Table Of ContentKANAZAWA-06-13
D Family Symmetry and Cold Dark Matter at LHC
6
Yuji Kajiyama
National Institute of Chemical Physics and Biophysics,
Ravala 10, Tallinn 10143, Estonia
Jisuke Kubo and Hiroshi Okada
Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan
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Abstract
1
We consider a non-supersymmetric extension of the standard model with a family symmetry
2
v based on D Z Z , where one of Z ’s is exactly conserved. This Z forbids the tree-level
6 2 2 2 2
2 × ×
7 neutrino masses and simultaneously ensures the stability of cold dark matter candidates. From
0
the assumption that cold dark matter is fermionic we can single out the D singlet right-handed
0 6
1
neutrino as the best cold dark mater candidate. We find that an inert charged Higgs with a mass
6
0 between 300 and 750 GeV decays mostly into an electron (or a positron) with a large missing
/
h energy, where the missing energy is carried away by the cold dark matter candidate. This will be
p
- a clean signal at LHC.
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e
h
PACS numbers: 14.60.Pq,95.35.+d,11.30.Hv,12.60.Fr, 12.15.Ff
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I. INTRODUCTION
It is now clear that the standard model (SM) has to be extended at least in two ways:
Neutrino masses [1] and cold dark matter (CDM) [2] have to be accommodated. Since
neutrinos are a part of dark matter [2], the nature of cold dark matter and neutrinos may
be somehow related. A natural possibility to connect apparently unrelated physics is a
symmetry. By introducing an unbroken discrete symmetry, for instance, we can make a
weakly interacting neutral particle stable so that it can become a CDM candidate, while
making this discrete symmetry responsible for the smallness of neutrino masses. Proposals
along this line of thought have been suggested in refs. [3, 4, 5, 6, 7]. The basic idea of refs.
[3, 4, 5] is to introduce an unbroken Z symmetry to forbid tree-level neutrino masses and to
2
assign an odd Z parity to CDM candidates. For this mechanism to work, one introduces an
2
additional SU(2) doublet Higgs, which does not acquire VEV [8], along with right-handed
L
neutrinos. We will adopt this idea in this paper. However, the introduction of an additional
Higgs doublet and the right-handed neutrinos into the SM introduces additional ambiguities
intheYukawasector. Becauseoftheseambiguities, itwouldbedifficult tomakequantitative
tests of this idea, except may be the existence of the additional Higgs particles which could
be found at LHC.
A natural guidance to constrain the Yukawa sector is a flavor symmetry 1. In this paper
we would like to consider a nonabelian discrete symmetry D , which is one of the dihedral
6
groupsD . ThesmallestdihedralgroupisD whichisisomorphictothesmallestnonabelian
N 3
group S . D has been used as a flavor symmetry in refs. [12, 13, 14], while D and D have
3 4 5 7
been considered in refs. [15] and [16], respectively. We will consider a non-supersymmetric
ˆ
extension of the SM, which possesses a flavor symmetry based on D Z Z , where Z is
6 2 2 2
× ×
exactly conserved and D Zˆ is spontaneously broken by the VEV of the SU(2) doublet
6 2 L
×
Higgs fields. The unbroken Z ensures the stability of the CDM candidate and at the same
2
time forbids the tree-level neutrino masses, while Zˆ is responsible for the suppression of
2
FCNCs in the quark sector. The D assignment is so chosen that the leptonic sector of
6
the model is made predictive as possible without having contradictions with experimental
observations in this sector: There are eight independent parameters to describe six lepton
masses, andthreemixing anglesandthreeCPviolatingphases oftheneutrino mixing matrix
V . This will be discussed in Sect. III, where we will calculate the radiative neutrino mass
MNS
matrix [17].
The µ eγ amplitude in our model does not vanish [18]. We will investigate it in Sect.
