JHEP10(2013)112

Published for SISSA by Springer

Received: August 3, 2013

Revised: September 30, 2013

Accepted: October 1, 2013

Published: October 18, 2013

Large θ13 in a SUSY SU(5)× T ′ model

Mu-Chun Chen,a Jinrui Huang,b K.T. Mahanthappac and Alexander M. Wijangcoa

aDepartment of Physics & Astronomy,

University of California, Irvine, CA 92697-4575, U.S.A.bTheoretical Division, T-2, MS B285,

Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A.cDepartment of Physics, University of Colorado at Boulder,

Boulder, CO 80309-0390, U.S.A.

E-mail: muchunc@uci.edu, jinruih@lanl.gov, ktm@pizero.colorado.edu,

awijangc@uci.edu

Abstract: In the model based on SUSY SU(5) combined with the T ′ family symmetry, it

is shown [1, 2] that fermion mass hierarchy and mixing angles can be naturally generated

in both lepton and quark sectors compatible with various experimental measurements. But

the predicted value for the non-zero lepton mixing angle of θ13 ' θc/3√

2 contradicts the

recent experimental results from Daya Bay and RENO. We propose to introduce one more

singlet flavon field in the neutrino sector, and with its additional contribution, we are able

to generate the large mixing angle θ13 ∼ 8◦−10◦. The analytical expressions of the mixing

angles and neutrino masses with the additional flavon field are derived. Our numerical

results show that a large region in the model parameter space is allowed for the normal

hierarchy case, while a much smaller region is allowed for the inverted hierarchy case. In

addition, the predicted value of the solar mixing angle is not affected by the additional

singlet contribution. On the other hand, there exists a correlation between θ13 value and

θ23 − π/4, as a result of the additional singlet contribution. We also make predictions for

various other variables that can be potentially tested in the future neutrino experiments

such as the Dirac CP phase, which is predicted to be in the range of δ ∼ (195◦ − 200◦),

and the neutrinoless double beta decay matrix element, 〈mββ〉. One numerical example

with the minimal χ2 fit is presented, and the model predictions are consistent with all

experimental results.

Keywords: Beyond Standard Model, Neutrino Physics, Discrete and Finite Symmetries,

GUT

ArXiv ePrint: 1307.7711

c© SISSA 2013 doi:10.1007/JHEP10(2013)112

JHEP10(2013)112

Contents

1 Introduction 1

2 The model 3

3 Quark and charged lepton masses and quark mixings 4

4 Neutrino masses, lepton mixings and CP phases 6

4.1 Neutrino masses and lepton mixings 6

4.2 Lepton Dirac CP phase δ, Jarlskog invariant J`, neutrinoless double beta

decay matrix element 〈mββ〉 12

5 Conclusion 15

1 Introduction

Quarks as well as leptons are known to mix, and the collective structure of these flavor

oscillations has been a standing puzzle for decades. Since the discovery of neutrino oscil-

lation, it is now well known that this phenomena could be described by a unitary matrix

of at least rank three. Recent hints from T2K [3], Minos [4], Double-Chooz [5], and dis-

covery from Daya Bay [6] and RENO [7] have now established a large value for the θ13

mixing angle in the neutrino sector. Existing global fits to a suite of neutrino oscillation

experiments are summarized in the table 1 for both normal hierarchy (NH) and inverted

hierarchy (IH) scenarios [8] (see also, [9–11] and [12–14]). The experimental observation

of a large lepton mixing angle θ13 ' 8◦ − 10◦ is clearly incompatible with the so-called

Tri-Bimaximal Mixing (TBM) pattern [15],

UTBM =

√

2/3√

1/3 0

−√

1/6√

1/3 −√

1/2

−√

1/6√

1/3√

1/2

, (1.1)

which yields the following values for the mixing angles,

sin2 θTBM23 = 1/2 , tan2 θTBM

12 = 1/2 , sin θTBM13 = 0 . (1.2)

There have been many attempts to address the problem of fermion mass hierarchy

and mixing. A particularly popular approach has been through the introduction of a

family symmetry, with which the number of free parameters in the Yukawa sector can be

reduced. The symmetries proposed include the continuous ones, such as SU(2) [16–18],

SU(3) [19] and U(1) [20–23], and the discrete ones [24, 25], such as A4 [26–30], S4 [31],

– 1 –

JHEP10(2013)112

parameter best fit ± 1σ 2σ

sin2 θ12 0.320+0.015−0.017 0.29-0.35

sin2 θ23 0.49+0.08−0.05(NH); 0.53+0.05

−0.07(IH) 0.41-0.62 (NH); 0.42-0.62 (IH)

sin2 θ13 0.026+0.003−0.004(NH); 0.027+0.003

−0.004(IH) 0.019-0.033 (NH); 0.020-0.034 (IH)

∆m221(10−5eV2) 7.62 ± 0.19 7.27-8.01

∆m231(10−3eV2) 2.53+0.08

−0.10(NH); −(2.40+0.10−0.07)(IH) 2.34-2.69 (NH); -(2.25-2.59) (IH)

Table 1. The current global fitting results on the neutrino mixing angles and mass splittings

from [8].

