Fermion masses and mixings and dark matter constraints in a model with radiative seesaw mechanism
Revised: March
masses and mixings and dark matter constraints in a model with radiative seesaw mechanism
Nicolas Bernal 0 1 4 5 6 7
A.E. Carcamo Hernandez 0 1 4 7
Ivo de Medeiros Varzielas 0 1 3 4 7
Sergey Kovalenko 0 1 4 7
0 Casilla 110V , Valpara so , Chile
1 91405 Orsay , France
2 47A15 , Bogota , Colombia
3 CFTP, Departamento de F sica, Instituto Superior Tecnico, Universidade de Lisboa
4 Carrera 3 Este
5 Laboratoire de Physique Theorique, CNRS, Universite ParisSud, Universite ParisSaclay
6 Centro de Investigaciones, Universidad Antonio Narin~o
7 Avenida Rovisco Pais 1 , 1049 Lisboa , Portugal
We formulate a predictive model of fermion masses and mixings based on a (27) family symmetry. In the quark sector the model leads to the viable mixing inspired texture where the Cabibbo angle comes from the down quark sector and the other angles come from both up and down quark sectors. In the lepton sector the model generates a predictive structure for charged leptons and, after radiative seesaw, an e ective neutrino mass matrix with only one real and one complex parameter. We carry out a detailed analysis of the predictions in the lepton sector, where the model is only viable for inverted neutrino mass hierarchy, predicting a strict correlation between mark point that leads to the best t values of 12, 13, predicting a speci c sin2 23 ' 0:51 (within the 3 range), a leptonic CPviolating Dirac phase doublebeta decay mee ' 41:3 meV. We turn then to an analysis of the dark matter candidates in the model, which are stabilized by an unbroken Z2 symmetry. We discuss the possibility of scalar dark matter, which can generate the observed abundance through the Higgs portal by the standard WIMP mechanism. An interesting possibility arises if the

Fermion
lightest heavy Majorana neutrino is the lightest Z2odd particle. The model can produce
a viable fermionic dark matter candidate, but only as a feebly interacting massive particle
(FIMP), with the smallness of the coupling to the visible sector protected by a symmetry
and directly related to the smallness of the light neutrino masses.
Masses and SM Parameters
ArXiv ePrint: 1712.02792
1 Introduction
(27) discrete group [6{26] has attracted a lot of attention
as a promising family symmetry for explaining the observed pattern of SM fermion masses
and mixing angles.
Another prominent issue in particle physics that motivates theories beyond the SM is
its lack of a viable Dark Matter (DM) candidate. In fact, there is compelling evidence for
the existence of DM, an unknown, nonbaryonic matter component whose abundance in
the Universe exceeds the amount of ordinary matter roughly by a factor of ve [27]. Still,
the nongravitational nature of DM remains a mystery [28{30]. Most prominent extensions
of the SM feature Weakly Interacting Massive Particles (WIMPs) as DM. WIMPs typically
have order one couplings to the SM and masses at the electroweak scale. The observation
that this theoretical setup gives the observed relic abundance is the celebrated WIMP
miracle [31]. In the standard WIMP paradigm, DM is a thermal relic produced by the
{ 1 {
freezeout mechanism. However, the observed DM abundance may have been generated
also out of equilibrium by the socalled freezein mechanism [32{37]. In this scenario, the
DM particle couples to the visible SM sector very weakly, so that it never enters chemical
equilibrium. Due to the small coupling strength, the DM particles produced via the
freezein mechanism have been called Feebly Interacting Massive Particles (FIMPs) [35]; see
ref. [37] for a recent review.
The solutions to the DM and the avour problems have indeed often been approached
separately in the literature. Nevertheless one could entertain the idea that they have a
common origin, whether because some residual avour symmetry stabilizes it [38{52], or
where there is a dark sector which communicates to the visible sector only through family
symmetry mediators [53, 54].
With respect to the avour problem, a viable form of the Yukawa structure for quarks
is the mixing inspired texture where the Cabibbo angle originates from the downquark
sector and the remaining (smaller) mixing angles come from the more hierarchical up quark
mixing [55]. We build a model based on the nonAbelian group
(27) which achieves a
generalisation of this mixing inspired texture for the quarks, and is therefore
phenomenologically viable. The model leads to a structure for the charged leptons which is diagonal
apart from an entry mixing the rst and third generations. The e ective neutrino mass
matrix arises through radiative seesaw and is in this case a very simple structure, a sum
of a democratic structure (all entries equal) plus a contribution only on the rst diagonal
entry. This predictive scenario for the leptons leads to a good t to all masses and mixing
angles with a correlation between 13 and 23, which depend only on the parameters of the
charged lepton sector. In addition to the
(27), we need to employ ZN symmetries that
constrain the allowed terms, and within these, a single Z2 symmetry remains unbroken
and stabilizes a DM, which can be either the lightest of the righthanded neutrinos (which
are the only Z2odd fermions) or a Z2odd scalar. The model can lead to the correct relic
abundance either under the WIMP or the FIMP scenarios.
