Trimaximal neutrino mixing from modular A 4 invariance with residual symmetries

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Trimaximal

neutrino

mixing

from

modular

A

₄

invariance

with

residual

symmetries

P.P. Novichkov

a

,

S.T. Petcov

a

,

b

,

∗

,

1

,

M. Tanimoto

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

b_{KavliIPMU(WIP),UniversityofTokyo,5-1-5Kashiwanoha,277-8583Kashiwa,Japan} c_{DepartmentofPhysics,NiigataUniversity,Niigata950-2181,Japan}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received23January2019

Receivedinrevisedform14April2019 Accepted16April2019

Availableonline24April2019 Editor:G.F.Giudice

We construct phenomenologically viablemodels of leptonmasses and mixing basedon modular A4

invariance broken to residual symmetries ZT

3 or Z3S T and ZS2 respectively in the charged lepton

and neutrino sectors. In these models the neutrino mixing matrix is of trimaximal mixing form. In addition to successfullydescribing thecharged leptonmasses, neutrino mass-squareddifferencesand theatmosphericandreactorneutrinomixinganglesθ23 andθ13,thesemodelspredictthevaluesofthe

lightestneutrino mass(i.e., theabsoluteneutrinomassscale), ofthe Diracand MajoranaCPviolation (CPV)phases,aswellastheexistenceofspecificcorrelationsbetweeni)thevaluesofthesolarneutrino mixingangleθ12 andtheangleθ13(whichdeterminesθ12),ii)thevaluesoftheDiracCPVphaseδand

oftheanglesθ23andθ13,iii)thesumoftheneutrinomassesandθ23,iv)theneutrinolessdoublebeta

decayeffectiveMajoranamassandθ23,andv)betweenthetwoMajoranaphases.

©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Understandingtheoriginoftheflavourstructureofquarksandleptons remainsoneoftheoutstandingproblemsinparticlephysics. Thepatternoftwolargeandonesmallneutrino(lepton)mixingangles,revealedbythedataobtainedinneutrinooscillationexperiments (see,e.g.,[1]),providesanimportantclueintheinvestigationsoftheleptonflavourproblem,suggestingtheexistenceofflavoursymmetry intheleptonsector. Theresults ofthe recentglobalanalyses oftheneutrinooscillationdatashow alsothat aneutrinomassspectrum withnormalordering(NO)isfavouredoverthespectrumwithinvertedordering(IO),aswellasapreferenceforavalueoftheDiracCP violation(CPV)phase

δ

closeto3

π

/2 (see,

e.g., [2]).

Theobserved3-neutrinomixingpatterncannaturally be explainedbyextending theStandardTheory(ST)withaflavour symmetry associatedwitha non-Abeliandiscrete symmetry group. Modelsbased on S3, A4, S4, A5 andother groups oflarger ordershave been

proposed and extensively studied (see, e.g., [3–9]). In particular, the A4 flavour model attracted considerable interest because the A4

groupistheminimaloneincludingatripletunitaryirreduciblerepresentation,whichallowsforanaturalexplanationoftheexistenceof threefamilies ofleptons[10–15].In allmodels basedonnon-Abeliandiscrete flavoursymmetry, theflavour symmetrymustbe broken inordertoreproducethemeasuredvaluesoftheneutrinomixingangles.ThisisachievedbyintroducingtypicallyalargenumberofST gaugesingletscalars- theso-called“flavons”- intheLagrangianofthetheory,whichhavetodevelopasetofparticularlyalignedvacuum expectationvalues(VEVs).Arrangingforsuchanalignmentrequirestheconstructionofratherelaboratescalarpotentials.

Anattractiveapproachtotheleptonflavourproblem,basedontheinvarianceunderthemodulargroup,hasbeenproposedinRef. [16], wherealsomodelsofthefinitemodulargroup



3



A4havebeenpresented.AlthoughthemodelsconstructedinRef. [16] arenotrealistic

andmakeuseofaminimalsetofflavonfields,thisworkinspiredfurtherstudiesofthemodularinvarianceapproachtotheleptonflavour problem.Themodulargroup includesS3, A4,S4,and A5 asitsprincipalcongruencesubgroups,



2



S3,



3



A4,



4



S4 and



5



A5

*

Correspondingauthor.

