Nuclear Structure Aspects of the Neutrinoless Double Beta Decay

0.16 MB·English

by  E. Caurier

Nuclear Structure Aspects of the Neutrinoless ββ-Decays
E. Caurier ∗, F. Nowacki ∗ and A. Poves +
(*) IPHC, IN2P3-CNRS/Universit´e Louis Pasteur BP 28, F-67037 Strasbourg Cedex 2, France
(+) Departamento de F´ısica Te´orica, C-XI. Universidad Aut´onoma de Madrid, E-28049, Madrid, Spain
In this article, we analyze some nuclear structure aspects of the 0ν double beta decay nuclear
matrixelements(NME).Wegiveresultsforthedecaysof48Ca,76Ge,82Se,124Sn,128Te,130Te,and
136Xe,usingimprovedeffectiveinteractionsandvalencespaces. Weexaminethedependenceofthe
NME’sontheeffectiveinteraction andthevalencespace, andanalyzetheeffectsoftheshort range
correlations and thefinitesize of thenucleon. Finally westudytheinfluenceofthedeformation on
thevalues of theNME’s.
PACSnumbers: 21.10.–k,27.40.+z,21.60.Cs,23.40.–s
Keywords: ShellModel,Doublebetadecaymatrixelements
8
0
0 I. INTRODUCTION stability ofthe NME’s under reasonablemodifications of
2
the nuclear structure inputs of the calculations.

n
The double beta decay is the rarest nuclear weak pro-
a
J cess. It takesplace betweentwoeven-evenisobars,when
II. DOUBLE BETA DECAYS
thedecaytotheintermediatenucleusisenergeticallyfor-
9
bidden or hindered by the large spin difference between
] the parent ground state and the available states in the As we have already mentioned, some nuclei, otherwise
h intermediate nuclei. It comes in three forms: The two- nearly stable, decayemitting two electrons andtwo neu-
-t neutrino decay ββ is just a second order process me- trinos (2ν ββ) by a second order process mediated by
2ν
cl diated by the weak interaction. It conserves the lepton the weak interaction,that has been experimentally mea-
u number and has been already observed in a few nuclei. sured in several favorable cases. The decay probability
n The secondmode, the neutrinoless decay ββ , needs an containsaphasespacefactorandthesquareofanuclear
0ν
[ extension of the standard model of the electroweak in- matrix element

2 teractions as it violates lepton number. A third mode,
v ββ isalsopossibleinsomeextensionsofthestandard
7 mo0dνe,χlandproceedsviaemissionofalightneutralboson, [T12/ν2]−1 =G2ν|MG2νT|2 (1)
7
aMajoronχ. Thelasttwomodes,notyetexperimentally
2 If the neutrinos are massive Majorana particles, the
observed,requiremassiveneutrinos–anissuealreadyset-
0 double beta decay can take place without emission of
tled by the recent measures by Super-Kamiokande [1],
. neutrinos(0ν ββ). Inthiscasethetransitionismediated
9
SNO [2] and KamLAND [3]. Interestingly, the double
bytermsthatgobeyondthe standardmodel. Thedecay
0
beta decay without emission of neutrinos would be the
7 probabilitycontainsaphasespacefactor[6],theeffective
only way to sign the Majorana character of the neutrino
0 electronneutrinomass(alinearcombinationofthe mass
: andtodistinguishbetweenthe differentscenariosforthe eigenvalueswhose coefficients are elements ofthe mixing
v
neutrino mass differences. In what follows we shall con-
i matrix) and the nuclear matrix element [7, 8].
