We propose a new model to explain the neutrino masses, the dark energy and
the baryon asymmetry altogether. In this model, neutrinos naturally acquire
small Majorana masses via type-II seesaw mechanism, while the
pseudo-Nambu-Goldstone bosons associated with the neutrino mass-generation
mechanism provide attractive candidates for dark energy. The baryon asymmetry
of the universe is produced from the Higgs triplets decay with CP-violation.Comment: 5 pages, 2 figures. Version accepted by PL

Gu, Pei-Hong

He, Hong-Jian

Sarkar, Utpal

Physics Letters B

English

arXiv

We propose a new model to explain the neutrino masses, the dark energy and the baryon asymmetry altogether. In this model, neutrinos naturally acquire small Majorana masses via type-II seesaw mechanism, while the pseudo-Nambu-Goldstone bosons associated with the neutrino massgeneration mechanism provide attractive candidates for dark energy. The baryon asymmetry of the universe is produced from the Higgs triplets decay with CP-violation

Publications of the IAS Fellows

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Neutrino Dark Energy and Baryon Asymmetry from Higgs Sector
Pei-Hong Gu1, Hong-Jian He2, and Utpal Sarkar3
1The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy
2Center for High Energy Physics, Tsinghua University, Beijing 100084, China
3Physical Research Laboratory, Ahmedabad 380009, India
We propose a new model to explain the neutrino masses, the dark energy and the baryon asym-
metry altogether. In this model, neutrinos naturally acquire small Majorana masses via type-II
seesaw mechanism, while the pseudo-Nambu-Goldstone bosons associated with the neutrino mass-
generation mechanism provide attractive candidates for dark energy. The baryon asymmetry of the
universe is produced from the Higgs triplets decay with CP-violation.
I. INTRODUCTION
The atmospheric, solar and laboratory neutrino oscil-
lation experiments [1] have confirmed that neutrinos have
tiny but nonzero masses, of the order of 10−2 eV. This
phenomenon is elegantly explained by the seesaw mech-
anism [2], in which neutrinos acquire small Majorana [2]
or Dirac [3] masses naturally. Furthermore, the observed
matter-antimatter asymmetry in the universe can be gen-
erated through leptogenesis [4] in the neutrino seesaw
scenario.
On the other hand, various cosmological observations
[1] provide strong evidence that the expansion of our uni-
verse is accelerating due to the existence of dark energy.
One possible explanation for the dark energy has its ori-
gin in a dynamical scalar field, such as the quintessence
[5] with an extremely flat potential. It was shown [6] that
the pseudo-Nambu-Goldstone boson (pNGB) provides an
attractive realization of the quintessence field.
A striking coincidence between the scales of neutrino
masses and dark energy (∼ (10−3 eV)4), inspires us to
consider them in a unified scenario, i.e., the neutrino dark
energy model which generically predicts neutrino-mass
variation [7]. There have been lots of recent activities
