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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

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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.

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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

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