Non-Minimal Flavored ${\bf S}_{3}\otimes {\bf Z}_{2} $ Left-Right Symmetric Model

0.53 MB·

by  Juan Carlos Gómez-Izquierdo

Non-Minimal Flavored S ⊗ Z Left-Right Symmetric Model
3 2
Juan Carlos Go´mez-Izquierdo1,2,3∗
1 Tecnologico de Monterrey, Campus Estado de Mexico, Atizapan de Zaragoza, Estado de
Mexico, Apartado Postal 52926, Mexico.
2 Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de M´exico, M´exico 3000,
D.F., M´exico.
7 3Instituto de F´ısica, Universidad Nacional Auto´noma de M´exico, M´exico 01000, D.F.,
1
M´exico.
0
2

n
a
J Abstract

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We propose a non minimal left-right symmetric model with Parity Symmetry where
1
the fermion mixings arise as result of imposing an S ⊗ Z flavor symmetry, and an
3 2
] extra Ze symmetry is considered to suppress some Yukawa couplings in the lepton sec-
h 2
p tor. As consequence, in the lepton sector, the effective neutrino mass matrix possesses
- approximatelytheµ−τ symmetry. Thebreakingofµ−τ symmetryinducessizablenon
p
e zero θ , and the deviation of θ from 45◦ is strongly controlled by an (cid:15) free parame-
13 23
h
ter and active neutrino masses. So that, we have constrained the allowed region for (cid:15)
[
parameter and the unknown neutrino masses, and at the same time, an exploration on

2
the extreme Majorana phases is done since these turn out being relevant to enhance or
v
7 suppress the reactor and atmospheric angles. As a result, the normal hierarchy is ruled
4 out since the reactor angle turns out too small, for any values of the Majorana phases.
7
For the inverted and degenerate hierarchy, there is one combination for the extreme
1
0 phases where the reactor and atmospheric angles may be compatible with their allowed
1. values but the parameters space is tight for the former ordering.
0
7
1
:
v
i
X
r
a
∗email: [email protected]
1 Introduction
Currently, we know that neutrinos oscillate and have a tiny mass. In the theoretical frame-
work of three active neutrinos, the difference of the squared neutrino masses for normal
(cid:0) (cid:1) (cid:0) (cid:1)
(inverted) hierarchy are given by ∆m2 10−5eV2 = 7.60+0.19, and |∆m2 | 10−3eV2 =
21 −0.18 31
2.48+0.05 (2.38+0.05). Additionally, we have the values of the mixing angles sin2θ /10−1 =
−0.07 −0.06 12
3.23±0.16,sin2θ /10−1 = 5.67+0.32 (5.73+0.25)andsin2θ /10−2 = 2.26±0.12(2.29±0.12)[1].
23 −1.24 −0.39 13
There is no yet solid evidence on the Dirac CP-violating phase.
A simplest route to include small neutrino masses and mixings to the Standard Model
(SM) is to add the missing right-handed neutrinos (RHN’s) states to the matter content, and
then invoking the see-saw mechanism [2–8]. However, we should point out that the RHN
mass scale is introduced by hand with no relation whatsoever to the Higgs mechanism that
gives mass to all other fields. Nonetheless, this problem may be alleviated if the minimal
extension of the SM is replaced by the left-right symmetric model (LRSM) [5,9–12] where
the RHN’s are already included in the matter content. Additionally, the see-saw mechanism
comes in rather naturally in the context of left-right symmetric scenarios; aside from other
nice features, as for instance the recovery of Parity Symmetry, and the appearance of right-
handed currents at high energy, which also makes such extensions very appealing. Recently,
the left-right scenarios have been revised [13–21] in order to make contact with the last
experimental data of LHC. Moreover, the dark matter problem [22,23] and the diphoton
excess anomaly [24–27] have been explored in this kind of scenarios.
Focusing in the peculiar neutrino mixing pattern (besides the CKM mixing matrix), the
mass textures have played an important role in trying to solve this puzzle [28]. Actually,
several discrete symmetries have been proposed [29–31] to get in an elegant way the mass
textures. In this line of thought, the S flavor symmetry, in particular, is a good candidate to
3
handle the Yukawa couplings for leptons and quarks; and this has been studied exhaustively
in different frameworks [32–53]. In most of these works, the meaning of the flavor has been
extended to the scalar sector such that three Higgs doublets are required to accommodate
the PMNS and CKM mixing matrices.