→
IV, making it possible to single out the best CDM candidate, once we assume that CDM is
fermionic. The fermionic CDM candidate is the D singlet right-handed neutrino. Its mass
6
and the masses of the additional Higgs fields are constrained by µ eγ and the observed
→
dark matter relic density. In Sect. V we will plot these masses from various aspects. It will
1
Recent flavor models are reviewed, for example, in [9, 10] and [11]
2
turn out, among other things, that the D singlet inert charged Higgs with a mass between
6
300 and 750 GeV decays mostly into an electron (or a positron) with a large missing energy,
where the missing energy is carried away by the CDM candidate. They are within the
accessible range of LHC [19]. Sect. V is devoted to summarizing our findings.
II. THE MODEL
A. D group theory
6
The dihedral groups, D (N = 3,4,...), are nonabelian finite subgroups ofSO(3), where
N
all the irreps. of D are real, and there exist only two- and one-dimensional irreps [20, 23]
N
2. The irreps of D are 2, 2′, 1, 1′, 1′′, 1′′′, and the group multiplication rules are given as
6
follows [20]:
1′ 1′ = 1′′ 1′′ = 1′′′ 1′′′ = 1, 1′′ 1′′′ = 1′,
× × × ×
1′ 1′′′ = 1′′, 1′ 1′′ = 1′′′ . (1)
× ×
The Clebsch-Gordan coefficients for multiplying the irreps are [23]
2 2 = 1′ + 1 + 2′
×
x y x y x y
1 1 1 1 2 2
= (x y x y ) (x y +x y ) − ,
1 2 2 1 1 1 2 2
x × y − x y +x y
2 ! 2 ! 1 2 2 1 !
2′ 2′ = 1 + 1′ + 2′
×
a b a b +a b
1 1 1 1 2 2
= (a b +a b ) (a b a b ) − ,
1 1 2 2 1 2 2 1
a × b − a b +a b
2 ! 2 ! 1 2 2 1 !
2 2′ = 1′′′ + 1′′ + 2
×
x a x a +x a
1 1 1 1 2 2
= (x a +x a ) (x a x a ) .
1 2 2 1 1 1 2 2
x × a − x a x a
2 ! 2 ! 1 2 2 1 !
−
In what follows we will use these multiplication rules to construct a flavor model.
B. The Lepton Yukawa interaction
The Yukawa sector of the SM contains a large number of independent parameters. We
would like to reduce this number to as few as possible by a symmetry argument and then to
2 The”coveringgroup”ofDN isQ2N [21],whichcontainscomplexirreps. Q4 andQ6 havebeenconsidered
in refs. [22] and [20, 23], respectively.
3
understand the flavor structure in terms of the symmetry. In the following discussions we
will concentrate on the leptonic sector of a non-supersymmetric model.
As we will see below, it is possible to construct a model with a family symmetry based on
D Zˆ Z , in which (1) the neutrino mass matrix contains only four real parameters,
6 2 2 ν
× × M
(2) the maximal mixing of atmospheric neutrinos follows from the family symmetry, and (3)
the absolute value of the e 3 element of the neutrino mixing matrix V can be expressed
MNS
−
in terms of the charge lepton masses. The leptons L,ec,n and SU(2) Higgs doublets φ,η
L
belong to irreducible representations of D , and we give the D Zˆ Z assignment in
6 6 2 2
× ×
Table I and II. Zˆ Z is an abelian factor which we impose on the model, where Z shall
2 2 2
×
remain unbroken after the spontaneous symmetry breaking of SU(2) U(1) . We use the
L Y
×
two component notation for the Weyl spinors in an obvious notation, where ec’s are the
charge-conjugate states of the right-handed electron family. Under Z (which plays the
2
L n ec L n ec
S S S I I I
SU(2) U(1) (2, 1/2) (1,0) (1,1) (2, 1/2) (1,0) (1,1)
L Y
× − −
D 1 1′′′ 1 2′ 2′ 2′
6
Zˆ + + + +
2
− −
Z + + + +
2
− −
TABLE I: The D Zˆ Z assignment for the leptons. The subscript S indicates a D singlet,
6 2 2 6
× ×
and the subscript I running from 1 to 2 stands for a D doublet. L’s denote the SU(2) -doublet
6 L
leptons, while ec and n are the SU(2) -singlet leptons.
