# Lepton flavor violation and leptogenesis in discrete flavor symmetric
scotogenic model

Bichitra Bijay Boruah<EMAIL_ADDRESS>Lavina Sarma<EMAIL_ADDRESS>Mrinal Kumar Das<EMAIL_ADDRESS>Department of Physics, Tezpur
University, Tezpur 784028, India

###### Abstract

We have studied the scotogenic model proposed by Ernest Ma, which is an
extension of the Standard Model by three singlet right handed neutrinos and a
scalar doublet. This model proposes that the light neutrinos acquire a non-
zero mass at 1-loop level. In this work, realisation of the scotogenic model
is done by using discrete symmetries $A_{4}\times Z_{4}$ in which the non-zero
$\theta_{13}$ is produced by assuming a non-degeneracy in the loop factor.
Considering different lepton flavor violating(LFV) proceses such as
$l_{\alpha}\longrightarrow l_{\beta}\gamma$ and $l_{\alpha}\longrightarrow
3l_{\beta}$, their impact on neutrino phenomenology is studied. We have also
analysed $0\nu\beta\beta$ and baryon asymmetry of the Universe(BAU) in this
work.

###### pacs:

12.60.-i,14.60.Pq,14.60.St

##  I Introduction:

Standard model(SM) has been the most successful model of particle physics
which is supported by a large amount of experimental evidences. But, SM fails
to provide proper explaination for some imporant problems of modern physics,
such as absolute neutrino massdeSalas:2017kay , baryon asymmetry of
universe(BAU)leptogenesis ; Hugle:2018qbw , dark matterbertone2005particle ;
Moore:1999nt  etc. The quest for these unexplained physical problems results
in the several SM extensions, aiming at a common explanation for these issues.
The scotogenic model proposed by Ernest MaMa:2006km  is one such attractive
framework where SM is minimally extended. In this framework, SM is extended by
a scalar doublet and three singlet fermions which are charged under $Z_{2}$
symmetry. This inbuilt discrete symmetry forbids the usual tree level
generation of neutrino mass. Neutrino mass is generated at 1-loop level within
this model. Phenomenology of scotogenic model has been adressed in various
literaturesBorah:2017dfn ; Borah:2018rca ; Merle:2015ica ; Lavina .

The scotogenic model is probably the simplest TeV scale model that can
simultaneously account for neutrino masses and dark matter. In this model,
lepton flavor violating processes, such as $\mu\rightarrow e\gamma$ and
$\mu\rightarrow 3e$, also takes place at 1-loop, via diagrams analogous to
those responsible for neutrino masses. Lepton flavor violation(LFV) in the
scotogenic model has already been studiedVicente:2014wga ; Takashi . However,
most of the literature primariliy foucses on $\mu\rightarrow e\gamma$ due to
its stringent experimental limit whereas other decays are rarely considered.
Here, in this work, we have LFV in details along with other phenomenologies.
We know that the baryon asymmetry of universe is one the most important
problems which cannot be adressed in SM as it fails to satisfy the Sakharov
conditions Sakharov:1967dj , which demands baryon number (B) violation, C and
CP violation, and departure from thermal equilibrium. We are therefore,
intersested to study BAU in the framework of scotogenic model. We can
incorporate baryogenesis via leptogenesis leptogenesis , where a net leptonic
asymmetry is generated first, which further gets converted into baryogenesis
through $(B+L)$ violating electroweak sphaleron phase transitions EWSphaleron
. As discussed in many literaturesDavidson:2002qv ; Buchmuller:2002rq , it is
known that there exists a lower bound of about 10TeV for the lightest of the
right handed neutrino in the Scotogenic model considering the vanilla
leptogenesis scenarioHugle:2018qbw ; Borah:2018rca . Thus, in our work, a
similar kind of leptogenesis takes place. One of the most important
significance of BSM frameworks is to study the origin of neutrino mass along
with charge lepton mixing and identifying possible symmetries related to this.
By incorporating symmetries to a model one can make a model more general and
predictive as it will corelate two or more free parameters of the model or
make them vanish. Discerete flavor symmetric realisation of sctogenic model is
done in a very few workAdulpravitchai:2009gi ; Pramanick:2019qpg .

In this work we realised the scotogenic model through $A_{4}\times Z_{4}$
discrete flavor symmetry. The implications of the discrete symmetry can be
seen as it contraints the Yukawa couplings of a particular model. Here, we
produce a realistic neutrino mixing to do an extensive analysis of lepton
flavor violating processes. Considering different lepton flavor violating(LFV)
proceses such as $l_{\alpha}\rightarrow l_{\beta}\gamma$ and
$l_{\alpha}\rightarrow 3l_{\beta}$, we analysed their impact on the neutrino
phenomenology as well. The most stringent bounds on LFV comes from the MEG
experimentTheMEG . The limit on branching ratio for the decay of
$\mu\rightarrow e\gamma$ from this experiment is obtained to be
Br($\mu\rightarrow e\gamma$)$<4.2\times 10^{-13}$. In case of
$l_{\alpha}\rightarrow 3l_{\beta}$ decay contraints comes from SINDRUM
experimentPerrevoort:2018cqi  is set to be $\rm BR(l_{\alpha}\rightarrow
3l_{\beta})<10^{-12}$. Neutrinoless double beta decay($0\nu\beta\beta$) is
also studied within the model by the consideration of the constraints from
KamLAND-Zen experiment. We have also incorporated BAU within the model and
have shown the viable parameter space satisfying the Planck bound.

The rest of the paper is organized as follows: in section(II) we introduce the
model whereas in section(III) we realise the scotogenic model using
$A_{4}\times Z_{4}$ symmetry. Section(IV) and (V) contains the discussions on
LFV processes and leptogenesis respectively. Phenomenological analysis of the
model is given in section(VI). Finally, we summarize our results and draw our
conclusions in section(VII) and present additional analytical results in
appendices A and B.

##  II Scotogenic Model

Scotogenic model is an extension of the IHDMHonorez:2010re ;
LopezHonorez:2006gr ; Arhrib:2012ia ; Bhattacharya:2019fgs ; BorahBR  and the
IHDM is nothing but a minimal extension of the SM by a Higgs field which is a
doublet under $SU(2)_{L}$ gauge symmetry with hypercharge $Y=1$ and a built-in
discrete $Z_{2}$ symmetry Ma:2006km ; Hambye:2009pw ; Dolle:2009fn ;
Honorez:2010re ; Gustafsson:2012aj . The necessity of this modification took
place as the inert Higgs doublet model(IHDM) could only accommodate dark
matter, whereas it failed in explaining the origin of neutrino masses at a
renormalizable level . In this model, three neutral singlet fermions $N_{i}$
with $i=1,2,3$ are added in order to generate neutrino masses and assign them
with a discrete $Z_{2}$ symmetry. Here, $N_{i}$ is odd under $Z_{2}$ symmetry,
whereas the SM fields remain $Z_{2}$ even. Symbolic transformation of the
particles under $Z_{2}$ symmetry is given by,

$N_{i}\longrightarrow-
N_{i},~{}\eta\longrightarrow-\eta,~{}\phi\longrightarrow\phi,~{}\Psi\longrightarrow\Psi,$
(1)

where $\eta$ is the inert Higgs doublet, $\phi$ is the SM Higgs doublet and
$\Psi$ denotes the SM fermions. The new leptonic and scalar particle content
can thereafter be represented as follows under the group of symmetries
$SU(2)\times U(1)_{Y}\times Z_{2}$:

$\begin{pmatrix}\nu_{\alpha}\\\
l_{\alpha}\end{pmatrix}_{L}\sim(2,-\dfrac{1}{2},+),~{}l^{c}_{\alpha}\sim(1,1,+),~{}\begin{pmatrix}\phi^{+}\\\
\phi^{0}\\\ \end{pmatrix}\sim(2,\frac{1}{2},+),$
$N_{i}\sim(1,1,-),~{}\begin{pmatrix}\eta^{+}\\\ \eta^{0}\\\
\end{pmatrix}\sim(2,1/2,-).$ (2)

The scalar doublets are written as follows :

$\eta=\begin{pmatrix}\eta^{\pm}\\\
\frac{1}{\sqrt{2}}(\eta^{0}_{R}+i\eta^{0}_{I})\\\
\end{pmatrix},\quad\phi=\begin{pmatrix}\phi^{+}\\\
\frac{1}{\sqrt{2}}(h+i\xi)\end{pmatrix}.$ (3)

Figure 1:  One- loop contribution of neutrino mass generation with the
exchange of right handed neutrino $N_{i}$ and the scalar $\eta_{0}$.

We have no Dirac mass term with $\nu$ and $N_{i}$, however, the similar
Yukawa-like coupling involving $\eta$ is allowed. Nevertheless, the scalar
cannot get a VEV. The neutrino mass can be generated through a one-loop
mechanism, which is based on the exchange of $\eta$ particle and a heavy
neutrino. In figure, we see two Higgs fields $\phi^{0}$ are involved. They
will not propagate but will acquire VEV after the EWSB. The lagrangian
involving the newly added field is :

$\mathcal{L}\supset\frac{1}{2}(M_{N})_{ij}N_{i}N_{j}+Y_{ij}\bar{L}\tilde{\eta}N_{j}+h.c$
(4)

where, the $1^{st}$ term is the Majorana mass term for the neutrino singlet
and the $2^{nd}$ term is the Yukawa interactions of the lepton. The new
potential on addition of the new inert scalar doublet is:

$\begin{split}V_{Scalar}=&m_{1}^{2}\phi^{+}\phi+m_{2}^{2}\eta^{+}\eta+\frac{1}{2}\lambda_{1}(\phi^{+}\phi)^{2}+\frac{1}{2}\lambda_{2}(\eta^{+}\eta)^{2}+\lambda_{3}(\phi^{+}\phi)(\eta^{+}\eta)\\\
&+\lambda_{4}(\phi^{+}\eta)(\eta^{+}\phi)+\frac{1}{2}\lambda_{5}[(\phi^{+}\eta)^{2}+h.c.]\end{split}$
(5)

All the parameters in Eq. (5) are real by hermicity of the Lagrangian, except
for $\lambda_{5}$. Since, the bilinear term $(\phi^{+}\eta)$ is forbidden by
the exact $Z_{2}$ symmetry, therefore one can always choose $\lambda_{5}$ real
by rotating the relative phase between $\phi$ and $\eta$. Furthermore, after
the spontaneous symmetry breaking like in the SM, we are left with one
physical Higgs boson h which resembles the SM Higgs boson, as well as four
dark scalars: one CP even($\eta^{0}_{R}$), one CP odd($\eta_{I}^{0}$) and a
pair of charged ones ($\eta^{\pm}$). The masses of these physical scalars are:

$\begin{split}m^{2}_{h}=&-m^{2}_{1}=2\lambda_{1}\textit{v}^{2},\\\
m^{2}_{\eta^{\pm}}=&m^{2}_{2}+\lambda_{3}\textit{v}^{2},\\\
m^{2}_{\eta_{R}^{0}}=&m^{2}_{2}+(\lambda_{3}+\lambda_{4}+\lambda_{5})\textit{v}^{2},\\\
m^{2}_{\eta_{I}^{0}}=&m^{2}_{2}+(\lambda_{3}+\lambda_{4}-\lambda_{5})\textit{v}^{2}.\end{split}$
(6)

It is clear from the above equations that all the scalar couplings are written
in terms of physical scalar masses and $m_{2}$ , thereby providing six
independent parameters of the model to be :
$\\{m_{2},m_{\textit{h}},m_{\eta_{R}^{0}},m_{\eta_{I}^{0}},m_{\eta^{\pm}},\lambda_{2}\\}{\large{\tiny}}$.
Here, $m_{\textit{h}}$ is the mass of SM-Higgs, $m_{\eta_{R}^{0}}$,
$m_{\eta_{I}^{0}}$ and $m_{\eta^{\pm}}$ are the masses of CP-even, CP-odd and
charged scalars of the inert doublet respectively. In this work, as we have
considered the CP-even scalar to be the lightest particle and a probable DM
candidate, so we consider $\lambda_{5}<0$ without any loss of generality.
Also, the limit $\lambda_{5}\rightarrow 0$ leads to the mass degeneracy of the
neutral components of the inert doublet. Following the ’t Hooft
scenariotHooft:1980xss  , the smallness of $\lambda_{5}$ to obtain the lepton
asymmetry, which would have been lost if considered to be zero, is acceptably
natural. We have a simplified diagram shown in Fig.(2) that can be split
further into two diagrams and from which the mass can be easily calculated by
considering mechanism after EWSB.

Figure 2:  One-loop diagram with exchange of $\eta_{R}^{0}$ and
$\eta_{I}^{0}$. $\nu_{i}$ and $\nu_{j}$ representing two different generations
of active neutrinos. $N_{i}$ is the right handed neutrino.

Calculation on the basis of one diagram is sufficient and considered as other
would be same except for $\eta_{R}^{0}$ replaced by $\eta_{I}^{0}$. The
neutrino mass matrix arising from the radiative mass model is given by :

$\begin{split}\textit{M}_{ij}^{\nu}=&\sum_{k}\frac{Y_{ik}Y_{jk}}{16\pi^{2}}M_{\textit{N}_{\textit{k}}}\left[\frac{m_{\eta_{R}^{0}}^{2}}{m_{\eta_{R}^{0}}^{2}-M_{\textit{N}_{\textit{k}}}^{2}}\>ln\frac{m_{\eta_{R}^{0}}^{2}}{M_{\textit{N}_{\textit{k}}}^{2}}-\frac{m_{\eta_{I}^{0}}^{2}}{m_{\eta_{I}^{0}}^{2}-M_{\textit{N}_{\textit{k}}}}^{2}\>ln\frac{m_{\eta_{I}^{0}}^{2}}{M_{\textit{N}_{\textit{k}}}}^{2}\right]\\\
\equiv&\sum_{k}\frac{Y_{ik}Y_{jk}}{16\pi^{2}}M_{\textit{N}_{\textit{k}}}[L_{k}(m^{2}_{\eta_{R}^{0}})-L_{k}(m^{2}_{\eta_{I}^{0}})],\end{split}$
(7)

where $M_{k}$ represents the mass eigenvalue of the mass eigenstate $N_{k}$ of
the neutral singlet fermion $N_{k}$ in the internal line with indices j=1,2,3
running over the three neutrino generation with three copies of $N_{k}$ and Y
is the Yukawa coupling matrix. The function $L_{k}(m^{2})$ used in Eq. (7) is
given by:

$L_{k}(m^{2})=\frac{m^{2}}{m^{2}-M^{2}_{\textit{N}_{\textit{k}}}}\ln\frac{m^{2}}{M^{2}_{\textit{N}_{\textit{k}}}}$
(8)

##  III Flavor Symmetric realisation of Scotogenic model:

In this section we have realised scotogenic model using $A_{4}\times Z_{4}$
flavor symmetry. Scotogenic model already has an inbuilt $Z_{2}$ symmetry
which provides the explanation of dark sector within this frameworkMa:2006km .
The particle content and respective charge corresponding to the discrete
symmetries are given in Table 1. The discrete symmetries, i.e $A_{4}\times
Z_{4}$ will impose constraints on the Yukawa coupling matrix, thereby,
constraining the model.

The Lagrangian for the charged lepton sector is given by:

$\mathcal{L}_{l}=\frac{y_{e}}{\Lambda}({l_{L}}\phi\chi_{T})l^{c}_{R_{1}}+\frac{y_{\mu}}{\Lambda}({l_{L}}\phi\chi_{T})l^{c}_{R_{2}}+\frac{y_{\tau}}{\Lambda}({l_{L}}\phi\chi_{T})l^{c}_{R_{3}}$
(9)

where, $\Lambda$ is the cut-off scale of the theory and
$y_{e}$,$y_{\mu}$,$y_{\tau}$ are the coupling constants. Terms in the first
parenthesis reperents the product of two triplets($l_{L}$ and $\chi_{T}$)
under $A_{4}$, each of these terms contracts with $A_{4}$ singlets
$1$,$1^{{}^{\prime}}$ and $1^{{}^{\prime\prime}}$ corresponding to $l_{R1}$,
$l_{R2}$ and $l_{R3}$ respectively. When the flavon $\chi_{T}$ gets vaccum
expectation value(vev), flavor symmetry will break and we will get the flavor
structure for lepton. Finally it sets the charged lepton coupling matrix as
the diagonal one, once the flavon vev as well as Higgs vev are inserted. For
the Dirac mass term, the effective Lagrangian can be written as:

$\mathcal{L}_{D}=\frac{\eta}{\Lambda}[Y^{{}^{\prime}}l_{L_{i}}N_{i}\chi_{S}+Y^{{}^{\prime\prime}}l_{L_{i}}N_{i}\chi].$
(10)

Again, the additional right handed neutral fermions are represented by the
Lagrangian:

$\mathcal{L}_{M_{N}}=MN_{i}N_{j}.$ (11)

Now, let us consider the vev allignment of the flavons as
followsAdulpravitchai:2009gi :
$<\chi_{T}>$= $v_{T}$(1,0,0), $<\chi_{S}>$= $v_{S}$(1,1,1), $<\chi>$= u. With
the above considerations, the charged leptonic mass matrix is given by:

$M_{l}=\frac{v_{T}<\phi>}{\Lambda}\begin{pmatrix}y_{e}&0&0\\\ 0&y_{\mu}&0\\\
0&0&y_{\tau}\\\ \end{pmatrix}$ (12)

also, the Yukawa coupling matrix takes the form:

$Y=\begin{pmatrix}2a+b&-a&-a\\\ -a&2a&b-a\\\ -a&b-a&2a\\\ \end{pmatrix}$ (13)

where, $a=Y^{{}^{\prime}}\frac{v_{S}}{\Lambda}$ and
$b=Y^{{}^{\prime\prime}}\frac{u}{\Lambda}$. From Eq.11 and by taking into
account the degenerate mass spectrum of the right handed neutrinos, we obtain
the mass matrix of the form:

$M_{N}=\begin{pmatrix}M&0&0\\\ 0&0&M\\\ 0&M&0\\\ \end{pmatrix}.$ (14)

Now, in order to transform the mass matrix of RHN to a diagonal one, we go to
a different basis with the help of an unitary matrix, U, which is represented
by:

$U=\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\\\ \end{pmatrix}.$ (15)

By this change in basis, we further obtain a change in the Yukawa coupling
matrix, which is given by:

$Y=\begin{pmatrix}-a&-a&2a+b\\\ b-a&2a&-a\\\ 2a&b-a&-a\\\ \end{pmatrix}.$ (16)

Field | $l_{L}$ | $l^{c}_{R_{1}}$ | $l^{c}_{R_{2}}$ | $l^{c}_{R_{3}}$ | $N_{i}$ | $\phi$ | $\eta$ | $\chi_{T}$ | $\chi_{S}$ | $\chi$
---|---|---|---|---|---|---|---|---|---|---
$A_{4}$ | $3$ | $1$ | $1^{{}^{\prime\prime}}$ | $1^{{}^{\prime}}$ | $3$ | $1$ | $1$ | $3$ | $3$ | $1$
$Z_{4}$ | $i$ | $i$ | $i$ | $i$ | $-1$ | $1$ | $1$ | $-1$ | $i$ | $i$
Table 1: Fields and their respective transformations under the symmetry group
of the model.

Also, it is seen that the charge lepton mass matrix remains diagonal in this
new basis:

$M_{l}^{{}^{\prime}diag}=U^{\dagger}.M_{l}.U=M_{l}^{diag}.$ (17)

As already mentioned, the realisation of scotogenic model is done through
$A_{4}\times Z_{4}$ flavor symmetry in this study. Within this model, a loop
contribution factor $\rm r_{i}$ is adressed via the relation $\rm
r_{i}\propto\frac{1}{M_{N_{i}}}$Soumita ; Pramanick:2019qpg . So, the
contribution of right handed neutrino can be given by $\rm
diag(r_{1},r_{2},r_{3})$. However, due to the degeneracy in the RHN masses,
the loop factor also becomes degenerate. Using Eq.7, the light neutrino mass
matrix arising from the loop diagram Fig.1 is of the following from:

$M_{\nu_{l}}=\begin{pmatrix}K_{1}&K_{2}&K_{2}\\\ K_{2}&K_{3}&K_{4}\\\
K_{2}&K_{4}&K_{3}\\\ \end{pmatrix}$ (18)

Eq.18 results in a $\mu-\tau$ symmetric light neutrino mass matrix. Therefore,
we concentrate on generation of realistic neutrino mixing i.e. non zero
$\theta_{13}$, which requires deviation from exact $\mu-\tau$ symmetric mass
matrix. To break the $\mu-\tau$ symmetry we have to consider the non-
degenerate right handed neutrino mass spectrum. Firstly we will take the
condition $\rm r_{1}\neq r_{2}=r_{3}=r$ and further split the degeneracy of
$N_{2}$ and $N_{3}$ by a small amount d, i.e $\rm r_{3}=r_{2}+\rm d$. Now the
structure of the light neutrino mass matrix given in eq(18) will deviate by
say $M^{0}$ which is proportional to $\rm d$. So the elements of light
neutrino mass matrix(M) after considering $\rm r_{1}\neq r_{2}=r_{3}=r$ are
given as follows:

$\displaystyle\rm
m_{11}=\frac{\pi^{2}}{16}\bigg{[}a^{2}r_{1}\bigg{\\{}\frac{-P\log(r_{1}^{2}P)}{r_{1}^{2}P-1}+\frac{Q\log(r_{1}^{2}Q)}{r_{1}^{2}Q-1}\bigg{\\}}+a^{2}r\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}$
(19)
$\displaystyle\rm+(2a+b)^{2}r\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}\bigg{]}$

$\displaystyle\rm
m_{12}=a\frac{\pi^{2}}{16}\bigg{[}-(-a+b)r_{1}\bigg{\\{}\frac{-P\log(r{1}^{2}P)}{r_{1}^{2}P-1}+\frac{Q\log(r_{1}^{2}Q)}{r_{1}^{2}Q-1}\bigg{\\}}-2ar\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}$
(20)
$\displaystyle\rm-(2a+b)r\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}\bigg{]}$

$\displaystyle\rm
m_{13}=a\frac{\pi^{2}}{16}\bigg{[}-2ar_{1}\bigg{\\{}\frac{-P\log(r_{1}^{2}P)}{r_{1}^{2}P-1}+\frac{Q\log(r_{1}^{2}Q)}{r_{1}^{2}Q-1}\bigg{\\}}-(-a-b)r\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}$
(21)
$\displaystyle\rm-(2a+b)r\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}\bigg{]}$
$\displaystyle\rm
m_{22}=\frac{\pi^{2}}{16}\bigg{[}(a-b)^{2}r_{1}\bigg{\\{}\frac{-P\log(r_{1}^{2}P)}{r_{1}^{2}P-1}+\frac{Q\log(r_{1}^{2}Q)}{r_{1}^{2}Q-1}\bigg{\\}}+4a^{2}r\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}$
(22)
$\displaystyle\rm+a^{2}r\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}\bigg{]}$

$\displaystyle\rm
m_{23}=a\frac{\pi^{2}}{16}\bigg{[}2(-a+b)r_{1}\bigg{\\{}\frac{-P\log(r_{1}^{2}P)}{r_{1}^{2}P-1}+\frac{Q\log(r_{1}^{2}Q)}{r_{1}^{2}Q-1}\bigg{\\}}+2(-a+b)r\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}$
(23)
$\displaystyle\rm+ar\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}\bigg{]}$
$\displaystyle\rm
m_{33}=\frac{\pi^{2}}{16}\bigg{[}4a^{2}r_{1}\bigg{\\{}\frac{-P\log(r_{1}^{2}P)}{r_{1}^{2}P-1}+\frac{Q\log(r_{1}^{2}Q)}{r_{1}^{2}Q-1}\bigg{\\}}+(a-b)^{2}r\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}$
(24)
$\displaystyle\rm+a^{2}r\bigg{\\{}\frac{-P\log(r^{2}P)}{r^{2}P-1}+\frac{Q\log(r^{2}Q)}{r^{2}Q-1}\bigg{\\}}\bigg{]}$

where, $\rm P=m^{2}+v^{2}(\lambda_{3}+\lambda_{4}-\lambda_{5})$ and $\rm
Q=m^{2}+v^{2}(\lambda_{3}+\lambda_{4}+\lambda_{5})$ with v signifying the vev
of the SM Higgs and $\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$, $\lambda_{5}$
are the quartic couplings.

Now the final light neutrino mass matrix after splitting the degereracy of
$N_{2}$ and $N_{3}$ with a small perturbation d can be written as:

$M_{\nu_{l}}=M+M^{0}$ (25)

where,

$M^{0}=\rm d\begin{pmatrix}0&0&x\\\ 0&x&0\\\ x&0&0\\\ \end{pmatrix}$ (26)

with $x=2\lambda_{5}v^{2}$.

##  IV Lepton flavor violating processes :

No experiment so far has observed a flavor violating process involving charged
leptons. However, many experiments are currently going on to set strong limits
on the most relevant LFV observables, in order to constraint parameter space
involved in many new physics models. In this section we will discuss various
lepton flavor violating processes (LFV) such as $l_{\alpha}\rightarrow
l_{\beta}\gamma$,$l_{\alpha}\rightarrow 3l_{\beta}$ and $\mu-e$ conversion in
nucleiLFV ; LFV2 . Currently muon decay experiments are most prominent in
nature which provides stringent limits for most models. The MEG
collaborationTheMEG  has been able to set the impressive bound on muon decay
$\rm BR(l_{\alpha}\rightarrow l_{\beta}\gamma)<4.2\times 10^{-13}$. This is
expected to improve as the experiment is upgraded to MEG II. In case of
$l_{\alpha}\rightarrow 3l_{\beta}$ decay, contraints comes from SINDRUM
experimentPerrevoort:2018cqi  to be $\rm BR(l_{\alpha}\rightarrow
3l_{\beta})<10^{-12}$ which is set long ago. The future Mu3e experiment
announces a sensitivity of $10^{-16}$, which would imply a 4 orders of
magnitude improvement on the current bound.

The neutrinoless $\mu-e$ conversion of muonic atom is the most interseting
developments regarding the LFV processesDinh:2012bp . There are many
experiments which will basically aim for the positive signal. DeeMeDeeMee ,
Mu2eMu2e , COMETcomet  and PRIMEprime  are such experiments primarily focusing
on $\mu-e$ conversion of muonic atom. The sensitivity of these experiments
will range from $10^{-14}$ to $10^{-18}$. The current limits on $\tau$
observables are less stringent, but will also get improved in the near future
by the LHC collaboration , as well as by B-factories such as Belle IIBelle2 .

Figure 3:  The 1-loop Feynman diagrams giving rise to
$l_{\alpha}\longrightarrow l_{\beta}\gamma$.

In this part we will discuss the analytical results of branching ratios of
different LFV processes such as $l_{\alpha}\rightarrow
l_{\beta}\gamma$,$l_{\alpha}\rightarrow 3l_{\beta}$ and $\mu-e$ conversion in
nuclei within the framework of scotogenic model.

In case of radiative lepton decay, the branching ratio of
$l_{\alpha}\rightarrow l_{\beta}\gamma$ is given byTakashi -

$\rm BR(l_{\alpha}\rightarrow
l_{\beta}\gamma)=\frac{3(4\pi^{3})\alpha_{em}}{4G_{F}^{2}}|A_{D}|^{2}BR(l_{\alpha}\rightarrow
l_{\beta}\nu_{\alpha}\bar{\nu{\beta}}).$ (27)

Here, $\rm G_{F}$ is the Fermi constant and
$\rm\alpha_{em}=\frac{e^{2}}{4\pi}$ is the electromagnetic fine structure
constant, with e the electromagnetic coupling. $\rm A_{D}$ is the dipole form
factor which is given by-

$A_{D}=\sum_{i=1}^{3}\frac{Y_{i\beta}^{*}Y_{i\alpha}}{2(4\pi)^{2}}\frac{1}{m_{\eta^{+}}^{2}}F_{2}(\rho_{i})$
(28)

where the parameter $\rho_{i}$ is defined as
$\rho_{i}=\frac{M_{N_{i}}^{2}}{m_{\eta^{+}}^{2}}$ and the loop function
$F_{2}(x)$ is given in appendix.

For three body decay process like $l_{\alpha}\rightarrow 3l_{\beta}$, the
branching ratio is given by-

$\displaystyle\rm BR(l_{\alpha}\rightarrow
3l_{\beta})=\frac{3(4\pi^{2})\alpha_{em}^{2}}{8G_{F}^{2}}\bigg{[}|A_{ND}|^{2}+|A_{D}|^{2}\bigg{(}\frac{16}{3}\log\bigg{(}\frac{m_{\alpha}}{m_{\beta}}\bigg{)}-\frac{22}{3}\bigg{)}$
(29)
$\displaystyle\rm+\frac{1}{6}|B|^{2}+\bigg{(}-2A_{ND}A_{D}^{*}+\frac{1}{3}A_{ND}B^{*}-\frac{2}{3}A_{D}B^{*}+h.c\bigg{)}\bigg{]}$
$\displaystyle\rm\times BR(l_{\alpha}\rightarrow
l_{\beta}\nu_{\alpha}\bar{\nu{\beta}}).$

Here, we have kept $m_{\beta}<<m_{\alpha}$ only in the logarithmic term, where
it avoids the appearance of an infrared divergence. The form factor $A_{D}$ is
generated by dipole photon penguins and is given in equation 17. Regarding the
other form factors $A_{ND}$ is given by-

$A_{ND}=\sum_{i=1}^{3}\frac{Y_{i\beta}^{*}Y_{i\alpha}}{6(4\pi)^{2}}\frac{1}{m_{\eta^{+}}^{2}}G_{2}(\rho_{i}).$
(30)

$\rm A_{ND}$ is generated by non-dipole photon penguins, whereas B, induced by
box diagrams is given by-

$\rm
e^{2}B=\frac{1}{(4\pi)^{2}m_{\eta^{+}}^{2}}\sum_{i,j=1}^{3}\bigg{[}\frac{1}{2}D_{1}(\rho_{i},\rho_{j})Y_{j\beta}^{*}Y_{j\beta}Y_{i\beta}^{*}Y_{i\alpha}+\sqrt{\rho_{i}\rho_{j}}D_{2}(\rho_{i},\rho_{j})Y_{j\beta}^{*}Y_{j\beta}^{*}Y_{i\beta}Y_{i\alpha}\bigg{]}.$
(31)

The loop functions $\rm G_{2}(x)$, $\rm D_{1}(x,y)$ and $\rm D_{2}(x,y)$ are
defined in appendix. Here, e Z-boson penguin contributions are negligible,
since in this model they are suppressed by charged lepton masses. Similarly,
Higgs-penguin contribution is also supressed which are not taken into
consideration.

Figure 4:  The penguin contributions to $l_{\alpha}\longrightarrow
l_{\beta}\gamma$, where the wavy lines depicts either a Z-boson or a photon.

