Texture zeroes and discrete flavor symmetries in light and heavy Majorana neutrino mass matrices: a bottom-up approach

0.33 MB·English

by  Amol Dighe

TIFR/TH/08-54
Texture zeroes and discrete flavor symmetries in light and heavy
Majorana neutrino mass matrices: a bottom-up approach
Amol Dighe1,∗ and Narendra Sahu2,†
1Tata Institute of Fundamental Research,
9
Homi Bhabha Road, Mumbai 400005, INDIA
0
0
2 2Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK

n
a Abstract
J

2 Texture zeroes in neutrino mass matrix M may give us hints about the symmetries involved in
ν

] neutrino mass generation. We examine the viability of such texture zeroes in a model independent
h
p
way through a bottom-up approach. Using constraints from the neutrino oscillation data, we
-
p
e develop an analytic framework that can identify these symmetries and quantify deviations from
h
[
them. We analyze the textures of Mν as well as those of MM, the mass matrix of heavy Majorana

2
v neutrinos in the context of Type-I seesaw. We point out how the viability of textures depends
5
9 on the absolute neutrino mass scale, the neutrino mass ordering and the mixing angle θ . We
13
6
0
also examine the compatibility of discrete flavor symmetries like µ–τ exchange and S permutation
3
.
2
1 with the current data. We show that the µ τ exchange symmetry for M can be satisfied for any
ν
8 −
0 value of the absolute neutrino mass, but for M to satisfy the S symmetry, neutrino masses have
ν 3
:
v
i to be quasi-degenerate. On the other hand, both these symmetries are currently allowed for MM
X
r for all values of absolute neutrino mass and both mass orderings.
a
PACS numbers: 14.60.Pq
∗Electronic address: [email protected]
†Electronic address: [email protected]
1
I. INTRODUCTION
The current low energy neutrino oscillation data [1] indicate that all three of the physical
left-handedneutrinos have different masses andthey mixamong themselves. Iftheneutrinos
are Majorana, the neutrino mass matrix M in the flavor basis is diagonalized by the unitary
ν
Pontecorvo-Maki-Nakagawa-Sakata matrix U [2, 3] through
PMNS
Mdiag = U† M U∗ , i.e. M = U MdiagUT . (1)
ν PMNS ν PMNS ν PMNS ν PMNS
For Majorana neutrinos, U is given by
PMNS
c c s c s e−iδ
12 13 12 13 13
U = U  s c c s s eiδ c c s s s eiδ s c  U , (2)
PMNS χ · − 12 23 − 12 23 13 12 23 − 12 23 13 23 13 · φ
 s s c c s eiδ c s s c s eiδ c c 
 12 23 12 23 13 12 23 12 23 13 23 13 
 − − − 
 
