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Double Beta Decay and Neutrino Masses Accuracy of the Nuclear Matrix Elements
Nuclear Physics B (Proc. Suppl.) 145 (2005) 213–218 www.elsevierphysics.com
Double Beta Decay and Neutrino Masses Accuracy of the Nuclear Matrix Elements Amand Faesslera a
Institute of Theoretical Physics, University of Tuebingen Auf der Morgenstelle 14, 72076 T¨ ubingen, Germany The neutrinoless double beta decay is forbidden in the standard model of the electroweak and strong interaction but allowed in most Grand Unified Theories (GUT’s). Only if the neutrino is a Majorana particle (identical with its antiparticle) and if it has a mass, the neutrinoless double beta decay is allowed. Apart of one claim that the neutrinoless double beta decay in 76 Ge is measured, one has only upper limits for this transition probability. But even the upper limits allow to give upper limits for the electron Majorana neutrino mass and upper limits for parameters of GUT’s and the minimal R-parity violating supersymmetric model. One further can give lower limits for the vector boson mediating mainly the right-handed weak interaction and the heavy mainly right-handed Majorana neutrino in left-right symmetric GUT’s. For that one has to assume that the specific mechanism is the leading one for the neutrinoless double beta decay and one has to be able to calculate reliably the corresponding nuclear matrix elements. In the present contribution, one discusses the accuracy of the present status of calculating the nuclear matrix elements and the corresponding limits of GUT’s and supersymmetric parameters.
1. Introduction The neutrinoless double beta decay is forbidden in the standard model but is allowed in most Grand Unified Theories (GUT’s) where the neutrino is a Majorana particle (identical with its antiparticle) and where the neutrinos have a mass. In GUT’s, each beta decay vertex in Fig. 1 can occure in eight different ways (see Fig. 2). The hadronic current changing a neutron into a proton can be left- or right-handed, the vector boson exchanged can be the light one W1 or the heavy orthogonal combination W2 mediating mainly a right-handed weak interaction and two different leptonic currents changing a neutrino into an electron. So the simple vertex on the left for the beta decay can occure in eight different ways. With two such vertices and the exchange of a light Majorana or a heavy mainly right-handed Majorana neutrino one already has 128 different matrix elements describing the neutrinoless double beta decay [1] (see Fig. 3). At the R-parity violating vertex (see Fig. 4)of the up and the down quarks and the SUSY elec0920-5632/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2005.04.009
76Se 34 42
e2ï
p p i
e1ï
Left Phase Space
ic
Left n
n
106 x 2 i``
76Ge 44 32
only for Majorana Neutrinos
i = ic
Figure 1. Neutrinoless double beta decay of 76 Ge through 76 As to the final nucleus 76 Se. The neutrino must be a Majorana particle that means identical with its antiparticle and must have a mass to allow this decay.
tron e˜, one has the R-parity violating coupling constant λ111 which is new compared to the standard model [2]. The formation of a pion by a down and a up quarks produces a long range zero neutrino double beta decay transition operator.
214
A. Faessler / Nuclear Physics B (Proc. Suppl.) 145 (2005) 213–218 p
p
p eï
+ W1
i
L/R
+
eï
i
p eï
Proton
l/r
u
W2
⎛
⎞
u
⎤
⎡
⎥ M12 cos ϑc ⎢⎢⎢ ⎥ GUT = GF√ ⎢1 · l · L + t g ϑ r · L + t g ϑ R · l + r · R⎥⎥⎥⎦ Hˆ Weak ⎢ M22 2 ⎣
eï 2
ic
l/r n
n
d
u Neutron
light i heavy N Neutrinos
Figure 3. Diagram of the double beta decay
Due to the short range Brueckner repulsive correlations between two nucleons, this is increasing the transition probability by a factor 10 000. The upper limit derived from upper limits of the zero neutrino transition probability for λ111 is therefore more stringent and one can derive from the zero neutrino transition probability in 76 Ge an upper limit for |λ111 | < 10−4 . To calculate the zero neutrino double beta decay transition probability, we use Fermi’s Golden Rule in second order.
