Neutrinoless double-beta decay and seesaw mechanism

Abstract

From the standard seesaw mechanism of neutrino mass generation, which is
based on the assumption that the lepton number is violated at a large
(~10exp(+15) GeV) scale, follows that the neutrinoless double-beta decay is
ruled by the Majorana neutrino mass mechanism. Within this notion, for the
inverted neutrino-mass hierarchy we derive allowed ranges of half-lives of the
neutrinoless double-beta decay for nuclei of experimental interest with
different sets of nuclear matrix elements. The present-day results of the
calculation of the neutrinoless double-beta decay nuclear matrix elements are
briefly discussed. We argue that if neutrinoless double-beta decay will be
observed in future experiments sensitive to the effective Majorana mass in the
inverted mass hierarchy region, a comparison of the derived ranges with
measured half-lives will allow us to probe the standard seesaw mechanism
assuming that future cosmological data will establish the sum of neutrino
masses to be about 0.2 eV.

Full text

Page 1

arXiv:1104.1952v3 [hep-ph] 11 Oct 2011
Neutrinoless double-beta decay and seesaw mechanism
Samoil M. Bilenky,1,2Amand Faessler,3Walter Potzel,1and FedorˇSimkovic2,4
1Physik-Department E15, Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany
2Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow region, Russia
3Institute of Theoretical Physics, University of Tuebingen, 72076 Tuebingen, Germany
4Department of Nuclear Physics and Biophysics, Comenius University,
Mlynsk´ a dolina F1, SK–842 15 Bratislava, Slovakia
(Dated: October 12, 2011)
From the standard seesaw mechanism of neutrino mass generation, which is based on the assump-
tion that the lepton number is violated at a large (∼ 1015GeV) scale, follows that the neutrinoless
double-beta decay (0νββ-decay) is ruled by the Majorana neutrino mass mechanism. Within this no-
tion, for the inverted neutrino-mass hierarchy we derive allowed ranges of half-lives of the 0νββ-decay
for nuclei of experimental interest with different sets of nuclear matrix elements. The present-day
results of the calculation of the 0νββ-decay nuclear matrix elements are briefly discussed. We argue
that if 0νββ-decay will be observed in future experiments sensitive to the effective Majorana mass
in the inverted mass hierarchy region, a comparison of the derived ranges with measured half-lives
will allow us to probe the standard seesaw mechanism assuming that future cosmological data will
establish the sum of neutrino masses to be about 0.2 eV.
PACS numbers: 98.80.Es,23.40.Bw; 23.40.Hc
I. INTRODUCTION
The observation of neutrino oscillations in atmospheric
[1], solar [2], reactor [3] and accelerator [4, 5] neutrino
experiments is the most important recent discovery in
particle physics. It is very unlikely that neutrino masses,
many orders of magnitude smaller than the masses of
quarks and leptons, are generated by the standard Higgs
mechanism. Small neutrino masses and neutrino mixing
are commonly considered as a signature of physics be-
yond the Standard Model (SM). Several beyond the SM
mechanisms of neutrino mass generation were proposed.
The most viable and plausible mechanism is the famous
seesaw mechanism which is based on the assumption that
the total lepton number L is violated at a scale much
larger than the electroweak scale v = (√2GF)−1/2≃ 246
GeV.
If the total lepton number is violated, neutrinos νiwith
definite masses are Majorana particles. After the discov-
ery of neutrino oscillations the problem of the nature of
neutrinos with definite masses (Majorana or Dirac?) be-
came the most pressing issue.
Information about the nature of neutrinos with def-
inite masses can not be obtained via the investigation
of neutrino oscillations [6]. In order to obtain such an
information it is necessary to study processes in which
the total lepton number L is violated. The investigation
of neutrinoless double-beta decay (0νββ-decay) of even-
even nuclei
(A,Z) → (A,Z + 2) + e−+ e−
is the most sensitive way to search for the effects of the
lepton number violation.
