a r X i v : n u c l  t h / 0 0 1 2 0 1 0 v 1 4 D e c 2 0 0 0
Neutrinoless double beta decay within SelfconsistentRenormalized Quasiparticle Random Phase Approximation andinclusion of induced nucleon currents
A. Bobyk, W. A. Kami´nski
Department of Theoretical Physics, Maria CurieSk lodowska University,Radziszewskigo 10, PL20031 Lublin, Poland.
F. ˇSimkovic
Department of Nuclear Physics, Comenius University,Mlynska dol., pav. F1, SK842 15 Bratislava, Slovakia
(February 8, 2008)
Abstract
The ﬁrst, to our knowledge, calculation of neutrinoless double beta decay(0
νββ
decay ) matrix elements within the selfconsistent renormalized Quasiparticle Random Phase Approximation (SRQRPA) is presented. The contribution from the momentumdependent induced nucleon currents to 0
νββ
decay amplitude is taken into account. A detailed nuclear structure studyincludes the discussion of the sensitivity of the obtained SRQRPA results for0
νββ
decay of
76
Ge to the parameters of nuclear Hamiltonian, twonucleonshortrange correlations and the truncation of the model space. A comparison with the standard and renormalized QRPA results is presented. We havefound a considerable reduction of the SRQRPA nuclear matrix elements, resulting in less stringent limits for the eﬀective neutrino mass. PACS numbers:23.40.Hc, 21.60.Jz, 27.50.+e, 27.60.+j
The neutrino mass and mixing of neutrinos are the main subject of the elementary1
particle physics nowadays. The experiments looking for oscillation of solar, atmospheric andterrestrial (LSNDexperiment) neutrinos constitute evidence for a new physics beyond theStandard Model [1]. The current constraints imposed by the results of neutrino oscillationexperiments allow to construct the spectrum of mass of neutrinos [2]. The predictions diﬀerfrom each other by diﬀerent input, structure of neutrino mixing and assumption, e.g., onthe phases and fundamental character of neutrinos.An important quantity to limit the space of possible neutrino mixing schemes is theeﬀective Majorana neutrino mass
m
ee
=
k
(
U
e
k
)
2
m
k
η
CP
k
(1)where
U
e
k
,
m
k
and
η
CP
k
are unitary mixing matrix elements, mass eigenstates and the relativeCPphases of neutrinos, respectively. The value of
m
ee
depends on the speciﬁc model of neutrino mixing and its predictions fall in the range of 10
−
3
eV to few eV [2]. The upperbound on
m
ee
can be deduced from the current experimental lower bound on the halflifeof neutrinoless double beta decay (0
νββ
decay) as follows [3 –5]:
m
ee
≤
m
e

M
m
ee

T
0
ν
−
exp1
/
2
G
01
.
(2)Here,
m
e
,
G
01
and
M
m
ee
are the mass of electron, the integrated kinematical factor [3,10]
and the 0
νββ
decay nuclear matrix element, respectively.The present most stringent lower bound on
T
0
ν
−
exp1
/
2
has been measured for
76
Ge byHeidelbergMoscow group and is equal to 5
.
7
×
10
25
years [6,7]. A condition for obtaining
reliable limit for fundamental particle physics quantity
m
ee
is that the nuclear matrix elements governing the Majorana mass mechanism of 0
νββ
decay can be calculated correctly[5]. However, the practical nuclear structure calculation always involves some approximations, which make it diﬃcult to obtain an unambiguous limit on
m
ee
. We note that thediﬀerence between the previous (
M
m
ee
= 4
.
18, i.e.,
m
ee
≤
0
.
18) [4] and the recent moreadvanced (
M
m
ee
= 1
.
67, i.e.,
m
ee
≤
0
.
46) [8] shell model calculations is signiﬁcant. Thereis also an other group of nuclear structure calculations which include the protonneutron2
Quasiparticle Random Phase Approximation (pnQRPA) [9] and its extensions [10,11]. Some
of them suggest the upper bound on
m
ee
to be in the 0.1 eV range [6].The aim of the present letter is to discuss the nuclear physics aspects of the pnQRPA[9], the renormalized pnQRPA (pnRQRPA) [12,13] and the selfconsistent pnRQRPA
(pnSRQRPA) [16,17] calculation of the nuclear matrix element
M
m
ee
for 0
νββ
decay of
76
Ge. We note that the pnSRQRPA results on 0
νββ
decay matrix elements have been notpresented in the literature till now.We shortly present the main diﬀerences between the above mentioned three QRPA approaches.The pnQRPA has been the most popular theoretical tool in description of the
β
and
ββ
decays of medium and heavy open shell nuclei. However, the pnQRPA develops acollapse beyond some critical value of the particleparticle interaction strength of nuclearHamiltonian close to its realistic value. This phenomena makes predictive power of theobtained results questionable.By implementing the Pauli exclusion principle (PEP) in an approximate way in the pnQRPA one gets the pnRQRPA [12,13], which avoids collapse within a physical range of
particleparticle force and oﬀers more stable solution. This fact has been conﬁrmed alsowithin the schematic models. In addition it was found that by restoring the PEP a betteragreement with the exact solution is obtained [14].The selfconsistent pnRQRPA (pnSRQRPA) [15,17,16], which is more complex version
of the RQRPA, is becoming increasingly popular to describe strongly correlated Fermionsystems. The pnSRQRPA goes a step further beyond the pnRQRPA. In the pnSRQRPAat the same time the mean ﬁeld is changed by minimizing the energy and ﬁxing the numberof particles in the correlated ground state instead of the uncorrelated BCS one as it is done inthe other versions of the QRPA (pnQRPA, pnRQRPA) [15,17,16]. Thus the pnSRQRPA
is closer to a fully variational theory.We proceed by writing the expression for the 0
νββ
decay nuclear matrix element. Wenote that the contribution to
M
m
ee
from the induced pseudoscalar term of the nucleon3
current were not considered before. Recently, it has been found that it is signiﬁcant andleads to a modiﬁcation of GamowTeller and new tensor contributions of
M
m
ee
[11]. Thus
M
m
ee
is given as sum of Fermi, GamowTeller and tensor contributions
M
m
ee
=
−
M
F
g
2
A
+
M
GT
+
M
T
(3)with
g
A
= 1
.
25. Expressed in relative coordinates and using the second quantization formalism
M
m
ee
takes the form
M
m
ee
=
J
π
pnp
′
n
′
mimf
J
(
−
)
j
n
+
j
p
′
+
J
+
J
(2
J
+ 1)
j
p
j
n
J j
n
′
j
p
′
J
×
p
(1)
,p
′
(2);
J
f
(
r
12
)
τ
+1
τ
+2
O
m
ee
(12)
f
(
r
12
)