n
Figure 4. Matrix element for the neutrinoless double beta decay within R-parity violating supersymmetrie
< f |H ˆ W |k >< k|H ˆ W |i > , Ei − Ek
T
=
Ei
= E0+ (76 Ge),
Ek T
= Ek (e− ) + Ek (ν) + EKern (76 As; k), = Mm < mν > +Mθ < tgϑ > M 2 2 MW R < > MR mp 2 MSUSY λ111 + MV R < >, MV R 2π 2 = |T | pf ≤ 4.4 · 10−33 [sec−1 ], (1) ¯h
k
+ l/r
i
d
n
Figure 2. The single beta decay of the change of a neutron into a proton by the emission of an electron and in the standard model of antineutrino has now the form indicated on the right hand side of the figure
e 1ï
e~
h’ 111
gV = 1 ; gA = 1.25
p
+ï ï
Neutron
L/R = p+ (gV ∓ gA σ ) n
p
/ e~
l/r = e¯ γ µ ⎝1 ∓ γ 5⎠ ν
Proton u
+ / ïï
n
n
eï
+ w
To calculate the nuclear matrix elements, the most reliable method has turned out to be the quasiparticle random phase approximation (QRPA) to calculate at the example of 76 Ge the wave function of the initial 76 Ge, the excited states of the intermediate nuclei 76 As and the ground state of the final nuclei 76 Se [3]. The QRPA approach describes the intermediate excited nuclear states for example in 76 As as a coherent superposition of two quasiparticle excitations and two quasiparticle annihilations relative to the initial ground state in our example of 76 Ge. a+ i A+ α
= =
u i c+ i − vi c¯i , + + [ai ak ]Jα ,
215
A. Faessler / Nuclear Physics B (Proc. Suppl.) 145 (2005) 213–218
|m > =
m [Xαm A+ α − Yα Aα ],
α Q+ m |g
>,
3.0
RQRPA (gA=1.25) RQRPA (gA=1.0) QRPA (gA=1.25) QRPA (gA=1.0)
2.5
(2)
The QRPA equation for the determination of Xαm and Yαm are derived from the many-body Schroedinger equation by using quasi-boson commutation relations for the quasiparticle pairs A†α . One uses therefore for deriving the QRPA equations that two quasiparticle states behave like bosons. This is a usual approximation one often is using in physics. For example, one describes a pair of quarks and antiquarks as a meson and treats it as a boson, although it consists out of a fermion pair. To test the quality of this approximation for the two neutrino and the zero neutrino double beta decay, we include the exact commutation relations of the fermion pairs at least as ground state expectation value. This is called renormalized QRPA (R-QRPA) [4], [5]. The R-QRPA includes the Pauli principle into the QRPA and reduces by that the number of quasiparticles in the ground state. RQRPA stabilises the intermediate nuclear wave functions compared to the QRPA approach. One is relatively close to a phase transition to stable protonneutron Gamow-Teller correlations. Since they are not included in the basis, the QRPA solution is collapsing if the 1+ proton-neutron nucleon-nucleon two-body matrix elements are increased. This one studies usually by multiplying all nucleon-nucleon particle-particle matrix elements of the Brueckner reaction matrix of the Bonn (or the Nijmegen or Argonne) potential with a factor gpp . With an increasing gpp the QRPA solution collapses like a spherical solution is collapsing if a nucleus gets deformed. The inclusion of the Pauli principle in the RQRPA approach moves this instability to much larger nucleon-nucleon matrix elements (larger factors gpp ) and the agreement for the two neutrino double beta decay with the experimental value is more in the range of gpp = 1. Fig. 5 shows the zero neutrino double beta decay matrix elements calculated for each initial nucleus indicated at the figure below in 36 different ways [6]. The approach includes three
2.0
0i
=
Q+ m
1.5
1.0 0.5 0.0 76
Ge
82
Se
96
Zr
100
Mo
116
Cd
128
Te
130
Te
136
Xe
Figure 5. Zero neutrino double beta decay matrix elements at the Majorana neutrino mass calculated for different initial double beta decay nuclei. Each point is calculated with three different forces (Bonn, Nijmegen and Argonne) and three different basis sets (a small basis with about two oscillator shells, an intermediate basis with about three oscillator shells and large basis with about five oscillator shells). The error bar indicated at each point is the 1σ error of these nine calculations plus the experimental error of the two neutrino double beta decay. In each calculation, the gpp is so adjusted as to reproduce the two neutrino double beta decay and this values gpp are used to multiply the nucleon-nucleon matrix elements. The values shown here are the arithmetic middle with the 1σ error of these nine values plus the experimental uncertainty of the two neutrino double beta decay probability. One can think of the values given as the theoretical ratio of the zero neutrino double beta decay probability divided by the calculated two neutrino double beta decay probability multiplied with the experimental value of the two neutrino transition probability. The four points calculated for each zero neutrino transition probability with the initial nuclei indicated below are for QRPA (open symbols) and the RQRPA (black symbols) which include the Pauli principle. The axial vector coupling constant gA = 1.25 (circles) and for the quenched gA = 1.0 (squares) are used.
