The 10 −3 eV frontier in neutrinoless double beta decay

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Letters

B

www.elsevier.com/locate/physletb

The

10

−

3

eV frontier

in

neutrinoless

double

beta

decay

J.T. Penedo

a

,

∗

,

S.T. Petcov

a

,

b

,

1 a_{SISSA/INFN,ViaBonomea265,34136Trieste,Italy}

b_{KavliIPMU(WPI),University ofTokyo,277-8583Kashiwa,Japan}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received15June2018

Receivedinrevisedform5September2018 Accepted18September2018

Availableonline17October2018 Editor: G.F.Giudice

Keywords:

Neutrinophysics Majorananeutrinos

Neutrinolessdoublebetadecay Normalordering

Theobservationofneutrinolessdoublebetadecaywouldallowtoestablishleptonnumberviolationand theMajorananatureofneutrinos.Therateofthisprocessinthecaseof3-neutrinomixingiscontrolled bytheneutrinolessdoublebetadecayeffectiveMajoranamass|m|.Foraneutrinomassspectrumwith normalordering,whichisfavouredoverthespectrumwithinvertedorderingbyrecentglobalfits,|m|

canbesignificantlysuppressed.Takingintoaccountupdateddataontheneutrinooscillationparameters, weinvestigatetheconditionsunderwhich|m|inthecaseofspectrumwithnormalorderingexceeds 10−3_(5×10−3_)eV:|m|

NO>10−3(5×10−3)eV.Weanalysefirstthegenericcasewithunconstrained

leptonicCPviolationMajoranaphases.Weshow,inparticular,thatifthesumofneutrinomassesisfound tosatisfy>0.10 eV,then|m|NO>5×10−3eV foranyvaluesoftheMajoranaphases.Weconsider

also caseswhere the values for thesephases are eitherCPconserving orare inline withpredictive schemescombiningflavourandgeneralisedCPsymmetries.

©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Despitetheirelusiveness,neutrinoshavegrantedusunique ev-idence for physics beyond the Standard Theory. Observations of flavour oscillations in experiments with solar, atmospheric, reac-tor,andacceleratorneutrinos(see,e.g., [1])implybothnon-trivial mixingin theleptonic sector and above-meV massesforatleast twoofthelightneutrinos.Neutrinooscillations,however,areblind totheabsolutescaleofneutrinomassesandtothenature –Dirac orMajorana –ofmassiveneutrinos[2,3].

InordertouncoverthepossibleMajorananatureofthese neu-tralfermions,searchesforthelepton-numberviolatingprocess of neutrinolessdoublebeta(

(ββ)0

ν -)decay areunderway (forrecent reviews,seee.g. [4,5]).Thisdecaycorresponds toatransition be-tweentheisobars

(

A

,

Z

)

and

(

A

,

Z

+

2

)

,accompaniedbythe emis-sionoftwoelectronsbut –unlikeusualdoublebetadecay – with-out the emission of two (anti)neutrinos. A potential observation of

(ββ)0

ν -decayisfeasible,inprinciple,wheneversinglebeta de-cayisenergeticallyforbidden,asisthecaseforcertaineven–even nuclei. The searches for

(ββ)0

ν -decay have a long history (see, e.g., [6]).Thebestlowerlimitsonthehalf-livesT₁0_/ν₂ ofthisdecay

*

Correspondingauthor.

E-mailaddress:jpenedo@sissa.it(J.T. Penedo).

1 _{Alsoat:InstituteofNuclearResearchandNuclearEnergy,BulgarianAcademyof}

Sciences,1784Sofia,Bulgaria.

have beenobtainedfortheisotopes ofgermanium-76, tellurium-130, andxenon-136: T₁0_/ν₂

(

76Ge

)

>

8

.

0

×

1025 yr reported by the GERDA-II collaboration [7], T₁0_/ν₂

(

130Te

)

>

1

.

5

×

1025 yr obtained fromthecombinedresultsoftheCuoricino,CUORE-0,andCUORE experiments [8],andT₁0ν_/2

(

136Xe

)

>

1

.

07

×

1026_{yr reached bythe} KamLAND-Zencollaboration [9],withalllimitsgivenatthe90%CL. InthestandardscenariowheretheexchangeofthreeMajorana neutrinos

ν

i

(

i

=

1

,

2

,

3

)

withmassesmi

<

10 MeV provides the dominant contribution to the decay rate, the

(ββ)0

ν -decay rate is proportional to the square of the so-called effective Majorana mass

|

m

|

. Given the present knowledge of neutrino oscillation data, the effectiveMajorana mass isbounded from belowin the caseofaneutrinomassspectrumwithinvertedordering(IO)[10],

|

m

|

IO

>

1

.

4

×

10−2 eV. Instead, in thecase ofa spectrum with normalordering(NO),

|

m

|

canbeexceptionallysmall:depending on the valuesof the lightest neutrinomass andof the CP viola-tion(CPV)Majoranaphaseswecanhave

|

m

|

NO



10−3 eV (see, e.g., [1]).Recentglobalanalysesshow apreferenceofthedatafor NO spectrumoverIOspectrumatthe2

σ

CL [11,12].Inthelatest analysisperformedin[13] thispreferenceisat3

.

1

σ

CL.

New-generation experimentsseektoprobe andpossiblycover the IO region of parameter space, working towards the

|

m

| ∼

10−2 eV frontier. Aside from upgrades to the ones mentioned above,suchexperimentsinclude(see,e.g., [4,5]):CANDLES(48_Ca), Majorana _{and LEGEND (}76_{Ge), SuperNEMO and DCBA (}82_Se

_,

150_{Nd), ZICOS (}96_{Zr), AMoRE and MOON (}100_{Mo), COBRA} https://doi.org/10.1016/j.physletb.2018.09.059

0370-2693/©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

(116_Cd

_,

130_{Te), SNO+ (}130_{Te), and NEXT, PandaX-III and nEXO} (136Xe). In case these searches produce a negative result, the next frontier in the quest for

(ββ)0

ν -decay will correspond to

|

m

| ∼

10−3_eV.