Therefore, we propose a non-minimal LRSM with Parity Symmetry where the fermion
mixings arise as result of imposing an S ⊗Z flavor symmetry, and an extra Ze symmetry is
3 2 2
considered to suppress some Yukawa couplings in the lepton sector (see [54,55] for flavored
LRSM).Additionally, anonconventionalassignmentisdoneforthemattercontentunderthe
S symmetry and this is the clear difference between the previous studies and this one. As
3
consequence, in the lepton sector, the effective neutrino mass matrix possesses approximately
the µ−τ symmetry [56–69]. The breaking of the µ−τ symmetry induces sizable non zero
θ , and the deviation of θ from 45◦ is strongly controlled by an (cid:15) free parameter and active
13 23
neutrino masses. So that, we have constrained the allowed region for (cid:15) parameter and the
unknown neutrino masses, and at the same time, an exploration on the extreme Majorana
phases is done since these turn out to be relevant to enhance or suppress the reactor and
atmospheric angle. As results, the normal hierarchy is ruled out since the reactor angle
comes out being tiny, for any values of the Majorana phases. For the inverted and degenerate
hierarchy, there is one combination for the extreme phases where the reactor and atmospheric
angles may be compatible with their allowed values but the parameter space is tight for the
former ordering. The quark sector will be discussed exhaustively in a future work, however,
1
some preliminary results will be commented. The paper is organized as follows: we present,
in Sec. II, the matter content of the model and also their respective assignment under the
S symmetry. In addition, we briefly explain the scalar sector and argue about the need to
3
include the Ze symmetry. In Sec. III, the fermion mass matrices are obtained and we put
2
attention on the lepton sector for getting the mixing matrices. We present, in Sec. IV, the
PMNS matrix that the model predicts. Finally, we present an analytic study on the mixing
angles and our results in Sec. V, and we close our discussion with a summary of conclusions.
2 Flavored Left-Right Symmetric Model
The left-right model we shall consider along our discussion is based on the usual, SU(3) ⊗
c
SU(2) ⊗SU(2) ⊗U(1) , gauge symmetry where Parity Symmetry, P, is assumed to be a
L R B−L
symmetry a high energy but it is broken at electroweak scale since there are no right-handed
currents. The matter fields and their respective quantum numbers (in parenthesis) under the
gauge symmetry are given by
(cid:18) (cid:19) (cid:18) (cid:19)
u ν
Q = ∼(3,(2,1),(1,2),1/3), (L,R)= ∼(1,(2,1),(1,2),−1),
(L,R) d (cid:96)
(L,R) (L,R)
(cid:32) φ0 φ(cid:48)+ (cid:33) (cid:32) δ+ δ++ (cid:33)
Φ= ∼(1,2,2,0); ∆ = 2 ∼(1,(3,1),(1,3),2). (1)
φ− φ(cid:48)0 (L,R) δ0 −δ+
2 (L,R)
On the other hand, the non-Abelian group S is the permutation group of three objects
3
and this has three irreducible representations: two 1-dimensional, 1 and 1 , and one 2-
S A
dimensional representation, 2 (for a detailed study see [29]). The multiplication rules among
them are
1S⊗1S =1S, 1S⊗1A=1A, 1A⊗1S =1A, 1A⊗1A=1A,
1S⊗2=2, 1A⊗2=2, 2⊗1S =2, 2⊗1A=2;
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
aa12 2⊗ bb12 2=(a1b1+a2b2)1S ⊕(a1b2−a2b1)1A⊕ aa11bb21+−aa22bb12 2. (2)
Having commented briefly the gauge group and the non-Abelian group, then we will consider
threeHiggsbidoubletsbutthreeleft-righttripletshavebeenaddedwiththepurposeofgetting
mixings in the lepton sector. In addition, the quark and lepton families have been assigned
in a different way under the irreducible representation of S . Explicitly, for the former and
3
the Higgs sector respectively, the first and second family have been put together in a flavor
doublet 2, and the third family is a singlet 1 . On the contrary, for the latter sector, the
S
first family is a singlet 1 and the second and third families are put in a doublet 2. The
S
advantage of making this choice is that the quark mass matrices may be put into two mass
textures fashion that fit the CKM matrix very well; in the lepton sector the appearance of
the approximated µ−τ symmetry in the effective neutrino mass matrix helps to obtain good
results for the mixings.