T ′ [32–36], A5 [37]. These discrete symmetries are particularly conducive to the TBM

mixing pattern; in particular it has been shown that an additional A4 symmetry can

generate lepton mixing matrix naturally [26–30]. However, there exists no simple model

featuring an A4 symmetry which gives rise to quark mixing. As such a model would likely

require additional mechanisms to be consistent with the quark sector [38–42], it is thus less

compiling to combine A4 with a grand unified theory (GUT).

In light of the discovery of a large θ13, various models based on discrete family symme-

tries, such as A4 [43–46], S4 [47], T ′ [48] and A5 [50] have been revisited lately, and more

examples can be found in the review [51]. It has been pointed out [52, 53] that within

the original A4 models, due to the presence of the Kahler potential terms induced by the

flavon vacuum expectation values (VEVs), there already exist ingredients that can generate

a large value of θ13 compatible with current experimental observation. By going beyond

the “minimal” setup with one triplet and one singlet flavon fields of the original A4 models,

it has also been shown [45] that the inclusion of additional A4 singlets can also give sizable

deviation from the TBM predictions, leading to phenomenologically viable predictions.

In the following, we consider the SUSY SU(5) GUT model combined with the T ′

family symmetry proposed in [1, 2]. This model is free [54] of T ′ part of the discrete

gauge anomalies [55] and can generate the fermion mass hierarchy and mixings in both

lepton and quark sectors with nine real parameters, addressing the structure of flavor in a

predictive way. In addition, leptogenesis can be realized by the geometrical CP phases [1,

2]. Unfortunately, while this model predicts a non-zero θ13 angle, the prediction is small

compared to the experimental results. However, featuring unification, flavor structure, and

a deterministic structure for CP violation, models of this class still merit consideration. The

goal would then be to include a mechanism that raises the predicted value of θ13, which is

the subject of this letter. Such efforts have been considered before; in a different SU(5)×T ′

GUT model it has been demonstrated that one can preserve TBM in the neutrino sector by

modifying the mixing in the charged lepton sector to generate an appropriate θ13 [48]. In

this letter, we take a different approach and introduce one more singlet scalar field coupling

to the neutrino sector [49]. Such an approach leads to deviations from TBM mixing, and

the lepton mixing consistent with experimental values are obtained with relatively fewer

parameters (namely ten, where all parameters are again real).

– 2 –

JHEP10(2013)112

T3 Ta F N H5 H ′5

∆45 φ φ′ ψ ψ′ ζ ζ′ ξ η η′′ S

SU(5) 10 10 5 1 5 5 45 1 1 1 1 1 1 1 1 1 1

T ′ 1 2 3 3 1 1 1′ 3 3 2′ 2 1′′ 1′ 3 1 1′′ 1

Z12 ω5 ω2 ω5 ω7 ω2 ω2 ω5 ω3 ω2 ω6 ω9 ω9 ω3 ω10 ω10 ω10 ω10

Z′12 ω ω4 ω8 ω5 ω10 ω10 ω3 ω3 ω6 ω7 ω8 ω2 ω11 1 1 1 ω2

Table 2. Field content of our model. The three generations of matter fields in 10 and 5 of SU(5)

are the T3, Ta (a = 1, 2) and F multiplets. The field N contains three generations of right-handed

neutrinos. Higgs fields that are needed to generate SU(5) invariant Yukawa interactions are H5,

H ′5

and ∆45. The flavon fields φ through ζ ′ are those that give rise to the charged fermion mass

matrices, while ξ through S are the ones that generate neutrino masses. The Z12 charges are given

in terms of the parameter ω = eiπ/6.

As the era of precision continues, it is the hope that the Dirac CP phase be measured

and perhaps some insights on the conjugate nature of the neutrino can be gained through

neutrinoless double beta decay searches. An excellent test of the model presented here can

occur at both experiments, and to this end the predictions for both the Dirac CP phase,

δ, and the neutrinoless double beta decay matrix element, 〈mββ〉, are presented as well.

This paper is organized as follows. In section 2, we present the model. The analytical

and numerical results of the quark and charged lepton masses and quark mixings are

discussed in the section 3. Both the analytical expressions and numerical results for the

neutrino masses, lepton mixings and the experimental variables that may be accessible in

the future experimental measurements are illustrated in the section 4. section 5 concludes

the paper.

2 The model

We minimally expand the model [1, 2] by adding a (non-trivial) singlet scalar field, η′′ ∼ 1′′.