The observation of the 0νββ-decay will prove that νi
are Majorana particles. In this paper we show that an ev-
idence of this process in future 0νββ-decay experiments
sensitive to the effective Majorana mass in the inverted
mass hierarchy region could allow us to obtain informa-
tion about the validity of the original seesaw idea [7] of
neutrino mass generation associated with a violation of
the total lepton number at GUT scale assuming that a
proper information about the lightest neutrino mass will
be available from future cosmological data.
II. SEESAW MECHANISM OF NEUTRINO
MASS GENERATION
The standard seesaw mechanism (type I seesaw) is
based on the assumption that there exist heavy Majo-
rana leptons Ni, singlets of the SUL(2) × U(1) group,
which have the following lepton number violating Yukawa
interactions with lepton and Higgs doublets
L = −√2
i,l
?
YliLlLNiR˜H + h.c..(1)
Here
LlL=
?νlL
lL
?
,H =
?H(+)
H(0)
?
(2)
are lepton and Higgs doublets,˜H = iτ2H∗, Yil are di-
mensionless constants and
Ni= Nc
i= C¯ NT
i
is the field of heavy Majorana leptons with mass Mi
which is much larger than v.
At electroweak energies for the processes with virtual
Nithe interactions (1) generate the effective Lagrangian
Leff= −1
Λ
?
l′,l,i
Ll′L˜H
?
i
(Yl′i
Λ
MiYli)C˜ HT(LlL)T+ h.c.,
(3)

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which does not conserve the total lepton number L and
is the only effective Lagrangian of the dimension five [8].
In (3), the parameter Λ has the dimension of mass and
characterizes the scale of new physics beyond the SM.
After spontaneous violation of the electroweak sym-
metry the Lagrangian (3) generates the left-handed Ma-
jorana mass term
LM= −1
2
?
l′,l
νl′LML
l′l(νlL)c+ h.c. = −1
2
?
i
mi¯ νiνi,
(4)
where
ML= Yv2
MYT= UmUT, (5)
νc
mass miand the flavor field νlLis given by the standard
mixing relation
i= νi is the field of the Majorana neutrino with the
νlL=
?
i
UliνiL. (6)
Here, Uli are the elements of the Pontecorvo-Maki-
Nakagawa-Sakata neutrino mixing matrix [9, 10]. The
size of neutrino masses is determined by the seesaw fac-
tor
Mi. From the existing data we can estimate that
Mi≃ (1014− 1015) GeV.
Let us stress that from the point of view of the stan-
dard seesaw approach small Majorana neutrino masses
are the only low energy signature of physics beyond the
SM at a GUT scale where the total lepton number is vi-
olated.1
The effective Lagrangian Leff and, consequently, the
left-handed Majorana mass term (4) can be generated
not only by the interaction (1) but also by an interaction
of lepton pairs and a Higgs pair with a triplet heavy
scalar boson ∆ (type II seesaw) and by an interaction of
lepton-Higgs pairs with heavy Majorana triplet fermion
Σ (type III seesaw) (see [12]).
From the previous discussion we can conclude that if
the total lepton number L is violated at a GUT scale due
to the existence of a heavy singlet (or triplet) Majorana
fermion or a heavy triplet scalar boson interacting with
standard lepton and Higgs doublets then
v2
• neutrinos have small, seesaw suppressed masses (in
accordance with the existing experimental data),
• neutrinos with definite masses are Majorana parti-
cles and the only mechanism of the 0νββ-decay is
the exchange of virtual Majorana neutrinos.
In this paper we will explore these just mentioned general
consequence of the standard seesaw mechanism.
1In the early Universe at very high temperatures heavy Majorana
leptons Ni can be produced. Their CP-violating decays could
lead to the baryon asymmetry of the Universe (see [11] and ref-
erences therein).
III.
0νββ-DECAY: NUCLEAR MATRIX
ELEMENTS
Neutrinoless double β-decay of even-even nuclei is a
process of second order in the Fermi constant GF with
the exchange of virtual Majorana neutrinos between n−
p−e−vertexes. The mixed neutrino propagator has the
form
?
i
U2
ei
?1 − γ5
2
?γ · p + mi
p2− m2
?1 − γ5
2
i
?1 − γ5
2
?
