Theoretical Status of the Double Beta Decay

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Topics in Astroparticle and Underground Physics (TAUP 2009) IOP Publishing Journal of Physics: Conference Series 203 (2010) 012058 doi:10.1088/1742-6596/203/1/012058

Theoretical status of the double beta decay

Amand Faessler University of Tuebingen, Institute for Theoretical Physics, Auf der Morgenstelle 14, 72076 Tuebingen, Germany E-mail: [email protected]

Abstract. The detection of the neutrinoless double beta decay would establish that the neutrino is a Majorana particle and it would allow to measure the absolute neutrino mass in the meV region. To extract from the neutrinoless double beta decay halflife the absolute neutrino mass, the transition matrix element is as important as the experimental data. The dramatic progress in the last years is reviewed to determine these transition matrix elements for the neutrinoless double beta decay. It is shown, that the values calculated in the Quasi- particle Random Phase Approach (QRPA) and the recent results in the interacting boson model agree with each other completely within the uncertainties of the theories, while the 76 1 shell model yields values which in the region of Ge are by a factor 2 smaller than QRPA. Possible reasons for the differences and an experimental way of measuring the Fermi part for the neutrinoless double beta decay transition matrix elements are discussed. This would allow to determine experimentally which of the two approaches is the correct one. Finally we show that the renormalization parameter gpp, which multiplies the nucleon-nucleon interaction matrix elements can be overconstraint together with the axial charge gA in three nuclei, where the ground state of the intermediate nucleus in the double beta decay system is a 1+ state and − apart of the 2νββ also the β decay for the second leg and the electron capture (EC) in the first leg is measured. It is shown, that this overconstraint determination of gpp and gA leads to the same result for the transition matrix elements as when one adjusts gpp to the 2νββ alone.

1. Introduction The neutrinoless double beta decay is presently the only method to determine an absolute scale for the neutrino mass in the meV region. For this the transition matrix elements for the neutrinoless double beta decay are as important as the data.

< f | HˆW | k >< k | HˆW | i > T = dEk(ν) 76 − 76 (1) Z Ee + G E e E ν E As k Xk 0 ( ) − [ k( 1 ) + k( ) + Kern( ; )] M1 2 ‘2 mp T = Mm < mν > +Mθ < tgϑ > +MWR < ( ) > +MSUSY λ111 + MVR < > +... (2) M2 MVR 2π − w = | T |2 ρ ≤ 4.4 · 1033[sec 1] (3) ¯h f The transition matrix element is calculated with Fermi’s golden rule in second order. In equation (1) the expression for the transition from 76Ge is given. The transition matrix element in equation (2) shows on the right ﬁrst the neutrino mass term, which is proportional to the Majorana double beta decay mass < mν >. The matrix element contains additional terms.

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< tgϑ > depends on the mixing angle ϑ of the vector bosons mediating the left-handed and vector bosons mediating the right-handed weak interaction. The term depending on the heavy vector meson mediating mainly the right-handed weak interaction with the mass M2 originates from ′ a left-right symmetric Grand unification, while the term with the coupling constant λ111 comes from R-parity violating supersymmetry. The last term in equation (2) describes the exchange of a heavy mainly right-handed neutrino. Equation (3) shows the transition probability according to Fermi’s golden rule and as long as we have no confirmed data for the neutrinoless double beta decay we have upper limits, like the one for 128T e in equation (3). The first term on the right-hand side in equation (2) is the so-called ”gold plated term”, which is proportional to the absolute scale of the Majorana neutrino mass for the double beta decay. It can be expressed by the three mass eigenvalues of the neutrinos and the transformation coefficients from the mass to the flavor eigenstates. So if one determines the transition matrix element T experimentally and one calculates the matrix elements Mm theoretically and one can assume that this is the leading term, one is able to determine the absolute scale of the Majorana double beta decay neutrino mass.