studying the neutrino dark energy models. A possible
connection between the neutrinos and the pNGB dark
energy was explored in the type-I seesaw scenario [8, 9].
In this paper, we propose a new neutrino dark en-
ergy model to simultaneously explain the generation of
neutrino masses and the origin of dark energy from the
Higgs sector. In particular, the pNGBs associated with
neutrino mass-generation provide the consistent candi-
dates of dark energy while the small neutrino masses de-
pending on the dark energy field are realized through the
type-II seesaw. Furthermore, the CP-violation and out-
of-equilibrium decays of the Higgs triplets produce the
baryon asymmetry in the universe.
II. THE MODEL
We extend the SU(2)L ×U(1)Y standard model (SM)
with triplet and singlet Higgs scalars,
ψLi =
 νLi
lLi
 (2,−1
2
) , H =
 H0
H−
 (2,−1
2
) ,
ξij ≡ ξji =
 1√2δ+ij δ++ijδ0ij − 1√2δ+ij
 (3, 1) ,
χij ≡ χji (1, 0) , (for i, j = 1, 2, 3), (1)
are left-handed lepton doublets, Higgs doublet, Higgs
triplets and Higgs singlets, respectively. The full La-
grangian is supposed to be invariant under a global
U(1)6 symmetry, generated by the independent phase
transformations of each Higgs triplet among the six ξij
fields. Transformations of the Higgs singlets χij under
this U(1)6 are determined by requiring the invariance of
the following scalar interactions,
χijH
T iτ2ξijH + h.c. , (2)
where we have defined the SM Higgs doublet H as a
singlet under U(1)6. The Higgs triplets have the Yukawa
couplings to the left-handed lepton doublets,
ψ cLiiτ2ξijψLj + h.c. , (3)
which explicitly break the U(1)6 down to its subgroup
U(1)3. So we have only three massless Nambu-Goldstone
bosons (NGBs) after the six χij ’s acquire their vacuum
expectation values (VEVs).
We write down the relevant Lagrangian for the Higgs
and lepton-Yukawa interactions,
L ⊃ −
∑
ij
(
µ2ij +
∑
kl
λij,klχ
†
klχkl
)
Tr
(
ξ†ijξij
)
−
∑
ij 6=kl
λ
′
ij,klχ
†
ijχklTr
(
ξ†ijξkl
)
−
∑
ij
(
1
2
yijψ
c
Liiτ2ξijψLj − hijχijHT iτ2ξijH
+ h.c. ) , (4)
2where λ
(′)
ij,kl ≡ λ(
′)
ji,kl ≡ λ(
′)
ij,lk , yij ≡ yji and hij ≡ hji are
dimensionless while µij has mass-dimension equal one.
After the Higgs singlets get their VEVs, 〈χij〉 ≡ 1√2fij ,
we can write
χij =
1√
2
(
fij + σij
)
exp
(
iϕij/fij
)
(5)
with σij ≡ σji , ϕij ≡ ϕji (i, j = 1, 2, 3) being the neutral
bosons and the NGBs, respectively. Among these six
NGBs, three of them will acquire nonzero masses via the
Coleman-Weinberg potential (due to the small explicit
breaking of global symmetries, U(1)6 → U(1)3) and thus
become pNGBs. The other three remain massless as the
result of spontaneous breaking of the subgroup U(1)3.
For convenience, we redefine the Higgs triplets as
exp
(
iϕij/fij
)
ξij → ξij . (6)
The mass matrix M for the physical triplet scalar fields
is now given by
Mij,kl ≡
[(
µ2ij+
1
2
∑
kl
λij,klf
2
kl
)
δij,kl +
1
2
λ
′
ij,klfijfkl
]1
2
. (7)
The VEVs of χij ’s generate the effective trilinear inter-
actions between the Higgs triplets and doublet,
1√
2
hijfijH
T iτ2ξijH + h.c. ≡ µijHT iτ2ξijH + h.c. , (8)
where the cubic couplings µij will be set as real after