C
≃ mββ
1
p2
?
C, (7)
where
mββ=
?
i
U2
eimi
(8)
is the effective Majorana mass.
Let us stress that
• due to the V − A structure of the weak charged
current and neutrino mixing the matrix elements
of the 0νββ-decay are proportional to mββ.
• the average momentum of the virtual neutrinos is
about 100 MeV. Thus, p2≫ m2
matrix elements do not depend on mi. As a result,
in the matrix elements of the 0νββ-decay, neutrino
properties and nuclear properties are factorized.
iand the nuclear
The inverted half-life of the 0νββ-decay is given by the
following general expression [13]
1
T0ν
1/2(A,Z)= |mββ|2|M(A,Z)|2G0 ν(E0,Z). (9)
Here M(A,Z) is the nuclear matrix element (NME) (ma-
trix element between states of the initial and the final
nuclei of the integrated product of two hadron charged
currents and the neutrino propagator) and G0ν(E0,Z) is
a known phase-space factor (E0is the energy release).2
For our calculations we will need the values of the
0νββ-decay NMEs for different nuclei of experimental in-
terest. We will briefly discuss here the present-day situa-
tion of the calculation of NMEs, compare existing meth-
ods, stress differences between them and present current
values of NMEs.
The calculation of the NME is a complicated nuclear
many-body problem. During many years two approaches
were used: the Quasiparticle Random Phase Approxi-
mation (QRPA)[15–17] and the Interacting Shell Model
(ISM)[18].There are substantial differences between
both approaches. The QRPA treats a large single particle
2For numerical values of G0 ν(E0,Z) see[14].

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TABLE I. The NME of the 0νββ-decay calculated in the framework of different approaches: interacting shell model (ISM)
[18], quasiparticle random phase approximation (QRPA) [15, 17], projected Hartree-Fock Bogoliubov approach (PHFB, PQQ2
parametrization) ) [20], energy density functional method (EDF) [22] and interacting boson model (IBM) [21]. The Miller-
Spencer Jastrow two-nucleon short-range correlations are taken into account. The EDF results are multiplied by 0.80 in order
to account for the difference between UCOM and Jastrow [16]. gA = 1.25 and R = 1.2A1/3are assumed.
Transition
M(A,Z)
ISM [18] QRPA [15, 17] IBM-2 [21] PHFB [20] EDF [22]
0.61
2.304.92 5.47
2.18 4.394.41
1.22
3.64 3.73
2.99
2.10
2.343.974.52
2.123.56 4.06
1.76 2.30
3.16 2.32
48Ca →48Ti
76Ge →72Se
82Se →82Kr
96Zr →96Mo
100Mo →100Ru
116Cd →116Sn
124Sn →124Te
128Te →128Xe
130Te →130Xe
136Xe →136Ba
150Nd →150Sm
1.91
3.70
3.39
4.54
4.08
3.80
3.87
3.30
4.12
3.38
1.37
2.78
6.55
3.89
4.36
3.16
model space, but truncates heavily the included configu-
rations. The ISM, by contrast, treats a small fraction of
this model space, but allows the nucleons to correlate in
many different ways. We note that the latest QRPA re-
sults of the Jyv¨ askyl¨ a-La Plata group [19] agree well with
those of the Tuebingen-Bratislava-Caltech group [15, 16]
by using the same way of adjusting the parameters of the
nuclear Hamiltonian [15].
In the last few years several new approaches have been
used for the calculation of the 0νββ-decay NMEs: the
angular momentum Projected Hartree-Fock-Bogoliubov
method (PHFB) [20], the Interacting Boson Model (IBM)
[21], and the Energy Density Functional method (EDF)
[22]. In the PHFB approach, the nucleon pairs different
from 0+in the intrinsic coordinate system are strongly
suppressed. In the framework of the ISM and the QRPA
approaches it was shown, however, that other neutron
pairs make significant contributions [23]. Let us notice
also that in the IBM approach only transitions of 0+and
2+neutron pairs into proton pairs are taken into account.