This additional singlet flavon field couples to the neutrino sector only, due to its charge

under the discrete symmetries of the model. The comprehensive particle content and their

SU(5), T ′ and Z12 × Z12 quantum numbers are summarized in table. (2). The charge

assignments lead to the following Yukawa superpotential,

WYuk =WTT +WTF +Wν , (2.1)

where

WTT = ytH5T3T3 +1

Λ2H5

[ytsT3Taψζ + ycTaTbφ

2]

+1

Λ3yuH5TaTbφ

′3 , (2.2)

WTF =1

Λ2ybH

′5FT3φζ +

1

Λ3

[ys∆45FTaφψζ

′ + ydH5′FTaφ

2ψ′]

, (2.3)

Wν = λ1NNS +1

Λ3

[H5FNζζ

′(λ2ξ + λ3η + λ4η

′′)], (2.4)

with Λ being the cutoff scale of the T ′ symmetry. The Yukawa couplings are real, which

can be achieved through appropriate redefinition of the flavor fields [1, 2]. At the scale Λ,

– 3 –

JHEP10(2013)112

the T ′ symmetry is broken spontaneously and the flavon fields acquire vacuum expectation

values (VEVs) along the following directions,

〈ξ〉 =

1

1

1

ξ0Λ,⟨φ′⟩

=

1

1

1

φ′0Λ, 〈φ〉 =

0

0

1

φ0Λ,

〈ψ〉 =

(1

0

)ψ0Λ,

⟨ψ′⟩

=

(1

1

)ψ′0Λ, (2.5)

〈ζ〉 = ζ0Λ ,⟨ζ ′⟩

= ζ ′0Λ, 〈η〉 = η0Λ,

〈S〉 = s0Λ,⟨η′′⟩

= η′′0Λ . (2.6)

We leave the study of the potential which gives rise to this alignment for future work. As

such, we will not consider inputs from this sector in the counting of parameters needed to

achieve the observed mixing, as these additional parameters in the flavon and Higgs sectors

(such as tanβ) will lead to additional predictions such as the flavon and Higgs masses.

Upon the T ′ symmetry breaking, the operators inWTT ,WTF , andWν give rise to the

masses of the up-type quarks, both the down-type quarks and charged leptons, and the

neutrinos respectively.

3 Quark and charged lepton masses and quark mixings

In terms of the T ′ and SU(5) component fields, the above superpotential gives the

following Yukawa interactions for the charged fermions in the weak charged current

interaction eigenstates:

−LYuk ⊃ UR,i(Mu)ijQL,j +DR,i(Md)ijQL,j + ER,i(Me)ij`L,j + h.c. , (3.1)

where QL denotes the quark doublets, UR and DR denotes the iso-singet up-type and down-

type quarks, and i and j represent the generation indices. Similarly, `L and ER denote the

iso-doublet and singlet charged leptons, respectively. The matrices Mu, Md, and Me, upon

the breaking of T ′ and the electroweak symmetry, are given by [1, 2],

Mu =

iyuφ′30 (1−i

2 )yuφ′30 0

(1−i2 )yuφ

′30 yuφ

′30 + ycφ

20 ytsψ0ζ0

0 ytsψ0ζ0 yt

vu , (3.2)

Md =

0 (1 + i)ydφ0ψ′0 0

−(1− i)ydφ0ψ′0 ysψ0ζ

′0v45vd

0

ydφ0ψ′0 ydφ0ψ

′0 ybζ0

vdφ0 , (3.3)

Me =

0 −(1− i)ydφ0ψ′0 ydφ0ψ

′0

(1 + i)ydφ0ψ′0 −3ysψ0ζ

′0v45vd

ydφ0ψ′0

0 0 ybζ0

vdφ0. (3.4)

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JHEP10(2013)112

The SU(5) symmetry in the model leads to the relation Md = MTe , up to the SU(5) CG’s.

Specifically, there is an additional factor of −3 in the (2,2) entry of Me due to the coupling

to ∆45. In addition, the Georgi-Jarlskog (GJ) relations require Me,d to be non-diagonal,

leading to corrections to the TBM pattern [1, 2]. Note that the complex coefficients in

the above mass matrices arise entirely from the CG coefficients of the T ′ group theory.

More precisely, these complex CG coefficients appear in couplings that involve the doublet

representations of T ′.

In total, the mass matrices in the charged fermion sector, Mu, Md, and Me, depend

on seven independent parameters,

Mu

ytvu=

ig 1−i2 g 0

1−i2 g g + h k

0 k 1

, (3.5)

Md, MTe

ybvdφ0ζ0=

0 (1 + i)b 0

−(1− i)b (1,−3)c 0

b b 1

, (3.6)

where

b ≡ ydyb

φ0ψ′0

ζ0, c ≡ ys

yb

ψ0ζ′0

ζ0

v45

vd, k ≡ yts

ytψ0ζ0, h ≡ yc

ytφ2

0, g ≡ yuytφ′30 . (3.7)