X centratemostlyinthe ββ mode. The uprisingofinter-
0ν
est in the observation of the ββ decay was somewhat
r 0ν
a obscuredby the analysisof the status of the calculations [T(0ν)(0+−>0+]−1 =
1/2
of the nuclear matrix elements, that enter in the lifetime
of the decay together with the effective neutrino mass, g 2 2 hm i 2
G M(0ν)− V M(0ν) ν (2)
made in ref. [4]. The authors concluded on a very pes- 0ν GT g F m
simistic note, sayingthat the spreadofthe available val- (cid:18) A(cid:19) ! (cid:18) e (cid:19)
ues for the nuclear matrix elements was such that there The 2ν matrix element can be written as:
wasnohopetotranslatetheexperimentalsignalintouse-
ful input for the theories beyond the standard model. A
criticalassessmentoftheresultsofthemanycalculations M(2ν) = hGD(J)|~σt−|1+i ih1+i |~σt−|Fi (3)
available in the framework of the quasi particle random GT ∆E
i
i
phase approximation (QRPA) has been made recently X
[5], with a much more optimistic conclusion. In this ar- Thenuclearstructureinformationenteringinthis ma-
ticle we want to continue improving upon the reliability trix element consists on the wave function of the ground
of the calculated nuclear matrix elements (NME) in the state of the father nucleus (F) and the wave function of
framework of large scale applications of the Interacting the state of the grand daughter, GD(J), nucleus (J=0,
ShellModel(ISM).Inaddition,weputforwardaprocess J=1, or J=2) to which the decay proceeds. In addi-
ofbenchmarkingthe differentapproaches,andstudy the tion, the wave functions and excitation energies of all
2
6
the 1+ states in the odd-odd daughter nucleus are in
principle necessary. The spin-isospin operators in the 5 82Se
nuclear medium are quenched by a factor q≈1/g , and 4
A
this quenching factor is also required to reproduce the 3
experimental data of the 2ν double beta decays. 2
To compute exactly the 0ν matrix element, we would 1
T
also need the wave function of the ground state of the G 0
M
father nucleus and of the ground state grand daughter,
-1
but in addition we would need all the wave functions
-2
and excitation energies of all the Jπ states in the odd-
-3
odd daughter nucleus. Most conveniently, the matrix
elements can be approximately obtained in the closure -4
approximation, that is good to better than 90% due to -5
thehighmomentumofthevirtualneutrinointhenucleus -6
(≈100 MeV) Hence, the matrix element can be written 0+ 0- 1+ 1- 2+2- 3+ 3- 4+ 4- 5+ 5- 6+ 6- 7+ 7- 8+ 8- 9+
as:
FIG. 1: The matrix element of the 82Se → 82Kr decay as a
MG(0Tν) =hGD|h(|r~1−r~2|)(~σ1·~σ2)(t−1t−2)|Fi (4) function of theJ of thedecaying pair
whereh(|r~ −r~|)istheneutrinopotential(≈1/r). Inthis
1 2
approximationno knowledgeofthe intermediate nucleus thepairingcontentofthenuclearwavefunctionsandthe
is directly implied. neutrinoless double beta decay operator.
Thetransitionoperatorsareusuallyobtainedfromthe
HamiltonianofDoietal. [8]However,accordingtorecent
claims [9], additional terms originating in the coupling
III. INTERACTING SHELL MODEL (ISM) AND
to the virtual pions, should give non-negligible contribu-
QRPA CALCULATIONS
tions. The finite size of the nucleon and the short range
correlationsneedtobetakenintoaccountinthe calcula-
In the quest for better wave functions to describe the
tionofthe twobody matrixelements ofthe0ν two-body
doublebetadecayprocesses,twomainavenueshavebeen
transition operators as well.
explored. Theinteractingshellmodelinlargerandlarger
Perhaps the most relevant issue for our mastering on
valencespaces,witheverimprovingeffectiveinteractions,
the neutrinoless double beta decays is to get a better
andthequasiparticleRPA.Ideally,bothmethodsshould
insightinthephysicalcontentofthetwobodytransition
be able to produce goodspectroscopyfor parent,daugh-
operator. Only with this knowledge could one decide
terandgrand-daughter,evenbetterifitextendstoafull
whether the nuclear wave functions that enter into the
mass region, correct total Gamow Teller strengths and
calculation of the NME’s contain the degrees of freedom
strengthfunctions, 2ν matrix elements, etc. In brief, the
thatcorrespondtothecorrelationstowhichtheoperator
goalistohaveadescriptionasclosetoperfectaspossible
couples dominantly.
of the dynamics of the nuclei involved in the transition.
