$$\Delta(27)$$与$$U(1)_\chi$$驱动的共生最大狄拉克中微子模型

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本文主要探讨了在标准模型的基础上，通过结合$SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_\chi$规范群，以及来自$SO(10)$到$SU(5) \times U(1)_\chi$的额外$U(1)_\chi$对称性，如何实现Scotogenic共生机制来解释轻子家族，特别是中微子的质量和混合特性。$U(1)_\chi$对称性在模型中起着关键作用，它通过暗物质的辐射过程自发破缺，从而驱动狄拉克中微子的质量生成。非阿贝尔离散对称性$\Delta(27)$被引入以描述三个轻子家族，并且以一种自然的方式被软性破缺。这种破缺导致了两种残留对称性：全局的$U(1)_L$轻子数守恒和一个暗对称性，可能表现为$Z_2$、$Z_3$或$U(1)_D$，具体取决于哪个标量粒子负责$U(1)_\chi$的破缺。这种理论框架的核心预测是出现了cobimaximal中微子混合模式，即$\theta_{13} \neq 0$、$\theta_{23} = \frac{\pi}{4}$以及$\delta_{CP} = \pm \frac{\pi}{2}$，这些都是当前实验中备受关注的中微子物理特性。作者Ernest Maa在论文《Eur.Phys.J.C (2019) 79:903》中，提出了一种创新的方法，通过这种模型，不仅能够解释中微子的非平凡性质，还可能为暗物质的起源提供线索。这篇Open Access的文章展示了理论物理学家对于中微子物理学的深入研究，以及他们如何利用高阶对称性和额外的规范群来挑战和扩展我们对基本粒子行为的理解。这篇论文是对标准模型的扩展和深化，为未来的实验验证提供了理论指导。

Eur. Phys. J. C (2019) 79:903

https://doi.org/10.1140/epjc/s10052-019-7440-x

Regular Article - Theoretical Physics

Scotogenic cobimaximal Dirac neutrino mixing from (27) and

U (1)

Ernest Ma

Physics and Astronomy Department, University of California, Riverside, CA 92521, USA

Received: 23 August 2019 / Accepted: 25 October 2019 / Published online: 8 November 2019

Abstract In the context of SU(3)

× SU(2)

×U (1)

U (1)

, where U (1)

comes from SO(10) → SU(5) ×

U (1)

, supplemented by the non-Abelian discrete (27)

symmetry for three lepton families, Dirac neutrino masses

and their mixing are radiatively generated through dark mat-

ter. The gauge U (1)

symmetry is broken spontaneously. The

discrete (27) symmetry is broken softly and spontaneously.

Together, they result in two residual symmetries, a global

U (1)

lepton number and a dark symmetry, which may be

, Z

,orU(1)

depending on what scalar breaks U (1)

Cobimaximal neutrino mixing, i.e. θ

= 0, θ

= π/4, and

=±π/2, may also be obtained.

1 Introduction

Whereas there exist powerful theoretical arguments that neu-

trinos are Majorana, there is yet no incontrovertible experi-

mental evidence that they are so, i.e. no deﬁnitive measure-

ment of a nonzero neutrinoless double beta decay [1], with

the best present experimental upper limit [2]ontheMajo-

rana electron neutrino mass in the range 61 to 165 meV. To

make a case for neutrinos to be Dirac, the existence of a right-

handed neutrino ν

must be justiﬁed, which is of course not

required in the standard model (SM) of quarks and leptons.

The canonical choice is to extend the SM gauge symme-

try SU(3)

× SU(2)

× U (1)

to the left-right symmetry

SU(3)

×SU(2)

×U (1)

(B−L)/2

[3]. In that case,

the SU(2)

doublet (ν, e)

is required, and the charged W

gauge boson is predicted along with a neutral Z



gauge boson.

A more recent choice is to consider U (1)

which comes

from SO(10) → SU(5) × U (1)

, with SU(5) breaking to

the SM at the same grand uniﬁed scale. Assuming that U (1)

survives to an intermediate scale, the current experimental

bound on the mass of Z

being about 4.1 TeV [4,5], then

must exist for the cancellation of gauge anomalies. Now

e-mail: ma@phyun8.ucr.edu

is a singlet and W

is not predicted. In this context, new

insights into dark matter [6,7] and Dirac neutrino masses

[8,9] have emerged. In particular, the breaking pattern was

fully discussed in Ref. [6].

To make sure that ν

itself does not have a Majorana mass,

the breaking of U (1)

should not come from a scalar which

couples to ν

. This simple idea was ﬁrst discussed [10]in

2013 in the case of singlet fermions charged under a gauge

U (1)

. If the latter is broken by a scalar with three units of

X charge, it is impossible for these fermions with one unit

of X charge to acquire Majorana masses. Hence the residual

symmetry is global U(1) in this case. It is straightforward then

to apply this idea to lepton number [11]. Another possible

residual lepton symmetry is Z

which was ﬁrst pointed out

in Ref. [12].

A second issue regarding Dirac neutrinos is that the cor-

responding Yukawa couplings linking ν

to ν

through the

SM Higgs boson must be very small. To avoid these tree-

level couplings, it is often assumed that some additional sym-

metry exists which forbids these dimension-four couplings,

but Dirac neutrino masses may be generated radiatively as

this symmetry is broken softly by dimension-three terms.

For a generic discussion, see Ref. [13], which is patterned

after that for Majorana neutrinos [14]. In some applications

[15–17], the particles in the loop belong to the dark sector.

This is called the scotogenic mechanism, from the Greek

‘scotos’ meaning darkness, the original one-loop example

[18] of which was applied to Majorana neutrinos.

Instead of the ad hoc extra symmetry which forbids the

tree-level couplings, exotic assignments of the gauge charges

of ν

maybeused[11,19–25] instead. However, a much

more efﬁcacious idea is to use a non-Abelian discrete fam-

ily symmetry, which is softly broken in the dark sector. In

this paper, (27) [26–30] is shown to be useful in achieving

the goal of having scotogenic Dirac neutrino masses with

a mixing pattern [31–33] called cobimaximal [34–39], i.e.

123

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