# Neutrino Mixing and Leptogenesis in a $L_{e}-L_{\mu}-L_{\tau}$ model

Giorgio Arcadia<EMAIL_ADDRESS>Simone Marcianob,c
<EMAIL_ADDRESS>Davide Meloni b<EMAIL_ADDRESS>a
Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze
della Terra, Universita degli Studi di Messina, Via Ferdinando Stagno
d’Alcontres 31, I-98166 Messina, Italy  b Dipartimento di Matematica e Fisica,
Università di Roma Tre, Via della Vasca Navale 84, 00146, Roma, Italy  c INFN
Sezione di Roma Tre, Via della Vasca Navale 84, 00146, Roma, Italy

###### Abstract

We present a simple extension of the Standard Model with three right-handed
neutrinos in a SUSY framework, with an additional U(1$)_{\text{F}}$ abelian
flavor symmetry with a non standard leptonic charge $L_{e}-L_{\mu}-L_{\tau}$
for lepton doublets and arbitrary right-handed charges. We show that the model
is able to reproduce the experimental values of the mixing angles of the PMNS
matrix and of the $r=\Delta m_{\text{sun}}^{2}/\Delta m_{\text{atm}}^{2}$
ratio, with only a moderate fine tuning of the Lagrangian free parameters. The
baryon asymmetry of the Universe is generated via thermal leptogenesis through
CP-violating decays of the heavy right-handed neutrinos. We present a detailed
numerical solution of the relevant Boltzmann equation accounting for the
impact of the distribution of the asymmetry in the various lepton flavors.

###### Contents

  1. I Introduction
  2. II The Model
    1. II.1 Charged lepton sector
    2. II.2 Neutrino sector
  3. III Leptogenesis
  4. IV Conclusions
  5. A  Full Boltzmann Equations

##  I Introduction

The Standard Model (SM) of particle physics has proven to be one of the most
accurate theories to explain microscopic interactions at an unprecedented
level. In spite of its many successes, it fails to account for relevant low
energy data, such as the structure of fermion masses and mixings (in
particular, the non-vanishing neutrino masses) and the value of the baryon
asymmetry of the Universe (BAU), which is commonly expressed by the parameter:

$\eta_{B}\equiv\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}}\bigg{|}_{0}\;,$ (1)

where $n_{B},n_{\bar{B}}$ and $n_{\gamma}$ are the number densities of the
baryons, antibaryons and photons, while the subscript ”0” stands for ”at
present time”. Latest observations provide a numerical value of
$\eta_{B}\approx 6.1\cdot 10^{-10}$ Aghanim  _et al._ (2020). In recent times,
an enormous experimental progress has been made in our knowledge of the
neutrino properties and it has been clearly shown that the lepton mixing
matrix contains two large and one small mixing angle, and that the two
independent mass-squared differences are both different from zero Esteban  _et
al._ (2020); de Salas  _et al._ (2021); Capozzi  _et al._ (2021). Although
several abelian and non-abelian symmetries acting on flavour space have been
proposed to explain such a pattern, not an unique framework emerged as the
optimal one Altarelli and Feruglio (2010). Thus, one is still motivated to
explore scenarios where different symmetries and/or field (charge) assignments
to the group representations are studied in details. In this context, a less
explored possibility (compared to the most famous discrete non-abelian
symmetries) is given by the $U(1)_{F}$ flavor symmetry with non-standard
leptonic charge $L_{e}-L_{\mu}-L_{\tau}$ for lepton doublets Petcov (1982) and
arbitrary right-handed charges Altarelli and Feruglio (2004). As it is well
known, in the limit of exact symmetry, the neutrino mass matrix assumes the
following structure:

$\displaystyle m_{\nu}=m_{0}\left(\begin{array}[]{ccc}0&1&x\\\ 1&0&0\\\
x&0&0\end{array}\right)\,,$ (5)

which leads to a spectrum of inverted type and to $\theta_{12}=\pi/4$,
$\tan\theta_{23}=x$ (_i.e._ large atmospheric mixing for $x\sim{\cal O}(1)$)
and $\theta_{13}=0$. While the previous texture can be considered a good
Leading Order (LO) result, it evidently fails to reproduce two independent
mass differences (two eigenvalues have the same absolute values) and, with the
exception of the atmospheric angle, also fails in the correct description of
the solar and reactor angles. Models based on the see-saw mechanism Lavoura
and Grimus (2000); Grimus and Lavoura (2005) have been proven to be
sufficiently realistic as to accommodate solar and atmospheric splittings but
either the solar angle was too large or the reactor angle was (almost)
vanishing. With the increasing precision in the measurement of oscillation
parameters, it turned out that both $\theta_{12}$ and $\theta_{13}$ were
substantially different from their LO results; the observation that
corrections of ${\cal O}(\lambda)$ ($\lambda$ being the Cabibbo angle) are
needed to bring both mixing angles to their experimental values, encouraged to
explore the contributions to the neutrino mixing matrix from the charged
lepton sector Petcov and Rodejohann (2005); Altarelli and Franceschini (2006);
in this context, a natural value of $r=\Delta m^{2}_{\it sol}/\Delta
m^{2}_{\it arm}\sim{\cal O}(\lambda^{2})$ was also obtained Meloni (2012),
thus showing that models based on $L_{e}-L_{\mu}-L_{\tau}$ are capable to
successfully describe low energy neutrino data. An important missing piece of
the previous constructions is the possibility to explain the value of the BAU
through leptogenesis. In Grimus and Lavoura (2005) it was clearly shown that
the baryon-to-photon ratio of the Universe $\eta_{B}$ is proportional to the
neutrino mass $m_{1}$ and, for vanishing lightest mass $m_{3}$,
$m_{1}\sim\sqrt{\Delta m^{2}_{\it atm}}$, thus producing an hopelessly small
$\eta_{B}$. Providing a quantitative leptogenesis analysis has become
sophisticated in recent years, due to the addition of many ingredients, such
as various washout effects Buchmuller  _et al._ (2005); Giudice  _et al._
(2004a) or thermal corrections to the particle masses Giudice  _et al._
(2004a); Comelli and Espinosa (1997). Also, the flavor effects can have a
significant impact on the final value of the baryon asymmetry, as widely shown
in Abada  _et al._ (2006); Nardi  _et al._ (2006); Fong  _et al._ (2012).
With the present paper we aim to go beyond the existing literature, assessing
whether see-saw models based on the $L_{e}-L_{\mu}-L_{\tau}$ quantum number
can simultaneously account for neutrino masses and mixing and explain the BAU
through thermal leptogenesis. The paper is structured as follows. In Sect.II
we describe our model and derive the analytic expressions for the mass ratio
$r$ and the mixing angles, showing that appropriate choices of the Lagrangian
parameters lead to a satisfactory description of low energy data; in Sect.III
we face the problem of reproducing the value of $\eta_{B}$, analyzing the
resonant and hierarchical scenarios and solving the related Boltzmann
equations. Our conclusions are drawn in Sect.IV.

##  II The Model

$l_{e}$ $l_{\mu}$ $l_{\tau}$ $l_{e}^{c}$ $l_{\mu}^{c}$ $l_{\tau}^{c}$ $F_{1}$
$F_{2}$ $H_{u}$ $H_{d}$ $N_{1}$ $N_{2}$ $N_{3}$ U(1$)_{\text{F}}$ +1 -1 -1
$-13$ $7$ 3 2 1/2 0 0 -1 1 0

Table 1: $U(1)_{F}$ charges for leptons, Higgses and flavon fields.

In the following, we summarize the relevant features of our see-saw flavor
model based on a broken $U(1)_{F}$ symmetry. In the proposed scenario, the
left handed lepton doubles have charge $L_{e}-L_{\mu}-L_{\tau}$ Altarelli and
Franceschini (2006) under the $U(1)_{F}$, while the right-handed $SU(2)$
singlets $l_{e,\mu,\tau}^{c}$ have the charges reported in Tab.1. The spectrum
of the theory also contains three heavy sterile neutrinos $N_{i=1,2,3}$,
needed for the generation of the light neutrino masses as well as for the
implementation of the leptogenesis process. The flavor symmetry is broken by
vacuum expectation values (vev’s) of $SU(2)$ singlet scalar fields (flavons)
suitably charged under the $U(1)_{F}$ symmetry. Non-vanishing vevs are
determined by the D-term potential Altarelli  _et al._ (2008):

$\displaystyle
V_{D}=\frac{1}{2}(M_{\text{FI}}^{2}-g_{F}|F_{1}|^{2}-g_{F}|F_{2}|^{2})\,,$ (6)

where $g_{F}$ denotes the gauge coupling constant of the $U(1)_{F}$ symmetry
while $M_{\text{FI}}$ is the Fayet-Iliopulos term. Non-zero vevs are obtained
by imposing the SUSY minimum $V_{D}=0$. Without loss of generality, we can
assume equal vevs for the two flavons and define $\lambda=\langle
F_{1}\rangle/M_{F}=\langle F_{2}\rangle/M_{F}$ the common ratio between the
vevs of the flavons and the scale $M_{F}$ at which the flavour symmetry is
broken.

###  II.1 Charged lepton sector

In the charged lepton sector, many operators of different dimensions enter the
Lagrangian; to avoid cumbersome expressions, we quote here the lowest
dimensional operators contributing to each entry of the mass matrix:

$\displaystyle\mathcal{L}=$ $\displaystyle
a_{11}l_{e}l_{e}^{c}\left(\frac{F_{1}}{M_{F}}\right)^{6}H_{d}+a_{12}l_{e}l_{\mu}^{c}\left(\frac{F_{1}^{\dagger}}{M_{F}}\right)^{4}H_{d}+a_{13}l_{e}l_{\tau}^{c}\left(\frac{F_{1}^{\dagger}}{M_{F}}\right)^{2}H_{d}+$
(7) $\displaystyle
a_{21}l_{\mu}l_{e}^{c}\left(\frac{F_{1}}{M_{F}}\right)^{7}H_{d}+a_{22}l_{\mu}l_{\mu}^{c}\left(\frac{F_{1}}{M_{F}}\right)^{3}H_{d}+a_{23}l_{\mu}l_{\tau}^{c}\left(\frac{F_{1}^{\dagger}}{M_{F}}\right)H_{d}+$
$\displaystyle
a_{31}l_{\tau}l_{e}^{c}\left(\frac{F_{1}}{M_{F}}\right)^{7}H_{d}+a_{32}l_{\tau}l_{\mu}^{c}\left(\frac{F_{1}}{M_{F}}\right)^{3}H_{d}+a_{33}l_{\mu}l_{\tau}^{c}\left(\frac{F_{1}^{\dagger}}{M_{F}}\right)H_{d}+\text{h.c.}\,,$

where all $a_{ij}$ coefficients are generic ${\cal O}(1)$ free parameters.
After flavor and electroweak symmetry breakings, the previous Lagrangian
generates a mass matrix whose elements have the general structure:

$(m_{l})_{ij}\sim a_{ij}l_{i}l_{i}^{c}{\left(\frac{\langle
F_{1}\rangle}{M_{F}}\right)}^{\alpha_{ij}}{\left(\frac{\langle
F_{2}\rangle}{M_{F}}\right)}^{\beta_{ij}}\langle H_{d}\rangle\,,$ (8)

where $\alpha_{ij}$ and $\beta_{ij}$ denote the appropriate powers of the
flavon fields needed to generate a singlet under $U(1)_{F}$. Factorizing out
the $\tau$ mass, the charged lepton mass matrix assumes the following form:

$m_{l}\sim
m_{\tau}\begin{pmatrix}a_{11}\lambda^{5}&a_{12}\lambda^{3}&a_{13}\lambda\\\
a_{21}\lambda^{6}&a_{22}\lambda^{2}\text{e}^{i\phi_{22}}&a_{23}\text{e}^{i\phi_{23}}\\\
a_{31}\lambda^{6}&a_{32}\lambda^{2}\text{e}^{i\phi_{32}}&1\end{pmatrix}\,.$
(9)

For $\lambda<1$, the following mass ratios
$m_{e}:m_{\mu}:m_{\tau}=\lambda^{5}:\lambda^{2}:1$ is found, which naturally
reproduces the observed pattern if $\lambda\sim 0.22$. It is not difficult to
derive the left-handed rotation $U_{l}$, which contributes to the total
neutrino mixing matrix $U_{PMNS}$ Maki  _et al._ (1962); Pontecorvo (1957),
diagonalizing the hermitean $m_{l}m_{l}^{\dagger}$ combination:

$U_{l}=\begin{pmatrix}-1&\dfrac{a_{13}\lambda}{\sqrt{1+a_{23}^{2}e^{2i\phi_{23}}}}&\dfrac{a_{13}\lambda}{\sqrt{1+a_{23}^{2}e^{2i\phi_{23}}}}\\\
-\dfrac{a_{13}a_{32}e^{i(\phi_{22}+\phi_{23})}\lambda}{-a_{23}a_{32}e^{i\phi_{22}}+a_{22}e^{i(\phi_{23}+\phi_{32})}}&\dfrac{a_{23}e^{i\phi_{23}}}{\sqrt{1+a_{13}^{2}e^{2i\phi_{23}}}}&\dfrac{a_{23}e^{i\phi_{23}}}{\sqrt{1+a_{13}^{2}e^{2i\phi_{23}}}}\\\
-\dfrac{a_{13}a_{22}e^{i(\phi_{23}+\phi_{32})}\lambda}{-a_{23}a_{32}e^{i\phi_{22}}+a_{22}e^{i(\phi_{23}+\phi_{32})}}&\dfrac{1}{\sqrt{1+a_{23}^{2}e^{2i\phi_{23}}}}&\dfrac{1}{\sqrt{1+a_{23}^{2}e^{2i\phi_{23}}}}\end{pmatrix}+\mathcal{O}(\lambda^{2})\,.$
(10)

As expected, the diagonal elements of $U_{l}$ are unsuppressed; also, the (23)
and (33) entries are of ${\cal O}(1)$, which indicates that the charged lepton
contribution to the atmospheric angle will be large. Notice also that, being
the (12) and (13) elements of ${\cal O}(\lambda)$, we expect a similar
corrections to the solar and reactor angles.

