Research Article Constraints on Light Neutrino Parameters ...Constraints on Light Neutrino...

Research ArticleConstraints on Light Neutrino Parameters Derived from theStudy of Neutrinoless Double Beta Decay

Sabin Stoica12 and Andrei Neacsu12

1 Horia Hulubei Foundation PO Box MG-12 Romania2Horia Hulubei National Institute of Physics and Nuclear Engineering PO Box MG-6 077125 Magurele Bucharest Romania

Correspondence should be addressed to Sabin Stoica stoicatheorynipnero

Received 7 February 2014 Accepted 28 April 2014 Published 28 May 2014

Academic Editor Abhijit Samanta

Copyright copy 2014 S Stoica and A Neacsu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

The study of the neutrinoless double beta (0]120573120573) decaymode can provide uswith important information on the neutrino propertiesparticularly on the electron neutrino absolute mass In this work we revise the present constraints on the neutrino mass parametersderived from the 0]120573120573 decay analysis of the experimentally interesting nuclei We use the latest results for the phase space factors(PSFs) and nuclear matrix elements (NMEs) as well as for the experimental lifetime limits For the PSFs we use values computedwith an improved method reported very recently For the NMEs we use values chosen from the literature on a case-by-case basistaking advantage of the consensus reached by the community on several nuclear ingredients used in their calculationThus we tryto restrict the range of spread of the NME values calculated with differentmethods and hence to reduce the uncertainty in derivinglimits for theMajorana neutrinomass parameter Our results may be useful to have an updated image on the present neutrinomasssensitivities associated with 0]120573120573 measurements for different isotopes and to better estimate the range of values of the neutrinomasses that can be explored in the future double beta decay (DBD) experiments

1 Introduction

Neutrinoless double beta decay is a beyond standard model(BSM) process by which an even-even nucleus transformsinto another even-even nucleus with the emission of twoelectronspositrons but no antineutrinosneutrinos in thefinal states Its study is very attractive because it would clarifythe question about the lepton number conservation decideon the neutrinos character (are they Dirac or Majorana par-ticles) and give a hint on the scale of their absolute massesMoreover the study of the 0]120573120573 decay has a broader potentialto search for other BSM phenomena The reader can findup-to-date information on these studies from several recentreviews [1ndash6] which also contain therein a comprehensive listof references in the domain

The scale of the absolute mass of neutrinos is a keyissue for understanding the neutrino properties It cannotbe derived from neutrino oscillation experiments which canonly measure the square of the neutrino mass differencesbetween different flavors [7ndash12] Analysis of 0]120573120573 decay and

cosmological data are at present the most sensitive ways toinvestigate this issue

The lifetime of the 0]120573120573 decay modes can be expressedin a good approximation as a product of a phase spacefactor (depending on the atomic charge and energy releasedin the decay 119876120573120573) a nuclear matrix element (related to thenuclear structure of the parent and daughter nuclei) anda lepton number violation (LNV) parameter (related to theBSM mechanism considered) Thus to extract reliable limitsfor the LNV parameters we need accurate calculations ofboth PSFs and NMEs as well as reliable measurements of thelifetime

The largest uncertainties in theoretical calculations forDBD are related to the NMEs values That is why there isa continuous effort in the literature to develop improvednuclear structure methods for their computation At presentthe NMEs are computed by several methods which dif-fer conceptually the most employed being the proton-neutron quasi-random phase approximation (pnQRPA) [13ndash20] interacting shell model (ISM) [21ndash25] interacting

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 745082 7 pageshttpdxdoiorg1011552014745082

2 Advances in High Energy Physics

boson approximation (IBA) [26ndash28] projected Hartree-Fock-Bogoliubov (PHFB) [29 30] and energy density func-tional (EDS) method [31] There are still large differencesbetween the NMEs values computed with different methodsand by different groups and these have been largely discussedin the literature (see eg [2 3]) On the other side there is aconsensus in the community on the way that several nucleareffects and nuclear parameters should be used in calculationsIn this work we take advantage of this consensus when wechose theNMEs values trying to restrict their range of spreadand consequently to reduce the uncertainty in deriving theneutrino Majorana mass parameters

Unlike the NMEs the PSFs have been calculated a longtime ago [32ndash39] and were considered to be computed withenough precision However recently they were recalculatedwithin an improved approach by using exact electron Diracwave functions (wf) taking into account the finite nuclearsize and electron screening effects [40] The authors founddifferences between their results and those calculated previ-ously with approximate electron wf especially for heaviernuclei We have also independently recalculated the PSFsby developing new routines for computing the relativistic(Dirac) electron wf by taking into account the nuclear finitesize and screening effects In addition we use a Coulombpotential derived from a realistic proton density distributionin the daughter nucleus [41 42] In this workwe use new PSFsvalues obtained by improving the numerical precision of ourroutines as compared with our previous works The obtainedvalues are very close to those reported in [40 41]

Finally for the lifetime limits we take the most recentresults found in the literature

The paper is organized as follows In the next sectionwe shortly recall the general formalism for the derivationof the neutrino mass parameters from 0]120573120573 decay analysishighlighting the nuclear ingredients involved in calculationsIn Section 3 we discuss the way of choosing the NME valuesand report our results for the light neutrino Majorana massparameters while in Section 4 we formulate the conclusionsof our work

2 Formalism

We shortly present the general formalism for the derivationof neutrino mass parameters from 0]120573120573 decay analysis Westart with the lifetime formula and then describe the mainsteps and ingredients used in the theoretical calculation oftheir components that is PSFs and NMEs

Assuming that the dominant mechanism of occurrencefor the 0]120573120573 decay mode is the exchange of Majorana left-handed light neutrinos between twonucleons from the parentnucleus the lifetime reads

(1198790]12)minus1= 1198660](119876120573120573 119885)