Next we consider the case of $\mu-e$ conversion in nuclei. The conversion
rate, normalized to the muon capture rate, can be expressed as -

$\displaystyle\rm
CR(\mu-e,Nucleus)=\frac{p_{e}E_{e}m_{\mu}^{3}G_{F}^{2}\alpha_{em}^{3}Z_{eff}^{4}F_{p}^{2}}{8\pi^{2}Z\Gamma_{capt}}\times\bigg{[}|(Z+N)(g_{LV}^{(0)}+g_{LS}^{(0)})+(Z-N)(g_{LV}^{(1)}+g_{LS}^{(1)})|^{2}$
(32)
$\displaystyle\rm+|(Z+N)(g_{RV}^{(0)}+g_{RS}^{(0)})+(Z-N)(g_{RV}^{(1)}+g_{RS}^{(1)})|^{2}\bigg{]}$

here, number of photon and neutron is given by Z and N. $\rm Z_{eff}$ is the
effective atomic charge, $F_{p}$ denotes the nuclear matrix element and total
muon capture rate is denoted by $\rm\Gamma_{capt}$. The values of these
parameter are different for different nucleus under consideration. Also, $\rm
p_{e}$ and $E_{e}$ are the momentum and energy of electron. The expression for
$g_{XK}^{(0)}$ and $g_{XK}^{(1)}$ (X=L,R and K=S,V) present in the above
equation is given by-

$g_{XK}^{(0)}=\frac{1}{2}\sum_{q=u,d,s}\bigg{(}g_{XK}(q)G_{K}^{q,p}+g_{XK}(q)G_{K}^{q,n}\bigg{)}$
(33)
$g_{XK}^{(1)}=\frac{1}{2}\sum_{q=u,d,s}\bigg{(}g_{XK}(q)G_{K}^{q,p}-g_{XK}(q)G_{K}^{q,n}\bigg{)}.$
(34)

Numerical values of $G_{K}$ are given in various literatures. Again the
effective couplings $g_{XK}(q)$ in scotogenic model has many contributions,
which are given below-

$g_{LV}(q)\approx g_{LV}^{\gamma}(q)$ (35)
$g_{RV}(q)=g_{LV}(q)|_{L\leftrightarrow R}$ (36) $g_{LS}(q)\approx 0$ (37)
$g_{RS}(q)\approx 0$ (38)

where $\rm g_{LV}^{\gamma}(q)$ stands for the contribution due to photon
penguins. Because of the inbuilt $Z_{2}$ symmetry in scotogenic model, there
is no box contribution to $\mu-e$ conversion of nuclei. This additional
symmetry forbids the coupling between scalars($\eta^{+}$ and $\eta^{-}$) and
the quark sector. Regarding the Z-boson penguins contributions, they turn out
to be suppressed by charged lepton masses. So the effective coupling can be
written as-

$g_{LV}^{\gamma}(q)=\frac{\sqrt{2}}{G_{F}}e^{2}Q_{p}(A_{ND}-A_{D}).$ (39)

The form factors $A_{ND}$ and $A_{D}$ have been already defined. Furthermore,
$Q_{p}$ is the electric charge of the corresponding quark.

##  V Leptogenesis

We study baryogenesis in the Scotogenic model realised by $A_{4}\times Z_{4}$
symmetry. We can produce observed baryogenesis via the mechanism of
leptogenesisleptogenesis  in our model. However, the leptogenesis process must
occur by the out of equilibrium decay of the RHN, in our case $N_{1}$. As
discussed in many literaturesDavidson:2002qv ; Buchmuller:2002rq , we now know
that there exists a lower bound of about 10TeV for the lightest of the
RHNs($M_{N_{1}}$) in the Scotogenic model considering the vanilla leptogenesis
scenarioHugle:2018qbw ; Borah:2018rca . For a heirarchical mass of RHN, i.e
$M_{N_{1}}<<M_{N_{2}},M_{N_{3}}$, the leptogenesis produced by the decay of
$N_{2}$ and $N_{3}$ are supressed due to the strong washout effects produced
by $N_{1}$ or $N_{2}$ and $N_{3}$ mediated interactionsBorah:2018rca .
Thereby, the lepton asymmetry is produced only by the virtue of $N_{1}$ decay
and this is further converted into the baryon asymmetry of the Universe(BAU)
by the electro-weak sphaleron phase transitionsDine:2003ax . Now for the
generation of BAU, we solve the simultaneous Boltzmann equations for $N_{1}$
decay and formation of $N_{B-L}$. The B-L calculation depends on the
comparison between the decay rates for $N_{1}\rightarrow
l\eta,\bar{l}\eta^{*}$ processes and the Hubble parameter, which causes a
certain impact on the asymmetry as well as on the CP-asymmetry parameter
$\epsilon_{1}$. In the calculation of leptogenesis, one important quantity
that differentiates between weak and strong washout regime is the decay
parameter. It is expressed as:

$K_{1}=\frac{\Gamma_{1}}{H(z=1)},$ (40)

where, $\Gamma_{2}$ gives us the total $N_{2}$ decay width, $H$ is the Hubble
parameter and $z=\frac{M_{N_{1}}}{T}$ with $T$ being the temperature of the
photon bath. We can express $H$ in terms of $T$ and the corresponding equation
is given by:

$H=\sqrt{\frac{8\pi^{3}g_{*}}{90}}\dfrac{T^{2}}{M_{Pl}}.$ (41)

In Eq.(41), $g_{*}$ stands for the effective number of relativistic degrees of
freedom and $M_{Pl}\simeq 1.22\times 10^{19}$ GeV is the Planck mass. We have
introduced a perturbation ($\xi$) in our work to generate non-zero
$\theta_{13}$ by breaking the $\mu-\tau$ symmetry. Thus we obtain a non-
degeneracy in the RHN masses as discussed in the above sectionIII. The mass of
the lightest RHN is fixed in the range $M_{N_{1}}=10^{4}-10^{5}$ GeV and that
of $N_{2}$ is $M_{N_{2}}=10^{8}-10^{9}$ GeV. The range of $M_{N_{3}}$ will be
evaluated considering the relation $M_{N_{3}}$= $M_{N_{2}}+d$. Now, by this
choice of RHN masses along with $m_{\eta^{0}_{R}}=400-800$ GeV and most
significantly the lightest active neutrino mass $m_{1}=10^{-13}-10^{-12}$ eV,
we fall on the weak washout regime. The Yukawa couplings obtained by solving
the model parameters are incorporated in the decay rate equation for $N_{1}$
which is given by,

$\Gamma_{1}=\frac{M_{N_{1}}}{8\pi}(Y^{\dagger}Y)_{11}\left[1-\Big{(}\frac{m_{\eta^{0}_{R}}}{M_{N_{1}}}\Big{)}^{2}\right]^{2}=\frac{M_{N_{1}}}{8\pi}(Y^{\dagger}Y)_{11}(1-\eta_{1})^{2}$
(42)

Again for the decays $N_{1}\rightarrow l\eta,\bar{l}\eta^{*}$, the CP
asymmetry parameter $\epsilon_{1}$ is given by,

$\epsilon_{1}=\frac{1}{8\pi(Y^{\dagger}Y)_{11}}\sum_{j\neq
1}Im[(Y^{\dagger}Y)^{2}]_{1j}\left[f(r_{j1},\eta_{1})-\frac{\sqrt{r_{j1}}}{r_{j1}-1}(1-\eta_{1})^{2}\right],$
(43)

where,

$f(r_{j1},\eta_{1})=\sqrt{r_{j1}}\left[1+\frac{(1-2\eta_{1}+r_{j1})}{(1-\eta_{1})^{2}}ln(\frac{r_{j1}-\eta_{1}^{2}}{1-2\eta_{1}+r_{j1}})\right],$
(44)

and $r_{j1}=\big{(}\frac{M_{N_{j}}}{M_{N_{1}}}\big{)}^{2}$,
$\eta_{1}\equiv\big{(}\frac{m_{\eta^{0}_{R}}}{M_{N_{1}}}\big{)}^{2}$.
The Boltzmann equations for the number densities of $N_{1}$ and $N_{B-L}$,
given by Davidson:2002qv ,

$\frac{dn_{N_{1}}}{dz}=-D_{1}(n_{N_{1}}-n_{N_{1}}^{eq}),$ (45)
$\frac{dn_{B-L}}{dz}=-\epsilon_{1}D_{1}(n_{N_{1}}-n_{N_{1}}^{eq})-W_{1}n_{B-L},$
(46)

respectively. $n_{N_{1}}^{eq}=\frac{z^{2}}{2}K_{1}(z)$ is the equilibrium
number density of $N_{1}$, where $K_{i}(z)$ is the modified Bessel function of
$i^{th}$ type and

$D_{1}\equiv\frac{\Gamma_{1}}{Hz}=K_{N_{1}}z\frac{K_{1}(z)}{K_{2}(z)}$ (47)

gives the measure of the total decay rate with respect to the Hubble rate, and
$W_{1}=\frac{\Gamma_{W}}{Hz}$ is the total washout rate. Again,
$W_{1}=W_{1D}+W_{\Delta L=2}$, i.e the total washout term is the sum of the
washout due to inverse decays $l\eta,\bar{l}\eta^{*}\rightarrow N_{1}$
($W_{1D}=\frac{1}{4}K_{N_{1}}z^{3}K_{1}(z)$) and the washout due to the
$\Delta L=2$ scatterings
$l\eta\leftrightarrow\bar{l}\eta^{*},ll\leftrightarrow\eta^{*}\eta^{*}$ which
is given by,

$W_{\Delta
L=2}\simeq\dfrac{18\sqrt{10}M_{Pl}}{\pi^{4}g_{l}\sqrt{g_{*}}z^{2}v^{4}}(\frac{2\pi^{2}}{\lambda_{5}})^{2}M_{N_{1}}\bar{m_{\varsigma}}^{2}.$
(48)

Here, $g_{l}$ is the internal degrees of freedom for the SM leptons, and
$\bar{m_{\varsigma}}$ is the effective neutrino mass parameter, defined by:

$\bar{m_{\varsigma}}^{2}\simeq
4\varsigma_{1}^{2}m_{1}^{2}+\varsigma_{2}m^{2_{2}}+\varsigma_{3}^{2}m_{3}^{2},$
(49)

where $m_{i}^{\prime}s$ is the light neutrino mass eigenvalues and
$\varsigma_{k}$ is as defined as:

$\varsigma_{k}=\Big{(}\frac{M^{2}_{\textit{N}_{\textit{k}}}}{8(m_{\eta_{R}^{0}}^{2}-m_{\eta_{I}^{0}}^{2})}[L_{k}(m^{2}_{\eta_{R}^{0}})-L_{k}(m^{2}_{\eta_{I}^{0}})]\Big{)}^{-1}$
(50)

The final B-L asymmetry $n_{B-L}^{f}$ is evaluated by numerically calculating
Eq.(45) and Eq.(46) before the sphaleron freeze-out. This is converted into
the baryon-to-photon ratio given by:

$n_{B}=\frac{3}{4}\frac{g_{*}^{0}}{g_{*}}a_{sph}n_{B-L}^{f}\simeq 9.2\times
10^{-3}n_{B-L}^{f},$ (51)

In Eq.(51), $g_{*}=110.75$ is the effective relativistic degrees of freedom at
the time when final lepton asymmetry was produced, $g_{*}^{0}=\frac{43}{11}$
is the effective degrees of freedom at the recombination epoch and
$a_{sph}=\frac{8}{23}$ is the sphaleron conversion factor taking two Higgs
doublet into consideration. The Planck limit 2018 gives a bound on the
observed BAU($n_{B}^{obs}$) to be $(6.04\pm 0.08)\times
10^{-10}$Aghanim:2018eyx . Therefore, in our work we have chosen the free
parameters appropriately so as to generate the observed BAU. The values of the
quartic coupling, $\lambda_{5}$ is taken in range $10^{-8}-10^{-4}$ for
successful generation of leptogenesis as well as to have significant results
for LFV.