different forces (Bonn, Nijmegen, Argonne) and three different basis sets (about two oscillator
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A. Faessler / Nuclear Physics B (Proc. Suppl.) 145 (2005) 213–218
Table 1 lower limits of the experimental lifetimes for the 0ν double beta decay indicated in the second row with the experimental references in the third row. The fourth row indicates the upper limit of the Majorana neutrino mass derived with the help of our QRPA matrix elements in an intermediate basis (about three oscillator shells). The fifth row gives the ratio of the proton mass over heavy Majorana neutrino mass for 76 Ge. This ratio is smaller than 0.79 which means that the heavy Majorana neutrino which is mainly right-handed, should have a mass larger than 1.2 GeV. The last row gives the upper limit of the R-parity violating coupling constant λ111 for the vertex with up and down quarks and the supersymmetric electron. The upper limit of about 10−4 derived from the data of 76 Ge is by one order of magnetude more stringent than other published values. 48 76 82 96 100 116 Nucleus Ca Ge Se Zr Mo Cd 21 25 22 21 22 T 1/2 (exp) [years] > 9.5 10 > 1.9 10 > 1.4 10 > 1.0 10 > 5.5 10 > 7.0 1022 Ref.: You Klapdor Elliott Arn. Ejiri Danevion < m > [eV ] < 22. < 0.47 < 8.7 < 40. < 2.8 < 3.8 ν ≈ m(p)/M (γ) < 200. < 0.79 < 15. < 79. < 6.0 < 7.0 λ111 [10−4 ] < 8.9 < 1.1 < 5.0 < 9.4 < 2.8 < 3.4
Table 2 second part of table 1 128 Nucleus Te T 1/2 (exp) [years] > 8.6 1022 Ref.: Ales. < m > [eV ] < 17. ν ≈ m(p)/M (γ) < 27. −4 λ111 [10 ] < 5.8
130
Te > 1.4 1022 Ales. < 3.2 < 4.9 < 2.4
134
Xe > 5.8 1022 Ber. < 27. < 38. < 6.8
136
Xe > 7.0 1023 Staudt < 3.8 < 3.5 < 2.1
150
Nd > 1.7 1021 Klimenk. < 7.2 < 13. < 3.8
A. Faessler / Nuclear Physics B (Proc. Suppl.) 145 (2005) 213–218
shells, about three oscillator shells and about five oscillator shells) and four different approaches QRPA and the renormalized QRPA (RQRPA) with two different axial vector coupling constants gA = 1.25 and with the quenched value gA = 1.0. In each of the 36 calculations for each nucleus, the factor gpp in front of the two nucleon-nucleon matrix element is adjusted to reproduce the experimental value of the two neutrino double beta decay. The error bars given in the figure include the 1σ error of the theory plus the experimental error of the two neutrino double beta decay transition probability. In several nuclei, like 76 Ge and 100 Mo the errors are as small as ±15 %. In most cases the error is less than ± 30 %. But for 96 Zr one has an error of the zero neutrino double beta decay matrix element of ± 100 %. The main part of the errors indicated in Fig. 5 are the experimental error of the two neutrino double beta decay probability. So when this measurements are improved, one is also automatically improving the prediction of the zero neutrino double beta decay matrix elements. The present calculation includes the higher order currents according to Towner and Hardy [9] and short range correlations in the zero neutrino double beta decay transition probability with a Jastrow factor. The results of other groups seem partially not to agree with our results given above: Muto’s results agree with our results if we include into in his calculations the higher order currents. The induced pseudoscalar current required by PCAC reduces his results by 30 %. In addition we have a nuclear radius R = 1.1 · A1/3 [fm] of the Woods-Saxon potential compared to R = 1.2 · A1/3 [fm] of Muto. This difference reduces his values further by 10 %. If we reduce his values for 76 Ge by 40 % his values M 0ν = 4.59 (QRPA) and 3.88 (RQRPA) are reduced to 2.76 (QRPA) and 2.34 (RQRPA) which agree nicely with our result of 2.68 (QRPA) and 2.41 (RQRPA). Civitarese and Suhonen have consistently higher values. Comparison of our results calculated in the QRPA approach with gA = 0.25 with the results of Civitarese and Suhonen [7]:
Nucleus 76 Ge 100 Mo 130 Te 136 Xe
CS (QRPA, 1.254) 3.33 2.97 3.49 4.64
217
our (QRPA, 1.25) 2.68 ± 0.12 1.30 ± 0.10 1.56 ± 0.47 0.90 ± 0.20
