Physics Letters B 788 (2019) 249–255
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Anomalous discrete flavor symmetry and domain wall problem
So Chigusa
a,∗
, Kazunori Nakayama
a,b
a
Department of Physics, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
b
Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
a r t i c l e i n f o a b s t r a c t
Article history:
Received
6 September 2018
Received
in revised form 31 October 2018
Accepted
16 November 2018
Available
online 20 November 2018
Editor:
J. Hisano
Discrete flavor symmetry is often introduced for explaining quark/lepton masses and mixings. However,
its spontaneous breaking leads to the appearance of domain walls, which is problematic for cosmology.
We consider a possibility that the discrete flavor symmetry is anomalous under the color SU(3) so that it
splits the energy levels of degenerate discrete vacua as a solution to the domain wall problem. We find
that in most known models of flavor symmetry, the QCD anomaly effect can only partially remove the
degeneracy and there still remain degenerate vacua.
© 2018 The Author(s). 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
Discrete flavor symmetry is often introduced in order to explain
the observed patterns of neutrino masses and mixings. Depend-
ing
on the choice of flavor symmetry and its way of spontaneous
breaking, various predictions have been obtained (for reviews, see
Refs. [1–4]). However, spontaneous breaking of a discrete sym-
metry
leads to the formation of domain walls [5]. Once formed,
domain walls are topologically stable and come to dominate the
universe, which is a disaster for cosmology. There are several solu-
tions.
First, if inflation happens after the discrete symmetry break-
ing,
domain walls are inflated away and no domain walls are left
in our observable universe and hence there is no problem. Second,
if the flavor symmetry is not exact but only an approximate one,
domain walls are unstable and collapse at some instance [6].
Whether
the first option works or not depends on the energy
scale of inflation, that of flavor symmetry breaking, and the re-
heating
temperature. If the flavor symmetry breaking scale is suffi-
ciently
larger than the inflation scale and maximum temperature
of the universe, the flavor symmetry may not be restored after
inflation. However, most concrete models introduce supersymme-
try
(SUSY) [7]in order to simplify the interaction among various
flavon fields and there often appear flat directions in the scalar
potential in SUSY [8]. These flat directions obtain masses of soft
SUSY breaking, which is usually much lower than the flavor sym-
metry
breaking scale. In this case, we need an inflation scale lower
than the SUSY breaking scale (e.g. O(TeV) for solving the hierarchy
*
Corresponding author.
E-mail
address: chigusa@hep-th.phys.s.u-tokyo.ac.jp (S. Chigusa).
problem) to solve the domain wall problem. Such a low-scale in-
flation
is not excluded, but rather unlikely. Thus we do not pursue
this option in this paper.
As
for the second option, it is always possible to introduce
small explicit symmetry breaking terms by hand unless the dis-
crete
symmetry is a remnant of some gauge symmetry. But there
are so many ways to introduce such breaking terms, which are not
under control and may (partly) destroy original motivation to con-
sider
flavor symmetry to explain observed data. Thus we restrict
ourselves to a special case: the discrete flavor symmetry is exact
at the classical level and only broken by the quantum anomaly.
The concept is the same as the Peccei–Quinn (PQ) symmetry for
solving the strong CP problem [9,10].
The
role of anomaly (in particular, anomaly under the color
SU(3)) for Abelian discrete symmetry to solve the domain wall
problem was explored in Ref. [11] and further applied to some
concrete models in Refs. [12–17]. On the other hand, the anomaly
of general (non-)Abelian discrete groups was extensively studied
in Refs. [18–27]. In the context of model-building, non-anomalous
discrete symmetry is often used in order to embed the discrete
symmetry into some continuous gauge group. Our attitude is com-
pletely
different: we want to make use of anomalous discrete fla-
vor
symmetry to find models without domain wall problem.
In
Sec. 2 we summarize the structure of the discrete anomaly in
general and examine all the non-Abelian discrete groups listed in
Ref. [2]to see whether the anomalous breaking of the symmetry is
allowed or not. We will find that none of them can be completely
anomalous and hence the domain wall problem is not solved solely
by the anomaly effect. In Sec. 3 we will check this statement with
some explicit examples. We conclude in Sec. 4.
https://doi.org/10.1016/j.physletb.2018.11.027
0370-2693/
© 2018 The Author(s). 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
.