{d}/\Lambda$
$\displaystyle+y_{\mu}(\Phi_{T3}\Phi_{\ell_{3}}+\Phi_{T1}\Phi_{\ell_{2}}+\Phi_{T2}\Phi_{\ell_{1}})\Phi_{\mu_{R}^{c}}\Phi_{d}/\Lambda$
$\displaystyle+y_{\tau}(\Phi_{T2}\Phi_{\ell_{2}}+\Phi_{T3}\Phi_{\ell_{1}}+\Phi_{T1}\Phi_{\ell_{3}})\Phi_{\tau_{R}^{c}}\Phi_{d}/\Lambda,$
$\displaystyle w_{d}^{T}$
$\displaystyle=-M(\Phi_{01}^{T}\Phi_{T1}+\Phi_{02}^{T}\Phi_{T3}+\Phi_{03}^{T}\Phi_{T2})$
$\displaystyle+\frac{2}{3}g\left[\Phi_{01}^{T}(\Phi_{T1}\Phi_{T1}-\Phi_{T2}\Phi_{T3})+\Phi_{02}^{T}(\Phi_{T2}\Phi_{T2}-\Phi_{T3}\Phi_{T1})+\Phi_{03}^{T}(\Phi_{T3}\Phi_{T3}-\Phi_{T1}\Phi_{T2})\right].$
(2)

Then, the Lagrangian $\mathcal{L}_{\Phi_{T}}$ for our model is given as

$\displaystyle\mathcal{L}_{\Phi_{T}}$ $\displaystyle=\int d^{2}\theta
w_{\Phi_{T}}+\int d^{2}\bar{\theta}\bar{w}_{\Phi_{T}}-V_{\Phi_{T}},$
$\displaystyle V_{\Phi_{T}}$ $\displaystyle=V_{\ell}+V_{T},$ (3)

where the scalar potential $V_{\ell}$ and $V_{T}$ are obtained from
$w_{d}^{T}$ and $w_{\ell}$ respectively. The scalar potential $V_{T}$ is
obtained as

$\displaystyle V_{T}$ $\displaystyle=\sum_{X}\left|\frac{\partial
w_{d}^{T}}{\partial X}\right|^{2}$
$\displaystyle=\left|-M\phi_{T1}+\frac{2}{3}g(\phi_{T1}\phi_{T1}-\phi_{T2}\phi_{T3})\right|^{2}$
$\displaystyle+\left|-M\phi_{T3}+\frac{2}{3}g(\phi_{T2}\phi_{T2}-\phi_{T3}\phi_{T1})\right|^{2}$
$\displaystyle+\left|-M\phi_{T2}+\frac{2}{3}g(\phi_{T3}\phi_{T3}-\phi_{T1}\phi_{T2})\right|^{2}$
$\displaystyle+\left|-M\phi_{01}^{T}+\frac{2}{3}g(2\phi_{01}^{T}\phi_{T1}-\phi_{02}^{T}\phi_{T3}-\phi_{03}^{T}\phi_{T2})\right|^{2}$
$\displaystyle+\left|-M\phi_{03}^{T}+\frac{2}{3}g(2\phi_{02}^{T}\phi_{T2}-\phi_{03}^{T}\phi_{T1}-\phi_{01}^{T}\phi_{T3})\right|^{2}$
$\displaystyle+\left|-M\phi_{02}^{T}+\frac{2}{3}g(2\phi_{03}^{T}\phi_{T3}-\phi_{01}^{T}\phi_{T2}-\phi_{02}^{T}\phi_{T1})\right|^{2},$
(4)

where $X=\phi_{T1}$, $\phi_{T2}$, $\phi_{T3}$, $\phi_{01}^{T}$,
$\phi_{02}^{T}$, and $\phi_{03}^{T}$. On the other hand, the scalar potential
$V_{\ell}$ is written by

$\displaystyle V_{\ell}$ $\displaystyle=\sum_{Y}\left|\frac{\partial
w_{\ell}}{\partial Y}\right|^{2}$
$\displaystyle=\left|(y_{e}\tilde{\phi}_{\ell_{1}}\tilde{\phi}_{e_{R}^{c}}+y_{\mu}\tilde{\phi}_{\ell_{2}}\tilde{\phi}_{\mu_{R}^{c}}+y_{\tau}\tilde{\phi}_{\ell_{3}}\tilde{\phi}_{\tau_{R}^{c}})h_{d}/\Lambda\right|^{2}$
$\displaystyle+\left|(y_{e}\tilde{\phi}_{\ell_{3}}\tilde{\phi}_{e_{R}^{c}}+y_{\mu}\tilde{\phi}_{\ell_{1}}\tilde{\phi}_{\mu_{R}^{c}}+y_{\tau}\tilde{\phi}_{\ell_{2}}\tilde{\phi}_{\tau_{R}^{c}})h_{d}/\Lambda\right|^{2}$
$\displaystyle+\left|(y_{e}\tilde{\phi}_{\ell_{2}}\tilde{\phi}_{e_{R}^{c}}+y_{\mu}\tilde{\phi}_{\ell_{3}}\tilde{\phi}_{\mu_{R}^{c}}+y_{\tau}\tilde{\phi}_{\ell_{1}}\tilde{\phi}_{\tau_{R}^{c}})h_{d}/\Lambda\right|^{2}$
$\displaystyle+\left|(y_{e}\phi_{T1}\tilde{\phi}_{e_{R}^{c}}+y_{\mu}\phi_{T2}\tilde{\phi}_{\mu_{R}^{c}}+y_{\tau}\phi_{T3}\tilde{\phi}_{\tau_{R}^{c}})h_{d}/\Lambda\right|^{2}$
$\displaystyle+\left|(y_{e}\phi_{T3}\tilde{\phi}_{e_{R}^{c}}+y_{\mu}\phi_{T1}\tilde{\phi}_{\mu_{R}^{c}}+y_{\tau}\phi_{T2}\tilde{\phi}_{\tau_{R}^{c}})h_{d}/\Lambda\right|^{2}$
$\displaystyle+\left|(y_{e}\phi_{T2}\tilde{\phi}_{e_{R}^{c}}+y_{\mu}\phi_{T3}\tilde{\phi}_{\mu_{R}^{c}}+y_{\tau}\phi_{T1}\tilde{\phi}_{\tau_{R}^{c}})h_{d}/\Lambda\right|^{2}$
$\displaystyle+\left|y_{e}(\phi_{T1}\tilde{\phi}_{\ell_{1}}+\phi_{T2}\tilde{\phi}_{\ell_{3}}+\phi_{T3}\tilde{\phi}_{\ell_{2}})h_{d}/\Lambda\right|^{2}$
$\displaystyle+\left|y_{\mu}(\phi_{T3}\tilde{\phi}_{\ell_{3}}+\phi_{T1}\tilde{\phi}_{\ell_{2}}+\phi_{T2}\tilde{\phi}_{\ell_{1}})h_{d}/\Lambda\right|^{2}$
$\displaystyle+\left|y_{\tau}(\phi_{T2}\tilde{\phi}_{\ell_{2}}+\phi_{T3}\tilde{\phi}_{\ell_{1}}+\phi_{T1}\tilde{\phi}_{\ell_{3}})h_{d}/\Lambda\right|^{2}$
$\displaystyle+\left|\left[y_{e}(\phi_{T1}\tilde{\phi}_{\ell_{1}}+\phi_{T2}\tilde{\phi}_{\ell_{3}}+\phi_{T3}\tilde{\phi}_{\ell_{2}})\tilde{\phi}_{e_{R}^{c}}+y_{\mu}(\phi_{T3}\tilde{\phi}_{\ell_{3}}+\phi_{T1}\tilde{\phi}_{\ell_{2}}+\phi_{T2}\tilde{\phi}_{\ell_{1}})\tilde{\phi}_{\mu_{R}^{c}}\right.\right.$
$\displaystyle+\left.\left.y_{\tau}(\phi_{T2}\tilde{\phi}_{\ell_{2}}+\phi_{T3}\tilde{\phi}_{\ell_{1}}+\phi_{T1}\tilde{\phi}_{\ell_{3}})\tilde{\phi}_{\tau_{R}^{c}}\right]/\Lambda\right|^{2},$
(5)

