Real and complex random neutrino mass matrices and theta13

0.42 MB·

by  Janusz Gluza

Real and complex random neutrino mass matrices and θ
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Janusz Gluza1 and Robert Szafron1
1Institute of Physics, University of Silesia, Uniwersytecka 4, PL-40007 Katowice, Poland
Recently it has been shown that one of the basic parameters of the neutrino sector, so called
θ angle is very small, but quite probably non-zero. We argue that the small value of θ can
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still be reproduced easily by a wide spectrum of randomly generated models of neutrino masses.
For that we consider real and complex neutrino mass matrices, also including sterile neutrinos. A
qualitative difference between results for real and complex mass matrices in the region of small θ
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valuesisobserved. Weshowthatstatisticallythepresentexperimentaldataprefersrandommodels
of neutrino masses with sterile neutrinos.
PACSnumbers: 14.60.Pq;02.10.Yn,14.60.Lm
2 Since their first appearance in modern science neutri- Probability
1 nos have played an important part in our understanding
0 Dirac
ofthelawsofparticlephysics. Toseehowimportantthe 0.25
2
weak interaction of neutrinos is in Nature lets just men-
n tion the mechanism in which the Sun is shining [1]. It 0.2 Majorana
a
J is also very well known that they influence evolution of 0.15 See(cid:45)Saw
the whole Universe as their tiny masses have an impact
3
0.1
on the dynamics of the expansion of the Universe [2, 3].

] Without any doubt the investigation of the properties of 0.05
h
these particles can reveal many interesting, hidden until
p
now physical phenomena or explain many hypothetical R
- 10(cid:45)5 10(cid:45)4 10(cid:45)3 10(cid:45)2 10(cid:45)1 1
p ideas.
e The theory of neutrino oscillations requires nonzero
h
neutrino masses as well as nonzero neutrino mixing an- FIG. 1: Frequency probability for R parameter obtained
[ gles. from 107 randomly generated three dimensional different
3 types of matrices. No constraints given by Eqs.3-7 are ap-
TheT2KCollaborationrecentlyannouncedthatanew
v plied.
measurement [4] has yielded a nonzero θ for CP con-
8 13
7 serving case with δCP = 0 (results at 90% confidence
2 level), i.e. Naturalness means that the neutrino mass matrices are
7
0.03≤ sin22θ ≤0.28, normal mass hierarchy, (1) generated randomly. Apart from neutrino physics, see
. 13
1 e.g. [11, 17], random matrices came to be an eminent
0.04≤ sin22θ ≤0.34, inverted mass hierarchy.(2)
1 13 tool used in many different fields of science, including
1
The Minos collaboration published similar results few day-to-day life problems [18].
1
days later [5]. These results with nonzero θ are of pri- Let us first consider a symmetric, three dimensional
: 13
v mary importance for the future of particle physics [6, 7]. matrixwithrandomelementsintherange[−1,1]. Phys-
i
X Present global fits of the data give [8] (at 1 σ C.L., icallythiscaserealizesMajoranatypeofneutrinos. Simi-
[34]) larly,weconsideralsoDiracneutrinos(generatedbygen-
r
a eral real random matrices) and so-called see-saw neutri-
7.05·10−5 eV2 ≤ ∆m2 ≤8.34·10−5 eV2, (3)
21 noswithmassesm definedthroughrelationmTM−1m
2.07·10−3 eV2 ≤ ∆m232 ≤2.75·10−3 eV2, (4) (hereagainmatriceνsmD andmRaregeneratedDranRdomlDy
0.39≤ sin2θ ≤0.5, (5) in the range [−1,1]). Next, we apply the singular value
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0.291≤ sin2θ ≤0.324, (6) decomposition theorem [19] to calculate the mixing ma-
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trixU whichdiagonalizestherandommatriceswestarted
0.008≤ sin2θ ≤0.036. (7)
13 from. Usingthestandardparametrisation,weareableto
There are many theoretical models which tune the linkexperimentalmixinganglestotheelementsofmatrix
space of possible neutrino parameters to get agreement U, e.g. sin2θ =|U |2.
