Interference Between Light and Heavy Neutrinos for $0\nu\beta\beta$ Decay in the Left-Right Symmetric Model

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by  Fahim Ahmed

Interference Between Light and Heavy Neutrinos for 0νββ
Decay in the Left-Right Symmetric Model
FahimAhmed,AndreiNeacsu,MihaiHoroi
DepartmentofPhysics,CentralMichiganUniversity,MountPleasant,Michigan48858,USA
7
1
0
2

n Abstract
a
J Neutrinoless double-beta decay is proposed as an important low energy phenomenon
thatcouldtestbeyondtheStandardModelphysics. Thereareseveralpotentiallycom-
1
1 peting beyond the Standard Model mechanisms that can induce the process. It thus
becomesimportanttodisentanglethedifferentprocesses. Inthepresentstudywecon-

] sidertheinterferenceeffectbetweenthelightleft-handedandheavyright-handedMa-
h
p jorana neutrino exchange mechanisms. The decay rate, and consequently, the phase-
- space factors for the interference term are derived, based on the left-right symmetric
p
model. The numerical values for the interference phase-space factors for several nu-
e
h clidesarecalculated,takingintoconsiderationtherelativisticCoulombdistortionofthe
[ electronwavefunctionandfinite-sizeofthenucleus. Thevariationoftheinterference

effectwiththeQ-valueoftheprocessisstudied.
1
v
7
7 1. Introduction
1
3 Neutrinolessdoublebetadecay(0νββ)isaprocesswhereanuclidedecaystoits
0 isobarhavingtwomoreprotons. Intheprocesstwoelectronsareemittedwithoutany
.
1 anti-neutrinos,
0
7 AZX→Z+A2X+2e−. (1)
1
: Unlikethetwoneutrinodoublebetadecay(2νββ)wheretwoanti-neutrinosarealso
v
emitted, in 0νββ the lepton number is violated by two units. In the Standard Model
i
X (SM) the lepton number is conserved. Thus the experimental observation of 0νββ
r wouldindicatephysicsbeyondtheStandardModel(BSM).Moreover,the0νββ pro-
a
cessrequirestheneutrinostobeMajoranafermions[1]. Allthesefeaturesmake0νββ
an exciting process for testing BSM physics, and several BSM mechanisms are pro-
posedforcontributingtothedecay[2,3].
Themoststudiedmechanisminthe0νββ decayliteratureisthesocalled‘standard
massmechanism’oflightleft-handed(LH)neutrinoexchange[2]. However,ifright-
handed(RH)V+Acurrentsarealsoconsideredalongsidetheleft-handedV Acur-
−
rents, thenseveralcompetingmechanismscouldcontributesignificantlytothe0νββ
Emailaddress:[email protected](FahimAhmed)
PreprintsubmittedtoElsevier January13,2017
decay rate [4]. The existence of V +A currents is well motivated by new physics
of BSM. One such popular BSM scenario is the left-right symmetric model (LRSM)
[5, 6], which is presently actively investigated at LHC [7]. RH currents are naturally
consideredinthismodel,sinceparityisrestoredathighenergies. Theextendedgauge
group for the electroweak sector of the model is SU(2) SU(2) U(1) , where
L R Y
× ×
RH neutrinos are incorporated as part of the SU(2) doublet. LRSM provides a nat-
R
uralframeworkfortypeI[5]andtypeII[8]seesawmechanisms, whichexplainsthe
smallness of neutrino mass. Apart from being crucial in determining absolute values
ofneutrinomasses,0νββ isanimportantlowenergyprobetostudyhighenergyBSM
mechanisms[9]. Invariablythisalsorequiresdisentanglingtheunderlyingmechanism
inducingtheprocess[10].
