n-p Interaction Effects on the Double Beta Decay Nuclear Matrix Elements for Medium Mass Nu

a r X i v :n u c l -t h /0410005v 1 1 O c t 2004n-p Interaction Effects on the Double Beta Decay NuclearMatrix Elements for Medium Mass NucleiP.K.Raina and A.Shukla Department of Physics and Meteorology,IIT Kharagpur-721302,India P.K.Rath,B.M.Dixit,K.Chaturvedi,and R.Chandra.Department of Physics,University of Lucknow,Lucknow-226007,India S.K.Dhiman and A.J.Singh.Department of Physics,H.P.University,Shimla-171002,India.Abstract The quality of HFB wave functions are tested by comparing the theoretically calculated results with the available experimental data for a number of spectroscopic properties like yrast spectra,reduced B(E2)transition probabilities,quadrupole moments and g-factors for the nuclei involved in 2νββdecay.It is observed that the np interactions vis-`a -vis the deformations of the intrinsic ground states of medium mass nuclei play a crucial role in the fine tuning of the nuclear matrix elements,M 2ν.It is well established by now that the implications of nuclear ββdecays are far reaching in nature in general.0νββdecay in particular is one of the very rare promising processes to test the physics beyond the standard model (SM)of fundamental particles.These aspects of nuclear ββdecay have been excellently elaborated in a number of review articles over the past years [1-8].The 2νββdecay,a second order process of weak interaction that conserves the lepton number exactly,is allowed in the SM.The half life of 2νββdecay is a product of accurately known phase space factor and appropriate nuclear transition matrix element M 2ν.The half lives of 2νββdecayhave been already measured for about ten nuclei and the values of M 2νcan be extracted directly.Consequently,the validity of different models employed for nuclear structure calculations can be tested by calculating the M 2ν.It is observed that in all cases the 2νββdecay matrix elements are sufficiently quenched.The main motive of all the theoretical calculations is to understand the physical mechanism responsible for the suppression of the M 2ν.The M 2νis calculated mainly in three types of models.One is the shell model and its variants.The second is the quasiparticle random phase approximation (QRPA)and extensions their of.The third type of models is classified as the alternative models.The details about these models -their advantages as well as shortcomings -have been discussed excellently by Suhonen and Civitarese [5]and Faessler and Simkovic [6].All the nuclei undergoing ββdecay are even-even type.Hence the pairing degrees of freedom play an important role.Moreover,it has been conjectured that the deformation can play a crucial role in ββdecay rates.Hence it is desirable to have a model that incorporates the pairing and deformation degrees of freedom on equal footing in its formalism.For this purpose the Projected Hartree-Fock-Bogoliubov (PHFB)model is one of the most natural choices.Coincidentally,most of the ββdecaying nuclei fall in medium mass region.The success of the PHFB model in explaining the observed experimental trends in this mass region has motivated us to apply the PHFB wave functions to the study of nuclear ββdecay as well.The mass region A ≈100provides us with a nice example of shape transitions [9],where,at one end,nuclei can be described in terms of shell model wave functions involving a small number ofconfigurations and,at the other end of this region,wefind good evidence of rotational collectivity. These nuclei lie between doubly magic132Sn and the strongly deformed100Zr,near which the structural changes are rather rapid with the addition of protons and neutrons.In the past there have been many attempts,[10-15]to explore the factors responsible for the structural changes in this mass region.Federman and Pittel[14]computed the deformation energy in the framework of Hartree-Fock-Bogoliubov(HFB)theory in conjunction with the surface delta interaction(SDI),suggesting that the neutron-proton(n-p)interaction in the spin-orbit partner(SOP)orbits-1g9/2and1g7/2in this case may be instrumental vis-`a-vis the onset of deformation in Mo isotopes with A>100.A systematic study of the behavior of the low-lying collective states of neutron-rich even Cd, Pd,Ru,and Mo isotopes has lead to the conclusion that these structural changes are related to the exceptionally strong n-p interaction in this region.It has also been observed that the n-p interactions among the SOP orbits have a deformation producing tendency and the systematics of low-lying states are intricately linked with the nature of n-p interaction.The sensitivity of the yrast spectra and the transition charge densities(TCD)to the neutron-proton interaction strength has lead to thefixing of these strengths very accurately and has been demonstrated[15]through the examples of110Cd and114Cd.We have adopted this method for fixing the p-n strength of QQ interaction by looking at the spectra of2+state of the nuclei involved inββdecay.A large number of theoretical as well as experimental studies of2νββdecay have already been done forβ−β−of96Zr,100Mo,110Pd,128,130Te nuclei and e+ββ(β+β+,β+EC and ECEC)in case of96Ru,106Cd,124Xe and130Ba nuclei over the past few years with more emphasis on100Mo and106Cd cases.Theββdecay is not an isolated nuclear process.The availability of data permits a rigorous and detailed critique of the ingredients of the microscopic models used to provide a description of these nuclei.We have studied the2νββdecay not isolatedly but together with other observed nuclear phe-nomena.This is in accordance