Neutrino observables from predictive flavour patterns

3.2 MB·

by  Luis M. Cebola

Prepared for submission to JHEP
Neutrino observables from predictive flavour patterns
6
1
0 Lu´ıs M. Cebola,a David Emmanuel-Costa,a and Ricardo Gonz´alez Felipeb,a
2
n aDepartamento de F´ısica and Centro de F´ısica Te´orica de Part´ıculas - CFTP, Instituto Superior
a T´ecnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal
J
bISEL - Instituto Superior de Engenharia de Lisboa, Instituto Polit´ecnico de Lisboa,
9
Rua Conselheiro Em´ıdio Navarro 1959-007 Lisboa, Portugal
2
E-mail: [email protected],
]
h [email protected], [email protected]
p
-
p Abstract: We look for predictive flavour patterns of the effective Majorana neutrino
e
massmatrixthatarecompatiblewithcurrentneutrinooscillationdata. Oursearchisbased
h
[ on the assumption that the neutrino mass matrix contains equal elements and a minimal

2 numberofparameters,intheflavourbasiswherethechargedleptonmassmatrixisdiagonal
v
and real. Three unique patterns that can successfully explain neutrino observables at the
0
5 3σ confidence level with just three physical parameters are presented. Neutrino textures
1
described by four and five parameters are also studied. The predictions for the lightest
6
0 neutrino mass, the effective mass parameter in neutrinoless double beta decays and the
.
1 CP-violating phases in the leptonic mixing are given.
0
6
1
:
v
i
X
r
a
Contents
1 Introduction 1
2 Minimal neutrino mass matrix textures 2
3 Textures with three parameters 6
4 Textures with four and five parameters 9
5 Conclusions 11
A Summary tables for classes with four and five parameters 12
B Extra correlations among the elements of m in class 2 1 0 19
ν 1 2 2
References 21
1 Introduction
Neutrino masses constitute a solid evidence of physics beyond the standard model. In the
absence of a compelling theory to explain the origin of the lepton flavour structure, one
expectsthenewphysicstoshedsomelightintheflavourpuzzle. Fromthetheoreticalpoint
ofview,anaturalapproachistorestrictthenumberoffreeparametersintheleptonflavour
sectorsothatthetheorybecomesmorepredictive. Onepossibleframeworkistoimposethe
existence of zero elements in the charged lepton and light neutrino mass matrices. Some of
such zeroes may simply appear as a result of weak basis transformations [1, 2] while others
could be obtained from a symmetry principle [3, 4].
Itiswellestablishedthatneutrinomassmatriceswithmorethantwoindependentzero
entries are not compatible with neutrino oscillation data, in the physical basis in which
the charged leptons are diagonal and ordered (e,µ,τ). On the other hand, there are several
Majorana neutrino patterns with two zeroes that are still viable [5–9]. The latter contain
eight real parameters of which five are physical (three unphysical phases can be removed).
Another example of predictive textures with five physical parameters are the so-called
hybrid textures [10], which have one texture zero and two equal nonzero elements in the
physical basis. Most of these hybrid textures are also compatible with current data [9, 11].
Following the same line of reasoning, it is interesting to look for other neutrino textures
which have the same predictability or which are even more predictive, i.e., contain less
than five physical parameters. Since increasing the number of zeroes is not permitted, a
straightforward possibility is to allow for correlations (equalities) of some elements in the
effective neutrino mass matrix.
– 1 –
Our knowledge of neutrino masses and leptonic mixing has expanded in the last years
thanks to solar, atmospheric, reactor and accelerator neutrino experiments [12–14], as well
as to cosmological observations [15]. Leptonic CP violation [16] remains an important
open question and it should be probed in near-future neutrino experiments [17]. For light
Majorana neutrinos, there are nine specific observables that are relevant in the low-energy
neutrino analysis: threeneutrino masses, threemixing angles, oneDirac CP phaseand two
Majorana phases. These parameters should be determined from the physical parameters
that characterise a given neutrino mass matrix.