proper phase rotations. From Eqs. (6), (7) and (8), we
derive the Lagrangian (4) as below,
L ⊃ −
∑
ij,kl
M2ij,kl Tr
(
ξ†ijξkl
)
−
∑
ij
[
1
2
yij exp
(−iϕij/fij)ψ cLiiτ2ξijψLj
− µijHT iτ2ξijH + h.c.
]
. (9)
We still have the freedom to redefine the phases of the
three lepton doublets, which can remove three of the
fields ϕij from the lepton-Higgs Yukawa interactions.
Without loss of generality, we choose the rephasing,
exp [−iϕii/(2fii)]ψLi → ψLi (10)
which transforms the Lagrangian (9) into a new form,
L ⊃ −
∑
ij,kl
M2ij,klTr
(
ξ†ijξkl
)
−
[
1
2
∑
i
yiiψ
c
Liiτ2ξiiψLi
+
1
2
∑
i6=j
yij exp(iφij/f)ψ
c
Liiτ2ξijψLj
−
∑
ij
µijH
T iτ2ξijH + h.c.
]
(11)
where
φij
f
= −ϕij
fij
+
1
2
ϕii
fii
+
1
2
ϕjj
fjj
, (12)
and f is of the order of fij . It is not possible to remove
φij (i 6= j) from Eq. (11) by any further transformations
and hence these φij ’s will become the pNGBs with tiny
masses and can naturally serve as the candidates of dark
energy. Note that from (6), (10) and the subsequent
phase rotations on the right-handed charged leptons, the
three massless NGBs, ϕii, will only have the derivative in-
teractions to other fields, but they are highly suppressed
by 1/fii and thus escape from experimental constraints
at low energy scales.
III. NEUTRINO MASS AND MIXING
The electroweak symmetry breaking takes place with
the VEV of the Higgs doublet, which induces small VEVs
to the triplets,
〈H〉 ≡ 1√
2
 v
0
, 〈ξij〉 ≡ 1√
2
 0 0uij 0
, (13)
with i, j = 1, 2, 3, where the VEVs of the Higgs triplets
ξij are deduced as
uij ≃
v2√
2
∑
kl
µkl
(
M−2
)
kl,ij
. (14)
Inspecting Eqs. (7) and (8), it is natural to take the
masses Mij,kl around the same order as the scalar cu-
bic couplings µij since they are both controlled by the
singlet VEVs (fij). Hence, the triplet VEVs in (14) are
seesaw-suppressed by the ratio of the electroweak scale v
over the heavy mass M , i.e., uij = O
(
v2/M
)≪ v.
These small VEVs will then generate the Majorana
masses for the neutrinos,
Lm = −1
2
∑
ij
(mν)ij ν
c
LiνLj + h.c. , (15)
where
(mν)ij =
{
mij , (for i = j),
mij exp
(
iφij/f
)
, (for i 6= j) , (16)
with mij ≡ 1√2 yijuij . With the 3× 3 symmetric mass-
matrix (16), we can readily realize the neutrino mass-
spectrum and mixings, consistent with the neutrino os-
cillation experiments. Moreover, the interactions of the
neutrinos with the pNGBs will induce a small long-range
force, which can have direct consequences in cosmology
[10, 11] and neutrino oscillation experiments [12].
IV. ORIGIN OF DARK ENERGY
Three of the NGBs, φij (i 6= j) as defined in (12),
will acquire small masses due to the Yukawa couplings
3νLi
νLi
νLj
νLj
−12mijeiφij/f
−12mijeiφij/f
FIG. 1: The one-loop diagram contributing to the Coleman-
Weinberg potential of the pNGBs.
between the left-handed lepton doublets and the Higgs
triplets, and thus become the pNGBs. As shown in Fig.
1, the leading loop diagram will contribute a Coleman-
Weinberg effective potential for φij . Similar to [8], we
explicitly calculate the potential,
V (φ12, φ23, φ31) = −
1
32pi2
3∑
k=1
m4k ln