###  II.2 Neutrino sector

In the neutrino sector, masses are generated through the standard type-I see-
saw mechanism; at the renormalizable level, the see-saw Lagrangian reads:

$\displaystyle\mathcal{L}^{LO}=$
$\displaystyle\frac{1}{2}\mathcal{M}W\overline{N}^{c}_{1}N_{2}+\frac{1}{2}\mathcal{M}Z\overline{N}^{c}_{3}N_{3}-a\overline{N}_{1}H_{u}l_{\mu}+$
(11)
$\displaystyle-b\overline{N}_{1}H_{u}l_{\tau}-c\overline{N}_{2}H_{u}l_{e}+\text{h.c.
}\,,$

where $\mathcal{M}$ is an overall Majorana mass scale while $W,Z,a,b,c$ are
dimensionless coefficients which will be regarded as free parameters. When
$H_{u}$ acquires a vev $v_{u}$, Majorana and Dirac mass matrices are
generated:

$M_{R}=\mathcal{M}\left(\begin{array}[]{ccc}0&W&0\\\ W&0&0\\\
0&0&Z\end{array}\right),\,\,\,m_{D}=v_{u}\left(\begin{array}[]{ccc}0&a&b\\\
c&0&0\\\ 0&0&0\end{array}\right)\,.$ (12)

Next-to-leading order (NLO) contributions are given by higher dimensional
operators suppressed by the large scale $M_{F}$; up to two-flavon insertions,
we get111Notice that we use the same overall scale $\mathcal{M}$ in the
Majorana mass terms.:

$\displaystyle\mathcal{L}^{NLO}=$
$\displaystyle\frac{1}{2}\mathcal{M}m_{11}\overline{N}_{1}^{c}N_{1}\left(\frac{F_{1}}{M_{F}}\right)+\frac{1}{2}\mathcal{M}m_{13}\overline{N}_{1}^{c}N_{3}\left(\frac{F_{2}}{M_{F}}\right)^{2}+\frac{1}{2}\mathcal{M}m_{22}\overline{N}_{2}^{c}N_{2}\left(\frac{F_{1}^{\dagger}}{M_{F}}\right)+$
(13)
$\displaystyle+\frac{1}{2}\mathcal{M}m_{23}\overline{N}_{2}^{c}N_{3}\left(\frac{F_{2}^{\dagger}}{M_{F}}\right)^{2}-d_{11}\overline{N}_{1}H_{u}l_{e}\left(\frac{F_{1}^{\dagger}}{M_{F}}\right)-d_{22}\overline{N}_{2}H_{u}l_{\mu}\left(\frac{F_{1}}{M_{F}}\right)+$
$\displaystyle-
d_{23}\overline{N}_{2}H_{u}l_{\tau}\left(\frac{F_{1}}{M_{F}}\right)-d_{31}\overline{N}_{3}H_{u}l_{e}\left(\frac{F_{2}^{\dagger}}{M_{F}}\right)^{2}-d_{32}\overline{N}_{3}H_{u}l_{\mu}\left(\frac{F_{2}^{\dagger}}{M_{F}}\right)^{2}+$
$\displaystyle-
d_{33}\overline{N}_{3}H_{u}l_{\tau}\left(\frac{F_{2}^{\dagger}}{M_{F}}\right)^{2}+\text{h.c.}\,.$

Their main effects is to fill the vanishing entries in eq.(12); however, as we
have numerically verified, some of the free parameters in eq.(13) needed to be
slightly adjusted to fit the low energy data. In particular, only a moderate
fine-tuning is necessary on $m_{11},\,m_{22},\,d_{11},\,d_{22}$ and $d_{23}$,
which we rescale according to:

$\displaystyle(m_{11},m_{22},d_{11},d_{22},d_{23})\rightarrow\lambda\;(m_{11},m_{22},d_{11},d_{22},d_{23})\,,\qquad(m_{11},m_{22},d_{11},d_{22},d_{23})\sim{\cal{O}}(1)\,.$

With the previous position (and reminding that $\lambda=\langle
F_{1}\rangle/M_{F}=\langle F_{2}\rangle/M_{F}$), the following Dirac and
Majorana mass matrices are obtained:

$Y=\frac{m_{D}}{v_{u}}\sim\overline{N}l\sim\begin{pmatrix}\lambda^{2}d_{11}&ae^{i\Sigma}&be^{i\Omega}\\\
ce^{i\Phi}&\lambda^{2}d_{22}&\lambda^{2}d_{23}e^{i\Theta}\\\
\lambda^{2}d_{31}&\lambda^{2}d_{32}&\lambda^{2}d_{33}\end{pmatrix}\,,$ (14)

and

$M_{R}\sim\overline{N}N\sim\mathcal{M}\begin{pmatrix}\lambda^{2}m_{11}&W&\lambda^{2}m_{13}\\\
W&\lambda^{2}m_{22}&\lambda^{2}m_{23}\\\
\lambda^{2}m_{13}&\lambda^{2}m_{23}&Z\end{pmatrix}\,.$ (15)

Notice that the Dirac mass matrix contains un-suppressed entries because of
the choice $Q_{N_{1}}=-Q_{N_{2}}$ for two of the right-handed neutrinos. The
four physical phases $\Sigma,\Omega,\Phi,\Theta$ in $Y$, obtained after a
suitable redefintion of the fermion fields, are the only source of CP
violation of our model and are not fixed by the symmetries of the Lagrangians.
For the sake of simplicity and without any loss of generality, we can assume
the parameters $m_{ij}\sim m$ and consider $m$ as a real quantity. From the
type-I seesaw master formula, $m_{\nu}\simeq-
v_{u}^{2}Y^{\text{T}}M_{R}^{-1}Y$, we get the following matrix for the light
SM neutrinos, up to $\mathcal{O}(\lambda^{2})$:

$\displaystyle m_{\nu}\simeq\frac{v_{u}^{2}}{\mathcal{M}}\times$ (16)
$\displaystyle\times\begin{pmatrix}\frac{ce^{i\Phi}\left(ce^{i\Phi}m-2d_{11}W\right)\lambda^{2}}{W^{2}}&\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}&\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\\\
-\frac{ace^{i(\Sigma+\Phi)}}{W}&\frac{ae^{i\Sigma}\left(ae^{i\Sigma}m-2d_{22}W\right)\lambda^{2}}{W^{2}}&\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\\\
-\frac{bce^{i(\Phi+\Omega)}}{W}&\frac{\left(abe^{i(\Sigma+\Omega)}m-ad_{23}e^{i(\Theta+\Sigma)}W-bd_{22}e^{i\Omega}W\right)\lambda^{2}}{W^{2}}&\frac{be^{i\Omega}\left(be^{i\Omega}m-2d_{23}e^{i\Theta}W\right)\lambda^{2}}{W^{2}}\end{pmatrix}\,.$

This mass matrix provides, as usual for models based on the
$L_{e}-L_{\mu}-L_{\tau}$ symmetry, an inverted mass spectrum.
From now on, we will distinguish two different scenarios, that will be further
elaborated when studying the BAU generated in our model, and identified by
different assumptions on the parameter $\mathcal{M}$. In order to better
understand this distinction, let us assume that all parameters in eq.(16) are
of $\mathcal{O}(1)$; thus, the light neutrino mass matrix can be recast in the
following form:

$m_{\nu}=m_{0}\begin{pmatrix}\lambda^{2}x_{1}&1&x\\\
1&x_{2}\lambda^{2}&x_{3}\lambda^{2}\\\
x&x_{3}\lambda^{2}&x_{4}\lambda^{2}\end{pmatrix}\,,$ (17)

where $m_{0}=v_{u}^{2}/\mathcal{M}\times{\cal O}(1)$ coefficients and
$(x,x_{i})$ are suitable combinations of the coefficients present in Dirac and
Majorana matrices in eqs.(14) and (15). At the leading order in $\lambda$,
$m_{\nu}$ has two degenerate eigenvalues $m_{1}=-m_{2}=\sqrt{1+x^{2}}$ and a
vanishing one, $m_{3}=0$; therefore, we can only construct the atmospheric
mass difference $\Delta m_{atm}^{2}=|m_{1}|^{2}-|m_{3}|^{2}$, which results
in:

$x^{2}=\frac{\Delta m_{atm}^{2}}{m_{0}^{2}}-1\,.$ (18)

To maintain $x\sim\mathcal{O}(1)$, we can choose the overall mass scale to
$m_{0}\sim\mathcal{O}(10^{-2})$ eV, which corresponds to the choice
$\mathcal{M}\sim 10^{15}$ GeV. Notice also that, taking into account the
corrections of $\mathcal{O}(\lambda^{2})$, the eigenvalue degeneracy is broken
and the solar mass difference can be accounted for, which results in:

$\Delta
m^{2}_{sol}=\dfrac{\lambda^{2}\left[x_{1}(1+x^{2})+x_{2}+2xx_{3}+x_{4}x^{2}\right]}{\sqrt{1+x^{2}}}\,.$
(19)

Since the masses of the three heavy right-handed neutrinos are simply given by
$M_{i}=\mathcal{M}\widetilde{M}_{i}$ with:

$\displaystyle\widetilde{M}_{1}\simeq$
$\displaystyle\;W+m\lambda^{2}+\mathcal{O}(\lambda^{3}),$ (20)
$\displaystyle\widetilde{M}_{2}\simeq$
$\displaystyle\;W-m\lambda^{2}+\mathcal{O}(\lambda^{3}),$
$\displaystyle\widetilde{M}_{3}\simeq$
$\displaystyle\;Z+\mathcal{O}(\lambda^{3})\,,$

and, in particular, the relation $W\simeq Z$ holds, we dubbed such a situation
as the resonant scenario. As expected, two mass eigenstates are degenerate, up
to corrections of order $\lambda^{2}$. The second possibility arises when
$\mathcal{M}\sim 10^{13}$ GeV and, consequently, $W$ should be around $10^{2}$
to maintain $m_{0}\sim\mathcal{O}(10^{-2})$ eV. With all other parameters
again of ${\cal O}(1)$, the light neutrino mass matrix is now as follows:

$m_{\nu}\simeq\frac{v_{u}^{2}}{\mathcal{M}W}\begin{pmatrix}-2ce^{i\Phi}d_{11}\lambda^{2}&-ace^{i(\Sigma+\Phi)}&-bce^{i(\Phi+\Omega)}\\\
-ace^{i(\Sigma+\Phi)}&-2ae^{i\Sigma}d_{22}\lambda^{2}&-\left(ad_{23}e^{i(\Theta+\Sigma)}+bd_{22}e^{i\Omega}\right)\lambda^{2}\\\
-bce^{i(\Phi+\Omega)}&-\left(ad_{23}e^{i(\Theta+\Sigma)}+bd_{22}e^{i\Omega}\right)\lambda^{2}&-2bd_{23}e^{i(\Theta+\Omega)}\lambda^{2}\end{pmatrix}\,.$
(21)

