Page 1
arXiv:1110.6838v1 [hep-ph] 31 Oct 2011
Light Sterile Neutrino in the Minimal Extended Seesaw
He Zhang1, ∗
1Max-Planck-Institut f¨ ur Kernphysik, Postfach 103980, 69029 Heidelberg, Germany
Motivated by the recent observations on sterile neutrinos, we present a minimal extension of the
canonical type-I seesaw by adding one extra singlet fermion. After the decoupling of right-handed
neutrinos, an eV-scale mass eigenstate is obtained without the need of artificially inserting tiny
mass scales or Yukawa couplings for sterile neutrinos. In particular, the active-sterile mixing is
predicted to be of the order of 0.1. Moreover, we show a concrete flavor A4 model, in which the
required structures of the minimal extended seesaw are realized. We also comment on the feasibility
of accommodating a keV sterile neutrino as an attractive candidate for warm dark matter.
INTRODUCTION
During the past decade, various neutrino oscillation
experiments have shown very solid evidence of non-
vanishing neutrino masses and lepton flavor mixing.
Apart from neutrino oscillations within three active fla-
vors, recent re-evaluations of the anti-neutrino spectra
suggest that there exists a flux deficit in nuclear reactors,
which could be explained if anti-electron neutrinos oscil-
late to sterile neutrinos [1]. Such a picture would require
one or more sterile states with masses at the eV scale,
together with sizable admixtures [i.e., O(0.1)] with ac-
tive neutrinos. Moreover, the light-element abundances
from precision cosmology and Big Bang nucleosynthesis
favor extra radiation in the Universe, which could be in-
terpreted with the help of additional sterile neutrinos [2].
From the theoretical side, it is unclear why the energy
scales related to electroweaksymmetry breaking and ster-
ile neutrinos are different by many order of magnitude.
In the canonical type-I seesaw [3], right-handed neutri-
nos could in principle play the role of sterile neutrinos if
their masses lie in the eV ranges. This could be techni-
cally natural since right-handed neutrinos are Standard
Model (SM) gauge singlets [4]. However, in such a case,
the Yukawa couplings relating lepton doublets and right-
handed neutrinos should be of the order 10−12(namely,
the Dirac mass should be at the sub-eV scale) for correct
mixings between active and sterile neutrinos. It is there-
fore more appealing to consider a natural and consistent
framework yielding both low-scale sterile neutrino masses
and sizable active-sterile mixings. In this respect, mod-
els simultaneously suppressing the Majorana and Dirac
mass terms have been proposed in the literature, e.g. the
split seesaw models in extra dimensions [5], the Froggatt-
Nielsen mechanism [6–9], and flavor symmetries [10].
Recall that the seesaw mechanism is among one of the
most popular theoretical attempts that gives a natural
way to understand the smallness of neutrino masses. This
motivates us to look for the possibility of generating eV-
scale sterile neutrino masses by using a similar approach.
Such an idea has been briefly mentioned in Ref. [7], in
which the type-I seesaw is extended by adding only one
singlet fermion [i.e., the minimal extended seesaw (MES)]
acting as a sterile neutrino, without the need of impos-
ing tiny Yukawa couplings or mass scales. A similar idea
was also employed in Ref. [11] to accommodate a sterile
neutrino of mass ∼ 10−3eV in order to explain the so-
lar neutrino problem. In this note, we exploit in detail
the properties of the MES. Especially, we will show that
the sterile neutrino mass is stabilized at the eV scale,
while a sizable active-sterile mixing accounting for the
rector neutrino anomaly is predicted. Furthermore, we
will discuss how to realize the MES structure in flavor
symmetries, i.e., a flavor model based on the tetrahedral
group A4. We also comment that the model could be gen-
eralized in order to include a keV sterile neutrino playing
the role of warm dark matter.