100381610038161003816100381610038161198720]10038161003816100381610038161003816

2(⟨119898]⟩

119898119890)

2

(1)

where 1198660] are the PSFs for this decay mode depending onthe energy decay 119876120573120573 and nuclear charge 119885 1198720] are thecorresponding NMEs depending on the nuclear structure ofthe parent and daughter nuclei involved in the decay 119898119890 is

electron mass and ⟨119898]⟩ is the light neutrino Majorana massparameter This parameter can be expressed as a (coherent)linear combination of the light neutrino masses

⟨119898]⟩ =

1003816100381610038161003816100381610038161003816100381610038161003816

3

sum119896=1

1198802119890119896119898119896

1003816100381610038161003816100381610038161003816100381610038161003816

(2)

where 119880119890119896 are the elements of the first row in the PMNS(Pontecorvo-Maki-Nakagawa-Sakata) neutrino matrix and119898119896 are the light neutrinomasses [56] From (1) the expressionof119898] reads

⟨119898]⟩ =119898119890

10038161003816100381610038161198720]1003816100381610038161003816

radic1198790] sdot 1198660] (3)

For deriving ⟨119898]⟩ we need accurate calculations ofboth PSFs and NMEs for each isotope for which there areexperimental lifetime limits The PSFs have been calculateda long time ago in some approximations [32ndash39] and wereconsidered until recently to be computed with enoughprecision However they were recalculated recently in [40ndash42] using more advanced approaches for the numericalevaluation of the Dirac wave functions with the inclusionof nuclear finite size and screening effects In addition in[41] the usual Coulomb spherical potential was replaced byanother one derived from a more realistic proton densitydistribution in the daughter nucleus These recent PSF cal-culations led to significant differences in comparison to theolder calculations especially for the heavier isotopes thatshould be taken into account for a precise derivation of theneutrino mass parameters

The computation of the NMEs is a subject of debate inthe literature for long time because they bring the largeuncertainties in the theoretical calculations for DBD Dif-ferent groups have developed several conceptually differentnuclear structure methods [13ndash31] as we have mentioned inthe previous section The expression of the NMEs can bewritten in general as a sum of three components

1198720]= 1198720]GT minus (

119892119881

119892119860)

2

sdot 1198720]119865 minus119872

0]119879 (4)

where 1198720]GT 1198720]119865 and 1198720]119879 are the Gamow-Teller (GT)

Fermi (119865) and Tensor (119879) components respectively Theseare defined as follows

1198720]120572 = sum119898119899

⟨0+1198911003817100381710038171003817120591minus119898120591minus119899119874

1205721198981198991003817100381710038171003817 0+119894 ⟩ (5)

where 119874120572119898119899 are transition operators (120572 = GT 119865 119879) andthe summation is performed over all the nucleon states Animportant part of the NME calculation is the computationof the reduced matrix elements of the two-body transitionoperators 119874

120572 Their calculation can be decomposed intoproducts of reduced matrix elements within the spin andrelative coordinate spaces Their explicit expressions are [423]

119874GT12 = 1205901 sdot 1205902119867(119903) 119874

11986512 = 119867 (119903)

11987411987912 = radic

2

3[1205901 times 1205902]

2sdot119903

119877119867 (119903) 119862

(2)(119903)

(6)

Advances in High Energy Physics 3

The most difficult is the computation of the radial partof the two-body transition operators which contains theneutrino potentials These potentials depend weakly on theintermediate states and are defined by integrals ofmomentumcarried by the virtual neutrino exchanged between the twonucleons [16]

119867120572 (119903) =2119877

120587intinfin

0119895119894 (119902119903)

ℎ120572 (119902)

120596

1

120596 + ⟨119864⟩1199022119889119902

equiv intinfin

0119895119894 (119902119903)119881120572 (119902) 119902

2119889119902

(7)

where 119877 = 119903011986013 fm (1199030 = 12 fm) 120596 = radic1199022 + 1198982]

is the neutrino energy and 119895119894(119902119903) is the spherical Besselfunction (i = 0 0 and 2 for GT F and T resp) Usuallyin calculations one uses the closure approximation whichconsists of a replacement of the excitation energies of thestates in the intermediate odd-odd nucleus contributing tothe decay by an average expression ⟨119864⟩ This approximationworks well in the case of 0]120573120573 decay modes and simplifiesmuch the calculations The expressions of ℎ120572 (120572 = GT 119865 119879)are

ℎ119865 = 1198662119881 (1199022) (8)

ℎGT (1199022) =

1198662119860 (1199022)

1198922119860[1 minus

2

3

1199022

1199022 + 1198982120587+1

3(

1199022

1199022 + 1198982120587)

2

]

+2

3

1198662119872 (1199022)

1198922119860

1199022

41198982119901

(9)

ℎ119879 (1199022) =

1198662119860 (1199022)

1198922119860[2

3

1199022

1199022 + 1198982120587minus1

3(

1199022

1199022 + 1198982120587)

2

]

+1

3

1198662119872 (1199022)

1198922119860

1199022

41198982119901

(10)

where119898120587 is the pion mass119898119901 is the proton mass and

119866119872 (1199022) = (120583119901 minus 120583119899)119866119881 (119902

2) (11)

with (120583119901 minus 120583119899) = 471The expressions (9)-(10) include important nuclear ingre-

dients that should be taken into account for a precisecomputation of the NMEs such as the higher order currentsin the nuclear interaction (HOC) andfinite nucleon size effect(FNS) Inclusion of HOC brings additional terms in the119867GTcomponent and leads to the appearance of the119867119879 componentin the expressions of the neutrino potentials FNS effect istaken into account through 119866119881 and 119866119860 form factors

119866119860 (1199022) = 119892119860(

Λ2119860

Λ2119860 + 1199022)

2

119866119881 (1199022) = 119892119881(

Λ2119881

Λ2119881 + 1199022)

