Neutrinoless double β decay within the self-consistent renormalized quasiparticle random phase approximation and inclusion of induced nucleon currents

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The first, to our knowledge, calculation of neutrinoless double beta decay ($0 uetaeta$-decay) matrix elements within the self-consistent renormalised Quasiparticle Random Phase Approximation (SRQRPA) is presented. The contribution from the

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    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 Self-consistentRenormalized Quasiparticle Random Phase Approximation andinclusion of induced nucleon currents A. Bobyk, W. A. Kami´nski Department of Theoretical Physics, Maria Curie-Sk lodowska University,Radziszewskigo 10, PL-20-031 Lublin, Poland. F. ˇSimkovic Department of Nuclear Physics, Comenius University,Mlynska dol., pav. F1, SK-842 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 self-consistent renormalized Quasi-particle Random Phase Approximation (SRQRPA) is presented. The con-tribution from the momentum-dependent 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, two-nucleonshort-range correlations and the truncation of the model space. A compari-son with the standard and renormalized QRPA results is presented. We havefound a considerable reduction of the SRQRPA nuclear matrix elements, re-sulting 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 (LSND-experiment) 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 relativeCP-phases 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 half-lifeof 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 byHeidelberg-Moscow 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 ele-ments governing the Majorana mass mechanism of 0 νββ  -decay can be calculated correctly[5]. However, the practical nuclear structure calculation always involves some approxima-tions, 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 proton-neutron2  Quasiparticle Random Phase Approximation (pn-QRPA) [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 pn-QRPA[9], the renormalized pn-QRPA (pn-RQRPA) [12,13] and the self-consistent pn-RQRPA (pn-SRQRPA) [16,17] calculation of the nuclear matrix element  M   m ee   for 0 νββ  -decay of  76 Ge. We note that the pn-SRQRPA 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 ap-proaches.The pn-QRPA has been the most popular theoretical tool in description of the  β   and ββ   decays of medium and heavy open shell nuclei. However, the pn-QRPA develops acollapse beyond some critical value of the particle-particle 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 pn-QRPA one gets the pn-RQRPA [12,13], which avoids collapse within a physical range of  particle-particle 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 pn-RQRPA (pn-SRQRPA) [15,17,16], which is more complex version of the RQRPA, is becoming increasingly popular to describe strongly correlated Fermionsystems. The pn-SRQRPA goes a step further beyond the pn-RQRPA. In the pn-SRQRPAat 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 (pn-QRPA, pn-RQRPA) [15,17,16]. Thus the pn-SRQRPA 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 Gamow-Teller and new tensor contributions of   M   m ee   [11]. Thus M   m ee   is given as sum of Fermi, Gamow-Teller 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 for-malism  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 short-range correlation function and  O  m ee  (12) represents the coordinateand spin dependent part of the two-body 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 qσ 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, axial-vector 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 multipo-larities. The form of the one-body 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 pn-RQRPA and the pn-SRQRPA 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 pn-QRPA 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 pn-QRPA studies of Ref. [9] (9-levels 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. Two-body G-matrix elements were calculatedfrom the Bonn one-boson exchange potential within within the Brueckner theory. In thepn-QRPA and the pn-RQRPA approaches pairing interactions have been adjusted to fit theempirical pairing gaps. In the pn-SRQRPA approach the pairing matrix elements of the NNinteraction have not been rescaled as the mean field is directly related to the excited states.The particle-particle and particle-hole channels of the G-matrix 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 pn-RQRPA and pn-SRQRPA approaches. The larger12-levels 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 pn-SRQRPA 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
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