THE MINIMAL EXTENDED SEESAW
Here we describe the basic structure of the MES, in
which three right-handed neutrinos and one singlet S
are introduced besides the SM particle content. We will
show that there could be a natural eV-scale sterile neu-
trino in this picture, without the need of inserting a small
mass term or tiny Yukawa couplings. Explicitly, the La-
grangian of neutrino mass terms is given by
−Lm= νLMDνR+ ScMSνR+1
2νc
RMRνR+ h.c.,(1)
where MDand MRdenote the Dirac and Majorana mass
matrices, respectively. Note that, MSis a 1 × 3 matrix,
since we only introduce one extra singlet. The full 7 × 7
neutrino mass matrix in the basis (νL,νc
R,Sc) reads
M7×7
ν
=
0MD
0
MT
0
DMR MT
MS
S
0
. (2)
Similar to the typical type-I seesaw model, MD is as-
sumed to be around the electroweak scale, i.e., 102GeV,
while the right-handed neutrino Majorana masses are
chosen to be not far away from the typical grand uni-
fication scale, MR ∼ 1014GeV. Furthermore, there is
no bare Majorana mass term assumed for S, while S is
only involved in the MSterm, which may originate from
Page 2
2
certain Yukawa interactions with right-handed neutrinos
and a SM singlet scalar. There is essentially no con-
straint on the scale of MS. In the remaining parts, we
will consider the interesting situation MR≫ MS.
In analogy to the type-I seesaw, the right-handed neu-
trinos are much heavier than the electroweak scale, and
thus they should be decoupled at low scales. Effectively,
one can block-diagonalize the full mass matrix M7×7
using the seesaw formula, and arrive at a 4 × 4 neutrino
mass matrix in the basis (νL,Sc), i.e.,
ν
by
M4×4
ν
= −
?
MDM−1
MS
?M−1
RMT
?TMT
D
MDM−1
DMSM−1
RMT
RMT
S
R
S
?
. (3)
One observes from Eq. (3) that there exist in total four
light eigenstates corresponding to three active neutrinos
and one sterile neutrino, and their masses are all sup-
pressed by a factor M−1
Rin the spirit of the seesaw mech-
anism. Moreover, M4×4
ν
is at most of rank 3, since
det?M4×4
ν
?
= det
?
MDM−1
RMT
D
?
det
?
−MSM−1
RMT
S
+ MSM−1
RMT
D
?
MDM−1
RMT
D
?−1MDM−1
M−1
R
− M−1
RMT
S
?
= det
?
MDM−1
RMT
D
?
det
?
MS
?
R
?
MT
S
?
= 0 ,
(4)
where we have assumed both MRand MDare invertible.
Therefore, at least one of the light neutrinos is massless.
We proceed to diagonalize M4×4
general three choices of the scale of MS: 1) MD∼ MS;
2) MD ≫ MS; 3) MD ≪ MS.
nearly democratic, indicating a maximal mixing between
active and sterile neutrinos, and therefore is not compat-
ible with neutrino oscillation data. In the second case,
active neutrinos are heavier than the sterile one. Such a
scenario results in more tension with cosmological con-
straints on the summation of light neutrino masses. We
will comment on this case later on. In what follows, we
shall concentrate on the third case, and study the prop-
erties of the sterile neutrino in detail.
Since in case 3 MSis larger than MDby definition, one
can apply the seesaw formula once again to Eq. (3), and
obtain at leading order the active neutrino mass matrix
ν
. There could be in
For case 1, M4×4
ν
is
mν ≃ MDM−1
− MDM−1
RMT
RMT
S
?MSM−1
RMT
S
?−1MS
?M−1
R
?TMT
D
(5)
D,
as well as the sterile neutrino mass
ms≃ −MSM−1
RMT
S. (6)
Note that the right-hand-side of Eq. (5) does not vanish
since MSis a vector rather than a square matrix; if MS
were a square matrix this would lead to an exact cancel-
lation between the two terms of Eq. (5). Here, mν can
be diagonalized by means of a 3 × 3 unitary matrix as
mν= Udiag(m1,m2,m3)UT, (7)
where mi (for i = 1,2,3) denote the masses of three
active neutrinos. The full neutrino mixing matrix then
takes a 4 × 4 form,
V ≃
?(1 −1
2RR†)U
−R†U
R
2R†R1 −1
?
, (8)
where R is a 3×1 matrix governing the strength of active-
sterile mixing,
R = MDM−1
RMT
S
?MSM−1
RMT
S
?−1. (9)
Essentially, the R matrix (i.e., Vα4 for α = e,µ,τ) is
suppressed by the ratio O(MD)/O(MS).