where $Y=\phi_{T1}$, $\phi_{T2}$, $\phi_{T3}$, $\tilde{\phi}_{\ell_{1}}$,
$\tilde{\phi}_{\ell_{2}}$, $\tilde{\phi}_{\ell_{3}}$,
$\tilde{\phi}_{e_{R}^{c}}$, $\tilde{\phi}_{\mu_{R}^{c}}$,
$\tilde{\phi}_{\tau_{R}^{c}}$, and $h_{d}$ with $\tilde{\phi}$ scalar fields
being slepton ones. The F-term contribution for our model is calculated as

$\displaystyle\int d^{2}\theta w_{\Phi_{T}}$
$\displaystyle=y_{e}(\phi_{T1}\ell_{1}+\phi_{T2}\ell_{3}+\phi_{T3}\ell_{2})e_{R}^{c}h_{d}/\Lambda$
$\displaystyle+y_{\mu}(\phi_{T3}\ell_{3}+\phi_{T1}\ell_{2}+\phi_{T2}\ell_{1})\mu_{R}^{c}h_{d}/\Lambda$
$\displaystyle+y_{\tau}(\phi_{T2}\ell_{2}+\phi_{T3}\ell_{1}+\phi_{T1}\ell_{3})\tau_{R}^{c}h_{d}/\Lambda$
$\displaystyle+y_{e}(\tilde{\psi}_{\phi_{T1}}\tilde{\phi}_{\ell_{1}}+\tilde{\psi}_{\phi_{T2}}\tilde{\phi}_{\ell_{3}}+\tilde{\psi}_{\phi_{T3}}\tilde{\phi}_{\ell_{2}})e_{R}^{c}h_{d}/\Lambda$
$\displaystyle+y_{\mu}(\tilde{\psi}_{\phi_{T3}}\tilde{\phi}_{\ell_{3}}+\tilde{\psi}_{\phi_{T1}}\tilde{\phi}_{\ell_{2}}+\tilde{\psi}_{\phi_{T2}}\tilde{\phi}_{\ell_{1}})\mu_{R}^{c}h_{d}/\Lambda$
$\displaystyle+y_{\tau}(\tilde{\psi}_{\phi_{T2}}\tilde{\phi}_{\ell_{2}}+\tilde{\psi}_{\phi_{T3}}\tilde{\phi}_{\ell_{1}}+\tilde{\psi}_{\phi_{T1}}\tilde{\phi}_{\ell_{3}})\tau_{R}^{c}h_{d}/\Lambda$
$\displaystyle+y_{e}(\tilde{\psi}_{\phi_{T1}}\ell_{1}+\psi_{\phi_{T2}}\ell_{3}+\psi_{\phi_{T3}}\ell_{2})\tilde{\phi}_{e_{R}^{c}}h_{d}/\Lambda$
$\displaystyle+y_{\mu}(\tilde{\psi}_{\phi_{T3}}\ell_{3}+\psi_{\phi_{T1}}\ell_{2}+\psi_{\phi_{T2}}\ell_{1})\tilde{\phi}_{\mu_{R}^{c}}h_{d}/\Lambda$
$\displaystyle+y_{\tau}(\tilde{\psi}_{\phi_{T2}}\ell_{2}+\psi_{\phi_{T3}}\ell_{1}+\psi_{\phi_{T1}}\ell_{3})\tilde{\phi}_{\tau_{R}^{c}}h_{d}/\Lambda$
$\displaystyle+y_{e}(\tilde{\psi}_{\phi_{T}1}\tilde{\phi}_{\ell_{1}}+\tilde{\psi}_{\phi_{T}2}\tilde{\phi}_{\ell_{3}}+\tilde{\psi}_{\phi_{T}3}\tilde{\phi}_{\ell_{2}})\tilde{\phi}_{e_{R}^{c}}\tilde{\psi}_{h_{d}}/\Lambda$
$\displaystyle+y_{\mu}(\tilde{\psi}_{\phi_{T}3}\tilde{\phi}_{\ell_{3}}+\tilde{\psi}_{\phi_{T}1}\tilde{\phi}_{\ell_{2}}+\tilde{\psi}_{\phi_{T}2}\tilde{\phi}_{\ell_{1}})\tilde{\phi}_{\mu_{R}^{c}}\tilde{\psi}_{h_{d}}/\Lambda$
$\displaystyle+y_{\tau}(\tilde{\psi}_{\phi_{T}2}\tilde{\phi}_{\ell_{2}}+\tilde{\psi}_{\phi_{T}3}\tilde{\phi}_{\ell_{1}}+\tilde{\psi}_{\phi_{T}1}\tilde{\phi}_{\ell_{3}})\tilde{\phi}_{\tau_{R}^{c}}\tilde{\psi}_{h_{d}}/\Lambda$
$\displaystyle+y_{e}(\phi_{T1}\ell_{1}+\phi_{T2}\ell_{3}+\phi_{T3}\ell_{2})\tilde{\phi}_{e_{R}^{c}}\tilde{\psi}_{h_{d}}/\Lambda$
$\displaystyle+y_{\mu}(\phi_{T3}\ell_{3}+\phi_{T1}\ell_{2}+\phi_{T2}\ell_{1})\tilde{\phi}_{\mu_{R}^{c}}\tilde{\psi}_{h_{d}}/\Lambda$
$\displaystyle+y_{\tau}(\phi_{T2}\ell_{2}+\phi_{T3}\ell_{1}+\phi_{T1}\ell_{3})\tilde{\phi}_{\tau_{R}^{c}}\tilde{\psi}_{h_{d}}/\Lambda$
$\displaystyle+y_{e}(\phi_{T1}\tilde{\phi}_{\ell_{1}}+\phi_{T2}\tilde{\phi}_{\ell_{3}}+\phi_{T3}\tilde{\phi}_{\ell_{2}})e_{R}^{c}\tilde{\psi}_{h_{d}}/\Lambda$
$\displaystyle+y_{\mu}(\phi_{T3}\tilde{\phi}_{\ell_{3}}+\phi_{T1}\tilde{\phi}_{\ell_{2}}+\phi_{T2}\tilde{\phi}_{\ell_{1}})\mu_{R}^{c}\tilde{\psi}_{h_{d}}/\Lambda$
$\displaystyle+y_{\tau}(\phi_{T2}\tilde{\phi}_{\ell_{2}}+\phi_{T3}\tilde{\phi}_{\ell_{1}}+\phi_{T1}\tilde{\phi}_{\ell_{3}})\tau_{R}^{c}\tilde{\psi}_{h_{d}}/\Lambda$
$\displaystyle-M(\tilde{\psi}_{\phi_{01}^{T}}\tilde{\psi}_{\phi_{T1}}+\tilde{\psi}_{\phi_{02}^{T}}\tilde{\psi}_{\phi_{T3}}+\tilde{\psi}_{\phi_{03}^{T}}\tilde{\psi}_{\phi_{T2}})$
$\displaystyle+\frac{2}{3}g\left[\phi_{01}^{T}(\tilde{\psi}_{\phi_{T1}}\tilde{\psi}_{\phi_{T1}}-\tilde{\psi}_{\Phi_{T2}}\tilde{\psi}_{\phi_{T3}})+\phi_{02}^{T}(\tilde{\psi}_{\phi_{T2}}\tilde{\psi}_{\phi_{T2}}-\tilde{\psi}_{\phi_{T3}}\tilde{\psi}_{\phi_{T1}})\right.$
$\displaystyle+\phi_{03}^{T}(\tilde{\psi}_{\phi_{T3}}\tilde{\psi}_{\phi_{T3}}-\tilde{\psi}_{\phi_{T1}}\tilde{\psi}_{\phi_{T2}})$
$\displaystyle+\tilde{\psi}_{\phi_{01}^{T}}(\phi_{T1}\tilde{\psi}_{\phi_{T1}}-\phi_{T2}\tilde{\psi}_{\phi_{T3}})+\tilde{\psi}_{\phi_{02}^{T}}(\phi_{T2}\tilde{\psi}_{\phi_{T2}}-\phi_{T3}\tilde{\psi}_{\phi_{T1}})+\tilde{\psi}_{\phi_{03}^{T}}(\phi_{T3}\tilde{\psi}_{\phi_{T3}}-\phi_{T1}\tilde{\psi}_{\phi_{T2}})$
$\displaystyle+\left.\tilde{\psi}_{\phi_{01}^{T}}(\tilde{\psi}_{\phi_{T1}}\phi_{T1}-\tilde{\psi}_{\phi_{T2}}\phi_{T3})+\tilde{\psi}_{\phi_{02}^{T}}(\tilde{\psi}_{\phi_{T2}}\phi_{T2}-\tilde{\psi}_{\phi_{T3}}\phi_{T1})+\tilde{\psi}_{\phi_{03}^{T}}(\tilde{\psi}_{\phi_{T3}}\phi_{T3}-\tilde{\psi}_{\phi_{T1}}\phi_{T2})\right],$
(6)

