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Physics Letters B 793 (2019) 411–414

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Scotogenic U (1)

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)

which

comes from SO(10) → SU(5) × U (1)

. 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.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

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

, 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)

× SU(2)

× U (1)

to the left-right sym-

metry

SU(3)

× SU(2)

× U (1)

(B−L)/2

. In that case, the

SU(2)

doublet (ν, e )

is required, and the charged W

gauge bo-

son

is predicted along with a neutral Z



gauge boson.

more recent choice is to consider U (1)

which comes from

SO(10) → SU(5) × U (1)

, with SU(5) breaking to the SM. As-

suming

that U (1)

survives to an intermediate scale, the current

experimental bound on the mass of Z

being about 4.1 TeV [1,2],

then ν

must exist for the cancellation of gauge anomalies. Now

is a singlet and W

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

to ν

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)

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)

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)

without breaking global lepton number. The fermions are

shown in Table 1 and scalars in Table 2.

New fermions N, N

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

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 

†

transforms exactly like 

, the linear combination  = (v



†





+ v

is the analog of the standard-model Higgs dou-

blet,

where φ

 = v

1,2

. An important Z

discrete symmetry is

imposed so that ν

is odd and all other SM ﬁelds are even, pre-

venting

thus the tree-level Yukawa coupling (νφ

− eφ

)ν

. This

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

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

which would violate lepton number,

hence only cases A and B will be considered. In case A, the quartic

scalar term ζ σ

is allowed. Hence the would-be dark U(1) sym-

metry

is reduced to Z

, i.e. ω for σ , N

and ω

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

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/

SCOAP

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