Inthepresentarticlewedeterminetheconditionsunderwhich theeffectiveMajoranamassinthecaseof3-neutrinomixingand NO neutrino mass spectrum exceeds the millielectronvolt value. Weconsiderboththegenericcase,wheretheMajoranaandDirac CPVphasesareunconstrained, aswellasa setofcases inwhich the CPV phases take particular values, motivated by predictive schemes combining generalised CP and flavour symmetries. Our study is a natural continuation and extension of the study per-formedin[14].

2. TheeffectiveMajoranamass

Taking the dominant contribution to the

(ββ)0

ν -decay rate,

0

ν , to be due to the exchange of three Majorana neutrinos

ν

i (mi

<

10 MeV; i

=

1

,

2

,

3), one can write the inverse of the de-cayhalf-life,

(

T₁0_/ν₂

)

−1

= 

0ν

/

ln 2,as



T0_1/ν₂



−1

=

G0ν

(

Q

,

Z

)



M0

ν

(

A

,

Z

)



2

|

m

|

2

,

(1) whereG0ν denotesthephase-spacefactor,whichdependsonthe Q -valueofthenucleartransition,and

M0

ν isthenuclearmatrix element (NME) ofthe decay. The former can be computedwith relativelygoodaccuracywhereas thelatterremains the predomi-nantsourceofuncertaintyintheextractionof

|

m

|

fromthedata (see,e.g., [4,15]).

TheeffectiveMajoranamass

|

m

|

isgivenby(see,e.g., [16]):

|

m

| =







3



i=1 U2_eimi







,

(2)

withU beingthePontecorvo–Maki–Nakagawa–Sakata(PMNS) lep-tonic mixing matrix. The first row of U is the one relevant for

(ββ)0

ν -decayandreads,inthestandardparametrization [1],

Uei

=



c12c13

,

s12c13eiα21/2

,

s13e−iδeiα31/2



i

.

(3)

Here,ci j

≡

cos

θ

i jandsi j

≡

sin

θ

i j,where

θ

i j

∈ [

0

,

π

/

2

]

arethe mix-ing angles, and

δ

and the

α

i j are the Dirac and Majorana CPV phases [2],respectively(

δ,

α

i j

∈ [

0

,

2

π

]

).

ThemoststringentupperlimitontheeffectiveMajoranamass wasreportedby theKamLAND-Zencollaboration.Usingthelower limit on the half-life of 136_{Xe obtained by the collaboration and} quoted in the Introduction, and taking into account the esti-mateduncertaintiesintheNMEsoftherelevantprocess,thelimit reads [9]:

|

m

| < (

0

.

061

−

0

.

165

)

eV

.

(4)

Neutrinooscillationdataprovidesinformationonmass-squared differences, but not on individual neutrino masses. The mass-squared difference



m2 responsible for solar

ν

e and very-long baseline reactor

ν

¯

e oscillations is much smaller than the mass-squared difference



m2

A responsible for atmospheric and ac-celerator

ν

μ and

ν

¯

μ and long baseline reactor

ν

¯

e oscillations,



m2

/

|

m2A

|

∼

1

/

30. At presentthe sign of



m2A cannot be de-terminedfrom the existing data.The two possible signs of



m2_A correspondto two typesof neutrinomass spectrum:



m2

A

>

0 – spectrum withnormalordering (NO),



m2_A

<

0 – spectrum with inverted ordering (IO). In a widely used convention we are also going to employ, the first corresponds to the lightest neutrino

Table 1

Rangesfortherelevantoscillationparametersinthecase ofanIOneutrinospectrum,at the3σ CL,taken from theglobalanalysisofRef. [13].AsinTable2, m2

A is

obtainedfromthequantitiesdefinedinRef. [13] using thebest-fitvalueofm2

21. m2 21 10−5_eV2 m2 23 10−3_eV2 sin2 θ12 10−1 sin2 θ13 10−2 6.92–7.91 2.38–2.58 2.64–3.45 1.95–2.43

being

ν

1, while the second corresponds to the lightest neu-trino being

ν

3. Combined with the fact that in this convention



m2

≡ 

m221

>

0 wehave:

•

m1

<

m2

<

m3,



m231

≡ 

m2A

>

0,forNO;and

•

m3

<

m1

<

m2,

−

m2₂₃

≡ 

m2_A

<

0,forIO, where



m2

i j

≡

m2i

−

m2j.For eitherordering,

|

m2A

|

=

max

(

|

m2i

−

m2_j

|)

, i

,

j

=

1

,

2

,

3.We alsodefine mmin

≡

m1

(

m3)inthe NO (IO) case. A NO or IO mass spectrum is additionally said to be nor-mal hierarchical(NH) orinverted hierarchical (IH) if respectively m1



m2,3 orm3



m1,2.Intheconverselimit ofrelativelylarge

mmin,mmin



0

.

1 eV,thespectrumis saidtobe quasi-degenerate (QD)andm1

m2

m3.Inthislastcase,thedistinctionbetween NOandIOspectraisblurredand



m 2 and

|

m2_A

|

canusuallybe neglectedwithrespecttom2

min.

In terms of the lightest neutrino mass, CPV phases, neutrino mixingangles,andneutrinomass-squareddifferences,theeffective Majoranamassreads:

|

m

|

NO

=





mminc122 c132

+





m2

+

m2_mins2₁₂c2₁₃eiα21

+





m2_A

+

m2_mins2₁₃eiα31



,

₍₅₎

|

m

|

IO

=







|

m2_A

| − 

m2

+

m2_minc₁₂2 c2₁₃

+



|

m2_A

| +

m2_mins2₁₂c2₁₃eiα21

+

_m mins213eiα  31



,

₍₆₎

wherewehavedefined

α

₃₁

≡

α

31

−

2

δ

.

Itprovesusefultorecast

|

m

|

NO and

|

m

|

IOgivenaboveinthe form

|

m

| =



_

m1

+ 

m2eiα21

+ 

m3eiα



31



 ,

₍₇₎

with



mi

>

0 (i

=

1

,

2

,

3).ItisthenclearthattheeffectiveMajorana massisthelength ofthevectorsumofthreevectorsinthe com-plexplane,whoserelativeorientationsaregivenbytheangles

α

21 and

α

₃₁.