The matter content of the model transforms in a not trivial way under the S symmetry
3
and this is displayed in the table below. Here, the Z symmetry has been added in order to
2
prohibit some Yukawa couplings in the lepton sector. Thus, the most general Yukawa mass
2
Matter Q Q (L ,R ) (L ,R ) Φ Φ ∆ ∆
I(L,R) 3(L,R) 1 1 J J I 3 I(L,R) 3(L,R)
S 2 1 1 2 2 1 2 1
3 S S S S
Z 1 1 1 −1 1 1 −1 1
2
Table 1: Non-minimal left-right model. Here, I = 1,2 and J = 2,3.
term, that respects the S ⊗Z flavour symmetry and the gauge group, is given as
3 2
−LY =y1q(cid:2)Q¯1L(Φ1Q2R+Φ2Q1R)+Q¯2L(Φ1Q1R−Φ2Q2R)(cid:3)+y2q(cid:2)Q¯1LΦ3Q1R+Q¯2LΦ3Q2R(cid:3)+y3q(cid:2)Q¯1LΦ1+Q¯2LΦ2(cid:3)Q3R
(cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105)
+y4qQ¯3L[Φ1Q1R+Φ2Q2R]+y5qQ¯3LΦ3Q3R+y˜1q Q¯1L Φ˜1Q2R+Φ˜2Q1R +Q¯2L Φ˜1Q1R−Φ˜2Q2R
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
+y˜2q Q¯1LΦ˜3Q1R+Q¯2LΦ˜3Q2R +y˜3q Q¯1LΦ˜1+Q¯2LΦ˜2 Q3R+y˜4qQ¯3L Φ˜1Q1R+Φ˜2Q2R +y˜5qQ¯3LΦ˜3Q3R+y1(cid:96)L¯1Φ3R1
+y2(cid:96)(cid:2)(L¯2Φ2+L¯3Φ1)R2+(L¯2Φ1−L¯3Φ2)R3(cid:3)+y3(cid:96)(cid:2)L¯2Φ3R2+L¯3Φ3R3(cid:3)+y˜1(cid:96)L¯1Φ˜3R1
(cid:104) (cid:105) (cid:104) (cid:105)
+y˜2(cid:96) (L¯2Φ˜2+L¯3Φ˜1)R2+(L¯2Φ˜1−L¯3Φ˜2)R3 +y˜3(cid:96) L¯2Φ˜3R2+L¯3Φ˜3R3 +y1LL¯1∆3LLc1+y2LL¯1[∆1LLc2+∆2LLc3]
+y3L(cid:2)L¯2∆1L+L¯3∆2L(cid:3)Lc1+y4L(cid:2)L¯2∆3LLc2+L¯3∆3LLc3(cid:3)+y1RR¯1c∆3RR1+y2RR¯1c[∆1RR2+∆2RR3]
+y3R(cid:2)R¯2c∆1R+R¯c∆2R(cid:3)R1+y4R(cid:2)R¯2c∆3RR2+R¯3c∆3RR3(cid:3)+h.c, (3)
where Φ˜ = −iσ Φ∗iσ . On the other hand, in here Parity Symmetry will be assumed in the
i 2 i 2
above Lagrangian. This requires that Ψ ↔ Ψ , Φ ↔ Φ† and ∆ ↔ ∆† for fermions and
iL iR i i iL iR
scalar fields, respectively. Thereby, the Yukawa couplings may reduce substantially and the
gauge couplings too. In here, we stress that an extra symmetry Ze is used to get a diagonal
2
charged lepton and Dirac neutrino mass matrix whereas the Majorana mass matrices retain
their forms. Explicitly, in the above Lagrangian, we demand that
L ↔−L , R ↔−R , ∆ ↔−∆ , ∆ ↔−∆ . (4)
3 3 3 3 2L 2L 2R 2R
so that some terms are absent in the lepton sector. On the other hand, in this kind of models
the spontaneous symmetry breaking is as follows: Parity Symmetry is broken at the same
scale where the ∆ right-handed scale acquires its vacuum expectation value (vev). At the
R
first stage, the RHN’s are massive particles, then the rest of the particles turn out massive
since the Higgs scalars get their vevs. Actually, the S flavor symmetry is broken by the Φ
3 i
three scalar bidoublets whose vev’s are given as
(cid:104)∆L,R(cid:105)=(cid:18) vL0,R 00 (cid:19), (cid:104)Φi(cid:105)=(cid:18) k0i k0i(cid:48) (cid:19), (cid:104)Φ˜i(cid:105)=(cid:18) k0i(cid:48)∗ k0i∗ (cid:19). (5)
We ought to comment that the scalar potential will not be analyzed for the moment, since
we are interested in masses and mixings for fermions. However, this study has to be done
eventually since is crucial for theoretical and phenomenological purpose.