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with these experimental data. The new data from the InFig.1werecoveroneoftheresultsdiscussedalready
T2K collaboration generated a new challenge for that. in [11], see Fig.1. Here R = ∆m2 /∆m2 is the ratio of
21 32
Frequently the pattern of a flavour symmetry is invoked, the smallest to the next smallest values of differences of
forrecenttheoreticaladjustmentsofthenonzeroθ mix- squared neutrino masses, m <m <m . Both in Fig.1
13 1 2 3
ing angle, see e.g. [10]. Here we face the problem differ- andnextplotsresultsaregeneratedbasedon107random
entlyaskinghownaturalaresmall,nonzerovaluesofθ . mass matrices. Hall et al. approach has an advantage of
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2
Probability Probability
0.07 Complex 0.07 Complex
Real Real
0.05 0.05
Data Data
0.03 0.03
0.01 0.01
Θ Θ
13 13
Π Π Π Π 5Π Π Π Π Π Π 5Π Π
12 6 4 3 12 2 12 6 4 3 12 2
FIG.2: Frequencyprobabilityforθ obtainedfrom107ran- FIG.3: Frequencyprobabilityforθ obtainedfrom107ran-
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domly generated real and imaginary three dimensional sym- domly generated real and imaginary three dimensional sym-
metric matrices (Majorana neutrinos). No constraints given metricmatrices(Majorananeutrinos). ConstraintsbyEqs.3-
by Eqs.3-6 are applied. Gray vertical band represents exper- 6 are applied at the 3σ level. Gray vertical band represents
imental values of θ , Eq.7 at the 1σ level. experimental values of θ , Eq.7 at the 3σ level.
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pure simplicity. A modified versions have been consid- try is violated (which is the case for randomly generated
ered in many papers, see e.g. [12–16]. complex mass matrices) then θ must be different from
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Let us note that the see-saw neutrino masses are triv- zero. We found that frequency probability for the phys-
ially not well defined if elements of M approach zero in ical CP violating phase δ which emerges from randomly
R
relation mTM−1m , which can happen as elements of generated mass matrices is constant (so this is a trivial
D R D
M andM aregeneratedinthe[−1,1]range. Thuswe plotandwedonotshowithere. Frompuremathematics
D R
willconsideronlyDiracandMajoranatypeofneutrinos. (a probability measure) follows that in such case a prob-
To disentangle these two types of neutrinos is of great ability to get δ equal to zero or π is zero, so probability
importance for neutrino physics [20]. If we connect the distributions should tend to zero for θ13 → 0, what can
mechanism of generation of their tiny masses with some be seen on histograms, see solid line in Fig. 2. If Eqs.3-6
heavy states, this issue starts to be important for high are applied, a number of mass matrices which cover the
energy colliders like e+e− [21, 22], e−e− [23] and LHC experimentally interesting values of θ13 decreases. For
[24]. instance, using 1σ cuts in Eqs.3-6, only 4886 real and
4686 complex matrices remain out of initial 107 random
We can extend this discussion to the case where el-
matrices. Using 3σ cuts we are left with 31853 real and
ements of the neutrino mass matrix involve complex
29141 complex matrices, see Fig.3. Comparing between
phases (random numbers with modulus in the interval
Dirac and Majorana cases more mass matrices in Majo-
[0,1] and a phase in a range [0;2π]). It might seem
ranacasesurvivethecuts. Themainconclusionremains
that real and imaginary matrices should produce qual-
however the same as in the case where no cuts Eqs.3-6
itatively the same results: situation differs only as an
areapplied: therearemorematriceswhichreproduceex-
additionaldegreeoffreedom(phase)entersineachentry
perimental data for real matrices and complex matrices
of the mass matrix. However, this is not true, as in fact
donotcondenseintheexperimentallyinterestingregion.
theydiffer. Thischangedoesnotaffecttheshapesofthe
eigenvalues distributions in a significant way but other Itisinterestingthatthelastexperimentaldataimplies
observablesdistributionslikeelementsofthemixingma- the presence of a fourth sterile neutrino [25–28]. Usually
trix may change. This is exactly the case for θ . the so called 3+1 and 3+2 models of sterile neutrinos
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Fig.2showstheresultsofnumericalpredictionsforθ , are considered [35]. We then inspect the mass matri-
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additional constraints given by Eqs.3-6 are not applied. ces of dimensions four and five. Here we focus on an
Wecanseethattherealandimaginaryrandommatrices element |U13|2 and define an effective angle such that
behave qualitatively differently in vicinity of small θ13 sin2θ13 =|U13|2, sowecancomparetheresultsobtained
and real matrices fit better. That frequency probability from models with different number of neutrinos.