2

(12)

For the vector and axial coupling constants the majority ofthe calculations take either the quenched value 119892119881 = 1or the unquenched one 119892119860 = 125 while the values of thevector and axial vector form factors are Λ119881 = 850MeVand Λ119860 = 1086MeV [1] respectively As one can see whenHOC and FNS corrections are included in the calculationsthe dependence of NMEs expression on 119892119860 is not trivialand the NMEs values obtained with the quenched or theunquenched value of this parameter cannot be obtained bysimply rescaling

To compute the radial matrix elements ⟨119899119897|119867120572|11989910158401198971015840⟩ an

important ingredient is the adequate inclusion of SRCsinduced by the nuclear interaction The way of introducingthe SRC effects has also been subject of debate ([16ndash18 20])The SRC effects are included by correcting the single particlewf as follows

120595119899119897 (119903) 997888rarr [1 + 119891 (119903)] 120595119899119897 (119903) (13)

The correlation function 119891(119903) can be parametrized in severalways There are three parameterizations which are usedMiller-Spencer (MS) UCOM and CCM (with CD-Bonnand AV18 potentials) The Jastrow prescription [34] for thecorrelation function is

119891 (119903) = minus119888 sdot 119890minus1198861199032

(1 minus 1198871199032) (14)

and includes all these parameterizations depending on valuesof the a b c parameters

Including HOC and FNS effects the radial matrix ele-ments of the neutrino potentials become

⟨1198991198971003816100381610038161003816119867120572 (119903)

1003816100381610038161003816 11989910158401198971015840⟩ = int

infin

01199032119889119903120595119899119897 (119903) 12059511989910158401198971015840 (119903) [1 + 119891(119903)]

2

times intinfin

01199022119889119902119881120572 (119902) 119895119899 (119902119903)

(15)

where ] is the oscillator constant and 119881120572(119902) is an expressioncontaining the 119902 dependence of the neutrino potentials

From (4)ndash(15) one can see that a set of approximationsand parameters are involved in the NMEs expressions asthe HOC FNS and SRC effects and 119892119860 1199030 (Λ119860 Λ 119861) ⟨119864⟩parameters Are there any recommendations on how shouldthey be included in the calculations At present there is ageneral consensus in the community in this respect that willbe discussed in the next section

3 Numerical Results and Discussions

The neutrino mass parameters are derived from (3) To get⟨119898]⟩ in the same units as119898119890 we take theNMEs dimensionlessand the PSFs (1198660]) in units of [yr]minus1

The PSF values were obtained by recalculating them withour code developed in [41] but with improved numericalprecision At this point we mention that the improved PSFvalues come on the one hand by the use of a Coulombpotential describing a more realistic proton charge density inthe daughter nucleus instead of the (usual) constant chargedensity one to solve the Dirac equations for obtaining the


electron wf On the other hand we got better precision ofour numerical routines that compute the PSFs by enhancingthe number of the interpolation points on a case-to-casebasis until the results become stationaryThe obtained valuesare very close to both those reported previously in [40 41]This gives us confidence on their reliability We mentionthat these PSFs values differ from older calculations as forexample those reported in [33 35ndash37] by up to 28 Suchdifferences are important for precise estimations and justifythe reactualization of the PSFs values in extracting Majorananeutrino mass parameters

For the experimental lifetime we took the most recentresults reported in the literature In particular we remark thenewest results for 76Ge fromGERDA [49] and for 136Xe fromKamLAND-Zen [54]

The largest uncertainty in the derivation of ⟨119898]⟩ comesfrom the values of the NMEs Fortunately at present there isa general consensus in the community on the employment ofthe different nuclear effects (approximations) and parameterswhich appear in the NMEs expressions (see (4)ndash(15)) [57]Thus one can restrict the range of spread of the NMEsvalues for a particular nucleus if one takes into account somerecommendations resulting from the analysis of many NMEscalculations For example one recommends the inclusion incalculation of the HOC FNS and SRC effects (although theireffects can partially compensate each other [43]) For SRCssofter parametrizations likeUCOM[17 18 20] andCCM[58ndash60] are recommended while the MS produces a too severecut of the wf for very short internucleon distances whichreflects into smaller NMEs values Concerning the nuclearparameters one recommends the use of an unquenched valuefor the 119892119860 axial vector constant the values specified above forthe vector and axial vector form factors (Λ119881 Λ119860) and a valueof 1199030 = 12 fm for the nuclear radius constant The value forthe average energy (⟨119864⟩) used in the closure approximationis a function of atomicmassA but the results are less sensitiveto changes within a few MeV The use in different ways ofthese ingredients can result in significant differences betweenthe NMEs values Hence a consensus is useful to approachthe results obtained by different groups Having agreementon these nuclear ingredients the differences in the NMEsvalues should be searched in the features of the differentnuclear structure methods These methods use differentways of building the wave functions and different specificmodel spaces and type of nucleon-nucleon correlations anduse some specific parameters [3 24 43] Unfortunately theuncertainties in the NMEs calculation associated with aparticular nuclear structure method cannot be easily fixedand they are still a subject of debate As a general featureShM calculations underestimate the NMEs values (due to thelimitations of the model spaces used) while the other meth-ods overestimate them There are however a few hints onhow to understandbring closer the NMEs results obtainedwith different methods One idea would be to analyze thestructure of the wave functions used in terms of the seniorityscheme [57] Another one is to (re)calculate the NME valuesas to reproduce sp occupancies numbers measured recentlyfor 76Ge and 82Se nuclei [61 62] For example when the

QRPA calculations have been modified with the sp energiesthat reproduce the experimental occupancies the newQRPANMEs values are much closer to the ShM ones