##  VI Numerical Analysis

In our work, we do a random scan for the free parameters of our model given
by:

$\rm M_{N_{1}},\rm M_{N_{2}},\eta_{R}^{0},\lambda_{5}$ (52)

The values of the above mentioned parameters for which we study the impact on
neutrino mass, LFV and BAU are given in the Table2 as follows:

Parameter | Parameter space
---|---
$\rm M_{N_{1}}$ |  $10^{4}$ GeV - $10^{5}$ GeV
$\rm M_{N_{2}}$ |  $10^{8}$ GeV - $10^{9}$ GeV
$\eta_{R}^{0}$ |  $400$ GeV- $800$ GeV
$\lambda_{5}$ |  $10^{-8}$\- $10^{-4}$
Table 2: Free parameters of the model and their respective parameter space.

Figure 5: Plots in the first-row shows baryon asymmetry as a function of RHN
($M_{N_{1}}$), the second-row shows baryon asymmetry as a function of the
quartic coupling ($\lambda_{5}$), in third-row baryon asymmetry as a function
of lightest neutrino mass eigenvalue($m_{l}$) is depicted and in the fourth
row a variation between leptonic Dirac CP phase($\delta$) and baryon asymmetry
is shown respectively. The black horizontal line gives the current Planck
limit for BAU.

Figure 6:  Effective mass as a function of lightest neutrino mass
eigenvalue($m_{l}$) for NH/IH. The horizontal(black) line is the upper limit
for the effective mass ($m_{\beta\beta}(eV)\sim 0.1(eV)$) of light neutrinos
obtained from KamLAND-Zen experiment and the vertical red line depicts the
Planck limit for the sum of the light neutrino masses.

Figure 7:  Baryon asymmetry of the Universe as a function of effective mass of
active neutrinos($m_{\beta\beta}$) for NH/IH. The horizontal(black) line is
Planck limit on BAU and the vertical red line depicts the upper bound on the
effective mass ($m_{\beta\beta}(eV)\sim 0.1(eV)$) of light neutrinos obtained
from KamLAND-Zen experiment.

Figure 8: Contour plots relating the model parameter a and b to the baryon
asymmetry of the Universe. The left panel is for NH and right panel for IH.

Figure 9: Contour plots relating the model parameter a and d(i.e. the
perturbation) to the baryon asymmetry of the Universe. The left panel is for
NH and right panel for IH.

We choose the parameter space in such a way so as to fulfill the constraints
coming from various phenomenologies. Considering the lightest RHN in TeV scale
is a significant characteristic for vanilla leptogenesis in Scotogenic
modelHugle:2018qbw ; Mahanta:2019gfe . A lower bound of about 10 TeV is set
for $\rm N_{1}$, which has been verified in many literaturesBorah:2018rca ;
Hugle:2018qbw . Again, an inert Higgs doublet cannot possibly produce the
observed relic density in the mass regime $\rm M_{W}<\rm M_{DM}\leq 550$ GeV,
also called the IHDM desert. Thus, we have considered the lightest of the
inert scalar doublet in the range given in Table.(2) in order to abide by the
bounds from Planck limit to be a probable dark matter candidate and also to
check its viability in the range 400-800 GeV. Again the charged
scalar($\eta_{+}$) of the inert doublet is taken to be ($\eta_{R}^{0}+5$) GeV,
following the constraints from LEP IILundstrom:2008ai . The choice of quartic
coupling between the SM Higgs and inert doublet $\lambda_{5}\neq 0$ is to
cause violation of the lepton number.

Now, we diagonalise the light neutrino mass matrix ($\rm M_{\nu l}$) by:

$\rm U^{T}_{PMNS}\rm M_{\nu l}\rm U_{PMNS}=\tilde{\rm M_{\nu
l}}=\begin{pmatrix}m_{1}&0&0\\\ 0&m_{2}&0\\\ 0&0&m_{3}\\\ \end{pmatrix}$ (53)

where,

$\rm
U_{\text{PMNS}}=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\end{array}\right)U_{\text{Maj}}$
(54)

is the PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix and
$c_{ij}=\cos{\theta_{ij}},\;s_{ij}=\sin{\theta_{ij}}$ and $\delta$ is the
leptonic Dirac CP phase. The diagonal matrix $\rm
U_{\text{Maj}}=\text{diag}(1,e^{i\alpha},e^{i(\beta+\delta)})$ contains the
Majorana CP phases $\alpha,\beta$.

The diagonal mass matrix of the light neutrinos can be written as, $\tilde{\rm
M_{\nu l}}=\text{diag}(m_{1},\sqrt{m^{2}_{1}+\Delta
m_{21}^{2}},\sqrt{m_{1}^{2}+\Delta m_{31}^{2}})$ for normal hierarchy and
$\tilde{\rm M_{\nu l}}=\text{diag}(\sqrt{m_{3}^{2}+\Delta m_{23}^{2}-\Delta
m_{21}^{2}},\sqrt{m_{3}^{2}+\Delta m_{23}^{2}},m_{3})$ for inverted hierarchy.
We then numerical solve the model parameter, thereby generating the light
neutrino mass matrix, the Yukawa coupling matrix and the neutrino mixing
matrix.
A significant experimental technique of detecting neutrino mass is by the
process of neutrinoless double beta decay ($0\nu\beta\beta$) Mohapatra:1986su
; Giunti:2004vv ; Barry:2013xxa ; Borgohain:2017inp . Some of the well known
experiments related to it are KamLAND-ZenKamland2 ; kamland , GERDAgerda ;
GERDA2 , KATRINkatrin2 ; KATRIN . We measure the effective neutrino mass $|\rm
m_{\beta\beta}|$ expressed by the formula,

$|\rm m_{\beta\beta}|=\sum_{k=1}^{3}\rm m_{k}\rm U_{ek}^{2}$ (55)

where, $\rm U_{ek}^{2}$ are the elements of the neutrino mixing matrix with
$k$ holding up the generation index. This eq.(55) can be further expressed as,

$|\rm m_{\beta\beta}|=|\rm m_{1}\rm U_{ee}^{2}+\rm m_{2}\rm U_{e\nu}^{2}+\rm
m_{3}\rm U_{e\tau}^{2}|.$ (56)

It is important to check if the model obeys the bound of the effective mass
with the lightest neutrino mass so that we can relate the current light
neutrino parameters giving correct hints to ongoing experiments and their
future sensitivity.

Figure 10: $Br(\mu\rightarrow e\gamma)$ and $Br(\mu\rightarrow 3e)$ as a
function of $\rho_{N}$ (where $\rho_{N}=(\frac{M_{N}}{m_{\eta^{+}}})^{2}$) for
NH and IH. The dashed horizontal lines are the recent upper bounds

.

Figure 11: $Br(\mu\rightarrow e\gamma)$ and $Cr(\mu\rightarrow e,Ti)$ as a
function of $\rho_{N}$ (where $\rho_{N}=(\frac{M_{N}}{m_{\eta^{+}}})^{2}$) for
NH and IH. The dashed horizontal lines are the recent upper bounds.

Figure 12: $Br(\mu\rightarrow e\gamma)$ and $Br(\mu\rightarrow 3e)$ as a
function of lightest neutrino mass eigenvalue($m_{l}$) for NH and IH. The
dashed horizontal lines are the recent upper bounds.

Figure 13: $R_{\mu e}$ as a function of lightest neutrino mass
eigenvalue($m_{l}$) for NH and IH. Here, $R_{\mu e}=\frac{Br(\mu\rightarrow
3e)}{Br(\mu\rightarrow e\gamma)}$ .

Figure 14: $R_{\mu e}$ as a function of $\rho_{N}$ for NH and IH. Here,
$R_{\mu e}=\frac{Br(\mu\rightarrow 3e)}{Br(\mu\rightarrow e\gamma)}$.

From Fig.(6), we see that the effective mass of the light neutrinos are
consistent with the KamLAND-Zen experiment for $\rm m_{l}=10^{-4}-10^{-2}$ eV
incase of both NH and IH. A variation between mass of the lightest RHN($\rm
M_{N_{1}}$) and BAU for NH/IH is depicted in the first row of Fig.(5). We see
that the parameter space taken for $\rm M_{N_{1}}$ is compatible for
generating the observed BAU for both NH/IH. The second row of Fig.(5) shows
that $\lambda_{5}=10^{-6}-10^{-4}$ has concentrated points satisfying the
Planck limit for BAU for NH/IH. However, we have fewer points below
$\lambda_{5}=10^{-6}$ which satisfy the bound for BAU. Thus, we can conclude
that the space $\lambda_{5}=10^{-6}-10^{-4}$ is consistent with the BAU. Again
from the variation plot between the lightest active neutrino mass
eigenvalue($\rm m_{l}$) and BAU, we get a constraint region of $\rm m_{l}$
which satisfies the Planck limit for BAU both for NH/IH. Also, interestingly
the constraint region of $\rm m_{l}$ falls within the bounds given by Planck
for the summation of the light neutrino masses. The fourth row of Fig.(5)
shows that the variation of CP violating phase($\delta$) with the baryon
asymmetry of the Universe. Here, we observe that $\delta$ value above
$10^{-2}$ satisfies the Planck limit for BAU for both NH/IH. A plot of BAU as
a function of effective neutrino mass is evaluated as can be seen from
Fig.(7). It is thereby observed that the effective neutrino mass ranging from
$10^{-2}-10^{-1}$ eV is successful in generating the correct BAU for NH,
similar to that incase of IH. In Fig.(8), we show a contour plot between the
model parameters a and b with the effective mass of light neutrinos. And from
Fig.(8), we see a co relation between a and b with the baryon asymmetry of the
Universe. From these plots, we obtain the range of the model parameters, i.e
$a\simeq 0.8\times 10^{-4}-0.13\times 10^{-4}$ and $b\simeq 0.2-0.4$ which
further gives rise to a constraint in the Yukawa coupling matrix with its
elements having values less than 3.We have considered a perturbation d in
order to break the degeneracy of the RHN as well as to deviate from the
$\mu-\tau$ symmetry. So, in Fig.9 we show a contour plot so as to constraint d
w.r.t the baryon asymmetry of the Universe.
We have studied the LFV processes within the model. LFV obervables are ploted
against different parameters defined within the parameter spaces of the model.
Relevant formulae for LFV proceses $l_{\alpha}\rightarrow l_{\beta}\gamma$ ,
$l_{\alpha}\rightarrow 3l_{\beta}$ and $\mu-e$ conversion are discussed in
section(IV). We have defined the ratio of branching ratio of two LFV decay
$l_{\alpha}\rightarrow l_{\beta}\gamma$ and $l_{\alpha}\rightarrow 3l_{\beta}$
as $R_{\mu e}$Takashi  i.e

$\rm R_{\mu e}=\frac{Br(l_{\alpha}\rightarrow
3l_{\beta})}{Br(l_{\alpha}\rightarrow l_{\beta}\gamma)}$ (57)

In case of $\mu-e$ conversion the Z-penguins gives a very little contribution
compared to $\gamma$-penguines. In this sutuation dipole operator will
dominate the conversion rate. So, the coversion rate will have a very simple
relation given below-

$\rm\frac{CR(\mu-e,,Nucleus)}{Br(\mu\rightarrow
e\gamma)}\approx\frac{f(Z,N)}{428}$ (58)

Where $\rm f(Z,N)$ is a funation which depends on the nucleus and ranges from
1.1 to 2.2 for nuclei of interest i.e titanium.