For the IO case, taking into account the 3

σ

ranges of



m2₃₂,



m2₂₁, sin2

θ12

, and sin2

θ13

summarised in Table 1, one finds that there is a hierarchy between the lengths of the three vec-tors, m



3

<

0

.

1



m2 and



m2

<

0

.

6m



1, which holds for all values of mmin. In particular,



m3

=

mmins213 can be neglected with re-spect to the other terms since s₁₃2



cos 2

θ12

.2 The above im-plies that extremal values of

|

m

|

IO are obtained when the three vectors are aligned (

α

21

=

α

₃₁

=

0,

|

m

|

IO is maximal) or when m



1 is anti-aligned with m



2,3 (

α

21

=

α

31

=

π

,

|

m

|

IO is minimal). It then follows that there is a lower bound on

|

m

|

IO for every value ofmmin [10].This bound reads:

|

m

|

IO



2 _{Itfollowsfromthecurrentdatathatcos 2}

Table 2

Rangesfor therelevantoscillation parametersinthecaseofaNO neu-trinospectrum,atthenσ (n=1,2,3)CLs,takenfromtheglobalanalysis ofRef. [13] (cf. Table1).

Parameter 1σ range 2σrange 3σ range

m2 21/ (10−5eV2) 7.20–7.51 7.05–7.69 6.92–7.91 m2 31/ (10− 3_eV2 ) 2.46–2.53 2.43–2.56 2.39–2.59 sin2_θ12/10−1 _{2.91–3.18 2.78–3.32 2.65–3.46} sin2θ13/10−2 _{2.07–2.23 1.98–2.31 1.90–2.39}



|

m2 A

| +

m2minc213cos 2

θ12

>



|

m2 A

|

c213cos 2

θ12

,

|

m

|

IO

>

1

.

4

×

10−2 eV, forvariations of oscillationparameters intheir respec-tive 3

σ

ranges. In the limit of negligible mmin (IH spectrum),

m2_min

 |

m2_A

|

,onehas

|

m

|

IO

∈ [

1

.

4

,

4

.

9

]

×

10−2 eV.

Before proceedingtothe analysisofthe NO case, let us com-mentonpresentconstraintsonthe absoluteneutrinomassscale. The “conservative” upper limit of Eq. (4),

|

m

|

max_exp

=

0

.

165 eV, whichisintherangeoftheQDspectrum,implies,asitisnot dif-ficulttoshow,thefollowingupperlimitontheabsoluteMajorana neutrinomass scale (i.e.,on the lightest neutrinomass): mmin

m1,2,3

<

0

.

60 eV,withmmin

 |

m

|

maxexp

/(

cos 2

θ12

−

s213

)

,takinginto account the3

σ

ranges ofcos 2

θ12

andsin2

θ13

. Measurementsof theend-pointelectronspectrumintritiumbetadecayexperiments constrain the combinationmβ

≡



i

|

Uei

|

2mi. The most stringent upperboundsonmβ,mβ

<

2

.

1 eV andmβ

<

2

.

3 eV, bothatthe

95%CL,aregivenbytheTroitzk [17] andMainz [18] collaborations, respectively.TheKATRINexperiment [19] isplannedtoeither im-prove this bound by an order of magnitude, or discover mβ

>

0

.

35 eV.Takingintoaccountthe3

σ

rangesfortherelevantmixing anglesandmass-squareddifferences,theTroitzkboundconstrains thelightestneutrinomasstobemmin

<

2

.

1 eV.Cosmologicaland astrophysicaldataconstrain insteadthesum



≡



imi. Depend-ing onthelikelihood functionanddata setused,the upperlimit on



reportedbythePlanckcollaboration [20] variesinthe inter-val



<

[

0

.

34

,

0

.

72

]

eV,95%CL.Includingdataonbaryonacoustic oscillationslowersthisboundto



<

0

.

17 eV,95%CL.Takinginto account the3

σ

ranges forthemass-squared differences, thislast bound implies mmin

<

0

.

05

(

0

.

04

)

eV in the NO (IO) case. One shouldnotethat thePlanckcollaboration analysisisbasedonthe



CDM cosmological model.The quotedboundsmaynot applyin nonstandardcosmologicalscenarios(see,e.g., [21]).

3. Thecaseofnormalordering

WehenceforthrestrictourdiscussiontotheeffectiveMajorana mass

|

m

|

NO,forwhichthereisnolowerbound.Infact,unlikein theIOcase,heretheorderingofthelengthsofthem



idependson the value ofmmin and cancellations in

|

m

|

NO are possible: one risks“falling”insidethe“wellofunobservability”.3

We summarise in Table 2 then

σ

(n

=

1

,

2

,

3) ranges forthe oscillation parameters relevant to

(ββ)0

ν -decay in the NO case, obtainedintherecentglobalanalysisofRef. [13].Considering vari-ations ofoscillation parameters in the corresponding 3

σ

ranges, formmin

≤

5

×

10−2 eV thereisan upperbound

|

m

|

NO

≤

5

.

1

×

10−2 _{eV (obtained for}

_α

21

=

α

31

=

0). In the limit of negligible

mmin,m2min

 |

m2A

|

,onehas

|

m

|

NO

∈ [

0

.

9

,

4

.

2

]

×

10−3 eV. From inspection of Eqs. (5) and (7), the vector lengths ex-plicitly read m



1

=

mminc212c213,



m2

=





m2

+

m2_mins2₁₂c2₁₃, and

3 _{Fortheconsequencesofnotobserving}

(ββ)0ν-decaywith|m|NO10−3 eV,

see [22].

Fig. 1. Lengthsmi ofthecomplex vectorsenteringtheexpression of|m|NO as

afunctionofthelightestneutrinomassmmin,forNOspectrum.Forcomparison,

thesumsm1+ m3+ |m|0andm2+ m3+ |m|0arealsoshown(seetext),with

|m|0=10−3 eV.Bands areobtained byvaryingthe mixingangles and

mass-squareddifferencesintheirrespective3σranges(seeTable2).Seetextfordetails.

