Radiative Lepton Model and Dark Matter with Global $U(1)'$ Symmetry

0.21 MB·

by  Seungwon Baek

KIAS-P14003, IPPP/14/07, DCPT/14/14
Radiative Lepton Model and Dark Matter
with Global U(1) Symmetry
′
Seungwon Baek,1,∗ Hiroshi Okada,1,† and Takashi Toma2,‡
1
School of Physics, KIAS, Seoul 130-722, Korea
2
Institute for Particle Physics Phenomenology University of Durham, Durham DH1 3LE, UK
4
1
We propose a radiative lepton model, in which the charged lepton masses are
0
2
generated at one-loop level, and the neutrino masses are induced at two-loop level.
r
a
M On the other hand, tau mass is derived at tree level since it is too heavy to generate
2 radiatively. Then we discuss muon anomalous magnetic moment together with the
1
constraint of lepton flavor violation. A large muon magnetic moment is derived due

]
h
to the vector like charged fermions which are newly addedto thestandard model. In
p
-
p addition, considering a scalar dark matter in our model, a strong gamma-ray signal
e
h is produced by dark matter annihilation via internal bremsstrahlung. We can also
[

obtain the effective neutrino number by the dark radiation of the Goldstone boson
3
v
1 coming from the imposed global U(1)′ symmetry.
2
9
6 PACS numbers:
.
1
0
4
1
:
v
i
X
r
a
∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
2
I. INTRODUCTION
Even though 26.8 % of energy density of our Universe is occupied by a non-baryonic dark
matter (DM) [1, 2], several current experiments are still under investigation of its nature
from various points of view such as direct and indirect searches. As for the direct detection
search, for example, XENON100 [3] and LUX [4] provides the most severe constraint on
spin independent elastic cross section with nuclei; that is, the cross sections is less than
around 10−46 cm2 at 100 GeV scale of DM mass. As for the indirect searches, AMS-02
has recently shown the positron excess with smooth curve in the cosmic ray, and reached
the energy up to 350 GeV [5]. This result has a good statistics and support the previous
experiment PAMELA [6]. On the other hand, the recent analysis of gamma-ray observed by
Fermi-LAT tells us that there may be some peak near 130 GeV [7, 8]. As for the neutrinos,
their small masses and mixing pattern call for new physics beyond the standard model (SM).
Plank, WMAP9 andground-baseddatarecently reportedapossibledeviationintheeffective
neutrino number, ∆N = 0.36 0.34 at 68 % confidential level [2, 9–11]. Compensating this
eff
±
deviation theoretically might come into one of the important issues. In this sense, radiative
seesaw models which support a strong correlation between DM and neutrinos come into an
elegant motivation. Many authors have proposed such kind of models in, e.g., ref. [12–46] 1.
In our paper, we propose a model that neutrino masses as well as charged lepton (muon
andelectron) massesaregeneratedbyradiativecorrection. Weobtainalargecontributionto
muon anomalous magnetic moment from the charged lepton sector as can be seen later. At
the same time, one should mind constraints from lepton flavor violations (LFVs) like µ eγ
→
since it is closely correlated with anomalous magnetic moment. Since neutrino masses are
generated at two-loop level, they are therefore naturally suppressed. As a result, unlike
the TeV scale canonical seesaw mechanism, extremely small parameters are not required to
lead the observed neutrino mass scale. Moreover the particles run in the loop can be DM
candidates. Our scalar DM interacts with vector like charged fermions, which are added to
theSM,andtheotherinteractionshouldbesuppressedtosatisfythedirectsearchconstraint.
Due to the interaction with the vector like charged fermions, a strong gamma-ray signal is
emitted by the DM annihilation via internal bremsstrahlung preserving consistency with the
1 Radiative models of the lepton mass are sometimes discussed with Non-Abelian discrete symmetries due
to their selection rules. See for example such kind of models: [47–51].