where c and s stand for cosθ and sinθ respectively. Here U = diag(eiφ1,eiφ2,1), with
ij ij ij ij φ
the Majorana phases φ and φ defined in such a way that the diagonal elements of Mdiag
1 2 ν
are given by
Mdiag = diag(m ,m ,m ). (3)
ν 1 2 3
Here m (i = 1,2,3) correspond to the neutrino masses, which are chosen to be real and
i
positive. The Dirac phase δ accounts for the charge parity (CP) violation in the lepton
number conserving processes. The phase matrix U diag(eiχe,eiχµ,eiχτ) consists of the
χ
≡
three“flavorphases”1 χ thatcorrespondtothemultiplicationofaneutrinoflavoreigenstate
α
ν byeiχα. Notethatoncethemixing anglesθ have beendefined tobeinthefirst quadrant,
α ij
the Dirac phase δ can take values in [0,2π) and the phases χ ,φ can take values between
α i
[0,π). All the angles and phases are then uniquely defined.
A global analysis of the current neutrino oscillation data at 3σ C.L. yields [1]
0.25 < sin2θ < 0.37 , 0.36 < sin2θ < 0.67 , sin2θ < 0.056 . (4)
12 23 13
1
These phases are often referred to in the literature as “unphysical phases”. Though these phases have
no relevance for the low energy neutrino phenomenology and cannot be determined through low energy
measurements, their values may be predictable within the context of specific models with new physics at
the high scale.
2
While the absolute mass scale of the neutrinos is not yet fixed, the two mass-squared differ-
ences have already been determined to a good degree of accuracy:
∆m2 m2 m2 = (7.06 8.34) 10−5 eV2 ,
⊙ ≡ 2 − 1 ··· ×
2
m +m
∆m2 m2 1 2 = (2.07 2.75) 10−3 eV2 . (5)
atm ≡ 3 − 2 ± ··· ×
(cid:18) (cid:19)
It is not known whether the neutrino mass ordering is normal (m < m < m ) or inverted
1 2 3
(m < m < m ). The Dirac phase δ and Majorana phases φ are completely unknown.
3 1 2 1,2
The absolute values of neutrino masses cannot be probed by oscillation experiments, the
direct limit on the neutrino mass scale m is obtained by the tritium beta decay experiments
0
[4] as m < 2.2 eV. The most stringent constraint on m however comes from cosmology:
0 0
the WMAP data implies [5]
m < 1 eV . (6)
i
∼
X
In this paper, we shall take the upper bound on each neutrino mass conservatively to be
m < 0.5 eV.
i
Given the absolute values of the neutrino masses and the complete matrix U , the
PMNS
neutrino mass matrix M in the flavor basis can be reconstructed through Eq. (1). The
ν
structure of this matrix may reveal the presence of flavor symmetries in the neutrino sector.
In this paper, we consider multiple texture zeroes of M [6, 7, 8] as well as symmetries like
ν
the µ τ exchange [9] and S permutation [10], which predict certain relations between the
3
−
elements of M .
ν
The symmetry-based relations among the elements of M and texture zeroes of M have
ν ν
been explored earlier mainly by adopting a top-down approach [11]. In this approach an
appropriate symmetry is imposed on the neutrino mass matrix, which in turn gives a pre-
diction for the neutrino mixing parameters that can then be checked against the available
data. We take the bottom-up approach, starting with our current knowledge about neutrino
masses andmixings, and checking if a certain texture zero combination or a symmetry-based
relation is allowed. This allows us to test in a model independent manner the symmetries
present in neutrino mass generation mechanisms. It also enables us to determine which
future measurements can act as tests of these symmetries.
In the present approach the elements of M are expressed as functions of the absolute
ν
neutrino masses, the mixing angles as well as the Dirac, Majorana and flavor phases. Our
3
current complete ignorance about these phases allows a lot of freedom for theelements of M
ν
in spite of the relatively well measured values of the masses and mixing angles. Even with
this freedom, some of the texture zero combinations andsymmetries areclearly forbidden, as
has been numerically verified [12]. We develop an analytical treatment, using perturbative
expansion in appropriate small parameters, and demonstrate the analytical rationale behind
the ruling out of some of these relations. This also leads us to the result that the additional
knowledge of the absolutemass scale of theneutrinos andthe mixing angleθ will be crucial
13
in testing for these relations in near future.