We have computed all the branching ratio of LFV decays and conversion ratio
taking consideration of contraints coming from the model. Variation of $\rm
Br(\mu\rightarrow e\gamma)$ and $\rm Br(\mu\rightarrow 3e)$ as a function of
$\rho_{N}$ (where $\rho_{N}=(\frac{M_{N}}{m_{\eta^{+}}})^{2}$) is depicted in
Fig.(10). In this case for both the mass orderings, we get $\rm
Br(\mu\rightarrow e\gamma)$ in the range $10^{-18}$ to $10^{-13}$ and $\rm
Br(\mu\rightarrow 3e)$ spanning from $10^{-33}$ to $10^{-23}$, which are
consistant with current and near future experimental limits. Similarly we have
ploted the variation of $\mu-e$ conversion ratio in Fig.(11) against
$\rho_{N}$ which is also in the experimental limits. Variation of lightest
neutrino mass eigenvalue for both the mass orderings with both the branching
ratio $\rm Br(\mu\rightarrow e\gamma)$ and $\rm Br(\mu\rightarrow 3e)$ is
given in Fig.(12). From this we can see that for both the mass ordering
results are consistant with experimental limit.

In Fig.(13) we have ploted the variation of $R_{\mu e}$ which is the ratio of
two LFV decays $l_{\alpha}\rightarrow l_{\beta}\gamma$ and
$l_{\alpha}\rightarrow 3l_{\beta}$ against the lightest neutrino mass
eigenvalue for both the mass orderings. From this we can infer that in case of
both NH/IH ,$l_{\alpha}\rightarrow l_{\beta}\gamma$ decay supress the
$l_{\alpha}\rightarrow 3l_{\beta}$ decay in our prameter spaces. Also, we can
see that the parameter space of lightest active neutrino mass for which we get
this kind of suppression is in the range $10^{-3}-1$eV. Further, we have also
generated a plot (Fig.(14)) depicting the co relation between $R_{\mu e}$ and
$\rho_{N}$ and it is observed that the viable range for $\rho_{N}$ is
$10^{10}-10^{12}$GeV. We have shown a common parameter space of the model
parameters, satisfying the baryon asymmetry of the Universe in Table3.

Parameter | NH | IH
---|---|---
a | $0.8\times 10^{-4}-0.13\times 10^{-4}$ | $0.6\times 10^{-4}-0.9\times 10^{-4}$
b | $0.2-0.4$ | $0.1-0.16$
d | $10^{-6}-0.15\times 10^{-4}$ | $0.9\times 10^{-6}-10^{-6}$
Table 3: Model parameters of the model and their respective parameter space
satisfying the Planck limit for baryon asymmetry of the Universe.

We see from Table.3 that a larger parameter space of a,b and d are consistent
with the Planck limit for BAU incase of NH as compared to that of IH.

##  VII Conclusion

In our work, we have basically realised Ernest Ma’s Scotogenic model with the
help of discrete symmetries $A_{4}\times Z_{4}$. A $\mu-\tau$ symmetric
neutrino mass matrix is obtained, which is thereafter broken by introducing a
perturbation to it. This mechanism is required for the generation of realistic
neutrino mixing i.e. non zero $\theta_{13}$, deviation of $\theta_{23}$ from
maximality and small correction in solar mixing angle $\theta_{12}$. As
$r_{i}\propto\frac{1}{M_{N_{i}}}$, we have broken the degenracy in the masses
of the RHN with the implementation of a perturbation which further breaks the
$\mu-\tau$ symmetry. We have taken four free parameters,
$M_{N_{1}},M_{N_{2}},\eta_{R}^{0}$ and $\lambda_{5}$ whose values are
mentioned above. By this choice of the parameter space, we have shown its
consistency with various experimental and cosmological bounds. The lightest of
the RHN, decays to produce lepton asymmetry which is further converted into
BAU. Thus, the parameter space taken into account for the generation of the
BAU are seen to follow the Planck limit for BAU. From the third row of
Fig.(5), we can see that the lightest active neutrino mass eigenvalue obtained
from the model satisfies the Planck limit for BAU and consecutively obeys the
Planck limit for the summation of light neutrino masses for NH/IH. Thus, the
model is viable in connecting BAU and $0\nu\beta\beta$, also satisfying the
bounds coming from neutrino oscillation data. We have also calculated the
effective mass of the light neutrinos and shown the results in Fig.(6). The
lightest active neutrino mass eigenvalue for both NH/IH is seen to satisfy the
KamLAND-Zen limit for effective mass of light neutrinos. Additional to this, a
contour plot co-relating the model parameters with BAU is also studied. The
conclusion we can draw from it is that the parameter space obtained from the
model falls within the experimental bounds, thereby constraining the Yukawa
coupling matrix. Furthermore, we have computed the branching ratios, $\rm
Br(\mu\rightarrow e\gamma)$ and $\rm Br(\mu\rightarrow 3e)$ of LFV decays
along with the $\mu-e$ conversion ratio. A variational plot between the
branching ratios and $\rho_{N}$ is shown in Fig.(10), which is conistent with
the current and the near future experimental bounds. Also, a similar plot for
conversion ratio is also shown in Fig.(11), which also satisfies the current
upper bound. We have also studied a co-relation plot between the branching
ratios and the lightest neutrino mass eigenvalue and see that it also obeys
the experimental upper bounds for the branching ratios. It is predicted well
from Fig.(13) and Fig.(14) that $l_{\alpha}\rightarrow l_{\beta}\gamma$ decay
supress the $l_{\alpha}\rightarrow 3l_{\beta}$ decay incase of both NH/IH. We
can say that the results hardly show any change depending on the mass
hierarchies. Overall, this realisation of the Scotogenic model by discrete
symmetries, with the considerations on the free parameters taken from various
bounds is viable for studying neutrino as well as cosmological
phenomenologies.

##  VIII APPENDIX:

###  VIII.1 $A_{4}$ product rules:

$A_{4}$ is discrete group of even permutation of four objects. It has three
inequivalent onedimensional representation 1, $1^{\prime}$
and$1^{\prime\prime}$ a irreducible three dimensional representation 3.
Product of the singlets and triplets are given by-

$1\otimes 1=1$ $1^{\prime}\otimes 1^{\prime}=1^{\prime\prime}$
$1^{\prime}\otimes 1^{\prime\prime}=1$ $1^{\prime\prime}\otimes
1^{\prime\prime}=1^{\prime}$ $3\otimes 3=1\oplus 1^{\prime}\oplus
1^{\prime\prime}\oplus 3_{A}\oplus 3_{S}$ (59)

where subscripts A and S stands for “asymmetric” and “symmetric” respectively.
If we have two triplets ($a_{1}$, $a_{2}$, $a_{3}$) and ($b_{1}$, $b_{2}$,
$b_{3}$), their products are given by

$1\approx a_{1}b_{1}+a_{2}b_{3}+a_{3}b_{2}$ $1^{\prime}\approx
a_{3}b_{3}+a_{1}b_{2}+a_{2}b_{1}$ $1^{\prime\prime}\approx
a_{2}b_{2}+a_{3}b_{1}+a_{1}b_{3}$
$3_{S}\approx\left(\begin{array}[]{c}2a_{1}b_{1}-a_{2}b_{3}-a_{3}b_{2}\\\
2a_{3}b_{3}-a_{1}b_{2}-a_{2}b_{1}\\\
2a_{2}b_{2}-a_{1}b_{3}-a_{3}b_{1}\end{array}\right)$
$3_{A}\approx\left(\begin{array}[]{c}a_{2}b_{3}-a_{3}b_{2}\\\
a_{1}b_{2}-a_{2}b_{1}\\\ a_{3}b_{1}-a_{1}b_{3}\end{array}\right)$

###  VIII.2 Loop functions:

We present in this appendix the loop functions relevant for the computation of
the LFV observables. These are

$F_{2}(x)=\frac{1-6x+3x^{2}+2x^{3}+6x^{2}\log(x)}{6-(1-x)^{4}}$ (60)
$G_{2}(x)=\frac{2-9x+18x^{2}-11x^{3}+6x^{3}\log(x)}{6-(1-x)^{4}}$ (61)

$D_{1}(x,y)=-\frac{1}{(1-x)(1-y)}-\frac{x^{2}\log(x)}{(1-x)^{2}(x-y)}-\frac{y^{2}\log(y)}{(1-y)^{2}(y-x)}$
(62)

$D_{2}(x,y)=-\frac{1}{(1-x)(1-y)}-\frac{x\log(x)}{(1-x)^{2}(x-y)}-\frac{y\log(y)}{(1-y)^{2}(y-x)}$
(63)

###  VIII.3 VEV ALIGNMENT OF THE FLAVONS

In this section we will evaluate the vev alignment of the flavons considered
in the model by minimizing the potential and solving it simultaneously. The
relevent potential can be written in terms of the field $\chi_{T}$, $\chi_{S}$
and their interaction terms. The interaction terms are forbidden in the model
due the additional $Z_{4}$ symmetry. The potential is given by-

$V=V(\chi_{T})+V(\chi_{S})+V_{int}$ (64)

with,

$V(\chi_{T})=-m^{2}_{\chi_{T}}(\chi^{\dagger}_{T}\chi_{T})+\lambda_{1}(\chi^{\dagger}_{T}\chi_{T})^{2}$
(65)

and

$V(\chi_{S})=-m^{2}_{\chi_{S}}(\chi^{\dagger}_{S}\chi_{S})+\lambda_{1}(\chi^{\dagger}_{S}\chi_{S})^{2}$
(66)

