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Heavy neutrino potential for neutrinoless double beta decay 7 1 0 2 n Yoritaka Iwata1,2 ∗ a 1InstituteofInnovativeResearch,TokyoInstituteofTechnology,Japan J 2DepartmentofMathematics,ShibauraInstituteofTechnology,Japan 6 2 E-mail: [email protected] ] h Heavy neutrino potential for neutrinoless double beta decay is studied with focusing on its sta- p - tistical property. The existence condition for heavy neutrinos is also presented. In particular p e sterileneutrinosarepossiblecandidateforheavynuetrinos. Thestatisticsprovideagrossviewof h understandingamplitudeofconstitutionalcomponentsofthenuclearmatrixelement. [ 1 v 4 9 8 7 0 . 1 0 7 1 : v i X r a The26thInternationalNuclearPhysicsConference 11-16September,2016 Adelaide,Australia ∗TheauthorwouldliketoexpresshissinceregratitudetoDr. J.Menéndezforfruitfulcomments.Numericalcal- culationswerecarriedoutattheworkstationsystemofInstituteofInnovativeResearch,TokyoInstituteofTechnology, andCOMAofUniversityofTsukuba.. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/ Heavyneutrinopotential... 1. Introduction Observation of neutrino oscillation has clarified the nonzero neutrino mass. The observation of neutrinoless double-beta decay, for whose existence nonzero neutrino mass plays a supportive role,isassociatedwithimportantphysics;e.g., • existenceofMajoranaparticle, • breakingofleptonicnumberconservation, • quantitativedeterminationofneutrinomass. Inthissenseneutrinolessdouble-betadecayisintriguingenoughtobringaboutanexampleexhibit- ing the physics beyond the standard model of elementary particle physics (for a review, see [1]). Although LSND [2] experiment has suggested the possible existence of heavy neutrinos (recog- nizedas“sterileneutrino”invariousliteratures),theoristsstartedtoaccountforsuchacontribution to the neutrinoless double beta decay half life only recently. In addition, regarding the motivation for sterile neutrinos, the GALLEX/SAGE experiments [3] and the reactor anomaly support such existences. AllthreeexperimentssuggestneutrinomassesontheeVscale. Anothermotivationis thatsterileneutrinoscouldbedarkmattercandidates,inthatcasethemassesareonthekeVscale. If heavy neutrinos exist, those neutrinos are mixed into the effective mass. As an example of relation between the half life of neutrinoless double-beta decay, the effective light neutrino mass (m ),andtheeffectiveheavyneutrinomass(η )isgivenby[4,5] ν N (cid:40) (cid:41) (cid:18)m (cid:19)2 [T1/2]−1=G |M0ν|2 ν +|M0N|2(η )2 , (1.1) 0ν m N e where G is the phase space factor (its value is obtained rather precisely), m is the electron mass e (its value is also precisely obtained), η denotes the effective mass relative to electron mass, and N M0ν and M0N are the nuclear matrix elements (NME, for short) for light and heavy neutrinos respectively. In this context light neutrinos mean already-observed ordinary neutrinos. Under the existence of heavy neutrino, we need to have half life observations for two different double-beta