An efficient and reliable approach to confront any neutrino pattern with the observa-
tional data is the χ2-analysis, which we shall use in this work. We shall look for predictive
flavour patterns with equal elements (and possibly additional zeroes) in the effective Ma-
jorana neutrino mass matrix, and test their compatibility with data at 3σ(99.73%) and
1σ(68.27%) confidence levels (CL).
The paper is organised as follows. In section 2, we describe the minimal neutrino mass
matrix textures and present their classification according to the number of physical param-
eters and correlated matrix elements. The number of solutions compatible with neutrino
oscillation data at the 1σ and 3σ CL, for normal (NO) and inverted (IO) mass ordering,
are also given. Section 3 is devoted to the search of maximally predictive textures. In
particular, it is shown that there exist three unique patterns that can successfully explain
neutrino observables at the 3σ CL with just three physical parameters. In section 4, neu-
trinotexturesdescribedbyfourandfivephysicalparametersarediscussed. Thepredictions
for the lightest neutrino mass, effective mass parameter in neutrinoless double beta decays
and CP-violating phases in the leptonic mixing are also given. Possible breaking paths of
the viable three-parameter textures into four- and five-parameter textures compatible with
neutrino oscillation data at 1σ are presented. Finally, our concluding remarks are given in
section 5.
2 Minimal neutrino mass matrix textures
We shall search for predictive flavour patterns of the effective Majorana neutrino mass
matrixm ,basedontheassumptionthatitcontainsequalmatrixelementsand,eventually,
ν
somevanishingelements. Ouranalysisisperformedinthephysicalbasiswherethecharged
lepton mass matrix is diagonal, real and ordered, i.e., m = diag(m ,m ,m ). Moreover,
(cid:96) e µ τ
we restrict our search to neutrino textures with a minimal number of physical parameters,
n < 6. By physical parameters we mean those that remain after removing all unphysical
phases.
In order to classify different textures according to the number of correlated matrix
elements, we shall use the notation N to specify that the matrix m contains N equal
nN ν
elements and that such correlation appears n times in the given texture. In this notation,
N
1 simplydenotesn non-vanishingarbitraryelements, whilethenotation0 willbeused
n1 1 n0
to indicate that the neutrino mass matrix has n independent texture zeroes. Since m
0 ν
is a 3 3 complex symmetric matrix, it has six independent entries. Clearly, any pattern
×
with more than four equal elements is unrealistic. Thus, in general, we identify any given
– 2 –
n Texture class No. of textures Solutions (1σ,3σ) Table in the text
NO IO
2 4 0 7∗ ( 15) - - -
1 2
⊂
3 2 0 21∗ ( 45) - - -
2 2
⊂
3 1 0 28∗ ( 60) - - -
1 1 2
⊂
3 2 0 60 (0,1) (0,2) 4
1 1 1
3 10 - - -
2
4 1 0 30 - - -
1 1 1
4 2 15 - - -
1 1
4 2 1 0 42∗ ( 90) (4,2) (1,3) 5
1 2 2
⊂
2 1 0 90 (15,3) (19,11) 6, 7
2 1 1
3 1 0 60 (9,3) (13,10) 8
1 2 1
3 2 1 60 (13,5) (13,5) 9
1 1 1
4 1 15 (5,0) (3,1) 10
1 2
5 1 0 7∗ ( 15) (6,0) (3,2) -
4 2
⊂
2 1 0 60 (31,4) (36,7) -
1 3 1
2 15 (4,0) (1,1) 11
3
3 1 20 (19,0) (18,0) 12
1 3
Table 1. Texture classification according to the number of physical parameters n. For each
class, the number of different physical textures within the class is given. The number of solutions
compatiblewithdataatthe1σ and3σ CLisalsopresented. Patternsmarkedwith areparticular
∗
cases of the well-known two-zero textures (denoted here as 1 0 ) and, therefore, only the subset
4 2
of potentially viable matrices are considered. The class 2 1 0 corresponds to the so-called hybrid
1 3 1
textures.
class of textures as 4 3 2 1 0 . The total number of textures for such a class can be
n4 n3 n2 n1 n0
determined through the combinatorial relation
6!