As a naive numerical example, for MD ∼ 102GeV,
MS ∼ 5 × 102GeV and MR∼ 2 × 1014GeV, one may
estimate that mν∼ 0.05 eV, ms∼ 1.3 eV together with
R ∼ 0.2. This is in very good agreement with the fitted
sterile neutrino parameters [12, 13].
The MES picture described above is a minimal exten-
sion of the type-I seesaw in the sense that one could al-
low for at most one extra singlet in order to account for
neutrino oscillation phenomena (see a recent analysis in
Ref. [14]). In other words, three heavy right-handed neu-
trinos can lead to at most three massive light neutrinos
(the “seesaw-fair-play-rule”[15]), out of which two are ac-
tive and needed to account for the solar and atmospheric
neutrino mixing.
Unfortunately, it is not possible to accommodate two
eV-scale sterile neutrinos in the MES, unless the num-
ber of right-handed neutrinos is increased. Furthermore,
without introducing addition fermions, this scenario can-
not be simply embedded into a grand unified theory
framework since it is not anomaly free. Apart from these
shortcomings, the MES possesses the following features:
• apart from the electroweak and seesaw scales, one
does not artificially insert small mass scales for ster-
ile neutrino masses. As in the canonical type-I see-
saw, one can take MS> MD∼ O(102GeV), while
MRcan be chosen close to the B −L scale, not far
from the grand unification scale;
• it is more predictive owing to the absence of one
active neutrino mass, while it does explain all the
experimental data. Neutrino-less double beta de-
cay is also allowed because not all of the neutrinos
are light.
• there exist heavy right-handed neutrinos that could
be responsible for thermal leptogenesis. Note that,
in the setup we considered, right-handed neutri-
nos would preferably decay to the sterile neutrino
since their coupling to S is larger than active neu-
trinos.However, this drawback could easily be
circumvented since S enters in the one-loop self-
energy diagram of the decay of right-handed neu-
trinos, which could compensate for this.
Page 3
3
We finally comment on the second case, i.e., MD ≫
MS. Now that M4×4
ν
possesses a hierarchical texture
along the inverted direction, one may still apply the see-
saw formula to Eq. (3), and obtain that, at leading order,
the active neutrino mass matrix is the same as that given
in the type-I seesaw, i.e., mν≃ −MDM−1
the sterile neutrino mass is vanishing. In viewing of the
experimental results on the active-sterile mass-squared
difference, one would expect all the three active neutrinos
to be located at the eV scale, which is however challenged
by standard cosmology since that leads to a large total
mass of neutrinos. Note also that, despite the cosmolog-
ical constraints, one can in principle add more singlets
since they do not affect active neutrino masses. Espe-
cially, in case of three additional singlets (S1,S2,S3), the
particle contents are analogous to those in the inverse
seesaw or double seesaw [16], although the mass matrix
structures are clearly different.
RMT
D, whereas
REALIZATION IN A FLAVOR A4 MODEL
In this section, we focus on a simple flavor A4model
giving rise to the exact mass structures depicted in
Eq. (1). In addition to the SM Higgs boson, we intro-
duce three sets of flavons ϕ, ξ and χ. An extra discrete
abelian symmetry Z4 has been introduced in order to
avoid interferences between the neutrino and charged-
lepton sectors. The particle assignments are shown in
Table I.
TABLE I: Particle assignments in the flavor A4 model.
Field ℓ eR µR τR H ϕ ϕ′ϕ′′ξ ξ′
χ νR1 νR2 νR3 S
SU(2) 2 1
A4
Z4
112 1 1
1 3 3
1 1 i −1 1 −1 −i
1 1 1
3 1 1′
1
1
1
1
1
1
1′
−i −1 i
1
1
1
13 1 1′′1′
1 111
At leading order, the A4⊗Z4invariant Lagrangian for
the lepton sector is given by
L =
ye
Λ
y1
Λ
1
2λ1ξνc
1
2ρχScνR1+ h.c. ,
?ℓHϕ?
?
1eR+yµ
Λ
?ℓHϕ?
?