The seesaw mechanism [13] is one of the most favored and explored mechanisms for
neutrino mass generation, which gives rise to light Majorana neutrinos that can satisfy the
low energy neutrino oscillation data, as well as to heavy Majorana neutrinos that may play
an important role in leptogenesis [14]. If the neutrino masses are generated from a Type-I
seesaw mechanism where three singlet heavy Majorana neutrinos are added to the Standard
Model (SM), then we have the effective neutrino mass matrix
M = m M−1(m )T , (7)
ν − D M D
where m is the Dirac mass matrix of neutrinos, and M is the Majorana mass matrix for
D M
the right-handed heavy Majorana neutrinos. If the heavy Majorana neutrinos are written in
a basis where m is diagonal, the texture zeroes as well as symmetry relations between ele-
D
ments ofM can berelated to those ofthe inverse neutrino matrix M−1 in a straightforward
M ν
manner [15]. The same analytical treatment developed for M can then be extended to test
ν
the symmetry relations for M in this basis. We perform this analysis, with a particular
M
emphasis on the dependence of these relations on the absolute masses of the light neutrinos.
The paper is organized as follows. In Sec. II, we introduce our formalism and set up the
analytical framework under which the symmetry relations may be examined. In Sec. III
and Sec. IV, we test the texture zeroes of M and M respectively, numerically as well as
ν M
analytically. In V and VI, we examine the µ τ exchange symmetry and S permutation
3
−
symmetry for M and M respectively. Sec. VII concludes.
ν M
4
II. THE ANALYTICAL FRAMEWORK
A. parameterization of neutrino masses and mixing
We parameterize the absolute values of neutrino masses in terms of three parameters m ,
0
ǫ and ρ as [16]
m = m (1 ρ)(1 ǫ) , m = m (1 ρ)(1+ǫ) , m = m (1+ρ) , (8)
1 0 2 0 3 0
− − −
where m sets the overall mass scale of neutrinos, while the dimensionless parameters ρ and
0
ǫ can be expressed in terms of the solar and atmospheric mass scales as
∆m2 ∆m2
ρ = atm , ǫ = ⊙ . (9)
4m2 4m2(1 ρ)2
0 0 −
Clearly, ρ is positive (negative) for normal (inverted) mass ordering of neutrinos. The sum
of neutrino masses may be expressed in terms of the above parameters as
ρ
m = 3m 1 < 1 eV . (10)
i 0
− 3
∼
Xi (cid:16) (cid:17)
The condition 0 < m < 0.5 eV then yields
i
m > 0.025 eV , 2.43 10−3 < ρ < 1 , 8 10−5 < ǫ < 1 . (11)
0
∼ × | | ×
The value of ρ approaches unity as m approaches its lowest allowed value. The value of ǫ
0
| |
can be > 0.01 only for normal mass ordering and m < 0.06 eV, whereas ǫ ρ everywhere
0
≪ | |
except for m 0.025 eV. Taking the best-fit values of solar and atmospheric neutrino
0
≈
masses, in Fig. 1 we show the values of ρ and ǫ as functions of m for normal as well as
0
inverted hierarchies. For the purpose of this paper, we divide the neutrino parameter space
into three scenarios:
(i) Normal mass ordering with hierarchical masses (NH), where m m m . The
1 2 3
≪ ≪
current data give ρ 0.85 and ǫ 0.92 in the extreme limit, however these values decrease
≈ ≈
ratherrapidlyasm increases, ascanbeseenfromFig.1. Thisscenario canthenbeanalyzed
0
through a perturbative expansion in the set of the small parameters ρ˜ 1 ρ and ǫ˜ 1 ǫ
≡ − ≡ −
in the extreme limit, however one has to be careful while treating quantities like (1 ρ2),
−
which stays higher than 0.3.
(ii) Inverted mass ordering with hierarchical masses (IH), such that m m < m . In this
3 1 2
≪
5
(a) (b)
1 1
−ρ
0.1 ρ 0.1
0.01 0.01
ε
ε
0.001 0.001
0.0001 0.0001
1e-05 1e-05
0.01 0.1 1 0.01 0.1 1
m (eV) m (eV)
0 0
FIG. 1: The parameters ρ and ǫ as functions of m for (a) normal ordering and (b) inverted
0
ordering of neutrino masses, for best-fit values of ∆m2 and ∆m2.
atm ⊙
case, ǫ 1 and 1+ρ 1, so we can use a perturbative expansion in the small parameters
≪ | | ≪
ρ 1+ρ and ǫ.
≡
(iii) Quasidegenerate neutrinos (QD), where m m m , with either mass ordering. In
1 2 3
≃ ≃
b
this case, ρ ,ǫ 1, so that we can use these two quantities as small parameters.
| | ≪
˜
In addition, at appropriate places we shall also consider θ and θ θ π/4 as small
13 23 23
≡ −
parameters in order to facilitate a perturbative expansion.
B. Elements in M , M−1, M and M
ν ν ν M
f
Let the low energy neutrino mass matrix in the flavor basis be written as
a b c
M =  b d e  . (12)
ν
 