The triplet flavons can be written in the form-

$<\chi_{T}>=(\chi_{T_{1}},\chi_{T_{2}},\chi_{T_{3}}),\\\
<\chi_{S}>=(\chi_{S_{1}},\chi_{S_{2}},\chi_{S_{3}})$ (67)

Considering $A_{4}$ product rule the potential terms for $\chi_{T}$ will be of
the form -

$\begin{split}V(\chi_{T})&=-\mu^{2}(\chi^{\dagger}_{T_{1}}\chi_{T_{1}}+\chi^{\dagger}_{T_{2}}\chi_{T_{3}}+\chi^{\dagger}_{T_{3}}\chi_{T_{2}})\\\
&+\lambda_{2}[(\chi^{\dagger}_{T_{1}}\chi_{T_{1}}+\chi^{\dagger}_{T_{2}}\chi_{T_{3}}+\chi^{\dagger}_{T_{3}}\chi_{T_{2}})^{2}+(\chi^{\dagger}_{T_{3}}\chi_{T_{3}}+\chi^{\dagger}_{T_{2}}\chi_{T_{1}}+\chi^{\dagger}_{T_{1}}\chi_{T_{2}})\\\
&\times(\chi^{\dagger}_{T_{2}}\chi_{T_{2}}+\chi^{\dagger}_{T_{1}}\chi_{T_{3}}+\chi^{\dagger}_{T_{3}}\chi_{T_{1}})+(2\chi^{\dagger}_{T_{1}}\chi_{T_{1}}-\chi^{\dagger}_{T_{2}}\chi_{T_{3}}-\chi^{\dagger}_{T_{3}}\chi_{T_{2}})^{2}\\\
&+2(2\chi^{\dagger}_{T_{3}}\chi_{T_{3}}-\chi^{\dagger}_{T_{1}}\chi_{T_{2}}-\chi^{\dagger}_{T_{2}}\chi_{T_{1}})\times(2\chi^{\dagger}_{T_{2}}\chi_{T_{2}}-\chi^{\dagger}_{T_{3}}\chi_{T_{1}}-\chi^{\dagger}_{T_{1}}\chi_{T_{3}})]\end{split}$
(68)

Taking derivative with respect to $\chi_{T_{1}}$, $\chi_{T_{2}}$ and
$\chi_{T_{3}}$ and equating it to zero will give the minimization condition.
After solving we will get three set of solutions which are basically the
possible alignments given by-

$\begin{split}(1)\hskip
5.69046pt&\chi_{T_{1}}\rightarrow\frac{\mu_{2}{}}{\sqrt{10\lambda_{2}}},\chi_{T_{2}}\rightarrow
0,\chi_{T_{3}}\rightarrow
0\implies<\chi_{T}>=\frac{\mu_{2}}{\sqrt{10\lambda_{2}}}(1,0,0)\\\ (2)\hskip
5.69046pt&\chi_{T_{1}}\rightarrow\frac{\mu_{2}{}}{2\sqrt{3\lambda_{2}}},\chi_{T_{2}}\rightarrow\frac{\mu_{2}{}}{2\sqrt{3\lambda_{2}}},\chi_{T_{3}}\rightarrow\frac{\mu_{2}{}}{2\sqrt{3\lambda_{2}}}\implies<\chi_{T}>=\frac{\mu_{2}}{2\sqrt{3\lambda_{2}}}(1,1,1)\\\
(3)\hskip
5.69046pt&\chi_{T_{1}}\rightarrow\frac{2\mu_{2}{}}{\sqrt{51\lambda_{2}}},\chi_{T_{2}}\rightarrow-\frac{\mu_{2}{}}{\sqrt{51\lambda_{2}}},\chi_{T_{3}}\rightarrow-\frac{\mu_{2}{}}{\sqrt{51\lambda_{2}}}\implies<\chi_{T}>=\frac{\mu_{2}}{\sqrt{51\lambda_{2}}}(2,-1,-1)\end{split}$
(69)

Similarly we get the solutions for $\chi_{S}$. We have used the first set of
solution to generate the charge lepton mass matrix and second set of solution
to get the Dirac neutrino mass.

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# Data-Efficiency with a Single GPU: An Exploration of Transfer Methods for
Small Language Models

Alon Albalak1, Akshat Shrivastava2, Chinnadhurai Sankar2, Adithya Sagar2, Mike
Ross2
1University of California, Santa Barbara 2Meta AI
<EMAIL_ADDRESS>
###### Abstract

Multi-task learning (MTL), instruction tuning, and prompting have recently
been shown to improve the generalizability of large language models to new
tasks. However, the benefits of such methods are less well-documented in
smaller language models, with some studies finding contradictory results. In
this work, we explore and isolate the effects of (i) model size, (ii) general
purpose MTL, (iii) in-domain MTL, (iv) instruction tuning, and (v) few-shot
fine-tuning for models with fewer than 500 million parameters. Our experiments
in the zero-shot setting demonstrate that models gain 31% relative
improvement, on average, from general purpose MTL, with an additional 37.6%
relative gain from in-domain MTL. Contradictory to prior works on large
models, we find that instruction tuning provides a modest 2% performance
improvement for small models.

##  1 Introduction

Many recent works have demonstrated the benefits of prompting for large
language models (see Liu et al. (2022) for an extensive survey). While prompts
started as simple task identifiers (eg. "topic" for topic classification) they
have expanded to include answer templates, examples, and instructions Raffel
et al. (2020); Albalak et al. (2022); Wei et al. (2022a); Mishra et al.
(2021); Ouyang et al. (2022). Studies on utilizing prompts have shown that as
model sizes scale up, the generalization abilities of a model increase (Brown
et al., 2020; Lester et al., 2021; Min et al., 2022). However, utilizing
models on the hundred-billion parameter scale is not accessible for most
researchers and practitioners. Furthermore, Wei et al. (2022b) show that
trends for large language models do not hold for smaller language models. For
this reason, it is crucial that we must empirically find the trends that occur
in smaller models and cannot rely on studies of larger models.

Interestingly, some findings on instruction tuning have been contradictory
across studies. For example, Wei et al. (2022a) find that models with fewer
than 8B parameters see decreases in generalization when utilizing
instructions, whereas Gupta et al. (2022) find consistent gains in models with
3B and fewer parameters. To conflate these results further though, Gupta et
al. (2022) only consider 2 situations: when inputs include instructions and
answer templates, or neither.

Simultaneously with the emergence of prompting, the explicit multi-task
learning (MTL) paradigm emerged, with works such as Muppet (Aghajanyan et al.,
2021) or T0 (Sanh et al., 2022) and their variants. Explicit MTL has been
demonstrated as a means of improving the downstream performance of pre-trained
language models in data-constrained settings. In this work we consider 2 types
of MTL: general purpose and in-domain. Specifically, general purpose MTL
consists of training across a wide variety of tasks and domains, whereas in-
domain MTL consists of training across a variety of tasks that all occur
within a domain.

One limitation of many previous works on prompting and multi-task learning is
that they focus on language models in the billion-parameter scale. For
situations with latency and memory limitations, small models may be the only
option. In this work, we study an example of such a domain; dialogue.

In this work we bridge the gap between previous studies by exploring the
effects of a variety of factors on the zero- and few-shot generalizability on
modestly sized language models (<500M parameters). Specifically, we run
experiments to find the effects of: (i) model size, (ii) general purpose MTL,
(iii) in-domain MTL, (iv) instruction tuning, and (v) fine-tuning with and
without instructions. Additionally, we perform a linguistic analysis on
instruction wording and analyze variations in performance across task
instructions.

In this study, we find that (1) In-domain multi-task learning (MTL) gives the
largest improvements to generalizability, up to 80% increased relative
performance, and 37.6% on average across all models (2) Increasing model size
alone has little effect on generalization, but when combined with in-domain
MTL leads to double the (already strong) performance improvement of in-domain
MTL (3) General purpose MTL can provide large gains (57% improvement) for
downstream tasks which closely resemble the MTL tasks, but still provides
modest gains (5%) even for tasks which are more dissimilar (4) Instruction
tuning during in-domain MTL provides modest gains of just over 2% performance,
regardless of model size.

##  2 Experiments

### Data

For this study we utilize 46 tasks from the Instructdial dataset (Gupta et
al., 2022). Each task is converted into a sequence-to-sequence format with an
answer template, allowing a single generative model to perform all tasks. Each
task contains 3 to 10 instructions, with 4.4 instructions on average. For our
zero-shot experiments, we use 3 splits of train/test tasks, where each split
contains 40 training tasks and 6 test tasks. For our few-shot experiments, we
use the first data split only. Tasks are divided into classification and
generation, where classification tasks are evaluated on accuracy and
generation tasks by Rouge-L scores. The full list of tasks and information on
train/test splits can be found in the section A of the Appendix.

### Models

In our experiments, we utilize 3 variants of BART model (Lewis et al., 2020):
BART-Base, BART-Large and BART0++ (Lin et al., 2022). BART0++ is a BART-Large
that has been multi-task trained on PromptSource (Bach et al., 2022) in the
same fashion as T0++ (Sanh et al., 2022).111All pre-trained models were
downloaded from the HuggingFace Transformers library.

Figure 1: Average performance on 10 zero-shot classification tasks (top) and 8
zero-shot generation tasks (bottom) comparing pre-trained models (Off the
shelf) with models explicitly multi-task trained on in-domain data with and
without instructions.  Figure 2: Absolute scores on 18 zero-shot tasks. Full
task names are found in section B of the Appendix.

### Experimental Setup

To study the effects of (i) model size, (ii) general purpose MTL, (iii) in-
domain MTL, and (iv) instruction tuning, we run a series of zero-shot
experiments. In order to measure the effect of (i) model size, we compare
performance between BART-Base (139M parameters) and BART-Large (406M
parameters). To measure the effect of (ii) general purpose MTL, we compare
performance between BART-Large and BART0++. To study the effect of (iii) in-
domain MTL, we compare each model trained with in-domain MTL against an off-
the-shelf version that is directly tested on each split. To measure the effect
of (iv) in-domain MTL with instructions, we include instructions in addition
to the answer template in the input sequences. All experiments were repeated
with 3 random seeds and reported scores are means.222Further training details
on hyperparameters are in section C of the Appendix

In addition to zero-shot experiments, we also consider the situation where we
have small quantities of data for fine-tuning. We run experiments with 10/10,
50/50, and 100/100 training/validation samples. For the few-shot experiments,
we study the effects of (i) model size, (ii) general-purpose MTL, (iv)
instruction tuning, and (v) fine-tuning with or without instructions.