The EDF approach is an improvement with respect to the
PHFB approach. Beyond-mean-field effects are included
within the generating coordinate method with particle
number and angular momentum projection for both ini-
tial and final ground states. But, the quality of the IBM
and the EDF many-body wave functions have not been
tested yet by the calculation of the 2νββ-decay half-lives.
In Table I, recent results of the different methods are
summarized. The presented numbers have been obtained
with the unquenched value of the axial coupling constant
(gA= 1.25)3, Miller-Spencer Jastrow short-range corre-
lations [24] (the EDF values are multiplied by 0.80 in or-
3
A modern value of the axial-vector coupling constant is gA=
1.269. We note that in the referred calculations of the 0νββ-
der to account for the difference between the unitary cor-
relation operator method (UCOM) and the Jastrow ap-
proach [16]), the same nucleon dipole form-factors, higher
order corrections to the nucleon current and the nuclear
radius R = r0A1/3, with r0= 1.2 fm (the QRPA values
[15] for r0= 1.1 fm are rescaled with the factor 1.2/1.1).
Thus, the discrepancies among the results of different
approaches are solely related to the approximations on
which a given nuclear many-body method is based.
From Table I we see that the smallest values of NMEs
are obtained in the ISM approach. They are by about a
factor of 2-3 smaller in comparison with results of other
methods. The largest values of NME are obtained in
the IBM (76Ge and128Te), PHFB (100Mo,130Te and
150Nd), QRPA (150Nd) and EDF (48Ca,96Zr,116Cd,
124Sn and136Xe) approaches. NMEs obtained by the
QRPA and IBM methods are in a good agreement (with
the exception of150Nd). It is remarkable that for130Te
the results of four different methods (QRPA, PHFB, IBM
and EDF) are close to each other.
The differences among the listed methods of NME cal-
culations for the 0νββ-decay are due to the following
reasons:
(i) The mean field is used in different ways. As a result,
single particle occupancies of individual orbits of various
methods differ significantly from each other [25].
(ii) The residual interactions are of various origin and
renormalized in different ways.
(iii) Various sizes of the model space are taken into ac-
count.
(iv) Different many-body approximations are used in the
diagonalization of the nuclear Hamiltonian.
decay NMEs the previously accepted value gA = 1.25 was as-
sumed.

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10-4
10-3
10-2
10-1
100
101
m0 [eV]
10-4
10-3
10-2
10-1
100
101
|mββ| [eV]
NS
cosmology
IS
disfavored by
FIG. 1.
|mββ| as function of the lightest neutrino mass m0 for the
cases of normal (NS, m0 = m1) and inverted (IS, m0 = m3)
spectrum of neutrino masses. ∆m2
[5], ∆m2
[3] and 0.03(0.04) < sin22θ13 < 0.28(0.34) [38] for NS (IS) are
considered. The current limit of?3
the sum of neutrino masses excludes values of m0 larger 0.084
eV.
(Color online) Effective Majorana neutrino mass
A= (2.43±0.13)×10−3eV2
S= (7.65+0.13
−0.20)×10−5eV2[37], tan2θ12 = 0.452+0.035
−0.033
i=1mi ≤ 0.28 eV [39] for
Each of the applied methods has some advantages and
disadvantages.
IV. POSSIBLE PROBE OF THE MAJORANA
NEUTRINO MASS MECHANISM OF THE
0νββ-DECAY
Many experiments on the search for 0νββ-beta decay
of different nuclei were performed (see [26]). No indica-
tions in favor of 0νββ-decay were obtained in these exper-
iments. There exist, however, a claim of the observation
of the 0νββ-decay of76Ge made by some participants
of the Heidelberg-Moscow collaboration [27]. Their es-
timated value of the effective Majorana mass (assuming
a specific value for the NME) is |mββ| ≃ 0.4 eV. This
result will be checked by an independent experiment rel-
atively soon. In the new germanium experiment GERDA
[28], the Heidelberg-Moscow sensitivity will be reached in
about one year of measuring time.