decayevents(forexample,decayofcalciumandxenon): (cid:40) (cid:41) (cid:18)m (cid:19)2 [T1/2]−1=G |M0ν|2 ν +|M0N|2(η )2 , (1.2) 0ν,I I I m I N e and (cid:40) (cid:41) (cid:18)m (cid:19)2 [T1/2]−1=G |M0ν|2 ν +|M0N|2(η )2 , (1.3) 0ν,II II II m II N e whereindicesIandIIidentifythekindofdecayingnuclei. Becausetherearetwounknownquan- tities: m and η , here we have two equations. In order to determine the neutrino mass, it is ν N necessarytocalculateM0ν,M0ν,M0N andM0N veryprecisely. Atthispoint,manycalculationsby I II I II varioustheoreticalmodelshavebeendedicatedtoNMEcalculations. Sincethedetailinformation on initial and final states (i.e., quantum level structure of these states) is necessary for the calculation of NMEs, it is impossible to have reliable NME without knowing nuclear structures. The 1 Heavyneutrinopotential... impact of precise NME calculations is expected to be large enough (e.g., for a large-scale shell modelcalculationforlightneutrinos, see Ref.[6]), andtheunknownleptonic mass-hierarchy and theMajorananatureofneutrinosareexpectedtobediscovered. In this article heavy neutrino potential for neutrinoless double beta decay (for the definition, see Eq. (3.7)) is studied from a statistical point of view. The results in this article are intended to be compared to light neutrino cases (that is, ordinary neutrino case) presented in Ref. [7]. The comparison clarifies the contribution of heavy neutrinos for neutrinoless double beta decay half- life. 2. Conditionfortheexistenceofheavyneutrino RoleofthenuclearmatrixelementisseenbysolvingEqs.(1.2)-(1.3). Underthevalidityof |M0N|2/|M0ν|2(cid:54)=|M0N|2/|M0ν|2, (2.1) I I II II theeffectiveneutrinomassforlightandheavyneutrinosarerepresentedby (cid:18)mν(cid:19)2= −|MI0IN|2[GIT01ν/,2I]−1+|MI0N|2[GIIT01ν/,2II]−1, (2.2) m |M0ν|2|M0N|2−|M0ν|2|M0N|2 e II I I II and |M0ν|2[G T1/2]−1−|M0ν|2[G T1/2]−1 (η )2= II I 0ν,I I II 0ν,II , (2.3) N |M0ν|2|M0N|2−|M0ν|2|M0N|2 II I I II respectively. The condition (2.1) is valid if nuclear structure effect on double beta decay is not so simple; indeed it is not true only if NMEs for different decay candidates are exactly the same in theirheavy-to-lightratios. ThisconditionwasexploredinRefs. [8,9]. AccordingtoEq.(2.2),the experimentally-confirmednonzeroneutrinoeffectivemasssuggeststhat |M0N|2G T1/2(cid:54)=|M0N|2G T1/2 (2.4) I I 0ν,I II II 0ν,II where note that Eq. (2.4) is written only by heavy neutrino NMEs and half lives. According to Eq. (2.3), |M0ν|2G T1/2=|M0ν|2G T1/2 (2.5) I I 0ν,I II II 0ν,II suggests that heavy neutrinos do not exist. The satisfaction of Eq. (2.5) means either one of the followingpossibilities: (i)thepresentframework(1.1)istoosimpletobevalid, (ii)heavyneutrinosdonotexist. Since Eq. (2.5) is written even without knowing anything about heavy neutrino, this condition is practicallyusedasthesufficientconditionfortheexistenceofheavyneutrinounderthevalidityof theframework(1.1)(i.e.,heavyneutrinoexistenceconditionforEq.(1.1)). Itisworthnotingthat, as discussed around Eq. (15) of Ref. [1], additional terms can be added to Eq. (1.1). Under the non-existenceofheavyneutrino(byapplyingEq.