##  3 Findings

Figure 1 shows the average zero-shot performance divided into classification
and generation tasks. Figure 2 shows the absolute scores for all models and
methods on each of the 18 test tasks.

### Effects of Model Size

When comparing off-the-shelf versions of BART-Base and BART-Large, we find
nearly identical performance across classification tasks, and slightly better
performance for BART-Base (11.2 vs. 10.2 Rouge-L) on generation tasks.
However, the benefits of model size are demonstrated once the models have been
further trained using in-domain MTL (Figure 1). We find that with in-domain
MTL the base model improves its average score by 6.5. but the large model
doubles that improvement, increasing it’s score by 13.3 points averaged across
all tasks.

Figure 3: Percentage of occurrences of a word that lead to better than average
performance for an instruction. Results calculated from BART-Base model and
only includes words that occur in more than 5 instructions.

### Effects of General Purpose Multi-task Learning

When comparing BART-Large and BART0++ we see improvements from general MTL on
14/18 tasks, and an average absolute improvement of 14.5 accuracy (57.1%
relative) on classification tasks, and more modest improvement of 0.6 Rouge-L
(5%) on generation tasks (Figure 1). This large discrepancy is likely due to
the distribution of tasks in the P3 dataset Bach et al. (2022) used to train
BART0++, which consists almost entirely of classification tasks with only
summarization as a generation task. Figure 2 shows that, indeed, an off-the-
shelf BART0++ outperforms all other methods on summarization, including in-
domain MTL.

### Effects of In-Domain Multi-Task Learning

We find that in-domain MTL (without instructions) contributes the largest
portion to the final generalization ability of each model. As shown in Figure
1, BART-Large gets the most improvement, with gains of 20.4 points in accuracy
(80% relative improvement) on classification tasks and 3 Rouge-L (29.3%) for
generation tasks. Bart-Base gets 41.8% and 11.5% relative improvements, and
BART0++ gets 37.7% and 25.3% relative improvements. Collectively, this
experiment and the previous experiments on general purpose MTL demonstrate the
importance of matching both the domain and the task distribution during MTL to
the downstream tasks and domain of interest. Additionally, as previously
mentioned, in-domain MTL combined with the increased capacity of a larger
model shows even greater improvements.

Figure 4: Average performance across 6 tasks when fine-tuning with 10, 50, or
100 samples. IDMTL refers to in-domain MTL and FT is fine-tuning. "w/" and
"w/out" refer to with and without instructions.

### Effects of Instruction Tuning on In-Domain Multi-Task Learning

Next, we compare the performance of in-domain MTL with and without
instructions. The benefits of instruction tuning on small models is less
prominent than the three previous variables, but is still beneficial overall.
Figure 1 shows that BART-Base improves by 3% on generation tasks, but loses 1%
accuracy on classification. To the contrary, BART-Large improves by 4% on
classification tasks, and loses 2% Rouge-L on generation tasks. Interestingly,
BART0++ sees no difference in performance on classification tasks and improves
by 5% on generation tasks. These results run counter to those of Wei et al.
(2022a), which found that instruction tuning can degrade performance of models
with fewer than 8B parameters by about 10%. This may be partly due to the in-
domain nature of the instructions utilized in our experiments (all
instructions are related to dialogue), suggesting that future works on
instruction tuning for small models should focus on (1) domain-specific
wording used in instructions, and (2) expanding the number of domains included
in instruction sets to see more general benefits.

### Findings on Sensitivity to Instructions

To better understand the importance of wording and draw insights, we take a
closer look at the tasks which had highest variance across instructions.
First, we find that Answer Selection is the task with highest variance (BART-
Base scores range from 27.3 to 63) and find that the three worst performing
instructions include variations of "select an option that can substitute
<MASK>". The three instructions including this phrase average an accuracy of
39, while the remaining seven average an accuracy of 60.1. This large
discrepancy is likely connected to the unnaturalness of the <mask> token, and
that it is unlikely to have appeared in a similar context in the pre-training
corpus, and only appears in 2/46 tasks in our in-domain dataset. The other
task which utilizes the <mask> token is the "Fill-in the Missing Utterance"
task, which also achieves very poor performance across all models and methods
(with and without instructions). This is a strong reminder that to create
generalizability in language models, it is crucial to match the downstream
task to the pre-training data.

Next, we analyze individual instruction words which most frequently give
better than mean performance (Figure 3). Interestingly, we find that "return"
(as used in "return a response to the conversation") almost always leads to
better than average performance (7/8 occurances for BART-Base and Large, and
8/8 for BART0++), although it only occurs in 3 tasks, and 8 instructions.

Finally, we look at the standard deviation between instructions, averaged
across all tasks and find very little difference between models, with slightly
increasing variation as models get larger, and are pretrained (BART-Base:
0.848, BART-Large: 0.867, BART0++: 0.882). At first glance, this seems to
suggest that BART-Base is most robust to wording in instructions, but this is
more likely due to the smaller number of tasks which BART-Base can
meaningfully perform, as seen in Figure 2.

### Few-shot Experiments

For our few-shot experiments we find that, similar to the zero-shot setting,
the combination of increasing model size and additional training data is a
large contributor to improved performance. On average, the relative
improvement from BART-Base to Large is 50%, 45.4%, and 31.7% for 10, 50, and
100 samples, respectively. On the other hand, the improvement due to general
purpose MTL is only 6.6%, 3.8%, and 3.5%. When considering our 3 methods of
including instructions, we find that always including instructions is
generally the best, outperforming the other 2 methods in 6/9 settings. We
expected that as additional fine-tuning samples are added, the benefits of
instruction-tuning will diminish due to the model learning an implicit mapping
of the task. Indeed, we find that for BART-Base, training with instructions is
beneficial with 50 or fewer fine-tuning samples, above which point training
without any instructions gives best performance. Interestingly, BART0++ gains
no performance from using instructions above 10 fine-tuning samples.

##  4 Limitations

While this study does it best to separately study each variable that
contributes to performance, there are other variables that may also have
significant impacts on generalization. For example, the BART-Base and Large
models are both trained on the same dataset, and while this is crucial to
determine the effect of model size isolated from other factors, it means that
we have only run experiments with 1 pre-training corpus. Ideally, to account
for the effect of pre-training corpus, we would train each of the BART models
from scratch with a different pre-training corpus, but that may be quite
costly.

Additionally, this study only considers a single model architecture: encoder-
decoder. The same experiments can be run with a decoder-only model and may
find different results. Running these experiments with a pre-trained decoder-
only model would allow us to study both a different architecture and pre-
training corpus, but would also confound the two variables.

In this work, we did not consider the situation where the in-domain dataset is
included during general purpose MTL, which may have provided the best
performance, based on previous works studying transfer learning Albalak et al.
(2022).

Finally, while we focus on small models, because there is no BART model
smaller than base, we were only able to go down to 139 million parameters.
However, since the conclusion of our study, the OPT model has been released
with many small sizes that would make for an even more in-depth study.

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##  Appendix A Training/Testing Data

The full list of tasks is:

Act Classification, Act Generation, Advice Generation, Advice Present, Answer
Generation, Answer Selection, Begins-with Controlled Generation, Belief State
Generation, Count Response Words, Database-based Generation, Deal Present,
Dialfact Classification, Document Grounded Generation, Edit Generation,
Emotion Generation, Emotion Tagging, Ends-with Controlled Generation,
Evaluation-Binary, Evaluation-Ranking, Fill-in the Missing Utterance, Find the
Incoherent Utterance, Graph-based Generation, Intent Classification, Intent
Present, Keyword Controlled Generation, Knowledge Grounded Generation, Natural
Language Inference, Non-Toxic Feedback Generation, Persona Grounded
Generation, Persuasion Generation, Persuasion Present, Persuasion Strategy,
Question Generation, Recovery Generation, Relation Classification, Relation
Present, Response Generation with n Words, Response Generation, Schema-based
Generation, Slot Present, Slot Tagging, Slot-Value Generation, Summarization,
Target Controlled Generation, Toxic Response Classification.

The first data split uses the following 6 tasks for testing, and the remainder
for training:

  * •

Act Classification

  * •

Document Grounded Generation

  * •

Intent Classification

  * •

Keyword Controlled Generation

  * •

Relation Classification

  * •

Slot Present

The second data split uses the following 6 tasks for testing, and the
remainder for training:

  * •

Answer Selection

  * •

Natural Language Inference

  * •

Persuasion Present

  * •

Schema-based Generation

  * •

Slot Tagging

  * •

Summarization

The third data split uses the following 6 tasks for testing, and the remainder
for training:

  * •

Advice Present

  * •

Answer Generation

  * •

Fill-in the Missing Utterance

  * •

Persona Grounded Generation

  * •

Response Generation

  * •

Toxic Response Classification

##  Appendix B Task Abbreviations

  * •

Act Cls - Act Classification

  * •

Adv Prsnt - Advice Present

  * •

Ans Gen - Answer Generation

  * •

Ans Sel - Answer Selection

  * •

DG Gen - Document Grounded Generation

  * •

Fill Utt - Fill-in the Missing Utterance

  * •

Int Cls - Intent Classification

  * •

KC Gen - Keyword Controlled Generation

  * •

NLI - Natural Language Inference

  * •

PG Gen - Persona Grounded Generation

  * •

Prs Prsnt - Persuasion Present

  * •

Rel Cls - Relation Classification

  * •

Resp Gen - Response Generation

  * •

SB Gen - Schema-based Generation

  * •

Slot Prsnt - Slot Present

  * •

Slot Tag - Slot Tagging

  * •

Summ. - Summarization

  * •

Toxic Cls - Toxic Response Classification

##  Appendix C Training Details

We train all models for a maximum of 3 epochs, and utilize validation based
early stopping. To determine the learning rate, we trained each model on a
single seed and validate the best learning rate in {1e-5, 5e-5, 1e-4}, then
train for 2 additional seeds using the best learning rate. We found for all
models that 5e-5 was the best learning rate. For all experiments we use the
AdamW optimizer.

##  Appendix D Additional Figures

Figure 5: Percentage of occurrences of a word that lead to better than average
performance for an instruction. Results calculated from BART-Large model and
only includes words that occur is more than 5 instructions. Figure 6:
Percentage of occurrences of a word that lead to better than average
performance for an instruction. Results calculated from BART0++ model and only
includes words that occur is more than 5 instructions. Figure 7: Performance
comparison against a baseline model that is trained without instructions.
Values shown are the absolute score difference between the designated method
and the baseline model. Figure 8: Performance comparison against an off-the-
shelf baseline model. Values shown are the absolute score difference between
the designated method and the baseline model.