m2k
Λ2
, (17)
where mk as a function of φij is the kth eigenvalue of the
neutrino mass matrix mν , and Λ is the ultraviolet cutoff.
A typical term in V that contributes to the potential of
a pNGB field Q has the form,
V (Q) ≃ V0 cos(Q/f) (18)
with V0 = O(m4ν). It is well known that with f of the
order of Planck mass MPl, the pNGB Q will acquire a
mass of order O(m2ν/MPl) and thus provides a consistent
candidate for the quintessence dark energy.
Finally, we also note that after the electroweak symme-
try breaking, the explicit breaking of the U(1)3 symme-
try is only generated by the dimension-3 soft mass term
in Eq. (15). So, in this model there is no dimension-4
term which explicitly breaks U(1)3, and thus there are
no higher order correction that could contribute a diver-
gent term to spoil the renormalizability and the original
symmetry of the theory. This point can be checked by
explicit calculations as well. For instance, at the two-loop
order, we have
V2 ≈
(
1
16pi2
)2
Λ2 Tr
(
mνm
†
ν
) [
Tr
(
yy†
)
+
(
1
2
+
1
4 cos2 θW
)
g2
]
.
Since Tr
(
mνm
†
ν
)
is independent on the pNGBs as well as
Tr
(
yy†
)
, we see that the two-loops have no contribution
to the effective potential for the pNGBs. Similarly, there
is no contribution from other higher loops.
V. BARYON ASYMMETRY
The decays of Higgs triplets,
ξij →
{
ψ cLi ψ
c
Lj , (L = − 2 ) ,
H∗H∗ , (L = 0 ) ,
(19)
can break the lepton number. As shown in Fig. 2,
the mass-mixings in (7) among different Higgs triplets
contribute the tree-level and one-loop diagrams that
interfere to generate CP asymmetry in these decays.
The decays of the triplet Higgs will then produce
enough lepton asymmetry before the electroweak phase
transition, which can successfully explain the observed
matter-antimatter asymmetry in the universe through
the sphaleron processes [13] which convert the lepton
asymmetry into the existing baryon asymmetry.
To calculate the CP-asymmetry, we define the Higgs
triplets in their mass-eigenbasis,
ξ̂a ≡
∑
ij
Ua,ijξij (20)
with the diagonalized mass-eigenvalues,
M̂a ≡
∑
ij,kl
Ua,ijMij,klUa,kl . (21)
Thus we can rewrite the Lagrangian (11) as
L ⊃ −
∑
a
M̂2aTr
(
ξ̂ †a ξ̂a
)
−
(
1
2
∑
a,ij
ŷaijψ
c
Liiτ2ξ̂aψLj
−
∑
a
µ̂aH
T iτ2ξ̂aH + h.c.
)
(22)
with the definition
µ̂a ≡
∑
ij
Ua,ijµij (23)
and
ŷaij ≡
{
Ua,ijyij , (for i = j) ,
Ua,ijyij exp
(
iφij/f
)
, (for i 6= j) . (24)
Here U is an orthogonal rotation matrix. The proper
CP-asymmetry parameter is then described by [14]
εa ≡ 2×
∑
ij
[
Γ
(
ξ̂∗a→ψLiψLj
)
− Γ
(
ξ̂a→ψ cLiψ cLj
)]
Γa
=
1
pi
∑
b6=a
Im
{
Tr
[(
ŷb
)†
ŷa
]}
µ̂bµ̂a
Tr
[
(ŷa)
†
ŷa
]
Mˆ2a + 4 µ̂
2
a
M̂2a
M̂2a − M̂2b
(25)
with
Γa =
1
8pi
{
1
4
Tr
[
(ŷa)† ŷa
]
+
µ̂2a
M̂2a
}
M̂a (26)
being the total decay width of ξ̂a or ξ̂
∗
a .
4ξij
ψ cLi
ψ cLj
+
ξij
H
H
ξkl ξij
ψ cLi
ψ cLj
FIG. 2: The Higgs triplets decay to the leptons at one-loop order.
For illustration, we use ξ̂a to denote the lightest Higgs
triplet and hence its contribution is expected to domi-
nate the final baryon asymmetry, which is given by the
approximate relation [15],
YB ≡
nB
s
≃ −28
79
×