. (2.1)
n4!(4!)n4n3!(3!)n3n2!(2!)n2n1!n0!
In table 1 we present the texture classification according to the number of physical
parameters n and the correlated matrix elements. For any given texture class, the number
of different physical textures within that class is also given. Patterns marked with are
∗
particular cases of the well-known two-zero textures [5] (denoted here as 1 0 ). Only the
4 2
subset of potentially viable matrices (i.e. patterns A , B and C in the notation
1,2 1,2,3,4
of [5]) has been considered [6–9]. Note also that the class 2 1 0 corresponds to the so-
1 3 1
called hybrid textures with two equal elements and one zero [10].
In the next sections we confront all textures in table 1 with the low-energy observa-
tional neutrino data. We recall that in the framework of the standard model with three
lightneutrinos,leptonicmixingisdescribedbythePontecorvo-Maki-Nakagawa-Sakatauni-
tary matrix U, which relates the mass eigenstate neutrinos ν (i = 1,2,3) to the flavour
i
eigenstate neutrinos ν (f = e,µ,τ) [18–20]. This matrix can be parametrised in terms of
f
three mixing angles θ , θ , and θ , a Dirac CP phase δ and two Majorana phases α ,
12 13 23 21
– 3 –
Parameter Best fit 1σ 3σ
±
∆m2 [10−5eV2] 7.60+0.19 7.11 – 8.18
21 −0.18
∆m2 [10−3eV2] (NO) 2.48+0.05 2.30 – 2.65
| 31| −0.07
(IO) 2.38+0.05 2.20 – 2.54
−0.06
sin2θ /10−1 3.23 0.16 2.78 – 3.75
12
±
sin2θ /10−1 (NO) 5.67+0.32 3.93 – 6.43
23 −1.24
(IO) 5.73+0.25 4.03 – 6.40
−0.39
sin2θ /10−2 (NO) 2.26 0.12 1.90 – 2.62
13
±
(IO) 2.29 0.12 1.93 – 2.65
±
δ/π (NO) 1.41+0.55 0.0 – 2.0
−0.40
(IO) 1.48 0.31 0.0 – 2.0
±
Table 2. Neutrinooscillationparametersatthe68.27%and99.73%CL[13]. Theupperandlower
rows in ∆m2 , sin2θ , sin2θ ,and δ correspond to NO and IO neutrino masses, respectively.
31 23 13
α [21]. The neutrino mass matrix m is then diagonalised by the unitary transformation
31 ν
U†m U∗ = diag(m ,m ,m ), (2.2)
ν 1 2 3
wherem arethelightneutrinophysicalmasses. Themixinganglesareeasilyreconstructed
i
from U as
U 2 U 2
sin2θ = | 12| , sin2θ = | 23| , sin2θ = U 2. (2.3)
12 1 U 2 23 1 U 2 13 | 13|
13 13
−| | −| |
From the rephasing-invariant quartet Q U U U∗ U∗ , the Dirac phase δ can be found
12 23 13 22
≡
through the relations
8Im(Q)
sinδ = , (2.4a)
cosθ sin2θ sin2θ sin2θ
13 12 13 23
Re(Q)+cos2θ sin2θ sin2θ sin2θ
13 12 13 23
cosδ = 8 . (2.4b)
cosθ sin2θ sin2θ sin2θ
13 12 13 23
Finally, the Majorana phases are obtained as
α = 2arg(U∗ U ) , α = 2arg(U∗ U )+2δ. (2.5)
21 11 12 31 11 13
Sincetheabsolutescaleofneutrinomassesisunknown,therearetwopossibleorderings
of m , namely, a normal ordering (NO) with m < m < m or an inverted ordering (IO)
i 1 2 3
with m < m < m . The mass spectrum may vary from hierarchical to quasi-degenerate,
3 1 2
yet cosmological observations place a stringent upper bound on the sum of the masses.