The Yukawa mass term is given by
1 1
−LY =q¯iL(Mq)ijqjR+(cid:96)¯iL(M(cid:96))ij(cid:96)jR+ 2ν¯iL(Mν)ijνjcL+ 2ν¯icR(MR)ijνjR+h.c. (6)
where the type I see-saw mechanism has been realized, M = −M M−1MT; so that the
ν D R D
M were neglected for simplicity. Therefore, the mass matrices can be read of Eq.(3) and
L
have the following form
aq+b(cid:48)q bq cq a(cid:96) 0 0  a(L,R) b(L,R) b(cid:48)(L,R)
Mq = bq aq−b(cid:48)q c(cid:48)q, M(cid:96)=0 b(cid:96)+c(cid:96) 0 , M(L,R)=b(L,R) c(L,R) 0 , (7)
fq fq(cid:48) gq 0 0 b(cid:96)−c(cid:96) b(cid:48)(L,R) 0 c(L,R)
3
where the q = u,d and (cid:96) = e,ν . Explicitly, the matrix elements for quarks and lepton
D
sector are given as
au=y2qk3+y˜2qk3(cid:48)∗, b(cid:48)u=y1qk2+y˜1qk2(cid:48)∗, bu=y1qk1+y˜1qk1(cid:48)∗, cu=y3qk1+y˜3qk1(cid:48)∗,c(cid:48)u=y3qk2+y˜3qk2(cid:48)∗, fu=y3†qk1+y˜3†qk1(cid:48)∗,
fu(cid:48) =y3†qk2+y˜3†qk2(cid:48)∗, gu=y5qk3+y˜5qk3(cid:48)∗, ad=y2qk3(cid:48) +y˜2qk3∗, b(cid:48)d=y1qk2(cid:48) +y˜1qk2∗, bd=y1qk1(cid:48) +y˜1qk1∗, cd=y3qk1(cid:48) +y˜3qk1∗;
c(cid:48)d=y3qk2(cid:48) +y˜3qk2∗, fd=y3†qk1(cid:48) +y˜3†qk1∗, fd(cid:48) =y3†qk2(cid:48) +y˜3†qk2∗, gu=y5qk3(cid:48) +y˜5qk3∗, aD =y1(cid:96)k3+y˜1(cid:96)k3(cid:48)∗, bD =y3(cid:96)k3+y˜3k3(cid:48)∗,
cD =y2(cid:96)k2+y˜2k2(cid:48)∗, ae=y1(cid:96)k3(cid:48) +y˜1(cid:96)k3∗, be=y3(cid:96)k3(cid:48) +y˜3k3∗, ce=y2(cid:96)k2(cid:48) +y˜2k2∗, a(L,R)=y1Rv1(L,R), b(L,R)=y2Rv2(L,R)
b(cid:48) =yRv , c =yRv . (8)
(L,R) 2 3(L,R) (L,R) 4 1(L,R)
where Parity symmetry has been considered. Moreover, we will end up having a complex
symmetric (diagonal) quark (lepton) mass matrix if the vev’s are complex; in the literature
this scenario is well known as pseudomanifest left-right symmetry. If the vev’s are real,
the quark (lepton) mass matrix is hermitian (real) and the number of CP phases are reduced,
this framework is known as manifest left-right symmetry. In this work, we will discuss
only the first framework and the second one will be studied in an extended version of the
model and its consequences on the quark sector.
3 Masses and Mixings
In principle, in the mass matrices, we can reduce a further the number of free parameters
considering certain alignment in the vev’s, see Eq.(8). So that, for the moment, we will
assume that the vev’s of Φ and Φ are degenerate. Explicitly, we demand that k = k ≡ k
1 2 1 2
and k(cid:48) = k(cid:48) ≡ k(cid:48). Additionally, v = v = v . Therefore, we have:
1 2 1R 2R R
Pseudomanisfest left-right theory.
aq+bq bq cq a(cid:96) 0 0  a(L,R) b(L,R) b(L,R)
Mq = bq aq−bq cq, M(cid:96)=0 b(cid:96)+c(cid:96) 0 , M(L,R)=b(L,R) c(L,R) 0 . (9)
cq cq gq 0 0 b(cid:96)−c(cid:96) b(L,R) 0 c(L,R)
Manifest left-right theory.
aq+bq bq cq a(cid:96) 0 0  a(L,R) b(L,R) b(L,R)
Mq = bq aq−bq cq, M(cid:96)=0 b(cid:96)+c(cid:96) 0 , M(L,R)=b(L,R) c(L,R) 0 . (10)
c∗q c∗q gq 0 0 b(cid:96)−c(cid:96) b(L,R) 0 c(L,R)
As was already commented the full analysis of the quark masses and mixings will be left
aside for this moment. However, we just make some comments. In the pseudomanifest
framework, the M mass matrix may be put into two mass textures fashion that fit the CKM
q
matrix very well. In similar way, the manifest framework is tackled. For this case, the quark
mixing matrix has fewer free parameters than the above framework since this is hermitian;
the study, and its predictions on the mixing angles is work in progress.