In Table 1 we display the NMEs values obtained withdifferent nuclear methods For uniformity and in agreementwith the consensus discussed above we chose those resultsthat were performed with the inclusion of HOC FNS andSRC (UCOM and CD-Bonn) effects and with unquenched119892119860 = 125 as nuclear ingredients We mention thatthe newest experimental determinations of this parameterreport values even larger (1269 1273) [63] However thedifferences between NMEs values obtained with 119892119860 = 125ndash1273 are not significant [43] The NMEs values for 76Geand 82Se written in parenthesis represent the adjusted NMEsvalues obtained with QRPA method by the Tuebingen andJyvaskyla groups when the sp energies were adjusted to theoccupancy numbers reported in [61 62] One remarks thatthe QRPA calculations with sp occupancies in accordancewith experiment get significantly close to the ShM resultswhich is remarkable In the future one expectsmeasurementsof the occupancy numbers for other nuclei as well Also itwould be interesting if other methods besides QRPA wouldtry to recalculate the NMEs by adjusting sp energies toexperimental occupancy numbers

We also make some remarks about the NMEs values ona case-by-case basis For 48Ca we appreciate that ShM cal-culations give more realistic results than the other methodsIn support of this claim we mention that in the case ofthis isotope ShM calculations are performed within the fullpf shell and using good effective NN interactions checkedexperimentally on other spectroscopic quantities [43 4464] Also we remark that NMEs values obtained with ShMfor this isotope were used to correctly predict 1198792]12 beforeits experimental measurement [65] For the isotopes withA = 96ndash130 there is a larger spread of the NMEs valuescalculated with different methods and consequently a largeruncertainty in predicting the ⟨119898]⟩ parameters For 136Xethere are new ShM large-scale calculations with inclusion ofa larger model space than the older calculations [45] For thisisotope the NMEs values are more grouped Corroboratedwith a quite good experimental lifetime from this isotope onegets the most stringent constraint for the ⟨119898]⟩ parameter

In Table 2 we present our results for the Majorananeutrino mass parameters (⟨119898]⟩) together with the values of119876120573120573 the PSFs (119866

0]) NMEs (1198720]) and experimental lifetime(1198790]12) for all the isotopes for which data exists Making asort of the NMEs values from the literature according tothe considerations presented we reduce the interval of theirspread to about a factor of 2 even less (with one exception)This results in reducing the uncertainty in deriving theconstraints on the light neutrino Majorana mass parameterswhile taking into account NME values obtained with all themain nuclear methods existent on the market One observesthat the stringent constraints are obtained from the 136Xeisotope followed by the 76Ge one This is due to boththe experimental sensitivity of the experiments measuringthese isotopes and the reliability of the PSFs and NMEstheoretical calculations The experiments measuring these


Table 1 The NMEs obtained with different methods The values are obtained using an unquenched value for 119892119860 and softer SRCparametrizations which are specified in the second column

Method SRC 48Ca 76Ge 82Se 96Zr 100Mo 116Cd 130Te 136Xe 150Nd[43] ShM CD-BONN 081 313 288[44] ShM CD-BONN 090 221 [45][24] ShM UCOM 085 281ndash352 264 265 219[27] IBM-2 CD-BONN 238 616 499 300 450 329 461 379 288[3] QRPA CD-BONN 593 (327) 530 (454) 219 467 372 480 300 316 [46][47] QRPA UCOM 536 (411) 372 312 393 479 422 280[31] GCM CD-BONN 237 460 422 565 508 472 513 420 171[29 30] PHFB CD-BONN 298 607 398 268

Table 2 Majorana neutrino mass parameters together with the other components of the 0]120573120573 decay halftimes the 119876120573120573 values theexperimental lifetime limits the phase space factors and the nuclear matrix elements

119876120573120573 [MeV] 1198790]120573120573exp [yr] 1198660]120573120573 [yrminus1] 1198720]120573120573 ⟨119898]⟩ [eV]48Ca 4272 gt58 1022 [48] 246119864 minus 14 081ndash090 lt[150ndash167]76Ge 2039 gt21 1025 [49] 237119864 minus 15 281ndash616 lt[037ndash082]82Se 2995 gt36 1023 [50] 101119864 minus 14 264ndash499 lt[170ndash321]96Zr 3350 gt92 1021 [51] 205119864 minus 14 219ndash565 lt[659ndash170]100Mo 3034 gt11 1024 [50] 157119864 minus 14 393ndash607 lt[064ndash099]116Cd 2814 gt17 1023 [52] 166119864 minus 14 329ndash479 lt[200ndash292]130Te 2527 gt28 1024 [53] 141119864 minus 14 265ndash513 lt[050ndash097]136Xe 2458 gt16 1025 [54] 145119864 minus 14 219ndash420 lt[025ndash048]150Nd 3371 gt18 1022 [55] 619119864 minus 14 171ndash316 lt[484ndash895]

isotopes are already exploring the quasi-degenerate scenariosfor the neutrino mass hierarchy (which is around 05 eV)With the ingredients presented in Table 2 (PSFs and NMEs)one can appreciate as well the sensitivities translated intoneutrino mass parameters of the future generation of DBDexperiments

4 Conclusions

We report new values of light Majorana neutrino massparameters from a 0]120573120573 decay analysis extended to all theisotopes for which theoretical and experimental data existsWe used the most recent results for the experimental lifetime1198790]12 as well as for the theoretical quantities 1198660] and 1198720]