With the numerical choice of

b = 0.00304, c = −0.0172, k = −0.0266, h = 0.00426, g = 1.45× 10−5 , (3.8)

the following mass ratios are obtained,

md : ms : mb ' θ4.6c : θ2.7

c : 1, mu : mc : mt ' θ7.5c : θ3.7

c : 1 , (3.9)

where θc '√md/ms ' 0.225 is the Cabbibo angle. These ratios in terms of powers of θc

agree with those given in [56]. We have also taken

yt/ sinβ = 1.25, ybφ0ζ0/ cosβ ' mb/mt ' 0.011, tanβ = 10 . (3.10)

The effects of the renormalization group corrections have been taken into account in our

numerical predictions, which are quoted at the electroweak scale. As a result of the GJ rela-

tions, realistic charged lepton masses are obtained. Making use of these input parameters,

the complex CKM matrix is, 0.974e−i25.4◦ 0.227ei23.1◦ 0.00412ei166◦

0.227ei123◦ 0.973e−i8.24◦ 0.0412ei180◦

0.00718ei99.7◦ 0.0408e−i7.28◦ 0.999

. (3.11)

For the three angles in the unitarity triangle, our model predictions are,

β = 23.6◦ (sin 2β = 0.734) , α = 110◦ , γ = δq = 45.6◦ , (3.12)

– 5 –

JHEP10(2013)112

(where δq is the CP phase in the standard parametrization), and they agree with the direct

measurements within 1σ of BaBar and 2σ of Belle (M. Antonelli et al in ref. [57–59]). Our

predictions for the Wolfenstein parameters are,

λ = 0.227 , A = 0.798 , ρ = 0.299 , η = 0.306 , (3.13)

which are very close to the global fit values except for ρ. The Jarlskog invariant is predicted

to be,

J ≡ Im(VudVcbV∗ubV

∗cd) = 2.69× 10−5 , (3.14)

in the quark sector and also agrees with the current global fit value. Potential direct

measurements at the LHCb for these parameters can test our predictions.

4 Neutrino masses, lepton mixings and CP phases

4.1 Neutrino masses and lepton mixings

The small neutrino masses are generated through the Type-I see-saw mechanism. The

right-handed Majorana mass matrix is given by,

MRR =

1 0 0

0 0 1

0 1 0

s0Λ

with the Dirac neutrino mass matrix being,

MD =

2ξ0 + η0 −ξ0 −ξ0 + η′′0−ξ0 2ξ0 + η′′0 −ξ0 + η0

−ξ0 + η′′0 −ξ0 + η0 2ξ0

ζ0ζ′0vu .

Without loss of generality, we have implicitly set λ1, 2, 3, 4 = 1. In the type-I seesaw

framework, after integrating out the right-handed neutrinos, the effective neutrino mass

matrix is given by,

M effν ' −MD ·M−1

RR ·MTD . (4.1)

Specifically for our model, the effective neutrino mass matrix is given in terms of the flavon

VEVs as,

M effν ' −

(ζ0ζ′0vu)2

s0Λ(4.2)

×

6ξ20 + 4ξ0η0 + η2

0 − 2ξ0η′′0 η′′20 − 3ξ2

0 + ξ0(η′′0 − 2η0) 2η0η′′0 − 3ξ2

0 + ξ0(η′′0 − 2η0)

η′′20 − 3ξ20 + ξ0(η′′0 − 2η0) ξ2

0 + 2(η0 − ξ0)(2ξ0 + η′′0) 6ξ20 + η2

0 + ξ0(η′′0 − 2η0)

2η0η′′0 − 3ξ2

0 + ξ0(η′′0 − 2η0) 6ξ20 + η2

0 + ξ0(η′′0 − 2η0) 4ξ0(η0 − ξ0) + (ξ0 − η′′0)2

,

where − (ζ0ζ′0vu)2

s0Λ is the overall neutrino mass scale. In the absence of the contribution from

the singlet scalar, η′′0 = 0, which corresponds to the exact model in [1, 2], the effective

neutrino mass matrix Mνeff is diagonalized by the TBM mixing matrix.

– 6 –

JHEP10(2013)112

With η′′0 6= 0, the neutrino diagonalization matrix deviates from the TBM matrix.

Therefore, in order to understand the structure of the diagonalization matrix, in particular

the analytic form of the deviations from TBM mixing pattern, we first multiply the effective

neutrino mass matrix by UTBM on both sides. The resulting mass matrix M effTBM = UTTBM ·

M effν · UTBM is simplified as,

M effTBM ' −

(ζ0ζ′0vu)2

s0Λ

(3ξ0 + η0 −η′′02 )2 − 3

4η′′20 0

√3

2 (2η0 − η′′0)η′′00 (η0 + η′′0)2 0

√3

2 (2η0 − η′′0)η′′0 0 −(3ξ0 − η0 +η′′02 )2 + 3

4η′′20

.