c e f
 
 
 
Since the neutrinos are Majorana, M is symmetric. In terms of the parameterization of
ν
neutrino masses in Sec. IIA, mixing angles and CP violating phases, the elements of M
ν
6
may be written as
a = m e2iχe e2iφ1( 1+ρ)( 1+ǫ)c2 c2
0 − − 12 13
(cid:20)
e2iφ2( 1+ρ)(1+ǫ)c2 s2 +e−2iδ(1+ρ)s2 ,
− − 13 12 13
(cid:21)
b = m ei(χe+χµ)c e−iδ(1+ρ)s s
0 13 23 13
(cid:20)
e2iφ1( 1+ρ)( 1+ǫ)c c s +eiδc s s
12 23 12 12 23 13
− − −
+e2iφ2( 1+ρ)(1+ǫ)s (cid:0)c c +eiδs s s (cid:1) ,
12 12 23 12 23 13
− −
(cid:21)
(cid:0) (cid:1)
c = m ei(χe+χτ)c e−iδ(1+ρ)c s
0 13 23 13
(cid:20)
e2iφ1( 1+ρ)( 1+ǫ)c s s +eiδc c s
12 12 23 12 23 13
− − − −
+e2iφ2( 1+ρ)(1+ǫ)s c(cid:0) s +eiδc s s ,(cid:1)
12 12 23 23 12 13
−
(cid:21)
(cid:0) (cid:1)
d = m e2iχµ (1+ρ)c2 s2
0 13 23
(cid:20)
+e2iφ1( 1+ρ)( 1+ǫ) c s +eiδc s s 2
23 12 12 23 13
− −
e2iφ2( 1+ρ)(1+ǫ) c(cid:0) c eiδs s s 2(cid:1) ,
12 23 12 23 13
− − −
(cid:21)
(cid:0) (cid:1)
e = m ei(χµ+χτ) (1+ρ)c c2 s
0 23 13 23
(cid:20)
e2iφ1( 1+ρ)( 1+ǫ) s s eiδc c s c s +eiδc s s
12 23 12 23 13 23 12 12 23 13
− − − −
+e2iφ2( 1+ρ)(1+ǫ) c(cid:0) s +eiδc s s (cid:1)c(cid:0) c eiδs s s (cid:1),
12 23 23 12 13 12 23 12 23 13
− −
(cid:21)
(cid:0) (cid:1)(cid:0) (cid:1)
f = m e2iχτ (1+ρ)c2 c2
0 23 13
(cid:20)
+e2iφ1( 1+ρ)( 1+ǫ) s s eiδc c s 2
12 23 12 23 13
− − −
e2iφ2( 1+ρ)(1+ǫ) c(cid:0) s +eiδc s s 2(cid:1) . (13)
12 23 23 12 13
− −
(cid:21)
(cid:0) (cid:1)
The inverse of the neutrino mass matrix, M−1, can be written as
ν
A B C
M 1
M−1 = B D E , (14)
ν Det(M ) ≡ Det(M )
ν ν
f C E F
 
 
 
where
Det(M ) = m3(1 ρ)2(1+ρ)(1 ǫ2)e2iPχ (15)
ν 0 − −
7
is the determinant of M , with χ χ +χ +χ . Here M is the adjoint neutrino mass
ν e µ τ
≡
matrix. From Eq. (14) it is obvioPus that texture zeroes in M−1 are the same as those in M.
fν
The elements of M can be written in terms of the masses and the elements of U U
PMNS
≡ f
matrix as
f
A = m2e2i(χµ+χτ) (1 ρ)2(1 ǫ2)(U U U U )2
0 − − 21 32 − 22 31
+(1 ρ2)(1(cid:2) ǫ)(U U U U )2 +(1 ρ2)(1+ǫ)(U U U U )2
21 33 23 31 22 33 23 32
− − − − −
B = m2ei(χe+χµ+2χτ) (1 ρ)2(1 ǫ2)(U U U U )(U U U U ) (cid:3)
0 − − 21 32 − 22 31 31 12 − 11 32
+(1 ρ2)(1 ǫ(cid:2))(U U U U )(U U U U )
31 13 33 11 21 33 23 31
− − − −
+(1 ρ2)(1+ǫ)(U U U U )(U U U U )
22 33 23 32 32 13 12 33
− − −
C = m2ei(χe+2χµ+χτ) (1 ρ)2(1 ǫ2)(U U U U )(U(cid:3) U U U )
0 − − 21 32 − 22 31 22 11 − 12 21
+(1 ρ2)(1 ǫ(cid:2))(U U U U )(U U U U )
11 23 13 21 21 33 31 23
− − − −
+(1 ρ2)(1+ǫ)(U U U U )(U U U U )
12 23 13 22 22 33 32 23
− − −
D = m2e2i(χe+χτ) (1 ρ)2(1 ǫ2)(U U U U )2 +(1(cid:3) ρ2)(1 ǫ)(U U U U )2
0 − − 11 32 − 12 31 − − 11 33 − 13 31
+(1 ρ2)(1(cid:2)+ǫ)(U U U U )2
12 33 32 13
− −
E = m2ei(2χe+χµ+χτ) (1 ρ)2(1 ǫ2)(U (cid:3)U U U )(U U U U )
0 − − 11 32 − 12 31 12 21 − 11 22
+(1 ρ2)(1 ǫ(cid:2))(U U U U )(U U U U )
11 33 13 31 21 13 11 23
− − − −
+(1 ρ2)(1+ǫ)(U U U U )(U U U U )
12 33 13 32 22 13 12 23
− − −
F = m2e2i(χe+χµ) (1 ρ)2(1 ǫ2)(U U U U )2 +(1(cid:3) ρ2)(1 ǫ)(U U U U )2
0 − − 11 22 − 12 21 − − 11 23 − 13 21
+(1 ρ2)(1(cid:2)+ǫ)(U U U U )2 . (16)
12 23 13 22
− −
(cid:3)
In order to analyze the heavy majorana neutrino mas matrix M , we invert Eq. (7) to
M
obtain
M = mTM−1m . (17)
M − D ν D
Following [15], we choose the “flavor” basis for heavy Majorana neutrinos in which the Dirac
mass matrix is real and diagonal,
m = diag(x,y,z) . (18)
D
8
In this basis, M may be written as
M
x2A xyB xzC
1
M = xyB y2D yzE . (19)
M
−Det(M )
ν
xzC yzE z2F
 