Assuming three species of degenerate massive neutrinos and a Λ cold dark matter model,
(cid:80)
the bound m < 0.23 eV at 95% CL has been obtained from a combined analysis of
i i
data[15]. Inwhatfollows,weshallassumethisboundinournumericalcalculations. Notice
also that in the physical basis (m diagonal) the absolute value of any matrix element of
(cid:96)
– 4 –
m is always smaller than the largest neutrino mass. Thus the above cosmological bound
ν
implies m (cid:46) 80 meV.
νij
| |
In our search for viable neutrino mass patterns at 3σ level, we shall consider five
observables, namely, the mass-squared differences ∆m2 and ∆m2 , and the mixing angles
21 31
sin2θ ,sin2θ ,andsin2θ . Whentestingtheviabilityofthepatternsat1σlevel,theDirac
12 23 13
CP phase δ is also included as an observable quantity. The current neutrino oscillation
data are presented in table 2 at the 1σ and 3σ CL [13]. If the predictions for the physical
observablesarewithintherangesgiveninthetable, atextureisconsideredtobeconsistent
with observations. For such cases, we look for the corresponding predictions for the Dirac
CP phase δ, the lightest neutrino mass m, the effective neutrino mass parameter m =
ββ
(cid:80) U2 m , andthetwoCP-violatingMajoranaphasesα ,α . Notethat, inthephysical
| i ei i| 21 31
basis, one has m = m (cid:46) 80 meV, for all textures considered in this work.
ββ νee
| |
Concerningtheatmosphericmixingangleθ , itisworthnoticingthatitsvalueisclose
23
to maximal (θ π/4). Starting from a given texture m , one can always define a new
23 ν
≈
texture m(cid:48), whose observables are directly related to those of m ,
ν ν
m(cid:48) = P m P . (2.6)
ν 23 ν 23
Here P is the permutation matrix acting in the sector (2,3). Said otherwise, m(cid:48) is
23 ν
obtained from m by interchanging its second and third rows and columns. The matrices
ν
m andm(cid:48) sharethesamemassspectrum,mixinganglesθ andθ ,andMajoranaphases
ν ν 12 13
α ,α , while the mixing angle θ and the Dirac phase δ are transformed as
21 31 23
π
θ θ , δ δ π. (2.7)
23 23
→ 2 − → −
One can then conclude from table 2 that, if the given texture m is viable at 3σ CL, the
ν
transformed m(cid:48) would also be compatible with neutrino oscillation data, provided that
ν
sin2θ 0.6 (i.e. θ 51◦) in both NO and IO cases.
23 23
≤ ≤
Weshallperformaχ2-analysisusingthestandarddefinitionoftheχ2-functioninterms
of the observable values and their deviations from the mean values. The minimisation of
the χ2 function is carried out with respect to the neutrino observables using the MINUIT
package [22], following the numerical strategy presented in [9]. This procedure allows us
to determine whether a given pattern is compatible at 1σ level. In order to obtain the
allowed ranges at 3σ CL for the predictable quantities, we then randomly scan the full
parameter space in the neutrino mass matrices. Since we do not impose any restriction on
the complex matrix elements m , our solutions exhibit an explicit symmetry on the CP
νij
phases:
δ δ, α α , α α . (2.8)
21 21 31 31
→ − → − → −
Since this symmetry does not change the value of any other observable or prediction, we
use it to restrict the Dirac phase to the range 0 δ π when presenting our results.
≤ ≤
Obviously, values of δ in the range π δ 2π are physical as well and may be even
≤ ≤
preferred by the experimental data [23] (cf. also table 2).