(4.3)

We immediately find that the resulting mass matrix M effTBM is diagonalizable by a further

rotation in the (1, 3)-plane

QT · UTϕ ·M effTBM · Uϕ ·Q = QT · diag(m1,m2,m3) ·Q = diag(|m1|, |m2|, |m3|) , (4.4)

with the rotation matrix Uϕ defined as,

Uϕ =

cosϕ 0 − sinϕ

0 1 0

sinϕ 0 cosϕ

, (4.5)

Q being a diagonal phase matrix, and |m1|, |m2|, and |m3| being the absolute effective

neutrinos masses. Defining the following variables,

−(ζ0ζ′0vu)2

s0Λ

((3ξ0 + η0 −

η′′02

)2 − 3η′′20

4

)≡ a ; (4.6)

−(ζ0ζ′0vu)2

s0Λ

(√3(2η0 − η′′0)

η′′02

)≡ b ; (4.7)

−(ζ0ζ′0vu)2

s0Λ

(−(3ξ0 − η0 +

η′′02

)2 +3η′′20

4

)≡ c , (4.8)

the neutrinos masses m1, m3 and the additional rotational angle ϕ can then be written as

functions of a, b and c. We find

m1 =

(a + c

2+

√(a− c)2 + 4b2

2

), (4.9)

m2 = (η0 + η′′0)2 , (4.10)

m3 =

(a + c

2−√

(a− c)2 + 4b2

2

), (4.11)

sin 2ϕ =2b√

(a− c)2 + 4b2, (4.12)

cos 2ϕ =a− c√

(a− c)2 + 4b2. (4.13)

As the overall phase for all masses is not physical, without losing any generality, we

choose the overall mass scale(ζ0ζ′0vu)2

s0Λ to be negative so that m2 is positive. For a+ c < 0,

– 7 –

JHEP10(2013)112

normal mass hierarchy is predicted; otherwise, it is inverted mass hierarchy. Furthermore,

m1 is always positive unless both a + c < 0 and ac > b2 are satisfied. Similarly, m3 is

always negative unless a + c > 0 and ac > b2 are satisfied simultaneously. The phase

matrix, Q is given by Q = diag(1, 1, eiπ) for inverted hierarchy, and Q = diag(eiπ, 1, eiπ) =

(eiπ)diag(1, e−iπ, 1) for the normal hierarchy. The overall coefficient (eiπ) for the normal

hierarchy case is an overall phase and has no physical meaning.

There are additional mass sum rules such as,

m1 +m2 +m3 = −[12ξ0

(η0 −

η′′02

)+(η0 + η′′0)2

](ζ0ζ

′0vu)2

s0Λ, (4.14)

and in terms of the absolute neutrino masses,

|m1|+ |m2|+ |m3|(ζ0ζ′0vu)2

s0Λ

(4.15)

=

12ξ0

(η0 −

η′′02

)+ (η0 + η′′0)2 ,

(m1 > 0, m3 > 0);

2√

81ξ40 + 9ξ2

0(2η20 − 2η0η′′0 − η′′20 ) + (η2

0 − η0η′′0 + η′′20 )2 + (η0 + η′′0)2 ,

(m1 > 0, m3 < 0);

−12ξ0

(η0 −

η′′02

)+ (η0 + η′′0)2 ,

(m1 < 0, m3 < 0).

which can be used to check against our numerical results. It is worth noting that some

previously found sum rules occur in models with two free parameters in the neutrino

sector. In this case, one of the masses is constrained against the other two. Our model

features three free parameters, which in turn constrains the sum of the masses to be some

combination in terms of the model parameters as shown in eq. 36. For examples of sum

rules in models with only two parameters in the neutrino sector, see [61, 62].

The full diagonalization matrix in the neutrino sector is given by UDν = UTBM ·Uϕ ·Q,

UDν =

√

23 cosϕ

√13 −

√23 sinϕ

−√

16 cosϕ−

√12 sinϕ

√13

√16 sinϕ−

√12 cosϕ

−√

16 cosϕ+

√12 sinϕ

√13

√16 sinϕ+

√12 cosϕ

·Q . (4.16)

The corresponding mixing angles are then given by,

tan2 θν13 =1− cos 2ϕ

2 + cos 2ϕ; (4.17)

tan2 θν12 =1

1 + cos 2ϕ; (4.18)

tan2 θν23 =2 + cos 2ϕ−

√3 sin 2ϕ

2 + cos 2ϕ+√

3 sin 2ϕ. (4.19)

– 8 –

JHEP10(2013)112

In the limit of ϕ = 0, these expressions coincide with the TBM predictions,

sin2 θ23 = 1/2, sin2 θ12 = 1/3, sin θ13 = 0 , (4.20)

as expected.