Comparison of the results of Civitarese and Suhonen with ours. The calculation of Civitarese and Suhonen [7] are different from ours in several respects: 1. They have fitted the factor gpp not to the two neutrino double beta decay but to selected known single beta decay transition probabilities. 2. They have included higher order terms of the weak nucleon current according to Suhonen, Kadhkikar and Faessler [8] which is based on simple quark model with a harmonic oscillator potential for the quarks leading to Gauss’schen form factors for the weak currents and leading to the weak magnetism as the main higher contribution. In our approach we included the higher order currents according to Towner and Hardy [9] who fitted the higher order currents to allowed beta decays, to first forbidden beta decays, to beta-gamma angular correlations, to beta-alpha angular correlations and to muon capture in nuclei. Towner and Hardy use dipole form factors and the induced pseudoscalar term and not weak magnetism yields the largest effect. Weak magnetism would reduce the zero neutrino double beta decay matrix elements by about 30 %. 3. Short range Brueckner correlations are not included in the calculation of Civitarese and Suhonen [7]. But they include the finite size of the nuclei. In our calculation, we include short range Brueckner repulsion between the nucleons for the zero neutrino transition operator in form of a Jastrow factor. Inclusion of the short range repulsion in the work of Civitarese and Suhonen would
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A. Faessler / Nuclear Physics B (Proc. Suppl.) 145 (2005) 213–218
reduce the matrix element for the zero neutrino transition probability and could bring it into agreement with our value if also a more accurat expression for the higher order weak currents and the different adjustment of gpp is included. 2. Conclusion In this contribution we investigated the accuracy of the zero neutrino matrix elements for the neutrinoless double beta decay. This accuracy determines exactly in the same way also the accuracy of the Majorana neutrino mass extracted from the neutrinoless double beta decay. We used the quasiparticle random phase approach (QRPA) and the renormalized QRPA (RQRPA) approximation which include the Pauli principle and reduces the ground state correlations and stabilises the wave function of the excited states of the intermediate nuclei. To get a feeling for the uncertainty we calculated the zero neutrino double beta decay for three different nucleon-nucleon forces (Bonn, Nijmegen, Argonne) and three different basis sets (small with about two oscillator shells, intermediate with about three oscillator shells and large with about five oscillator shells). In addition we performed the calculations with QRPA, with RQRPA (including the Pauli principle) and for two different axial vector coupling constants gA = 1.25 and gA = 1.00. In each of the 36 calculations for every nucleus we adjust a factor gpp multiplying the nucleon-nucleon twobody matrix elements of the order between 0.85 and 1.1 to fit the experimental two neutrino transition probabilities. We give the theoretical error of the zero neutrino transition probability as the 1σ error of the nine calculations for each method plus the experimental error of the two neutrino transition probability. In many cases the theoretical error of the zero neutrino nuclear matrix element is less than ± 20 % but in one case 96 Zr it is ± 100 %. But the main part of the error as indicated in Fig. 5 comes from the experimental error of the two neutrino transition probability. An improvement of the measurement of the two neutrino transition probability would even further reduce the uncertainty and by that further
reduce the error of the Majorana neutrino mass which can be determined if the zero neutrino double beta decay is measured. Acknowledgement: I would like to thank Dr. F. Simkovic, Dr. V. Rodin and Prof. P. Vogel for important contributions to the results presented here. REFERENCES 1. T. Tomoda, A. Faessler; Phys. Lett. B199 (1987) 475 2. A. Faessler, S. Kovalenko, F. Simkovic, J. Schwieger; Phys. Rev. Lett. 78 (1997) 183 3. O. Civitarese, A. Faessler, T. Tomoda; Phys. Lett. B194 (1987) 11 4. J. Toivanen, J. Suhonen; Phys. Rev. C55 (1997) 2314 5. J. Schwieger, F. Simkovic, A. Faessler; Nucl. Phys. A600 (1996) 179 F. Simkovic, J. Schwieger, M. Veselsky, G. Pantis, A. Faessler; Phys. Lett. B393 (1997) 267 6. V.A. Rodin, A. Faessler, F. Simkovic, P. Vogel; Phys. Rev. C68 (2003) 044302, to be published 7. O. Civitarese, J. Suhonen; Nucl. Phys. A729 (2003) 867 8. J. Suhonen, S.B. Khadkikar, A. Faessler; Nucl. Phys. A529 (1991) 727 9. I.S. Towner, J.C. Hardy; nucl-th/9504015 v1 (12 Apr 1995)