2. The transition matrix elements Figure 1 shows as the points with the error bars, the theoretical transition matrix elements calculated in Tuebingen in the QRPA [1, 2]

Figure 1. Dimensionless transition matrix elements of the neutrinoless double beta decay. The circles with error bars are calculated [1] in the QRPA. The error bars indicate the highest and the lowest value from a series of theoretical calculations with three different sizes of basis sets, two different short range correlations: Jastrow [3] and UCOM [4], for an unquenched axial charge gA = 1.25 and the quenched value gA = 1.00 and for QRPA and RQRPA for the realistic force Bonn CD. The nucleon-nucleon matrix elements are determined by solving a relativistic version of the Bethe-Goldstone equation. The error bar indicates the highest and the lowest of these 24 calculations, while the circle is put into the middle. The triangles are the result of shell model calculations from Strassburg and Madrid [5], while the squares indicate values calculated in the interacting boson model, treating protons and neutrons separately IBM 2 [6]. The point for 150Nd for QRPA (Tuebingen) is calculated spherically. But the reduction factor 0.6 due to different deformations β(150Nd) = 0.183 and β(150Sm) = 0.114 fitted to the measured quadrupole momenta is included.

2 Topics in Astroparticle and Underground Physics (TAUP 2009) IOP Publishing Journal of Physics: Conference Series 203 (2010) 012058 doi:10.1088/1742-6596/203/1/012058

Figure 2. Contributions to the neutrinoless double beta decay matrix element in the decay 82Se to 82Kr for different angular momenta of the pairs of two neutrons which change into pairs of two protons. The left bar is calculated for the small shell model basis: 1f5/2, 2p3/2, 2p1/2, 1g9/2. The second bar includes in addition the spin-orbit partners 1f7/2 and 1g7/2. This guarantees that the Ikeda sum rule is fulfilled. The addition of the spin-orbit partners changes the results appreciably. The addition of the whole sd-shell does not change much the final results. The small single particle basis for the shell model calculations seems to be the cause for the smaller values for the SM than in the QRPA approach.

The shell model is handicapped by the limitation on the number of single particle states which can be included. Figure 2 shows the results for the 82Se decay. The first bar is calculated with QRPA in the shell model basis: 1f5/2, 2p3/2, 2p1/2, 1g9/2. The second bar is calculated including the spin orbit partners 1f7/2 and 1g7/2. This guarantees that the Ikeda sum rule is fulfilled. Inclusion of the full s, d shell does not change too much anymore. Figure 2 indicates the shell model yields to small values, at least in 76Ge and in 82Se due to the small single particle basis. But figure 2 gives also an information on the result in the interacting boson model [6]. This model contains only S and D pairs of two neutrons, which change into two protons. Figure 2 shows that also higher multipole pairs are involved. The very good agreement between QRPA [1] and the IBM 2 comes from the fact that the contribution of the s-bosons is smaller than for the S pairs of neutrons in QRPA. But QRPA has a stronger cancellation due to higher neutron angular momentum pairs which change into proton-proton pairs. In view of this the agreement between QRPA and IBM2 shown in figure 1 is surprisingly good. In [2] for the first time the transition matrix elements are calculated fully consistently: A realistic force Bonn CD and Argonne V18 is used to solve a relativistic approach to the Bethe-Goldstone equation in the so-called coupled-cluster method (CCM) or the exponential S approach. For the first time one uses the same realistic NN force to calculate the NN interaction matrix elements, the pairing correlations, the nuclear structure problem by QRPA and the short range correlation [2]. Under the assumption that Klapdor and coworkers [7] observed the 0νββ decay in 76Ge the CCM yields for the Bonn CD a double beta decay Majorana

3 Topics in Astroparticle and Underground Physics (TAUP 2009) IOP Publishing Journal of Physics: Conference Series 203 (2010) 012058 doi:10.1088/1742-6596/203/1/012058 neutrino mass < mν >= 0.22 ± 02(exp) ± 0.01(theor)[eV ] and for the Argonne V18 potential < Mν >= 0.24 ± 0.02(exp) ± 0.01(theor)[eV ].