To obtain the full leptonic mixing matrix, the diagonalization matrix in the charged

lepton sector must be taken into account. Given the SU(5) relation, Me = MTd (up to the

SU(5) CG’s) in our model, the charged lepton mixing matrix is strongly constrained by

the quark sector. The charged lepton mixing matrix Ue,L is fixed by the SU(5) symmetry

to be,

Ue,L =

0.997ei177◦ 0.0823ei131◦ 1.31× 10−5e−i45◦

0.0823ei41.8◦ 0.997ei176circ 0.000149e−i3.58◦

1.14× 10−6 0.000149 1

, (4.21)

Up to field rephasing, the diagonalization matrix Ue,L can be approximated, at the leading

order, as

Ue,L '

1 θce−iδe/3 0

θceiδe/3 1 0

0 0 1

. (4.22)

By multiplying the neutrino mixing matrix UDν , we obtain the full lepton mixing matrix

U ` = U †e,L · UDν , which is given in the standard parametrization as follows,

UPMNS =

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13e

iδ s23c13

s12s23 − c12c23s13eiδ −c12s23 − s12c23s13e

iδ c23c13

1 0 0

0 eiα1/2 0

0 0 eiα2/2

,

(4.23)

where cij ≡ cos θij , sij ≡ sin θij are the lepton mixing angles, δ is the Dirac CP phase, and

α1, α2 are the Majorana CP phases.

To the first order, the lepton mixing angle θ13 can be approximated by,

Ue3 = sin θ13e−iδ ' θce

−iδe

3√

2cosϕ+

[θν13 + κ

θce−iδe

3

](4.24)

=θce−iδe

3√

2cosϕ+

[−√

2

3+θce−iδe

3√

6

]sinϕ ,

where θc/3√

2 arises from the charged lepton sector while θν13 = −√

2/3 sinϕ and

κ =√

1/6 sinϕ are contributions from the neutrino sector. The later two terms are the

additional contributions due to the additional singlet η′′ in our present modified model.

For small ϕ, an expansion around ϕ = 0 leads to the following analytic expressions for θ13

and δ:

θ13 ' |Ue3| 'θc

3√

2+ ϕ

[θc − 6 cos(δe)

3√

6

]+O(ϕ2, θ2

c ) , (4.25)

δ = −Arg

[Ue3

]+ π ' δe + ϕ

[2√

3 sin(δe)

θc

]+ π +O(ϕ2, θ2

c ) . (4.26)

– 9 –

JHEP10(2013)112

Similarly, the sum rule for the solar mixing angle θ12 and θc is found to be,

tan θ212 '

1

2+θc3

cos(δe) + ϕ

[θc cos(δe)

2√

3

]+O(ϕ2, θ2

c ) . (4.27)

With the full charged lepton diagonalization matrix Ue,L given in eq. (4.21), the full lepton

mixing matrix U ` = U †e,L ·UDν can be expanded as a function of cϕ ≡ cosϕ and sϕ ≡ sinϕ,

U ` '

−(0.838 + 0.0202i)cϕ (−0.539− 0.0618i) −(0.0434− 0.0388i)cϕ

−(0.0434− 0.0388i)sϕ +(0.838 + 0.0202i)sϕ(0.362− 0.0223i)cϕ (−0.605− 0.0760i) (0.703 + 0.0492i)cϕ

+(0.703 + 0.0492i)sϕ −(0.362− 0.0223i)sϕ−0.408cϕ + 0.707sϕ 0.577 0.707cϕ + 0.408sϕ

·Q .

(4.28)

The lepton mixing angles can then be expressed as the following,

tan2 θ`13 '1.010− 1.001c2ϕ − 0.102s2ϕ

1.853 + c2ϕ + 0.101s2ϕ; (4.29)

tan2 θ`12 '0.843

1.010 + c2ϕ + 0.102s2ϕ; (4.30)

tan2 θ`23 '1.887 + 1.098c2ϕ − 1.522s2ϕ

2.001 + c2ϕ + 1.733s2ϕ. (4.31)

where c2ϕ ≡ cos 2ϕ and s2ϕ ≡ sin 2ϕ. By applying the various constraints from the mixing

angles summarized in table. (1) up to 2σ confidence level, we obtain the following range of

allowed values for the the angle ϕ. For the normal mass hierarchy (NH) and inverted mass

hierarchy (IH), the range of allowed values for tan 2ϕ are, respectively,

− 0.337 . tan 2ϕ . −0.218(NH); −0.344 . tan 2ϕ . −0.228(IH) . (4.32)

These correspond to the following values for the angle ϕ,

− 9.305◦ . ϕ . −6.155◦(NH); −9.501◦ . ϕ . −6.414◦(IH) . (4.33)

We scan the parameter space of the neutrino flavon VEVs (ξ0, η0, η′′0), and find a

volume satisfying the lepton mixing angles and the mass squared splittings up to the 2σ

accuracy. These allowed regions are shown in the following two figures (figure 1). The

left panel corresponds to the normal mass hierarchy while the right one illustrates the

allowed range for the inverted mass hierarchy. Therefore, both NH and IH are allowed

in this model, which is different from the prediction in the model [1, 2] where only NH

exists. However, from the figure 1, we notice that it is much easier to satisfy all of the

experimental observations in the NH than the IH in this model. More interestingly, we

notice that roughly the η′′0 parameter depends on the parameter ξ0 linearly when η0 � 1

and it is due to the constraints on the ϕ rotational angle. It can be understood from the

relation tan 2ϕ = (√

3(2η0− η′′0)η′′0)/(18ξ20 + 2η2

0 − 2η0η′′0 − η′′20 ) while η0 → 0, which means,

(18 tan 2ϕ)ξ20 = (tan 2ϕ−

√3)η′′20 . (4.34)