The right-handed neutrino spectrum will feature two, almost degenerate states
with mass $M_{1}\simeq M_{2}\sim 10^{15}$ GeV, and a lighter one with mass
$M_{3}\simeq Z\mathcal{M}\sim 10^{13}$ GeV; we call this scenario as the
hierarchical scenario. Even in this case, the left-handed neutrino mass matrix
can be recast in the form (17), with the obvious redefinition of $(x,x_{i})$
in terms of the parameters appearing in eq.(21). Thus, in both resonant and
hierarchical scenarios, we get the same structure of the diagonalizing matrix
$U_{\nu}$:

$U_{\nu}=\begin{pmatrix}-\dfrac{1}{\sqrt{2}}&\dfrac{1}{\sqrt{2}}&0\\\
\dfrac{1}{\sqrt{2(1+x^{2})}}&\dfrac{1}{\sqrt{2(1+x^{2})}}&-\dfrac{x}{\sqrt{2(1+x^{2})}}\\\
\dfrac{x}{\sqrt{2(1+x^{2})}}&\dfrac{x}{\sqrt{2(1+x^{2})}}&-\dfrac{1}{\sqrt{1+x^{2}}}\end{pmatrix}+\mathcal{O}(\lambda^{2}).$
(22)

Oscillation parameters | Best fits (IH)
---|---
$\theta_{12}/^{\circ}$ | $33.45^{+0.78}_{-0.75}$
$\theta_{13}/^{\circ}$ | $8.61^{+0.12}_{-0.12}$
$\theta_{23}/^{\circ}$ | $49.3^{+1.0}_{-1.3}$
$\delta_{cp}/^{\circ}$ | $287^{+27}_{-32}$
$\frac{\Delta m_{21}^{2}}{10^{-5}\,\,eV^{2}}$ | $7.42^{+0.21}_{-0.20}$
$\frac{\Delta m_{31}^{2}}{10^{-3}\,\,eV^{2}}$ | $-2.497^{+0.028}_{-0.028}$
Table 2: Best fits and 1$\sigma$ ranges for the oscillation parameters, from
Esteban  _et al._ (2020).

The final expressions of the neutrino mixing angles (from the relation
$U_{PMNS}=U_{l}^{\dagger}U_{\nu}$) and the ratio $r=\Delta m^{2}_{sol}/\Delta
m^{2}_{atm}$ in terms of the model parameters are quite cumbersome. Thus, we
prefer to report here the order of magnitude in the expansion parameter
$\lambda$, which are valid for both scenarios considered in this paper:

$\displaystyle r\sim$ $\displaystyle\,\mathcal{O}(\lambda^{2}),$ (23)
$\displaystyle\tan\theta_{12}\sim$ $\displaystyle\,1+\mathcal{O}(\lambda),$
$\displaystyle\sin\theta_{13}\sim$ $\displaystyle\,\mathcal{O}(\lambda),$
$\displaystyle\tan\theta_{23}\sim$ $\displaystyle\,1\,.$

Our order of magnitude estimates are in good agreement with their measured
values, reported in Tab.2. This conclusion has been further strengthened by a
succesful numerical scan Marciano (2021) over the model free parameters, with
moduli extracted flat in the intervals $[\lambda,5]$ and all the phases in
$[-\pi,\pi]$. Notice that the model provides a CP conserving leptonic
phase222We preferred not to report here its analytic cumbersome expression.
(still compatible with the data at less than 3$\sigma$).

##  III Leptogenesis

As stated above, our study of leptogenesis will be performed within two
reference scenarios, identified by different mass patterns for the heavy
right-handed neutrinos:

  * •

the resonant scenario, with $M_{1}\simeq M_{2}\simeq M_{3}$ around $10^{15}$
GeV, and the mass difference comparable to the decay width $\Delta
M_{ij}\sim\Gamma_{i}$ ;

  * •

the hierarchical scenario, with $M_{3}\ll M_{1,2}\simeq 10^{15}$ GeV .

The three Majorana neutrinos decay in the early Universe creating a lepton
asymmetry, which is consequently conversed in a baryon asymmetry through non
perturbative processes, known as _sphaleron processes_ Giudice  _et al._
(2004a); Buchmuller  _et al._ (2005). As we will clearify in the following, a
different Majorana neutrino mass spectrum can lead to different CP-violating
parameters, affecting the final amount of baryon asymmetry in the Universe.

#### Resonant Scenario

We are interested in computing the BAU via thermal leptogenesis. To facilitate
the understanding of the numerical results, we will first provide analytical
(approximated) expressions of all relevant quantities entering our
computations, which will be validated against a full numerical solution of an
appropriate system of Boltzmann’s equations. Being mass degenerate, we expect
that all the three right-handed neutrinos contribute to the leptogenesis
process. Following Davidson  _et al._ (2008), we write the baryon asymmetry
as:

$\eta_{B}\simeq 7.04\cdot
10^{-3}\sum_{i}\varepsilon_{i}\eta_{i}\quad\text{with i=1,2,3}\,,$ (24)

with $\varepsilon_{i}$ being the CP-asymmetries produced in the decay of the
i-th neutrino and $\eta_{i}$ the corresponding efficiency factors. As will be
clarified below, the generation of the lepton asymmetry in the resonant
scenario occurs in an unflavored regime; the final asymmetry is consequently
just the sum of the contributions associated to the individual neutrinos. The
efficiency factors can be written in terms of decay parameters $K_{i}$ Fong
_et al._ (2012):

$\eta_{i}=\frac{1}{K_{i}},\,\,\,\,\,K_{i}\equiv\frac{\widetilde{m}_{i}}{m^{\ast}}\,,$
(25)

with:

$\displaystyle\widetilde{m}_{i}=$
$\displaystyle\frac{Y^{\ast}_{ii}Y_{ii}v_{u}^{2}}{M_{i}}\,,$ (26)
$\displaystyle m_{\ast}=$ $\displaystyle 1.1\times 10^{-3}\text{ eV}\,.$

As it is well known, different values of $K_{i}$ define different washout
regimes, namely _strong_ ($K_{i}\gg 1$), _intermediate_ ($K_{i}\simeq 1$) and
_weak_ ($K_{i}\ll 1$). In terms of the parameters of the Dirac and Majorana
neutrino mass matrices of eq.(14), their expressions up to ${\cal
O}(\lambda^{4})$ are as follows:

$\displaystyle K_{1}\simeq$
$\displaystyle\dfrac{27.7}{W}\bigg{\\{}c^{2}+\left(d_{11}^{2}+d_{31}^{2}\right)\lambda^{4}+\frac{2}{(W-Z)}\left(\frac{c^{2}m^{2}}{(W-Z)}+\sqrt{2}\,c\,d_{31}\,m\cos(\Phi)\right)\lambda^{4}\bigg{\\}}\,,$
(27) $\displaystyle K_{2}$
$\displaystyle\simeq\dfrac{27.7}{W}\big{\\{}a^{2}+(d_{22}^{2}+d_{33}^{2})\lambda^{4}\big{\\}},\quad
K_{3}\simeq\dfrac{27.7}{Z}\big{\\{}b^{2}+(d_{23}^{2}+d_{33}^{2})\lambda^{4}\big{\\}}\,.$

Being all the above parameters of ${\cal O}(1)$, we straightforwardly conclude
$K_{i}\sim{\cal O}(10)$, implying a intermediate/strong washout regime.
We are now in the position to discuss the CP asymmetry. Since the heavy
neutrinos are close in mass, the CP-asymmetry can be resonantly enhanced
Davidson  _et al._ (2008); Pilaftsis and Underwood (2004); Dev  _et al._
(2018). To check whether such an enhancement occurs, and hence properly
evaluate the CP asymmetry parameters, a good rule of thumb consists in
computing the ratios $\Delta M_{ij}/\Gamma_{i}$ between the mass splittings
and the decay widths of the right-handed neutrinos, verifying that the
resonance condition $\Delta M_{ij}\sim\Gamma_{i}$ is satisfied. In the
scenario under scrutiny, with degenerate masses and not strongly hierarchical
Yukawa couplings, we can assume $\Gamma_{1}\sim\Gamma_{2}\sim\Gamma_{3}$, so
that:

$\displaystyle\frac{\Delta M_{12}}{\Gamma_{1}}$
$\displaystyle\simeq\frac{\Delta
M_{12}}{\Gamma_{2}}\simeq\frac{32\,m\pi}{a^{2}\,W}\lambda^{2}+{\cal
O}(\lambda^{4})\;,$ (28) $\displaystyle\frac{\Delta M_{23}}{\Gamma_{2}}$
$\displaystyle\simeq\frac{\Delta
M_{23}}{\Gamma_{3}}\simeq\frac{16\,(Z-W)\,\pi}{b^{2}+\,Z}+{\cal
O}(\lambda^{2})\;,$ $\displaystyle\frac{\Delta M_{13}}{\Gamma_{1}}$
$\displaystyle\simeq\frac{\Delta
M_{13}}{\Gamma_{3}}\simeq\frac{16\,(Z-W)\,\pi}{b^{2}+\,W}+{\cal
O}(\lambda^{2})\;,$

where we have used $\Gamma_{i}=M_{i}\left(Y^{\dagger}Y\right)_{ii}/(16\pi)$.
As evident, for $W-Z\simeq O(0.1)$, $\Delta M_{i3}\simeq\Gamma_{i}$.
Consequently, leptogenesis occurs in the resonant regime. In such a case, the
self energy contribution dominates the CP violation parameters. Furthermore,
as shown in De Simone and Riotto (2007a), the asymmetry parameters are time
dependent. Following De Simone and Riotto (2007a, b), we rewrite the latter
as:

$\varepsilon_{i}(z)\simeq\sum_{j\not=i}\frac{\text{Im}\left[\left(Y^{\dagger}Y\right)^{2}_{ij}\right]}{(Y^{\dagger}Y)_{ii}(Y^{\dagger}Y)_{jj}}\frac{\Delta
M_{ij}/\Gamma_{j}}{1+\left(\Delta
M_{ij}/\Gamma_{j}\right)^{2}}\left[f_{ij}^{\text{mix}}(z)+f_{ij}^{\text{osc}}(z)\right]\,,$
(29)

where $z=M/T$, with $M$ the sterile neutrino mass. The coefficient in front of
the squared parenthesis is the (constant) usual CP-asymmetry and it is
resonantly enhanced for $\Delta M\sim\Gamma$, while the second one is the sum
of two $z$ (and hence time) dependent functions:

$\displaystyle f^{\text{mix}}_{ij}(z)=$ $\displaystyle 2\sin^{2}\left(\Delta
M_{ij}t\right)=+2\sin^{2}\left[\frac{K_{i}z^{2}\Delta
M_{ij}}{4\Gamma_{i}}\right],$ (30) $\displaystyle f^{\text{osc}}_{ij}(z)=$
$\displaystyle-\frac{\Gamma_{j}}{\Delta M_{ij}}\sin\left(\Delta
M_{ij}t\right)=-\frac{\Gamma_{j}}{\Delta
M_{ij}}\sin\left[\frac{K_{i}z^{2}\Delta M_{ij}}{2\Gamma_{i}}\right],$

In the case $\Delta M_{ij}t\equiv\frac{K_{i}z^{2}\Delta
M_{ij}}{4\Gamma_{i}}\gg 1$, eq.(30) is a strongly oscillating function. Making
the average over a generic time interval (or, equivalently, in $z$)
$t\in[0,\tau]$, we have that:

$\left\langle 2\sin^{2}\Delta M_{ij}t-\frac{\Gamma_{ij}}{\Delta
M_{ij}}\sin\Delta M_{ij}t\right\rangle=1-\frac{\sin 2\Delta
M_{ij}\tau}{2\Delta M_{ij}\tau}-\frac{\Gamma_{i}}{\Delta
M_{ij}}\frac{1-\cos\Delta M_{ij}\tau}{\Delta M_{ij}\tau}$ (31)

As evident, the average tends to 1 if $\Delta M_{ij}\tau\gg 1$. In such a
limit we have that the CP asymmetry averages to the following constant value:

$\langle\epsilon(z)\rangle=\sum_{j\not=i}\frac{\text{Im}\left[\left(Y^{\dagger}Y\right)^{2}_{ij}\right]}{(Y^{\dagger}Y)_{ii}(Y^{\dagger}Y)_{jj}}\frac{\Delta
M_{ij}/\Gamma_{j}}{1+\left(\Delta M_{ij}/\Gamma_{j}\right)^{2}}\,.$ (32)

In the case of a large decay parameter, the regime $\Delta M_{ij}t\gg 1$
occurs for small values of $z$. In good approximation the whole leptogenesis
process can be described by replacing the time dependent CP-asymmetry with its
average. Being, in our case, $K_{i}\simeq{\cal{O}}(10)$, we can adopt this
approach and, consequently, propose an analytic estimate of the baryon
asymmetry neglecting the time dependence of the $\varepsilon_{i}$’s. The
$\eta_{B}$ will be nevertheless compared against a complete numerical
treatment. The time-independent piece of the CP-asymmetry parameters in
eq.(29) can be expressed in power series of the small parameter $\lambda$ as
follows:

$\displaystyle\varepsilon_{1}=$
$\displaystyle+\frac{16\,\pi\,(W-Z)\,Z}{c^{2}\left(256\pi(W-Z)^{2}+b^{4}Z^{2}\right)}\big{[}c^{2}d_{23}^{2}\sin\left[2\left(\Theta-\Phi\right)\right]+$
(33)
$\displaystyle+bd_{11}\left(bd_{11}\sin\left[2\Omega\right]+2cd_{23}\sin\left[\Theta-\Phi+\Omega\right]\right)\big{]}\lambda^{4}+\mathcal{O}(\lambda^{5})$
$\displaystyle\varepsilon_{2}=$
$\displaystyle-\frac{16\left(b^{2}\pi(W-Z)Z\sin\left[2(\Sigma-\Omega)\right]\right)}{256\pi^{2}(W-Z)^{2}+b^{4}Z^{2}}+$
$\displaystyle+\frac{16b^{2}m\pi
Z\left(256\pi^{2}(W-Z)^{2}-b^{4}Z^{2}\right)\sin\left[2\left(\Sigma-\Omega\right)\right]\lambda^{2}}{\left(256\pi^{2}(W-Z)^{2}+b^{4}Z^{2}\right)^{2}}+\mathcal{O}(\lambda^{4})$
$\displaystyle\varepsilon_{3}=$
$\displaystyle+\frac{16\left(b^{2}\pi(W-Z)Z\sin\left[2(\Sigma-\Omega)\right]\right)}{256\pi^{2}(W-Z)^{2}+b^{4}Z^{2}}+$
$\displaystyle-\frac{16b^{2}m\pi
Z\left(256\pi^{2}(W-Z)^{2}-b^{4}Z^{2}\right)\sin\left[2\left(\Sigma-\Omega\right)\right]\lambda^{2}}{\left(256\pi^{2}(W-Z)^{2}+b^{4}Z^{2}\right)^{2}}+\mathcal{O}(\lambda^{4})\,.$

$a$ $b$ $c$ $d_{11}$ $d_{22}$ $d_{23}$ $d_{31}$ $d_{32}$ 1.28 1.49 1.46 1.07
1.13 1.45 1.70 1.01 $d_{33}$ $m$ $\Sigma$ $\Omega$ $\Theta$ $\Phi$ $Z$ $W$
1.50 1.50 0.007 0.007 -0.003 -0.004 1.38 1.27

Table 3: Set of parameter assignments which lead to the final baryon asymmetry
value shown in fig. (1). The parameters are selected close to 1 as to avoid
particular effects of enhancement or suppression in the $\varepsilon_{i}$.

By inspecting the analytical expressions above we see that, for
$W-Z\simeq\mathcal{O}(0.1)$, the parameters $\varepsilon_{2,3}$ tend
approximately to $\mathcal{O}(0.1)$, while $\varepsilon_{1}$ is of order
$10^{-4}$. Therefore, the main contribution to the final baryon asymmetry of
the Universe is carried by $\varepsilon_{2,3}$, leading to an $\eta_{B}$ value
which exceeds the experimentally favoured one $\eta_{B}\simeq 6.1\cdot
10^{-10}$ by several orders of magnitude. It is nevertheless possible to
suppress the values of the $\varepsilon_{i}$ parameters by an ad-hoc
assignations of the phases to trigger a destructive interference among them.
First of all, by taking $\Sigma-\Omega\rightarrow 0$, the leading order
contributions to $\varepsilon_{2,3}$ go to zero and, as for $\varepsilon_{1}$,
all $\varepsilon$’s are of $\mathcal{O}$$(\lambda^{4})$:

$\displaystyle\varepsilon_{2}=$
$\displaystyle\frac{32bd_{22}d_{23}\pi(W-Z)Z\sin\left[\Theta\right]\lambda^{4}}{a\left(256\pi^{2}(W-Z)^{2}+b^{4}Z^{2}\right)}+\mathcal{O}(\lambda^{5})$
(34) $\displaystyle\varepsilon_{3}=$ $\displaystyle\frac{16\pi
W(Z-W)}{b^{2}}\bigg{[}\frac{2abd_{22}d_{23}\sin\left[\Theta\right]}{a^{4}W+256\pi^{2}(W-Z)^{2}}+$
$\displaystyle+$
$\displaystyle\frac{b^{2}d_{11}\sin\left[2\Sigma\right]+cd_{23}\left(cd_{23}\sin\left[2(\Theta-\Phi)\right]+2bd_{11}\sin\left[\Theta+\Sigma-\Phi\right]\right)}{c^{4}W^{2}+256\pi^{2}(W-Z)^{2}}\bigg{]}\lambda^{4}+\mathcal{O}(\lambda^{5})\,.$

Upon numerical check, we have found that the $\lambda^{4}$ suppression was not
enough to guarantee viable values of the $\varepsilon_{i}$. Consequently, we
need to further assume individually suppressed $\Theta$, $\Phi$ and $\Sigma$
phases. For the benchmark values reported in Tab.3, the efficiency factors
turn to be:

$\displaystyle\eta_{1}\simeq 1.0\cdot 10^{-2},\quad\eta_{2}\simeq 2.2\cdot
10^{-2},\quad\eta_{3}\simeq 1.6\cdot 10^{-2}\,,$ (35)

while the CP-asymmetry parameters are:

$\displaystyle\varepsilon_{1}\simeq 1.7\cdot
10^{-5},\quad\varepsilon_{2}\simeq 3.2\cdot
10^{-6},\quad\varepsilon_{3}\simeq-1.1\cdot 10^{-5}\,.$ (36)

Using the approximated formula in eq.(24), we can estimate the final baryon
asymmetry, obtaining $\eta_{B}\approx 5.20\cdot 10^{-10}$, in good agreement
with the observations. We conclude our study of the resonant scenario with a
numerical validation of the analytical results presented above. We have hence
solved the following set of coupled Boltzmann’s equations Buchmuller  _et al._
(2005):

$\displaystyle\frac{dN_{i}}{dz}=$
$\displaystyle-\left(D_{i}+S_{i}\right)\left(N_{i}-N_{i}^{\text{eq}}\right)\quad\;i=1,2,3$
(37) $\displaystyle\frac{dN_{B-L}}{dz}=$
$\displaystyle\sum_{i}^{3}\varepsilon_{i}D_{i}\left(N_{i}-N_{i}^{\text{eq}}\right)-W_{i}N_{B-L}\,,$

where $N_{i}$ stands for number density of the RH sterile neutrinos, while
$N_{B-L}$ is the amount of $B-L$ asymmetry, both normalized by comoving volume
Buchmuller  _et al._ (2005). $\varepsilon_{i}=\varepsilon_{i}(z)$ are the full
time dependent asymmetry parameters as given in eqs.(29-30). $D_{i}$ and
$S_{i}$ indicate, respectively, inverse decay and scattering contributions to
the production of the right-handed neutrinos while the $W_{i}$ represent the
total rate of Wash-out processes including both inverse decay and $\Delta
L\neq 0$ scattering contributions (see Appendix A for further details). From
the benchmark values of Tab.3, the following entries of the Yukawa matrix are
obtained:

$\hat{\text{Y}}=\begin{pmatrix}+0.969-0.004i&-0.343-0.002i&-0.405+0.002i\\\
-1.102+0.004i&-0.408-0.002i&-0.491+0.002i\\\
+0.02851+0.00005i&+1.168-0.008i&+1.355-0.009i\end{pmatrix}\,,$ (38)

where the symbol _hat_ refers to the Yukawa evaluated in the physical mass
basis of the Majorana neutrinos, _i.e._ $\hat{Y}=UY$, where $U$ is the unitary
matrix such that $U^{T}M_{R}^{-1}U=\left(M_{R}^{\text{diag}}\right)^{-1}$.

Figure 1: $B-L$ asymmetry and neutrino abundance evolution during the
expansion of the Universe in case of zero initial neutrino abundance. The blue
line refers to the full solution of the Boltzmann’s equations, obtained
retaining the time dependence of the CP-violation parameter. The red line
refers to the solution of the analogous system but adopting a time-constant
value of the asymmetry parameters. Finally, the green line represents the
abundance of the right-handed neutrinos, as given by the solution of the
system. For reference, the latter is compared with the function $N_{N}^{\rm
eq}$ (dashed line) rwhich represents a thermal equilibrium abundance for
right-handed neutrinos.

Plugging the latter values in the interaction rates appearing in the
Boltzmann’s equations, we have solved the system assuming null initial
abundance for the right-handed neutrinos in the primordial plasma. The B-L
yield $N_{B-L}$ as a function of $z$ is shown with a blue line in fig.1, while
the abundance of the right-handed neutrinos is displayed with a green line.
For reference, we have reported the corresponding equilibrium function as a
dashed orange line. To better pinpoint the impact on our result of the time
dependency of the CP-asymmetry, we have shown, as red line in fig.1, solution
of the same system of equation but retaining a constant CP violating parameter
as given by eq.(36). As evident, there is a nice agreement between the curves,
justifying the assumption of neglecting the time dependence of the asymmetry
parameter in our analytical treatment. Some more comments are in order.
Starting from a negligible abundance, the yield of the right-handed neutrino
is driven by inverse decays toward the equilibrium value which is reached for
$z_{\rm eq}<1$. For $z>z_{\text{eq}}$, the decays dominate and the neutrino
abundance decreases, until it becomes almost zero around $z\simeq 10$. This
means that the leptogenesis processes is completed at temperatures above
$10^{13}\,\mbox{GeV}$. This justifies our assumption of neglecting flavor
effects since the latter are relevant only if leptogenesis occurs at
temperatures below $10^{12}\,\mbox{GeV}$. The shape of the $N_{B-L}$ also
clearly evidences the time dependency of the $\varepsilon_{i}$ parameters. The
solution of the system $N_{B-L}(\infty)$ can be related to $\eta_{B}$ through
the relation $\eta_{B}=(a_{\text{sph}}/f)N_{\text{B-L}}(\infty)$. Here
$a_{\text{sph}}=28/79$ Buchmuller  _et al._ (2005) is the fraction of $B-L$
asymmetry converted into a baryon asymmetry by the sphaleron processes while
$f=N_{\gamma}^{\text{rec}}/N_{\gamma}^{\ast}=2387/86$ is the dilution factor
calculated assuming standard photon production from the onset of leptogenesis
till recombination. The values of the latter parameters, as obtained from the
numerical solution of the system are:

$\displaystyle N_{B-L}(\infty)=$ $\displaystyle 2.11\times 10^{-8},$ (39)
$\displaystyle\eta_{B}=$ $\displaystyle 3.01\times 10^{-10}.$

We conclude that, after the phase suppression discussed above, it is possible
to provide a viable leptogenesis in the resonant scenario.