BecauseoftheexistenceofRHcurrentstherewillbeseveralcompetingcontribu-
tionstothedecayrateof0νββ intheLRSMscenario[11]. Additionally,theseesaw
mechanismrequirestheexistenceofheavy,mostlysterileneutrinos[8]. Neutrinomix-
ingschemeswouldthennaturallyincorporateheavymasseigenstatesforbothLHand
RHneutrinos(seebelowfordetails). Theinversehalf-lifeformulafor0νββ hasthe
followinggeneralstructure,
(cid:12) (cid:12)2
(cid:104)T0ν(cid:105)−1=(cid:12)(cid:12)(cid:88)(cid:16)NPP&PPP(cid:17) (cid:16)PSF(cid:17)1/2 (cid:16)NME(cid:17)(cid:12)(cid:12) , (2)
1/2 (cid:12)(cid:12) i× i × i(cid:12)(cid:12)
i
whereNPPandPPParetheneutrinoandparticlephysicsparametersarisingfromthe
BSMphysics. Thephase-spacefactors(PSF)takeintoaccountthekinematicalfactors
of the two outgoing electrons. The summation i is over all possible mechanisms that
couldinducethe0νββ process.Becauseofthemodulussquared,interferencebetween
differentmechanismsalsocontributetothetotaldecayrateoftheprocess. Foralong
timethePSFweregenerallyconsideredasbeingreliablycalculated[4,12],butrecent
re-evaluations[13,14,15,16,17]haveshowndiferencesofupto100%inthecaseof
heaviernuclei(e.g. G0ν of150Nd),whencomparingtotheolderresults. Presently,the
08
uncertaintiesinthePSFaremaintainedundercontrolandnewresultsareinagreement
toeachother. Lastly,theNMEaretherelevantnuclearmatrixelementsforthenuclear
transitionbetweentheinitialandfinalnuclei. Thereareseveralmethodscomonlyused
tocalculatetheNME,ofwhichthemostemployedonesaretheinteractingshellmodel
(ISM)[18,19,20,2,21,22,23,24], thequasi-randomphaseapproximation(QRPA)
[12,25,26,27,28,29],theinteracting-bosonapproach(IBM2)[30,31],theprojected
Hartree-Fock-Bogoliubov(PHFB)[32],andthegeneratorcoordinatormethod(GCM)
[33,34]. ThecurrentsituationoftheNMEisnotyetsettled,withdifferencesofupto
300%amongtheresultsofdifferentgroupsandmethods(SeeFig.6andFig.7of[35]).
Also,relativelyfewNMEresultsotherthanthoseforthe’standardmassmechanism’
exist.
If two competing mechanisms are considered for the 0νββ decay, as is the case
forourpresentstudy,thenthemagnitudeofthecontributionoftheinterferencetothe
half-lifewillprimarilydependsonthePSF.Thephasedifferencearisingfromthetwo
differentcomplexneutrinophysicsparameterswillactasthephaseoftheinterference
oscillation. Thechiralstructureoftheoutgoingelectronsdeterminesthenatureofthe
PSF for the individual mechanisms. This also results in a different form of the PSF
2
fortheinterference. Thetotaldecayratefortheprocessisobtainedbyintegratingthe
PSF over the energies of the outgoing electrons. Several effects must be considered
inordertoaccuratelydeterminenumericalvaluesforthePSF.TheCoulombattraction
ofthedaughternucleuswilldistortthewavefunctionsoftheoutgoingelectrons[36].
Moreover,thefinitesizeofthenucleuswillaffectthedistortedelectronwavefunction
[13,16].
Thesubjectofinterferencebetweenpairsof0νββ mechanismsisobscureinrecent
literature, withthelatestpaperdevotedtothissubjectbeing34yearsold[37]. Since
thereisgreatinteresttodisentanglethedifferentunderlyingmechanismsof0νββ,itis
importantthattheinterferencebetweenmechanismsisalsoconsidered. Inthepresent
paperwere-considertheinterferencebetweenthelightandheavyMajorananeutrinos.
Wecalculatethedecayrateof0νββ forlightandheavyneutrinoexchangeinLRSM.