(2.5)toEq.(2.2)), (cid:18)mν(cid:19)2= [GIT01ν/,2I]−1|MI0Iν|2|MI0N|2−|MI0ν|2|MI0IN|2 = 1 [T01ν/,2I]−1 (2.6) m |M0ν|2 |M0ν|2|M0N|2−|M0ν|2|M0N|2 G |M0ν|2 e I II I I II I I 2 Heavyneutrinopotential... triviallyfollows. Ifthesquaredmassesarepositive, (cid:16) (cid:17)(cid:16) (cid:17) |M0N|2[G T1/2]−1−|M0N|2[G T1/2]−1 |M0ν|2[G T1/2]−1−|M0ν|2[G T1/2]−1 ≤0, II I 0ν,I I II 0ν,II II I 0ν,I I II 0ν,II (2.7) mustbesatisfied(i.e.,realmasscondition). 3. Neutrinopotential 3.1 Nuclearmatrixelement Nuclearmatrixelementindoublebetadecayisinvestigatedundertheclosureapproximation. Itapproximatesallthedifferentvirtualintermediateenergiesbyasingleintermediateenergy(i.e., with the averaged energy called closure parameter). For neutrinoless double beta decay, nuclear matrixelementforlightandheavyneutrinosarewrittenby M0ν =M0ν−gV2M0ν +M0ν (3.1) F g2 GT T A and M0N =M0N−gV2M0N+M0N (3.2) F g2 GT T A respectively,whereg andg denotevectorandaxialcouplingconstants,andα ofM0ν istheindex V A α for the double beta decay of three kinds: α = F, GT, T (Fermi, Gamow-Teller, and tensor parts). AccordingtoRef.[10], eachpartisfurtherrepresentedbythesumoftwo-bodytransitiondensity (TBTD)andanti-symmetrizedtwo-bodymatrixelements. M0x=(cid:104)0+|O0x|0+(cid:105) α f α i =∑TBTD(n(cid:48)l(cid:48) j(cid:48)t(cid:48),n(cid:48)l(cid:48) j(cid:48)t(cid:48),n l j t ,n l j t ;J) (3.3) 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 (cid:104)n(cid:48)l(cid:48) j(cid:48)t(cid:48),n(cid:48)l(cid:48) j(cid:48)t(cid:48);J|O0x(r)|n l j t ,n l j t ;J(cid:105) 1 1 1 1 2 2 2 2 α 1 1 1 1 2 2 2 2 AS whereO0x(r)aretransitionoperatorsofneutrinolessdoublebetadecay,and0+ and0+ denoteini- α i f tialandfinalstates,respectively(xiseitherν orN). Thesumistakenoverindices(nl jt,n(cid:48)l(cid:48)j(cid:48)t(cid:48)) i i i i j j j j withi,j=1,2, wheren, l, j andt meanprincipal, angularmomentumandisospinquantumnum- bers, respectively, j and j (or j(cid:48) and j(cid:48)) arecoupled toJ (orJ), similarlyl andl (orl(cid:48) andl(cid:48)) 1 2 1 2 1 2 1 2 arecoupledtoλ (orλ(cid:48)),andt =t =1/2, t(cid:48) =t(cid:48) =−1/2isvalidifneutronsdecayintoprotons. 1 2 1 2 Thetwo-bodymatrixelementbeforetheanti-symmetrizationisrepresentedby (cid:104)n(cid:48)l(cid:48) j(cid:48)t(cid:48),n(cid:48)l(cid:48) j(cid:48)t(cid:48);J|O0x(r)|n l j t ,n l j t ;J(cid:105) 1 1 1 1 2 2 2 2 α 1 1 1 1 2 2 2 2 =2 ∑ (cid:112)j(cid:48) j(cid:48)S(cid:48)λ(cid:48)(cid:112)j j Sλ (cid:104)l(cid:48)l(cid:48)λ(cid:48)S(cid:48);J|S |l l λS;J(cid:105)(cid:104)n(cid:48)l(cid:48)n(cid:48)l(cid:48);J|H (r)|n l n l (cid:105) 1 2 1 2 1 2 α 1 2 1 1 2 2 α 1 1 2 2 S,S(cid:48),λ,λ(cid:48) (3.4)     l(cid:48) 1/2 j(cid:48) l 1/2 j  1 1  1 1 l(cid:48) 1/2 j(cid:48) l 1/2 j 2 2 2 2 λ(cid:48) S(cid:48) J  λ S J  whereH (r)istheneutrinopotential,S denotesspinoperators,SandS(cid:48)meanthetwo-bodyspins, α α and {·} including nine numbers denotes the 9j-symbol. By implementing the Talmi-Moshinsky transforms: (cid:104)nl,NL|n l ,n l (cid:105) (cid:104)n(cid:48)l(cid:48),N(cid:48)L(cid:48)|n(cid:48)l(cid:48),n(cid:48)l(cid:48)(cid:105) (3.5) 1 1 2 2 λ 1 1 2 2 λ(cid:48) 3 Heavyneutrinopotential... hF hGT hT 0.0012 0.0012 0.0003 0 10 0 10 0 10 0 0 r [fm] 0 0 r [fm] 0 0 r[fm] q [MeV] q [MeV] q [MeV] 1600 1600 1600 (a)n=n(cid:48)=0andl=l(cid:48)=3 hF hGT hT 0.0012 0.0012 0.0003 0 10 0 10 0 10 0 0 r [fm] 0 0 r [fm] 0 0 r[fm] q [MeV] q [MeV] q [MeV] 1600 1600 1600 (b)n=n(cid:48)=1andl=l(cid:48)=0 Figure 1: (Color online) Integrands of Eq. (3.10) are depicted for n=n(cid:48) =0 and l =l(cid:48) =3 in panel a, √ and for n=n(cid:48) =1 and l =l(cid:48) =0 in panel b. The plots are made for r= 2ρ =0 to 10 fm and q=0 to 1600 MeV. The closure parameter (cid:104)E(cid:105) is fixed to 0.5 MeV, which is suggested by the calculation without usingclosureapproximation[18]. theharmonicoscillatorbasisistransformedtothecenter-of-masssystem. (cid:104)l(cid:48)l(cid:48)λ(cid:48)S(cid:48);J|S |l l λS;J(cid:105)(cid:104)n(cid:48)l(cid:48)n(cid:48)l(cid:48);J|H (r)|n l n l (cid:105) 1 2 α 1 2 1 1 2 2 α 1 1 2 2 √ (3.6) = ∑ (cid:104)nl,NL|n l ,n l (cid:105) (cid:104)n(cid:48)l(cid:48),N(cid:48)L(cid:48)|n(cid:48)l(cid:48),n(cid:48)l(cid:48)(cid:105) (cid:104)l(cid:48)Lλ(cid:48)S(cid:48);J|S |lLλS;J(cid:105)(cid:104)n(cid:48)l(cid:48)|H ( 2ρ)|nl(cid:105), 1 1 2 2 λ 1 1 2 2 λ(cid:48) α α mos2 √ where ρ =r/ 2 is the transformed coordinate of center-of-mass system, and “mos2” means that the sum is taken over (n,n(cid:48),l,l(cid:48),N,N(cid:48)) [10]. In this article, in order to have a comparison to the precedingresults[7],wefocusontheneutrinopotentialeffectarisingfrom √ (cid:104)n(cid:48)l(cid:48)|H ( 2ρ)|nl(cid:105). (3.7) α This part is responsible for the amplitude of each transition from a state with n, l to another state withn(cid:48),l(cid:48),whilethecancellationisdeterminedbyspin-dependentpart. Forcalculationsofheavy- neutrinoexchangematrixelements,seeRefs.[11,12,13,14,15]. 3.2 Neutrinopotentialrepresentedinthecenter-of-masssystem Undertheclosureapproximationneutrinopotential[16,17,18]isrepresentedby √ (cid:90) ∞ √ H ( 2ρ)= 2R f ( 2ρq)√ h√α(q) q2 dq. (3.8) α π 0 α q2+m2ν( q2+m2ν+(cid:104)E(cid:105)) 4 Heavyneutrinopotential... Fermi Gamow-Teller Tensor ents60 total:112 60 total:112 400 total:630 v E of 40 sum 40 sum sum er l= l' = 0 l= l' = 0 200 l= l' = 0 mb20 20 u N 0 0 0 0 0.06 0.12 0.18 0.24 0 0.06 0.12 0.18 0.24 -0.04 0 0.04 0.08 0.12 0.16 0.2 Value of neutrino potential part √ Figure2: (Coloronline)Frequencydistributionof(cid:104)n(cid:48)l(cid:48)|H ( 2ρ)|nl(cid:105)isshownlimitedtononzerocases. α Cases with n,n(cid:48) =0,1,···,3 and l,l(cid:48) =0,1,···,6 are taken into account, where note that l (cid:54)=l(cid:48) results in √ (cid:104)n(cid:48)l(cid:48)|H ( 2ρ)|nl(cid:105)=0 in Fermi and Gamow-Teller cases [10]. The total number of events with nonzero α √ (cid:104)n(cid:48)l(cid:48)|H ( 2ρ)|nl(cid:105)isshownineachpanel. α where q is the momentum of virtual neutrino, m is the effective neutrino mass, R denotes the ν radius of decaying nucleus, and f is a spherical Bessel function (α =0,2), In particular (cid:104)E(cid:105) is α called the closure parameter, which means the averaged excitation energy of virtual intermediate state. InEq.