Fig. 2. RangesofmminforaNOspectrumandforoscillationparametersinsidetheir

nσ (n=1,2,3)intervals(seeTable2)forwhich:ingreen,a)|m|NO>|m|0=

10−3 _{eV forallvaluesof}

θi j, m2i j, andα

()

i j fromthe correspondingallowedor

definingintervals;ingrey,b) thereexistvaluesofθi j,m2i jfromthe1σ,2σ and

3σ allowedintervalsandvaluesofα()

i j suchthat|m|NO<|m|0=10−3eV;and

inred,c) forallvaluesofθi j andm2i j fromthecorrespondingallowedintervals

thereexistvaluesofthephasesα21andα31 forwhich|m|NO<|m|0=10−3eV.



m3

=





m2

A

+

m2mins213.InFig.1,theselengthsareplottedas func-tionsofmminfor3

σ

variationsofoscillationparameters.

The requirement ofhaving theeffective Majoranamass above a referencevalue

|

m

|

0 is geometrically equivalent to not being able to form a quadrilateral with sides m



1, m



2, m



3, and

|

m

|

0. This happens whenever one of the lengths exceeds the sum of theother three.Ifhowever

|

m

|

0

>



i



mi,itfollowsthat

|

m

| ≤



i



mi

<

|

m

|

0. Thus, forvalues ofmmin and oscillation parame-ters forwhich



m2

>

m



1

+ 

m3

+ |

m

|

0 or



m1

>

m



2

+ 

m3

+ |

m

|

0 (see Fig.1) one is guaranteed to have

|

m

|

NO

>

|

m

|

0 indepen-dentlyofthechoiceofCPVphases

α

21and

α

₃₁ .Thereareinstead valuesofmminforwhichtheconditions



m2

<

m



1

+ 

m3

+|

m

|

0and



m1

<

m



2

+ 

m3

+ |

m

|

0holdindependentlyofthevaluesof oscilla-tionparameterswithinagivenrange.Insuchacase,valuesof

α

21 and

α

₃₁suchthat

|

m

|

NO

<

|

m

|

0 aresuretoexist.

We summarise in Fig. 2 the ranges of mmin for which these differentconditionsapply(seecaption).Wevaryoscillation param-etersintheirrespectiven

σ

(n

=

1

,

2

,

3)intervalsandfocusonthe millielectronvolt “threshold”,

|

m

|

0

=

10−3 eV. We find that, for 3

σ

variations ofthe sin2

θ

i j and



m2i j,one isguaranteed tohave

Fig. 3. ThesameasinFig.2,butforthereferencevalue|m|0=5×10−3eV:in

green,a)|m|NO>|m|0=5×10−3eV forallvaluesofθi j,m2i j,andα

()

i j from

thecorrespondingallowedordefiningintervals;ingrey,b) thereexistvaluesof

θi j,m2i jfromthe1σ,2σ and3σ allowedintervalsandvaluesofα

()

i j suchthat

|m|NO<|m|0=5×10−3eV;andinred,c) forallvaluesofθi jandm2i jfrom

thecorrespondingallowedintervalsthereexistvaluesofthephasesα21andα31

forwhich|m|NO<|m|0=5×10−3 eV.Inthedarkergreyrangesd) ofmmin,

onehas|m|NO<|m|0=5×10−3eV independentlyofthevaluesofoscillation

parametersandCPVphases.

|

m

|

NO

>

10−3 eV ifmmin

>

1

.

10

×

10−2 eV.Thiscorresponds to thelower bound



>

0

.

07 eV onthesumofneutrinomasses.For 2

σ

variations,havingmmin

<

2

×

10−4eV ormmin

>

9

.

9

×

10−3eV isenoughtoensure

|

m

|

NO

>

10−3eV.

If one takes instead the higher value

|

m

|

0

=

5

×

10−3 eV and allows the relevant oscillation parameters to vary in their respective 3

σ

ranges,

|

m

|

NO

>

|

m

|

0 is guaranteed provided

mmin

>

2

.

3

×

10−2 eV, which corresponds to the lower bound



>

0

.

10 eV on the sum of neutrino masses. This lower bound on



practically coincides with min

()

in the case of IO spec-trum. Thus, if



is found to satisfy



>

0

.

10 eV, that would imply that

|

m

|

exceeds 5

×

10−3 eV, unless there exist addi-tionalcontributions to the

(ββ)0

ν -decay amplitude which cancel atleastpartially thecontribution dueto the3 light neutrinos.If insteadmmin

<

1

.

4

×

10−2 eV, forall (3

σ

allowed) values of os-cillation parameters there is a choice of

α

21 and

α

₃₁ such that

|

m

|

NO

<

|

m

|

0

=

5

×

10−3 eV. Theseresults are shown graphi-callyinFig.3.

Let us briefly remark on the dependence of

|

m

|

NO on the CPV phases.4 For the present discussion, 3

σ

variations of oscil-lationparameters areconsidered. Forallvalues of

α

₃₁ and

>

0 thereexistvaluesof

α

21andmminsuchthat

|

m

|

NO

<

,i.e. such that

|

m

|

NO isarbitrarily small.Thisisa consequenceofthefact that,foranyfixed oscillationparameters and

α

₃₁ ,thereisalways apointm∗_minatwhich

|

m1(m∗_min

)

+ 

m3(m∗_min

)

eiα₃₁

_|

_{= }

_m2(m∗

min

)

. In-stead,thereare valuesof

α

21 and

>

0 forwhich,independently of

α

31andmmin,onehas

|

m

|

NO

>

,i.e. forwhich

|

m

|

NOcannot bearbitrarily small.This conclusionmay beanticipated fromthe graphicalresults of Ref. [28], where the structure of the

|

m

|

NO “well”hasbeenstudiedasafunctionofmminand

α

21.Infact,we findthat for

α

21



0

.

81

π

or

α

21



1

.

19

π

,

|

m

|

NO cannotbezero attree-levelsince

|

m1

+ 

m2eiα21

|

>



m3,strictly.

In Fig. 4 we highlight the region of the

(

mmin

,

α

21) plane in which

|

m

|

NOisguaranteedtosatisfy

|

m

|

NO

>

5

×

10−3eV, inde-pendentlyof

α

₃₁andofvariationsofoscillationparametersinside their3

σ

ranges.