Thefocusofourpaperistoinvestigatethecontributionofinterferenceeffects. Were-
visitthecalculationsandincluderelativisticFermifunctionandfinite-sizeeffectofthe
nucleusinthephase-spacefactors(PSF)oftheinterferenceterms.Wenumericallycal-
culatetheinterferencePSFfordifferentnuclidesandcompareourresultswiththoseof
Ref. [37]. Althoughthenumericalresultsof[37]aresimilartoours,theformulaeused
in Ref. [37] seems to contain typos that make them difficult to use. In addition, our
analysis is extended to more isotopes than previously considered and for transitions
to excited states. The paper is organized as follows: Section II presents the general
formalism for 0νββ considering both LH and RH currents. We derive the exact ex-
pressionsforthePSFs. SectionIIIpresentsthenumericalresultsforthePSFsandthe
effectofinterferencefordifferentnuclidesconsideringtransitionstothegroundstate
(g.s.) andtothefirstexcited0+stateofthedaughternucleus.
2. Formalismfor0νββ decay
Thederivationofthedecayratefor0νββ beginswiththeeffectivecurrent-current
HamiltonianincorporatingtheRHleptonicandhadroniccurrents[4],
(cid:18)G cosθ (cid:19) (cid:104)(cid:16) (cid:17)(cid:16) (cid:17)
HIβ = F√2 c 2 e¯Lγανe(cid:48)L JL†α+κJR†α
(cid:16) (cid:17)(cid:16) (cid:17) (cid:105)
+ e¯Rγανe(cid:48)R ηJL†α+λJR†α +H.c. . (3)
Here e and ν are the LH(RH) electron and electron-neutrino fields, respec-
L(R) e(cid:48)L(R)
tively. J isthechargedLH(RH)hadroniccurrent. λ,η andκ aretherespective
L(R)α
coupling constants between (V+A)(V+A), (V+A)(V A) and (V A)(V+A) in-
− −
teractions. G and θ are the Fermi constant and Cabbibo angle, respectively. H.c.
F C
denotestheHermitianconjugate.
3
V A V+A
dL − uL dR uR
W W
L R
V−A e−L V+A e−R
m m
ν N
V−A e−L V+A e−R
W W
L R
dL V A uL dR V+A uR
−
(a)Lightneutrinoexchange. (b)Heavyneutrinoexchange.
Figure1: Lightandheavyneutrinoexchangemechanismsfor0νββ inLRSM.
Intheleft-rightsymmetricmodel(LRSM)theelectron-neutrinomixingisgivenas
[21]:
light heavy
(cid:88) (cid:88)
νe(cid:48)L= UekνkL+ SekNkcR, (4)
k k
light heavy
(cid:88) (cid:88)
νe(cid:48)R= Te∗kνkcL+ Ve∗kNkR. (5)
k k
Heretheflavoreigenstatesν arelinearcombinationsoflight(ν)andheavy(N)mass
(cid:48)
eigenstates.ThemixingmatricesU andV arealmostunitarywhileSandT aretakento
besmall. ThustheLHfieldsarepredominantlylightneutrinos,whereastheRHfields
are predominantly heavy, ν (cid:80)lightU ν and ν (cid:80)heavyV N . Thus under
e(cid:48)L ≈ k ek kL e(cid:48)R ≈ k e∗k kR
theseassumptions,thetwodominatingdiagramsforthesecondorderprocessof0νββ
arisingfromtheHamiltonianofEq.(3)areshowninFigure1.Figure1aisthestandard
mass mechanism of light neutrino, m , exchange. In Figure 1b, 0νββ is induced by
ν
theexchangeofheavyneutrino, m . Thecouplingconstantbetween(V+A)(V+A)
N
interaction, λ, is shown to be (m /m )2 in the LRSM. Here m is the mass of
WL WR WL/R
theLH(RH)Wboson.
ApplyingsecondorderperturbationtheorytoourinteractionHamiltonian,Eq.(3),
wegettheS-matrixfor0νββ processasfollows,
(cid:104)f|S(2)|i(cid:105)0ν =iA0(cid:2)ηLνELMν+ηRNERMN(cid:3)2πδ(Ef+E1+E2−Ei). (6)
The first and the second terms inside the square brackets correspond to the light and
heavyneutrinos,respectively. ThedimensionfulconstantA0isdefinedas:
g2m2(cid:18)G cosθ (cid:19)2 1
A0= A e F c , (7)
− 2π √2 rA
where g = 1.27 is the axial vector constant [38], m is the electronic mass, and
A e
1
rA =mer0A3 with r0 =1.2 fm, is the nuclear radius in the units of me, A being the
4
massnumberofthenuclide. Theneutrinophysicsparametersforthelightandheavy
neutrinosaredefinedrespectivelyasfollows:
light
1 (cid:88)
ην = (U )2m ξ , (8)
L m ek k k
e
k
heavy
ηRN =λ2 (cid:88)(Ve∗k)2mMpΞk. (9)
k
k
Herem andM arethemassesofthelightandheavyneutrinos,respectively.Similarly,
k k
the Majorana phase factors for light and heavy neutrinos are ξ and Ξ , respectively.