2
Z6
The model
The model we propose is an extension of the SM that incorporates the
(27)
Z2
Z5
Z10
Z16 discrete symmetry and a particle content extended with the SM singlets:
scalars , 1
, 2
, ,
, , ' and two right handed Majorana neutrinos N1; 2R. All the
nonSM
elds are charged under the above mentioned discrete symmetry. All the discrete
groups are spontaneously broken, except for the Z2 under which only ' and N1; 2R are odd.
In this setup the light active neutrino masses arise at oneloop level through a radiative
seesaw mechanism, involving two right handed Majorana neutrinos and the Z2 odd scalars
that do not acquire VEVs.
Our model reproduces a predictive mixing inspired textures where the Cabibbo mixing
arises from the downtype quark sector whereas the remaining mixing angles receive
contributions from both up and down type quark sectors. These textures describe the charged
fermion masses and quark mixing pattern in terms of di erent powers of the Wolfenstein
parameter
= 0:225 and order one parameters. The full symmetry G of the model exhibits
{ 2 {
the following spontaneous breaking:
G = SU(3)C
SU (2)L
U (1)Y
(27)
Z2
Z5
Z6
Z10
Z16
+
+ v
SU(3)C
SU (2)L
U (1)Y
Z2
SU(3)C
U (1)Q
Z2 ;
(2.1)
we assume to be much larger than the electroweak symmetry breaking scale v = 246 GeV.
The assignments of the scalars and the fermions under the
(27)
Z2
Z5
Z10
Z16 discrete group are listed in tables 1 and 2, where the dimensions of the
irreducible representations are speci ed by numbers in boldface and di erent charges are
written in the additive notation. It is worth mentioning that all the scalar elds of the
model acquire nonvanishing VEVs, except for the SM singlet scalar eld ', which is the
only scalar charged under the preserved Z2 symmetry.
Yukawa terms arise: With the above particle content, the following quark, charged lepton and neutrino
6
5
4
2
Z6
(27)
(2.2)
(2.3)
(2.4)
(2.5)
8
7
6
4
5
3
LY
(U) = y1(U1)q1L eu1R 8 + y1(U2)q1L eu2R 6 + y1(U3)q1L eu3R 4
+ y2(U1)q2L eu1R 6 + y2(U2)q2L eu2R 4 + y2(U3)q2L eu3R 2
+ y3(U1)q3L eu1R 4 + y3(U2)q3L eu2R 2 + y3(U3)q3L eu3R + h:c;
LY
(D) = y1(D1)q1L d1R 7 + y1(D2)q1L d2R 2
6 + y1(D3)q1L d3R 7
+ y2(D1)q2L d1R 5 + y2(D2)q2L d2R
4 51 + y2(D3)q2L d3R 5
5
+ y3(D1)q3L d1R 3 + +y3(D2)q3L d2R
+ y3(D3)q3L d3R 3 + h:c;
4
7
2
3
L(Yl) = y3(l3) lL
+ y2(l2) lL
LY
( ) = y1( ) lL e
2
10;1 l3R 3 + y1(l3) lL
10;2 l2R
10;0
3
5 + y1(l1) lL
'
N1R 2 + y2( ) lL e
+ mN1R N 1RN1CR + mN2R N 2RN2CR + h:c;
constrain through a t to the observed fermion masses and mixings parameters.
In addition to these terms, the symmetries unavoidably allow terms in L(Yl) where the
contraction lL
is replaced with lL
y . For example, in addition to lL
l3R 3
2
(27)
10;0
10;0
10;0
10;1
q3L
10;0
u1R
10;0
u2R
10;0
u3R
10;0
d1R
10;0
d2R
10;0
d3R lL
10;0
Z2
Z5
Z6
Z10
Z16
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
2
0
0
0
0
1
10;0
0
0
0
5
0
Z2
0
0
0
0
0
0
0
0
0
3
10;0
0
0
0
0
8
'
10;0
1
0
0
0
0
l2R
10;0
0
1
0
0
3
the SM SU(2) Higgs doublet. The ZN charges, q, shown in the additive notation so that the group
element is ! = e2 i q=N . For the (27) representations and the notations see appendix A.
the following term is allowed: lL
y
l3R 5
2
. These terms have two additional
suph i
pressions of h i
=
and can be safely neglected if there is a mild hierarchy between h i and
. This hierarchy in the VEVs is consistent is also consistent with the mild hierarchy
obtained for the masses of the light e ective neutrinos after seesaw.
As indicated by the current low energy quark
avour data encoded in the Standard
parametrization of the quark mixing matrix, the complex phase responsible for CP violation
in the quark sector is associated with the quark mixing angle in the 13 plane. Consequently,
in order to reproduce the experimental values of quark mixing angles and CP violating
phase, the Yukawa coupling in eq. (2.2) y1(U3) is required to be complex.
An explanation of the role of each discrete group factor of our model is provided in
the following. The
(27), Z5, Z6 and Z10 discrete groups are crucial for reducing the
number of model parameters, thus increasing the predictivity of our model and giving rise
to predictive and viable textures for the fermion sector, consistent with the observed pattern
of fermion masses and mixings, as will be shown later in sections 3 and 4. The
(27),
Z5, Z6 and Z10 discrete groups, which are spontaneously broken, determine the allowed
entries of the quark mass matrices as well as their hierarchical structure in terms of di erent
powers of the Wolfenstein parameter, thus giving rise to the observed SM fermion mass
and mixing pattern. In particular the Z5 discrete symmetry is crucial for explaining the
tau and muon charged lepton masses as well as the Cabbibo sized value for the reactor
mixing angle, which only arises from the charged lepton sector. The Z6 discrete group
allows us to get a predictive texture for the light active neutrino sector. This symmetry
forbids mixings between the two right handed Majorana neutrinos N1R and N2R. The Z10
{ 4 {
discrete symmetry allows to get the right hierarchical in the second column of the down
type quark mass matrix crucial to successfully reproduce the right values of the strange
quark mass and the Cabbibo angle with O(1) parameters.
