Physics Letters B 803 (2020) 135294
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
A modular A
4
symmetric scotogenic model
Takaaki Nomura
a,∗
, Hiroshi Okada
b,c
, Oleg Popov
d
a
School of Physics, KIAS, Seoul 02455, Republic of Korea
b
Asia Pacific Center for Theoretical Physics (APCTP), Headquarters San 31, Hyoja-dong, Nam-gu, Pohang 790-784, Republic of Korea
c
Department of Physics, Pohang University of Science and Technology, Pohang 37673, Republic of Korea
d
Institute of Convergence Fundamental Studies, Seoul National University of Science and Technology, Seoul 139-743, Republic of Korea
a r t i c l e i n f o a b s t r a c t
Article history:
Received 7 September 2019
Received in revised form 20 January 2020
Accepted 10 February 2020
Available online 14 February 2020
Editor: A. Ringwald
We propose a minimal extension of the Standard Model where neutrino masses are generated radiatively
at one-loop level via Scotogenic scenario. The model is augmented with A
4
modular symmetry as a
scotogenic and flavor symmetry. With minimal number of parameters, the model makes predictions for
neutrino oscillation data, Majorana and Dirac phases, dark matter characteristics, and neutrinoless double
beta decay.
© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Particle physics experiments and observations have been suc-
cessfully
confirming the standard model (SM) of particle physics.
On the other hand, there are some issues indicating an existence
of physics beyond the SM such as existence of dark matter (DM),
non-zero tiny neutrino masses and origin of flavor structure. In
describing these issues, symmetry would play an important role
like guaranteeing stability of DM, forbidding neutrino mass at tree
level and restricting flavor structure. It is thus interesting to con-
struct
a model of physics beyond the SM adopting a new symme-
try.
Modular flavor symmetries have been recently proposed by [1,
2]to
provide more predictions to the quark and lepton sector due
to Yukawa couplings with a representation of a group. Their typ-
ical
groups are found in basis of the A
4
modular group [2–11],
S
3
[12–15], S
4
[16–18], A
5
[19,20], larger groups [21], multi-
ple
modular symmetries [22], and double covering of A
4
[23]in
which masses, mixings, and CP phases for quark and lepton are
predicted.
1
Also, a systematic approach to understand the origin
of CP transformations has been recently achieved by ref. [32]. In
particular a model with modular A
4
symmetry is discussed in
ref. [6,11] where neutrino mass is generated at one-loop level. For
model in ref. [6], inert doublet and singlet scalar fields are intro-
duced
to generate neutrino mass using modular form of weight 2
which is a triplet of A
4
and the basis for constructing other mod-
*
Corresponding author.
E-mail addresses: nomura@kias.re.kr (T. Nomura), hiroshi.okada@apctp.org
(H. Okada),
opopo001@ucr.edu (O. Popov).
1
Several reviews are helpful to understand the whole idea [24–31].
ular forms with higher weight. On the other hand, for model in
ref. [11], vector-like leptons are introduced to get positive contri-
bution
to muon g − 2 and triplet modular form with weight 4 is
used.
In this paper, we apply modular A
4
symmetry in minimal Sco-
togenic
model [33]in which neutrino mass is generated at one-
loop
level and DM candidates are contained. We find that the
minimal Scotogenic model can be realized by use of modular forms
with higher weight which are constructed by that of weight 2.
Thus field contents and the structure of model are much simpler
than previous models [6,11]. In our construction the right-handed
neutrinos are introduced as a triplet of A
4
assigning modular
weight −k =−1. Also, non-zero modular weight is assigned to in-
ert
Higgs doublet as −k =−3. Interestingly we find that additional
Z
2
symmetry is not necessary to realize structure of Scotogenic
model due to the nature of modular form. Then numerical analy-
sis
for neutrino mass matrix is carried out to show predictions of
our model as a result of modular A
4
symmetry.
Manuscript is organized as follows. In Sec. 2, we give our model
set up under A
4
modular symmetry. We discuss right-handed neu-
trino
mass spectrum, lepton flavor violation (LFV) and generation
of the active neutrino mass at one-loop level in Sec. 3. Numerial
analysis is presented in Sec. 4. Finally, we conclude and discuss in
Sec. 5.
2. Model
In this section we introduce our model, which is based on A
4
modular symmetry. Leptonic and scalar fields of the model and
their representations under A
4
symmetry and modular weights are
given by Table 1, while the ones of Yukawa couplings are given
https://doi.org/10.1016/j.physletb.2020.135294
0370-2693/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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