#### Hierarchical scenario

This alternative scenario is obtained by lowering the mass scale $\mathcal{M}$
down to a value of the order of $10^{13}\,\mbox{GeV}$, which brings to a
hierarchical mass spectrum for the sterile neutrinos, with two heavy, almost
degenerate, states with $M_{1}\simeq M_{2}\sim 10^{15}$ GeV, and a lighter one
with $M_{3}\simeq 10^{13}$ GeV. In this setup, the baryon asymmetry of the
Universe can be generated through the conventional thermal leptogenesis only
via the out-of-equilibrium decay of the lightest heavy neutrino. Similarly to
the previous case, no flavor effects need to be accounted for. As discussed in
Davidson  _et al._ (2008), the latter becomes relevant when the rate of
processes mediated by $\tau$, $\mu$ and $e$ exceeds the Hubble expansion rate.
This occurs when the temperature of the Universe drops below, respectively:

$\displaystyle T_{\tau}\simeq 5\cdot 10^{11}\text{ GeV },\quad T_{\mu}\simeq
2\cdot 10^{9}\text{ GeV },\quad T_{e}\simeq 4\cdot 10^{4}\text{ GeV }\,.$ (40)

In the scenario under consideration we have $M_{3}\gg T_{\tau}\gg T_{\mu,e}$.
Thus, we can work in the so-called _unflavored regime_ , in which the lightest
neutrino decays via flavor blind processes. The baryon asymmetry can then be
parametrized as:

$\eta_{B}\simeq 7.04\cdot 10^{-3}\,\varepsilon_{3}\,\eta_{3}\,,$ (41)

with $\eta_{i}$ as in eq.(25) and the unflavored CP-asymmetry as Blanchet and
Di Bari (2012); Flanz  _et al._ (1995); Covi  _et al._ (1996); Buchmuller and
Plumacher (1998):

$\varepsilon_{i}=\frac{1}{8\pi\left(Y^{\dagger}Y\right)_{ii}}\sum_{j\not=i}\text{Im}\left[\left(Y^{\dagger}Y\right)_{ij}^{2}\right]f\left[\frac{M_{j}^{2}}{M_{i}^{2}}\right],$
(42)

where $x_{j}=M_{j}^{2}/M_{i}^{2}$ and the loop function is:

$f[x]=\sqrt{x}\left[1-(1+x)\log\left(1+\frac{1}{x}\right)+\frac{1}{1-x}\right]\,,$
(43)

that can be approximated to $f[x]\simeq-3/(2\sqrt{x})$ in the limit $x\gg 1$.
Expressing the elements of the Yukawa matrices in terms of the model
parameters, we have:

$\displaystyle\varepsilon_{3}=\;f\left[\frac{M_{2}^{2}}{M_{3}^{2}}\right]\frac{1}{8b^{4}\pi}\cdot\bigg{\\{}$
$\displaystyle\sin\left[2(\Sigma-\Omega)\right]\;\left(a^{2}b^{2}-2a^{2}(d_{23}^{2}+d_{33}^{2})\right)+$
(44)
$\displaystyle-2a\,b\,d_{22}\,d_{23}\sin\left[\Theta-\Sigma+\Omega\right]+$
$\displaystyle+2\,a\,b\,d_{32}\,d_{33}\sin\left[\Sigma-\Omega\right]+\mathcal{O}(\lambda^{4})\bigg{\\}}.$

Using the values in Tab.4, and noticing that
$f\left[M_{2}^{2}/M_{3}^{2}\right]\simeq 10^{-2}$, we obtain
$\varepsilon_{3}\simeq 3.24\cdot 10^{-4}$.

$a$ $b$ $c$ $d_{11}$ $d_{22}$ $d_{23}$ $d_{31}$ $d_{32}$ 1.18 1.07 1.65 1.99
1.95 1.88 1.82 1.71 $d_{33}$ $m$ $\Sigma$ $\Omega$ $\Theta$ $\Phi$ $Z$ $W$
1.83 1.50 0.39 1.02 4.10 5.11 1.41 159.97

Table 4: Set of parameter assignments which lead to a final baryon asymmetry
close to the cosmological observations. See text for further details.

The corresponding efficiency factor $\eta_{3}$ can be simply approximated to:

$\eta_{3}\simeq 3.6\cdot
10^{-4}\left(\frac{Z}{b^{2}}+O(\lambda^{4})\right)\simeq 4.3\cdot 10^{-4}\,,$
(45)

for the assignations in Tab.4. The efficiency factor can be converted to the
baryon asymmetry parameter $\eta_{B}$:

$\eta_{B}\simeq 7.04\times 10^{-3}\left[\varepsilon_{3}\eta_{3}\right]\approx
9.5\times 10^{-10}\,,$ (46)

very close to the experimentally favoured value. Also in this case, we have
verified the goodness of our analytical approximations by solving a suitable
set of Boltzmann equations, which take the form:

$\displaystyle\frac{dN_{3}}{dz}=$
$\displaystyle-\left(D_{3}-S_{3}\right)\left(N_{3}-N_{3}^{\text{eq}}\right)\,,$
(47) $\displaystyle\frac{dN_{B-L}}{dz}=$
$\displaystyle\;\varepsilon_{3}D_{3}\left(N_{3}-N_{3}^{\text{eq}}\right)-W_{3}N_{B-L}\,.$

From the parameter choices in Tab.4, we get the following Yukawa matrix :

$\hat{\text{Y}}=\begin{pmatrix}0.39-1.07i&-0.71-0.32i&-0.44-0.70i\\\
-0.52+1.07i&-0.84-0.32i&-0.36-0.59i\\\
+0.08825-0.00004i&+0.0821-0.0003i&+0.0881-0.0007i\end{pmatrix}\,.$ (48)

Notice that the imaginary parts of the third row is suppressed by a factor
proportional to $Z/W$.
This choice corresponds to the CP-violation parameter
$\varepsilon_{3}=4.3\cdot 10^{-4}$.

Figure 2: Evolution of the $B-L$ symmetry in the hierarchical scenario. The
color code is the same as in fig.1.

For the benchmark values under consideration, the numerical solution of the
Boltzmann’s equations is shown in fig.2 and gives:

$\displaystyle N_{B-L}(\infty)=$ $\displaystyle\;2.75\cdot 10^{-8}\,,$ (49)
$\displaystyle\eta_{B}=$ $\displaystyle\;3.96\cdot 10^{-10}\,,$

confirming what we expected from the simplified analytical analysis. It is
interesting to notice that, contrary to the resonant regime, in the
hierarchical scenario it is possible to find a parameter assignation leading
to viable leptogenesis without imposing a fine tuning on the CP violating
phases.

##  IV Conclusions

In this paper we have provided a proof of existence about the possibility of
contemporary achieving viable masses and mixing patterns for the SM neutrinos
and a value of the BAU, via leptogenesis, compatible with the experimental
determination, in models based on the abelian flavor symmetry
$L_{e}-L_{\mu}-L_{\tau}$. Given the large parameter space of the model, we
have identified two reference scenarios. The first one, dubbed the resonant
scenario, provides a viable light neutrino mass spectrum and assures the
existence of three degenerate right-handed neutrinos at a mass scale of
$10^{15}\,\mbox{GeV}$. In this scenario, the generation of the lepton
asymmetry is resonantly enhanced so that a baryon asymmetry exceeding the
experimentally favored value is generically predicted. This problem can be
overcome by invoking an ad-hoc suppression of the CP-violating phases in the
Yukawa matrix. In the second scenario, that we called the hierarchical
scenario, one of the right-handed masses is lowered down to
$10^{13}\,\mbox{GeV}$ without destroying the good agreement with the lepton
masses and mixing. The BAU is generated via the conventional thermal
leptogenesis. We have verified that it is possible to find parameter
assignations leading to the correct value of the BAU without invoking specific
assignations for the CP violating phases.

##  Appendix A
Full Boltzmann Equations

In this Appendix we discuss in greater detail the Boltzmann’s equations solved
in the main text. For the sake of simplicity, we will write them as a single
equation for the right-handed neutrino specie $N_{1}$. In the case of the
resonant regime, we just need to consider multiple copies of the same
equation. The relevant processes in the thermal plasma are:

  * •

$N_{1}$ decays (D) and inverse decays (ID) into leptons and Higgs bosons
$N_{1}\rightarrow\phi l$, and into anti-leptons and anti-Higgs bosons
$N_{1}\rightarrow\bar{\phi}\bar{l}$;

  * •

$\Delta L=2$ scattering processes mediated by the heavy Majorana neutrinos,
$l\phi\leftrightarrow\bar{l}\bar{\phi}$ (N) and
$ll\leftrightarrow\bar{\phi}\bar{\phi}$,
$\bar{l}\bar{l}\leftrightarrow\phi\phi$ (Nt);

  * •

$\Delta L=1$ scattering, with an intermediate Higgs boson field $\phi$,
involving the top quark. The s-channel $N_{1}l\leftrightarrow\bar{t}q$,
$N_{1}\bar{l}\leftrightarrow t\bar{q}$ ($s$) and the t-channel
$N_{1}t\leftrightarrow\bar{l}q$, $N_{1}\bar{t}\leftrightarrow l\bar{q}$ ($t$);

In brackets we have indicated a short-hand notation for such processes, to be
used later on. Considering $z=M/T$, where $M$ is the mass of the decaying
neutrino, the Boltzmann’s equations read:

$\displaystyle\frac{dN_{1}(z)}{dz}=$
$\displaystyle-\big{(}D(z)+S(z)\big{)}\big{(}N_{1}(z)-N_{1}^{eq}(z)\big{)}\,,$
(50) $\displaystyle\frac{dN_{B-L}(z)}{dz}=$
$\displaystyle\varepsilon(z)D(z)\big{(}N_{1}(z)-N_{1}^{eq}(z)\big{)}-W(z)N_{B-L}(z)\,,$

where $\varepsilon$ is the usual CP-violation parameter. Instead of the number
density $n_{X}$ of the particle species, it is useful to consider their number
$N_{X}$ in some portion of the comoving volume, in such a way to automatically
take into account the expansion of the Universe. In the literature, the
comoving volume $R^{3}_{\ast}(t)$ is usually chosen such that it contains one
photon at the time $t_{\ast}$, before the onset of the leptogenesis Buchmuller
_et al._ (2002):

$N_{X}(t)=n_{X}(t)R^{3}_{\ast}(t)\,,$ (51)

with

$R_{\ast}(t_{\ast})=\left(n_{\gamma}^{eq}(t_{\ast})\right)^{-1/3}\,,$ (52)

so that $N_{\gamma}(t_{\ast})=1$. Differently, one could even choose to
normalize the number density to the entropy density $s$ considering then
$Y_{X}=n_{x}/s$, as widely done in literature, _e.g._ , Buchmuller  _et al._
(2005); however, if the entropy is conserved during the Universe evolution,
the two normalizations are related by a constant. Introducing the thermally
averaged dilation factor $\braket{1/\gamma}$ as the ratio of the modified
Bessel functions of the second type:

$\left\langle\frac{1}{\gamma}\right\rangle=\frac{\mathcal{K}_{1}}{\mathcal{K}_{2}}\,,$
(53)

we can write the decay term $D(z)$ as Kolb and Turner (1990):

$D(z)=Kz\left\langle\frac{1}{\gamma}\right\rangle.$ (54)

$K$ is the decay parameter, which is introduced in the context of the GUT
baryogenesis Kolb and Turner (1990), to control whether the decays of the
sterile neutrinos are in equilibrium or not. This parameter depends on the
_effective neutrino mass_ $\widetilde{m}_{1}$ Plumacher (1997):

$\widetilde{m}_{1}=\frac{\left(Y^{\dagger}Y\right)_{11}v_{u}^{2}}{M_{1}}\,,$
(55)

where $Y$ is the Dirac neutrino Yukawa matrix of eq.(II.2), $v_{u}$ is the
vacuum expectation value of the $H_{u}$ doublet field and $M_{1}$ is the mass
of the decaying neutrino.
This effective mass has to be compared with the _equilibrium neutrino mass_
Buchmuller  _et al._ (2005):

$m^{\ast}\simeq 1.08\cdot 10^{-3}\text{ eV}.$ (56)

The decay parameter turns out to be:

$K=\frac{\widetilde{m}_{1}}{m^{\ast}}\;.$ (57)

From the parameter $D(z)$ we can obtain the inverse decay parameter
$W_{ID}(z)$, which contributes to the washout of the lepton asymmetry. Indeed,
the inverse decay parameter can be written as Buchmuller  _et al._ (2005):

$W_{ID}(z)=\frac{1}{2}D(z)\frac{N_{1}^{eq}(z)}{N_{l}^{eq}}$ (58)

with

$N_{1}^{eq}(z)=\frac{3}{8}z^{2}\mathcal{K}_{2}(z)\,,\quad
N_{l}^{eq}=\frac{3}{4}\,.$ (59)

Therefore, the contribution of the inverse decays to the final washout is:

$W_{ID}(z)=\frac{1}{4}Kz^{3}\mathcal{K}_{1}(z)\,.$ (60)