where $\tilde{\psi}$ fields correspond to flavinos. The VEV alignments of
$\phi_{T}$ and $\phi_{0}^{T}$ are obtained from the potential minimum
condition, $V_{T}=0$, as in Ref. [50];

$\langle\phi_{T}\rangle=v_{T}(1,0,0),\qquad
v_{T}=\frac{3M}{2g},\qquad\langle\phi_{0}^{T}\rangle=(0,0,0).$ (7)

The charged lepton masses are then obtained after $h_{d}$ developing its VEV,
$\langle h_{d}\rangle=v_{d}$, as
$m_{\\{e,\mu,\tau\\}}=y_{\\{e,\mu,\tau\\}}v_{d}v_{T}/\Lambda$. After taking
the VEV alignments, e.g. the $A_{4}$ triplet flavon ($\phi_{T1}$, $\phi_{T2}$,
$\phi_{T3}$)=($v_{T}+\varphi_{T1}$, $\phi_{T2}$, $\phi_{T3}$), we find the
mass terms of flavons and flavinos as follows,

$\displaystyle\mathcal{L}_{\Phi_{T}}\supset\ $ $\displaystyle
M^{2}|\varphi_{T1}|^{2}+4M^{2}|\phi_{T2}|^{2}+4M^{2}|\phi_{T3}|^{2}+M^{2}|\phi_{01}^{T}|^{2}+4M^{2}|\phi_{02}^{T}|^{2}+4M^{2}|\phi_{03}^{T}|^{2}$
$\displaystyle-M\tilde{\psi}_{\phi^{T}_{01}}\tilde{\psi}_{\phi_{T1}}-2M(\tilde{\psi}_{\phi_{02}^{T}}\tilde{\psi}_{\phi_{T3}}+\tilde{\psi}_{\phi_{03}^{T}}\tilde{\psi}_{\phi_{T2}})+h.c.\,.$
(8)

Notice that pairs of chiral fermion components
$\\{\tilde{\psi}_{\phi^{T}_{01}},\tilde{\psi}_{\phi_{T1}}\\}$,
$\\{\tilde{\psi}_{\phi_{02}^{T}},\tilde{\psi}_{\phi_{T3}}\\}$ and
$\\{\tilde{\psi}_{\phi_{03}^{T}},\tilde{\psi}_{\phi_{T2}}\\}$ construct Dirac
mass terms. We also find that the mass ratio of flavons(flavinos) is $1:2:2$,
and flavino and flavon masses are degenerate without considering SUSY breaking
effects. In this work the DM candidate is the lightest flavino which is Dirac
fermion with mass $M$. We rewrite it as

$X_{R}\equiv\tilde{\psi}^{c}_{\phi^{T}_{01}},\qquad
X_{L}\equiv\tilde{\psi}_{\phi_{T1}},$ (9)

where $"c"$ indicates charge conjugation and $X$ is identified as Dirac
fermion. Here we focus on interactions among the DM $X$ and the lightest
flavons $\varphi_{T1}$ and $\phi^{T}_{01}$ since it provide dominant
contributions in the relic density estimation as we discuss in next section.
The relevant interactions are summarized by

$\displaystyle\mathcal{L}_{\Phi_{T}}\supset\ $
$\displaystyle\frac{M}{v_{T}}\left[2\varphi_{T1}\overline{X_{R}}X_{L}+\phi_{01}^{T}\overline{X^{c}_{L}}X_{L}+h.c.\right]$
$\displaystyle\frac{M^{2}}{v_{T}}\left[\varphi_{T1}\varphi_{T1}^{*}\varphi_{T1}^{*}+c.c.\right]-\frac{2M^{2}}{v_{T}}\left[\phi_{01}^{T}\phi_{01}^{T*}\varphi_{T1}^{*}+c.c.\right].$
(10)

These interactions induce DM annihilation processes that determine the relic
density.

##  3 Dark matter physics

In this section we discuss the relic density of flavino DM calculating its
annihilation cross section. Other DM phenomenologies such as direct/indirect
detections and collider searches are also discussed briefly.

###  3.1 DM annihilation cross sections

In our scenario the flavinos mainly annihilate into the lightest flavons,
$\\{\varphi_{T},\phi_{01}^{T}\\}$, due to small couplings among flavino and
the SM leptons. There are many possible processes of such annihilation
processes as

$\displaystyle\overline{X}X\rightarrow\\{\varphi_{T1}\varphi_{T1},\overline{\varphi_{T1}}\
\overline{\varphi_{T1}},\overline{\phi^{T}_{01}}\phi^{T}_{01},\overline{\varphi_{T1}}\varphi_{T1}\\},$
$\displaystyle XX\rightarrow\overline{\phi^{T}_{01}}\
\overline{\varphi_{T1}},\quad\overline{X}\
\overline{X}\rightarrow\phi^{T}_{01}\varphi_{T1}.$ (11)

Although many diagrams can be drawn from the Lagrangian, we find the dominant
contributions are
$\overline{X}X\rightarrow\\{\varphi_{T1}\varphi_{T1},\overline{\varphi_{T1}}\
\overline{\varphi_{T1}}\\}$ and $XX\rightarrow\overline{\phi^{T}_{01}}\
\overline{\varphi_{T1}}(\overline{X}\
\overline{X}\rightarrow\phi^{T}_{01}\varphi_{T_{1}})$ where the other
processes are suppressed by $p^{2}/M^{2}$ factor with $p$ being magnitude of
DM momentum. In the following we summarize the cross sections of the dominant
processes.

(i) $X\bar{X}\longrightarrow\varphi_{T1}\varphi_{T1}$:
In this process, all channels are present and relevant matrix elements are

(a) s-channel

(b) t-channel

(c) t-channel

Figure 1: Diagrams for the process $\bar{X}X\to\varphi_{T1}\varphi_{T1}$.