k k
The proton mass m is introduced for dimensional consistency. The nuclear matrix
p
elements(NME)forthelight(Mν)andheavy(MN)neutrinosarethecombinedFermi,
Gamow-Tellerandtensormatrixelementsasdefinedin[39]:
Mν/N =(cid:104)Mν/N (cid:18)gV(cid:19)2Mν/N+Mν/N(cid:105). (10)
GT − g F T
A
Thedeltafunctionimposestheenergyconservation,E andE aretheenergiesofthe
i f
initial and final nuclei, respectively. E1 and E2 are the energies of the two outgoing
electrons. E and E correspond to the two final state electron current for light and
L R
heavyneutrinoexchange,respectively:
(cid:104) (cid:105)
EL= u¯(p1s1)(1+γ5)Cu¯T(p2s2) , (11)
(cid:104) (cid:105)
ER= u¯(p1s1)(1 γ5)Cu¯T(p2s2) , (12)
−
where the free particle spinor u is defined for the two outgoing electrons having mo-
menta p1, p2 and spins s1, s2, respectively. The presence of the γ5 matrices makes
thechiralstructureofthetwocurrentsmanifestlydifferent,namelyboththeelectrons
areLHforthecaseoflightneutrinoexchange,whereasbothneutrinosareRHforthe
heavyneutrinoexchange(seeFigure1). Theamplitudeoftheprocessis:
Mf0iν =A0ηLνELMν+A0ηRNERMN =Mfliight+Mfheiavy. (13)
Squaring the amplitude one gets the interference terms between the light and heavy
neutrinoexchangemechanisms,
(cid:12)(cid:12)Mf0iν(cid:12)(cid:12)2=(cid:12)(cid:12)(cid:12)Mfliight(cid:12)(cid:12)(cid:12)2+(cid:12)(cid:12)(cid:12)Mfheiavy(cid:12)(cid:12)(cid:12)2+MfliightMfheiavy∗+Mfliight∗Mfheiavy. (14)
Thefirstandsecondtermsaretheamplitudesquaresforthelightandheavyneutrinos,
respectively. The third and fourth terms are the interference terms between the light
andheavyneutrinos. Thespinaveragedamplitudesquareisthen,
(cid:12) (cid:12)2 (cid:88)(cid:104)
(cid:12)M0ν(cid:12) =A2 ην 2 Mν 2 E 2+ ηN 2 MN 2 E 2
(cid:12) fi (cid:12) 0 | L| | | | L| | R| | | | R|
spin
5
(cid:105)
+ηLν(ηRN)∗Mν(MN)∗ELER∗+(ηLν)∗ηRN(Mν)∗MNEL∗ER . (15)
For0νββ weonlyneedtosumoverthetwooutgoingelectrons. Calculatingthespin
sumsusingCasimir’strickandtracetheorems[40]weget:
(cid:18) (cid:19)
(cid:88) EL 2=(cid:88) ER 2=8E1E2 1 |p1||p2|cosθ , (16)
e1,e2| | e1,e2| | − E1E2
spin spin
(cid:88) (cid:88)
ELER∗= EL∗ER=−8m2e. (17)
e1,e2 e1,e2
spin spin
Herep1andp2arethethree-momentaofthetwoelectronsemittedatanangleθ. Thus
weseethatthespinaveragefortheinterferencetermintheamplitudesquareisdifferent
fromthatofthetermsfortheindividualmechanisms.Fortheindividualterms,thespin
averagegivesacontribution,whichdependsontheenergiesoftheelectronswhilefor
theinterferenceterm, thespinaverageisindependentoftheelectronicenergies. The
phase-space factors for the individual and interference terms will be different due to
thisdifferenceinthespinaverage.