3.1 Lepton Sector
The M mass matrix is complex and diagonal then one could identify straight the physical
e
masses, however, we will make a similarity transformation in order to prohibit a fine tuning
in the free parameters. What we mean is the following, the M mass matrix is diagonalized
e
4
by U = S P and U = S P†, this is, Mˆ = diag.(|m |,|m |,|m |) = U† M U =
eL 23 e eR 23 e e e µ τ eL e eR
P†m P† with m = ST M S . After factorizing the phases, we have m = P m¯ P where
e e e 23 e 23 e e e e
 
1 0 0
me=diag.(me,mµ,mτ), S23=0 0 1, Pe=diag.(eiηe,eiηµ,eiητ) (11)
0 1 0
As result, one obtains |m | = |a |, |m | = |b −c | and |m | = |b +c |.
e e µ e e τ e e
On the other hand, the M effective neutrino mass matrix is given as
ν
 Xa2D −aDY(bD+cD) −aDY(bD−cD) X −Y −Y
Mν =−aDY(bD+cD) W(bD+cD)2 Z(b2D−c2D)  where M−R1≡−Y W Z  (12)
−aDY(bD−cD) Z(b2D−c2D) W(bD−cD)2 −Y Z W
Now as hypothesis, we will assume that b is larger than c , in this way the effective mass
D D
matrix can be written as
 
Aν −Bν(1+(cid:15)) −Bν(1−(cid:15))
Mν ≡−Bν(1+(cid:15)) Cν(1+(cid:15))2 Dν(1−(cid:15)2) (13)
−Bν(1−(cid:15)) Dν(1−(cid:15)2) Cν(1−(cid:15))2
where A ≡ Xa2 , B ≡ Ya b , C ≡ Wb2 and D ≡ Zb2 are complex. Besides, (cid:15) ≡ c /b
ν D ν D D ν D ν D D D
is complex too. Here, we want to stress that the last parameter will be considered as a
perturbationtotheeffectivemassmatrixsuchthat|(cid:15)| ≪ 1. Tobemorespecific,0 ≤ |(cid:15)| ≤ 0.3
inordertobreaksoftlytheµ−τ symmetry. Sothat, hereafter, wewillneglectthe(cid:15)2 quadratic
terms in the above matrix. Having done this, we go back to the effective neutrino mass
matrix. In order to cancel the S contribution that comes from the charged lepton sector,
23
we make the following to M . We know that Mˆ = diag.(m ,m ,m ) = U†M U∗, then
ν ν ν1 ν2 ν3 ν ν ν
U = S U where the latter mixing matrix will be obtained below. Then, Mˆ = U†M U∗
ν 23 ν ν ν ν ν
with
 
Aν −Bν(1−(cid:15)) −Bν(1+(cid:15))
Mν ≈−Bν(1−(cid:15)) Cν(1−2(cid:15)) Dν  (14)
−Bν(1+(cid:15)) Dν Cν(1+2(cid:15))
When the (cid:15) parameter is switched off, the effective mass matrix, which is denoted by M0,
ν
possesses the µ−τ symmetry and this is diagonalized by
cosθν ei(ην+π) sinθν ei(ην+π) 0 
U0= −si√nθν co√sθν −√1  (15)
ν  2 2 2
−si√nθν co√sθν √1
2 2 2
where the M0 matrix elements are fixed in terms of the complex neutrinos physical masses,
ν
the θ free parameter and the η Dirac CP phase. To be more explicit,
ν ν
√
Aν =(m0ν1cos2θν+m0ν2sin2θν)e2i(ην+π), 2Bν =−cosθνsinθν(m0ν2−m0ν1)ei(ην+π);
1 1
Cν = 2(m0ν1sin2θν+m0ν2cos2θν+m0ν3), Dν = 2(m0ν1sin2θν+m0ν2cos2θν−m0ν3). (16)
Including the (cid:15) parameter we can write the effective mass matrix as M = M0 +M(cid:15); the
ν ν ν
second matrix contains the perturbation, then, when we apply U0 one gets M = U0†(M0+
ν ν ν ν
M(cid:15))U0∗. Explicitly
ν ν
Mν =Diag.(m0ν1,m0ν2,m0ν3)+−sinθν(m000ν3+m0ν1)(cid:15) cosθν(m0ν03+m0ν2)(cid:15) −cosisnθθνν(m(m0ν00ν33++mm0ν0ν21)(cid:15))(cid:15) (17)
5
The contribution of second matrix to the mixing one is given by
 
N1 0 −N3sinθr1(cid:15)
Uν(cid:15) ≈ 0 N2 N3cosθνr2(cid:15) (18)
N1sinθνr1(cid:15) −N2cosθνr2(cid:15) N3
where we have defined the complex mass ratios r ≡ (m0 + m0 )/(m0 − m0 ).