As a result of the (27)
Z2
Z5
Z6
Z10
Z16 charge assignment for scalars and
quarks given in tables 1 and 2, the Cabibbo mixing will arise from the down type quark
sector, whereas the remaining mixing angles will receive contributions for both up and
down type sectors. The preserved Z2 symmetry allows the implementation of the one loop
level radiative seesaw mechanism for the generation of the light active neutrino masses as
well as provides a viable DM particle candidate.
We assume the following VEV pattern for the
(27) triplet SM singlet scalars
which is consistent with the scalar potential minimization equations for a large region of
parameter space as shown in detail in ref. [56].
Besides that, as the hierarchy among charged fermion masses and quark mixing angles
emerges from the breaking of the
(27)
Z2
Z5
Z6
Z10
Z16 discrete group, we
set the VEVs of the SM singlet scalar
elds with respect to the Wolfenstein parameter
= 0:225 and the model cuto
, as follows:
v
v 1
v 2
v
v
; v
3=2 :
We require a mild hierarchy between the VEVs of the two
(27) triplet scalars
and
(merely a factor of two), which is su cient to suppress the e ect of unavoidable terms in
the charged lepton sector, which could otherwise spoil the phenomenology of the model
discussed in section 4. The model cuto scale
can be thought of as the scale of the UV
completion of the model, e.g. the masses of FroggattNielsen messenger elds. It is
straightforward to show that the assumption regarding the VEV size of the SM singlet scalars given
by eq. (2.7) is consistent with the scalar potential minimization. That assumption given
by that equation can be justi ed by considering
quartic scalar couplings of the same order of magnitude.
2 <
2
2
1
2
2
2
2 and the
(2.6)
(2.7)
3
Quark masses and mixings
From the quark Yukawa terms of eqs. (2.2) and (2.3), inserting the VEV magnitudes of
the scalars with respect to
we rewrite it in term of e ective parameters
LY
(Q) =
+ a(3D1)q3Ld1R 3 + a(3D2)q3Ld2R 3 + a(3D3)q3Ld3R 3
a(1D1)q1Ld1R 7 + a(1D2)q1Ld2R 6 + a(1U3 )q1Ld3R 7
+ h:c:
(3.1)
{ 5 {
Then it follows that the quark mass matrices take the form:
0 a(1U1 ) 8 a(1U2 ) 6 a(1U3 ) 4 1
MU = BBB a(2U1 ) 6 a(U) 4 a(2U3 ) 2 CCC pv ;
2 2
0 a(1D1) 7 a(1D2) 6 a(1D3) 7 1
MD = BBB a(2D1) 5 a(2D2) 5 a(2D3) 5 CCC pv ; (3.2)
2
where ai(jU) and ai(jD) (i; j = 1; 2; 3) are O(1) parameters. Here
= 0:225 is the Wolfenstein
parameter and v = 246 GeV the scale of electroweak symmetry breaking. The SM quark
mass textures given above indicate that the Cabibbo mixing emerges from the down type
quark sector, whereas the remaining mixing angles receive contributions from both up and
down type quark sectors. Indeed, this texture is a generalisation of the particular case
referred to as the mixing inspired texture [55], in which the two small quark mixing angles
would arise solely from the up type quark sector. Besides that, the low energy quark
avour data indicates that the CP violating phase in the quark sector is associated with
the quark mixing angle in the 13 plane, as follows from the Standard parametrization of
the quark mixing matrix. Consequently, in order to get quark mixing angles and a CP
violating phase consistent with the experimental data, we adopt a minimalistic scenario
where all the dimensionless parameters given in eq. (3.2) are real, except for a(1U3 ), taken to
be complex.
The obtained values for the physical quark mass spectrum [57, 58], mixing angles and
Jarlskog invariant [59] are consistent with their experimental data, as shown in table 3,
starting from the following benchmark point that would correspond to the limit of the
mixing inspired texture [55]:1
a(1U1 ) ' 1:266; a(2U2 ) ' 1:430; a(3U3 ) ' 0:989; a(1U3 ) '
0:510
1:262i; a(2U3 ) ' 0:806;
a(1D1) ' 0:550; a(2D2) ' 0:554; a(3D3) ' 1:411; a(1D2) ' 0:565:
(3.3)
In table 3 we show the model and experimental values for the physical observables of the
quark sector. We use the MZ scale experimental values of the quark masses given by ref. [57]
(which are similar to those in ref. [58]). The experimental values of the CKM parameters
are taken from ref. [60]. As indicated by table 3, the obtained quark masses, quark mixing
angles, and CP violating phase can be tted to the experimental low energy quark avour
data. We note that the values (3.3) of the parameters ai(U; D) are compatible with O(1).