From the most precise experiments on the search for
0νββ-decay the following bounds were inferred [29–31]:
|mββ| < (0.20 − 0.32) eV (76Ge),
< (0.30 − 0.71) eV (130Te),
< (0.50 − 0.96) eV (130Mo).
These bounds we obtained using the 0νββ-decay NMEs
of [32] calculated with Brueckner two-nucleon short-
range correlations.
(10)
In future experiments,
SuperNEMO
CUORE[30],
[35],
EXO[33],
SNO+ MAJORANA[34],
Kamland-ZEN and others [26], a sensitivity
[36],
|mββ| ≃ a few 10−2eV(11)
is planned to be reached.
The value of the effective Majorana mass strongly de-
pends on the character of the neutrino mass spectrum.
For the case of three neutrinos two types of mass spectra
are allowed by the neutrino oscillation data:
1. Normal spectrum (NS)
m1< m2< m3,∆m2
12≪ ∆m2
23, (12)
2. Inverted spectrum (IS)
m3< m1< m2,∆m2
12≪ |∆m2
13|. (13)
In the case of NS, we have for the neutrino masses
m2=
?
m2
0+ ∆m2
S,m3≃
?
m2
0+ ∆m2
A.(14)
For IS we find
m1=
?
m2
0+ ∆m2
A,m2≃
23(|∆m2
?
m2
0+ ∆m2
A.(15)
Here ∆m2
solar and atmospheric neutrino mass-squared differences,
respectively, and m0= m1(m3) is the lightest neutrino
mass for NS(IS).
In Fig. 1 the effective Majorana mass |mββ| is plot-
ted as a function of m0for the cases of the NS and the
IS4. The lowest value for the sum of the neutrino masses,
which can be reached in future cosmological measure-
ments [39–41], is about (0.05-0.1) eV. The corresponding
values of m0are in the region, where the IS and the NS
predictions for |mββ| differ significantly from each other.
Future experiments on the search for the 0νββ-decay
will probe the region of the inverted mass hierarchy5
12= ∆m2
Sand ∆m2
13|) = ∆m2
Aare the
m3≪ m1< m2.(16)
In this case we have
m1≃ m2≃
?
∆m2
A,m3≪
?
∆m2
A.(17)
Neglecting small contribution of the term m3|Ue3|2, for
the effective Majorana mass in the case of the inverted
mass hierarchy (16) we obtain the following expression
|mββ| ≃
?
∆m2
Acos2θ13(1 − sin22θ12sin2α12)
1
2,
(18)
4Notice that for NS this figure differs significantly from analogous
figures published in the literature. This is connected with the fact
that we use new T2K data [38] for the values of the parameter
sin22θ13.
5Let us note that the 0νββ-decay in the case of the inverted hi-
erarchy was considered in detail in a recent paper [42].

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where α12 = α2− α1 is the difference of the Majorana
phases of the elements Ue2and Ue1with Uei= |Ue1|eiαi
(i=1,2).
The phase difference α12is the only unknown parame-
ter in the expression for |mββ|. From (18) we obtain the
following inequality
?
∆m2
Acos2θ13 cos2θ12 ≤ |mββ| ≤
?
∆m2
Acos2θ13,
(19)
where upper and lower bounds correspond to the case
of the CP invariance in the lepton sector (upper (lower)
bound corresponds to the same (opposite) CP-parities of
ν1and ν2).
From (19) we find
1.5 · 10−2eV ≤ |mββ| ≤ 5.0 · 10−2eV,
where we used the MINOS value ∆m2
10−3eV2[5], the solar-KamLAND value tan2θ12 =
0.452+0.035
−0.033[3] and the recent T2K observation θ13:
0.04 < sin22θ13< 0.34 [38].
From (17) follows that in the case of inverted mass
hierarchy we have for the sum of the neutrino masses
(20)
A= (2.43±0.13)×
3
?
i=1
mi≃ 2
?
∆m2
A≃ 10−1eV.(21)
As is well known, the quantity?
from the measurement of the distribution of galaxies and
other cosmological observations. At present from cosmo-
logical data the bound?