4. CPandgeneralisedCP

Giventhe strong dependence of

|

m

|

on

α

21 and

α

₃₁ , some principle which determines these phases are welcome. The

re-4 _{TheMajoranaphasesα}

21andα31 heredefinedarerelatedto

parameterisation-independentcharged-leptonrephasinginvariantproductsU_ei∗Ue j [23–27] through

α21=2arg U∗e1Ue2andα31=2arg U∗e1Ue3.

Fig. 4. Regions inthe (mmin,α21) plane wheredifferent conditions on |m|NO

apply. Inthe green(dark grey)region, |m|NO satisfies |m|NO>5×10−3 eV

(|m|NO<5×10−3eV)forallvaluesofθi j,m2i j,andα13fromthe

correspond-ing3σ ordefiningintervals.Intheredandgreyregions,conditionsanalogousto thosedescribedinthecaptionofFig.2applyandareindicated.Thisfigureistobe contrastedwithFig.3,wherethedependenceonα21isnotexplicit.

quirementofCPinvarianceconstrainsthevaluesoftheCPVphases

α

21,

α

31, and

δ

to integer multiples of

π

[29–31], meaning the relevantCP-conservingvaluesare

α

21,

α

31

=

0

,

π

.Non-trivial pre-dictions for the leptonic CPV phases mayinstead arise fromthe breaking ofadiscrete symmetrycombined withageneralised CP (gCP) symmetry. We focus on schemes with large enough resid-ualsymmetrysuchthatthePMNSmatrixdependsatmostonone real parameter

θ

[32] and realisations thereofwhere the predic-tions fortheCPV phasesareunambiguous, i.e. independentof

θ

. For symmetry groups with less than 100 elements, aside from the aforementioned CP-conserving values, the non-trivial values

α

21,

α

₃₁

=

π

/

2

,

3

π

/

2 arepossiblepredictions [33–39].

In what follows, we analyse the behaviour of

|

m

|

NO and

|

m

|

IO for each of 16 different

(

α

21,

α

31

)

pairs, with the rele-vantphasestakinggCP-compatiblevalues:

α

21,

α

₃₁

∈ {

0

,

π

/

2

,

π

,

3

π

/

2

}

.5 _{As can be seen from Eq. (7), some pairs are} redun-dant as they lead to the same values of

|

m

|

. We are left with 10 inequivalent pairs of phases:

(

α

21,

α

₃₁

)

= (

π

/

2

,

0

)

∼

(

3

π

/

2

,

0

)

,

(

π

/

2

,

π

)

∼ (

3

π

/

2

,

π

)

,

(

0

,

π

/

2

)

∼ (

0

,

3

π

/

2

)

,

(

π

,

π

/

2

)

∼

(

π

,

3

π

/

2

)

,

(

π

/

2

,

π

/

2

)

∼ (

3

π

/

2

,

3

π

/

2

)

, and

(

π

/

2

,

3

π

/

2

)

∼

(

3

π

/

2

,

π

/

2

)

.

The 2

σ

-allowed values of the effective Majorana mass

|

m

|

are presented in Fig. 5 as a function of mmin, for both order-ings. Regionscorrespondingtodifferentpairs

(

α

21,

α

₃₁

)

with CP-conserving phases,

α

21,

α

₃₁

=

0

,

π

, are singled out. The predic-tions for the remaining pairs of fixed phases, containing at least one phase which is gCP-compatible but not CP-conserving, are shown inFig. 6 forIO and inFigs. 7 and 8for NO (for one CP-conserving phase and for no CP-conservingphases, respectively). Allowedvaluesof

|

m

|

arefoundbyconstructinganapproximate

χ

2 _{function fromthe sumof theone-dimensional projections in} Ref. [13],andvaryingmixinganglesandmass-squareddifferences

5 _{Givenourscopeandtheavailableliterature,wefindthatifα}

31=π/2,3π/2,

thennecessarilyα21=π/2,3π/2 ispredicted.Weneverthelesstakeall16pairsof

Fig. 5. TheeffectiveMajoranamass|m|asafunctionofmmin,forbothorderings,allowingforvariationsofmixinganglesandmass-squareddifferencesatthe2σ CL(see

text).Thephasesα21andα31=α31−2δarevariedintheinterval[0,2π].Blueandgreenbandscorrespondto(theindicated,withk=0,1)CP-conservingvaluesofthe

phases(α21,α31 ),forIOandNOneutrinomassspectra,respectively,whileinredregionsatleastoneofthephasestakesaCP-violatingvalue.Bluehatchingisusedto

locateCP-conservingbandsinthecaseofIOspectrumwheneverIOandNOspectraregionsoverlap.TheKamLAND-ZenboundofEq. (4) isindicated.Seealso[1,14].

Fig. 6. TheeffectiveMajoranamass|m|asafunctionofmmin,forIOspectrum,

allowingforvariationsofmixinganglesandmass-squareddifferencesatthe2σCL (seetext).Yellowbandscorrespondto(theindicated,k=0,1,2,3)gCP-compatible butnotCP-conservingvaluesofthephases(α21,α31 ).Bluebandscorrespondto

CP-conservingphases(seeFig.5)andhatchingindicatesoverlapwithsuchregions, whileredregionsarenotgCP-compatible(forthemodelsunderconsideration,see text).TheKamLAND-ZenboundofEq. (4) isalsoindicated.

whilekeeping

χ

2

_(θ

i j

,



m2i j

)



9

.

72 (2

σ

CL,forjointestimationof 4parameters).

FromFigs.5–8,oneseesthatforeachvalueofmminthereexist valuesoftheeffectiveMajoranamasswhichareincompatiblewith CP conservation.Some of thesepoints maynonetheless be com-patible withgCP-based predictive models. ForIO, one sees there issubstantial overlapbetweenthe bandswith

(

α

21,

α

₃₁

)

= (

0

,

0

)

and

(

0

,

π

)

,betweenthoseof

(

π

,

0

)

and

(

π

,

π

)

,andbetweenthe four bands

(

π

/

2

,

k

π

/

2

)

, with k

=

0

,

1

,

2

,

3. In the case of NO, it isinteresting to note that, fora fixed, gCP-compatible butnot

Table 3

Lowerboundson|m|NOgivenatthe3σ (2σ)CL,whereapplicable,fordifferent

fixedvaluesofthephasesα21andα31.A tildedenotesequivalencebetweencases.