In the next sections, we seek for minimal neutrino textures that fulfil our criteria, i.e.,
that contain equal or vanishing elements and simultaneously have a minimal number of
– 5 –
physical parameters. Since m is symmetric, we shall use the following sequential notation
ν
to label the position of the matrix elements of a given neutrino texture T,
 
1 2 3
2 4 5. (2.9)
 
3 5 6
We denote the position of any vanishing element labeled i with a superscript, i.e. Ti,
while equal elements with labels i,j,...,k are listed as subscripts, T . For instance, the
ij...k
notations T5,6 and T6 correspond to the textures
12,34 123,45
   
a a b a a a
T5,6 = a b 0 and T6 = a b b , (2.10)
12,34   123,45  
b 0 0 a b 0
respectively.
3 Textures with three parameters
Followingourχ2 numericalstrategy,wehaveanalysedallpossibletexturesbelongingtothe
classes listed in table 1. The number of solutions compatible with neutrino oscillation data
at the 1σ and 3σ CL are reported in this table.1 From our search of the different pattern
classes, no solution with only two physical parameters (class 4 0 ) was found, neither for
1 2
NO nor for IO neutrino spectrum. Thus, we conclude that under our assumptions (equal
or vanishing elements) the minimal number of free physical parameters is 3. Moreover,
from the table, one can see that the only viable class with n = 3 is 3 2 0 . Out of the 60
1 1 1
textures within this class, only the patterns T2 for NO neutrino spectrum and T4
136,45 123,56
and T6 for IO spectrum are compatible with data at the 3σ level. None of the textures
123,45
passed the 1σ CL test.
For the texture T2 the effective neutrino mass matrix m has the following form
136,45 ν
 
a 0 a
m 0 b b . (3.1)
ν ≡  
a b a
The minimum of χ2 for this solution is χ2 = 6.68. An example that leads to this value is
min
given by the choice of a = 10.5731 meV and b = 29 meV, argb = 0.08284π. This in turn
| |
yields the following low-energy neutrino parameters:
∆m2 = 7.61 10−5eV2, ∆m2 = 2.47 10−3eV2,
21 31
× | | × (3.2)
sin2θ = 3.14 10−1, sin2θ = 6.31 10−1, sin2θ = 2.44 10−2.
12 23 13
× × ×
1Notethatthenumberofsolutionsat3σ referstothenumberoftexturesthatareconsistentwithdata
at 3σ but not at 1σ CL.
– 6 –
It also predicts
m = 12meV, m = 15meV, m = 51meV, m = 11meV,
1 2 3 ββ
(3.3)
δ = 0.41π, α = 1.5π, α = 0.70π.
21 31
Similarly, for the texture T4 corresponding to
123,56
 
a a a
m a 0 b , (3.4)
ν ≡  
a b b
the minimum of χ2 is given by χ2 = 5.96. This minimum is obtained if we take, for
min
instance, a = 28.56 meV and b = 18.8744 meV, argb = 1.00845π, which lead to the
| |
low-energy neutrino parameters
∆m2 = 7.59 10−5eV2, ∆m2 = 2.38 10−3eV2,
21 31
× | | × (3.5)
sin2θ = 3.25 10−1, sin2θ = 5.74 10−1, sin2θ = 2.57 10−2,
12 23 13
× × ×
and the predictions
m = 50meV, m = 51meV, m = 10meV, m = 29meV,
1 2 3 ββ
(3.6)
δ = 0.32π, α = 1.3π, α = 0.49π.
21 31
Finally, for the texture T6 one has
123,45
 
a a a
m a b b . (3.7)
ν ≡  
a b 0
In this case, the minimum of χ2 corresponds to a somewhat larger value, χ2 = 20.00,
min
which can be reached, e.g., with a = 28.56 meV and b = 18.8746 meV, argb = 0.99155π,
| |
leading to the following low-energy neutrino observables:
∆m2 = 7.59 10−5eV2, ∆m2 = 2.38 10−3eV2,
21 31
× | | × (3.8)
sin2θ = 3.25 10−1, sin2θ = 4.26 10−1, sin2θ = 2.57 10−2.