We can now move to the $\Delta L=1$ and $\Delta L=2$ scattering processes. The
latter contributes to the washout of the lepton asymmetry, while the former
counts towards both the production of the right-handed sterile neutrinos and
the final washout.
In general, the scattering terms $S_{x}(z)$, where the subscript $x$ indicates
the different processes to be considered, are:

$S_{x}(z)=\frac{\Gamma_{x}(z)}{Hz}\,,$ (61)

with $H$ being the evolution Hubble constant. $\Gamma_{x}$ are the scattering
rates, defined as Hahn-Woernle  _et al._ (2009):

$\Gamma_{x}(z)=\frac{M_{1}}{24\zeta(3)g_{N}\pi^{2}}\frac{\mathcal{I}_{x}}{\mathcal{K}_{2}(z)z^{3}}\,,$
(62)

where $g_{N}=2$ is the number of degrees of freedom of the right-handed
neutrinos. The quantity $\mathcal{I}_{x}$ is the integral:

$\mathcal{I}_{x}=\int_{z^{2}}^{\infty}d\Psi\hat{\sigma}_{x}(\Psi)\sqrt{\Psi}\mathcal{K}_{1}(\sqrt{\Psi})\,,$
(63)

of the reduced cross section $\hat{\sigma}_{x}$ given in Buchmuller  _et al._
(2002). In particular, for the scattering processes mediated by the three
Majorana neutrinos, _i.e._ the $\Delta L=2$ scatterings, the reduced cross
section reads Plümacher (1998):

$\hat{\sigma}_{N,Nt}(x)=\frac{1}{2\pi}\left[\sum_{i}\left(Y^{\dagger}Y\right)^{2}_{ii}f_{ii}^{N,Nt}(x)+\sum_{i<j}\mathcal{R}e\left(Y^{\dagger}Y\right)^{2}_{ij}f_{ij}^{N,Nt}(x)\right]\,,$
(64)

with

$f_{ii}^{N}(x)=1+\frac{a_{i}}{D_{i}(x)}+\frac{xa_{i}}{2D_{i}^{2}(x)}-\frac{a_{i}}{x}\left[1+\frac{x+a_{i}}{D_{i}}\right]\ln\left(1+\frac{x}{a_{i}}\right),$
(65) $\displaystyle f_{ij}^{N}(x)=\sqrt{a_{i}a_{j}}\bigg{[}$
$\displaystyle\frac{1}{D_{i}(x)}+\frac{1}{D_{j}(x)}+\frac{x}{D_{i}(x)D_{j}(x)}+$
(66)
$\displaystyle+\left(1+\frac{a_{i}}{x}\right)\left(\frac{2}{a_{j}-a_{i}}-\frac{1}{D_{j}(x)}\right)\ln\left(1+\frac{x}{a_{i}}\right)+$
$\displaystyle+\left(1+\frac{a_{j}}{x}\right)\left(\frac{2}{a_{i}-a_{j}}-\frac{1}{D_{i}(x)}\right)\ln\left(1+\frac{x}{a_{j}}\right)\bigg{]},$
$f_{ii}^{Nt}(x)=\frac{x}{x+a_{i}}+\frac{a_{i}}{x+2a_{i}}\ln\left(1+\frac{x}{a_{i}}\right),$
(67) $\displaystyle
f_{ij}^{Nt}(x)=\frac{\sqrt{a_{i}a_{j}}}{(a_{i}-a_{j})(x+a_{i}+a_{j})}\bigg{[}(2x+3a_{i}+a_{j})\ln\left(1+\frac{x}{a_{j}}\right)+$
(68)
$\displaystyle-(2x+3a_{j}+a_{i})\ln\left(1+\frac{x}{a_{i}}\right)\bigg{]}.$

Here $a_{j}\equiv M_{j}^{2}/M_{1}^{2}$ and
$1/D_{i}(x)\equiv(x-a_{i})/[(x-a_{i})^{2}+a_{i}c_{i}]$ is the off-shell part
of the $N_{i}$ propagator with
$c_{i}=a_{i}(Y^{\dagger}Y)^{2}_{ii}/(8\pi)^{2}$.
On the other hand, for the $\Delta L=1$ scattering processes, it is convenient
to write the rates $S_{s,t}(z)$ in term of the functions $f_{s,t}(z)$ defined
as Buchmuller  _et al._ (2002):

$f_{s,t}(z)=\frac{\int_{z^{2}}^{\infty}d\Psi\chi_{s,t}(\Psi/z^{2})\sqrt{\Psi}\mathcal{K}_{1}(\sqrt{\Psi})}{z^{2}\mathcal{K}_{2}(z)}\,,$
(69)

with the functions $\chi_{s,t}(x)$ as follows:

$\chi_{s}(x)=\left(\frac{x-1}{x}\right)^{2},$ (70)
$\chi_{t}(x)=\frac{x-1}{x}\left[\frac{x-2+2a_{h}}{x-1+a_{h}}+\frac{1-2a_{h}}{x-1}\log\left(\frac{x-1+a_{h}}{a_{h}}\right)\right]\,,$
(71)

where $a_{h}=m_{\phi}/M_{1}$ is the infrared cut-off for the t-channel and
$m_{\phi}$ is the mass of the Higgs boson which receives contributions from
the thermal bath and its value can be written as $m_{\phi}\simeq 0.4\;T$
Giudice  _et al._ (2004b). In this a way, the $\Delta L=1$ scattering terms
are:

$S_{s,t}(z)=\frac{K_{s}}{9\zeta(3)}f_{s,t}(z)\,,$ (72)

with $K_{s}=\widetilde{m}_{1}/m_{\ast}^{s}$ and Buchmuller  _et al._ (2005):

$m_{\ast}^{s}=\frac{4\pi^{2}}{9}\frac{g_{N}}{h_{t}^{2}}m_{\ast}\,,$ (73)

where $h_{t}$ is the top quark Yukawa coupling evaluated at $T\simeq M_{1}$.
The total scattering term $S(z)$, related to the production of the sterile
neutrinos, is given by $S(z)=2S_{s}(z)+4S_{t}(z)$, where the s-channel
acquires a factor of 2 stemming from the diagram involving the antiparticles,
and an additional factor of 2 in the t-channel comes from the u-channel
diagram.
Therefore, the parameter $S(z)$ can finally be written as:

$S(z)=\frac{2K_{s}}{9\zeta(3)}(f_{s}(z)+2f_{t}(z))\,.$ (74)

Thus, for the total washout term $W(z)$, which includes contributions from the
inverse decay processes and the $\Delta L=1,2$ scatterings, we have Hahn-
Woernle  _et al._ (2009), Buchmuller  _et al._ (2002):

$W(z)=W_{ID}(z)\left[1+\frac{2}{D(z)}\left(2S_{t}(z)+\frac{N_{1}(z)}{N_{1}^{eq}(z)}S_{s}(z)+2S_{N}(z)+2S_{Nt}(z)\right)\right].$
(75)

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Pruned Lightweight Encoders for Computer Vision

Jakub Žádník, Markku Mäkitalo, Pekka Jääskeläinen
Faculty of Information Technology and Communication Sciences

Tampere University, Finland

Latency-critical computer vision systems, such as autonomous driving or drone control, require fast image or video compression when offloading neural network inference to a remote computer.
To ensure low latency on a near-sensor edge device, we propose the use of lightweight encoders with constant bitrate and pruned encoding configurations, namely, ASTC and JPEG XS.
Pruning introduces significant distortion which we show can be recovered by retraining the neural network with compressed data after decompression.
Such an approach does not modify the network architecture or require coding format modifications.
By retraining with compressed datasets, we reduced the classification accuracy and segmentation mean intersection over union (mIoU) degradation due to ASTC compression to 4.9–5.0 percentage points (pp) and 4.4–4.0 pp, respectively.
With the same method, the mIoU lost due to JPEG XS compression at the main profile was restored to 2.7–2.3 pp.
In terms of encoding speed, our ASTC encoder implementation is 2.3x faster than JPEG.
Even though the JPEG XS reference encoder requires optimizations to reach low latency, we showed that disabling significance flag coding saves 22–23% of encoding time at the cost of 0.4–0.3 mIoU after retraining.

Image Compression, Computer Vision, Texture Compression, Low Latency, JPEG XS, ASTC

§ INTRODUCTION

Automated systems that make fast decisions based on visual input, such as autonomous driving, drone control, or smart factories, rely on a very short response time to prevent damage or injury.
Low-latency network transmission enabled by recent development in networking technologies, such as 5G, allows edge devices with low computing power to offload expensive DNN inference of the vision task to a nearby server.

Figure <ref> depicts a model scenario of an obstacle suddenly emerging in a trajectory of a self-driving car.
Considering the car's speed of 100 km/h, if the end-to-end latency of the brake control system increased by 40 ms, for example, due to slow compression, the car would travel an additional 1.1 meters, potentially hitting the obstacle instead of stopping in front of it.

The reaction time of an autonomous vehicle control system determines whether it can avoid hitting an obstacle.

Fast compression of source images is necessary to ensure low latencies over a transfer channel, and one way to decrease the latency is to reduce the codec complexity.
Removing coding features, however, typically results in decreased vision performance at the same bitrate.
Since DNN have the ability to learn from the input data, it is possible to retrain them on the compressed dataset (after decompressing it) to overcome the coding efficiency lost by pruning the coding features.
At the same time, pruning the existing codecs allows to reuse existing hardware support and retraining does not modify the neural network architecture since only the weights change.

Several types of low-complexity codecs exist.
Real-time texture compression can reach very high encoding speeds compared to other methods at the expense of rather low coding efficiency [1, 2, 3].
A “mezzanine compression” family of codecs is designed specifically to meet ultra-low latency requirements, with JPEG XS [4] as the newest standard in this family.
The recently standardized HTJ2K [5] simplifies the otherwise complex JPEG 2000 [6] with the goal of 10x throughput improvement.
However, it does not offer such a precise rate allocation as JPEG XS.
Traditional hybrid video codecs, such as HEVC [7] or VVC [8] offer advanced coding features and great coding efficiency.
However, the additional complexity of, for example, the inter or intra prediction and advanced entropy coding can be prohibitive in resource-constrained devices.

JPEG [9] shares the core transform coding features with hybrid video codecs, but without the additional complexity.
At the same time, it can deliver sufficient quality for computer vision applications, as shown in the paper.

In this paper, we explore the idea of pruning the encoding configurations to reduce the encoding time and latency and compensating the lost vision performance by retraining the vision model with the compressed dataset.
As two case studies we chose reducing the configuration space of the otherwise very complex ASTC [10] format and JPEG XS that was designed specifically as a lightweight, low-latency codec.

Both codecs operate in a constant bitrate mode which is important for ensuring predictable latency.

We evaluate the effect of ASTC compression artifacts on the image classification accuracy of ShuffleNet [11] V2 and both ASTC and JPEG XS on the semantic segmentation of LR-ASPP-MobileNetV3 [12].
We compare both to JPEG, and in the case of the segmentation task also to HTJ2K and JPEG 2000.

The contributions of this paper are:

* We propose a lightweight ASTC encoder[<github.com/cpc/simple-texcomp>] that is approximately $2.3\times$ faster than JPEG on a Samsung S10 smartphone.

* We study how pruning JPEG XS encoding configurations impacts latency and computer vision performance.

* We demonstrate that the quality vs. latency tradeoff can be alleviated by retraining the classification and segmentation models with the compressed datasets.

§ BACKGROUND AND RELATED WORK

§.§ Adaptive Scalable Texture Compression

ASTC is the newest and most flexible texture compression format adopted as an OpenGL extension by the Khronos Group.
Like other texture compression formats, it quantizes the input block's color space and represents its pixels as indices pointing at one of the quantized colors.

Compared to the older BCn formats, ASTC supports many configuration options: scaling the input block size, partitioning, different color endpoint modes (CEM), endpoint and weight quantizations, dual-plane encoding, and BISE.
An important property of texture compression is a fixed compression ratio and random access: The individual pixels are randomly addressable from the compressed representation without decompressing.

Modern GPU have texture fetch units that can perform the decompression online during rendering with a negligible overhead which further enhances the low-latency potential of texture compression.

§.§ JPEG XS

JPEG XS is a wavelet-based mezzanine codec designed primarily for low complexity, low latency, high bandwidth, and high-quality video delivery.

The minimal coding unit of JPEG XS is one precinct whose size can range from less than one pixel line up to several lines of the image.