AllboundsaregiveninmeV.

α21 α31 0 π/2 π 3π/2 0 3.1(3.3) 2.4(2.4) 1.0(1.1) ∼ (0,π/2) π/2 2.4(2.4) 3.1(3.3) 2.1(2.2) ∼ (3π/2,π/2) π no bounda ₀ .91(0.95)b no boundc _{∼ (π} ,π/2) 3π/2 ∼ (π/2,0) 1.0(1.1) ∼ (π/2,π) ∼ (π/2,π/2) a _|m|

NO>1 meV ifmmin>5.8(mmin∈ [/ 0.1,5.3])meV. b _{Onlyboundedcasewhere|m|}

NOisnotstrictlyatorabovethemeVvalue,for

mmin∈ [2.9,5.9]([3.2,5.3])meV.|m|NO>1 meV ife.g. sin2θ13>2.04×10−2. c _|m|

NO>1 meV ifmmin∈ [/ 3.1,11.5]([3.4,10.6])meV.

CP-conserving pair

(

α

21,

α

31

)

,

|

m

|

NO isbounded frombelowat the 2

σ

CL, with the lower bound at or above the meV value,

|

m

|

NO



10−3 eV.WecollectinTable3informationonthelower boundon

|

m

|

NO foreachpairofphases.

5. Conclusions

The observationof

(ββ)0

ν -decaywouldallow toestablish lep-tonnumberviolationandtheMajorananatureofneutrinos.Inthe standardscenarioofthreelightneutrinoexchangedominance,the rate ofthis process is controlled by the effectiveMajorana mass

|

m

|

. In the case of neutrino mass spectrum with inverted or-dering (IO) the effective Majorana mass is bounded from below,

|

m

|

IO

>

1

.

4

×

10−2 eV,wherethislowerboundisobtainedusing the current 3

σ

allowed ranges of the relevant neutrino oscilla-tionparameters –thesolarandreactorneutrinomixingangles

θ12

and

θ13

,andthetwoneutrinomass-squareddifferences



m2

21and



m2₂₃.IntheNO case,theeffectiveMajoranamass

|

m

|

NO,under certain conditions,can be exceedingly small,

|

m

|

NO



10−2 eV, suppressingthe

(ββ)0

ν -decayrate.

Currentlytakingdataandnext-generation

(ββ)0

ν -decay experi-mentsseektoprobeandpossiblycovertheIOregionofparameter space,workingtowardsthe

|

m

| ∼

10−2 _{eV frontier.Incasethese}

Fig. 7. TheeffectiveMajorana mass|m|asafunctionofmmin,forNOspectrum,allowingfor variationsofmixinganglesandmass-squareddifferencesat the2σ CL

(seetext).Yellowbandscorrespondto(theindicated)pairs(α21,α31 )ofphases,withoneCP-conserving,theotherbeing gCP-compatiblebutnotCP-conserving.Green

bandscorrespondtoCP-conservingphases(seeFig.5)andhatchingindicatesoverlapwithsuchregions,whileredregionsarenotgCP-compatible(forthemodelsunder consideration,seetext).

Fig. 8. The same as in Fig.7, with yellow bands corresponding to pairs(α21,α31 )with both phases being gCP-compatible but not CP-conserving.

searchesproducea negative result, thenext frontier inthequest for

(ββ)0

ν -decaywillcorrespondto

|

m

| ∼

10−3eV.

Takinginto account updated global-fit data on the3-neutrino mixinganglesandtheneutrinomass-squareddifferences,wehave determined the conditions under which the effective Majorana massintheNO case

|

m

|

NO exceeds the10−3 eV (5

×

10−3 eV) value.The effective Majoranamass

|

m

|

NO ofinterest, asis well known, depends on the solar and reactor neutrino mixing an-gles

θ12

and

θ13

, on the two neutrino mass-squared differences



m2₂₁ and



m2₃₁, on the lightest neutrino mass mmin as well as on the CPV Majorana phase

α

21 and on the Majorana–Dirac phasedifference

α

₃₁

=

α

31

−

2

δ

. Forvariations of

θ12

,

θ13

,



m221

and



m2₃₁in theirn

σ

(n

=

1

,

2

,

3) intervals, wehavedetermined the ranges of the lightest neutrino mass mmin such that (see Figs.2–4):

a)

|

m

|

NO

>

10−3

(

5

×

10−3

)

eV independentlyofthe valuesof

α

21 and

α

₃₁ ;

|

m

|

NO

>

5

×

10−3 eV isfulfilledwhenmmin

>

2

.

3

×

10−2eV (for3

σ

variations),

b) for some values of the

θ

i j and



m2i j there are choices of the CPV phases

α

21 and

α

31 such that

|

m

|

NO

<

10−3

(

5

×

10−3

₎

_eV,

c) forallvaluesofthe

θ

i j and



m2i j therearechoicesoftheCPV phases

α

21 and

α

31 such that

|

m

|

NO

<

10−3

(

5

×

10−3

)

eV, and

d)

|

m

|

NO

<

5

×

10−3 eV independently of the values of

α

21 and

α

₃₁ .

We have shown, in particular, that if the sumof the three neu-trinomassesisfoundtosatisfythelowerbound



>

0

.

10 eV,that wouldimplyinthecaseofNOneutrinomassspectrum

|

m

|

NO

>

5

×

10−3 eV foranyvaluesoftheCPVphases

α

21and

α

31,unless thereexistadditionalcontributionstothe

(ββ)0

ν -decayamplitude whichcancelatleastpartially thecontributionduetothe3light neutrinos.

We have additionally studied the predictions for

|

m

|

IO and

|

m

|

NO incases wheretheleptonic CPV phasesare fixedto par-ticularvalues,

α

21,

α

31

−

2

δ

∈ {

0

,

π

/

2

,

π

,

3

π

/

2

}

,whichareeither CPconserving(seeFig.5)ormayariseinpredictiveschemes com-bininggeneralisedCPandflavoursymmetries(seeFigs.6–8,lower boundson the effectivemass

|

m

|

NO for such choicesof phases aregiveninTable3).We findthat

|

m

|

NO



10−3 eV forall gCP-compatiblebutnotCP-conservingpairsoftherelevantphases.