L
φ φ η η
S I S I
SU(2) U(1) (2, 1/2) (2, 1/2) (2, 1/2) (2, 1/2)
L Y
× − − − −
D 1 2′ 1′′′ 2′
6
Zˆ + + +
2
−
Z + +
2
− −
TABLE II: The D Zˆ Z assignment for the SU(2) Higgs doublets.
6 2 2 L
× ×
rolle of R parity in the MSSM), only the right-handed neutrinos n ,n and the extra Higgs
S I
η ,η are odd. The quarks are assumed to belong to 1 of D with (+,+) of Zˆ Z so that
S I 6 2 2
×
the quark sector is basically the same as the SM, where the D singlet Higgs φ with (+,+)
6 S
ˆ
of Z Z plays the rolle of the SM Higgs in this sector. No other Higgs can couple to
2 2
×
the quark sector at the tree-level. In this way we can avoid tree-level FCNCs in the quark
ˆ
sector. So, Z is introduced to forbid tree-level couplings of the D singlet Higgs φ with the
2 6 S
leptons and simultaneously to forbid tree-level couplings of φ ,η and η with the quarks.
I I S
4
The most general renormalizable D Zˆ Z invariant Yukawa interactions in the
6 2 2
× ×
leptonic sector can be described by
1 1
= Yed(L iσ φ )ec +Yνd(η†L )n M n n M n n +h.c.(2)
LY ab a 2 d b ab d a b − 2 1 I I − 2 S S S
a,b,Xd=1,2,S h i IX=1,2
The Yukawa matrices Y’s are given by
y 0 y 0 y 0
2 5 2
−
Ye1 = 0 y 0 , Ye2 = y 0 y , YeS = 0, (3)
2 2 5
y 0 0 0 y 0
4 4
h 0 0 0 h 0
2 2
−
Yν1 = 0 h 0 , Yν2 = h 0 ,
2 2
h 0 0 0 h 0
4 4
0 0 0
YνS = 0 0 0 . (4)
0 0 h
3
C. The scalar potential and the η masses
The most general renormalizable Higgs potential, invariant under D Zˆ Z , is given
6 2 2
× ×
by 3
V(φ,η) = V [φ;µφ,µφ,λφ,...,λφ]+V [η;µη,µη,λη,...,λη]+V [φ,η], (5)
1 1 2 1 7 1 1 2 1 7 2
where
V [φ;µ ,µ ,λ ,...,λ ]
1 1 2 1 7
= µ2(φ†φ ) µ2(φ†φ )+V [(φ†φ),(φ†φ);λ ,λ ,λ ]
− 1 I I − 2 S S 3 1 2 3
+λ (φ†φ )(φ†φ )+λ (φ†φ )(φ†φ )+[λ (φ†φ )2 +h.c.]+λ (φ†φ )2, (6)
4 S S I I 5 S I I S 6 S I 7 S S
V [φ,η]
2
= V [(φ†φ),(η†η);κ ,κ ,κ ]
3 1 2 3
† † † † † †
+κ (φ φ )(η η )+κ (φ φ )(η η )+κ (φ φ )(η η )
4 S S I I 5 I I S S 6 S S S S
+V [(φ†η),(η†φ);κ ,κ ,κ ]
3 7 8 9
+κ (φ†η )(η†φ )+κ (φ†η )(η†φ )+κ (φ†η )(η†φ )
10 S I I S 11 I S S I 12 S S S S
+ V [(φ†η),(φ†η);κ ,κ ,κ ]
3 13 14 15
+κ(cid:8) (φ†η )(φ†η )+κ (φ†η )(φ†η )+κ (φ†η )(φ†η )+h.c. , (7)