This fact supports the desired feature of the model that the hierarchy of masses and mixing
angles are encoded in the powers of
and texture zero of the mass matrices eq. (3.2), which
in its turn is the consequence of the particular avour symmetry of the model.
4
Lepton masses and mixings
We can expand the contractions of the
(27) (anti)triplets lL,
and
according to
the scalar VEV directions in eq. (2.6). Then we have lL
10;0 / l1L, lL
10;2 / l2L,
1This limit corresponds to a(1U2 ) = a(2U1 ) = a(3U1 ) = a(3U2 ) = 0, a(1D3) = a(2D1) = a(2D3) = a(3D1) = a(3D2) = 0.
{ 6 {
mu [MeV]
mc [MeV]
mt [GeV]
md [MeV]
ms [MeV]
mb [GeV]
sin 1(q2)
sin 2(q3)
sin 1(q3)
speci ed in eq. (2.7), we rewrite eqs. (2.4) and (2.5) in the form
lL
10;1 / l3L, and lL
10;0 / l1L + l2L + l3L . Taking into account v
v
(v + H)
p
p
2
2
From eq. (4.1) we nd the charged lepton mass matrix
0 a(l) 9
1
0
0
a(l) 5
2
0
0
4
a(l) 4 1
0
a(l) 3
3
v
CA p2 ;
where a(kl) (k = 1;
charged lepton sector to the PMNS matrix, U (l) consists in a rotation by a single
non; 4) are O(1) dimensionless parameters. The contribution from the
vanishing angle 1(l3) which depends crucially on a(4l).
The e ective neutrino mass matrix M
arises after radiative seesaw, from the Yukawa
terms (which we expanded in eq. (4.2)) with scalar ' (which does not acquire a VEV)
and the masses of the righthanded neutrinos. The mechanism is associated with the loop
diagrams in gure 1. Considering these diagrams and the Dirac couplings in eq. (4.2) with
', which we represent in the matrix form Y' :
(4.2)
(4.3)
(4.4)
Y' = p
v
0 v y( ) v y( ) 1
1
0
0
2
2
2
v y( ) C ;
v y( )
A
{ 7 {
one reads o there will be a democratic contribution associated with the y2( ) coupling
lling each entry in M equally (due to the coupling to the combination l1L + l2L + l3L )
Re ϕ0, Im ϕ0
νiL
v× v×Ξ
v×
×v
×
N2R N2R
v×
×v
×
N1R N1R
Re ϕ0, Im ϕ0
ν1L
v×Φ ×v
of the mass mNnR .
whereas the y1( ) coupling is responsible for a contribution solely to the 11 entry of M .
Thus we write the e ective neutrino mass matrix in the form
where the dimensionful parameters A( ) and A( ) follow from the loop functions of the
diagrams in gure 1.
with k = 1, 2. We note that ' needs to be a complex scalar otherwise the loop functions
vanish, and further the real and imaginary parts of ' must not have degenerate masses.
The structure of M
is such that it has an eigenvector (0; 1; 1)=p2 with a vanishing
eigenvalue, corresponding therefore to a massless neutrino. This means the neutrino sector's
contribution to the PMNS matrix, U ( ), has one direction which is (0; 1; 1)=p2, meaning
1(3) = 0 and 2(3) =
=4. This gets modi ed by the contribution from the charged lepton
sector such that the reactor angle is nonzero, but given that the associated state is the
massless state this structure is viable for the inverted hierarchy of neutrino masses (but
not for the normal hierarchy). Indeed, we nd that for our model the normal hierarchy
scenario leads to a too large reactor mixing angle, thus being ruled out by the current data
on neutrino oscillation experiments.
{ 8 {
(4.9)
masses, the reactor mixing parameter sin2 13 6= 0 and the deviation sin2 23
A( )
1 , A( ) and arg hA( )i.
2 1
2
1
1
2
In turn, A( ) and A( ) are dimensionful parameters crucial to determine the neutrino mass
squared splittings as well as the solar angle sin2 12. For the sake of simplicity and proving
these leptonic structures are viable, we assume that the parameters al(l) (l = 1;
A( ) are real whereas A( ) is taken to be complex. We have checked numerically that
; 4),
the simplest scenario of all lepton parameters (al(l) (l = 1;
; 4), A( ) and A(2 )) being
1
real leads to a solar mixing parameter sin2 12 close to about 0:2, which is below its 3
experimental lower bound.
In order to reproduce the experimental values of the physical observables of the lepton
sector, i.e. the three charged lepton masses, two neutrino mass squared splittings and the
three leptonic mixing parameters, we proceed to t the parameters a(kl) (k = 1;
; 4),
For the case of inverted neutrino mass hierarchy we nd the following best t result
a(l)
1 ' 1:936;
A( )
1
' 69:7 meV;
a(l)
2 ' 1:025;
A( )
2
' 20:6 meV;
a(l)
3 ' 0:864;
a(l)
4 ' 0:813;
arg hA( )i
1
'
The small hierarchy between e ective parameters A( )
1 , A(2 ) is consistent with the mild
hierarchy between h i and h i.
