#### Connecting neutrino masses and dark matter by high-dimensional lepton number violation operator

JHE
Connecting neutrino masses and dark matter by high-dimensional lepton number violation operator
F. del Aguila 0 2 7 8 9
A. Aparici 0 2 7 8 9
S. Bhattacharya 0 2 7 8 9
A. Santamaria 0 2 7 8 9
J. Wudka 0 2 7 8 9
Effective 0 2 7 8 9
0 Hsinchu , 300 Taiwan
1 Department of Physics, Tsinghua University
2 Chao-Qiang Geng
3 Collaborative Innovation Center of Quantum Matter
4 Department of Physics, National Tsing Hua University
5 Physics Division, National Center for Theoretical Sciences
6 Chongqing University of Posts & Telecommunications
7 Open Access , c The Authors
8 Beijing 100084 , China
9 Chongqing , 400065 , China
We propose a new model with the Majorana neutrino masses generated at two-loop level, in which the lepton number violation (LNV) processes, such as neutrinoless double beta decays, are mainly induced by the dimension-7 LNV effective operator O7 = ¯lc γμLL(DμΦ)ΦΦ. Note that it is necessary to impose an Z2 symmetry in order R that O7 dominates over the conventional dimension-5 Weinberg operator, which naturally results in a stable Z2-odd neutral particle to be the cold dark matter candidate. More interestingly, due to the non-trivial dependence of the charged lepton masses, the model predicts the neutrino mass matrix to be in the form of the normal hierarchy. We also focus on a specific parameter region of great phenomenological interests, such as electroweak precision tests, dark matter direct searches along with its relic abundance, and lepton flavor violation processes.
Neutrino Physics; Global Symmetries; Effective field theories
1 Introduction 2
Two-loop Majorana neutrino masses
Neutrinoless double beta decay
Phenomenological constraints
Electroweak precision tests
Lepton flavor violation
Numerical results
Introduction
The presence of the tiny neutrino masses and mixings between different neutrino flavors
have been established by many neutrino oscillation experiments [1–7], while more and more
evidences are accumulated for the existence of dark matter (DM) over the last several
decades, with the most precise measurement of its relic abundance by PLANCK [8, 9].
Both phenomena cannot be explained within the Standard Model (SM), thus providing
us with two windows towards new physics beyond it. An interesting idea is to connect
neutrinos and DM in a unified framework as many existing attempts in the literature (see
e.g. refs. [10–13]). We would like to push this connection further in the present paper.
In order to understand the mass hierarchy problem in the neutrino sector, there are
many models in the literature to naturally generate small Majorana neutrino masses such
as the traditional Seesaw [14–26] and radiative mass generation mechanisms [10–13, 27–29].
Most of them can be summarized as a specific realization of the conventional dimension-5
Weinberg operator. However, the generation of Majorana neutrino masses only requires
the lepton number violation (LNV) by two units, and there exist many other equally
legitimate LNV effective operators [30–38], which are composed of the SM fields but with
higher scaling dimensions. From the effective field theory perspective, it is generically
believed that these high-dimensional effective operators are subdominated by the Weinberg
operator due to the suppression from the corresponding high powers of the large cutoff.
In order for these operators to show up as the leading contributions, one usually needs
to impose an additional symmetry on the model to break the usual scaling arguments.
Furthermore, if this symmetry is kept unbroken, then the lightest symmetry-protected
neutral particle would provide a perfect DM candidate. In this way, the symmetry connects
l−
Di−
l−
one-loop realization of O7, and (c) the two-loop neutrino mass generation.
Majorana neutrino masses and DM physics by the high-dimensional effective operators.
Such a connection has been already exemplified by some recent three-loop neutrino mass
with the DM embedded in the loop.
[32, 34, 37] and construct a UV complete model with an unbroken Z2 symmetry to
accomplish the above general arguments. In this model, Majorana neutrino masses arise
by a new “long-range” contribution,1 as the results of the existence of O7, while DM can
also be embedded naturally as the lightest Z2-odd neutral state.