3
thermal relic density of DM[52, 53]. Inparticular it ispossible to adapt with the gamma-ray
anomaly found in the Fermi data at around 130 GeV. The neutrino effective number is also
led without conflicting with the other parts of DM physics.
This paper is organized as follows. In Sec. II, we show our model building for the lepton
sector, and discuss Higgs sector, muon anomalous magnetic moment, and LFV. In Sec. III,
DMphenomenology such asrelic density, strong gamma-raysignal and theneutrino effective
number is discussed. We summarize and conclude in Sec. IV.
II. THE MODEL
A. Model setup
Particle L ec ec e′ e′c n′ n′c Nc
i j 3 i i j j
(SU(2) ,U(1) ) (2, 1/2) (1,1) (1,1) (1, 1) (1,1) (1,0) (1,0) (1,0)
L Y
− −
U(1)′ Z2 (ℓ, ) (0, ) ( ℓ, ) (ℓ,+) ( ℓ,+) (ℓ,+) ( ℓ,+) (0, )
× − − − − − − −
TABLE I: The particle contents and the charges for fermions. The i,j are generation indices:
i = 1,2,3, j = 1,2.
Particle Φ η χ Σ
(SU(2) ,U(1) ) (2,1/2) (2,1/2) (1,0) (1,0)
L Y
U(1)′ Z2 (0,+) (0, ) ( ℓ, ) (ℓ,+)
× − − −
TABLE II: The particle contents and the charges for bosons.
We construct a radiative lepton model with global U(1)′ symmetry, in which charged
lepton sector is obtained through one-loop level, and two-loop level for neutrino sector. In
the model, only tau mass is generated at tree level, but electron and muon masses are
generated at one-loop level. This is because tau mass is too heavy to generate radiatively.
The particle contents are shown in Tab. I and Tab. II. The quantum number ℓ(= 0) in the
6
tables is arbitrary. Here L and ec (i = 1,2,3) are the SM left-handed and right-handed
i i
lepton fields. For right-handed charged leptons ec (i = 1,2,3), different charges of U(1)′
i
are assigned to the first, second generation and the third generation in order to distinguish
4
the mass generation mechanism. We add three generation of SU(2) singlet vector like
L
charged fermions e′ and e′c (i = 1,2,3), two generation of vector like neutral fermions n′
i i j
and n′c (j = 1,2), a singlet Majorana fermion Nc2. For new bosons, we introduce SU(2)
j L
doublet scalar η and singlet scalars χ and Σ in addition to the SM Higgs doublet Φ. The
SM Higgs Φ should be neutral under U(1)′ not to couple quarks to Goldstone boson through
chiral anomaly to be consistent with the axion particle search3. We assume that only the
SM Higgs doublet Φ and the SM singlet Σ have vacuum expectation values. Otherwise the
Z
symmetry which guarantees DM stability is spontaneously broken.
2
The renormalizable Lagrangian for Yukawa sector and scalar potential are given by
M
LY = ynηn′cLη +ynχNcn′χ+ 2NNcNc +Mn′n′cn′ +h.c.
+yτΦΦ†ec3L+yℓηη†e′cL+yℓχec1,2e′χ+Me′e′e′c +h.c. (II.1)
= m2Φ†Φ+m2η†η +m2Σ†Σ+m2χ†χ+λ (Φ†Φ)2 +λ (η†η)2 +λ (Φ†Φ)(η†η)
V 1 2 3 4 1 2 3
+λ (Φ†η)(η†Φ)+λ [(Φ†η)2 +h.c.]+λ′[(Σ†χ)2 +h.c.]+λ′′[(Σχ)2 +h.c.]