16 S I S I 17 I S I S 18 S S S S
o
3 See also for instance [24].
5
and I runs from 1 to 2. Here V is defined as
3
V [(A†B),(C†D);κ ,κ ,κ ]
3 1 2 3
† † † †
= κ (A B )(C D )+κ (A (iσ ) B )(C (iσ ) D )
1 I I J J 2 I 2 IJ J K 2 IJ L
+κ [ (A†(σ ) B )(C† (σ ) D )+(A†(σ ) B )(C† (σ ) D ) ], (8)
3 I 1 IJ J K 1 IJ L I 3 IJ J K 3 IJ L
where A,B,C and D are SU(2) doublets and belong to 2′ of D , and (A†B) is an SU(2)
L 6 L
invariant product. As we can see from (7) and (8), an exact lepton number U(1)′ invariance
L
emerges in the absence of κ κ , where the right-handed neutrinos n and n are
13 18 I S
∼
neutral under U(1)′ in contrast to the conventional seesaw models [25]. This U(1)′ forbids
L L
the neutrino masses, so that the smallness of the neutrino masses has a natural meaning.
Therefore, the radiative neutrino masses will be proportional to these Higgs couplings.
We assume that (µη)2,(µη)2 < 0 so that
1 2
< η >=< η >= 0 (9)
S I
corresponds to a local minimum of the scalar potential (5). This is the essence of refs. [4, 5]
to connect the neutrino masses and the nature of CDM, because Z remains unbroken. We
2
further observe that the scalar potential (5) has an accidental symmetry S :
2
φ ,η φ ,η , (10)
1 1 2 2
↔
while the D singlets Higgs fields φ and η remain unchanged. This symmetry ensures that
6 S S
if (µφ)2,(µφ)2 > 0,
1 2
v /2 v /√2
D S
< φ >=< φ > = and < φ >= (11)
1 2 S
0 0
! !
can correspond to a local minimum of the potential, suggesting an appropriate field redefi-
nition 4
1 1
φ = (φ φ ) , η = (η η ). (12)
± 1 2 ± 1 2
√2 ± √2 ±
A consequence of (9) is that η′s do not mix with φ′s in the mass matrix. Furthermore,
because of the absence of the φ φ η η type couplings, η and η dot not mix with each
S I S I ± S
other. Keeping these observations in mind, we can write down the mass terms for η’s as
1 1
= m2η(+)η(−) + mR 2η(0)η(0) + mI 2χ(0)χ(0)
LMη − a a a 2 a a a 2 a a a
a=+,−,S(cid:20)
X (cid:0) (cid:1) (cid:0) (cid:1)
+ mRI 2η(0)χ(0) , (13)
a a a
i
(cid:0) (cid:1)
4 The tree-level W boson mass constraint is (v2 +v2)=v2 (246GeV)2.
D S
≃
6
where η(−) (= (η(+))∗),η(0) and χ(0) are SU(2) components of η, i.e.,
L
(η(0) +iχ(0))/√2
η = . (14)
η(−)
!
Since the neutrino masses will be proportional to the U(1)′ violating Higgs couplings κ
L 13 ∼
κ , we may assume that they are small. In the absence of these Higgs couplings, there will
18
be no mixing of the scalar and pseudo scalar components of the neutral η′s, i.e., mRI = 0,
a
and mR = mI. In general, even though it is small, there exists mixing:
a a
η(0) = cosγ ηˆ +sinγ χˆ , χ(0) = sinγ ηˆ +cosγ χˆ , (a = ,S), (15)
a a a a a a − a a a a ±
where ηˆ and χˆ are mass eigenstates. We denote their masses by mη and mχ, respectively.
a a a a
The difference (mη)2 (mχ)2 will be proportional to the U(1)′ violating couplings, and we
a − a L
find that
∆m2 = (mη)2 (mχ)2
a a − a
= [(mR)2 (mI)2)]cos2γ 2(mRI)2sin2γ , (16)
a − a a − a a
where
2[(Rev )2 (Imv )2](Reκ +Reκ )+2[(Rev )2 (Imv )2]Reκ
D D 13 15 S S 16
− −
+4Rev Imv (Imκ +Imκ )+4Rev Imv Imκ
D D 13 15 S S 16
..................................................................