$\displaystyle\mathcal{M}_{s}$
$\displaystyle=-\left(\frac{2M}{v_{T}}\right)\left(\frac{2M^{2}}{v_{T}}\right)\frac{1}{s-M^{2}}\bar{v}_{(p_{2})}P_{R}u_{(p_{1})}$
$\displaystyle\mathcal{M}_{t}$
$\displaystyle\thicksim{\left(\frac{M}{v_{T}}\right)}^{2}\frac{1}{M}\bar{v}_{(p_{2})}P_{R}u_{(p_{1})}$
$\displaystyle\mathcal{M}_{u}$
$\displaystyle\thicksim{\left(\frac{M}{v_{T}}\right)}^{2}\frac{1}{M}\bar{v}_{(p_{2})}P_{R}u_{(p_{1})},$
(12)

where $s=(p_{1}+p_{2})^{2}$ with $p_{1(2)}$ being flavino $X(\bar{X})$
momentum in the initial state. Here, we ignored other Mandelstam variables
assuming $t,u\ll M^{2}$. Since we consider non-retivistic limit, we
approximate $s\thicksim 4M^{2}$. Thus, the matrix element of the process is

$\displaystyle\mathcal{M}\thicksim\frac{20M}{3{v_{T}}^{2}}\bar{v}_{(p_{2})}P_{R}u_{(p_{1})}.$

The cross section is

$\displaystyle\sigma_{X\bar{X}\rightarrow\varphi_{T1}\varphi_{T1}}\thicksim\frac{25}{72\pi}\frac{M^{2}}{{v_{T}}^{4}}\frac{s-2M^{2}}{s}.$
(13)

(ii)$X\bar{X}\longrightarrow\overline{\varphi_{T1}}\,\overline{\varphi_{T1}}$:
This process are just conjugate of
$X\bar{X}\longrightarrow\varphi_{T1}\varphi_{T1}$. Thus, cross section is

$\displaystyle\sigma_{X\bar{X}\rightarrow\overline{\varphi_{T1}}\,\overline{\varphi_{T1}}}$
$\displaystyle=\sigma_{X\bar{X}\rightarrow\varphi_{T1}\varphi_{T1}}$
$\displaystyle\thicksim\frac{25}{72\pi}\frac{M^{2}}{{v_{T}}^{4}}\frac{s-2M^{2}}{s}.$
(14)

(iii)$XX\longrightarrow\overline{\phi^{T}_{01}}\,\overline{{\phi}_{T1}}$:
In this prosess, all channels are present. Matrix elements are

(a) s-channel

(b) t-channel

(c) t-channel

Figure 2: Diagrams for the process
$XX\to\overline{\varphi_{T1}}\overline{\phi^{T}_{01}}$.

$\displaystyle\mathcal{M}_{s}$
$\displaystyle=\left(\frac{M}{v_{T}}\right)\left(\frac{2M^{2}}{v_{T}}\right)\frac{1}{s-M^{2}}\bar{v}_{(p_{2})}P_{L}u_{(p_{1})}$
$\displaystyle\mathcal{M}_{t}$
$\displaystyle\thicksim\mathcal{M}_{u}\thicksim\left(\frac{2M}{{v_{T}}^{2}}\right)\bar{v}_{(p_{2})}P_{L}u_{(p_{1})}$
(15)

This process’s matrix element is

$\displaystyle\mathcal{M}\thicksim\left(\frac{14M}{{3v_{T}}^{2}}\right)\bar{v}_{(p_{2})}P_{L}u_{(p_{1})}.$

The cross section is

$\displaystyle\sigma_{XX\rightarrow\overline{\phi^{T}_{01}}\overline{\phi_{T1}}}\thicksim\left(\frac{49}{144\pi
s}\right)\left(\frac{{M}^{2}}{{v_{T}}^{4}}\right)\left(s-2M^{2}\right).$ (16)

(iv)$\bar{X}\bar{X}\longrightarrow{\phi}^{T}_{01}{\varphi}_{T1}$:
This process are just conjugate of
$XX\longrightarrow\overline{\phi^{T}_{01}}\overline{\varphi_{T1}}$. Thus,
cross section is

$\displaystyle\sigma_{\bar{X}\bar{X}\rightarrow{\phi}^{T}_{01}{\varphi}_{T1}}$
$\displaystyle=\sigma_{XX\rightarrow\overline{\varphi^{T}_{01}}\overline{\varphi_{T1}}}$
$\displaystyle\thicksim\left(\frac{49}{144\pi
s}\right)\left(\frac{{M}^{2}}{{v_{T}}^{4}}\right)\left(s-2M^{2}\right).$ (17)

We apply these cross sections in DM relic density calculation.

###  3.2 Relic density

Relic density of flavino DM is obtained by solving Boltzmann equation for
number density $n_{X}$ of DM $X$,

$\dot{n}_{X}+3Hn_{X}=\langle\sigma v\rangle(n_{X_{\rm eq}}^{2}-n_{X}^{2}),$
(18)

where $H$ is Hubble parameter, $\langle\sigma v\rangle$ is thermal average of
DM annihilation cross section and $n_{X_{eq}}$ is density of $X$ in
equilibrium. The annihilation cross section is the sum of cross sections
discussed in the previous subsection. Relic density $\Omega_{X}h^{2}$ can be
approximately estimated as [52, 53]

$\displaystyle\Omega_{X}h^{2}\simeq\frac{1.07\times
10^{9}}{\sqrt{g^{*}(x_{f})}M_{Pl}J(x_{f})\ {\rm[GeV]}},$ (19)

where $x_{f}=M/T_{f}$ with $T_{f}$ being freeze out temperature,
$g^{*}(x_{f})$ is effective relativistic degrees of freedom at $T_{f}$, and
$M_{Pl}\simeq 1.22\times 10^{19}$ is the Planck mass. The factor
$J(x_{f})\equiv\int_{x_{f}}^{\infty}dx\frac{\langle\sigma v\rangle}{x^{2}}$ is
written by

$J(x_{f})=\int_{x_{f}}^{\infty}\left[\frac{\int_{4M^{2}}^{\infty}ds\sqrt{s-4M^{2}}(\sigma
v)K_{1}\left(\frac{\sqrt{s}}{M}x\right)}{16M^{5}x[K_{2}(x)]^{2}}\right],$ (20)

where $K_{1,2}$ denote the modified Bessel functions of the second kind of
order 1 and 2.

Figure 3: Red points shows the parameters that provide observed relic density
$\Omega_{X}h^{2}\simeq 0.12$. The gray region is disfavored since coupling $g$
is beyond the perturbative limit. Relic density is over abundant in the light
gray region. The light blue region is excluded by the constraint from flavor
violating $\tau$ decay.

In estimating relic density we apply micrOMEGAs code [54] implementing
relevant interactions in the model; we compared the numerical results with
analytical calculation with Eq. (19) and confirmed consistency. In Fig. 3, we
show parameters that provide observed amount of relic density
$\Omega_{X}h^{2}\simeq 0.12$ [55] by red points where relic density is over
abundant in the light gray region and smaller than observed amount below red
points. Also the gray region is disfavored since coupling $g$ becomes non-
perturbative value. In addition, the light blue region is excluded by the
constraint from flavor violating $\tau$ decay [48] (see Appendix B). We thus
find that the range of the lightest flavino(flavon) mass $[6,6\times 10^{4}]$
[GeV] can accommodate observed relic density. Also allowed region of $v_{T}$
is roughly 30 [GeV] to $2\times 10^{4}$ [GeV].

###  3.3 Comments on ohter flavino DM phyiscs

Here we provide brief discussion regarding some phenomenology of flavino DM.