The searches for lepton number non-conservation performed by the neutrinoless double beta decay experiments are part of thesearches fornew physics beyondthat predictedby the Stan-dard Theory. They are of fundamental importance – as impor-tant as the searches for baryon number non-conservation in the form of, e.g., proton decay. Therefore if current and next-generation

(ββ)0

ν -decay experiments seekingtoprobethe IO re-gion ofparameter space produce a negative result, the quest for

(ββ)0

ν -decay should continue towards the

|

m

| ∼

5

×

10−3 eV andpossiblythe

|

m

| ∼

10−3 eV frontier.

Acknowledgements

We would like to thank F. Capozzi, E. Lisi, A. Marrone and A. Palazzoforkindlysharingtheirdatafileswithone-dimensional

χ

2 _{projections.ThisworkwassupportedinpartbytheINFN} pro-gramonTheoreticalAstroparticlePhysics(TASP),bytheEuropean Union Horizon 2020 research and innovation programme under theMarieSklodowska-Curiegrants 674896and690575 (J.T.P. and S.T.P.),andbytheWorldPremierInternationalResearchCenter Ini-tiative(WPIInitiative),MEXT,Japan(S.T.P.).

References

[1] C.Patrignani,etal.,ParticleDataGroup,Reviewofparticlephysics,Chin.Phys. C 40 (10) (2016) 100001, https://doi.org/10.1088/1674-1137/40/10/100001, and2017update.

[2] S.M.Bilenky,J.Hosek,S.T.Petcov,OnoscillationsofneutrinoswithDiracand Majorana masses, Phys. Lett.B 94 (1980)495–498, https://doi.org/10.1016/ 0370-2693(80)90927-2.

[3] P.Langacker,S.T. Petcov, G.Steigman,S. Toshev,Onthe Mikheev–Smirnov– Wolfenstein (MSW) mechanism of amplification ofneutrino oscillations in matter, Nucl. Phys. B 282 (1987) 589–609, https://doi.org/10.1016/0550 -3213(87)90699-7.

[4] J.D.Vergados,H.Ejiri,F.Šimkovic,Neutrinolessdoublebetadecayand neu-trinomass,Int.J.Mod.Phys.E25 (11)(2016)1630007,https://doi.org/10.1142/ S0218301316300071,arXiv:1612.02924.

[5] S. Dell’Oro,S.Marcocci,M.Viel, F.Vissani, Neutrinolessdoublebetadecay: 2015review,Adv.HighEnergyPhys.2016(2016)2162659,https://doi.org/10. 1155/2016/2162659,arXiv:1601.07512.

[6] A.S. Barabash, Double beta decay: historical review of 75 years of research, Phys. At. Nucl. 74 (2011) 603–613, https://doi.org/10.1134/ S1063778811030070,arXiv:1104.2714.

[7] M.Agostini,etal.,Improvedlimitonneutrinolessdouble-βdecayof76_Gefrom

GERDAphaseII,Phys. Rev.Lett. 120 (13)(2018)132503,https://doi.org/10. 1103/PhysRevLett.120.132503,arXiv:1803.11100.

[8] C.Alduino,etal.,FirstresultsfromCUORE:asearchforleptonnumber viola-tionvia0νββdecayof130_{Te,Phys.Rev.Lett.120 (13)(2018)132501,https://}

doi.org/10.1103/PhysRevLett.120.132501,arXiv:1710.07988.

[9] A.Gando,et al.,Searchfor Majorana neutrinosnearthe invertedmass hi-erarchyregionwith KamLAND-Zen,Phys. Rev.Lett.117 (8) (2016) 082503, https://doi.org/10.1103/PhysRevLett.117.109903,arXiv:1605.02889;Addendum: Phys.Rev.Lett.117 (10)(2016)109903,https://doi.org/10.1103/PhysRevLett. 117.082503.

[10] S.Pascoli,S.T.Petcov, TheSNOsolarneutrinodata,neutrinolessdoublebeta decayandneutrinomassspectrum,Phys.Lett.B544(2002)239–250,https:// doi.org/10.1016/S0370-2693(02)02510-8,arXiv:hep-ph/0205022.

[11] F.Capozzi,E.DiValentino,E.Lisi,A.Marrone,A.Melchiorri,A.Palazzo,Global constraintsonabsoluteneutrinomassesandtheirordering,Phys.Rev.D95 (9) (2017)096014,https://doi.org/10.1103/PhysRevD.95.096014,arXiv:1703.04471. [12] I.Esteban,M.C.Gonzalez-Garcia,M.Maltoni,I.Martinez-Soler,T.Schwetz, Up-datedfittothreeneutrinomixing:exploringtheaccelerator–reactor comple-mentarity,J.HighEnergyPhys.01(2017)087;andNuFIT3.2,www.nu-fit.org, arXiv:1611.01514,https://doi.org/10.1007/JHEP01(2017)087,2018.

[13]F.Capozzi,E.Lisi,A.Marrone,A.Palazzo,Currentunknownsinthethree neu-trinoframework,arXiv:1804.09678.

[14] S.Pascoli, S.T.Petcov, Majorananeutrinos,neutrinomassspectrumand the |m|∼10−3 _{eV frontierinneutrinoless doublebetadecay,Phys.Rev.D77}

(2008)113003,https://doi.org/10.1103/PhysRevD.77.113003,arXiv:0711.4993. [15]F.Iachello,J.Kotila,J.Barea,QuenchingofgA anditsimpactindoublebeta

decay,PoSNEUTEL2015(2015)047.

[16] S.M.Bilenky,S.T.Petcov,Massiveneutrinosandneutrinooscillations,Rev.Mod. Phys.59(1987)671,https://doi.org/10.1103/RevModPhys.59.671;Erratum:Rev. Mod.Phys.61(1989)169;Erratum:Rev.Mod.Phys.60(1988)575. [17] V.N.Aseev,etal.,AnupperlimitonelectronantineutrinomassfromTroitsk

experiment,Phys.Rev.D84(2011)112003,https://doi.org/10.1103/PhysRevD. 84.112003,arXiv:1108.5034.