This paper is organized as follows. In section 2, we first describe the particle content
and write down the relevant part of the Lagrangian for the model. We then calculate the
the emphasis on their relations to the high-dimensional effective operator O7. In section 3,
the constraints on the model are addressed from the electroweak (EW) precision tests,
dark matter searches, and lepton flavor violating (LFV) processes. Finally, we give the
conclusions in section 4.
Figure 1a shows the neutrino mass generation induced by the one-loop diagram with O7. In
SU(2)L × U(1)Y . A Z2 symmetry is also imposed, in which only the new particles carry
odd charges. The relevant new parts of the Lagrangian are given by
pectation value v ≡
eigenstates with their masses, given by
MS21 =
MS22 =
M H2 = Mχ2 + (λ4 + λ5)v2 , MA2 = Mχ2 + (λ4 − λ5)v2 .
The mass splittings between the charge and neutral components of the inert fermion
doublets can only be induced by loop corrections with values around a few hundred MeV [44].
In this paper, we will characterize the model by using the physical quantities:
which are less relevant in our discussion.
Two-loop Majorana neutrino masses
As seen in figure 1b, the effective operator O7 can be induced by the one-loop diagram,
whereas the Weinberg operator cannot.2 Consequently, Majorana neutrino masses appear
2There is a similar realization of O7 in ref. [37], in which a triplet replaced the singlet s of our model.
A fundamental distinction of their paper from the present one is that O7 does not give the dominant
contribution to Majorana neutrino masses in ref. [37].
X (ml ζil ξil′ + ml′ ζil′ ξil)[Ii1 − Ii2] , (2.8)
Iij ≡
2(1 − 3x)
x(1 − x)
Z 1−x3
Z 1−x2−x3
6y1(2 − x)
(1 − x)2
log(mi2j ) + −2y1(2 − x) M W2
x(1 − x)
mi2j ≡ y1[x1M H2 + x2MA2 + x3M W2 ] + y2x(1 − x)MS2j
+(1 − y1 − y2)x(1 − x)M D2i ,
x = x1 + x2 .
where m1,2,3 are three neutrino mass eigenvalues, which can have the normal ordering,
Nakagawa-Sakata mixing matrix [47, 48]. Without loss of generality, V can be written as
the standard parametrization by appropriate rephasing in LL’s and lR’s, given by [49]
V = −s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ
From eq. (2.8), we can get two important features for this mass generation mechanism.
will make all neutrinos massless. Secondly, the neutrino masses are positively correlated
3Similar topology with one W ± exchange in a two-loop neutrino mass model can also be found in
refs. [45, 46], in which a different high-dimensional effective operator is realized without DM.
which is shown in refs. [49, 50] to rule out the inverted ordering of neutrino mass spectrum at
Note that in the limit of eq. (2.13), ref. [49] even shows that the lightest neutrino mass m1
can only be located within the range 0.001eV . m1 . 0.01eV. Moreover, the smallness of
fitting for the neutrino oscillation data [49], the corresponding mass matrices for Textures
A, B, C, and D (TA, TB, TC and TD) are given by
TA : Mν = 0.12 1.9 2.7 (10−2) eV ,
−0.24 2.7
(10−2) eV ,
−1.1
−2.3
−2.1 (10−2) eV ,
−0.055 −2.1
−3.1
−0.90 −0.24
−1.1 −0.055
0.12 0.052
TA : ζ ∝ 0.12 0.89 0.14 ,
0.052 0.14 0.068
−1.
−1. −0.0031
We now search for possible coupling matrix forms to realize the above four CP
conserv
−0.013 0.14
−0.84 −0.013
−0.081 0.062
−0.086 1.1
−2.6
−2.2
−2.2 (10−2) eV ,
−2.9
−0.0031 −0.11 −0.088
−1.1
−0.11 ,
−1.2
−0.12 ,
−0.12 −0.081
insertion on the propagator, (b) the contribution involving O7, and (c) the one-loop construction
which realizes O7. For (b) and (c), the corresponding upside-down diagrams also need be considered.