The two body decay operators can be written generi-
Untilratherrecently,ISMcalculationsthatcouldencom-
cally as:
passmostoftherelevantdegreesoffreedomofthenuclei
of interest, were only available for the lighter emitters,
Mˆ(0ν) = MJ ((a†a†)J (a a )J)0 (5) and,asaconsequence,mostsystematicstudieswereper-
i,j,k,l i j k l
formed in the QRPA framework. As we will discuss in
J i,j,k,l
X X the next section, all the potential double beta emitters
In words, what the operator does is to annihilate two are now within the reach of the ISM except 150Nd, that,
neutronsinthe parentnucleusandtocreatetwoprotons being deformed, is also out of the reach of the spherical
The NME is the overlap of the resulting object with the QRPA calculations.
grand daughter ground state. In the ISM approach, the valence spaces contain a
The contributions to the 0ν matrix element as a func- numberoforbitsthatis”small”comparedtotheQRPA,
tionoftheJoftheofthedecayingpairhaveaverytelling however,allthe possiblewaysofdistributing the valence
structure as canbe seen in Figure 1. The dominantcon- particlesamongthevalenceorbitsaretakenintoaccount.
tribution corresponds to the J=0+ pairs, while all the In the QRPAcalculations,the number ofactive orbits is
other pairs have much smaller contributions, but all of larger, but only 1p-1h and 2p-2h excitations from the
them of sign opposite to the leading term. If the ini- normal filling are considered (and not all of them).
tial and final wave functions had seniority zero, only the The effective interactionsused inthe ISMcalculations
leading term would contribute and the matrix element areusuallyG-matriceswhosemonopolebehaviorisfitted
would be maximal. This is a first indication, relating tothespectroscopicpropertiesofalargeregionofnuclei,
3
ingeneralthosecomprisedbetweentomagic closuresfor
TABLE I:Update of theISM 0ν results
theneutronsandfortheprotons. Insomecasestheinter-
actions are plainly fitted to a set of experimental masses hmνi for T1 = 1025 y. MG(0Tν) 1-χF
2
and excitation energies. In the QRPA description, the
starting point is provided by realistic or schematic in- 48Ca 0.85 0.67 1.14
76Ge 0.90 2.35 1.10
teractions, however, both the particle-particle and the
82Se 0.42 2.26 1.10
particle-hole channels of the interaction are affected of
124Sn 0.45 2.11 1.13
strength parameters dubbed g and g that are fitted
ph pp 128Te 1.92 2.36 1.13
to selected experimental data. In particular, it has been
130Te 0.35 2.13 1.13
shown that the double beta decay matrix elements de-
136Xe 0.41 1.77 1.13
pendcriticallyofg . Therearedifferentopinionsamong
pp
theQRPApractitionersaboutthebestchoiceofthevalue
of g . The two preferred options being to fit the exper-
pp
imental 2ν matrix elements, or to fit the Gamow Teller
All these spaces are accessible to large scale ISM
strength functions. In a sense, none is really safe, be-
descriptions. The Strasbourg-Madrid [10] Shell-Model
cause the pure ~σt± channel plays a rather minor role in
codes can deal with problems involving bases of O(1010)
the 0ν decay.