For the PSFs we use newly obtained values recalculated withan approach described in [41] but with improved numericalaccuracy We use exact electron wf obtained by solving aDirac equation when finite nuclear size and screening effectsare included and in addition a Coulomb potential derivedfrom a realistic proton distribution in the daughter nucleushas been employed For choosing the NMEs we take advan-tage of the general consensus in the community on severalnuclear ingredients involved in the calculations (HOC FHSand SRCs effects values of several nuclear input parameters)and restrict the range of spread of the NMEs values reportedin the literature This in turn reduces the uncertainty inderiving constraints on the light Majorana neutrino mass

parameters while taking into account NME values obtainedwith all the main nuclear methods The stringent constraintsare obtained from the 136Xe and 76Ge isotopes due to boththe experimental sensitivity and the reliability of the PSFsand NMEs calculations The experiments measuring theseisotopes are already exploring the quasi-degenerate scenariosfor the neutrino mass hierarchy which is around 05 eV Ourresults may be useful for having an up-to-date image onthe current neutrino mass sensitivities associated with 0]120573120573measurements for different isotopes and to better estimatethe range of the neutrino masses that can be explored in thefuture DBD experiments

Note After the submission of this paper Exo-2000 publishedtheir 2-year new limits for the neutrinoless double beta decayof 136Xe which is less than the value that we used for derivingthe neutrinomass parameter in this case For this isotope thepresently stringent limit for the lifetime is 19 times 10

25 y fromKamLand-Zen experiment [33] Hence we use this valuewhich shifts the interval of the neutrino mass parameter to[023ndash044] which represents a small change in our resultsfor this isotope

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

This work was done with the support of the MEN andUEFISCDI through the project IDEI-PCE-3-1318 Contractno 5828102011 and Project PN-09-37-01-022009

References

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[2] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 Article ID124002 2012

[3] A Faessler V Rodin and F Simkovic ldquoNuclearmatrix elementsfor neutrinoless double-beta decay and double-electron cap-turerdquo Journal of Physics G Nuclear and Particle Physics vol 39no 12 Article ID 124006 2012

[4] J Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquo Reports on Progress in Physics vol 75 no10 Article ID 106301 2012

[5] H Ejiri ldquoDouble beta decays and neutrino nuclear responsesrdquoProgress in Particle and Nuclear Physics vol 4 no 2 pp 249ndash257 2010

[6] W Rodejohann ldquoNeutrinoless double-beta decay and neutrinophysicsrdquo Journal of Physics G Nuclear and Particle Physics vol39 no 12 Article ID 124008 2012

[7] F Kaether W Hampel G Heusser J Kiko and T KirstenldquoReanalysis of the Gallex solar neutrino flux and source exper-imentsrdquo Physics Letters B vol 685 no 1 pp 47ndash54 2010

[8] A Gando Y Gando K Ichimura et al ldquoConstraints on 12057913from a three-flavor oscillation analysis of reactor antineutrinosat KamLANDrdquo Physical Review D vol 83 Article ID 0520022011

[9] K Abe Y Hayato T Iida et al ldquoSolar neutrino results in Super-Kamiokande-IIIrdquo Physical Review D vol 83 Article ID 0520102010

[10] Y Abe C Aberle T Akiri et al ldquoIndication of reactor ]119890disappearance in the double chooz experimentrdquoPhysical ReviewLetters vol 108 Article ID 131801 2012

[11] F P An J Z Bai A B Balantekin et al ldquoObservationof electron-antineutrino disappearance at daya bayrdquo PhysicalReview Letters vol 108 Article ID 171803 2012

[12] B Aharmim S N Ahmed A E Anthony et al ldquoLow-energy-threshold analysis of the phase I and phase II data sets of theSudbury Neutrino Observatoryrdquo vol 81 Article ID 0555042010

[13] V A Rodin A Faessler F Simkovic and P Vogel ldquoUncertaintyin the 0]120573120573 decay nuclear matrix elementsrdquo Physical Review Cvol 68 Article ID 044302 2003

[14] V A Rodin A Faessler F Simkovic and P Vogel ldquoErratumto ldquoAssessment of uncertainties in QRPA 0]120573120573-decay nuclearmatrix elementsrdquo [Nucl Phys A 766 (2006) 107]rdquo NuclearPhysics A vol 793 no 1ndash4 pp 213ndash215 2007

[15] S Stoica and H V Klapdor-Kleingrothaus ldquoCritical view ondouble-beta decay matrix elements within Quasi RandomPhase Approximation-based methodsrdquo Nuclear Physics A vol694 no 1-2 pp 269ndash294 2001

[16] F Simkovic A Faessler H Muther V Rodin and M Staufldquo0]120573120573-decay nuclear matrix elements with self-consistent

short-range correlationsrdquo Physical Review C vol 79 Article ID055501 2009

[17] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters B vol 647 no 2-3 pp 128ndash132 2007

[18] M Kortelainen and J Suhonen ldquoImproved short-range corre-lations and 0]120573120573 nuclear matrix elements of 76Ge and 82SerdquoPhysical Review C vol 75 Article ID 051303 2007

[19] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC vol 77 Article ID 045503 2008

[20] M Kortelainen and J Suhonen (Jyvaskyla U) ldquoNuclear matrixelements of neutrinoless double beta decay with improvedshort-range correlationsrdquo Physical Review C vol 76 Article ID024315 2007

[21] E Caurier A P Zuker A Poves and G Martinez-Pinedo ldquoFullpf shell model study of 119860 = 48 nucleirdquo Physical Review C vol50 p 225 1994

[22] J Retamosa E Caurier and F Nowacki ldquoNeutrinoless doublebeta decay of 48Cardquo Physical Review C vol 51 p 371 1995

[23] MHoroi and S Stoica ldquoShell model analysis of the neutrinolessdouble-120573 decay of 48Cardquo Physical Review C vol 81 Article ID024321 2010

[24] J Menendez A Poves E Caurier and F Nowacki ldquoDisassem-bling the nuclear matrix elements of the neutrinoless 120573120573 decayrdquoNuclear Physics A vol 818 no 3-4 pp 139ndash151 2009

[25] E Caurier J Menendez F Nowacki and A Poves ldquoInfluenceof pairing on the nuclear matrix elements of the neutrinoless120573120573 decaysrdquo Physical Review Letters vol 100 Article ID 0525032008