Neutrinoless double beta decay
In the previous section, we have built a two-loop neutrino mass model in which the LNV
operator O7 provides the leading contribution. The next step is to study the LNV effect in
this model induced by this high-dimensional operator. The most sensitive smoking gun for
For the Majorana neutrino masses, there always exists the traditional long-range decay
process by the exchange of neutrinos with a pair of left-handed electrons emitted as shown
in figure 2a. Note that there is a chirality flipping on the internal neutrino propagator,
which leads to the proportionality of the amplitude to the neutrino mass matrix element
does not give the main part of this process, and one should have a prior consideration on
the effects of other new diagrams. For example, a class of neutrino models [51–53] that can
well studied in refs. [51, 52, 54–56], and the new contribution is much larger than that
from figure 2a by orders of magnitude of 108. For our model or those with O7 as the main
1 s2θξζ ee (M H2 − MA2 )(I1′ − I2′) ,
Ij′ =
Z 1−x3
Z 1−x2−x3
x1M H2 +x2MA2 +x3MS2j +(1−x1 −x2 −x3)M D2 ,
GERDA-1(76Ge) [59]
KamLAND-Zen(136Xe) [60]
NEMO-3(150Nd) [62]
CUORICINO(130Te) [63]
NEMO-3(82Se) [64, 65]
NEMO-3(100Mo) [65]
4.4 × 10−9
8.3 × 10−9
2.9 × 10−7
2.3 × 10−8
1.5 × 10−8
3.5 × 10−8
3.6 × 10−4
2.8 × 10−4
1.5 × 10−3
4.2 × 10−4
1.5 × 10−3
5.6 × 10−4
which is originated from the one-loop generation for O7 (in figure 2c).
By using the
numerical results therein and also in ref. [58], we find that the contribution proportional
Finally, we end this section by mentioning that the role of the Z2 symmetry is to make
the dimension-7 operator O7 become the dominant contributions to the LNV processes and
singlet (s) scalars are the same as those in the Zee’s model [27], the Majorana neutrino
masses would be mainly generated by the corresponding one-loop diagrams related to the
conventional Weinberg operator if the Z2 symmetry is absent.
However, with the Z2
symmetry, O7 is singled out at 1-loop level, while other LNV effective operators, especially
the dimension-5 Weinberg operator, are much suppressed since they would be only induced
by higher-loop diagrams. In this way, the Z2 symmetry breaks the conventional effective
would also change the leading modes accordingly.
Phenomenological constraints
Electroweak precision tests
As discussed previously, in order to have the two-loop neutrino masses in our model, the
=
=
EW gauge quantum numbers. Both effects could change the values of the EW oblique S
and T parameters. In particular, the T parameter should yield a stronger constraint on
this model. The deviation of T from the SM is given by [13]
with the function F defined by
Fx,y =
− Mx2 − My2 log
The value of Fx,y becomes zero when Mx → My, and it increases with the mass splitting
between Di and the SM leptons nor tree-level mass splitting among Di, while the deviation
for the S parameter can also be ignored [66]. The formulae of eq. (3.1) is a general result
for the models with the mixings between the inert doublet and singlet scalars. The global
MS2 can not be too small, since there exists a lower bound on MS2 located within 70 to
90 GeV [67] from the LEP experiments.
In this model, the lightest of the extra neutral particles: H0, A0, and D10,2,3 could be
a DM candidate, whose stability is guaranteed by the imposed Z2 symmetry. In the
following, we will focus on the case that DM is constituted solely by H0 with a small
studied inert doublet model [68–70]. Furthermore, we concentrate on the low DM mass
region with 50 GeV 6 MH 6 80 GeV [69, 70], in which a large H0-A0 mass splitting can
be allowed for the generation of the right two-loop neutrino masses. In addition, the mass
of S2± should be higher than 90 GeV in order to escape the LEP bounds [67], so that the
co-annihilation channels, such as H0-A0 and H0-S2±, would be strongly suppressed and
We use the package micrOMEGAs [71] to accurately calculate the relic abundance
Higgs resonance in the s-channel would become prominent, which is characterized by the
cross section to accommodate the DM relic abundance [69, 70], which is omitted in figure 4.