n
(1)
,n
′
(2);
J×
0
+
f
[
c
+
p
′
˜
c
n
′
]
J
J
π
m
f
J
π
m
f

J
π
m
i
J
π
m
i
[
c
+
p
˜
c
n
]
J
0
+
i
.
(4)Here,
f
(
r
12
) is the shortrange correlation function and
O
m
ee
(12) represents the coordinateand spin dependent part of the twobody 0
νββ
decay transition operator
O
m
ee
(12) =
−
H
F
(
r
12
)
g
2
A
+
H
GT
(
r
12
)
σ
12
+
H
T
(
r
12
)
S
12
,
(5)where
σ
12
=
σ
1
·
σ
2
,
S
12
= 3(
σ
1
·
ˆ
qσ
2
·
ˆ
q
)
−
σ
12
,
and
H
K
(
r
12
) = 2
πg
2
A
Rr
12
∞
0
sin(
qr
12
)
q
+
E
m
(
J
)
−
(
E
i
+
E
f
)
/
2
h
K
(
q
2
)
dq,
(
K
=
F,GT,T
) (6)with
h
F
(
q
2
) =
g
2
V
(
q
2
)
g
2
A
,h
GT
(
q
2
) =
g
2
A
(
q
2
) + 13
g
2
P
(
q
2
)
q
4
4
m
2
p
−
23
g
A
(
q
2
)
g
P
(
q
2
)
q
2
2
m
p
,h
T
(
q
2
) = 23
g
A
(
q
2
)
g
P
(
q
2
)
q
2
2
m
p
−
13
g
2
P
(
q
2
)
q
4
4
m
2
p
.
(7)Here,
R
is the mean nuclear radius [11].
E
i
,
E
f
and
E
m
(
J
) are respectively the energies of the initial, ﬁnal and intermediate nuclear state with angular momentum
J
. The momentumdependence of the vector, axialvector and pseudoscalar formfactors (
g
V
(
q
2
),
g
A
(
q
2
) and
g
P
(
q
2
)) is given in Ref. [11].4
We note that the ﬁrst sum on the r.h.s of Eq. (4) represents summation over all multipolarities. The form of the onebody transition densities [see Eq. (4)] to excited states

J
π
m
i
and

J
π
m
f
generated from the initial (A,Z) and the ﬁnal (A,Z+2) ground states

0
+
i
and0
+
f
within the pnRQRPA and the pnSRQRPA is the same and can be found together withother details of the nuclear structure in Refs. [13,11,17]. The diﬀerence consists only in the
calculated value of the renormalized coeﬃcients
D
, which in the case of the pnQRPA is just equal to unity. The overlap factor entering the expression (4) can be ﬁnd in Ref. [18].
The calculation of 0
νββ
decay matrix elements of
76
Ge are performed within two modelspaces both for protons and neutrons as follows: i) The model space I (m.s. I) consists of the full 3
−
4¯
hω
major oscillator shells and has been considered in the pnQRPA studies of Ref. [9] (9levels model space). ii) The model space II (m.s. II) comprises the full 2
−
5¯
hω
major shells (12 levels model space).The single particle energies were obtained by using a Coulomb–corrected Woods–Saxonpotential with Bertsch parameterization. Twobody Gmatrix elements were calculatedfrom the Bonn oneboson exchange potential within within the Brueckner theory. In thepnQRPA and the pnRQRPA approaches pairing interactions have been adjusted to ﬁt theempirical pairing gaps. In the pnSRQRPA approach the pairing matrix elements of the NNinteraction have not been rescaled as the mean ﬁeld is directly related to the excited states.The particleparticle and particlehole channels of the Gmatrix interaction of the nuclearHamiltonian
H
are renormalized by introducing the parameters
g
pp
and
g
ph
, respectively.In Fig. 1 the calculated partial matrix elements
M
F
,
M
GT
and
M
T
are plotted asfunction of
g
pp
(
g
ph
= 1
.
0) for the pnRQRPA and pnSRQRPA approaches. The larger12levels model space is considered. One ﬁnd a strong dependence of
M
F
and
M
GT
on
g
pp
.The smallest
M
T
matrix element is rather insensitive to this parameter. In general, thebehavior of plotted matrix elements is similar for both approaches. Nevertheless, the Fermiand GT matrix elements of the pnSRQRPA one reach zero value inside the physical rangeof
g
pp
parameter (0
.
8
≤
g
pp
≤
1
.
2).In Fig. 2 the multipole decomposition (according to intermediate multipoles
J
π
) of 5