– 10 –

JHEP10(2013)112

Figure 1. The allowed parameter spaces of the neutrino flavon VEVs (ξ0, η0, η′′0 ) consistent with

the global fits of the lepton mixing angles and the mass squared splittings within the 2σ range. The

left and right panel correspond to the normal neutrino mass hierarchy and inverted neutrino mass

hierarchy respectively.

We can notice the linear relationship between ξ0 and η′′0 immediately with a fixed tan 2ϕ,

which is consistent with the eq. (4.32) where tan 2ϕ is close to a constant. The other section

shown in figure 1 can be understood as a conical section with an elliptical projection. For

fixed ϕ, one can also find the relation:

18 tan(2ϕ)ξ20 +

3

4

(√3− 4√

3− tan(2ϕ)+ 3 tan(2ϕ)

)η2

0

=(

tan(2ϕ)−√

3)(

η′′0 +tan(2ϕ) +

√3

2(tan(2ϕ)−√

3)η0

)2

. (4.35)

Restrictions on these surfaces are determined by mass splitting constraints.

Given that, with a fixed Ue,L, all three leptonic mixing angles and the Dirac phase are

determined by one variable, ϕ, there exist correlations among the three mixing angles and

phase. The correlations are almost identical for NH and IH, and we show the results found

in the parameter scan in figure 2. Specifically, figure 2 shows the correlations between θ12

and θ23 versus θ13.

In figure 2, the blue and red point in the left and right panels demonstrate the relations

between the mixing angle θ12, θ23 and θ13 respectively. Blue (red) points represent the a

normal (inverted) hierarchy solution. The solid black lines in the left and right panels

correspond to 35.26◦ and 45◦ respectively while the dotted line are the 2σ limits on the

mixing angles θ12 and θ23 from the global fit. From figure 2, we find that for both NH

and IH, the mixing angle θ12 tends to be smaller than 35.26◦ and as the mixing angle

θ13 increases, the deviation decreases, even though the variation in θ12 is very suppressed.

However, the trend of the mixing angle θ23 behaves in an opposite way, and it is higher

than 45◦ and the deviation increases as θ13 is bigger.

– 11 –

JHEP10(2013)112

Figure 2. The distribution of the deviation from TBM mixing vs θ13 this model. Points in the

left panel are θ12 vs. θ13 and the points in the right panel are θ23 vs θ13, where blue a blue point

indicates a NH solution and a red point indicates an IH solution. The solid black lines correspond

to 35.26◦ and 45◦ and the dotted lines are the 2σ limits on θ12 and θ23 from the global fit in the

left and right panel respectively.

Among the large parameter space, we illustrate one numerical example with minimal

χ2 = 2.643. The VEVs in the neutrino sector are fixed to be,

ξ0 = 0.185; η0 = −0.865; η′′0 = 0.200 , (4.36)

leading to the following leptonic mixing angles,

θ12 = 33.351◦, θ23 = 49.162◦, θ13 = 9.189◦. (4.37)

In addition, we choose the overall mass scale of the neutrinos, − (ζ0ζ′0vu)2

s0Λ = 0.0217 eV, and

therefore the neutrino masses are,

m1 = 0.00398 eV, m2 = 0.00959 eV, m3 = −0.0505 eV , (4.38)

corresponding to normal neutrino mass hierarchy with,

∆m2atm = 2.53× 10−3 eV2, ∆m2

� = 7.62× 10−5 eV2 . (4.39)

So all mixing angles and mass squared splittings are consistent with the experimental

measurements in the 2σ region. Furthermore, the neutrino mass sums are,

|m1|+ |m2|+ |m3| = 0.0640 eV , m1 +m2 +m3 = −0.0369 eV . (4.40)

We find that our numerical results for the neutrino masses, mixing angles as well as the

mass sum rules are consistent with the analytical expressions derived above.

4.2 Lepton Dirac CP phase δ, Jarlskog invariant J`, neutrinoless double beta

decay matrix element 〈mββ〉

There are still many fundamental properties of the neutrinos that are unknown at present.