As follows from eqs. (4.6){(4.8), the obtained numerical values given above for the
neutrino parameters A( )
1 , A(2 ) and arg hA( )i can be obtained from the following benchmark
1
point:
mN1 = 500 GeV;
= 2:41 105 TeV;
mN2 = 2 TeV;
jy1 j = 1:12;
mRe' = 900 GeV;
mIm' = 600 GeV;
y2 = 0:61;
arg [y1 ] '
37:4 : (4.11)
The benchmark point given above is one out of the many similar solutions that yields
physical observables for the neutrino sector consistent with the experimental data. We
have numerically checked that for a
xed mass splittings between the masses of the real
and imaginary components of ', the cuto
scale has a low sensitivity with the masses of
the scalar and fermionic seesaw mediators. In addition, we have checked that lowering the
mass splitting between Re' and Im' leads to a decrease of the cuto scale. In particular
lowering this mass splitting from 50% up to 0:1% of the mass of Im' leads to a decrease of
the cuto scale from
108 GeV up to
107 GeV. From table 4, it follows that the reactor
sin2 13 and solar sin2 12 leptonic mixing parameters are in excellent agreement with the
experimental data, whereas the atmospheric sin2 23 mixing parameter is deviated 3 away
from its best t value. Figure 2 shows the correlation between the solar mixing parameter
sin2 12 and the Jarlskog invariant for the case of inverted neutrino mass hierarchy. We
found a leptonic Dirac CP violating phase of 281:6 and a Jarlskog invariant close to about
3:3
10 2 for the inverted neutrino mass hierarchy.
{ 9 {
m221 [10 5eV2] (IH)
m213 [10 3eV2] (IH)
me [MeV]
m
[MeV]
2 range
0:487
102:8
1:75
7:20
2:41
182
0:289
0:404
0:556
0:0006
0:0006
7:95
2:57
347
0:359
0:456
0:625
3 range
0:487
102:8
1:75
7:05
2:37
0
142
0:273
0:388
0:0009
0:0009
8:14
2:61
31
360
0:379
0:638
0:0197
0:0230
0:0189
0:0239
splittings and leptonic mixing parameters for the inverted (IH) mass hierarchy. The model values
for CP violating phase are also shown. The experimental values of the charged lepton masses are
taken from ref. [57], whereas the range for experimental values of neutrino mass squared splittings
and leptonic mixing parameters, are taken from ref. [61].
HJEP05(218)3
0.35
0.34
0.33
0.31
2
1
θ
2
isn 0.32
0.30
0.0340
0.0335
0.0330
J
0.0325
0.0320
the case of inverted neutrino mass hierarchy. The horizontal lines are the minimum and maximum
values of the solar mixing parameter sin2 12 inside the 1 experimentally allowed range.
Let us consider the e ective Majorana neutrino mass parameter
mee =
X Ue2km k ;
j
mee ' 41:3 meV :
where Uej and m k are the PMNS leptonic mixing matrix elements and the neutrino
Majorana masses, respectively. The neutrinoless double beta (0
) decay amplitude is
proportional to mee. From eq. (4.5) it follows that in our model there is a massless neutrino.
It is well known that in this case, independently of the other parameters, one expects for
the inverted neutrino mass hierarchy 15 meV < mee < 50 meV. With the model best t
values in table 4 we nd
(4.12)
(4.13)
(5.1)
(5.2)
(5.3)
(5.4)
0
T1=2 (136Xe)
(N is nucleon),
reach [64].
5
Scalar potential
This is within the declared reach of the nextgeneration bolometric CUORE
experiment [62] or, more realistically, of the nexttonextgeneration tonscale 0
decay
experiments. The current most stringent experimental upper limit mee
160 meV is set by
1:1
1026 yr at 90% C.L. from the KamLANDZen experiment [63].
In theory, Lepton Flavour Violation processes are expected from this kind of model.
However, in realisations such as these the new scale
associated with family symmetry
breaking scale is very high. Thus, the rate of muon conversion processes such as
N ! eN
! eee,
! e is several orders of magnitude beyond experimental
In this section we consider the scalar potential. As can be seen in table 1, the scalar content
of the model has many degrees of freedom. We assume that all scalars except for
and
' get their VEVs at the family symmetry breaking scale, which should be near the cuto
scale , much greater than the electroweak breaking scale de ned by the VEV of h i
(we can check the selfconsistency of this assumption in the benchmark point in eq. (4.11)).
v
Due to this, the family symmetry breaking scalars decouple, such that we have at the TeV
scale the e ective potential V ( ; '). We divide it into separate parts for convenience, and
use without loss of generality the mass eigenstates Re ', Im ' instead of ', ' :
V ( ; ') = V ( ) + V ( ; ') + V (')
where
V ( ) =
2
y
+
is simply the SM potential (one Higgs doublet) and
V ( ; ') = 1
y
Re '2 + 2
Im '2 + h:c;
has only quartic interactions between the doublet
and the Z2odd scalar '. The term
V (') =
m12 Re '2
m22 Im '2 + 1 (Re ')4 + 2 (Im ')4 + 3 Re '2 Im '2 +h:c;
+ h:c;
has the masses and quartic interactions that involve only the Z2 odd scalar. Given this,
the masses of the real and imaginary parts of ' will not be degenerate. As the symmetry is
enhanced in the limit of degeneracy (a U(1) symmetry instead of the preserved Z2), if the
splitting between their masses is small it remains small, and a small splitting is technically
natural in that sense as it is protected by an approximate symmetry.