ThetotaldecayrateoftheprocessisthenobtainedusingtheFermi’sGoldenrule
[40](h¯ =1,c=1),andintegratingoverthephase-spaceofthetwoelectrons,
1 (cid:90) d3p d3p (cid:12) (cid:12)2
Γ0ν = 2!2π (2π)321E1(2π)322E2(cid:12)(cid:12)Mf0iν(cid:12)(cid:12) δ(Ef+E1+E2−Ei). (18)
Carryingouttheangularintegrationforthe3Dmomentumintegralinpolarcoordinates
we can then re-express p in terms of relativistic energy E =p2+m2, where k=
| k| k k e
1,2. The E2 integral then can be carried out through the δ-function. Due to energy
conservation, we can express E2 in terms of E1. Since the kinetic energy is shared
between the two electrons, the E1 integral is carried out between the limits me and
Q+m . ThetwooutgoingelectronsareattractedbytheCoulombfieldofthedaughter
e
nucleus. The free electron wavefunction is thus modified by the Fermi factor F0 as
giveninreference[41]. Thetotaldecayratefor0νββ becomes:
g4(G cosθ )4m9(cid:90) T+1
Γ0ν = A (F2π)5rA2c e 1 dε1p˜1p˜2F0(Zs,ε1)F0(Zs,ε2)
(cid:34) (cid:35)
(cid:104) (cid:105) (cid:104) (cid:105)
× ε1ε2 |ηLν|2|Mν|2+|ηRN|2|MN|2 − ηLνηRN∗MνMN∗+ηLν∗ηRNMν∗MN , (19)
wherewehaveswitchedtoenergies(ε )andmomenta(p˜ )inthem units. Similarly,
k k e
T istheQ-valueinthem units. Thehalflifeisthenrelatedtothetotaldecayrateby:
e
(cid:104) (cid:105) 1 (cid:104) (cid:105)
T0ν − =g4G0ν ην 2 Mν 2+ ηN 2 MN 2
1/2 A 1 | L| | | | R| | |
(cid:104) (cid:105)
−g4AG10ν(cid:48) ηLνηRN∗MνMN∗+ηLν∗ηRNMν∗MN , (20)
6
wheretheintegratedphase-spacefactors(PSF)aredefinedby:
g0ν(cid:90) T+1
G01ν = r2 dε1F0(Zs,ε1)F0(Zs,ε2)p˜1p˜2ε1ε2δ(ε2+ε1−T−2), (21)
A 1
g0ν(cid:90) T+1
G01ν(cid:48) = r2 dε1F0(Zs,ε1)F0(Zs,ε2)p˜1p˜2δ(ε2+ε1−T−2), (22)
A 1
withthecommondimensionfulconstanthavingthevalue,
(G cosθ )4m9
g0ν = F(2π)5lnc2 e =2.8×10−22 yr−1. (23)
The PSF for the light and heavy neutrino exchange are the same, G0ν of Eq. (21).
1
As Eq. (21) and (22) show, the PSF for the interference term, G0ν(cid:48), has a different
1
structure than the PSF of the individual mechanisms, G0ν. Because of the absence
1
ofelectronenergies,theinterferencePSFissuppressedconsiderablycomparedtothe
‘non-interference’PSF.
3. Results
The accuracy of PSF calculations depends on certain assumptions and methods.
References [13] and [16] considered the effect of finite nuclear size and screening of
nuclear charge due to atomic electrons in calculating the non-interference PSF. An
easytouse,fasterandsufficientlyaccuratemethodwasrecentlyintroducedin[17]by
considering a screening factor S to the charge of the final nucleus Z , still retaining
f f
the original assumption of point-like nuclear charge. This modification to the charge
(Z = Sf Z )replicatestheeffectsoffinitenuclearsizeandelectronscreeningtogood
s 100 f
accuracy.ForourcaseweconsiderthevalueS =94.5%forG0ν (seetableIVof[17]).
f 1
7
Table1:NumericalvaluesofPSFofdifferentnucleiforg.s. g.s. transitions. Column
→
4 lists the values of the interference PSF Eq. (22) and column 5 is the ratio to the
individualPSFin%(seeEq.(21)). (Q valuesaretakenfrom[16]).