 
 
Again, the texture zeroes of M are same as the texture zeroes of M−1, and hence those
M ν
of M, which have relatively tractable analytical expressions. The discrete symmetries like
µ τ exchange or S , on the other hand, also depend on the values of the Dirac masses.
3
−f
However even in that case, we can test for certain relations between elements of M that
M
are independent of these Dirac masses, as we shall see in Sec. VI.
C. Quantifying deviation from exact symmetry in the bottom-up approach
In the traditional top-down approach, discrete flavor symmetries like µ τ exchange,
−
S -permutation, etc. are assumed in the neutrino mass matrix, which predict the mixing
3
parameters measured in the low energy neutrino oscillation data. The main purpose of these
symmetries is to understand why the (1-3) family mixing is small, while the (2-3) family
mixingisalmostmaximalandthe(1-2)familymixingislargebutnotmaximal. Traditionally
these symmetries are employed to set U = 0. The dynamical breaking of these symmetries
13
then may predict a non-zero U , which is then compared with the experimental value. Since
13
the current low energy neutrino oscillation data have large uncertainties, the data allow an
enormous freedom to propose such discrete flavor symmetries in the top-down approach.
However it is also crucial to examine, by starting with the available low energy neutrino
oscillation data, the parameter space of neutrino mass matrix where such discrete flavor
symmetries can be realized. We call it the bottom-up approach. In this approach, since the
low energy data have intrinsic uncertainties, the symmetry relations can only be said to be
satisfied approximately. One therefore needs to quantify when one may declare the relevant
symmetry to be allowed.
In M , the magnitudes of all the elements are expected to be m , as can be seen from
ν 0
∼
Eqs. (13), taking into account that the sine and cosine of θ ,θ (1) and also assuming
12 23
∼ O
thatφ (1)in theabsence ofany symmetry principle. If anelement M (i,j) is m , it
i ν 0
∼ O | | ≪
is either an accidental cancellation or the signature of a discrete symmetry at work. We take
the position that for a sufficiently small value of ξ, the observation M (i,j) /m < ξ would
ν 0
| |
9
indicate that the symmetry that would make M (i,j) = 0 is present. We choose ξ = 10−2,
ν
which is motivated by the accuracy to which the mixing angles are currently known. It
also indicates the extent to which we tolerate the breaking of the discrete symmetries under
consideration. In other words, when M (i,j) /m < 10−2, we consider M (i,j) to be
ν 0 ν
| |
effectively zero. Thus, we declare a texture zero viable if
M (i,j)
Min | ν | < 10−2, (20)
m
(cid:18) 0 (cid:19)
where the minimization is over all the allowed values (3σ) of the mixing parameters at a
particular value of m . If this condition is not satisfied, then the symmetry that would lead
0
to M (i,j) = 0 is ruled out.
ν
Similarly, from Eq. (16) one can see that the elements of M are expected to be m2 in
∼ 0
the absence of any cancellations. Hence if
f
M(i,j)
Min | | < 10−2, (21)
m2
0 !
f
then we conclude that the symmetry that requires M(i,j) = 0 is still allowed. In the case
of the discrete symmetries like µ τ exchange or S , certain ratios are expected to be equal
3
− f
to unity. Here, we demand that the deviation of such ratios from unity to be less than 10−2
for the symmetry to be acceptable. Note that the right hand side of Eqs. (20) and (21) can
be changed to any small number of one’s choice, depending on how much deviation from the
exact symmetry one is willing to allow. Our numerical results cover the complete relevant
range, so the required numbers can be read off from our figures.
Note that our criteria give the necessary conditions for a particular symmetry to hold.
Further considerations may disallow some of the symmetry relations that are permitted by
conditions in Eqs. (20) and (21).
III. TEXTURE ZEROES IN M
ν
A. Individual zeroes in M
ν
Whether a particular element in the neutrino mass matrix M can potentially vanish
ν
can be checked analytically from Eqs. (13). To simplify the expressions, we define three
10