of random complex matrices tends to zero with θ → 0 In a case of more than three dimensional mass matri-
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can be understood in the following way. For real matri- ces relations among the elements of the mixing matrix
cesCPcomplexphaseiszeroorπ andCPisnotviolated and experimentally defined mixing angles are more in-
when θ =0 and therefore there is no direct restriction volved, therefore we will consider this case without cuts
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on values of θ . Reversing the argument, if CP symme- driven by Eqs.3-7 and without a discussion of the im-
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3
Probability Probability
Complex
3x3
0.07
4x4 Real
0.07
0.05 5x5 Data
0.05
0.03
0.03
0.01
0.01
Θ
Π Π Π Π 5Π Π 13 0.2 0.4 0.6 0.8 1 (cid:200)U14(cid:200)
12 6 4 3 12 2
FIG. 4: Shifts in the direction of small values of θ with FIG. 5: Probability distributions for |U | elements of the
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increasing number of sterile neutrinos. No constraints given neutrino mixing matrix when the constraint Eq.7 for θ is
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by Eqs.3-6 are applied. applied. The gray vertical band stands for 3σ values which
follow from global fits [32].
pact of additional mixing matrix parameters of four and Probability
five dimensional mass matrices. We proceed as in the Complex
previous case and generate random matrices and then
Real
calculatethemixingmatrix. InFig.4weshowresultsfor 0.07
Data
Majorana (symmetric) real matrices of different dimen-
sions. We observe that the more neutrinos we have the 0.05
bigger number of random matrices reproduce the exper-
imental value of the θ parameter. Similar results give 0.03
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Dirac neutrinos.
We have no clear explanation why the probability of 0.01
getting small values for θ13 grows with dimensionality of 0.2 0.4 0.6 0.8 1 (cid:200)U24(cid:200)
matrices. There is a mathematical law of Wigner [29]
connected with real symmetric random matrices which
FIG. 6: Probability distributions for |U | elements of the
states that their eigenvalues accumulate around zero 24
neutrino mixing matrix when the constraint Eq.7 for θ is
(Wigner’s semicircle law [30] is exact in the limit of infi- 13
applied. The gray vertical band stands for upper bounds de-
nite dimensions of matrices). To our knowledge, there is
rived by MINOS [33].
no relation connecting the distributions of the values of
elements of eigenvectors with the dimensionality of real
symmetric matrices. distribution argument suggests.) It is interesting that
Nonetheless,letustrytounderstandtheresultweob- analyses of recent experiments predict non-zero values
tained. Intheanarchicalmodelofneutrinomasses,when forthem[31,32]. InFig.5andFig.6theplotsaregiven
no cuts on parameters are applied, mixing angles have for real and complex neutrino Majorana mass matrices
very similar probability distributions. This is because (Diracpatternsareverysimilar). Wecanseethatproba-
every element of the mass matrix has exactly the same bilitydistributionsforbothrealandcomplexcasesreach
probability distribution. Therefore, if the dimension of a maximum at larger values of |U | (|U | (cid:39) 0.7
14 14 max
the mass matrix grows, the unitarity of the mixing ma- for a real case, |U | (cid:39) 0.58 for a complex case)
14 max
trixconstrainssumsofsquaredmodulusofmixingmatrix when compared with |U | (|U | (cid:39) 0.1, real case;
24 24 max
elements,andanaveragevalueofeachofelementsofthe |U | (cid:39) 0.4, complex case). This is in agreement
14 max
mixingmatrixwilldecreasewithincreasingdimensional- with fits discussed in [32] (|U |2 < |U |2 for both low
µ4 e4
ity. Thisforcestheprobabilitydistributiontotakehigher and high energy neutrino experimental fits). In Fig.5 we
values for small angles as the dimension grows, a kind of can see that the most frequent values of |U | do not co-
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pattern obtained in Fig.4. incide with its experimentally preferable region, though
We can make a step further and consider for the first in this experimental region they can also be substantial.
time implications of nonzero θ on values of elements More involved analyses including other experimental
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|U | and |U | in the 3+1 sterile model when the con- constraints and various mass hierarchies in 3 + 1 and
14 24
straint Eq.7 for θ is applied. (Without cuts, probabil- 3+2modelsgobeyondthisBriefReportandareleftfor
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ity distributions for |U | and |U | are practically the a future work.
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same as for θ in Fig.2, as the just discussed equal- In conclusion, though random matrices can not solve
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4
fundamentalproblemsinneutrinophysics,theygenerate 702 (2011) 220 [arXiv:1104.0602 [hep-ph]].
intriguinghintsonthenatureofneutrinomassmatrices. [16] S. -F. Ge, D. A. Dicus and W. W. Repko,
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manuscript. Work supported in part by the Research [19] SVDNash,J.C.TheSingular-ValueDecompositionand
Its Use to Solve Least-Squares Problems
Executive Agency (REA) of the European Union un-
Ch. 3 in Compact Numerical Methods for Computers:
dertheGrantAgreementnumberPITN-GA-2010-264564
LinearAlgebraandFunctionMinimisation,2nded.Bris-
(LHCPhenoNet),bythePolishMinistryofScienceunder
tol, England: Adam Hilger, pp. 30-48, 1990.
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