ββ
Nuclei Q G0ν G0ν(cid:48) (cid:15)= G10ν(cid:48)
ββ 1 1 G0ν
[MeV] [ 10 14yr 1] [ 10 15yr 1] 1
− − − − [%]
× ×
48Ca 4.27226 2.45462 1.09027 4.44168
76Ge 2.03904 0.228003 0.28987 12.7134
82Se 2.99512 0.996509 0.757323 7.59976
96Zr 3.35037 2.04454 1.32482 6.47976
100Mo 3.03440 1.57402 1.17824 7.48553
110Pd 2.01785 0.465953 0.603145 12.9443
116Cd 2.8135 1.65694 1.38323 8.34808
124Sn 2.28697 0.886628 0.979603 11.0486
128Te 0.86795 0.0554 0.017355 31.3251
130Te 2.5269 1.4104 1.36624 9.68692
136Xe 2.45783 1.44863 1.45738 10.0604
150Nd 3.37138 6.60043 4.27367 6.47483
Table2: SameasTable1,fortransitionsfromg.s. 1st 0+excitedstates. (Q values
→ ββ
aretakenfrom[17]).
Nuclei Q G0ν G0ν(cid:48) (cid:15)= G01ν(cid:48)
ββ 1 1 G0ν
[MeV] [ 10 15yr 1] [ 10 16yr 1] 1
− − − − [%]
× ×
48Ca 1.275 0.292636 0.633955 21.6636
76Ge 0.917 0.185314 0.552101 29.7927
82Se 1.508 0.906993 1.64925 18.1837
96Zr 2.202 4.43543 5.12937 11.5646
100Mo 1.904 3.03925 4.221 13.8883
110Pd 0.5472 0.120312 0.531163 44.1487
116Cd 1.057 0.720768 1.89277 26.2605
124Sn 1.12 0.953037 2.36911 24.8585
130Te 0.7335 0.358299 1.28598 35.8911
136Xe 0.879 0.651285 2.0187 30.9956
150Nd 2.631 27.5308 25.3183 9.19636
The effect of the interference term was considered in [37] where the numerical
valuesofthesuppressionfactorsofthetermfordifferentnucleiwerecalculated. We
trytoverifythevaluesclaimedin[37]basedonourderivationoftheinterferenceterm
PSF(seeEq.(22)). WeanalyzethecontributionofthePSFformaximuminterference
so as to make our conclusions as general as possible. This is in anticipation of its
8
smallness,aswasalreadyclaimedin[37]and[10]. Tomaketheanalysisformaximum
interference more transparent we introduce certain assumptions and notations. The
NMEaretakentoberealMν =Mν,MN =MN.TheLNVparametersarenonetheless
∗ ∗
treatedascomplexhavingthephases,
ην = ην eiφ1 , ηN =(cid:12)(cid:12)ηN(cid:12)(cid:12)eiφ2. (24)
L | L| R R
(cid:12) (cid:12)
Furthermore we make the assumptions ην Mν (cid:12)ηN(cid:12)MN ηM. Thus the inverse
| L| ≈ R ≈
half-lifeofEq.(20)canberewrittensoastogivemaximuminterference,
(cid:104) (cid:105) 1 (cid:104) (cid:104) (cid:105) (cid:104) (cid:105)(cid:105)
T10/2ν −≈g4A G01ν |η|2|M|2+|η|2|M|2 −G01ν(cid:48) |η|2|M|2e(φ1−φ2)+|η|2|M|2e(φ2−φ1)
(cid:104) (cid:105)
=2g4G0ν η 2 M 2 1 G0ν(cid:48)/G0νcosφ =2g4G0ν η 2 M 2[1 (cid:15)cosφ], (25)
A 1 | | | | − 1 1 A 1 | | | | −
wherewehavedefinedthephasedifferencebetweenthetwoLNVparametersasφ =
φ1 φ2 and the ratio between the two phase space factors of Eq. (21), (22) as (cid:15)=
G0ν−(cid:48)/G0ν. Depending on the value of (cid:15), the (cid:15)cosφ term will mostly determine the
1 1
contributionoftheinterference. ThenumericalvaluesaretabulatedinTables1and2
fortheg.s. g.s.andg.s. 0+(1stexcited)transitions,respectively.Thefirstexcited
→ →
0+ transitions are also considered because of their potential experimental relevance.