Direct detection:
The interactions from superpotential do not contribute to flavino-nucleon
scattering at tree and one-loop level since they only have coupling terms
among flavino, flavon, Higgs(Higgsinos) and leptons(sleptons); there is no
flavino-flavino-Higgs interaction. Thus these interactions are not constrained
by the direct detection experiments; flavino can also interact with electron
via flavon $\varphi_{T1}$ exchange but cross section is tiny since flavon-
electron coupling is proportional to electron mass. In fact flavino-nucleon
scattering is possible via Higgs portal interactions when flavon and Higgs
bosons mix through SUSY-breaking terms. In this work we assume effect of SUSY
breaking is small for flavon sector to avoid current experimental bounds, e.g.
XENON1T [56], PandaX-4T [57] and LUX-ZEPLIN [58].

Indirect detection:
In the scenario cross sections of flavino annihilation into flavons are
suppressed at current universe due to mass degeneracy between the lightest
flavino and flavon. Flavino can also annihilate into charged leptons via
flavon exchanging processes, $X\bar{X}\to\varphi_{T1}\to\ell^{+}\ell^{-}$, but
cross section of the processes are much smaller than order of $\sim
10^{-26}$cm${}^{2}/$s, since flavon-lepton coupling is small as
$m_{\ell}/v_{\varphi}$. Thus the model is safe from indirect detection
constraints where the strongest bound on the annihilation cross section is
given by Fermi-LAT data [59, 60].

Collider search:
Flavino DM can be searched for at collider experiments such as the LHC as the
preferred scale of flavino mass is $\mathcal{O}$(10) GeV to $\mathcal{O}$(10)
TeV. One possible process is slepton pair production followed by slepton decay
$\tilde{\ell}\to\ell X$ when a slepton is the next to lightest SUSY particle.
The signal of this case is the same as slepton decaying into neutralino DM
[40] and it is difficult to distinguish. We can expect more specific signals
of the model when heavier flavinos and flavons are lighter than sleptons. In
such a case we would have cascade decay of slepton, e.g.
$\tilde{\ell}\to\ell\overline{\tilde{\psi}_{T2}}\to\ell\overline{X}\phi_{T3}(\to\bar{\ell}^{\prime}\ell^{\prime})$,
inducing multi-leptons with missing-energy signal. For these signals we may be
able to reconstruct flavon mass and it helps us to confirm the model. Detailed
analysis of collider signals is beyond the scope of this paper and we left it
in future work.

##  4 Summary

We have studied the relic density of the flavino DM in modified AF $A_{4}$
model which is respecting the $SU(2)_{L}\times A_{4}\times Z_{3}\times
U(1)_{R}$ symmetry. First of all, we have calculated the Lagrangian from the
superpotential in the modified AF $A_{4}$ model. In estimating relic density,
we have considered the relevant interactions from the Lagrangian that realize
the VEV alignments and charged lepton masses where we assume SUSY breaking
effects are small for flavon sector. As a result, we have found the degenerate
masses between the lightest flavon and flavino, and only two parameters in the
potential determines the relic density. Then allowed parameter space of these
parameters have been estimated from relic density calculation taking a
constraint from lepton flavor violation (LFV) into account. We thus have found
that the range of lightest flavino(flavon) mass $[6,6\times 10^{4}]$ [GeV]
could accommodate observed relic density. Also allowed region of $v_{T}$ has
been roughly 30 [GeV] to $2\times 10^{4}$ [GeV]. We also briefly have
discussed other DM physics such as direct detection, indirect detection and
collider search.

Acknowledgement

We thank T. Toma for useful discussions. The work of T. N. was supported by
the Fundamental Research Funds for the Central Universities. The work of Y. S.
was partially supported by JSPS KAKENHI Grant No. 20H01898.

## Appendix

##  Appendix A Multiplication rule of $A_{4}$ group

We use the multiplication rule of the $A_{4}$ triplet as follows:

$\displaystyle\begin{pmatrix}a_{1}\\\ a_{2}\\\ a_{3}\end{pmatrix}_{\bf
3}\otimes\begin{pmatrix}b_{1}\\\ b_{2}\\\ b_{3}\end{pmatrix}_{\bf 3}$
$\displaystyle=\left(a_{1}b_{1}+a_{2}b_{3}+a_{3}b_{2}\right)_{\bf
1}\oplus\left(a_{3}b_{3}+a_{1}b_{2}+a_{2}b_{1}\right)_{{\bf 1}^{\prime}}$
$\displaystyle\oplus\left(a_{2}b_{2}+a_{1}b_{3}+a_{3}b_{1}\right)_{{\bf
1}^{\prime\prime}}$
$\displaystyle\oplus\frac{1}{3}\begin{pmatrix}2a_{1}b_{1}-a_{2}b_{3}-a_{3}b_{2}\\\
2a_{3}b_{3}-a_{1}b_{2}-a_{2}b_{1}\\\
2a_{2}b_{2}-a_{3}b_{1}-a_{1}b_{3}\end{pmatrix}_{{\bf
3}}\oplus\frac{1}{2}\begin{pmatrix}a_{2}b_{3}-a_{3}b_{2}\\\
a_{1}b_{2}-a_{2}b_{1}\\\ a_{3}b_{1}-a_{1}b_{3}\end{pmatrix}_{{\bf 3}\ }\ .$
(21)

More details are shown in the review [5, 6].

##  Appendix B LFV constraint in the model

In the model some LFV charged lepton decay processes are forbidden by residual
$Z_{3}$ symmetry [48]. In particular LFV muon decays are forbidden and the
strongest constraint is obtained from $\tau\to\mu\mu\bar{e}$ process. The
branching ratio of the process is given by

$\displaystyle{\rm BR}(\tau\to\mu\mu\bar{e})$
$\displaystyle=\tau_{\tau}\frac{m_{\tau}^{5}}{3027\pi^{3}}\left(\left|\frac{m_{\tau}m_{\mu}}{v_{T}^{2}m^{2}_{\phi_{T2}}}\right|^{2}+\left|\frac{m_{\mu}m_{e}}{v_{T}^{2}m^{2}_{\phi_{T3}}}\right|^{2}\right)$
$\displaystyle\simeq$ $\displaystyle\frac{2.9\times 10^{6}\,{\rm
GeV}^{8}}{v_{T}^{4}(2M)^{4}}$ (22)

where $\tau_{\tau}$ is the lifetime of tau lepton. It is compared with the
current constraint by measurement at Belle experiment
BR$(\tau\to\mu\mu\bar{e})<1.7\times 10^{-8}$ [61] to constrain $\\{M,v_{T}\\}$
parameter space.

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# Structured Generations: Using Hierarchical Clusters to guide Diffusion
Models

Jorge da Silva Gonçalves  Laura Manduchi  Moritz Vandenhirtz  Julia E. Vogt

###### Abstract

This paper introduces Diffuse-TreeVAE, a deep generative model that integrates
hierarchical clustering into the framework of Denoising Diffusion
Probabilistic Models (DDPMs). The proposed approach generates new images by
sampling from a root embedding of a learned latent tree VAE-based structure,
it then propagates through hierarchical paths, and utilizes a second-stage
DDPM to refine and generate distinct, high-quality images for each data
cluster. The result is a model that not only improves image clarity but also
ensures that the generated samples are representative of their respective
clusters, addressing the limitations of previous VAE-based methods and
advancing the state of clustering-based generative modeling.