2[(Rev )2 (Imv )2](Reκ +Reκ )+2[(Rev )2 (Imv )2]Reκ
(mR)2 (mI)2 = D − D 14 15 S − S 16
a − a +4RevDImvD(Imκ14 +Imκ15)+4RevSImvSImκ16
..................................................................
2[(Rev )2 (Imv )2]Reκ +4Rev Imv Imκ
D D 17 D D 17
−
+2[(RevS)2 −(ImvS)2]Reκ18 +4RevSImvSImκ18
2Rev Imv (Reκ +Reκ )+2Rev Imv Reκ
D D 13 15 S S 16
[(Rev )2 (Imv )2](Imκ +Imκ ) [(Rev )2 (Imv )2]Imκ
− D − D 13 15 − S − S 16
.................................................................
2Rev Imv (Reκ +Reκ )+2Rev Imv Reκ
(mRI)2 = D D 14 15 S S 16
a −[(RevD)2 −(ImvD)2](Imκ14 +Imκ15)−[(RevS)2 −(ImvS)2]Imκ16
.................................................................
2Rev Imv Reκ +2Rev Imv Reκ
D D 17 S S 18
−[(RevD)2 −(ImvD)2]Imκ17 −[(RevS)2 −(ImvS)2]Imκ18
for a = +, ,S. As we will see below, the absence of the mixing among η and η is
± S
−
responsible for the reason that the neutrino masses and mixing of the present model have
the same structure as that of the S model of ref. [27].
3
7
III. LEPTON MASSES AND MIXING
A. CP phases
Let us first figure out the structure of CP phases. To this end, we introduce phases
explicitly as follows:
y eipyay (a = 2,4,5) (17)
a a
→
for the Yukawa couplings, where we assume that the possible phases coming from the VEVs
of φ′s are absolved into the Yukawa couplings. The y’s on the right-hand side are supposed
to be real and π/2 p′s π/2, and similarly for the fields
− ≤ ≤
LI → eipLLI , LS → eipLSLS , ecI → eipeecI , ecS → eipeSecS,
nI eipnnI , nS eipnSnS. (18)
→ →
The phases of the right-handed neutrinos are used to absorb the phases of their Majorana
masses M and M . The phases of y ,y and y can be rotated away if
1 S 2 4 5
0 = p +p +p , 0 = p +p +p , 0 = p +p +p (19)
L y2 e LS y4 e L y5 eS
are satisfied. So, only one free phase is left, which we assume to be p . We will use this
L
freedom to make certain entries of the one-loop neutrino mass matrix real. After that no
further phase rotation which does not change physics is possible.