[18] C.Kraus,etal.,FinalresultsfromphaseIIoftheMainzneutrinomasssearchin tritiumbetadecay,Eur.Phys.J.C40(2005)447–468,https://doi.org/10.1140/ epjc/s2005-02139-7,arXiv:hep-ex/0412056.

[19] K. Eitel,Direct neutrinomass experiments, Nucl. Phys. B, Proc. Suppl.143 (2005)197–204,https://doi.org/10.1016/j.nuclphysbps.2005.01.105.

[20] N.Aghanim,etal.,Planckintermediateresults.XLVI.Reductionoflarge-scale systematiceffectsinHFIpolarizationmapsand estimationofthe reioniza-tionopticaldepth,Astron.Astrophys.596(2016)A107,https://doi.org/10.1051/ 0004-6361/201628890,arXiv:1605.02985.

[21] S.M.Koksbang,S.Hannestad,Constrainingdynamicalneutrinomass genera-tionwithcosmologicaldata,J.Cosmol.Astropart.Phys.1709 (09)(2017)014, https://doi.org/10.1088/1475-7516/2017/09/014,arXiv:1707.02579.

[22] S.-F. Ge,M.Lindner, ExtractingMajoranaproperties fromstrongbounds on neutrinolessdoublebeta decay,Phys.Rev.D95 (3)(2017) 033003,https:// doi.org/10.1103/PhysRevD.95.033003,arXiv:1608.01618.

[23] J.F.Nieves,P.B.Pal,MinimalrephasinginvariantCPviolatingparameterswith DiracandMajoranafermions,Phys.Rev.D36(1987)315,https://doi.org/10. 1103/PhysRevD.36.315.

[24] P.J.O’Donnell,U.Sarkar,NeutrinolessdoublebetadecayandCPviolation,Phys. Rev.D52(1995)1720–1721,https://doi.org/10.1103/PhysRevD.52.1720,arXiv: hep-ph/9305338.

[25] J.A.Aguilar-Saavedra,G.C.Branco,Unitaritytrianglesandgeometrical descrip-tionofCPviolationwithMajorananeutrinos,Phys.Rev.D62(2000)096009, https://doi.org/10.1103/PhysRevD.62.096009,arXiv:hep-ph/0007025. [26] S.M.Bilenky,S.Pascoli,S.T.Petcov, Majorananeutrinos,neutrinomass

spec-trum,CPviolationandneutrinolessdoublebetadecay.1.Thethreeneutrino mixingcase,Phys.Rev.D64(2001)053010,https://doi.org/10.1103/PhysRevD. 64.053010,arXiv:hep-ph/0102265.

[27] J.F.Nieves,P.B.Pal,RephasinginvariantCPviolatingparameterswithMajorana neutrinos,Phys.Rev.D64(2001)076005,https://doi.org/10.1103/PhysRevD.64. 076005,arXiv:hep-ph/0105305.

[28] Z.-z.Xing,Z.-h.Zhao,Theeffectiveneutrinomassofneutrinolessdouble-beta decays:howpossibletofallinto awell, Eur.Phys.J. C77 (3)(2017)192, https://doi.org/10.1140/epjc/s10052-017-4777-x,arXiv:1612.08538.

[29] L. Wolfenstein,CPproperties ofMajorana neutrinosand doublebetadecay, Phys.Lett.B107(1981)77–79,https://doi.org/10.1016/0370-2693(81)91151-5. [30] B.Kayser,CPT,CPandcphasesandtheireffectsinMajoranaparticleprocesses,

Phys.Rev.D30(1984)1023,https://doi.org/10.1103/PhysRevD.30.1023. [31] S.M.Bilenky,N.P.Nedelcheva,S.T.Petcov,SomeimplicationsoftheCP

invari-anceformixingofMajorananeutrinos,Nucl.Phys.B247(1984)61–69,https:// doi.org/10.1016/0550-3213(84)90372-9.

[32] F.Feruglio,C.Hagedorn, R.Ziegler,Leptonmixingparametersfrom discrete andCPsymmetries,J.HighEnergyPhys.07(2013)027,https://doi.org/10.1007/ JHEP07(2013)027,arXiv:1211.5560.

[33] G.-J.Ding,Y.-L.Zhou,Predictingleptonflavormixingfrom(48)and gener-alizedC P symmetries,Chin.Phys.C39 (2)(2015)021001,https://doi.org/10. 1088/1674-1137/39/2/021001,arXiv:1312.5222.

[34] G.-J.Ding,Y.-L.Zhou,Leptonmixingparametersfrom(48)familysymmetry andgeneralisedCP,J.HighEnergyPhys.06(2014)023,https://doi.org/10.1007/ JHEP06(2014)023,arXiv:1404.0592.

[35] S.F. King, T. Neder, Lepton mixing predictions including Majorana phases from(6n2_{)flavoursymmetryandgeneralisedCP,Phys.Lett.B736(2014)}

308–316,https://doi.org/10.1016/j.physletb.2014.07.043,arXiv:1403.1758. [36] G.-J.Ding,S.F.King,GeneralizedC P and(96)familysymmetry,Phys.Rev.D

89 (9)(2014)093020,https://doi.org/10.1103/PhysRevD.89.093020,arXiv:1403. 5846.

[37] C.Hagedorn,A.Meroni,E.Molinaro,Leptonmixingfrom(3n2_)and(6n2₎

andCP,Nucl.Phys.B891(2015)499–557,https://doi.org/10.1016/j.nuclphysb. 2014.12.013,arXiv:1408.7118.

[38] G.-J.Ding,S.F.King,T.Neder,GeneralisedCPand(6n2_{)familysymmetryin}

semi-directmodelsofleptons,J.HighEnergyPhys.12(2014)007,https://doi. org/10.1007/JHEP12(2014)007,arXiv:1409.8005.

[39] G.-J. Ding,S.F.King,GeneralizedCPand (3n2_{)familysymmetryfor}

semi-directpredictionsofthePMNSmatrix,Phys.Rev.D93(2016)025013,https:// doi.org/10.1103/PhysRevD.93.025013,arXiv:1510.03188.