[26] J Barea and F Iachello ldquoNeutrinoless double-120573 decay in themicroscopic interacting boson modelrdquo Physical Review C vol79 Article ID 044301 2009

[27] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 Article ID 0143152013

[28] J Barea J Kotila and F Iachello ldquoLimits on neutrino massesfrom neutrinoless double-120573 decayrdquo Physical Review Letters vol109 Article ID 042501 2012

[29] P K Rath R Chandra K Chaturvedi P K Raina and J GHirsch ldquoUncertainties in nuclear transition matrix elementsfor neutrinoless 120573120573 decay within the projected-Hartree-Fock-Bogoliubov modelrdquo Physical Review C vol 82 Article ID064310 2010

[30] P K Rath R Chandra K Chaturvedi P Lohani P K Rainaand J G Hirsch ldquoNeutrinoless 120573120573 decay transition matrix ele-ments within mechanisms involving light Majorana neutrinosclassical Majorons and sterile neutrinosrdquo Physical Review Cvol 88 Article ID 064322 2013

[31] T R Rodriguez and G Martinez-Pinedo ldquoEnergy densityfunctional study of nuclear matrix elements for neutrinoless 120573120573decayrdquo Physical Review Letters vol 105 Article ID 252503 2010

[32] H Primakov and S P Rosen ldquoDouble beta decayrdquo Reports onProgress in Physics vol 22 p 121 1959

[33] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998

[34] T Tomoda ldquoDouble beta decayrdquo Reports on Progress in Physicsvol 54 no 1 p 53 1991

[35] M Doi T Kotani H Nishiura and E Takasugi ldquoDouble betadecayrdquo Progress ofTheoretical Physics vol 69 no 2 pp 602ndash6351983


[36] M Doi and T Kotani ldquoNeutrino emittingmodes of double betadecayrdquo Progress of Theoretical Physics vol 87 no 5 pp 1207ndash1231 1992

[37] M Doi and T Kotani ldquoNeutrinoless modes of double betadecayrdquo Progress of Theoretical Physics vol 89 no 1 pp 139ndash1591993

[38] W C Haxton and G J Stephenson Jr ldquoDouble beta decayrdquoProgress in Particle and Nuclear Physics vol 12 pp 409ndash4791984

[39] M Doi T Kotani and E Takasugi ldquoDouble beta decay andmajorana neutrinordquo Progress ofTheoretical Physics Supplementsvol 83 pp 1ndash175 1985

[40] J Kotila and F Iachello ldquoPhase-space factors for double-120573decayrdquo Physical Review C vol 85 Article ID 034316 2012

[41] S Stoica and M Mirea ldquoNew calculations for phase spacefactors involved in double-120573 decayrdquo Physical Review C vol 88Article ID 037303 2013

[42] T E Pahomi A Neacsu M Mirea and S Stoica ldquoPhase spacecalculations for beta- beta- decays to final excited 2 + 1 statesrdquoRomanian Reports in Physics vol 66 2 p 370 2014

[43] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014

[44] M Horoi ldquoShell model analysis of competing contributions tothe double-120573 decay of 48Cardquo Physical Review C vol 87 ArticleID 014320 2013

[45] M Horoi and B A Brown ldquoShell-model analysis of the 136Xedouble beta decay nuclear matrix elementsrdquo Physical ReviewLetters vol 110 Article ID 222502 2013

[46] D L Fang A Faessler V Rodin and F Simkovic ldquoNeutrinolessdouble-120573 decay of 150Nd accounting for deformationrdquo PhysicalReview C vol 82 Article ID 051301(R) 2010

[47] O Civitarese and J Suhonen ldquoNuclear matrix elements fordouble beta decay in the QRPA approach a critical reviewrdquoJournal of Physics Conference Series vol 173 no 1 Article ID012012 2009

[48] S Umehara T Kishimoto I Ogawa et al ldquoNeutrino-lessdouble-120573 decay of 48Ca studied by CaF2(Eu) scintillatorsrdquoPhysical Review C vol 78 Article ID 058501 2008

[49] C Macolino and GERDA Collaboration ldquoResults on neutrino-less double-beta decay from GERDA phase Irdquo Modern PhysicsLetters A vol 29 no 1 Article ID 1430001 20 pages 2014

[50] A S Barabash andV B Brudanin ldquoInvestigation of double-betadecay with the NEMO-3 detectorrdquo Physics of Atomic Nuclei vol74 no 2 pp 312ndash317 2011

[51] J Argyriades R Arnold C Augier et al ldquoMeasurement ofthe two neutrino double beta decay half-life of Zr-96 with theNEMO-3 detectorrdquoNuclear Physics A vol 847 no 3-4 pp 168ndash179 2010

[52] F A Danevich A S Georgadze V V Kobychev et al ldquoSearchfor 2120573 decay of cadmium and tungsten isotopes final results ofthe Solotvina experimentrdquo Physical Review C vol 68 Article ID035501 2003

[53] E Andreotti C Arnaboldi and FT Avignone III ldquo130Te neu-trinoless double-beta decay with CUORICINOrdquo AstroparticlePhysics vol 34 no 11 pp 822ndash831 2011

[54] A Gando Y Gando H Hanakago et al ldquoLimit on neutrinoless120573120573 decay of 136Xe from the first phase of KamLAND-Zen andcomparison with the positive claim in 76Gerdquo Physical ReviewLetters vol 110 Article ID 062502 2013

[55] J Argyriades R Arnold C Augier et al ldquoMeasurement of thedouble-120573 decay half-life of Nd150 and search for neutrinolessdecay modes with the NEMO-3 detectorrdquo Physical Review Cvol 80 Article ID 032501(R) 2009

[56] S F King andC Luhn ldquoNeutrinomass andmixingwith discretesymmetryrdquo Reports on Progress in Physics vol 76 no 5 ArticleID 056201 2013