B. Charged fermion masses
The charged lepton masses are generated from the S invariant VEVs (11), and the mass
2
matrix becomes
m m m
2 2 5
−
M = m m m , (20)
e 2 2 5
m m 0
4 4
where
m = v y /2,m = v y /2,m = v y /2. (21)
2 D 2 4 D 4 5 D 5
| | | | | |
All the mass parameters appearing in (20) can be assumed to be real. Diagonalization of
the mass matrices is straightforward. The mass eigen values are approximately given by [27]
(m m )2
m2 = 4 5 +O((m )4), (22)
e (m )2 +(m )2 4
2 5
m2 = 2(m )2 +(m )2 +O((m )4), (23)
µ 2 4 4
(m m )2
m2 = 2[ (m )2 +(m )2 ]+ 4 2 +O((m )4). (24)
τ 2 5 (m )2 +(m )2 4
2 5
8
Concrete values are given as m /m 0.00041 and m /m 0.0596 and m 1254 MeV to
4 5 2 5 5
≃ ≃ ≃
obtain m = 0.51 MeV, m = 105.7 MeV and m = 1777 MeV. The diagonalizing unitary
e µ τ
matrices (i.e., UT M U ) assume a simple form in the m 0 limit, which is equivalent to
eL e eR e →
the m 0 limit. We find that U can be approximately written as [26]
4 eL
→
ǫ (1 ǫ2) (1/√2)(1 ǫ2 +2ǫ2ǫ2) 1/√2
e − µ − − e e µ
U = ǫ (1+ǫ2) (1/√2)(1 ǫ2 2ǫ2ǫ2) 1/√2 , (25)
eL − e µ − e − e µ
1 ǫ2 √2ǫ √2ǫ ǫ2
− e e e µ
where ǫ = m /m and ǫ = m /(√2m ). In the limit m = 0, the unitary matrix U
µ µ τ e e µ e eL
becomes
0 1/√2 1/√2
−
0 1/√2 1/√2 , (26)
1 0 0
which is the origin of a maximal mixing of the atmospheric neutrinos.
C. Radiative neutrino masses
η(0),χ(0)
νL N× νL
FIG. 1: One-loop radiative neutrino mass.
The neutrino mass matrix is generated from the one-loop diagram fig. 1 [4] and is given
by
( ) = (Yνa)∗ (Yνa)∗ Γa(M ), (27)
Mν ij ik jk k
a=±,Sk=1,2,S
X X
9
where
h h 0
2 2
1 1 −
Yν+ = ( Yν1 +Yν2 ) = h h 0 , (28)
√2 √2 2 2
h h 0
4 4
h h 0
2 2
1 1 − −
Yν− = ( Yν1 Yν2 ) = h h 0 , (29)
√2 − √2 − 2 2
h h 0
4 4
−
M (mη/M)2ln(mη/M )2 (mχ/M )2ln(mχ/M )2
Γa(M ) = k exp( i2γ ) a k a k a k a k (30)
k 8π2 − a 1 (mη/M )2 − 1 (mχ/M )2
(cid:20) − a k − a k (cid:21)
M 1 (mη/M )2 +ln(mη/M )2
k exp( i2γ )∆m2 − a k a k . (31)
≃ 8π2 − a a (1 (mη/M )2)2
a k
−
The Yukawa matrices, mη,χ, γ and ∆m2 are defined in (3), (4), (13), (15) and (16), respec-
a a a
tively. (Recall that M = M because of the D symmetry.)
1 2 6
Using the explicit form of the Yukawa matrices we obtain
G+(M )h2 0 0
1 2
= 0 G+(M )h2 G−(M )h h , (32)
Mν 1 2 1 2 4
0 G−(M )h h G+(M )h2 +ΓS(M )h2
1 2 4 1 4 S 3
where
G±(M ) = Γ+(M ) Γ−(M ). (33)
1 1 1
±
At this stage we recall that the phase p was left undetermined (see (19) ), and so we use
L
it to make the (1,1) and (2,2) entries of real. Then (2,3) and (3,2) entries of can
ν ν
M M
be made real by multiplying a diagonal phase matrix
1 0 0
P = 0 1 0 (34)
0 0 exp(ip)
with from left and right, and we can rewite the neutrino mass matrix as
ν
M
2(ρ )2 0 0
2
P P = M = 0 2(ρ )2 2ρ ρ , (35)
ν ν 2 2 4
M
0 2ρ ρ 2(ρ )2 +(ρ )2expi2ϕ
2 4 4 3 3
where p is an independent parameter and enters into the definition of the neutrino mixing
matrix V , and the ρ’s in (35) are real numbers. One can convince oneself that M can
MNS ν
be diagonalized as [11, 27]
m eiφ1−iφν 0 0
ν1
UTM U = 0 m eiφ2+iφν 0 , (36)
ν ν ν ν2
0 0 m
ν3
10