12 23 13
× × ×
The predicted mass spectrum and CP-violating phases are
m = 50meV, m = 51meV, m = 10meV, m = 29meV,
1 2 3 ββ
(3.9)
δ = 0.68π, α = 0.67π, α = 1.5π.
21 31
For each texture, we have also performed a random scan of the input parameters a
and b in order to determine the viable ranges at 3σ for the quantities δ, m, m , α and
ββ 21
α . The results are collected in table 4 of appendix A for the textures T2 , T4
31 136,45 123,56
and T6 . From the table one sees that neutrino observables are quite constrained in the
123,45
three cases. In particular, for the texture T2 the Majorana phase α is predicted to
136,45 21
be approximatively 3π/2.
– 7 –
50° 60°
40° 50°
2 3
1 2
θ θ
30° 40°
20° 30°
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
δ/π δ/π
10° 2.0
1.6
9°
1.2
π
3 /
θ1 31
α 0.8
8°
0.4
7° 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.2 1.6 2.0
δ/π α /π
21
Figure1. Predictionsfor3-parametertexturesT2 (NO,redpoints),T4 (IO,greenpoints)
136,45 123,56
and T6 (IO, blue points).
123,45
In figure 1, we show the correlation of the Dirac phase δ with the three mixing angles,
as well as the correlation between the Majorana phases. Notice that in virtue to the
symmetry (2.8) we restrict the values of δ to [0,π]. The effective mass parameter m as
ββ
a function of the lightest neutrino mass m is plotted in figure 2(a). As can be seen in the
figure, the predicted values for m are not far away from the values expected in the future
ββ
CUORE [24] and SuperNEMO [25] 0νββ experiments.
Notice that the textures T4 and T6 are related through the transformation
123,56 123,45
given in equation (2.6) and, therefore, their predictions are similar, except for the mixing
angle θ and the Dirac phase δ that are transformed according to equation (2.7). We
23
remark that for the Majorana phases α and α this conclusion may not be obvious in
21 31
table 4 and the numerical examples presented above, since after bringing the value of δ to
the range [0,π], the Majorana phases flip their signs. For the case of the texture T2 ,
136,45
one has necessarily sin2θ > 0.6, i.e., θ > 51◦ (see figure 1) and thus the transformed
23 23
texture obtained from equation (2.6) does not lead to a viable neutrino pattern.
– 8 –
100 100
IO IO
10 10
V] V]
e e
m m
[β NO [β NO
mβ mβ
1 T2 1
136,45
T4
123,56
T6
123,45
0.1 0.1
0.1 1 10 100 0.1 1 10 100
m[meV] m[meV]
(a) Class 3 2 0 (b) Class 4 1
1 1 1 1 2
100 100
IO IO
10 10
V] V]
e e
m m
[β NO [β NO
mβ mβ
1 1
0.1 0.1
0.1 1 10 100 0.1 1 10 100
m[meV] m[meV]
(c) Class 2 1 0 (d) Class 2
1 2 2 3
Figure 2. Predictions for m for different texture classes with NO (red colour) and IO (blue
ββ
colour). The inner dark bands correspond to the 1σ neutrino oscillation parameters, while the
lighter outer bands include the 3σ uncertainties. The green horizontal band corresponds to the
expected sensitivity in near future of 0νββ experiments.
4 Textures with four and five parameters
In this section, we consider the remaining classes in table 1 and look for solutions with
four and five physical parameters compatible with low-energy neutrino data at 3σ CL, and
eventually consistent with data at 1σ. As it turns out, all classes (with n = 4,5) contain
viable solutions at 1σ CL. In appendix A, the predictions of the corresponding neutrino
textures are summarised in tables 5–10 for n = 4 and tables 11 and 12 for n = 5. As
before, the values of δ are restricted to [0,π]. We remark that the five-parameter textures
belonging to the classes 1 0 and 2 1 0 are not considered, since they have been studied
4 2 1 3 1
in [9].
From the tables, we conclude that textures within the same class do not share neces-
sarily the same predictions. Remarkably, some of the classes predict a quite constrained
– 9 –