3. Overconstraint determination of gpp and gA Due to the finite size of the single particle basis set, the strength of the nucleon-nucleon (NN) interaction must be slightly renormalized by a factor gpp. This factor gpp is usually adjusted to the two neutrino double beta decay transition probability. But in three nuclei 100Mo,116 Cd and 128T e we know in addition to the 2νββ halflives also the β− decay probability from the intermediate 1+ state to the final ground state and the electron capture (EC) from the intermediate 1+ ground state to the initial 0+ ground state nucleus. With this three data we can overconstrain two parameters, the renormalization constant gpp and the axial charge, which has in nuclei not his vacuum value gA = 1.25 but is quenched already in the pf-shell to about gA = 1.00 and to smaller values in medium heavy and heavy nuclei.

Figure 3. Overconstraint determination of the renormalization factor gpp and the axial charge − gA from the 2νββ decay and the β transition from the 1+ ground state of the intermediate nucleus for the double beta decay system 116cd to 116Sn.

The overconstraint renormalization factor of the NN-interaction gpp and the axial charge gA in the three nuclei 100Mo,116 Cd and 128T e are used in ﬁgure 4 to calculate the neutrinoless double beta decay transition matrix element and are compared with the usual adjustment of gpp to the 2νββ transition probability. The ﬁnal results do not change within the uncertainties of the calculations.

4. Summary The QRPA calculations in Tuebingen [1,2] and the calculations in Jyväskylä[9] obtain now in QRPA the same result. These results are also confirmed by Barea and Iachello [6] within the uncertainty of the calculation (see figure 1), while the shell model calculation gives in the 76Ge region results, which are by a factor 2 smaller. Recently the Tuebingen group [10] proposed a method, how one can at least for the Fermi part of the 0νββ transition probability determine experimentally which result is correct. At the moment the score is 2 : 1 in favor of the QRPA result.

4 Topics in Astroparticle and Underground Physics (TAUP 2009) IOP Publishing Journal of Physics: Conference Series 203 (2010) 012058 doi:10.1088/1742-6596/203/1/012058

Figure 4. Neutrinoless double beta decay matrix elements for 100Mo,116 Cd and 128T e calculated with the overconstraint renormalization factor gpp and axial charge gA. The black circle is the overconstraint calculation in a large basis, while the black square gives the same results in a smaller basis. The open circle are the old results, where only gpp is adjusted to 2νββ transitions. The ﬁnal results did not change.

I would like to thank Dr. Vadim Rodin and Prof. Fedor Simkovic for the collaboration. The work in Tuebingen was supported by the Sonderforschungsbereich Transregio 27: ”Neutrinos and beyond”, by the International Graduiertenkolleg GRK 683 Basel - Graz - Tuebingen and the EU project ILIAS N6, WP1: ”Double beta decay and related topics”.

References [1] F. Simkovic, A. Faessler, V. Rodin, P. Vogel, Jonathan Engel, Phys. Rev.C77, 045503 (2008) [2] F. Simkovic, A. Faessler, H. Muether, V. Rodin, M. Stauf, Phys. Rev.C79, 055501 (2009) [3] G. A. Miller, J. E. Spencer, Arm. Phys. (N. Y) 100, 562 (1976) [4] R. Roth, H. Hergert, P. Papakoutantinou, T. Neﬀ, H. Feldmeier, Phys. Rev.C72, 034002(2005) [5] E. Courier, J. Menendez, F. Nowacki, A. Poves, Phys. Rev. Lett. 100, 052503 (2008) [6] J. Barea, F. Iachello, Phys. Rev. C79, 044301 (2009) [7] Klapdor et al. Mod. Phys. Lett.A21, 1547 (2006) [8] A. Faessler, G. L. Fogli, E. Lisi, V. Rodin, A. M. Rotunno, F. Simkovic, J. Phys.G 35, 075104 (2008) [9] J. Suhonen, O. Civitarese, Phys. Lett.B668, 277 (2008) [10] V. Rodin, A. Faessler, http://Arxiv.org/abs/0906.1759