Machine Learning, ICML

##  1 Introduction

Generative modeling and clustering represent two fundamental and distinct
approaches within the field of machine learning. Generative modeling aims to
approximate the underlying distribution of data, thereby enabling the
generation of new samples (Kingma & Welling, 2014; Goodfellow et al., 2014).
Clustering, conversely, seeks to identify meaningful and interpretable
structures within data. This is achieved through the unsupervised detection of
intrinsic relationships and dependencies (Ezugwu et al., 2022), which can
enhance data visualization and interpretation. TreeVAE (Manduchi et al., 2023)
was recently proposed to combine these two research directions by integrating
hierarchical dependencies into a deep latent variable model. TreeVAE models
the distribution of data by learning the optimal tree structure of latent
variables. The resulting latent embeddings are automatically organized into a
hierarchical structure that mimics the hierarchical clustering process. As a
result, it can generate new data via conditional sampling and perform
hierarchical clustering. However, its generative performance falls short of
state-of-the-art deep generative methods, and it exhibits common issues
associated with VAEs, such as generating blurry images (Bredell et al., 2023).
In contrast, diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020)
have recently gained significant attention for their image-generation
capabilities.

Our work aims to bridge this gap by (a) improving the architectural design of
TreeVAE, and (b) integrating a second-stage Denoising Diffusion Probabilistic
Model (DDPM) that is conditioned on the cluster-specific representations
learned by TreeVAE. Our proposed approach, Diffuse-TreeVAE, generates high-
quality, distinct, and representative cluster-specific images; i.e. images for
each leaf of the learned tree. The proposed generation process (depicted in
Figure 1) goes as follows: Diffuse-TreeVAE samples the root embedding of its
tree, then propagates the generations through the leaves, and, finally, it
produces high-quality leaf-specific images by guiding the reverse process of
the DDPM on both the leaf reconstructions and the corresponding path in the
tree. The resulting leaf-specific images share common general properties
(which are sampled at the root) and differ by cluster-specific features.

Our main contributions are as follows: We provide (i) a holistic approach to
clustering-based generative modeling, and (ii) a novel method for controlling
image synthesis in diffusion models. We show that our approach (a) overcomes
previous generative limitations of VAE-based clustering methods, and (b)
produces newly generated samples that are more representative of the
respective clusters in the data and closer to the true image distribution.

Figure 1: Schematic overview of the Diffuse-TreeVAE model: The reverse model
of the DDPM (bottom) is conditioned on both the reconstruction and the index
of the selected leaf $l$ obtained from the associated, pre-trained TreeVAE.
The denoising function of the DDPM learns to refine the TreeVAE-based
reconstructions.

##  2 Diffuse-TreeVAE

We propose Diffuse-TreeVAE111The code is publicly available at
https://github.com/JoGo175/diffuse-treevae, a two-stage generative framework
that is composed of a VAE-based generative hierarchical clustering model
(TreeVAE), followed by a cluster-conditional denoising diffusion probabilistic
model (DDPM). This novel combination of VAEs and diffusion models extends the
generator-refiner framework introduced by DiffuseVAE (Pandey et al., 2022) to
hierarchical clustering tasks. Here, TreeVAE (Manduchi et al., 2023) serves as
the generator, while a DDPM (Ho et al., 2020), conditioned on the TreeVAE
leaves, refines the generated images. Figure 1 illustrates the workflow of
Diffuse-TreeVAE.

The first part of Diffuse-TreeVAE involves an adapted version of the TreeVAE
model (Manduchi et al., 2023). TreeVAE is a generative model that
intrinsically learns to hierarchically separate data into clusters via a
latent tree. During training, the model grows a binary tree structure
${\mathcal{T}}$. The set ${\mathbb{V}}$ represents the nodes of the tree. Each
node corresponds to a stochastic latent variable, denoted as
${\mathbf{z}}_{0},\dots,{\mathbf{z}}_{V}$. The parameters of these latent
variables are determined by their parent nodes through neural networks called
transformations. The set of leaves ${\mathbb{L}}$, where
${\mathbb{L}}\subset{\mathbb{V}}$, represents the clusters present in the
data. Starting from the root node, ${\mathbf{z}}_{0}$, a given sample
traverses the tree to a leaf node, ${\mathbf{z}}_{l}$, in a probabilistic
manner. The probabilities for whether to go to the left or right child at each
internal node are determined by neural networks termed routers. Thus, the
latent tree encodes sample-specific probability distributions of paths. Each
leaf embedding, ${\mathbf{z}}_{l}$ for $l\in{\mathbb{L}}$, represents the
learned data representations, and leaf-specific decoders use these embeddings
to reconstruct or generate new images, i.e. given a dataset ${\bm{X}}$,
TreeVAE reconstructs $\hat{{\bm{X}}}=\\{\hat{{\bm{X}}}^{(l)}\mid
l\in{\mathbb{L}}\\}$.

In this work, we improve the architectural design of the TreeVAE model. In the
original TreeVAE, an initial encoder projects the images to flattened
representations at the start of the bottom-up process, with the remaining
components of the model relying on MLP layers. We adapted our TreeVAE method
to use convolutional layers throughout the model structure instead of MLP
layers. Thus, our adaptation avoids flattening the representations and instead
utilizes lower-dimensional representations with multiple channels throughout
the model. Additionally, we incorporated residual connections to enhance the
training and performance of the model. These modifications aim to preserve
spatial information and enable more efficient learning, contributing to the
overall effectiveness of our model. However, it is important to note that this
model suffers from the typical VAE issue of producing blurry image generations
(Bredell et al., 2023). Despite this limitation, the reconstructed images and
learned clustering still provide meaningful representations of the data, which
are utilized in the second stage of our proposed Diffuse-TreeVAE framework.

Figure 2: (Top) Samples from the CIFAR-10 test set. (Middle) Reconstructions
from the CNN-TreeVAE model. (Bottom) Refined reconstructions from the Diffuse-
TreeVAE model, conditioned on the CNN-TreeVAE reconstructions and the
corresponding leaves.

Our model leverages the cluster assignments and image generation capabilities
of TreeVAE to guide a second-stage diffusion model, specifically a DDPM (Ho et
al., 2020). We adopt and adapt the generator-refiner framework (Pandey et al.,
2022), where the VAE generates the initial, typically blurred images and the
conditioned DDPM refines these reconstructions to produce sharper, higher-
quality images. Instead of employing a conventional VAE, our model integrates
TreeVAE. During training and inference, the selected leaf is randomly sampled
based on the leaf probabilities learned by TreeVAE. The selected leaf
reconstruction, along with the leaf index as the cluster signal, conditions
the DDPM reverse process, as depicted in Figure 1. Formally, given the input
data ${\bm{X}}$, here denoted as ${\bm{X}}_{0}$, we define a sequence of $T$
noisy representations of the input ${\bm{x}}_{0}$, yielding ${\bm{x}}_{1:T}$.
The forward process, $q\left({\bm{x}}_{1:T}\mid{\bm{x}}_{0}\right)$, that
gradually destroys the structure of each data sample follows the standard DDPM
process (Ho et al., 2020). The reverse process, on the other hand, is
conditioned on the TreeVAE reconstructions
$\hat{{\bm{x}}}_{0}=\\{\hat{{\bm{x}}}_{0}^{(l)}\mid l\in{\mathbb{L}}\\}$ and
on the leaf assignments:

$\displaystyle\begin{split}&l\sim p(l|{\bm{x}}_{0}),\\\
&p_{\psi}\\!\\!\left({\bm{x}}_{0:T}\\!\mid\\!\hat{{\bm{x}}}_{0}^{(l)},l\right)=p\left({\bm{x}}_{T}\right)\prod_{t=1}^{T}p_{\psi}\\!\\!\left({\bm{x}}_{t-1}\\!\mid\\!{\bm{x}}_{t},\hat{{\bm{x}}}_{0}^{(l)},l\right)\\!,\\\
\end{split}$ (1)

where $p(l|{\bm{x}}_{0})$ is the probability that the sample ${\bm{x}}_{0}$ is
assigned to leaf $l$. This method ensures that leaves with smaller assignment
probabilities are considered, encouraging the DDPM to perform effectively
across all leaves. Consequently, our approach addresses the distinct clusters
inherent in TreeVAE, allowing the model to adapt specifically to different
clusters and encouraging cluster-specific refinements in the images. This
guidance in the image generation process assists the denoising model in
learning cluster-specific image reconstructions. On the other hand, the
forward noising process remains unconditional. Diffuse-TreeVAE directly
utilizes the reconstructions instead of the latent embeddings for
conditioning, as there exists a deterministic relationship between leaf
embeddings and leaf reconstruction, provided by the leaf-specific decoder.