with the basic philosophy of nuclear many body theory,which is to explain all the observed properties of nuclei in a coherent manner.Hence as a test of the reliability of the wave functions,we have calculated the yrast spectra,reduced B(E2)transition probabilities, static quadrupole moments and g-factors and compared with the available experimental data.The theoretical formalism to calculate the half life of2νββdecay mode has been given by Haxton and Stephenson[1],Doi et al[2,3]and Tomoda[4].Very brief outlines of the calculation of nuclear transition matrix elements of theββdecay in the PHFB model are presented here. Details of expressions used in calculation of spectroscopic properties can be found in Dixit et al [16].The half-life of2νββdecay for0+→0+transition is given byT2ν1/2(0+−→0+) −1=G2ν|M2ν|2(1) whereM2ν= N 0+ στ+ 1+N 1+N στ+ 0+E N −(M I+M F)/2=−2M2νGTM2νGT=14 αβγδ αβ σ1.σ2τ+τ+ γδ εη 1+F(ν)Z,N(θ)f(ν)Z−2,N+2 −1εα(f(ν)Z−2,N+2)εβ× 1+F(π)Z,N(θ)f(π)Z−2,N+2 −1γη(F(π)Z,N)ηδsinθdθ(5) wheren J=π{det[1+F(π)(θ)f(π)†]}1/2×{det[1+F(ν)(θ)f(ν)†]}1/2d J00(θ)sin(θ)dθ(6)andn(Z,N),(Z−2,N+2)(θ)={det[1+F(π)Z,N (θ)f(π)†Z−2,N+2]}1/2×{det[1+F(ν)Z,N(θ)f(ν)†Z−2,N+2]}1/2(7)Theπ(ν)represents the proton(neutron).PHFB calculations are summarized by the coeffi-cients(U im,V im)and C ij,m and their matrices(F N,Z(θ))αβand(f N,Z).The details can be found in reference[16].In the present calculations we treat the doubly even nucleus76Sr(Z=N=38)as an inert core and the valence space is spanned by the orbits1p1/2,2s1/2,1d3/2,1d5/2,0g7/2,0g9/2and0h11/2 for protons and neutrons.The set of single particle energies(SPE’s)but for theε(0h11/2)which is slightly lowered,employed here is same as used in a number of successful shell model as well as variational model[10-16]calculations for nuclear properties in the mass region A=100.The effective two-body interaction is the PPQQ type[17].Table1.Variation in excitation energies in MeV of Jπ=2+,4+and6+yrast states for100Mo and100Ru nuclei with change inχpn keepingfixed G p=-0.30MeV,G n=-0.20MeV andε(0h11/2)=8.6MeV.Nucleusχpn0.018260.018660.019060.019460.01986Exp.[21]χpn0.017580.017980.018380.018780.01918comparable to those suggested by Arima on the basis of an empirical analysis of the effective two-body interactions[20].Table2.Experimental half-lives T2ν1/2,and corresponding nuclear matrix elements M2ν,along with the theoretical values in different models for0+→0+2νββdecay of100Mo.The numbers corresponding to(a)and(b)are calculated for g A=1.25and1.0respectively.ExperimentRef Projects T2ν1/2|M2ν|(1018yrs)[22]UC-Irvin 6.82+0.38−0.53±0.68a)0.125+0.012−0.009[24]SRPA(WS)0.05930.474.3[23]NEMO9.5±0.4±0.9a)0.106+0.008−0.007[25]SU3(DEF)0.1089.0922.2 [8]Average8.0±0.7Mode†Ref Model|M2ν|(T2ν1/2)†a)b)[27]>2.4×1020∗∗[28]>1.0×1019∗[29]>9.2×1017[27]>4.1×1020[28]>0.66×1019∗[29]>2.6×1017ECEC[31]PHFB0.23811.2428.42 1020yrs[33]SU(4)0.19317.0042.04[24-26].In Table2,we have compiled some of the latest available experimental and the theoretical results along with our calculated M2νand the corresponding half-life T2ν1/2.We have used a phase space factor G2ν=9.434×10−18yr−1given by Doi et al[2]and an energy denominator E d= 11.2MeV given by Haxton et al[1].In column4of Table2,we have presented the M2νextractedfrom the experimentally observed T2ν1/2using the phase space factor given above.The phase spaceintegral has been evaluated for g A=1.25by Doi et al[2].However in heavy nuclei it is more justified to use the nuclear matter value of g A around1.0.Hence,the experimental M2νas well asthe theoretical T2ν1/2are calculated for g A=1.0and1.25.The present calculation and that of Hirschet al using SU3(SPH)[25]give nearly identical value.They are close to the experimental result given by De Silva et al[22]for g A=1.25while for g A=1.0,the above two M2νare in agreement with the results of NEMO.The calculated values given by Stoica using SRPA(WS)[24]are too low and those from Suhonen et al[26]are slightly on higher side.Further the value M2νgiven by Hirsch et al using SU3(DEF)[25]favors the results of NEMO[23]for g A=1.25.Another example we take for the0+→0+positronββ(β+β+,β+EC and ECEC)decay of 106Cd→106Pd.This transition has also been investigated by many experimental groups and in different theoretical frameworks.In Table3,we have compiled some of the latest available experi-mental[27-30]and the theoretical results[31-33]along with our calculated M2νand correspondinghalf-lives T2ν1/2.We have used phase space factors given by Doi and Kotani[3]and the averageenergy from Haxton and Stephenson[1].Our calculated values are nearly half of the recently given QRPA results of Suhonen and Civitaresse[32]for all the three modes.The theoretical values of PHFB and SU(4)[33]are in better agreement(factor of roughly two third)for theβ+EC and ECEC modes.From the above discussions,it is clear that the validity of nuclear models presently employed to calculate the M2νcannot be uniquely established due to error bars in experimental results as well as uncertainty in g A.Further work is necessary both on the experimental and theoretical front to judge the relative applicability,success and failure of various models used so far for the study of double beta decay processes.As an example to see quantitatively the effect of deformation on M2νvis-a-vis the variation of the strength of pn part of the QQ interaction,the results are displayed in Fig.1for100Mo and 106Cd cases.It is observed that the deformations of the HFB intrinsic states play an importantrole in the calculations of M2νand hence on the half life.To 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