The JPEG XS rate allocation can predict the bitrate precisely, unlike HTJ2K where a precise rate allocation would require a significant additional complexity [4].

The latest version of the standard also supports direct Bayer data compression [13] which can be used to bypass the traditional image processing pipeline at the sensor side and thus save latency.

To the best of our knowledge, no publicly available JPEG XS encoder currently exists.
Therefore, we utilized the reference JPEG XS reference software, version 1.4.0.4 (ISO/IEC 21122-5:2020).

In the literature, [14] developed a JPEG XS codec capable of running at 60 FPS at 8K resolution on a 64-core AMD EPYC processor.

§.§ Compression for Computer Vision

Some previous works optimize the perceptual model of JPEG for computer vision [15, 16], leading to significant quality improvements.
[17] optimized the global JPEG XS encoding parameters (gains and priorities) to better capture the characteristics of a computer vision target.
Our approach of retraining the vision model with the compressed dataset is complementary to codec parameter optimizations.

[18] used retraining to recover object detection and semantic segmentation performance of BC1 and YCoCg-BC3 [1] texture compression.
To the best of our knowledge, no prior work implements a minimal-subset ASTC in the context of computer vision.

[19] proposed a modified loss function of a DNN to achieve more efficient restoration of classification accuracy lost to compression artifacts.
They achieved a minor but consistent gain of up to 0.79 pp validation accuracy compared to a simple retraining method used in this work.

The recent exploration of VCM by MPEG is an effort to develop a coding scheme with both machine and human perception in mind [20].
The current development is being built on top of VVC which is a more complex format than what we target in this paper.
Furthermore, our use case considers only the computer vision performance without the human in the loop.
JPEG AI [21] also explores compression for both human and computer vision targets, but focuses on utilizing learning-based coding methods.

Yet another approach to adapting compression for computer vision is “feature compression” which encodes intermediate neural features [22].
Feature compression, however, requires computing some of the convolutional layers on the encoding device which contrasts with our approach of decreasing the encoding complexity.

§ IMPLEMENTATION OF PRUNED CODECS

§.§ ASTC

Due to the ASTC complexity, exhaustively searching for encoding parameters is not feasible in real time, and such, heuristics must be used to prune the configuration space.
In our work, we reduce the configuration space to only one configuration: 5-bit color endpoint and 2-bit and weight quantization with a weight grid of $8\times5$.

The selected configuration showed the lowest per-pixel distortion measured as PSNR on a sample dataset from adjacent configurations without requiring BISE.

Since the only way to scale the ASTC bitrate is to modify the input block size, we implemented both $12\times12$ and $8\times8$ input block sizes, implying a CR of 27:1 and 12:1 ($0.\overline{8}$ and 2.0 bpp), respectively.

The encoding of a block starts by selecting the endpoints with a small inset similar to [1].
Then, “ideal weights” are selected by orthogonally projecting the input pixels onto the line defined by the endpoints.
Lastly, the “ideal weights” are bilinearly downsampled to the $8\times5$ grid and quantized into two bits.

§.§ JPEG XS

The long encoding time of the reference encoder is caused by the rate allocation algorithm exhaustively computing the bit budget for each precinct at all quantization levels and using all possible coding methods.
We reduced the number of searched quantizations and coding methods to 13 and 5, respectively, without losing quality as the other combinations were unused in our tests.

To reduce the number of rate allocation passes further, we disabled the significance flag coding.
Significance flag coding detects a run of all-zero “significance groups” (groups of 8 adjacent coefficients) that can be encoded with a single flag and requires an additional rate allocation pass.
Disabling this method brings the number of utilized coding methods from 5 to 3 and removes the need for “refresh” passes,  significantly reducing the encoding time.
However, it also reduces the coding efficiency, which we try to recover by retraining with the compressed dataset.
We kept the coefficient prediction from a previous line, as disabling it would prevent the encoder meet the target bitrate.

§ EXPERIMENTAL SETUP

§.§ Implementation Details

The ASTC encoder uses ARM NEON intrinsics to vectorize the most significant loops using 8-bit fixed-point representation.

For a fair runtime comparison with JPEG, we chose the SIMD-optimized  library[<libjpeg-turbo.org> (version 2.1.1)] and developed a wrapper encoder application around the library.
The JPEG coding parameters were chosen to match the defaults of the  command-line utility: YCbCr color space with 4:2:0 subsampling and no restart intervals.
The quality parameter (Q) 45 was selected so that the bitrate of a random sample of 10000 ImageNet images after encoding is the highest possible at or below the rate of ASTC $12\times12$.
Both ASTC and JPEG were evaluated on a single core (A76) of a Samsung S10 smartphone.

The JPEG XS encoder was compiled only for the x86 instruction set and evaluated on a single thread of Intel i7-8650U laptop CPU at a base frequency of 2.1 GHz with disabled frequency scaling.

For runtime comparison we chose two open source encoders: [<github.com/GrokImageCompression/grok> (version 9.7.7)] for JPEG 2000 and [<github.com/aous72/OpenJPH> (version 0.9.0)] for HTJ2K, both using irreversible DWT.

§.§ Vision Tasks

We evaluated the image classification accuracy of ShuffleNet V2[<pytorch.org/hub/pytorch_vision_shufflenet_v2>] in $0.5\times$ and $1.0\times$ sizes trained on the ImageNet dataset [23] with training hyperparameters derived from [24] and [11].

We also evaluated a semantic segmentation task with LR-ASPP-MobileNetV3[<github.com/ekzhang/fastseg> (commit 91238cd)] in both large and small versions trained on the Cityscapes dataset [25].

The Cityscapes images for training were cropped to $\sqrt{2}$ of the original size in each dimension to avoid running out of GPU memory.
Both networks were retrained with the dataset compressed with ASTC to recover the vision performance lost by compression artifacts.
Unfortunately, the JPEG XS encoder was not able to encode some of the ImageNet images at the bitrate of $0.\overline{8}$ bpp.
Therefore, we evaluated only the segmentation task with this codec.

All encoders mentioned in the previous subsection were used for quality evaluations, along with [<github.com/ARM-software/astc-encoder> (version 3.7)] at the fastest profile for quality evaluations on the Cityscapes dataset.

JPEG-compressed images were used only for retraining the ShuffleNet V2 network.
JPEG, JPEG 2000, and HTJ2K reach segmentation mIoU within 1.5% below the uncompressed result, and retraining is expected to bring the results on par with the uncompressed results, therefore, we did not retrain with these codecs.

§ RESULTS

§.§ Quality

*Image Classification

Table <ref> summarizes the highest achieved classification accuracies of ShuffleNet V2 under different conditions.
When the compressed data is used as an input to the network trained on uncompressed data (the “orig.” column), the ASTC compression degrades the accuracy by more than 15 pp, while the difference caused by JPEG compression is only 1.3 and 0.6 pp.
However, when retrained with the compressed dataset (the “retr.” column), the accuracy decrease for ASTC with $12\times12$ block size is only 4.9 and 5.0 pp for the $0.5\times$ and $1.0\times$ network sizes, respectively.
The ASTC $8\times8$ achieves higher quality than $12\times12$: only 2.3–1.8 pp accuracy decrease compared to the uncompressed result.
Retraining with the JPEG-compressed dataset brings a 0.2 pp increase in the classification accuracy of the smaller network.
The results show that retraining the larger network does not improve the already high accuracy for JPEG.

Validation top-1 accuracy of ShuffleNet V2 on ImageNet validation set with JPEG and the proposed ASTC compression with and without retraining.

2c| 0.5x      2c| 1.0x

compression     bpp     orig.     retr.     orig.     retr.

uncompressed         24.0     2c|54.4%     2c|64.3%

ASTC 12x12           0.89     -16.8     -4.9     -15.1     -5.0

JPEG Q45         0.89      -1.3     -1.1      -0.6     -0.7

ASTC 8x8             2.00      -6.4     -2.3      -6.6     -1.8

*Semantic Segmentation

Figure <ref> compares rate-distortion curves of multiple encoders according to three metrics: PSNR, SSIM, and validation mIoU of LR-ASPP-MobileNetV3 (large vesion) trained on an uncompressed dataset.
The small version of the model shows similar relations between the mIoU curves to the large version, therefore, we omitted it for brevity.
The plots show that despite significant PSNR and SSIM differences between JPEG 2000, JPEG, and HTJ2K, the mIoU difference is relatively small.
Disabling significance flag coding of JPEG XS (denoted as “no-sf”) shows a consistent decrease of all metrics in both main and subline profiles.
Similarly, when compared to a full-featured  encoder at the fastest preset, our pruned implementation achieves significantly lower quality.
Both SSIM and mIoU metrics decrease rapidly with JPEG XS at lower bitrates ($0.\overline{8}$ and 1.0 bpp), especially the subline profile.

dataline = [color=black, dashed, mark options=solid, mark=+]

yshift           = -11cm,

xlabel           = bpp,
xlabel near ticks,
ylabel           = PSNR (dB),

ylabel near ticks,
grid             = both,
major grid style = dotted,black!90,
minor grid style = dotted,gray!60,
legend columns    = 3,
legend style      =  at = (0.433, 1.05), anchor = south, font = ,
legend cell align =  left ,
xmin = 0,
xmax = 4,
ymin = 30,
ymax = 55,
ytick distance = 10,
minor x tick num = 9,
minor y tick num = 9,

[dataline, black, mark=x]
table[x=bpp_mean, y=psnr_mean, col sep=comma] data/generated/grok_jp2_irev.csv;

[dataline, blue]
table[x=bpp_mean, y=psnr_mean, col sep=comma] data/generated/openjph.csv;
[dataline, green!75!black]
table[x=bpp_mean, y=psnr_mean, col sep=comma] data/generated/turbojpeg.csv;
[dataline, solid, black!40!purple!70!green]
table[x=bpp_mean, y=psnr_mean, col sep=comma] data/generated/astc_u8.csv;
[dataline, black!40!purple!70!green]
table[x=bpp_mean, y=psnr_mean, col sep=comma] data/generated/astcenc_fastest.csv;
[dataline, red]
table[x=bpp_mean, y=psnr_mean, col sep=comma] data/generated/jxs_p3.csv;
[dataline, orange!50!brown]
table[x=bpp_mean, y=psnr_mean, col sep=comma] data/generated/jxs_p5.csv;
[dataline, solid, red]
table[x=bpp_mean, y=psnr_mean, col sep=comma] data/generated/jxs_nosf_p3.csv;
[dataline, solid, orange!50!brown]
table[x=bpp_mean, y=psnr_mean, col sep=comma] data/generated/jxs_nosf_p5.csv;

J2K (grok), HTJ2K (OpenJPH), JPEG (libjpeg-turbo), ASTC (ours),
ASTC (astcenc), JXS main, JXS subline, JXS main no-sf, JXS subline no-sf

yshift           = -16.6cm,
xlabel           = bpp,
xlabel near ticks,
ylabel           = SSIM (-),

ylabel near ticks,
grid             = both,
major grid style = dotted,black!90,
minor grid style = dotted,gray!60,
xmin = 0,
xmax = 4,
ymin = 0.89,
minor x tick num = 9,
minor y tick num = 4,

[dataline, black, mark=x]
table[x=bpp_mean, y=ssim_mean, col sep=comma] data/generated/grok_jp2_irev.csv;

[dataline, blue]
table[x=bpp_mean, y=ssim_mean, col sep=comma] data/generated/openjph.csv;
[dataline, green!75!black]
table[x=bpp_mean, y=ssim_mean, col sep=comma] data/generated/turbojpeg.csv;
[dataline, solid, black!40!purple!70!green]
table[x=bpp_mean, y=ssim_mean, col sep=comma] data/generated/astc_u8.csv;
[dataline, black!40!purple!70!green]
table[x=bpp_mean, y=ssim_mean, col sep=comma] data/generated/astcenc_fastest.csv;
[dataline, red]
table[x=bpp_mean, y=ssim_mean, col sep=comma] data/generated/jxs_p3.csv;
[dataline, orange!50!brown]
table[x=bpp_mean, y=ssim_mean, col sep=comma] data/generated/jxs_p5.csv;
[dataline, solid, red]
table[x=bpp_mean, y=ssim_mean, col sep=comma] data/generated/jxs_nosf_p3.csv;
[dataline, solid, orange!50!brown]
table[x=bpp_mean, y=ssim_mean, col sep=comma] data/generated/jxs_nosf_p5.csv;

yshift           = -22.2cm,

xlabel           = bpp,
xlabel near ticks,
ylabel           = mIoU (-),

ylabel near ticks,
grid              = both,
major grid style  = dotted,black!90,
minor grid style  = dotted,gray!60,
xmin              = 0,
xmax              = 4,
minor x tick num  = 9,
minor y tick num  = 9,

[dataline, black, mark=x]
table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] data/generated/grok_jp2_irev.csv;

[dataline, blue]
table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] data/generated/openjph.csv;
[dataline, green!75!black]
table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] data/generated/turbojpeg.csv;
[dataline, solid, black!40!purple!70!green]
table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] data/generated/astc_u8.csv;
[dataline, black!40!purple!70!green]
table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] data/generated/astcenc_fastest.csv;
[dataline, red]
table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] data/generated/jxs_p3.csv;
[dataline, orange!50!brown]
table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] data/generated/jxs_p5.csv;
[dataline, solid, red]
table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] data/generated/jxs_nosf_p3.csv;
[dataline, solid, orange!50!brown]
table[x=bpp_mean, y=fastseg_large_miou, col sep=comma] data/generated/jxs_nosf_p5.csv;
[mark=none, black, domain=0:4] 0.66539;

(0, 260) node [right] uncompressed;

Mean intersection over union (mIoU) of a FastSeg large network (bottom), SSIM (middle) and PSNR (top) of a Cityscapes validation set compressed with different methods.