[57] A Giuliani and A Poves ldquoNeutrinoless double-beta decayrdquoAdvances in High Energy Physics vol 2012 Article ID 85701638 pages 2012

[58] C Giusti H Muther F D Pacati and M Stauf ldquoShort-rangeand tensor correlations in the 16O(ee1015840pn) reactionrdquo PhysicalReview C vol 60 Article ID 054608 1999

[59] H Muther A Polls ldquoCorrelations derived from modernnucleon-nucleon potentialsrdquo Physical Review C vol 61 ArticleID 014304 1999

[60] H Muther ldquoTwo-body correlations in nuclear systemsrdquoProgress in Particle and Nuclear Physics vol 45 no 1 pp 243ndash334 2000

[61] J P Schiffer S J Freeman J A Clark et al ldquoNuclear structurerelevant to neutrinoless double 120573 decay 76Ge and 76Serdquo PhysicalReview Letters vol 100 Article ID 112501 2008

[62] B P Kay J P Schiffer S J Freeman et al ldquoNuclear structurerelevant to neutrinoless double 120573 decay the valence protons in76Ge and 76Serdquo Physical Review C vol 79 Article ID 021301(R)2009

[63] J Liu M P Mendenhall A T Holley et al ldquoDeterminationof the axial-vector weak coupling constant with ultracoldneutronsrdquo Physical Review Letters vol 105 Article ID 1818032010

[64] J Menendez A Poves E Caurier and F Nowacki ldquoOccupan-cies of individual orbits and the nuclear matrix element of the76Ge neutrinoless 120573120573 decayrdquo Physical Review C vol 80 ArticleID 048501 2009

[65] E Caurier A Poves and A P Zuker ldquoA full View 0 h120596description of the 2]120573120573 decay of 48Cardquo Physics Letters B vol252 no 1 pp 13ndash17 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



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Superconductivity


Statistical MechanicsInternational Journal of


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Solid State PhysicsJournal of

Computational Methods in Physics

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AerodynamicsJournal of

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ThermodynamicsJournal of


boson approximation (IBA) [26ndash28] projected Hartree-Fock-Bogoliubov (PHFB) [29 30] and energy density func-tional (EDS) method [31] There are still large differencesbetween the NMEs values computed with different methodsand by different groups and these have been largely discussedin the literature (see eg [2 3]) On the other side there is aconsensus in the community on the way that several nucleareffects and nuclear parameters should be used in calculationsIn this work we take advantage of this consensus when wechose theNMEs values trying to restrict their range of spreadand consequently to reduce the uncertainty in deriving theneutrino Majorana mass parameters

Unlike the NMEs the PSFs have been calculated a longtime ago [32ndash39] and were considered to be computed withenough precision However recently they were recalculatedwithin an improved approach by using exact electron Diracwave functions (wf) taking into account the finite nuclearsize and electron screening effects [40] The authors founddifferences between their results and those calculated previ-ously with approximate electron wf especially for heaviernuclei We have also independently recalculated the PSFsby developing new routines for computing the relativistic(Dirac) electron wf by taking into account the nuclear finitesize and screening effects In addition we use a Coulombpotential derived from a realistic proton density distributionin the daughter nucleus [41 42] In this workwe use new PSFsvalues obtained by improving the numerical precision of ourroutines as compared with our previous works The obtainedvalues are very close to those reported in [40 41]

Finally for the lifetime limits we take the most recentresults found in the literature

The paper is organized as follows In the next sectionwe shortly recall the general formalism for the derivationof the neutrino mass parameters from 0]120573120573 decay analysishighlighting the nuclear ingredients involved in calculationsIn Section 3 we discuss the way of choosing the NME valuesand report our results for the light neutrino Majorana massparameters while in Section 4 we formulate the conclusionsof our work

2 Formalism

We shortly present the general formalism for the derivationof neutrino mass parameters from 0]120573120573 decay analysis Westart with the lifetime formula and then describe the mainsteps and ingredients used in the theoretical calculation oftheir components that is PSFs and NMEs

Assuming that the dominant mechanism of occurrencefor the 0]120573120573 decay mode is the exchange of Majorana left-handed light neutrinos between twonucleons from the parentnucleus the lifetime reads

(1198790]12)minus1= 1198660](119876120573120573 119885)

100381610038161003816100381610038161198720]10038161003816100381610038161003816

2(⟨119898]⟩

119898119890)

2

(1)

where 1198660] are the PSFs for this decay mode depending onthe energy decay 119876120573120573 and nuclear charge 119885 1198720] are thecorresponding NMEs depending on the nuclear structure ofthe parent and daughter nuclei involved in the decay 119898119890 is

electron mass and ⟨119898]⟩ is the light neutrino Majorana massparameter This parameter can be expressed as a (coherent)linear combination of the light neutrino masses

⟨119898]⟩ =

1003816100381610038161003816100381610038161003816100381610038161003816

3

sum119896=1

1198802119890119896119898119896

1003816100381610038161003816100381610038161003816100381610038161003816

(2)

where 119880119890119896 are the elements of the first row in the PMNS(Pontecorvo-Maki-Nakagawa-Sakata) neutrino matrix and119898119896 are the light neutrinomasses [56] From (1) the expressionof119898] reads

⟨119898]⟩ =119898119890

10038161003816100381610038161198720]1003816100381610038161003816

radic1198790] sdot 1198660] (3)

For deriving ⟨119898]⟩ we need accurate calculations ofboth PSFs and NMEs for each isotope for which there areexperimental lifetime limits The PSFs have been calculateda long time ago in some approximations [32ndash39] and wereconsidered until recently to be computed with enoughprecision However they were recalculated recently in [40ndash42] using more advanced approaches for the numericalevaluation of the Dirac wave functions with the inclusionof nuclear finite size and screening effects In addition in[41] the usual Coulomb spherical potential was replaced byanother one derived from a more realistic proton densitydistribution in the daughter nucleus These recent PSF cal-culations led to significant differences in comparison to theolder calculations especially for the heavier isotopes thatshould be taken into account for a precise derivation of theneutrino mass parameters