εa
g∗
, (for K ≪ 1) ,
0.3 εa
g∗K (lnK)
0.6 , (for K ≫ 1) ,
(27)
with the factor 28/79 being the value of B/(B − L) and
the parameter K defined by
K ≡ Γa
2H(T )
∣∣∣∣
T=cM
a
(28)
as a measurement of the departure from equilibrium.
Here H(T ) = (8pi3g∗/90)
1
2 T 2/MPl is the Hubble con-
stant with the Planck mass MPl ≃ 1.2 × 1019 GeV and
the relativistic degrees of freedom g∗ ≃ 106.75. For
simplicity, we further consider
rba ≡
M̂b
M̂a
=
µ̂b
µ̂a
≫ 1 and ŷb ≡ ŷae−iδba , (29)
where δba is the relative phase between ŷ
b and ŷa. We
thus neglect the contribution from the heavier triplets
to the neutrino masses and conveniently express K as
K =
[
(4pi)5g∗
45
]− 1
2 (
BψBH
)− 1
2
MPlm
v2
. (30)
Here the quadratic mean of the neutrino masses (m) is
defined by
m2 ≡
3∑
k=1
m2k ≡ Tr
(
m†νmν
) ≃ 1
4
Tr
[
(ŷa)
†
ŷa
] µ̂2av4
M̂4a
= (8pi)2BψBHΓ
2
a
v4
M̂4a
(31)
and (Bψ , BH) are the branching ratios of the tree-level
decays of ξ̂a into the lepton and Higgs doublets, which
always hold the relationship,
Bψ +BH ≡ 1 , =⇒ BψBH 6
1
4
. (32)
Then, we compute the CP-asymmetry (25) as
εa = −
1
pi
Tr
[
(ŷa)
†
ŷa
]
µ̂2a
Tr
[
(ŷa)
†
ŷa
]
M̂2a + 4µ̂
2
a
∑
b6=a
rba sin δba
r2ba − 1
≃ − 1
pi
(
BψBH
)1/2 M̂am
v2
sin δba
rba
. (33)
Inputting v ≃ 246GeV, BψBH = 1/4, M̂a = 4 × 1012
GeV, m = 0.1 eV, sin δba = 0.1 and rba = 10, we derive
the sample predictions: εa ≃ −1.1× 10−5 and K ≃ 46 .
Finally, we deduce, nB/s ≃ 10−10, consistent with the
cosmological observations.
VI. CONCLUSION AND DISCUSSION
In this paper, we propose a new model to unify the neu-
trino dark energy and baryon asymmetry by extending
only the SM Higgs sector with triplet and singlet Higgs
scalars. The Higgs triplets naturally acquire tiny VEVs
and give small Majorana masses for the neutrinos. The
model contains three pNGBs associated with the neu-
trino mass-generation, which provide consistent candi-
dates for the quintessence field. The matter-antimatter
asymmetry in the universe is produced by the out-of-
equilibrium decays of the Higgs triplets with CP-violating
couplings. We can readily accommodate the dark matter
as well in our construction by adding a darkon field [16]
or an inert Higgs doublet [17].
In our model, the neutrino masses are functions of the
dark energy field. Being a dynamical component, the
dark energy will evolute with time and/or in space. Ac-
cordingly, the neutrino masses will vary instead of being
constants. The prediction of the neutrino-mass variation
could be verified in the present and future experiments,
such as the observations on the cosmic microwave back-
ground and the large scale structures [10], the measure-
ment of the extremely high-energy cosmic neutrinos [11],
and the analysis of the neutrino oscillation data [12].
Acknowledgments: P.H.G. would like to thank Jun-
Bao Wu, Wander G. Ney and Alexei Yu. Smirnov for
helpful discussions.
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Neutrino dark energy and baryon asymmetry from Higgs sector

AbstractWe propose a new model to explain the neutrino masses, the dark energy and the baryon asymmetry altogether. In this model, neutrinos naturally acquire small Majorana masses via type-II seesaw mechanism, while the pseudo-Nambu–Goldstone bosons associated with the neutrino mass-generation mechanism provide attractive candidates for dark energy. The baryon asymmetry of the universe is produced from the Higgs triplets decay with CP-violation

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Neutrino dark energy and baryon asymmetry from Higgs sector 

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Neutrino Dark Energy and Baryon Asymmetry from Higgs Sector

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