1′µR+yτ
Λ
?ℓHϕ?
1′′τR
+
ℓ˜ Hϕ
?
1νR1+y2
Λ
ℓ˜ Hϕ′?
1′′νR2+y3
Λ
?
ℓ˜ Hϕ′′?
1νR3
+
R1νR1+1
2λ2ξ′νc
R2νR2+1
2λ3ξνc
R3νR3
+
(10)
where Λ denotes the cut-off scale and˜H ≡ iτ2H.
we choose the real basis for A4, along with the flavon
If
alignments1
?ϕ? = (v,0,0) ,
?ξ? = ?ξ′? = v ,
?ϕ′? = (v,v,v) ,
?χ? = u ,
?ϕ′′? = (0,−v,v) ,
(11)
then the charged-lepton mass matrix is diagonal2, i.e.,
mℓ=?H?v
Λ
diag(ye,yµ,yτ) , (12)
while the Dirac mass term is given by
MD=?H?v
Λ
y1 y2
0 y2
0 y2 −y3
0
y3
. (13)
Due to the additional Z4 symmetry, the right-handed
neutrino mass matrix is diagonal as well, viz.
MR= diag(λ1v,λ2v,λ3v) .(14)
Furthermore, the singlet fermion S does not acquire a
Majorana mass term, at least at leading order. The cou-
pling matrix between S and right-handed neutrinos reads
MS=?ρu 0 0?
. (15)
As a rough numerical example, we assume the follow-
ing mass scales: v ≃ 1013GeV, Λ ≃ 1014GeV, and u ≃
102GeV. One can then estimate that, by assuming order
1 Yukawa couplings, the condition MR≫ MS> MDcan
be satisfied. Compared to Eq. (5), we obtain
mν= −?H?2v
Λ2
y2
λ2
y2
λ2
y2
λ2
2
y2
λ2
2
y2
λ2
2
2
y2
2λ3+y2
λ2λ3
y2
λ2λ3
3λ2
y2
2λ3−y2
λ2λ3
y2
λ2λ3
3λ2
2
2λ3−y2
3λ2
2λ3+y2
3λ2
. (16)
It is straightforward to see that mνfeatures a µ−τ sym-
metry, which generally predicts a maximal mixing in the
2 − 3 sector and a vanishing θ13. Indeed, mν can be an-
alytically diagonalized by using the tri-bimaximal mix-
ing [18] matrix VTBas
mν= −VTBdiag
?
0,3y2
2?H?2v
λ2Λ2
,2y2
3?H?2v
λ3Λ2
?
VT
TB,(17)
with VTBbeing
VTB=
2
√6
−1
−1
1
√3
1
√3
1
√3−1
0
1
√6
√6
√2
√2
.(18)
1We do not expand our discussions on how the vacuum alignment
of flavons is achieved, whereas we refer readers to Ref. [17], in
which the same flavon vacuum alignment is acquired by assuming
a radiative symmetry breaking mechanism.
2Note that, the hierarchies between charged-lepton masses can be
obtained by using the Froggatt–Nielsen mechanism, viz., assign-
ing different U(1)FNcharges to the right-handed fields.
Page 4
4
Therefore, the normal mass ordering (m1≪ m2≪ m3)
together with the tri-bimaximal mixing pattern are ob-
tained. Taking for example y3 = 0.91, y2 = 0.31, and
λ2= λ3 = 1, one obtains ∆m2
∆m2
global-fit data of neutrino mass-squared differences [19].
The sterile neutrino mass is obtained from Eq. (7) as
21≃ 7.6 × 10−5eV2and
31≃ 2.5 × 10−3eV2, being consistent with current
ms≃ρ2u2
λ1v
.(19)
Fitting to the sterile neutrino mass from a recent best-fit
given in [12], one can get ms≃ 1.2 eV (corresponding to
∆m2
y1= 1 and inserting the above parameters to Eq. (9), we
arrive at the active-sterile mixing, i.e.,
41≃ 1.5 eV2) for ρ = 1.1 and λ1= 1. By choosing
R ≃
?
y1?H?v
ρuΛ
0 0
?T
≃?0.16 0 0?T,(20)
corresponding to |Ve4|2≃ 0.025 together with |Vµ4| =
|Vτ4| = 0, in good agreement with the best-fit value
of active-sterile mixing [20] in the four neutrino mixing
scenario. Therefore, in this simple model, both the tri-
bimaximal mixing pattern in the active neutrino sector
and a sizable active-sterile neutrino mixing are predicted
without the need of fine-tuning the Yukawa couplings.