Double Beta Decay and Neutrino Masses Accuracy of the Nuclear Matrix Elements Amand Faesslera a
Institute of Theoretical Physics, University of Tuebingen Auf der Morgenstelle 14, 72076 T¨ ubingen, Germany The neutrinoless double beta decay is forbidden in the standard model of the electroweak and strong interaction but allowed in most Grand Unified Theories (GUT’s). Only if the neutrino is a Majorana particle (identical with its antiparticle) and if it has a mass, the neutrinoless double beta decay is allowed. Apart of one claim that the neutrinoless double beta decay in 76 Ge is measured, one has only upper limits for this transition probability. But even the upper limits allow to give upper limits for the electron Majorana neutrino mass and upper limits for parameters of GUT’s and the minimal R-parity violating supersymmetric model. One further can give lower limits for the vector boson mediating mainly the right-handed weak interaction and the heavy mainly right-handed Majorana neutrino in left-right symmetric GUT’s. For that one has to assume that the specific mechanism is the leading one for the neutrinoless double beta decay and one has to be able to calculate reliably the corresponding nuclear matrix elements. In the present contribution, one discusses the accuracy of the present status of calculating the nuclear matrix elements and the corresponding limits of GUT’s and supersymmetric parameters.
1. Introduction The neutrinoless double beta decay is forbidden in the standard model but is allowed in most Grand Unified Theories (GUT’s) where the neutrino is a Majorana particle (identical with its antiparticle) and where the neutrinos have a mass. In GUT’s, each beta decay vertex in Fig. 1 can occure in eight different ways (see Fig. 2). The hadronic current changing a neutron into a proton can be left- or right-handed, the vector boson exchanged can be the light one W1 or the heavy orthogonal combination W2 mediating mainly a right-handed weak interaction and two different leptonic currents changing a neutrino into an electron. So the simple vertex on the left for the beta decay can occure in eight different ways. With two such vertices and the exchange of a light Majorana or a heavy mainly right-handed Majorana neutrino one already has 128 different matrix elements describing the neutrinoless double beta decay [1] (see Fig. 3). At the R-parity violating vertex (see Fig. 4)of the up and the down quarks and the SUSY elec0920-5632/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2005.04.009
76Se 34 42
e2ï
p p i
e1ï
Left Phase Space
ic
Left n
n
106 x 2 i``
76Ge 44 32
only for Majorana Neutrinos
i = ic
Figure 1. Neutrinoless double beta decay of 76 Ge through 76 As to the final nucleus 76 Se. The neutrino must be a Majorana particle that means identical with its antiparticle and must have a mass to allow this decay.
tron e˜, one has the R-parity violating coupling constant λ111 which is new compared to the standard model [2]. The formation of a pion by a down and a up quarks produces a long range zero neutrino double beta decay transition operator.
214
A. Faessler / Nuclear Physics B (Proc. Suppl.) 145 (2005) 213–218 p
p
p eï
+ W1
i
L/R
+
eï
i
p eï
Proton
l/r
u
W2
⎛
⎞
u
⎤
⎡
⎥ M12 cos ϑc ⎢⎢⎢ ⎥ GUT = GF√ ⎢1 · l · L + t g ϑ r · L + t g ϑ R · l + r · R⎥⎥⎥⎦ Hˆ Weak ⎢ M22 2 ⎣
eï 2
ic
l/r n
n
d
u Neutron
light i heavy N Neutrinos
Figure 3. Diagram of the double beta decay
Due to the short range Brueckner repulsive correlations between two nucleons, this is increasing the transition probability by a factor 10 000. The upper limit derived from upper limits of the zero neutrino transition probability for λ111 is therefore more stringent and one can derive from the zero neutrino transition probability in 76 Ge an upper limit for |λ111 | < 10−4 . To calculate the zero neutrino double beta decay transition probability, we use Fermi’s Golden Rule in second order.