6
Dark matter constraints
In this section we consider the possibilities o ered by the model to provide a viable DM
candidate. The Z2 symmetry, under which only the scalar eld ' and the fermions N1R
and N2R are charged, remains unbroken and stabilizes the lightest Z2odd mass eigenstate.
Scalar dark matter scenario
The rst scenario considered is the one where one component of the scalar eld ' is
the lightest Z2odd particle. In this case, DM is produced in the early Universe via the
vanilla WIMP paradigm. If Im ' is the lightest Z2 odd state, it can annihilate into
a pair of SM particles via the schannel exchange of a Higgs boson. Additionally, the
annihilation into Higgs bosons also occurs via the contact interaction and the mediation
by an Im ' in the t and u channels. Finally, DM could also annihilate into a pair
SM neutrino/antineutrino via the t and uchannel exchange of a N1. However the latter
channel is typically very suppressed by the tiny e ective neutrino Yukawa coupling y1
Hence, the DM relic abundance is mainly governed by the DM mass mIm ' and the quartic
coupling 2, between two DM particles and two Higgs bosons. The freezeout of heavy DM
particles (mIm ' > mh) is largely dominated by the annihilations into Higgs bosons,2 with
a thermallyaveraged crosssection given by:
gure 3 it is shown the parameter space ( 2; mIm ') giving rise to the observed DM
relic abundance.
The black thick line corresponds to the full computation using
micrOMEGAs [65{68], whereas the red line to the analytical case given by eq. (6.1). The
vertical dashed blue line corresponds to mIm ' = mh. The direct detection constraints are
obtained by comparing the spinindependent cross section for the scattering of the DM o
of a nucleon,
SI =
22 m4N f 2
8 m4h mI2m '
;
(6.1)
(6.2)
to the latest limits on SI provided by PandaXII [69]. Here mN is the nucleon mass and
f ' 1=3 corresponds to the form factor [
70, 71
]. Again, the analytical result is in good
agreement with the numerical computation by micrOMEGAs. Figure 3 also presents the
2For mIm ' = 200 GeV, annihilations into Higgses correspond to
80% and into tt to
20%. When
mIm ' = 10 TeV, the annihilation into a pair of Higgses constitutes almost 100%.
100 h
10− 2
102
P
a
n
d
a
X
I
I
103
mIm ϕ [GeV]
104
(6.3)
(6.4)
abundance via the WIMP mechanism, using the full annihilation crosssection (thick black line)
and the only the annihilation into Higgs bosons (thin red line). The light blue region is in tension
with the latest PandaXII results.
DM spinindependent direct detection exclusion region, that sets strong tension for the
model if the DM is lighter than
Fermionic dark matter scenario
The second case corresponds to the scenario where N1R is the lightest Z2odd particle.
DM can annihilate into a pair of SM neutrinos via the tchannel exchange of the real
and the imaginary parts of '. This comes from an e ective neutrino Yukawa coupling
y1
jy1 j
v produced by eq. (2.5) or its expanded version, eq. (4.2):
L
y( ) l1L N1R ' h ei :
1
The DM relic abundance is then governed by the DM mass mN1 , the mediator masses
mRe ' and mIm ', and the e ective Yukawa coupling y1 . The thermallyaveraged
annihilation crosssection is given by:
h vi ' 32
9 y14
m2N1
2m2N1 + m2Re ' + mI2m '
2
m2N1 + m2Re '
m2N1 + mI2m '
2
2
:
Figure 4 shows the required e ective coupling y1 in order to reproduce the observed DM
relic abundance via the standard thermal WIMP paradigm, and assuming mRe ' = mIm '.
As expected for WIMP DM, the e ective coupling has to be of the order of O(1), if DM
is heavier than
100 GeV. For the DM production this is perfectly viable, however we
also want to generate the neutrino masses. In what follows we proceed to scan for the
required to reproduce the values of the neutrino parameters A( )
CP odd scalar mass mIm' and e ective neutrino Yukawa coupling y1 = jy1 j
1 , A( ) and arg hA( )i
2 1
v needed
3Furthermore, one has to take into account astrophysical uncertainties [72{81] when interpreting the
results of the DM searches.
observed relic abundance via the WIMP mechanism, assuming m'
mRe '
mIm '.
2 1χ = 0.2
102
observed relic abundance via the FIMP mechanism, assuming m'
mRe '
mIm '.
shown in eq. (4.10). Fixing the right handed Majorana neutrino masses to typical values
mN1
500 GeV, mN2
2 TeV, mIm'
the values of the neutrino parameters A( )
1 TeV and
108 GeV, required to reproduce
1 , A( ) and arg hA( )i, the e ective neutrino
2 1
Yukawa coupling y1 has to be of the order of 10 7 to 10 4. Values in this ballpark are
too small to reproduce the observed DM relic abundance via the WIMP mechanism, which
requires O(1) e ective Yukawa coupling y1
as indicated by
gure 4. Consequently the
fermionic DM scenario of our model can not be produced via the usual WIMP paradigm.