4 5 5 5
+λ (Σ†Σ)2 +λ′(χ†χ)2 +λ′′ Σ†Σ χ†χ +λ (Σ†Σ)(Φ†Φ)+λ′(χ†χ)(Φ†Φ)
6 6 6 7 7
+λ (Σ†Σ)(η†η)+λ′(χ†χ)(η†(cid:0)η)+(cid:1)a(cid:0)(η†Φ(cid:1))(Σχ)+h.c. + a′(Φ†η)(Σχ)+h.c. ,
8 8
(cid:2) (cid:3) (cid:2) (cid:3)(II.2)
where λ , λ′, λ′′, and one of a and a′ can be chosen to be real without any loss of generality
5 5 5
by absorbing the phases to scalar bosons. The Φ†e′cL term which might generate mixing
i
between e′c and ec is not allowed by the Z symmetry. The Yukawa interaction Φ†ec L
i 3 2 1,2
which gives the tree level masses of electron and muon is forbidden by U(1)′ symmetry. The
term NcLη which induces one-loop neutrino masses [12] is also excluded by U(1)′ symmetry.
The couplings λ , λ , λ and λ′ have to be positive to stabilize the Higgs potential. Inserting
1 2 6 6
the tadpole conditions; m2 = λ v2 λ v′2/2 and m2 = λ v′2 λ v2/2, the resulting mass
1 − 1 − 7 3 − 6 − 7
matrix of the neutral component of Φ and Σ defined as
v +φ0(x) v′ +σ(x)
Φ0 = , Σ = eiG(x)/v′, (II.3)
√2 √2
2 Multi-component vector like fermions are required to produce the observed charged lepton masses and
neutrino oscillation data. There are other patterns of particle content to derive proper lepton masses.
3 If Φ is charged under U(1)′, its breaking scale should be very large (& 1012 GeV), which is inconsistent
with the observed value 246 GeV.
∼
5
is given by
2λ v2 λ vv′ cosα sinα m2 0 cosα sinα
m2(φ0,σ) = 1 7 = h − ,
 λ vv′ 2λ v′2   sinα cosα  0 m2  sinα cosα 
7 6 − H
     (II.4)
where h implies SM-like Higgs with the mass of 125 GeV and H is an additional CP-even
Higgs mass eigenstate. The mixing angle α is given by
λ vv′
7
tan2α = . (II.5)
λ v′2 λ v2
6 1
−
The Higgs bosons φ0 and σ are rewritten in terms of the mass eigenstates h and H as
φ0 = hcosα+Hsinα,
σ = hsinα+Hcosα. (II.6)
−
A Goldstone boson G appears due to the spontaneous symmetry breaking of the global
U(1)′ symmetry. This massless particle would be dark radiation contributing to the effective
neutrino number we will discuss later [54].
The resulting mass matrix of the neutral component of η and χ defined as
η +iη χ +iχ
η0 = R I, χ = R I, (II.7)
√2 √2
is given by
m2 m2 cosβ sinβ m2 0 cosβ sinβ
m2(η ,χ ) = ηR ηRχR = R R h′R R − R ,
R R
m2 m2   sinβ cosβ  0 m2  sinβ cosβ 
ηRχR χR − R R HR′ R R
     (II.8)
for CP even mass eigenstates where h′ and H′ are mass eigenstates of inert Higgses. The
R R
imaginary part of these inert Higgses (CP odd states) is defined by replacing the index R
into I, hereafter. The mixing angle β is given by
R
2m2
tan2β = ηRχR . (II.9)