By using the generator-refiner framework, Diffuse-TreeVAE maintains the same
clustering performance as the underlying TreeVAE. The DDPM refines the
generated output samples without influencing the cluster assignments in the
TreeVAE model. This is achieved through a two-stage training strategy, where
the conditional DDPM is trained using a pre-trained CNN-TreeVAE model. Thus,
Diffuse-TreeVAE combines the effective clustering of TreeVAE with the superior
image generation capabilities of diffusion models.

##  3 Results

We evaluate the generative performance of our model on the MNIST (Lecun et
al., 1998), FashionMNIST (Xiao et al., 2017), and CIFAR-10 (Krizhevsky, 2009)
datasets. Our analysis compares three models: the original MLP-based TreeVAE,
referenced as MLP-TreeVAE (Manduchi et al., 2023); our CNN-based adaptation,
referred to as CNN-TreeVAE; and our novel proposal, the Diffuse-TreeVAE model.
The latter model is conditioned on reconstructions and clusters derived from
the CNN-TreeVAE, enhancing its capacity for generative performance.
Reconstruction performance is assessed through the FID score (Heusel et al.,
2017), calculated for the reconstructed images sourced from the 10,000 samples
within the test set. Similarly, generation performance is evaluated using the
FID score, this time computed for 10,000 newly generated images.

Table 1: Test set generative performances of different TreeVAE models. Lower FID scores indicate better performance. Means and standard deviations are computed across 10 runs with different seeds. The best result for each dataset is marked in bold. Dataset | Method | FID (rec) | FID (gen)
---|---|---|---
MNIST | MLP-TreeVAE | $25.8\,\text{{\footnotesize$\pm\,0.4$}}$ | $25.3\,\text{{\footnotesize$\pm\,1.0$}}$
| CNN-TreeVAE | $24.9\,\text{{\footnotesize$\pm\,1.1$}}$ | $22.8\,\text{{\footnotesize$\pm\,1.4$}}$
| Diffuse-TreeVAE |  1.5 $\pm\,0.1$ |  16.2 $\pm\,5.7$
Fashion | MLP-TreeVAE | $44.7\,\text{{\footnotesize$\pm\,0.6$}}$ | $46.8\,\text{{\footnotesize$\pm\,0.9$}}$
| CNN-TreeVAE | $36.5\,\text{{\footnotesize$\pm\,0.6$}}$ | $39.0\,\text{{\footnotesize$\pm\,0.8$}}$
| Diffuse-TreeVAE |  4.1 $\pm\,0.6$ |  4.2 $\pm\,0.5$
CIFAR10 | MLP-TreeVAE | $225.5\,\text{{\footnotesize$\pm\,3.3$}}$ | $237.0\,\text{{\footnotesize$\pm\,4.0$}}$
| CNN-TreeVAE | $190.5\,\text{{\footnotesize$\pm\,2.0$}}$ | $200.9\,\text{{\footnotesize$\pm\,2.5$}}$
| Diffuse-TreeVAE |  15.4 $\pm\,0.3$ |  22.3 $\pm\,0.3$

Figure 3: Diffuse-TreeVAE model trained on FashionMNIST. For each cluster,
random newly generated images are displayed. Below each set of images, a
normalized histogram (ranging from 0 to 1) shows the distribution of predicted
classes from an independent, pre-trained classifier on FashionMNIST for all
newly generated images in each leaf with a significant probability of reaching
that leaf.

Figure 4: Diffuse-TreeVAE model trained on CIFAR-10. For each cluster, random
newly generated images are displayed. Below each set of images, a normalized
histogram (ranging from 0 to 1) shows the distribution of predicted classes
from an independent, pre-trained classifier on CIFAR-10 for all newly
generated images in each leaf with a significant probability of reaching that
leaf.

Table 1 illustrates the generative performance across the various datasets. In
every case, the CNN-TreeVAE demonstrates improvements compared to the original
model. However, despite being lower, its FID scores remain at a similar level.
Hence, the CNN-TreeVAE model continues to generate visibly blurry images. On
the other hand, the Diffuse-TreeVAE significantly enhances the generative
capabilities of the model, yielding much lower FID scores, often by an order
of magnitude. This improvement is evident in the quality of the generated
images, as depicted in Figure 2. Here, we visually compare the reconstructions
generated by the Diffuse-TreeVAE model with those produced by the underlying
CNN-TreeVAE model, which was used to condition the Diffuse-TreeVAE along with
the cluster signal. Specifically, it can be observed that the reconstructions
generated by the Diffuse-TreeVAE are notably sharper and thus exhibit closer
adherence to the true distribution of the test data. The model demonstrates
enhanced capability in reproducing fine details within the images while
preserving the overall color and structure. However, it is important to note
that these improvements may introduce some inconsistencies, resulting in
reconstructions that appear more realistic but deviate slightly from the
original image being reconstructed.

To assess the quality of the newly generated images, we train a classifier on
the original dataset using the training data and then utilize it to classify
the newly generated images from our Diffuse-TreeVAE. Specifically, we classify
the newly generated images for each cluster separately. Ideally, the majority
of generated images from a cluster are classified into one or very few classes
from the original dataset. The more generations from a cluster that are
classified into one class only, the “purer” or “less ambiguous” we consider
the generations to be. For this classification task, we utilize a ResNet-50
model (He et al., 2016) trained on each dataset.

Figure 5: Image generations from each leaf of (top) a CNN-TreeVAE, (middle) a
cluster-unconditional Diffuse-TreeVAE, and (bottom) a cluster-conditional
Diffuse-TreeVAE, all trained on CIFAR-10. Each row displays the generated
images from all leaves of the specified model, starting with the same sample
from the root. The corresponding leaf probabilities are shown at the top of
the image and are by design the same for all models.

In Figure 3, we present randomly generated images from a Diffuse-TreeVAE model
trained on FashionMNIST. Notably, the model in this instance has identified
only seven clusters instead of the expected ten. These clusters tend to group
various clothing items together, such as “Shirt”, “T-Shirt”, and “Pullover”.
Below the generated images, normalized histograms depict the distribution of
the predicted classes by the classifier on the newly generated images. For
instance, clusters representing trousers and bags appear to accurately and
distinctly capture their respective classes, as all their generated images are
classified into one group only. Conversely, certain clusters manifest a
mixture of classes, indicating that they are grouped together. This
observation is further supported by the histograms. Similar results can be
observed for the CIFAR-10 or MNIST data, as shown in Figure 4 and Figure 6
respectively.