Alternatively, very suppressed couplings between the visible and the dark sectors are
characteristic in nonthermal scenarios where the DM relic abundance is created in the
early Universe via freezein [32{37]. Figure 5 shows the e ective couplings required in
order to produce FIMP DM. As expected for this kind of scenarios, y1 is in the range
10 8 to
10 11. The light blue region is disregarded because N1 is not the lightest
particle of the dark sector.
Finally, to close this section, we discuss the splitting between the masses of the real and
imaginary parts of '. To start, we note that a small scalar mass splitting of 10 3 times
the mass of the imaginary part of ' (which is required in order to have fermionic DM
through the FIMP mechanism) may look unnatural, but it is actually technically natural
in the sense that it is protected by a symmetry: in the limit where the RH neutrino masses
and the splitting of the ' masses vanish, the symmetry of the Lagrangian is enlarged from
the Z2 to a U(1) symmetry. The nontrivial U(1) charges of the RH neutrinos and of '
under this U(1) would forbid Majorana terms for the RH neutrinos and force the masses
of the real and imaginary parts of ' to be the same. Considering this, if the U(1) is broken
only by the Majorana terms (but not in the scalar potential), the splitting of the masses
is no longer protected by the symmetry and is generated, but only radiatively. In such a
scenario, the splitting would be naturally small.
Although we do not consider this scenario in great detail, we propose also some more
explicit mechanisms that can explain the splitting between the masses of the real and
imaginary parts of ' when starting from the symmetry limit where the splitting vanishes.
The
rst possibility consists in extending our model by adding an extra spontaneously
broken Z3 discrete symmetry under which ' is assumed to have a charge +1 (in additive
notation). In addition, an extra SM scalar singlet, i.e. , with Z3 charge +1 has to be
added. The remaining scalar and fermions are neutral under Z3. Consequently no new
contributions to the quarks, charged leptons and neutrino Yukawa terms originate from the
extra eld
and the Z3 discrete symmetry. The splitting between the masses of Re ' and
Im ' will arise from the trilinear scalar interaction A'2
which preserves both this added
Z3 and the existing Z2. The invariance of the neutrino Yukawa interactions under the Z3
discrete symmetry requires that the right handed Majorana neutrinos N1R and N2R should
have a Z3 charge equal to +1, such that their masses will need to arise from the Yukawa
interactions N 1RN1CR
and N 2RN2CR
after the spontaneous breaking of the Z3 discrete
group. This is an explicit realization of the mechanism described above, showing there is a
relation between the ' mass splitting and the NiR masses. If this Z3 is broken at the TeV
scale the right handed Majorana neutrinos are within the LHC reach and there is a viable
fermionic DM candidate through the FIMP mechanism.
A di erent mechanism to generate the splitting by replacing the SM scalar singlet
' with an inert SU(2) scalar doublet charged under the preserved Z2 symmetry. That
scenario was proposed for the rst time in ref. [38]. In that scenario, the splitting between
the masses of Re ' and Im ' (in that scenario ' is a SU(2) scalar doublet) will arise form
the quartic scalar interaction
y ' 2, as explained in detail in ref. [38]. In this case, the
coupling between righthanded neutrinos and ' does not include the Higgs .
7
Conclusions
We have built a viable family symmetry model based on the
discrete group, which leads to a mixing inspired texture for the quarks and to similarly
predictive structures for the leptons. For the quarks, the down sector parameters control
the Cabibbo angle, and the up and down sector parameters control the remaining angles.
For the leptons, the e ective neutrino parameters that arise after radiative seesaw control
the solar angle, and the charged lepton parameters control the reactor angle, which is also
correlated to the deviation of the atmospheric angle from its maximal value. The model
is only viable for inverted hierarchy and after tting to the best t values of the solar and
reactor angle, predicts sin2 23 ' 0:51,
' 281:6 and mee = 41:3 meV.
Additionally, the model has viable DM candidates, stabilized by an unbroken Z2
symmetry, which we analyze quantitatively. A simple possibility is that there is scalar WIMP
DM, which is produced through the Higgs portal. An alternative scenario is when we
consider fermionic DM, which in our model would be the lightest righthanded neutrino.
In order for it to be a WIMP and to obtain the right abundance, its e ective coupling
to the visible sector is too large to be consistent with what is required by the e ective
neutrino masses. Instead, if our fermionic DM candidate is a FIMP, the e ective coupling
needs to be quite small. This is consistent with obtaining the required neutrino masses but
requires a very small splitting of the real and imaginary components of the Z2odd scalar
(the splitting divided by the mass scale would be at the per mille level). The smallness of
the splitting is technically natural as when the splitting goes to zero, the symmetry of the
theory is enhanced.
This model addresses the avour problem while providing a viable DM candidate
(scalar or fermionic), and is a novel example of the interplay of constraints coming from
the observed DM abundance to a family symmetry model, namely by relating the DM
abundance to the light neutrino masses.
Acknowledgments
IdMV acknowledges funding from Fundac~ao para a Ci^encia e a Tecnologia (FCT) through
the contract IF/00816/2015, partial support by Fundac~ao para a Ci^encia e a Tecnologia
(FCT, Portugal) through the project CFTPFCT Unit 777 (UID/FIS/00777/2013) which
is partially funded through POCTI (FEDER), COMPETE, QREN and EU, and partial
support by the National Science Center, Poland, through the HARMONIA project under
contract UMO2015/18/M/ST2/00518. IdMV thanks Universidad Tecnica Federico Santa
Mar a for hospitality, where this work was
nished. The visit of IdMV to Universidad
Tecnica Federico Santa Mar a was supported by Chilean grant Fondecyt No. 1170803.