(1,2) ν3 ν(1,2) ν3 ν(1,2)
Besides, N , N and N are the normalization factors which are given as
1 2 3
N1=(cid:0)1+sin2θν|r1(cid:15)|2(cid:1)−1/2, N2=(cid:0)1+cos2θν|r2(cid:15)|2(cid:1)−1/2, N3=(cid:0)1+sin2θν|r1(cid:15)|2+cos2θν|r2(cid:15)|2(cid:1)−1/2. (19)
Finally, the effective mass matrix given in Eq.(13) is diagonalized approximately by U ≈
ν
S U0U(cid:15). Therefore, the theoretical PMNS mixing matrix is written as V = U† U =
23 ν ν PMNS eL ν
P†U0U(cid:15).
e ν ν
4 PMNS Mixing Matrix
The PMNS mixing matrix is given explicitly as
 cosθνN1ei(ην+π) sinθνN2ei(ην+π) sin2θνN23(r2−r1)(cid:15)ei(ην+π)
VPMNS =P†e−si√n2θνN1(1+r1(cid:15)) co√s2θνN2(1+r2(cid:15)) −N√32[1−(cid:15)r3]  (20)
−si√n2θνN1(1−r1(cid:15)) co√s2θνN2(1−r2(cid:15)) N√32[1+(cid:15)r3]
where r ≡ r cos2θ + r sin2θ . On the other hand, comparing the magnitude of entries
3 2 ν 1 ν
V with the mixing matrix in the standard parametrization of the PMNS, we obtain
PMNS
the following expressions for the lepton mixing angles
sin2θ13=|V13|2= sin242θνN32|(cid:15)|2 |r2−r1|2, sin2θ23= 1−|V|V231|23|2 = N2321|1−−si(cid:15)n2r3θ|123,
sin2θ12= 1−|V|V121|23|2 = 1N−22ssiinn22θθ1ν3. (21)
We have to point out that the reactor and atmospheric angles depend strongly on the active
neutrino masses ratios. At the same time, the reactor angle does not depend on the phase
of the parameter (cid:15), but on the other hand, the atmospheric one has a clear dependency on
this phase.
5 Analytic Study and Results
In order to make an analytic study on the above formulas, let us emphasize that we are
working in a perturbative regime which means that 0 ≤ |(cid:15)| ≤ 0.3. Then N normalization
i
factors should be the order of 1 so that, as is usual in models where the µ−τ symmetry is
broken softly, the solar angle is directly related to the free parameter θ , as can be seen in
ν
Eq. (21). Therefore, at the leading order we have that
sin2θ = sin2θ , then, θ = θ . (22)
12 ν 12 ν
Additionally, we will analyze the extreme Majorana phases for the complex neutrino masses
for each hierarchy. What we mean by extreme phases is that these can be either 0 or π.
6
Explicitly, m0 = ±|m0 |, for i = 1,2,3, where |m0 | is the absolute mass. As we will see,
νi νi νi
these phases can be relevant to enhance or suppress the reactor and atmospheric angles.
Depending of the spectrum hierarchy there is one neutrino mass which is undetermined so
that this will be constrained.
Normal hierarchy. From experimental data, the absolute neutrino masses are |m0 | =
(cid:113) (cid:113) ν3
∆m2 +|m0 |2 and |m0 | = ∆m2 +|m0 |2. Now, the mass ratios r , r and r can be
31 ν1 ν2 21 ν1 1 2 3
approximated as follows
m0 m0 m0
r1≈1+2m0ν1 ≈1, r2≈1+2mν02, r3≈1+2mν02 cos2θν (23)
ν3 ν3 ν3
as results of this, we obtain
sin2θ13≈sin22θν|(cid:15)|2(cid:32)mm0ν0ν32(cid:33)2, sin2θ23≈ 12|1−(cid:15)(cid:0)1+21(m−0νs2in/2mθ0ν133)(cid:1)cos2θν)|2. (24)
As can be noticed, if the strict normal hierarchy is assumed then the reactor angle comes
out being very small since m0 /m0 ≈ ∆m2 /∆m2 , and |(cid:15)| ≪ 1. This holds for any extreme
ν2 ν3 21 31
Majorana phases in the neutrino masses and this result does not change substantially if the
m0 is non-zero. Therefore, the normal spectrum is ruled out for 0 ≤ |(cid:15)| ≤ 0.3.