n
Figure 4. Matrix element for the neutrinoless double beta decay within R-parity violating supersymmetrie
< f |H ˆ W |k >< k|H ˆ W |i > , Ei − Ek
T
=
Ei
= E0+ (76 Ge),
Ek T
= Ek (e− ) + Ek (ν) + EKern (76 As; k), = Mm < mν > +Mθ < tgϑ > M 2 2 MW R < > MR mp 2 MSUSY λ111 + MV R < >, MV R 2π 2 = |T | pf ≤ 4.4 · 10−33 [sec−1 ], (1) ¯h
k
+ l/r
i
d
n
Figure 2. The single beta decay of the change of a neutron into a proton by the emission of an electron and in the standard model of antineutrino has now the form indicated on the right hand side of the figure
e 1ï
e~
h’ 111
gV = 1 ; gA = 1.25
p
+ï ï
Neutron
L/R = p+ (gV ∓ gA σ ) n
p
/ e~
l/r = e¯ γ µ ⎝1 ∓ γ 5⎠ ν
Proton u
+ / ïï
n
n
eï
+ w
To calculate the nuclear matrix elements, the most reliable method has turned out to be the quasiparticle random phase approximation (QRPA) to calculate at the example of 76 Ge the wave function of the initial 76 Ge, the excited states of the intermediate nuclei 76 As and the ground state of the final nuclei 76 Se [3]. The QRPA approach describes the intermediate excited nuclear states for example in 76 As as a coherent superposition of two quasiparticle excitations and two quasiparticle annihilations relative to the initial ground state in our example of 76 Ge. a+ i A+ α
= =
u i c+ i − vi c¯i , + + [ai ak ]Jα ,
215
A. Faessler / Nuclear Physics B (Proc. Suppl.) 145 (2005) 213–218
|m > =
m [Xαm A+ α − Yα Aα ],
α Q+ m |g
>,
3.0
RQRPA (gA=1.25) RQRPA (gA=1.0) QRPA (gA=1.25) QRPA (gA=1.0)
2.5
(2)
The QRPA equation for the determination of Xαm and Yαm are derived from the many-body Schroedinger equation by using quasi-boson commutation relations for the quasiparticle pairs A†α . One uses therefore for deriving the QRPA equations that two quasiparticle states behave like bosons. This is a usual approximation one often is using in physics. For example, one describes a pair of quarks and antiquarks as a meson and treats it as a boson, although it consists out of a fermion pair. To test the quality of this approximation for the two neutrino and the zero neutrino double beta decay, we include the exact commutation relations of the fermion pairs at least as ground state expectation value. This is called renormalized QRPA (R-QRPA) [4], [5]. The R-QRPA includes the Pauli principle into the QRPA and reduces by that the number of quasiparticles in the ground state. RQRPA stabilises the intermediate nuclear wave functions compared to the QRPA approach. One is relatively close to a phase transition to stable protonneutron Gamow-Teller correlations. Since they are not included in the basis, the QRPA solution is collapsing if the 1+ proton-neutron nucleon-nucleon two-body matrix elements are increased. This one studies usually by multiplying all nucleon-nucleon particle-particle matrix elements of the Brueckner reaction matrix of the Bonn (or the Nijmegen or Argonne) potential with a factor gpp . With an increasing gpp the QRPA solution collapses like a spherical solution is collapsing if a nucleus gets deformed. The inclusion of the Pauli principle in the RQRPA approach moves this instability to much larger nucleon-nucleon matrix elements (larger factors gpp ) and the agreement for the two neutrino double beta decay with the experimental value is more in the range of gpp = 1. Fig. 5 shows the zero neutrino double beta decay matrix elements calculated for each initial nucleus indicated at the figure below in 36 different ways [6]. The approach includes three
2.0
0i
=
Q+ m
1.5
1.0 0.5 0.0 76
Ge
82
Se
96
Zr
100
Mo
116
Cd
128
Te
130
Te
136
Xe
Figure 5. Zero neutrino double beta decay matrix elements at the Majorana neutrino mass calculated for different initial double beta decay nuclei. Each point is calculated with three different forces (Bonn, Nijmegen and Argonne) and three different basis sets (a small basis with about two oscillator shells, an intermediate basis with about three oscillator shells and large basis with about five oscillator shells). The error bar indicated at each point is the 1σ error of these nine calculations plus the experimental error of the two neutrino double beta decay. In each calculation, the gpp is so adjusted as to reproduce the two neutrino double beta decay and this values gpp are used to multiply the nucleon-nucleon matrix elements. The values shown here are the arithmetic middle with the 1σ error of these nine values plus the experimental uncertainty of the two neutrino double beta decay probability. One can think of the values given as the theoretical ratio of the zero neutrino double beta decay probability divided by the calculated two neutrino double beta decay probability multiplied with the experimental value of the two neutrino transition probability. The four points calculated for each zero neutrino transition probability with the initial nuclei indicated below are for QRPA (open symbols) and the RQRPA (black symbols) which include the Pauli principle. The axial vector coupling constant gA = 1.25 (circles) and for the quenched gA = 1.0 (squares) are used.