NB is partially supported by the Spanish MINECO under Grants FPA201454459P and
FPA201784543P. This project has received funding from the European Union's Horizon
2020 research and innovation programme under the Marie SklodowskaCurie grant
agreements 674896 and 690575; and from Universidad Antonio Narin~o grant 2017239. AECH
and SK were supported by Chilean grants Fondecyt No. 1170803, No. 1150792 and
CONICYT PIA/Basal FB0821, ACT1406 and the UTFSM internal grant PI M 17 5. A.E.C.H
is very grateful to the Instituto Superior Tecnico for hospitality.
1C1
1C1(1)
1C1(2)
3C1(0;1)
3C1(0;2)
C(1;p)
3
C(2;p)
3
1
1
3
3
3
3
1(r;s)
1
1
1
!s
!2s
!r+sp
!2r+sp
3
3!2
3!
0
0
0
0
3
3!
3!2
0
0
0
0
(27) discrete group is a subgroup of SU(3), has 27 elements divided into 11 conjugacy
classes. Then the
(27) discrete group contains the following 11 irreducible representations:
two triplets, i.e. 3[0][1] (which we denote by 3) and its conjugate 3[0][2] (which we denote
by 3) and 9 singlets, i.e. 1k;l (k; l = 0; 1; 2), where k and l correspond to the Z3 and Z03
charges, respectively [1]. The
(27) discrete group, which is a simple group of the type
(3n2) with n = 3, is isomorphic to the semidirect product group (Z03
is worth mentioning that the simplest group of the type
(3n2) is
(3)
Z030) o Z3 [1]. It
Z3. The next
group is
(12), which is isomorphic to A4. Consequently the
(27) discrete group is the
simplest nontrivial group of the type
(3n2). Any element of the
can be expressed as bkama0n, being b, a and a0 the generators of the Z3, Z03 and Z030 cyclic
groups, respectively. These generators ful ll the relations:
a3 = a03 = b3 = 1;
aa0 = a0a;
bab 1 = a 1a0 1;
ba0b 1 = a:
(A.1)
The characters of the
(27) discrete group are shown in table 5. Here n is the number
of elements, h is the order of each element, and ! = e 3 =
unity, which satis es the relations 1 + ! + !2 = 0 and !3 = 1. The conjugacy classes of
2 i
p
12 + i 23 is the cube root of
(27) are given by:
C1 :
C1(1) :
C1(2) :
3
3
C(0;1) :
C(0;2) :
feg;
fa; a02g;
fa2; a0g;
fa02a02g;
fa02; a2; aa0g;
C(1;p) : fbap; bap 1a0p 2a02g; h = 3;
3
C(2;p) : fbap; bap 1a0p 2a02g; h = 3:
3
h = 1;
h = 3;
h = 3;
h = 3;
h = 3;
The multiplication rules between
(27) singlets and
(27) triplets are given by [1]:
The tensor products of
(27) singlets 1k;` and 1k0;`0 take the form [1]:
0
x(1; 1)
1
( 1;0)
0
x(2; 2)
1
( 2;0)
x
x
(z)1k;l =
(z)1k;l =
B !r x(0;2)z C
0
B
0
A
1
A
:
;
1k;`
1k0;`0 = 1k+k0 mod 3;`+`0 mod 3:
(A.2)
(A.3)
(A.5)
(A.6)
(A.7)
B x0;1 C
1
A
x
1;0
0
x2; 2
B x0;2 C
1
A
x
2;0
0
x1; 1
B x0;1 C
1
A
x
1;0
B y0;1 C
A
1
y
1;0
0
y2; 2
B y0;2 C
A
1
y
2;0
0
y
1;1
1
y1;0
= B
1;0y
1;0
x
1
0
Bx
1;0y1; 1
x1; 1y0;1
1;0
A
1
C
A
0
= B
2;0y
2;0
0
1
Bx
2;0y0;2
2;0y2; 2 +x2; 2y
x2; 2y0;2 +x0;2y2; 2
1
2;0CA
1
Bx
2;0
x
2;0y0;2
x2; 2y
x0;2y2; 2
2;0CA
1
0
1
Bx
1;0 +x
1;0y0;1
1;0y1; 1 +x1; 1y
x1; 1y0;1 +x0;1y1; 1
1
1;0CA
x
1;0y0;1
x1; 1y
x0;1y1; 1
1
1;0CA
;
;
=
X(x1; 1y
1;1 +!2r x0;1y0; 1 +!
r
x
1;0y1;0)1(r;0)
r
r
r
X(x1; 1y0; 1 +!2r x0;1y1;0 +!
X(x1; 1y1;0 +!2r x0;1y
1;1 +!
r
r
x
x
1;0y
1;1)1(r;1)
1;0y0; 1)1(r;2)
:
(A.4)
The tensor products between
(27) triplets are described by the following relations [1]:
From the equation given above, we obtain explicitly the singlet multiplication rules of the
(27) group, which are given in table 6.
Singlets
101
102
110
111
112
120
121
122
Attribution License (CCBY 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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