Slater determinants, using relatively modest computa-
The dominant correlations in the nuclei are due to
tional resources.
the pairing and quadrupole-quadrupole terms of the nu-
cleon nucleon interaction in the medium. In the ISM,
the pairing correlations are treated exactly within the
V. UPDATE OF THE ISM 0ν RESULTS
valence space. Proton and neutron numbers are exactly
conserved. Proton-proton,neutron-neutron,andproton-
In the valence spaces r -g (76Ge, 82Se) andr -h
neutron (iso-vector and iso-scalar) pairing channels are 3 9/2 4 11/2
(124Sn, 128−130Te, 136Xe) we have obtained high quality
includedonequalfooting. Onthecontrary,inthespher-
effective interactions by carrying out multi-parametrical
icalQRPAonlyproton-protonandneutron-neutronpair-
fits [11] whose starting point is given by realistic G-
ing terms are considered. They are treated in the BCS
matrices [12]. With these interactions the beta decay
approximation. Protonandneutronnumbersarenotex-
properties of a large set of nuclei are well reproduced.
actly conserved. From the spherical shell model point
The 2ν double beta decay half-lives are found in rea-
of view, this approach is a seniority zero approximation
sonably good agreement with the experimental results
at the BCS level. When the RPA correlations are taken
as well. In the valence space proposed for 96Zr, 100Mo,
into account, higher seniority components appear in the
110Pd and 116Cd, our results are still preliminary and
ground states of the father and grand daughter nuclei,
subject to further improvement both on the interaction
however,for the RPA to be valid, their amplitudes must
sideandontheremovalofthe yetnecessarytruncations.
decrease rapidly with increasing seniority.
Ourresultsareobtainedinthe closureapproximation,
The multipole correlations and the eventual intrinsic
with the short range and finite size corrections mod-
deformation can be properly described in the laboratory
eled as described in [13]; using r =1.2 fm to make the
frame in the ISM description. Angular momentum con- 0
matrix element dimensionless; with g =1.25; and with-
servation is exactly preserved. In the QRPA the multi- A
out higher order contributions to the nuclear current. A
pole correlations of the ground state are treated at the
preliminary estimation of the higher order contributions
RPA level. The posibility of having permanent intrinsic
gives a reduction of the ISM NME’s in the range of 10%
deformation is not contemplated yet.
. Our present best values are collected in table I. χ is
F
defined as:
IV. THE ISM VALENCE SPACES RELEVANT
FOR ββ DECAYS g 2
χ = V M(0ν) (6)
F g F
Classical 0~ω ISM valence spaces are the p, sd and (cid:18) A(cid:19)
pf shells. These correspond to the harmonic oscillator Except in the case of doubly magic 48Ca, whose NME
major shells of principal quantum number p equal to is severely quenched, all the other values cluster around
1, 2, and 3. We use the following notation; in shell p, a value M(0ν)=2.5. The limits of the effective neutrino
the orbit j=p+1/2 is denoted by the spectroscopic label mass for a half life limit of 1025 y, that incorporate the
of l=p and the remaining ones by r . The space r g is phase space factors, show a mild preference for some of
p 3
adequatefor76Ge,82Seandtheirdescendants. Withthe the potential emitters.
r h space for the neutrons and the g-r for the protons, In table II we compare the ISM results with the most
4 4
we aim to understand the decays of 96Zr, 100Mo,110Pd, recentQRPAcalculationsincludingthehigherordercor-
and 116Cd. Finally the r -h space is a natural choice for rections discussed before. The range of values of the
4
thedescriptionofthe124Sn,128−130Te,and136Xedecays. NME’s shown in the table is that given by the authors,
4
TABLE II: Comparison of the present ISM results with the TABLE III: Comparison of the ISM results for the 48Ca de-
range of values proposed in different QRPA calculations; (a) caysusingtheeffectiveinteractionsKB3,FPD6,andGXPF1
ref.[5,14],(b)ref[15]. Thevaluesofcolumn(a)havebeenin-
creasedby10%withrespecttothenumbersgiveninref.[14],
KB3 FPD6 GXPF1
forapropercomparison with theothers,becauseoftheiruse
ofr0=1.1fm. Inthesamevein,theISMvaluesshouldalsobe MGT(2ν) 0.08 0.10 0.11
reduced by ∼10% to account approximately for the absence MGT(0ν) 0.67 0.73 0.62
of higher order corrections to thenuclear current.