R m2 m2
χR − ηR
The η and χ are rewritten in terms of the mass eigenstates h′ and H′ as
R R R R
η = h′ cosβ +H′ sinβ ,
R R R R R
χ = h′ sinβ +H′ cosβ . (II.10)
R − R R R R
6
Each mass component is defined as
1 1
m2 m2(η±) = m2 + λ v2 + λ v′2, (II.11)
η ≡ 2 2 3 2 8
1 1
m2 m2(η ) = m2 + λ v′2 + (λ +λ +2λ )v2, (II.12)
ηR ≡ R 2 2 8 2 3 4 5
1 1
m2 m2(η ) = m2 + λ v′2 + (λ +λ 2λ )v2, (II.13)
ηI ≡ I 2 2 8 2 3 4 − 5
1 1 1
m2 m2(χ ) = m2 + λ′′v′2 + λ′v2 +λ′v′2 +λ′′v′2 , (II.14)
χR ≡ R 3 2 2 6 2 7 5 5
(cid:18) (cid:19)
1 1 1
m2 m2(χ ) = m2 + λ′′v′2 + λ′v2 λ′v′2 λ′′v′2 , (II.15)
χI ≡ I 3 2 2 6 2 7 − 5 − 5
(cid:18) (cid:19)
1 1
m2 = vv′(a+a′), m2 = vv′(a a′). (II.16)
ηRχR 4 ηIχI 4 −
We note that we need mass splitting between η (χ ) and η (χ ) which is required to gen-
R R I I
erate the non-zero lepton masses. The tadpole conditions for η and χ, which are given by
∂ /∂η = 0, ∂ /∂χ = 0, 0 < ∂2 /∂η2 and 0 < ∂2 /∂χ2 tell us that
V |VEV V |VEV V |VEV V |VEV
v2 v′2 v2 v′2
0 < m2 + (λ +λ +2λ )+ λ , 0 < m2 + λ′ + (λ′ +λ′′ +λ′′), (II.17)
2 2 3 4 5 2 8 4 2 7 2 5 5 6
to satisfy the condition η = 0 and χ = 0 at tree level, respectively. In order to avoid
h i h i
that Φ = Σ = 0 be a local minimum, we require the following condition:
h i h i
2
λ λ λ < 0. (II.18)
7 1 6
− 3
p
To achieve the global minimum at η = χ = 0, we find the following condition
h i h i
2
0 < λ′ λ λ′. (II.19)
8 − 3 2 6
p
Finally, if the following conditions
2 2 2
0 < λ + λ λ , 0 < λ + λ λ , 0 < λ′ + λ λ′,
3 3 1 2 7 3 1 6 7 3 1 6
2p 2p 2p
0 < λ + λ λ , 0 < λ′ + λ λ′, 0 < λ′′ + λ λ′, (II.20)
8 3 2 6 8 3 2 6 6 3 6 6
p p p
are satisfied, the Higgs potential Eq. (II.2) is bounded from below.
B. Charged lepton and neutrino mass matrix
The tau mass is given at tree level, after the spontaneous symmetry breaking as m =
τ
yΦv/√2. On the other hand, the electron and muon masses are generated at one-loop, as
τ
7
FIG. 1: Radiative generation of charged lepton masses.
FIG. 2: Radiative generation of neutrino masses.
can be seen in Fig. 1 as follows:
(m ) = (yℓη)αi(yℓχ)iβMe′isin2βR F m2h′R F m2HR′ +(R I), (II.21)
ℓ αβ 4(4π)2 M2 − M2 →
i " e′i! e′i!#
X
where F(x) = xlogx/(1 x). The total mass matrix is diagonalized by bi-unitary matrix.
−
From the mass formula, for example, the Yukawa coupling (yηyχ) 1 is required for muon
ℓ ℓ ∼
mass and (yℓηyℓχ) ∼ 0.01 for electron mass when Me′ ∼ 500 GeV, sin2βR(I) ∼ 0.1 and O(1)
of the loop function. The Yukawa coupling yχ should be (1) to obtain the observed DM
ℓ O
relic density as we will see in Sec. III.