different forces (Bonn, Nijmegen, Argonne) and three different basis sets (about two oscillator
216
A. Faessler / Nuclear Physics B (Proc. Suppl.) 145 (2005) 213–218
Table 1 lower limits of the experimental lifetimes for the 0ν double beta decay indicated in the second row with the experimental references in the third row. The fourth row indicates the upper limit of the Majorana neutrino mass derived with the help of our QRPA matrix elements in an intermediate basis (about three oscillator shells). The fifth row gives the ratio of the proton mass over heavy Majorana neutrino mass for 76 Ge. This ratio is smaller than 0.79 which means that the heavy Majorana neutrino which is mainly right-handed, should have a mass larger than 1.2 GeV. The last row gives the upper limit of the R-parity violating coupling constant λ111 for the vertex with up and down quarks and the supersymmetric electron. The upper limit of about 10−4 derived from the data of 76 Ge is by one order of magnetude more stringent than other published values. 48 76 82 96 100 116 Nucleus Ca Ge Se Zr Mo Cd 21 25 22 21 22 T 1/2 (exp) [years] > 9.5 10 > 1.9 10 > 1.4 10 > 1.0 10 > 5.5 10 > 7.0 1022 Ref.: You Klapdor Elliott Arn. Ejiri Danevion < m > [eV ] < 22. < 0.47 < 8.7 < 40. < 2.8 < 3.8 ν ≈ m(p)/M (γ) < 200. < 0.79 < 15. < 79. < 6.0 < 7.0 λ111 [10−4 ] < 8.9 < 1.1 < 5.0 < 9.4 < 2.8 < 3.4
Table 2 second part of table 1 128 Nucleus Te T 1/2 (exp) [years] > 8.6 1022 Ref.: Ales. < m > [eV ] < 17. ν ≈ m(p)/M (γ) < 27. −4 λ111 [10 ] < 5.8
130
Te > 1.4 1022 Ales. < 3.2 < 4.9 < 2.4
134
Xe > 5.8 1022 Ber. < 27. < 38. < 6.8
136
Xe > 7.0 1023 Staudt < 3.8 < 3.5 < 2.1
150
Nd > 1.7 1021 Klimenk. < 7.2 < 13. < 3.8
A. Faessler / Nuclear Physics B (Proc. Suppl.) 145 (2005) 213–218
shells, about three oscillator shells and about five oscillator shells) and four different approaches QRPA and the renormalized QRPA (RQRPA) with two different axial vector coupling constants gA = 1.25 and with the quenched value gA = 1.0. In each of the 36 calculations for each nucleus, the factor gpp in front of the two nucleon-nucleon matrix element is adjusted to reproduce the experimental value of the two neutrino double beta decay. The error bars given in the figure include the 1σ error of the theory plus the experimental error of the two neutrino double beta decay transition probability. In several nuclei, like 76 Ge and 100 Mo the errors are as small as ±15 %. In most cases the error is less than ± 30 %. But for 96 Zr one has an error of the zero neutrino double beta decay matrix element of ± 100 %. The main part of the errors indicated in Fig. 5 are the experimental error of the two neutrino double beta decay probability. So when this measurements are improved, one is also automatically improving the prediction of the zero neutrino double beta decay matrix elements. The present calculation includes the higher order currents according to Towner and Hardy [9] and short range correlations in the zero neutrino double beta decay transition probability with a Jastrow factor. The results of other groups seem partially not to agree with our results given above: Muto’s results agree with our results if we include into in his calculations the higher order currents. The induced pseudoscalar current required by PCAC reduces his results by 30 %. In addition we have a nuclear radius R = 1.1 · A1/3 [fm] of the Woods-Saxon potential compared to R = 1.2 · A1/3 [fm] of Muto. This difference reduces his values further by 10 %. If we reduce his values for 76 Ge by 40 % his values M 0ν = 4.59 (QRPA) and 3.88 (RQRPA) are reduced to 2.76 (QRPA) and 2.34 (RQRPA) which agree nicely with our result of 2.68 (QRPA) and 2.41 (RQRPA). Civitarese and Suhonen have consistently higher values. Comparison of our results calculated in the QRPA approach with gA = 0.25 with the results of Civitarese and Suhonen [7]:
Nucleus 76 Ge 100 Mo 130 Te 136 Xe
CS (QRPA, 1.254) 3.33 2.97 3.49 4.64
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our (QRPA, 1.25) 2.68 ± 0.12 1.30 ± 0.10 1.56 ± 0.47 0.90 ± 0.20