(3.8)neutrinopotentialsincludethedipoleformfactors(notjusttheformfactors)that takeintoaccountthenucleonsize. Themasslessneutrinolimit(m →0)ofneutrinopotentialis ν √ (cid:90) ∞ √ H ( 2ρ)= 2R f ( 2ρq)hα(q) qdq, (3.9) α π α q+(cid:104)E(cid:105) 0 andtheheavymasslimit(m >>(cid:104)E(cid:105), m2 >>q2)ofneutrinopotentialis ν ν √ (cid:90) ∞ √ H ( 2ρ)= 1 2R f ( 2ρq)h (q)q2 dq, (3.10) α m2 π α α ν 0 Forordinarylightneutrinos, theneutrinopotentialinthemasslesslimitcanbeutilized. Simkovic √ unit is exploited for heavy neutrino case, in which the value of m2H ( 2ρ) is divided by proton √ ν α andelectonmasses(i.e. thevalueof(m2/m m )H ( 2ρ)isshowninthisarticle). Followingthe ν p e α corresponding study on massless limit cases [7], this article is devoted to investigate heavy mass limitcases. Therepresentationofneutrinopotentialsare h (q2)= gV2 F (1+q2/Λ2)4 V (cid:18) (cid:19) h (q2)= 2 q2 (µ −µ )2 gV2 + 1−2 q2 +1(cid:16) q2 (cid:17)2 g2A GT 34m2p p n (1+q2/ΛV2)4 3q2+m2π 3 q2+m2π (1+q2/Λ2A)4 (3.11) (cid:18) (cid:19) h (q2)= 1 q2 (µ −µ )2 gV2 + 2 q2 −1(cid:16) q2 (cid:17)2 g2A T 34m2p p n (1+q2/ΛV2)4 3q2+m2π 3 q2+m2π (1+q2/Λ2A)4 where µ and µ aremagneticmomentssatisfying µ −µ =4.7,m andm areprotonmassand p n p n p π pionmass,andΛ =850MeV,Λ =1086MeVarethefinitesizeparameters. V A Figure 1 shows the integrand of Eq. (3.10). In any case ripples of the form: qρ = const. can be found if q and ρ are relatively large. The upper-value of the integral range should be at least equal to or larger than q = 1600 MeV. In our research including our recent publication [6], we take q=2000 MeV and r=10 fm as the maximum value for numerical integration of Eq. (3.10) (massless neutrino cases). We noticed that, for the convergence, q of the integral should be max ratherlargerfortheheavycasescomparedtothelightcases. 5 Heavyneutrinopotential... Table1: Largecontributionsarelistedfrom1stto10thlargestones. Twosymmetriccasesresultinginan equivalentvalueareshowninthesameposition. Fermi Gamow-Teller Tensor Ranking (nl n(cid:48) l(cid:48)) Value (nl n(cid:48) l(cid:48)) Value (nl n(cid:48) l(cid:48)) Value 1 (0000) 0.261 (0000) 0.230 (0000) 0.125 2 (1010) 0.232 (1010) 0.217 (1010) 0.099 3 (2020) 0.210 (2020) 0.207 (0010) 0.088 (1000) 4 (0010) 0.193 (3030) 0.198 (2020) 0.083 (1000) 5 (3030) 0.192 (1020) 0.1809 (1020) 0.080 (2010) (2010) 6 (1020) 0.190 (2030) 0.1806 (0001) 0.072 (2010) (3020) (0100) 7 (2030) 0.179 (0010) 0.171 (3030) 0.0714 (3020) (1000) 8 (1030) 0.161 (1030) 0.160 (0011) 0.0710 (3010) (3010) (1100) 9 (0020) 0.157 (0020) 0.145 (2030) 0.070 (2000) (2000) (3020) 10 (0030) 0.132 (0030) 0.130 (0020) 0.065 (3000) (3000) (2000) 4. Statistics Since actual quantum states are represented by the superposition of basic states such as |nl(cid:105) in the shell-model treatment, the contribution of neutrino potential part can be regarded as the superposition: √ ∑ k (cid:104)n(cid:48)l(cid:48)|H ( 2ρ)|nl(cid:105). n,n(cid:48),l,l(cid:48) α (4.1) n,n(cid:48),l,l(cid:48) usingasuitablesetofcoefficients{k }determinedbythenuclearstructureofgrandmotherand n,n(cid:48),l,l(cid:48) daughter nuclei. Accordingly, in order to see the difference between the light and heavy neutrino contributions, it is worth investigating the statistical property of neutrino potential part (3.7) at heavymasslimit. Frequencydistributionofneutrinopotentialpart(3.7)isshowninFig.2. Thevaluesarealways positiveforFermiandGamow-Tellerparts, whilethetensorpartincludesnon-negligiblenegative values. Indeed,thesumofpositiveandnegativecontributionsoftensorpartsuggeststhattotalsum 9.458 is obtained by the cancellation between +9.943 and −0.485 (i.e., 9.458=9.943−0.485). Theorderofthemagnitudeisdifferentonlyforthetensorpart. Indeed,theaverageofthenonzero components is 0.0526 for the Fermi part, 0.0485 for the Gamow-Teller part, and 0.0063 for the tensor part. Contributions with l =l(cid:48) =0 (sum) cover 49.6% of the total contributions (sum) for 6 Heavyneutrinopotential... 0.053 0.053 r 0.12 e 0.2 l el r T o - 0.08 w s n o e m 0.1 T a 0.04 G 0.049 0.006 0 0 0 0.1 0.2 0 0.1 0.2 Fermi Fermi Figure 3: (Color online) Correlation between Eq. (3.10) values are examined by assuming l =l(cid:48). [Left] Correlation between Eq. (3.10) values for Fermi and Gamow-Teller parts, where the condition l =l(cid:48) does notbringaboutanylimitationsforFermiandGamow-Tellerparts. [Right]CorrelationbetweenEq.(3.10) values for Fermi and tensor parts, where values for the tensor part is always positive if l =l(cid:48) is assumed. Forbothpanels,allthetop10contributionslistedinTable1areincludedindotted-bluerectangles,andthe averageofallthenonzerocontributionsareshowningreendashedlines. Fermi, 52.3% for Gamow-Teller parts, and 12.8% for tensor part. Since the corresponding values inlightordinaryneutrinocasesare27.1%forFermipart,27.1%forGamow-Tellerpart,and7.2% fortensorpart[7],l=l(cid:48)=0componentisclarifiedtoplayamoredominantrole(roughlyequalto twice)inheavyneutrinocase. Large contributions for Fermi, Gamow-Teller and tensor parts are summarized in Table 1. Contribution labeled by (n l n(cid:48) l(cid:48)) = (0 0 0 0) (i.e. transition between 0s orbits) provides the largest contribution in any part. Roughly speaking, we see that s-orbit is remarkably significant in heavy neutrino cases. Indeed, all the top 10 contributions of Fermi and Gamow-Teller parts are completely filled with s-orbit contributions. As seen in the top 10 list the order of the kind (n l n(cid:48) l(cid:48)) are similar for Fermi and Gamow-Teller parts, where note that the order of Fermi and Gamow-Teller parts is exactly the same for ordinary light neutrino case as far as the top 10 list is concerned[7]. Tenlargestcontributions(sum)cover49.6%ofthetotalcontributions(sum)forthe Fermi part, 52.3% for the Gamow-Teller part, and 13.4% for the tensor part. The minimum value forthetensorpartis-0.0086achievedby(nl n(cid:48) l(cid:48))=(3034)and(3430). CorrelationbetweenthevaluesofEq.