M0ν ISM QRPA(a) QRPA(b)
76Ge 2.58 3.81 - 4.96 4.06 - 5.26 TABLEIV:Influenceoftheshortrangecorrelations(SR)and
82Se 2.49 3.20 - 4.42 2.72 - 3.60 ofthenucleonfinitesize(FS)onthenuclearmatrixelements
128Te 2.67 2.79 - 4.00 3.37 - 4.42 of the neutrinoless doublebeta decays
130Te 2.41 2.57 - 3.59 3.00 - 4.22
136Xe 2.00 1.39 - 2.32 1.92 - 2.92 MGT(0ν) bare +FS +SR +FS+SR
48Ca →48Ti 1.21 0.93 0.74 0.67
76Ge →76Se 3.55 2.88 2.57 2.35
82Se→82Kr 3.33 2.72 2.43 2.26
andderivesfromthe differentchoicesofg andg used 130Te →130Xe 3.34 2.70 2.32 2.13
pp A
in the calculations, as well as from the use or not of a 136Xe →136Ba 2.76 2.24 1.93 1.77
renormalized version of the QRPA. In addition, in order
to make the comparison more transparent, we have se-
lected the results that treat the short range correlations
by means of a Jastrow factor. Overall, the two sets of ISM description, once the finite size of the nucleon is
QRPA calculations are now compatible. The ISM pre- takenintoaccountbymeansofthe standarddipole form
dictions of the NME’s are systematically smaller than factor[20],theeffectoftheshortrangecorrelations,that
the QRPA central values, except in the case of 136Xe. wemodelbyaJastrowansatzfollowingref. [21],isabout
This nucleus is semi-magic, and the present experimen- one half of what it would be without it. In fact, the
tal limit on its 2ν decay half life is surprisingly large, ISM corrections for finite size and short range effects
perhaps indicating that some subtle cancelation mecha- with the Jastrow prescription are quite close in per-
nism is at work. For the others, a plausible explanation centagetotheQRPAvalues. Shortrangeandfinitesize
of the discrepancy, relating it to the implicit seniority corrections proceed mainly through the reduction of the
truncations present in the spherical QRPA calculations, Jπ=0+ pair contribution. Preliminary ISM calculations
will be presented elsewhere. usingasofterprescriptionfortheshortrangecorrelations
donotchangethispictureatall. Infact,therelativeval-
ues of the ISM NME’s with respect to the QRPA ones
seem to be independent of the choice of the (common)
VI. EXPLORING THE DEPENDENCES OF
prescription used for the the short range correlations.
THE ISM RESULTS
A. Effective interaction
C. The influence of deformation
The ISM results depend only weakly on the effective
Changing adequately the effective interaction used in
interactionsprovidedthey arecompatible with the spec-
thecalculations,wecanincreaseordecreasethedeforma-
troscopy of the region. For instance, in the pf shell we
tion of parent, grand-daughter, or both, at will. In this
have three interactions that work properly, KB3 [16],
manner,wecangaugetheeffectofthesevariationsonthe
FPD6 [17] and GXPF1 [18]. Their predictions for the
decays. We have artificially changed the deformation of
2ν and the neutrinoless modes are quite close to each
48Tiand48CraddinganextraλQ·Qtermtotheeffective
other(seetableIII). Similarly,inther gandr hspaces,
3 4 interaction. The results are presented in Fig. 2. Positive
thevariationsamongthepredictionsofspectroscopically
valuesofλincreasethedeformationwhilenegativevalues
tested interactions are small (10-20%).
reduceit. Forzerovaluesofbothλ’s,48Crisalreadywell
deformed,while48Tiistransitional. Thecircletotheup-
per left corresponds to the spherical-spherical situation,
B. Finite size and short range corrections the square to the upper right to equally deformed Tita-
niumandChromium,andthediamondtothebottomleft
There has been a certaindebate among QRPA practi- toasphericalTitaniumandaverydeformedChromium.