The DM H0 in this low mass region could be constrained by the DM direct detection
experiments. Since we need a relatively large Higgs-mediation annihilation channel to
generate DM relic abundance, the Higgs exchange channel can also give rise to sizeable
spin-independent signals, with the corresponding DM-nucleon cross section as follows [69]:
Currently, the most stringent bound on the spin-independent DM-nucleon cross section is
for a DM mass of 33 GeV. It is shown in figure 4 that the LUX experiment has already
probed some parameter space required by the DM relic abundance. Especially, the low
DM mass region with MH 6 52 GeV is actually ruled out, as indicated by the shaded area
in the plot. However, most parameter spaces are still allowed by LUX.
Lepton flavor violation
The current experimental constraints on LFV processes, such as the radiative decays l →
usually constrains a model in the most stringent way. In our model, we have
1 KH′ +
KA
dance. The gray shaded area is excluded by the LUX experiment, and the black dot represents the
where the loop integrals Kx and Kx′ are defined as
Kx =
2z3 + 3z2 − 6z + 1 − 6z2 log z
6(1 − z)4Mx2
Kx′ = −
z3 − 6z2 + 3z + 2 + 6z log z
6(1 − z)4Mx2
constant combination
next-generation experiments in the future.
Numerical results
Based on the above constraints from the LFV processes, EW precision tests, direct searches
of DM with the required relic abundance, we find a benchmark point from the allowed
parameter space, given by:
MH = 70 GeV ,
MA = 95 GeV ,
MS1 = 310 GeV ,
MS2 = 90 GeV ,
MD1 = MD2 = MD3 = MD = 200GeV ,
red, green, and yellow curves correspond to the neutrino textures TA, TB, TC , and TD, respectively,
while the black dot is the benchmark point.
=
=
0.005 0.0022
0.0022 0.0061 0.0029
benchmark point by a black dot in figures 5a, 3b, and 4, where the experimental results
from LFV processes, oblique parameters, and DM searches are well satisfied, respectively.
the self-consistence of the perturbation theory.
Conclusions
We have tried to make the connection between neutrino physics and dark matter searches.
In particular, we have emphasized that every effective operator, which violates the lepton
number by two units, can give an equally good mechanism to generate Majorana neutrino
masses. The problem lies in the fact that the new high-dimensional operators might be
buried by the overwhelming effects from the conventional Weinberg operator which possess
the smallest scaling dimension. One way to break this effective field theory ordering is
to impose some symmetry which would protect the lightest neutral symmetry-protected
states to be the DM particle.
We have explicitly realized this connection by constructing a UV complete model with
contribution are closely related to O7. Especially, the neutrino mass matrix is predicted
to be of the normal ordering due to the hierarchy in the charged lepton masses, and the
we impose an additional CP symmetry in the lepton sector, we can even determine the
form of the neutrino mass matrix completely. We have also focused on a specific parameter
region with a small mixing between charged scalars, and considered the constraints from
the electroweak precision tests, dark matter searches, and LFV processes.
This work was supported by National Center for Theoretical Sciences, National Science
Council (Grant No.
NSC-101-2112-M-007-006-MY3) and National Tsing Hua Univer
sity (Grant No. 104N2724E1), the National Science Foundation of China (NSFC Grant
No. 11475092), the Tsinghua University Initiative Scientific Research Program (Grant No.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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