Alternatively, the flavor model described above can be
slightly changed in order to admit the inverted mass or-
dering of active neutrinos (i.e., m2 ? m1 ≫ m3). For
this purpose, one could instead take the VEV alignment
?ϕ′? = (2v,−v,−v), which retains the tri-bimaximal mix-
ing in the active neutrino mixing, and leads to a vanishing
mass m3= 0. Note that, in case of the inverted mass or-
dering, the next-to-leading seesaw corrections [21] should
be included in the diagonalization of M4×4
the degeneracy between m1and m2.
As mentioned in the previous section, in this model,
both active and sterile neutrinos may mediate the
neutrino-less double beta decay processes, and their con-
tributions to the effective mass are not cancelled. Con-
cretely, we have ?m?ee≃ |m2V2
the normal mass ordering case, and ?m?ee ≃ |m1V2
m2V2
ordering case.In addition, effects from right-handed
neutrinos are negligibly small since they are highly sup-
pressed by MR(see e.g. Ref. [22] for detailed discussions).
This is a very distinctive feature, in particular compared
to models with only eV-scale right-handed neutrinos, in
which neutrino-less double beta decays are forbidden.
One may also wonder if the model could be modified
to allow for a keV sterile neutrino warm dark matter can-
didate. Indeed, the sterile neutrino mass can be chosen
at the keV ranges by setting, e.g. u ∼ 4 TeV. Using the
same Yukawa coupling parameters in the previous discus-
sions, one then arrives at ms≃ 1.9 keV. Unfortunately,
the active-sterile mixing θs= R11≃ 4 × 10−3turns out
ν
, because of
e2+ msV2
e4| ≃ ms|Ve4|2in
e1+
e2+ msV2
e4| ≃ |m1+ msV2
e4| in the inverted mass
to exceed the current X-ray constraint [23],
θ2
s? 1.8 × 10−5
?1 keV
ms
?5
.(21)
In order to keep θssmall enough, we need a mild tuning of
the Yukawa coupling, i.e., y1< 0.2. For example, taking
s≃ 1.2 × 10−5?
the bound in Eq. (21).3
y1 = 0.15, we get θ2
1 keV
ms
?5
, satisfying
CONCLUSION
In this note, we have studied a minimal extension of
the type-I seesaw, which contains an extra singlet fermion
coupled purely to the right-handed neutrinos. In such a
framework, both active and sterile neutrino masses are
suppressed via the seesaw mechanism, and thus, an eV-
scale sterile neutrino together with sizable active-sterile
mixing is accommodated without the need of artificially
inserting small mass scales or Yukawa couplings. Fur-
thermore, we have presented a flavor A4model, in which
both the MES structures and the tri-bimaximal mixing
pattern are realized. In particular, for a sterile neutrino
with mass being around the eV scale, the active-sterile
mixing (i.e., |Ve4|) is found to be of the order of 0.1, in
good agreement with current experimental observations.
The model may also be modified to take keV sterile neu-
trino warm dark matter into account. We hope this note
serves as a useful guide for future model building works
on low-scale sterile neutrinos.
We would like to thank Werner Rodejohann and
James Barry for helpful discussions and comments on the
manuscript. This work was supported by the ERC un-
der the Starting Grant MANITOP and by the Deutsche
Forschungsgemeinschaft in the Transregio 27 “Neutrinos
and beyond – weakly interacting particles in physics, as-
trophysics and cosmology”.
∗Electronic address: he.zhang@mpi-hd.mpg.de
[1] G. Mention, M. Fechner, T. Lasserre, T. Mueller,
D. Lhuillier, et al., Phys.Rev. D83, 073006 (2011),
1101.2755; P. Huber, Phys.Rev. C84, 024617 (2011),
1106.0687.