tioners about the amount of the reduction of the NME’s Our conclusion is that a large mismatch of deformation
due to the short range correlations [19]. The debate has can reduce the ββ matrix elements by factors as large
become milder after the publication of ref. [14]. In the as 2-3. This exercise indicates that the effect of defor-
5
2.5 λ (Cr)=-2 stant, even if the total Gamow-Teller strengths, (GT+)
qq and (GT-), increase from 0.15 to 0.34 and from 20.5 to
λ (Cr)=0
qq 26.9. ThelargerGT+strengthissomehowcompensated
2 λ (Cr)=+2
qq by a larger cancellation among the contributions of the
differentintermediatestatesandbyashiftofthecentro¨ıd
of the strength toward higher energy.
ν 1.5
0
MGT We have also computed the 136Xe decay in the r4h
spaceincluding2p-2hexcitationsfromthe0g9protonor-
2
1 bit and the matrix element increases less than 10%. In
another set of calculations, we have included 2p2h neu-
tron excitations toward the 0h9 and 1f7. The occupan-
0.5 2 2
cies that we obtain are relatively large (0.25 neutrons in
each orbit) and the effect is to increase the matrix ele-
-2 -1 0 1 2
λ (Ti) ment by 15%. It is interesting to note that the increase
qq
withthetwoorbitssimultaneouslyactiveisequivalentto
that obtained including one or another orbit separately.
FIG. 2: Influence of deformation in the (fictitious) decay of Therefore there is no pile-up of the contributions of the
48Ti. smallcomponentsofthewavefunction. Asapreliminary
conclusion,theISMresultsseemtoberobustagainstthe
inclusion of small components of the wave function.
mation is very important and cannot be overlooked. We
havereachedsimilarconclusionsforheavieremittersand
a systematic exploration in realistic cases is under way.
VIII. CONCLUSIONS
VII. ISM CALCULATIONS IN QRPA-LIKE
Large scale shell model calculations with high quality
VALENCE SPACES
effectiveinteractionsareavailableorwillbeintheimme-
diatefutureforallbutoneoftheneutrinolessdoublebeta
The ISM valence space for the 76Ge and 82Se decays emitters. The theoretical spread of the values of the nu-
has traditionally encompassed the orbits 1p3, 0f5, 1p1, clearmatrixelementsenteringinthelifetimecalculations
2 2 2
and 0g9. In the QRPA, two major oscillator shells are is greatly reduced if the ingredients of each calculation
2
taken into account; 0f7, 1p3, 0f5, 1p1, 0g9, 1d5, 0g7, are examined critically and only those fulfilling a set of
2 2 2 2 2 2 2
2s1, and 1d3. qualitycriteriaareretained. Aconcertedeffortofbench-
A2 s a first2step toward a more complete benchmark- markingbetweenISMandQRPApractitionerswouldbe
ing, we have evaluated the influence of the 2p-2h jumps of utmost importance to increase the reliability and pre-
fromthe1f orbit–56Nicoreexcitations–inourresults cision of the nuclear structure input for the double beta
7/2
for the 82Se decay. Similar calculations for the 76Ge de- decay processes.
cay are under way. The calculation in the full r g space
3
plus 2p-2h proton excitations from the 0f27 orbit gives a Acknowledgments This work is supported by the
20% increase of M0ν, but probably we overestimate the Spanish Ministry of Education and Science with a grant
amountofcoreexcitations. Our0f7 protonoccupancies, FIS2006-1235 and by the IN2P3(France) CICyT(Spain)
2
7.71and7.69in82Se and82Kraresmallerthanthe BCS collaboration agreements. Some of the results of this
occupancies of Rodin et al. [5], 7.84 and 7.84. There- work have been obtained in the framework of the ILIAS
fore the above20%must be takenas a veryconservative integrating activity (Contract R113-CT-2004-506222)as
upperbound. The2νmatrixelementremainsnearlycon- partoftheEUFP6programmeinAstroparticlePhysics.
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