Comparison of the results of Civitarese and Suhonen with ours. The calculation of Civitarese and Suhonen [7] are different from ours in several respects: 1. They have fitted the factor gpp not to the two neutrino double beta decay but to selected known single beta decay transition probabilities. 2. They have included higher order terms of the weak nucleon current according to Suhonen, Kadhkikar and Faessler [8] which is based on simple quark model with a harmonic oscillator potential for the quarks leading to Gauss’schen form factors for the weak currents and leading to the weak magnetism as the main higher contribution. In our approach we included the higher order currents according to Towner and Hardy [9] who fitted the higher order currents to allowed beta decays, to first forbidden beta decays, to beta-gamma angular correlations, to beta-alpha angular correlations and to muon capture in nuclei. Towner and Hardy use dipole form factors and the induced pseudoscalar term and not weak magnetism yields the largest effect. Weak magnetism would reduce the zero neutrino double beta decay matrix elements by about 30 %. 3. Short range Brueckner correlations are not included in the calculation of Civitarese and Suhonen [7]. But they include the finite size of the nuclei. In our calculation, we include short range Brueckner repulsion between the nucleons for the zero neutrino transition operator in form of a Jastrow factor. Inclusion of the short range repulsion in the work of Civitarese and Suhonen would
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reduce the matrix element for the zero neutrino transition probability and could bring it into agreement with our value if also a more accurat expression for the higher order weak currents and the different adjustment of gpp is included. 2. Conclusion In this contribution we investigated the accuracy of the zero neutrino matrix elements for the neutrinoless double beta decay. This accuracy determines exactly in the same way also the accuracy of the Majorana neutrino mass extracted from the neutrinoless double beta decay. We used the quasiparticle random phase approach (QRPA) and the renormalized QRPA (RQRPA) approximation which include the Pauli principle and reduces the ground state correlations and stabilises the wave function of the excited states of the intermediate nuclei. To get a feeling for the uncertainty we calculated the zero neutrino double beta decay for three different nucleon-nucleon forces (Bonn, Nijmegen, Argonne) and three different basis sets (small with about two oscillator shells, intermediate with about three oscillator shells and large with about five oscillator shells). In addition we performed the calculations with QRPA, with RQRPA (including the Pauli principle) and for two different axial vector coupling constants gA = 1.25 and gA = 1.00. In each of the 36 calculations for every nucleus we adjust a factor gpp multiplying the nucleon-nucleon twobody matrix elements of the order between 0.85 and 1.1 to fit the experimental two neutrino transition probabilities. We give the theoretical error of the zero neutrino transition probability as the 1σ error of the nine calculations for each method plus the experimental error of the two neutrino transition probability. In many cases the theoretical error of the zero neutrino nuclear matrix element is less than ± 20 % but in one case 96 Zr it is ± 100 %. But the main part of the error as indicated in Fig. 5 comes from the experimental error of the two neutrino transition probability. An improvement of the measurement of the two neutrino transition probability would even further reduce the uncertainty and by that further
reduce the error of the Majorana neutrino mass which can be determined if the zero neutrino double beta decay is measured. Acknowledgement: I would like to thank Dr. F. Simkovic, Dr. V. Rodin and Prof. P. Vogel for important contributions to the results presented here. REFERENCES 1. T. Tomoda, A. Faessler; Phys. Lett. B199 (1987) 475 2. A. Faessler, S. Kovalenko, F. Simkovic, J. Schwieger; Phys. Rev. Lett. 78 (1997) 183 3. O. Civitarese, A. Faessler, T. Tomoda; Phys. Lett. B194 (1987) 11 4. J. Toivanen, J. Suhonen; Phys. Rev. C55 (1997) 2314 5. J. Schwieger, F. Simkovic, A. Faessler; Nucl. Phys. A600 (1996) 179 F. Simkovic, J. Schwieger, M. Veselsky, G. Pantis, A. Faessler; Phys. Lett. B393 (1997) 267 6. V.A. Rodin, A. Faessler, F. Simkovic, P. Vogel; Phys. Rev. C68 (2003) 044302, to be published 7. O. Civitarese, J. Suhonen; Nucl. Phys. A729 (2003) 867 8. J. Suhonen, S.B. Khadkikar, A. Faessler; Nucl. Phys. A529 (1991) 727 9. I.S. Towner, J.C. Hardy; nucl-th/9504015 v1 (12 Apr 1995)