(3.10)fordifferentpartsareexaminedinFig.3. Com- parison between Fermi and Gamow-Teller parts shows that they provide almost the same values, althoughtheFermipartgenerallyshowsslightlylargervaluecomparedtotheGamow-Tellerpart. SuchaquantitativesimilaritybetweenFermiandGamow-Tellerpartsisnottrivialsincewecanfind essentiallydifferentmathematicalrepresentationsatleastintheirformfactors(cf.Eq.(3.11)). The tensor part is positively correlated with the Fermi part (therefore Gamow-Teller part). The l =l(cid:48) componentsofthetensorpartcontributions(sum)cover28.9%ofthetotaltensorpartcontributions 7 Heavyneutrinopotential... (sum). 5. Summary There are components of the two kinds in the nuclear matrix element; one is responsible for theamplitudeandtheotherisforthecancellation. Asacomponentresponsiblefortheamplitude, neutrino potential part (i.e., Eq. (3.7)) is investigated in this article. The presented results are valid not only to a specific double-beta decay candidates but also to all the possible candidates within n,n(cid:48) =0,1,···,3 and l,l(cid:48) =0,1,···,6. Note that, in terms of the magnitude, almost 40% smaller values are applied for the Gamow-Teller part in calculating the nuclear matrix element since(g /g )2=(1/1.27)2∼0.62(cf. Eq.(3.2)). V A Among several results on heavy neutrino cases, positive correlation of the values between Fermi, Gamow-Teller and tensor parts has been clarified. This property is common to the light ordinaryneutrino cases. Apart fromthetensorpart values, almostahalf ofthetotalcontributions has been shown to be occupied only by 10 largest contribution in which 10 largest contribution is exactly the same as l = l(cid:48) = 0 contributions. As a result the enhanced dominance of s-wave contributionisnoticedforheavyneutrinocases. TheothercomponentsoftheNMEsalsoresponsibleforthecancellationwillbestudiedinthe nextopportunity. References [1] J.EngelandJ.Menéndez,arXiv:1610.06548. [2] S.Dodelson,A.Melchiorri,A.Slosar,Phys.Rev.Lett.97,04301(2006). [3] J.N.Bahcall,Phys.Rev.C56,3391(1997). [4] J.D.Vergados,H.Ejiri,andF.Simkovic,Rep.Prog.Phys.75,106301(2012). [5] M.Horoi,Phys.Rev.C87,014320(2013). [6] Y.Iwata,N.Shimizu,T.Otsuka,Y.Utsuno,J.Menéndez,M.Honma,andT.Abe,Phys.Rev.Lett. 116112502(2016). [7] Y.Iwata,toappearinNucl.Phys.Lett.;arXiv:1609.03118. [8] A.Faessler,G.L.Fogli,E.Lisi,A.M.RotunnoandF.Simkovic,Phys.Rev.D83,113015(2011). [9] E.Lisi,A.RotunnoandF.Simkovic,Phys.Rev.D92,093004(2015). [10] Y.Iwata,J.Menéndez,N.Shimizu,T.Otsuka,Y.Utsuno,M.Honma,andT.Abe,CNSannualreport. CNS-REP-94,71(2016). [11] M.Blennow,E.Fernandez-Martinez,J.Lopez-Pavon,andJ.Menéndez,JHEP07,096(2010). [12] A.Faessler,M.González,S.Kovalenko,andF.Simkovic,Phys.Rev.D90,096010(2014). [13] J.Barea,J.Kotila,andF.Iachello,Phys.Rev.D92,093001(2015). [14] J.HyvarinenandJ.Suhonen,Phys.Rev.C91,024613(2015). [15] M.HoroiandA.Neacsu,Phys.Rev.C93,024308(2016). 8 Heavyneutrinopotential... [16] T.Tomoda,Rep.Prog.Phys.5453(1991). [17] M.HoroiandS.Stoica,Phys.Rev.C81,024321(2010). [18] R.A.Sen’kovandM.Horoi,Phys.Rev.C88,064312(2013). 9