The Dirac neutrino mass matrix at one-loop level as depicted in the left hand side of
Fig. 2 is given by
(m ) = (ynχ)i(ynη)iβMn′isin2βR F m2h′R F m2HR′ (R I), (II.22)
D iβ 4(4π)2 M2 − M2 − →
i " n′i! n′i!#
X
With the Dirac neutrino mass matrix, the active neutrino mass matrix is obtained by canon-
8
ical seesaw mechanism as
1
(m ) = mTm . (II.23)
ν1 αβ −M D D αβ
N
In addition, there is another contribution to the n(cid:0)eutrino(cid:1)masses coming from the right hand
side of Fig. 2. The mass matrix is expressed as [34]
(m ) = (ynη)iα(ynχ)i(ynχ)k(ynη)kβMn′iMNFloop, (II.24)
ν2 αβ 16(4π)4Mn′k ik
i k
XX
where the loop function Floop is given by
ik
δ(x+y +z 1)
Floop = d3x −
ik y(y 1)
Z −
M2 m2 M2 m2
sin22β G ih′R, h′R G ih′R, HR′ +(h′ H′ )
× R M2 M2 − M2 M2 R ↔ R
"( n′k n′k! n′k n′k!! )
M2 m2 M2 m2
sin2β sin2β G ih′R, h′I G ih′R, HI′ (h′ H′ )
− R I M2 M2 − M2 M2 − R ↔ R
( n′k n′k! n′k n′k!! )
+(R I) , (II.25)
↔
#
with
x(1 y)logx+y(1 x)logy
G(x,y) = − − − , (II.26)
(1 x)(1 y)(x y)
− − −
and
xm2 +yM2 +zm2
M2 n′i N a (II.27)
ia ≡ y(y 1)
−
where a = h′ ,H′ ,h′,H′. Whole neutrino mass matrix is sum of the two contributions as
R R I I
m = m +m . From the neutrino mass formula, (yχyη) 0.01 is needed to obtain the
ν ν1 ν2 n n ∼
proper neutrino mass scale by assuming Mn′ 500 GeV, MN 1 TeV, (0.1) of the loop
∼ ∼ O
functions.
C. The muon anomalous magnetic moment and Lepton Flavor Violation
The muon anomalous magnetic moment, (g 2) , has been measured at Brookhaven
µ
−
National Laboratory. The current average of the experimental results [55] is given by
aexp = 11659208.0(6.3) 10−10,
µ ×
which has a discrepancy from the SM prediction with 3.2σ [56] to 4.1σ [57] as
∆a = aexp aSM = (29.0 9.0 to 33.5 8.2) 10−10.
µ µ − µ ± ± ×
9
In our model, there are several contributions to the (transition) magnetic moment µ
αβ
which is coefficient of the operator µ ℓ σµνℓ F . The muon anomalous magnetic moment
αβ α β µν
is identified as ∆a = µ . The largest contribution comes from photon attaching to vector
µ µµ
like charged fermions since it is proportional to mα/Me′ where mα is charged lepton mass.
On the other hand, the other contributions are proportional to m2/M2. The contributions
α e′
coming from the loop of η+ and n′ in neutrino sector are also proportional to m2/M2. Thus
α n′
these are neglected in our calculation, and the (transition) magnetic moment is calculated
as
µ 2 sin2βR mα (yη)αi(yχ)iβ +(yη∗)βi(yχ∗)iα H m2h′R +H m2HR′ +(R I)
αβ ≃ Xi=1 2(4π)2 Me′i(cid:16) ℓ ℓ ℓ ℓ (cid:17)"− Me2′i! Me2′i!# →
1 4x+3x2 2x2lnx
with H(x) = − − . (II.28)
2(1 x)3
−
More preciously, the unitary matrices which diagonalize the charged lepton mass matrix
should be multiplied from left and right. It is understood by replacing Yukawa couplings
yη, yχ to yη′, yχ′. This expression of the (transition) magnetic moment is closely related
ℓ ℓ ℓ ℓ
with radiative induced charged lepton masses Eq. (II.21). To reproduce the muon mass, for
example, sin2θR(I) and Me′i are taken to be (10−2) and (1) TeV, respectively. Thus we
O O
obtain ∆a = (10−9), when (yη)(yχ)[H(m2 /M2 ) H(m2 /M2 )] is roughly 0.1.
