Physics Letters B 793 (2019) 411–414
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Scotogenic U (1)
χ
Dirac neutrinos
Ernest Ma
Physics and Astronomy Department, University of California, Riverside, CA 92521, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received
5 February 2019
Received
in revised form 5 April 2019
Accepted
2 May 2019
Available
online 6 May 2019
Editor:
A. Ringwald
The standard model of quarks and leptons is extended to include the gauge symmetry U (1)
χ
which
comes from SO(10) → SU(5) × U (1)
χ
. The radiative generation of Dirac neutrino masses through dark
matter is discussed in two examples. One allows for light Dirac fermion dark matter. The other allows for
self-interacting scalar dark matter with a light scalar mediator which decays only to two neutrinos.
© 2019 The Author. 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
Whereas neutrinos are usually assumed to be Majorana, there
is yet no experimental evidence, i.e. no definitive measurement
of a nonzero neutrinoless double beta decay. To make a case for
neutrinos to be Dirac, the first is to justify the existence of a right-
handed
neutrino ν
R
, which is not necessary in the standard model
(SM) of quarks and leptons. An obvious choice is to extend the SM
gauge symmetry SU(3)
C
× SU(2)
L
× U (1)
Y
to the left-right sym-
metry
SU(3)
C
× SU(2)
L
× SU(2)
R
× U (1)
(B−L)/2
. In that case, the
SU(2)
R
doublet (ν, e )
R
is required, and the charged W
±
R
gauge bo-
son
is predicted along with a neutral Z
gauge boson.
A
more recent choice is to consider U (1)
χ
which comes from
SO(10) → SU(5) × U (1)
χ
, with SU(5) breaking to the SM. As-
suming
that U (1)
χ
survives to an intermediate scale, the current
experimental bound on the mass of Z
χ
being about 4.1 TeV [1,2],
then ν
R
must exist for the cancellation of gauge anomalies. Now
ν
R
is a singlet and W
±
R
is not predicted. In this context, new in-
sights
into dark matter [3,4] and Dirac neutrino masses [5]have
emerged. In particular, it helps with the following second issue re-
garding
a Dirac neutrino mass. Since neutrino masses are known
to be very small, the corresponding Yukawa couplings linking ν
L
to ν
R
through the SM Higgs boson must be very small. To avoid
using such a small coupling, a Dirac seesaw mechanism [6,7]is
advocated in Ref. [5]. The alternative is to consider radiative mech-
anisms,
especially through dark matter, called scotogenic from the
Greek ‘scotos’ meaning darkness. Whereas the original idea [8]was
applied to Majorana neutrinos, one-loop [9,10] and two-loop [11]
examples
for Dirac neutrinos already exist in the context of the
SM. For a generic discussion of Dirac neutrinos, see Ref. [12], which
is patterned after that for Majorana neutrinos [13]. Here two new
E-mail address: ma@phyun8.ucr.edu.
U (1)
χ
examples are shown. One allows for light Dirac fermion
dark matter. The other allows for self-interacting scalar dark matter
with a light scalar mediator which decays only to two neutrinos.
2. First scotogenic U (1)
χ
model
The particle content follows that of Ref. [5]except for the addi-
tion
of ζ ∼ (1, 15) from the 672 of SO(10). This is used to break
U (1)
χ
without breaking global lepton number. The fermions are
shown in Table 1 and scalars in Table 2.
New fermions N, N
c
belonging to 126
∗
, 126 respectively are
added per family , as well as a Higgs doublet from 144 and a sin-
glet
from 16. Note that their Q
χ
charges are fixed by the SO(10)
representations from which they come. It should also be clear
that incomplete SO(10) and SU(5) multiplets are considered here
(which is the case for all realistic grand unified models). Since
†
1
transforms exactly like
2
, the linear combination = (v
1
†
1
+
v
2
2
)/
v
2
1
+ v
2
2
is the analog of the standard-model Higgs dou-
blet,
where φ
0
1
,2
= v
1,2
. An important Z
2
discrete symmetry is
imposed so that ν
c
is odd and all other SM fields are even, pre-
venting
thus the tree-level Yukawa coupling (νφ
0
− eφ
+
)ν
c
. This
Z
2
symmetry is respected by all dimension-four terms of the La-
grangian.
It will be broken softly by the dimension-three trilinear
term μσ
†
η (in cases B and D) or the m
N
NN
c
mass term (in
cases A and C). This allows the one-loop diagram of Fig. 1 to
generate a radiative Dirac neutrino mass. Cases C and D allow
the Yukawa coupling ζ ν
c
N
c
which would violate lepton number,
hence only cases A and B will be considered. In case A, the quartic
scalar term ζ σ
3
is allowed. Hence the would-be dark U(1) sym-
metry
is reduced to Z
3
, i.e. ω for σ , N
c
and ω
2
for η, N, where
ω = exp(2π i/3). In case B, it is forbidden, so the model possesses
a dark U(1) symmetry, i.e. D is 1 for σ , N
c
and −1for η, N. In
either case, there is still a conserved lepton symmetry, i.e. L = 1
https://doi.org/10.1016/j.physletb.2019.05.006
0370-2693/
© 2019 The Author. 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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