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1 SUPPRESSION OF 0ν2β DECAY FROM CP VIOLATION
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MAREKGO´Z´DZ´ andWIESL AWA.KAMIN´SKI
a
J Department of Informatics, Maria Curie-Sk lodowska University,
pl. Marii Curie–Sk lodowskiej 5, 20-031 Lublin, Poland
5
[email protected]
] [email protected]
h
p
Received(receiveddate)
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p Accepted (DayMonthYear)
e
h
Theobservedphenomenonofneutrinooscillationsisinterpretedastheproofthatneutri-
[
nosmusthavemass.Asthisistruefortheneutrinosinthemassbasis,themassmatrix
1 in the flavor (weak) basis may still contain zeros. This can happen if the CP violating
v phases, usually neglected, come into play and result in suppression of processes which
6 half-lifedependsonthemassesofνe,νµ,orντ.Inthepresentpaperweinvestigatethe
3 possibilityofsuchsuppressionoftheneutrinolessdoublebetadecay(0ν2β).
2
1
1. Introduction
.
1
1
0 The oscillations of neutrinos have been confirmed by a number of experiments.
2 The usual interpretation of this fact within standard quantum mechanics implies
1 that the quantities ∆m2 = m2 m2 must be non-zero, thus the neutrinos (with
: ij | i − j|
v the possible exception of the lightest one) must be massive. This calls for need of
i
X extendingthestandardmodelofparticlesandinteractions(SM),inwhichneutrinos
r are assumed to be massless.
a
Neutrinosrevealalsoonemorecharacteristicfeature,namely,the particlesthat
take part in weak interactions, like the β decay, are not the same particles that
propagatethroughspace.Inanotherwords,theν , ν ,andν neutrinos,whichare
e µ τ
producedin weakprocessesare saidto be in the interaction(or flavor)basis,while
neutrinos that propagate (eg. from the Sun to Earth, from radioactive source to
the detector etc.) are in the physical basis and are labeled by ν , ν , and ν . The
1 2 3
statesν ,i=1,2,3,aremasseigenstates,sothatonecanspeakabouttheirmasses
i
m . The weak eigenstates ν , α=e,µ,τ, are linear combinations of ν ,
i α i
ν =U ν , (1)
α αi i
anddo notpossessdefinite masses.The neutrinophenomenologicalmassmatrix in
the flavor basis, , is usually written as
M
=U∗diag(m ,m ,m )U†, (2)
1 2 3
M
where U in Eqs. (1) and (2) is the unitary Pontecorvo-Maki-Nakagawa-Sakata
(PMNS) neutrino mixing matrix.For massiveMajorananeutrinos it canbe conve-
1
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2 M. G´o´zd´z and W. A. Kamin´ski
niently parameterized by three mixing angles θ , θ , and θ , two Majorana CP
12 13 23
violating phases α , and α , and one Dirac CP violating phase δ:
12 13
c c s c s e−iδ
12 13 12 13 13
U = s12c23 c12s23s13eiδ c12c23 s12s23s13eiδ s23c13
− − −
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
diag(1,eiα12,eiα13), (3)
×
where s sinθ , c cosθ . Three mixing angles θ vary between 0 and π/2.
ij ij ij ij ij
≡ ≡
The CP violating phases vary between 0 and 2π.
The investigation of neutrino oscillations provides pieces of information about
themixinganglesandthedifferencesofmassessquared∆m2 .Itgivesus,however,
ij
no information about the absolute masses and the possible CP phases, which do
not affect the oscillation probabilities. To obtain the absolute values of neutrino
3
masses,one hasto lookfor the neutrinolessdouble beta decay(0ν2β), investigate
4
the end-pointofthe beta decayspectruminTritium, orrelayonthe ratherrough
cosmological models of supernova explosions, large scale structure of the Universe
5
etc. The aforementioned 0ν2β process is at present the only one known, capable
ofdistinguishingbetweenMajoranaandDiracneutrinos,whichmakesthe searches
for 0ν2β particularly important.
Another problem, which is not resolved by the oscillation experiments, is the
hierarchyofmasses.Fromthe differences of massessquaredone cannot deduce the
actualalignmentofm .In agreementwith the presentexperimentalknowledgeare
i
two scenarios:
(i) the so-called normal hierarchy scenario (NH), which is defined by
m m m ,
1 2 3
≪ ≪
(ii) and the inverted hierarchy (IH), in which m is the lightest one,
3
m m <m .
3 1 2
≪
At present we have no hint which of the hierarchies is realized in nature, so both
of them have to be considered.
The neutrinoless double beta decay is a second order process, forbidden in the
standard model due to lepton number violation (∆L = 2). It may occur only
if the neutrinos are Majorana particles and some mechanism of lepton number
violation is introduced. These conditions are fullfilled by many extensions of the
SM, like the R-parity violating MSSM and others. Therefore the 0ν2β process,
as the possible source of information about the physics beyond SM, is intensively
investigated theoretically and searched for in experiments. The half-life of 0ν2β is
proportionaltotheso-calledeffectiveneutrinomass,whichisjustthe element
ee
M
ofthe neutrino mass matrix Eq.(2). It is a commonbelief that the confirmationof
neutrino oscillations gives a strong back-up for the 0ν2β decay. However, in such
discussions one usually forgets about the possible CP phases. In the present paper
January 6, 2012 1:13 WSPC/INSTRUCTION FILE art12
Suppression of 0ν2β Decay from CP Violation 3
we show, that the CP phases may in fact completely suppress the 0ν2β process,
while still being in agreement with all the oscillation data.