To assess whether the additional conditioning on the selected leaf index helps
create more cluster-specific representations, we perform an ablation study.
This study compares the generations of two Diffuse-TreeVAE models which only
differ in one aspect: one conditioned only on the reconstructions and the
other conditioned on both the reconstructions and the leaf index. For this
ablation, we use the previously defined independent classifier to create
histograms for each leaf to evaluate how cluster-specific the newly generated
images are. As previously mentioned, ideally, the majority of generated images
from a cluster should be classified into one or very few classes from the
original dataset. To quantify this, we compute the average entropy for all
leaf-specific histograms. Lower entropy indicates less variation in the
histograms, and thus more leaf-specific generations. Table 2 presents the
results for the unconditional and conditional Diffuse-TreeVAEs across all
datasets. The conditional model consistently shows lower mean entropy,
indicating that additional conditioning on the leaf indices indeed helps guide
the model to generate more distinct and representative images for each leaf.
Figure 5 visually presents the leaf generations for one sample of these models
alongside the underlying CNN-TreeVAE generations, which were used to condition
both models. Further examples can be found in A.2. It can be observed that
both the unconditional and conditional models exhibit a significant
improvement in image quality. However, the images in the cluster-conditional
model are more diverse, demonstrating greater adaptability for each cluster.
This is evident as the images clearly show indications of multiple true
CIFAR-10 classes, with recognizable features such as horses, ships, or cars.
Notably, across all models, the leaf-specific images share common properties
sampled at the root while varying in cluster-specific features from leaf to
leaf within each model.

Table 2: Cluster-specificity of Diffuse-TreeVAE generations for cluster-unconditional and cluster-conditional reverse models, measured by mean entropy. Lower entropy indicates more cluster-specific generations. Mean entropy is computed across all leaf-specific histograms of the predicted classes for newly generated images. The best result for each dataset is marked in bold. Dataset | Method | Mean Entropy
---|---|---
MNIST | unconditional | $1.24$
| conditional | 0.13
Fashion | unconditional | 0.66
| conditional | 0.66
CIFAR10 | unconditional | $1.12$
| conditional | 0.82

##  4 Conclusion

In this work, we present Diffuse-TreeVAE, a novel approach to integrate
hierarchical clustering into diffusion models. By enhancing TreeVAE with a
Denoising Diffusion Probabilistic Model conditioned on the cluster-specific
representations, we have developed a model capable of generating distinct,
high-quality images that faithfully represent their respective data clusters.
This approach not only improves the visual fidelity of generated images but
also ensures that these representations are true to the underlying data
distribution. Diffuse-TreeVAE offers a robust framework that bridges the gap
between clustering precision and generative performance, thereby expanding the
potential applications of generative models in areas requiring detailed and
accurate visual data interpretation.

## Acknowledgments and Disclosure of Funding

Laura Manduchi is supported by the SDSC PhD Fellowship #1-001568-037. Moritz
Vandenhirtz is supported by the Swiss State Secretariat for Education,
Research and Innovation (SERI) under contract number MB22.00047.

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##  Appendix A Appendix

###  A.1 Generations on MNIST and CIFAR-10

Figure 6 presents an additional plot similar to those in Figure 3 and Figure 4
from the main text. This plot illustrates the generated images of the Diffuse-
TreeVAE model when trained on the MNIST dataset. In each of these plots, we
display randomly generated images for each cluster. Below each set of leaf-
specific images, we provide a normalized histogram showing the distribution of
predicted classes by an independent ResNet-50 classifier that has been pre-
trained on the training data of the respective dataset. This visualization
helps in understanding how well the model can generate distinct and meaningful
clusters in the context of different datasets.

Figure 6: Diffuse-TreeVAE model trained on MNIST. For each cluster, random
newly generated images are displayed. Below each set of images, a normalized
histogram (ranging from 0 to 1) shows the distribution of predicted classes
from an independent, pre-trained classifier on MNIST for all newly generated
images in each leaf with a significant probability of reaching that leaf.

###  A.2 Additional Generation Examples for Conditional vs. Unconditional
Diffuse-TreeVAE

Figure 7 presents three additional examples similar to Figure 5, comparing the
leaf-specific generations of the conditional and unconditional Diffuse-TreeVAE
models, alongside the underlying CNN-TreeVAE generations.

Figure 7: For each example, we show image generations from each leaf of (top)
a CNN-TreeVAE, (middle) a cluster-unconditional Diffuse-TreeVAE, and (bottom)
a cluster-conditional Diffuse-TreeVAE, all trained on CIFAR-10. Each row
displays the generated images from all leaves of the specified model, starting
with the same sample from the root. The corresponding leaf probabilities are
shown at the top of the image and are by design the same for all models.


construct a suitable candidate list for exhaustive multi-object spectroscopic
follow up. Although time intensive, such an approach will be required in order
to determine the recent star formation rate within the GC - and consequently
the production rate of relativistic remnants - as well as the rate of feedback
of mechanical energy and chemically enriched material via stellar winds and
SNe. This is essential to our understanding of the relative contributions of
massive stars and the supermassive black hole Sgr A∗ to the energy budget and
ecology of the GC, in terms of regulating the progression of star formation,
the generation of very high energy (GeV) $\gamma$-ray emission, the inflation
of both X-ray and radio out-of-plane bubbles centred on the GC, and the
production of PeV cosmic rays.

###### Acknowledgements.

This research was supported by the Science and Technology Facilities Council.
F.N. acknowledges financial support through Spanish grants
ESP2017-86582-C4-1-R and PID2019-105552RB-C41 (MINECO/MCIU/AEI/FEDER) and from
the Spanish State Research Agency (AEI) through the Unidad de Excelencia
“MarÍa de Maeztu”-Centro de AstrobiologÍa (CSIC-INTA) project No.
MDM-2017-0737. L.R.P. acknowledges support from the Generalitat Valenciana
through the grant PROMETEO/2019/041

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##  Appendix A Incomplete exploitation of the Pa$\alpha$ excess survey

After excluding clusters members and those isolated stars with spectroscopic
observations, 27 objects from the primary list of 152 Paschen $\alpha$ excess
sources of Dong et al. (dong11 (2011)) remain unobserved. Of these six have
colours indicative of foreground stars ($(H-K)<1$ \- P1, 27, 55, 116, 145, and
149) while a further eight lack photometry (P14, 32, 33, 37 93, 138, 146, and
152). This leaves a total of 13 stars with colours consistent with membership
of the CMZ (P24, 26, 44, 45, 46, 51, 52, 54, 61, 76, 115, 139, and 148);
however of these only P44, 76, and 115 have $K<13$ which, excluding classical
Be stars, characterise the majority of massive stars in our survey (Sect. 3).

The list of 189 secondary Pa$\alpha$ excess targets is much more poorly
characterised. 101 of the isolated stars from this roster lack both
spectroscopic and photometric observations so we may not assign them to the
CMZ, let alone classify them. Of the isolated stars currently lacking
spectroscopic observations but with near-IR photometric data, 31 appear to
have colours consistent with foreground objects (S48, 50, 51, 53, 54, 68, 70,
71, 72, 76, 77, 83, 84, 86, 89, 92, 93, 95, 97,99, 101, 109, 118, 121, 128,
145, 148, 149, 150, 154, and 170) leaving 33 possible CMZ members (S47, 49,
52, 60, 64, 65, 78, 79, 88, 91, 103, 106, 110, 113, 114, 117, 119, 123, 125,
136, 146, 151, 153, 158, 161, 168, 169, 173, 178, 182, 183, 185, 187). Of the
latter, ten have $K<13$ and hence are encouraging candidates for massive stars
(S47, 52, 88, 103, 114, 117, 119, 125, 151, and 169).

To summarise: in comparison to our spectroscopic census, 13 stars appear to be
strong massive star candidates, 26 fainter objects have photometric properties
consistent with classical Be stars located within the GC, while a large cohort
of 109 stars currently lack the photometric data required to assess of their
location and nature.