ν1 (cid:113)
Inverted hierarchy. In this case, we have that |m0 | = ∆m2 +∆m2 +|m0 |2 and
ν2 13 21 ν3
(cid:113)
|m0 | = ∆m2 +|m0 |2. The mass ratios r , r and r are written approximately as
ν1 13 ν3 1 2 3
r(1,2)≈−(cid:32)1+2mm00ν3 (cid:33), r2−r1≈2m0ν3(cid:34)mm0ν20−mm00ν1(cid:35), r3≈−(cid:34)1+2m0ν3(cid:0)m0ν1coms20θνm+0 mν2sin2θν(cid:1)(cid:35) (25)
ν(1,2) ν2 ν1 ν2 ν1
Due to the mass difference m0 −m0 in the factor r −r , the reactor angle can be small or
ν2 ν1 2 1
large since the relative signs in these two masses may conspire to achieve it. There are four
independent cases where the signs in the masses can affect substantially the mixing angles:
• Case A. If m > 0.
νi
r2−r1≈2|m0ν3|(cid:34)|m|m0ν20|−||m|m00ν|1|(cid:35), r3≈−(cid:34)1+2|m0ν3|(cid:0)|m0ν2||smin02θ||νm+0|m| ν1|cos2θν(cid:1)(cid:35) (26)
ν2 ν1 ν2 ν1
• Case B. If m0 > 0 and m0 < 0.
ν(2,1) ν3
r2−r1≈−2|m0ν3|(cid:34)|m|m0ν20|−||m|m00ν|1|(cid:35), r3≈−(cid:34)1−2|m0ν3|(cid:0)|m0ν2||smin02θ||νm+0|m| ν1|cos2θν(cid:1)(cid:35) (27)
ν2 ν1 ν2 ν1
• Case C. If If m0 > 0 and m0 < 0.
ν(3,2) ν1
r2−r1≈−2|m0ν3|(cid:34)|m|m0ν20|+||m|m00ν|1|(cid:35), r3≈−(cid:34)1−2|m0ν3|(cid:0)|mν2||smin02θ||νm−0|m| 0ν1|cos2θν(cid:1)(cid:35) (28)
ν2 ν1 ν2 ν1
• Case D. If m0 > and m0 < 0.
ν2 ν(3,1)
r2−r1≈2|m0ν3|(cid:34)|m|m0ν20|+||m|m00ν|1|(cid:35), r3≈−(cid:34)1+2|m0ν3|(cid:0)|mν2||smin02θ||νm−0|m| 0ν1|cos2θν(cid:1)(cid:35) (29)
ν2 ν1 ν2 ν1
7
Noticing, ifthestrictinvertedhierarchywererealized, r −r = 0andr = −1, wewouldhave
2 1 3
that sin2θ = 0 and sin2θ = N2|1+(cid:15)|2/2, which is not compatible with the observations.
13 23 3
Nonetheless, this strict ordering allows us to infer that the |(cid:15)|eiα(cid:15) parameter magnitude has to
be small in order to deviate sufficiently the atmospheric angle from 45◦; and the associated
phase determines if we are above or below of 45◦. On the contrary, if the constraint, on
the lightest neutrino mass, is relaxed, the reactor angle comes out being non zero; and the
atmospheric one has an extra contribution, r , which can enlarge or reduce the |(cid:15)| magnitude
3
since this may be greater or minor than 1. So that, the factor |(cid:15) r | might deviate drastically
3
the atmospheric angle beyond of 45◦. Roughly speaking, the reactor angle turns out being
equal for the Cases A and B, and also, for C and D. The key difference among them comes
from the atmospheric angle as can be seen in Eq.(26-29).