[43],[39]. It is expected that in the future various cosmo-
logical observables will be sensitive to?
(6 · 10−3− 10−1) eV (see, for example, [41]). Thus, the
inverted neutrino mass hierarchy (16) will be tested by
future precision cosmology.
We will now discuss a possibility to check the Majorana
mass mechanism for the case that the 0νββ-decay will
be observed in future experiments sensitive to the region
(20) of the inverted hierarchy.
From (9) and (19) we find the following inequalities for
the half-life of 0νββ-decay
imi can be inferred
imi<
∼0.5 eV was obtained
imiin the range
Tmin
1/2(A,Z) ≤ T0ν
1/2(A,Z) ≤ Tmax
1/2(A,Z) (22)
with
Tmin
1/2(A,Z) =
1
∆m2
A|M(A,Z)|2G0 ν(E0,Z),
Acos22θ12|M(A,Z)|2G0 ν(E0,Z).Tmax
1/2(A,Z) =
1
∆m2
(23)
In Fig.2 we present ranges of 0νββ-decay half-lives of dif-
ferent nuclei. If the measured half-life of the 0νββ-decay
is in the range given by Eq. (22) this will be an evidence
in favor of the Majorana neutrino mass mechanism (as-
suming inverted mass hierarchy).
Two remarks are in order:
1. It is seen from Fig.1 that the horizontal band de-
termined by the inequality (20) is restricted by the
condition m0≤
responds to?
mological data will establish this range for m0and
the measured 0νββ-decay half-lives will be within
the range given by inequality (22) this will be an
evidence in favor of the Majorana mass mechanism.
It is obvious that without knowledge of the value
of the lightest neutrino mass it is impossible to de-
termine in which region we are (IS or NS).
?∆m2
A≃ 5 · 10−2eV (which cor-
imi≤ 1.9 · 10−1eV). If future cos-
2. In addition to the Majorana mass mechanism, also
other mechanisms of the 0νββ-decay, caused by a
possible violation of the total lepton number L at a
scale which is much smaller than the standard see-
saw GUT scale, were discussed in the literature. If
L is violated at ∼ TeV scale a contribution of these
additional mechanisms to the matrix element of the
0νββ-decay can be comparable with the Majorana
mass contribution.6. Thus, a significant violation
of the inequalities (22) could happen. Examples of
new mechanisms are the exchange of heavy Majo-
rana neutralino or gluino in SUSY models with the
violation of the R-parity (see recent papers [45–
48]), the exchange of right-handed Majorana neu-
trinos with mass at the electroweak scale (see [49]),
the exchange of heavy Majorana right-handed neu-
trinos in L − R models (see [44]), etc.
New mechanisms are characterized by parameters
which are not connected with neutrino oscillation
parameters. In addition, the NMEs of the Majo-
rana mass mechanism and these possibly additional
mechanisms are not connected and are different.
Thus, it requires fine tuning for such mechanisms to
contribute to the relatively narrow Majorana mass
region (20) and mimic the effect of the Majorana
phase difference. It is natural to expect that con-
tributions to the matrix element of the 0νββ-decay
of the Majorana mass mechanism and mechanisms
connected with the violation of the lepton number
at TeV scale could be quite different (see, for exam-
ple, [49, 50]). Moreover, such mechanisms will be
checked in LHC experiments by the search for ef-
fects of a violation of the total lepton number (like
production of the same-sign lepton pairs in p − p
collisions, etc., see [50]).
6In fact, the matrix element of the 0νββ-decay in the case
of the Majorana neutrino mass mechanism is proportional to
(GF
p2 , where p ≃ 100 MeV is the average momentum of
the virtual neutrino. On the other side, the contribution to the
matrix element of the 0νββ-decay by an exchange of a heavy
Majorana lepton with a mass Mχ ∼ Λ (Λ is the scale of new
physics) is proportional to (GF
W-boson) [44]. If we assume that |mββ| ≃ 10−2eV we conclude
that both contributions are comparable, if Mχ≃ 2 TeV.
√2)2mββ
√2)2(M4
W
Λ5) (MW is the mass of the