[2] J. Hamann, S. Hannestad, G. G. Raffelt, I. Tamborra,
and Y. Y. Wong, Phys.Rev.Lett. 105, 181301 (2010),
1006.5276; Y. Izotov and T. Thuan, Astrophys.J. 710,
3Note that various mechanisms for the production of sterile neu-
trino warm dark matter have been proposed in the literature,
which, however, will not be discussed in this note.
Page 5
5
L67 (2010), 1001.4440;
E. D. Skillman, JCAP 1005, 003 (2010), 1001.5218;
J. Hamann, S. Hannestad, G. G. Raffelt, and Y. Y. Wong
(2011), 1108.4136.
[3] P. Minkowski, Phys. Lett. B67, 421 (1977); T. Yanagida,
in Proc. Workshop on the Baryon Number of the Universe
and Unified Theories, edited by O. Sawada and A. Sug-
amoto (1979), p. 95; M. Gell-Mann, P. Ramond, and
R. Slansky, in Supergravity, edited by P. van Nieuwen-
huizen and D. Freedman (1979), p. 315; R. N. Mohapatra
and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).
[4] A. de Gouvea, Phys.Rev. D72, 033005 (2005), hep-
ph/0501039.
[5] A. Kusenko, F. Takahashi, and T. T. Yanagida, Phys.
Lett. B693, 144 (2010), 1006.1731; A. Adulpravitchai
and R. Takahashi, JHEP 09, 127 (2011), 1107.3829.
[6] C. Froggatt and H. B. Nielsen, Nucl.Phys. B147, 277
(1979).
[7] J. Barry, W. Rodejohann, and H. Zhang, JHEP 1107,
091 (2011), 1105.3911.
[8] A. Merle and V. Niro,
1105.5136.
[9] J. Barry,W. Rodejohann,
1110.6382.
[10] R. Mohapatra, Phys.Rev. D64, 091301 (2001), hep-
ph/0107264; M. Shaposhnikov, Nucl.Phys. B763, 49
(2007), hep-ph/0605047; M. Lindner, A. Merle, and
V. Niro, JCAP 1101, 034 (2011), 1011.4950.
E. Aver, K. A. Olive, and
JCAP 1107, 023 (2011),
and H. Zhang (2011),
[11] E. Chun, A. S. Joshipura, and A. Smirnov, Phys.Lett.
B357, 608 (1995), hep-ph/9505275.
[12] J. Kopp, M. Maltoni, and T. Schwetz, Phys.Rev.Lett.
107, 091801 (2011), 1103.4570.
[13] C. Giunti and M. Laveder (2011), 1109.4033.
[14] A. Donini, P. Hernandez, J. Lopez-Pavon, and M. Mal-
toni, JHEP 1107, 105 (2011), 1106.0064.
[15] Z.-Z. Xing, Chin.Phys. C32, 96 (2008), 0706.0052.
[16] R. Mohapatra, Phys.Rev.Lett. 56, 561 (1986); R. N. Mo-
hapatra and J. W. F. Valle, Phys. Rev. D34, 1642 (1986).
[17] S. F. King and M. Malinsky, Phys. Lett. B645, 351
(2007), hep-ph/0610250.
[18] P. F. Harrison, D. H. Perkins, and W. G. Scott, Phys.
Lett. B530, 167 (2002), hep-ph/0202074; Z.-Z. Xing,
Phys. Lett. B533, 85 (2002), hep-ph/0204049; P. F. Har-
rison and W. G. Scott, Phys. Lett. B535, 163 (2002),
hep-ph/0203209.
[19] T. Schwetz, M. Tortola, and J. W. F. Valle (2011),
1108.1376.
[20] T. Schwetz, M. Tortola, and J. W. F. Valle, New J. Phys.
13, 063004 (2011), 1103.0734.
[21] H. Hettmansperger, M. Lindner, and W. Rodejohann,
JHEP 1104, 123 (2011), 1102.3432.
[22] W. Rodejohann, Int. J. Mod. Phys. E20, 1833 (2011),
1106.1334.
[23] A. Boyarsky, O. Ruchayskiy, and M. Shaposhnikov,
Ann.Rev.Nucl.Part.Sci. 59, 191 (2009), 0901.0011.