On the other hand, we observe that
|m0ν2|−|m0ν1|≈2|m0ν1|R1, |m0ν2|+|m0ν1|≈2|m0ν1|[1+R1], |m0ν1||m0ν2|≈|m0ν1|2[1+2R1], (30)
where, R ≡ ∆m2 /4|m0 |2 ≈ O(10−3), if the |m0 | lightest neutrino mass is tiny. Therefore,
1 21 ν1 ν3
we can conclude that the first two scenarios are ruled out since the reactor angle is propor-
tional to the small quantity (|m0 |R /|m0 |)(cid:15), where |(cid:15)| ≪ 1. For the Case C ( Case D)
ν3 1 ν1
the corresponding sign is the upper (lower), then the mixing angles are given as
sin2θ13≈4sin22θν|(cid:15)|2(cid:32)||mm0ν0ν13||(cid:33)2(1−R1)2, sin2θ23≈ 21|1+(cid:15)(cid:2)1±2(1|m−0νs3i|n/2|mθ10ν31|)cos2θν(cid:3)|2. (31)
From these formulas, in general, an |(cid:15)| large value will be needed to compensate the |m0 |
ν3
lightest neutrino mass to get the allowed region for the reactor angle. But, the atmospheric
angle prefers an |(cid:15)| small values so that the complex parameter phase is taken to be α = 0
(cid:15)
in order to increase this angle for reaching its central value. In order to fix ideas, with
√
sinθ ≈ 1/ 3, which is approximately correct to the solar angle, we obtain for the Case
ν
C: (a) if |(cid:15)| ≈ 0.3, it is required that |m0 |/|m0 | ≈ 0.26, to obtain sin2θ ≈ 0.0229. As
ν3 ν1 13
consequence, we get sin2θ ≈ 0.92 which is too large; (b) if |(cid:15)| ≈ 0.1, then we need that
23
|m0 |/|m0 | ≈ 0.8 to get sin2θ ≈ 0.0229, and therefore, sin2θ ≈ 0.62, which is still large in
ν3 ν1 13 23
comparison to the central value. Here, the latter mass ratio could be wrong for the inverted
hierarchy since this implies that |m0 | ∼ |m0 |. Now, in figure 1 the parameter space is shown
ν3 ν1
where we have considered the exact formulas given in Eq. (21), the central values for the θ
12
solar angle, ∆m2 and ∆m2 . As can be seen, this case is tight since there is no common
21 13
region for the parameter space.
√
For Case D, with sinθ ≈ 1/ 3, the reactor angle has approximately the same values
ν
for |(cid:15)| ≈ 0.3, |(cid:15)| ≈ 0.1 and their respective |m0 |/|m0 | mass ratios as above case. Taking
ν3 ν1
the former and latter values of |(cid:15)|, we obtain sin2θ ≈ 0.78 and sin2θ ≈ 0.55, respectively.
23 23
Bothvaluesareapproachingthealloweddata, then, thiscaseseemstobefavoredsincealarge
contribution of |(cid:15)| is suppressed by r which is minor than 1 and the reactor angle prefers an
3
|(cid:15)| large values. In figure 2, one can see that there is a small region in the parameter space
where the reactor and atmospheric angles can be accommodated. In this plot, as in the figure
1, we have considered the exact expressions for the mixing angles and the same values for the
observables. Figures 1 and 2 confirm our theoretical predictions on the mixing angles, the
atmospheric angle prefers small values for |(cid:15)| whereas the reactor one needs a large value.
Degenerated hierarchy. In this case, |m0 | (cid:117) |m0 | (cid:117) |m0 | (cid:117) m , with m (cid:38)
ν3 ν2 ν1 (cid:112)0 0
0.1 eV. Then, the absolute neutrino masses can be written as |m0 | = ∆m2 +m2 ≈
ν3 31 0
8
Sin2(θ13) Sin2(θ23)
0.20 0.20
0.15 0.15
ϵ0.10 ϵ0.10
0.05 0.05
0.00 0.00
0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10
mν3(eV) mν3(eV)
Figure 1: Case C: Allowed region for sin2θ and sin2θ , respectively. The dash and thick
13 23
lines stand for 1 σ and 3 σ for each case
Sin2(θ13) Sin2(θ23)
0.20 0.20
0.15 0.15
ϵ0.10 ϵ0.10
0.05 0.05
0.00 0.00
0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10
mν3(eV) mν3(eV)
Figure 2: Case D: Allowed region for sin2θ and sin2θ . The dash and thick lines stand
13 23
for 1 σ and 3 σ for each case
(cid:112)
m (1+∆m2 /2m2) and |m0 | = ∆m2 +m2 ≈ m (1+∆m2 /2m2). As in the inverted
0 31 0 ν2 21 0 0 21 0
case, there are four independent cases for the signs which are shown below.
• Case A. If m0 > 0,
νi
r1A= ||mm0ν0ν33||−+mm00, r2A= ||mm0ν0ν33||−+||mm0ν0ν22||, r3A=r2Acos2θν+r1Asin2θν. (32)
• Case B. If If m0 > 0 and m0 < 0,
ν(2,1) ν3
r1B = ||mm0ν0ν33||−+mm00 = r11A, r2B = ||mm0ν0ν33||−+||mm0ν0ν22|| = r12A, r3B =r2Bcos2θν+r1Bsin2θν. (33)
• Case C. If m0 > 0 and m0 = −m ,
ν(3,2) ν1 0
r1C = ||mm0ν0ν33||−+mm00 = r11A, r2C = ||mm0ν0ν33||−+||mm0ν0ν22|| =r2A, r3C =r2Ccos2θν+r1Csin2θν. (34)
9