The computation of the NMEs is a subject of debate inthe literature for long time because they bring the largeuncertainties in the theoretical calculations for DBD Dif-ferent groups have developed several conceptually differentnuclear structure methods [13ndash31] as we have mentioned inthe previous section The expression of the NMEs can bewritten in general as a sum of three components

1198720]= 1198720]GT minus (

119892119881

119892119860)

2

sdot 1198720]119865 minus119872

0]119879 (4)

where 1198720]GT 1198720]119865 and 1198720]119879 are the Gamow-Teller (GT)

Fermi (119865) and Tensor (119879) components respectively Theseare defined as follows

1198720]120572 = sum119898119899

⟨0+1198911003817100381710038171003817120591minus119898120591minus119899119874

1205721198981198991003817100381710038171003817 0+119894 ⟩ (5)

where 119874120572119898119899 are transition operators (120572 = GT 119865 119879) andthe summation is performed over all the nucleon states Animportant part of the NME calculation is the computationof the reduced matrix elements of the two-body transitionoperators 119874

120572 Their calculation can be decomposed intoproducts of reduced matrix elements within the spin andrelative coordinate spaces Their explicit expressions are [423]

119874GT12 = 1205901 sdot 1205902119867(119903) 119874

11986512 = 119867 (119903)

11987411987912 = radic

2

3[1205901 times 1205902]

2sdot119903

119877119867 (119903) 119862

(2)(119903)

(6)



119867120572 (119903) =2119877

120587intinfin

0119895119894 (119902119903)

ℎ120572 (119902)

120596

1

120596 + ⟨119864⟩1199022119889119902

equiv intinfin

0119895119894 (119902119903)119881120572 (119902) 119902

2119889119902

(7)

where 119877 = 119903011986013 fm (1199030 = 12 fm) 120596 = radic1199022 + 1198982]


ℎ119865 = 1198662119881 (1199022) (8)

ℎGT (1199022) =

1198662119860 (1199022)

1198922119860[1 minus

2

3

1199022

1199022 + 1198982120587+1

3(

1199022

1199022 + 1198982120587)

2

]

+2

3

1198662119872 (1199022)

1198922119860

1199022

41198982119901

(9)

ℎ119879 (1199022) =

1198662119860 (1199022)

1198922119860[2

3

1199022

1199022 + 1198982120587minus1

3(

1199022

1199022 + 1198982120587)

2

]

+1

3

1198662119872 (1199022)

1198922119860

1199022

41198982119901

(10)


119866119872 (1199022) = (120583119901 minus 120583119899)119866119881 (119902

2) (11)



119866119860 (1199022) = 119892119860(

Λ2119860

Λ2119860 + 1199022)

2

119866119881 (1199022) = 119892119881(

Λ2119881

Λ2119881 + 1199022)

2

(12)




120595119899119897 (119903) 997888rarr [1 + 119891 (119903)] 120595119899119897 (119903) (13)


119891 (119903) = minus119888 sdot 119890minus1198861199032

(1 minus 1198871199032) (14)



⟨1198991198971003816100381610038161003816119867120572 (119903)

1003816100381610038161003816 11989910158401198971015840⟩ = int

infin

01199032119889119903120595119899119897 (119903) 12059511989910158401198971015840 (119903) [1 + 119891(119903)]

2

times intinfin

01199022119889119902119881120572 (119902) 119895119899 (119902119903)

(15)





















4 Conclusions









Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119867120572 (119903) =2119877

120587intinfin

0119895119894 (119902119903)

ℎ120572 (119902)

120596

1

120596 + ⟨119864⟩1199022119889119902

equiv intinfin

0119895119894 (119902119903)119881120572 (119902) 119902

2119889119902

(7)

where 119877 = 119903011986013 fm (1199030 = 12 fm) 120596 = radic1199022 + 1198982]


ℎ119865 = 1198662119881 (1199022) (8)

ℎGT (1199022) =

1198662119860 (1199022)

1198922119860[1 minus

2

3

1199022

1199022 + 1198982120587+1

3(

1199022

1199022 + 1198982120587)

2

]

+2

3

1198662119872 (1199022)

1198922119860

1199022

41198982119901

(9)

ℎ119879 (1199022) =

1198662119860 (1199022)

1198922119860[2

3

1199022

1199022 + 1198982120587minus1

3(

1199022

1199022 + 1198982120587)

2

]

+1

3

1198662119872 (1199022)

1198922119860

1199022

41198982119901

(10)


119866119872 (1199022) = (120583119901 minus 120583119899)119866119881 (119902

2) (11)



119866119860 (1199022) = 119892119860(

Λ2119860

Λ2119860 + 1199022)

2

119866119881 (1199022) = 119892119881(

Λ2119881

Λ2119881 + 1199022)

2

(12)




120595119899119897 (119903) 997888rarr [1 + 119891 (119903)] 120595119899119897 (119903) (13)


119891 (119903) = minus119888 sdot 119890minus1198861199032

(1 minus 1198871199032) (14)



⟨1198991198971003816100381610038161003816119867120572 (119903)

1003816100381610038161003816 11989910158401198971015840⟩ = int

infin

01199032119889119903120595119899119897 (119903) 12059511989910158401198971015840 (119903) [1 + 119891(119903)]

2

times intinfin

01199022119889119902119881120572 (119902) 119895119899 (119902119903)

(15)





















4 Conclusions









Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics


















4 Conclusions









Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Research Article Constraints on Light Neutrino Parameters ...Constraints on Light Neutrino...

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Transcript of Research Article Constraints on Light Neutrino Parameters ...Constraints on Light Neutrino...