µ O ℓ ℓ h′R e′i − HR′ e′i
It is the common fact that muon g 2 and Lepton Flavor Violation tend to conflict each
−
other. In LFV processes, µ eγ especially gives the most stringent bound. The upper limit
→
of the branching ratio is given by Br(µ eγ) 5.7 10−13 at 95% confidence level from
→ ≤ ×
the MEG experiment [58]. In our model, the diagonal Yukawa matrix yη and yχ is required
ℓ ℓ
not to conflict with Lepton Flavor Violating processes such as µ eγ. Nevertheless, the
→
contribution to µ eγ still comes from the neutrino sector, and it is calculated as
→
2
3α M2
Br(µ eγ) = em (yη) (yη)∗ F n′i , (II.29)
→ 64πG2m4 n iµ n ie 2 m2
F η (cid:12)(cid:12)Xi (cid:18) η (cid:19)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where α =1/137 is the fine structure constant, G is the Fermi constant and F (x) is the
em (cid:12) F (cid:12) 2
loop function defined in ref. [59]. From the Eq. (II.29), we obtain a rough estimation for
the Yukawa coupling ynη . 0.05 by setting mη = Mn′ ∼ 500 GeV. This estimation does not
contradict with the discussion of neutrino masses.
10
III. DARK MATTER
We have two DM candidates: vector like fermion n′, the lightest eigenstate of η0 and
χ (one of h′ , H′ , h′, H′). One may think the scalar DM candidate decays into the SM
R R I I
Z
particles since the SM leptons also have odd charge under the imposed symmetry in
2
our model. However, the decay of the DM candidate is forbidden by Lorentz invariance.
Namely, this means that the scalar DM candidate can decay into only even number of
fermion, however such a decay process is not allowed in the model.
We identify h′ is DM here since it has interesting DM phenomenology. The mixing angle
R
sinβ isneeded to besmall enough since tiny neutrino masses areproportional tothemixing
R
angle. Note that in the limit of sinβ 0, there is no contribution from h′ and H′ to the
R → R R
charged lepton and neutrino masses as one can see from the previous section. However we
still have the contribution of h′ and H′. The neutrino masses are generated from h′ and
I I I
H′. The parameter relation a a′ is required to construct such a situation as one can
I ≈ −
see Eq. (II.16). In this case, the DM candidate h′ corresponds to just χ . Thus we regard
R R
χ as DM hereafter. The couplings λ′, λ′′, λ′′ and λ′ in the scalar potential also should be
R 5 5 6 7
suppressed not to have large elastic cross section with nuclei. Otherwise elastic scattering
occurs via Higgs exchange and it is excluded by direct detection experiments of DM such as
XENON [3] or LUX [4]. The spin independent elastic cross section with proton in the limit
of sinβ 0 is given by
R
→
Cµ2m2 µ cosα µ sinα 2
χ p χχh χχH
σ = + , (III.1)
p πm2 v2 m2 m2
χR (cid:18) h H (cid:19)
where µ is reduced mass defined as µ = (m + m−1)−1, m = 938 MeV is the proton
χ χ χR p p
mass and C 0.079. The couplings µ and µ are given by
χχh χχH
≈
λ′′ λ′
µ = λ′ +λ′′ + 6 v′sinα+ 7vcosα, (III.2)
χχh − 5 5 2 2
(cid:18) (cid:19)
λ′′ λ′
µ = λ′ +λ′′ + 6 v′cosα+ 7vsinα. (III.3)
χχH 5 5 2 2
(cid:18) (cid:19)
The elastic cross section is strongly constrained by LUX as σ . 7.6 10−46 cm2 at m
p × χR ≈
33 GeV. Thus the couplings λ′, λ′′, λ′′ and λ′ are required to be (0.001) in order to satisfy
5 5 6 7 O
the constraint when v′ 1 TeV and sinα 1.
∼ ∼
Due to the strong constraint from direct detection of DM, the annihilation cross section
for the process χ χ ff via Higgs s-channel is extremely suppressed. The cross section
R R
→