To recover the large mIoU degradation of ASTC and JPEG XS at low bitrates, we retrained the network with the compressed datasets.
Table <ref> summarizes the mIoU improvements of retraining LR-ASPP-MobileNetV3 in both small and large variants.

For ASTC $12\times12$, retraining brought an improvement of 1.1 and 8.6 pp for the small and large networks, respectively.
Retraining JPEG XS in the main profile resulted in mIoU around 2.3–2.6 pp lower than the uncompressed result.
The subline profile of JPEG XS shows a sharp decline in the vision performance without retraining.
Retraining allows recovering most of the quality back.
However, the results still do not reach the quality of ASTC $12\times12$.

Mean intersection over union (mIoU) of LR-ASPP-MobileNetV3 with Cityscapes validation set and the JPEG XS compression.

2c| small      2c| large

compression                  bpp       orig.     retr.     orig.     retr.

uncompressed                 24.0      2c|61.2%     2c|66.5%

JPEG XS (main, sf)           0.89     -4.9      -2.3      -6.5      -2.6

JPEG XS (main, no-sf)        0.89     -5.8      -2.7      -7.4      -2.3

JPEG XS (subline, sf)        0.89     -14.4     -5.4      -20.5     -5.3

JPEG XS (subline, no-sf)     0.89     -17.3     -6.7      -25.2     -6.8

ASTC 12x12                   0.89     -5.5      -4.4      -12.6     -4.0

ASTC 8x8                     2.00     -3.5      -2.8      -6.4      -1.7

Figure <ref> shows segmentations of two challenging scenes after the retrained model inference using compressed images to illustrate the effect of compression on the segmentation result.
The first image shows shape deformations caused by the ASTC and JPEG XS main profile (“p3”) while the network trained with JPEG XS subline profile (“p5”) fails to detect the people at all.
In the second image, all cases detect the person in the foreground but fail to detect some, or all, the people in the distance.
It should be emphasized that the MobileNet networks were trained with images containing approximately half of the pixels of the full-resolution images due to GPU memory limitations.
Therefore, the examples do not correspond to the best predictions achievable with these networks.

Visual comparison of LR-ASPP-MobileNetV3 (small version) segmentation of two Cityscapes images. From left to right: Original image (brightened), ground truth, segmentation of the model trained by datasets compressed by the ASTC ($12\times12$), pruned JPEG XS (main profile) and pruned JPEG XS (subline profile) encoders.

§.§ Runtime

*ASTC Encoder

Table <ref> compares the average encoding time of a Cityscapes image (resolution $2048\times1024$) of our ASTC encoder with the block size $12\times12$ and $8\times8$ to JPEG with the quality parameters Q 85 and 96.
The Q parameters were determined by the same procedure as in Subsection <ref> to ensure approximately the same bitrate as ASTC on a random subset of 100 images.
For the $12\times12$ block size, the images were padded to a resolution divisible by a block size of 12 before ASTC encoding.

The results show our simple ASTC $12\times12$ encoder is approximately $2.3\times$ faster than the JPEG encoder based on .

The JPEG decoding was slightly slower than the encoding.

While we did not conduct ASTC decoding measurements, in [18], we measured BC1 and YCoCg-BC3 decoding time of an 8K frame as less than 1 ms on a desktop GPU.
ASTC decoding is more complicated, but the decoding overhead is still expected to be close to negligible in comparison to the encoding.

Encoding time of the proposed ASTC encoder and  encoder and decoder (ARM A76 single-core).

0.89     2.0

ASTC                      5.8         7.0

JPEG (enc, libjpeg-turbo)     13.3        16.7

JPEG (dec, libjpeg-turbo)     13.6        21.5

For comparison, we also measured the encoding and decoding time of JPEG at quality parameters 0 and 100 as 11 and 22 ms, and 7 and 32 ms, respectively.
These numbers establish the encoding speed bounds of this format.

On a single core of Intel i7-8650U laptop CPU, the AVX2-optimized  encoder compressed one Cityscapes image at approximately 44 and 61 ms (block sizes $12\times12$ and $8\times8$, respectively) at the fastest preset, suggesting that both JPEG and our pruned ASTC encoders are faster than a traditional ASTC encoder even with the latter evaluated on a more powerful CPU.

*JPEG XS Encoder

Table <ref> compares JPEG XS with three encoders: JPEG, JPEG 2000, and HTJ2K.
The HTJ2K quantization was determined by a similar procedure as in Subsection <ref>.
The results show that by disabling the significance flag coding, the encoding time of one Cityscapes frame improved by 22–23%, and is only about 9–20% slower than JPEG 2000.
HTJ2K encoding by OpenJPH was 3.8–$6.2\times$ faster than JPEG XS without the significance coding flags.
JPEG by  brought this difference further by almost another order of magnitude.
Kakadu HTJ2K is not publicly available, therefore, we used results published in [26] and extrapolated them to our setup.
More specifically, we scaled the result by the number of pixels from 4K to $2048\times1024$, multiplied by 4 since the original result was obtained on a 4-core machine, and finally scaled to our frequency of 2.1 GHz from the original 3.4 GHz.

Based on this rough estimation, it seems likely the HTJ2K encoder is capable of reaching encoding throughput close to JPEG.

Encoding times of JPEG XS, JPEG, JPEG 2000, and HTJ2K encoders on a single core of i7-8650U CPU

0.89     2.0            3.3

JPEG XS (main, sf)                        625         654               683

JPEG XS (main, no-sf)                     477         503               531

JPEG (libjpeg-turbo)                      11.3        13.5              16.0

JPEG 2000 (grok)                          437         439               441

HTJ2K (OpenJPH)                           73.7        105               138

HTJ2K (Kakadu [26])     -           14.4*     -

* extrapolated from the result in the publication

Table <ref> shows JPEG XS encoding time and latency at two different bitrates and three profiles: high, main, and subline.
The “precinct” column denotes the number of lines that form one presinct.
The “enc” column shows the total frame encoding time and the “latency” column shows the time until the first precinct is done encoding and thus represents the minimum theoretical achievable latency.
While the overall frame encoding time does not differ dramatically between presets, the precinct size has a major impact on latency: The precinct of the high profile consisting of three lines shows a latency of 32–33% of the total encoding time, while the latency of a half-line precinct of the subline profile is three times smaller portion of the encoding time.
Thus, even without reaching a high throughput, it is possible to achieve latency almost an order of magnitude lower than the encoding time.
It should also be noted that the latency includes the wavelet transform over the whole frame and can be further reduced by pipelining it with the rest of the computation.

Latency and throughput comparison of the pruned JPEG XS reference encoder without significance flag coding.

precinct     bpp      enc      2c|latency

profile       (lines)               (ms)     (ms)     (%)

high        3            0.89     502      167      33%

high        3            2.0      528      166      31%

main        2            0.89     477      136      29%

main        2            2.0      504      137      27%

subline     0.5          0.89     443      53       12%

subline     0.5          2.0      469      53       11%

§ DISCUSSION

Retraining with the compressed dataset showed the largest improvements when the vision performance without retraining was very low, such as the classification with the ASTC $12\times12$ and segmentation with the subline JPEG XS encoders.
On the segmentation task, the overall quality decrease without retraining was smaller, because the images are less noisy and less prone to compression artifacts.
The vision performance could be further enhanced by improving the retraining process and optimizing encoding parameters for computer vision using one of the methods introduced in Subsection <ref>.

The lightweight ASTC encoder achieves a higher encoding speed than JPEG, making it the fastest encoder evaluated.
On the other hand, the pruned JPEG XS reference encoder does not achieve sufficient runtime performance.
However, its low complexity and results from literature [14] suggest a fast implementation is possible.

The rate allocation can be further optimized by, for example, replacing the exhaustive search with a binary search, in combination with other heuristics.

The second most expensive operation in the reference encoder is the wavelet transform which we did not modify.
In the reference encoder, the wavelet transform is performed over the whole frame before the coding of individual precincts.
However, it is possible to interleave the wavelet transform with the precinct coding as the latency of the wavelet transform ranges from a few pixels to 6 lines [4].

To put the results into a practical perspective, let us consider a scenario of compressing a Cityscapes image and sending it over a 500 Mbit/s commercially available 5G network and an embedded transceiver capable of 2 Mbit/s.
Assuming a 1 ms latency budget for encoding and network transfer, the ASTC at the bitrate of $0.\overline{8}$ would require a latency of 145 and 3.2 lines, respectively, assuming the encoding speed of 5.8 ms/frame.
While the first case allows partitioning the image into larger chunks, in the second case, as shown in Figure <ref>, lowering the latency of JPEG XS comes at a significant quality loss and thus necessitates the compensation by retraining.

§ CONCLUSION

We explored decreasing an image encoder complexity to achieve lower latency.
Namely, we evaluated a lightweight implementation of an ASTC encoder and a pruned version of a JPEG XS reference encoder.

The ASTC encoder outperforms JPEG in terms of encoding speed by approximately $2.3\times$ at the same bitrate.
When retrained with the dataset compressed with ASTC at the lowest bitrate of $0.\overline{8}$, the classification accuracy was about 5 pp, and the segmentation mIoU 4.4–4.0 pp lower than the output of the networks trained and evaluated without any compression.

The pruned JPEG XS reference encoder is not nearly as fast as ASTC and needs more optimizations to be usable for real-time tasks.
Nevertheless, we show that disabling significance flag coding decreases the number of required rate allocation passes, and boosts the encoding speed by 22–23% at the cost of only 0.4–0.3 pp of segmentation mIoU after retraining.

HTJ2K and JPEG outperform both the tested codecs in terms of vision performance.
However, ASTC still holds the advantage of the fastest coding speed, while JPEG XS, if sufficiently optimized, is suitable for applications requiring ultra-low latencies.
To improve the quality, it is possible to apply computer vision–specific encoding parameter optimizations or improve the retraining process.

§ ACKNOWLEDGMENT

The work was financially supported by the Tampere University ITC Graduate School. It was also supported by European Union’s Horizon 2020 research and innovation program under Grant Agreement No 871738 (CPSoSaware) and in part by the Academy of Finland under Grant 325530.

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JPEGjoint photographic experts group
PSNRpeak signal-to-noise ratio
ASTCadaptive scalable texture compression
ASICapplication-specific integrated circuit
CPUcentral processing unit
GPUgraphics processing unit
HEVChigh efficiency video coding
AVCadvanced video coding
bppbits per pixel
FPSframes per second
mIoUmean intersection over union
SIMDsingle instruction multiple data
QPquantization parameter
NNneural network
DNNdeep neural network
VCMvideo coding for machines
VVCversatile video coding
pppercentage points
CEMcolor endpoint mode
BISEbounded integer sequence encoding
DSPdigital signal processing
ROIregion of interest
CRcompression ratio
HTJ2Khigh throughput JPEG 2000
DWTdiscrete wavelet transform
EBCOTembedded block coding with optimized truncation
FBCOTfast block coding with optimized truncation
SSIMstructural similarity index
MPEGmoving picture experts group