2. Calculations and Results
1
The best-fit neutrino oscillation parameters may be summarized as follows:
∆m2 =2.1 10−3 eV2, sin22θ =1.00,
23 × 23
∆m2 =7.1 10−5 eV2, tan2θ =0.40.
12 × 12
For the angle θ only the upper bound is known. From exclusion plot obtained
13
2
from the data of the reactor experiment CHOOZ we have
sin2θ 0.05 (90% c.l.),
13
≤
with zero being the best-fit value.
Theelementsofthe neutrinomassmatrix becomenowarathercomplicated
M
functions of the phases. Let us start with the case, in which we take sin2θ = 0.
13
Now, directly from Eqs. (2) and (3) one can obtain the following expressions:
=c2 m +s2 m e−i2α12 (5)
Mee 12 1 12 2
1
= = c s m +m e−i2α12 , (6)
eµ eτ 12 12 1 2
M −M √2 −
(cid:0) (cid:1)
1
= = s2 m +c2 m e−i2α12 +m e−i2α13 , (7)
Mµµ Mττ 2 12 1 12 2 3
(cid:0) (cid:1)
1
= s2 m +c2 m e−i2α12 m e−i2α13 . (8)
Mµτ −2 12 1 12 2 − 3
(cid:0) (cid:1)
The elements are complex so the physically relevant quantities are .
αβ αβ
M |M |
Thestrategynowis,thatwewanttofindsuchcombinationofthelightestneutrino
mass (m in NH and m in IH) and CP phases to obtain = 0. Because
1 3 αβ
|M |
= [ ( )2+ ( )2]1/2, where and stand for real and imaginary
αβ αβ αβ
|M | ℜ M ℑ M ℜ ℑ
parts, respectively, we are looking for solutions of the set of equations:
( )=0
αβ
ℜ M
(cid:26) ( )=0
αβ
ℑ M
for each element in the case of NH and IH.
Our results are presented in Tab. 1. As one sees, in the case of NH only the
Table1.Resultsforthecases13=0.
Element Hierarchy Mass
|Mee|=0 NH m1=3.557×10−3 eV
|Mµµ|=0 IH m3≥2.271×10−2 eV
|Mττ|=0 IH m3≥2.271×10−2 eV
|Mµτ|=0 IH m3≥2.272×10−2 eV
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4 M. G´o´zd´z and W. A. Kamin´ski
1
0.1
0.01
V) 0.001
e
M| (ee
| 1e-04
1e-05
1e-06
1e-06 1e-05 1e-04 0.001 0.01 0.1 1
m1 (eV)
Fig.1.|Mee|asthefunctionoflightestneutrinomassm1(normalhierarchyscenario).Thedashed
linecorrespondstosinθ13=0.05,thesolidlinetosinθ13=0.
element may be zero, and this situation requires a precisely fine-tuned value
ee
M
of m . The corresponding phase α is given by the condition cos(2α ) = 1.
1 12 12
−
Other elements, of less interest for us, has been calculated as well. It turned out
that it is impossible to find zerosolutions for the eµ andeτ elements, while forthe
remaining ones there are solutions in the case of IH only. One has to bear in mind,
that the lower bounds on m come out from our calculations but do not take into
3
accountotherconstraintsonneutrinomasses.Forexampletheglobalastrophysical
and cosmological limit on the sum of all three masses m is roughly given by
i
1 eV, which means that m cannot exceed this value. NoPtice also that, for obvious
3
reasons, it is impossible to have all diagonal elements being zero at one time.
In the case in which s = 0, the expressions for become very compli-
13 αβ
6 6 M
cated functions. Some of them depend simultaneously on all three phases which
makestheiranalysisnearlyimpossible,alsonumerically.Fortunately,theeeelement
remains relatively simple:
=c2 c2 m +c2 s2 m e−i2φ2 +s2 m e−i2(α13−δ). (9)
Mee 13 12 1 13 12 2 13 3
Assuming the maximal allowed value s2 = 0.05 one obtains the zero solution for
13
NH and
8.047 10−5 eV m 7.473 10−3 eV.
1
× ≤ ≤ ×
Onenoticesthat,asexpected,thissolutioncontainstheoneobtainedinthesimpler
case s =0. The possible values of are depicted on Fig. 1.
13 ee
|M |
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Suppression of 0ν2β Decay from CP Violation 5
3. Conclusions
We have shown that the popular practice of neglecting CP phases in the neutrino
mass matrix may have severe consequences. In particular, for certain combination
of parameters (phases and the mass of the lightest neutrino), weak processes like
the neutrinoless double beta decay may be suppressed. However, the inclusion of
CP phases in the calculations is a highly non-linear problem, which may be a
big obstacle. It is thus desirable to find a method of determining the possible CP
violation in the neutrino sector experimentally.
Itisinterestingtonote,thattheobtainedvaluesofm whichgivezerosolutions
1
for the element are within the range of experimental and theoretical bounds
ee
M
(roughly speaking they do not exceed the limit of 1 eV).
Acknowledgements
Thefirstauthorisgratefulforthe financialsupportfromtheFoundationforPolish
Science. He would like also to express his gratitude to prof. A. Faessler for his
hospitality at the University of Tu¨bingen during the Summer 2006.
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