Thefifthcolumninbothtablesliststhevaluesofthefactorformaximuminterference,
(cid:15)in%. Fromthevalueswecanclearlyseethattheinterferenceeffectdecreaseswith
theQ values.FromTable1oneobservesquiteclearlytheextremecasesfor48Caand
ββ
128Te. A4%increasetothe0νββ decayrateduetoconsiderationoftheinterference
effectforaQ valueof4.27MeVfor 48Ca,whereasfor 128Tethecontributionisas
ββ
highas31%foraQ of0.868MeV.Thevariationof(cid:15)withQ isplottedinFigure
ββ ββ
2, obtained by fixing the mass number A=76 for Ge and the proton number of the
final nucleus Z =34. As the plot suggests, for decay modes of smaller Q values
f ββ
theeffectofinterferencewillbestronger. Theeffectcanbeashighas 50%ascan
be seen for the first excited 0+ transition for 110Pd (see Table 2 and F≈igure 2). We
alsostudiedthevariationof(cid:15)withmassnumberAandprotonnumberZ forthefinal
f
nucleus. NosignificantdependenceonAandZ wasfound.
f
9
60
)
%
(
ϵ
e, 50
c
en △110Pd(2nd0+)
r
rfe 40
nte △128Te(g.s.)
mi 30 □130Te(2nd0+)
u
m △124 +
Sn(2nd0 )
xi
ma 20 △82Se(2nd0+)
r
o
orf 10 76Ge(g.s.) □ 116Cd(g.s.)
act □ □ 48Ca(g.s.)
F 150Nd(g.s.) □
0
0 1 2 3 4 5
Qvalue,Q (MeV)
ββ
Figure2:Variationofthefactorformaximuminterference(cid:15)withQ .Plotisobtained
ββ
byvaryingtheQ-value,whilekeepingthemassfixedto76Geandthechargeofthefinal
nucleusZ =34of 76Se. Differentnucleiareaddedintheplotfortheg.s. andfirst0+
f
excitedstatesQ values.
ββ
Theexchangeofheavyneutrinosforinducing0νββ wasfirstconsideredin[42].
Theinterferencebetweenlightandheavyneutrinoswasthenconsideredin[37]. Itwas
foundthatthePSFfortheinterferencetermwastypicallyfactorof10smallerthanthe
PSF of the individual mechanisms [37, 10]. The suppression factors of the interfer-
encetermwerecalculatedforseveralnuclidesandwasfoundtobelessthan10%. The
smallness of the contribution of the interference is mainly due to the opposite chiral
structuresoftheoutgoingelectrons. DuetotheCoulombattractionofthedaughternu-
cleus,theoutgoingelectronwavefunctionsaredistorted. ThereforethePSFhastobe
calculated(numerically)takingthisintoaccountbyintroducingFermifactor[36]. In
[37]thiswasdonebyusingnon-relativisticFermifactor,whichisindependentofthe
Q-valueoftheprocess. Moreovertheelectronswereassumedtobeultra-relativisticin
theanalysisinordertoarriveatthenumericalvaluesofthesmallsuppressionfactors
oftheinterferenceterm. Thisisincontrasttotheconsiderationofthenon-relativistic
Fermi factor for electron wave function used in Ref. [37]. For our analysis we have
correctlyconsideredtherelativisticFermifactor. Inadditionwehavealsoconsidered
theeffectoffinitenuclearsize. Althoughthenumericalresultsobtainedinourcurrent
studyareveryclosetothevaluesinreferences[37]and[10],ourresultsaremoregen-
eral since the assumption of ultra-relativistic electrons can be relaxed. Consideration
forrelativisticFermifactorandfinitenuclearsizeextendtheanalysisandallowsusto
predicttheQ valuesforwhichtheeffectofinterferencecanbeobservable.
ββ
10