These include the Dirac CP phase, the two Majorana phases (if neutrinos are Majorana

fermions), the Dirac versus Majorana nature, and the mass hierarchy. With the large value

– 12 –

JHEP10(2013)112

for the θ13 lepton mixing angle, the measurement of the Dirac CP phase is attainable in

the future. The parametrization independent CP violation measure, the Jarlskog invariant

J`, in the lepton sector, is related to the Dirac CP phase and is given by,

J` ≡ Im[U `µ3U`∗e3U

`e2U

`∗µ2] = 3.338× 10−5 − 0.00967c2ϕ + 0.00554s2ϕ . (4.41)

J` is a function of variable ϕ, our additional rotational matrix in the neutrino sector. Within

the range that satisfies the mixing angle measurements at 2σ level we derived above, the

Jarlskog invariant J` for both NH and IH is restricted to the region,

− 0.0109 . J` . −0.0106 (NH/IH). (4.42)

From the Jarlskog invariant J`, we can further determine the Dirac CP phase. Comparing

with the original Dirac CP phase derived in [60], the additional rotational matrix Uϕ does

not modify the Dirac CP phase significantly, especially our ϕ rotational angle is small,

therefore, the Dirac CP phase is still expected to be close to 5/4π (225◦). Including

the effects from the ϕ rotational angle, within the 2σ allowed range for ϕ, the Dirac CP

phase can be obtained via J` = (sin(2θ13) sin(2θ23) sin(2θ12) cos θ13 sin δ)/8 and we find it

is restricted to,

195.900◦ . δ . 200.139◦ (NH); 195.703◦ . δ . 199.692◦ (IH). (4.43)

As this model assumes neutrinos are Majorana fermions, in addition to the Dirac CP

phase there are two additional Majorana CP phases. One way to test it is through the

neutrinoless double beta decay ((ββ)0ν-decay). The rate is proportional to the variable,

neutrinoless double beta decay matrix element defined as,

〈mββ〉 =

∣∣∣∣∣3∑i=1

(U `ei)2|mi|

∣∣∣∣∣ . (4.44)

In this model, it can be written as,

〈mββ〉 '∣∣∣(f1 + f2c2ϕ + f3s2ϕ)|m1|+ f4|m2|+ (f1 − f2c2ϕ − f3s2ϕ)|m3|

∣∣∣ (4.45)

with the following coefficients defined as,

f1 = (0.351 + 0.0153i); f2 = (0.351 + 0.0186i); (4.46)

f3 = (0.0371− 0.0316i); f4 = (0.287− 0.0667i).

From the eq. (4.45), the coefficients in front of the masses mi (i = 1, 2, 3) are constants or

close to constants due to the narrow allowed region on the rotational angle ϕ, therefore,

a nearly linear dependence on the neutrinos masses is expected. Shown in figure 3 is the

neutrinoless double beta decay matrix element 〈mββ〉 for the normal mass hierarchy, and

each point corresponds to the data point in the left panel of figure 1 which satisfies all

neutrino mixing angles and mass squared splittings experimentally. Note that the variable

〈mββ〉 roughly depends on the neutrino masses linearly. In addition, we realize there

– 13 –

JHEP10(2013)112

Figure 3. The neutrinoless double beta decay matrix element verse 〈mβ〉 ≡√∑3

i=1 |Uei|2|mi|2,

M ≡ |m1|+ |m2|+ |m3| and mmin ≡ min(|m1|, |m2|, |m3|). The blue points are predictions for the

normal mass hierarchy, while the red points are predictions for the inverted hierarchy. The black

line is the projected reach of the Majorana Demonstrator [64].

are many data points with 〈mββ〉 ∼ 10−2 eV, which is still below the sensitivity range of

various neutrinoless double beta decay experiments [63]. The black solid line in the figure 3

represents the experimental reach of 〈mββ〉 ' 3×10−2 eV with 3 year data at the Majorana

Demonstrate [64].

One specific example with minimal χ2 predicts the Dirac CP phase to be,

δ = 197.734◦ . (4.47)

In addition, the leptonic Jarlskog is calculated to be J` = −0.0108 and the neutrinoless

double beta decay matrix element 〈mββ〉 = 0.00662 eV.

– 14 –

JHEP10(2013)112

5 Conclusion

We modify the SU(5)×T ′ GUT model proposed in [1, 2] with the additional singlet field η′′,

and a larger value for the lepton mixing angle θ13 ' 8◦ − 10◦ can be accommodated. The

analytical expressions of the lepton mixing angles and neutrino masses as well as various

experimental observables including the Dirac CP phase δ, Jarlskog invariant J` and the

neutrinoless double beta decay matrix element 〈mββ〉 are derived. We present the numerical

results in this letter and find large parameter space can satisfy all current experimental

constraints. We also learn that it is easier to realize normal neutrino mass hierarchy than

inverted neutrino mass hierarchy in this model. The illustrated numerical results of the

model with the minimum χ2 fitting are consistent with our analytical expressions.

Acknowledgments

We thank Michael Ratz for useful comments. The work of M-CC and A.M.W. was sup-

ported, in part, by the National Science Foundation under Grant No. PHY-0970173. The

work of KTM was supported, in part, by the Department of Energy under Grant No. DE-

FG02-04ER41290. JH is supported by the DOE Office of Science and the LANL LDRD

program. A.M.W. would also like to acknowledge the hospitality of the TASI Summer

School, where part of this work was completed.

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