E-mailaddresses:pavel.novichkov@sissa.it(P.P. Novichkov),petcov@sissa.it(S.T. Petcov),tanimoto@muse.sc.niigata-u.ac.jp(M. Tanimoto). 1 _{AlsoatInstituteofNuclearResearchandNuclearEnergy,BulgarianAcademyofSciences,1784Sofia,Bulgaria.}

https://doi.org/10.1016/j.physletb.2019.04.043

0370-2693/©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

[17].However,thereisasignificant differencebetweenthemodelsbasedonthemodular S3, A4, S4 etc.symmetryandthosebasedon

theusualnon-Abeliandiscrete S3, A4, S4 etc.flavour symmetry.The constantsofatheory basedonthefinitemodularsymmetry, such

asYukawacouplingsand, e.g.,the right-handedneutrinomassmatrix intype Iseesawscenario, alsotransformnon-triviallyunderthe modularsymmetryandarewrittenintermsofmodularformswhichareholomorphicfunctionsofacomplexscalarfield- themodulus

τ

. Atthesametimethemodularformstransformundertheusualnon-Abeliandiscreteflavoursymmetries.Inthemosteconomicalversions ofthemodelswithmodularsymmetry, theVEVofthe modulus

τ

istheonlysourceofsymmetry breakingwithouttheneedofflavon fields.

InRef. [18] a realistic modelwithmodular



2



S3 symmetrywas built withthehelp ofa minimal setof flavonfields.Arealistic

model ofthe chargedlepton and neutrino massesand of neutrinomixing without flavons, in which the modular



4



S4 symmetry

was used,was constructedin [19]. Subsequently,lepton flavour modelswithoutflavons,based onthemodular symmetry



3



A4 was

proposedinRefs. [20,21].Acomprehensiveinvestigationofthesimplestviablemodelsofleptonmassesandmixing,basedonthemodular

S4 symmetry, was performed inRef. [22]. Necessary ingredients for constructing flavour models based, in particular, on the modular

symmetries

(96)

and

(384)

havebeenobtainedin[23],whileformodelsbasedon A5 symmetrytheyhavebeenderivedin[24].

Ifoneofthe subgroupsofthe consideredfinitemodulargroup ispreserved, thisresidualsymmetry fixes

τ

to aspecific value (see, e.g.,[22]).Phenomenologicallyviable modelsbasedon themodular S4 and A5 symmetries,brokenrespectively toresidual

Z

3 and

Z

5

symmetriesinthechargedleptonsectorandtoa

Z

2 symmetryintheneutrinosector,werepresentedinRefs. [22,24].Sofar,apartfrom

thesetwostudies,theimplicationsofresidualsymmetrieshavebeeninvestigatedonlyintheframeworkoftheusualnon-Abeliandiscrete symmetryapproachtothelepton(andquark)flavourproblem.Ithasbeenshownthatthey lead,inparticular,tospecificexperimentally testablecorrelationsbetweenthevaluesofsomeoftheneutrinomixinganglesand/orbetweenthevaluesoftheneutrinomixingangles andoftheDiracCPviolationphaseintheneutrinomixing[25–31].

In the present articlewe construct phenomenologically viable models of lepton masses and mixingbased on residual symmetries resultingfromthe breakingof the A4 modularsymmetry. It is foundthat the weight4 modularforms are requiredto obtaincharged

lepton andneutrinomassmatricesleading to lepton massesandmixingwhich areconsistent withthe experimental dataon neutrino oscillations.WealsofindthatinthesemodelsthePMNSmatrix[32–34] ispredictedtobeofthetrimaximalmixingform[35,36].

Thepaperisorganizedasfollows.Insection2,wegiveabriefreviewonthemodularsymmetry.Insection3,wediscusstheresidual symmetriesof A4 andtheirmodularforms.Insection4,wepresenttheleptonmassmatricesintheresidualsymmetry.Insection5,we

presentmodelsandtheirnumericalresults.Section6isdevotedtoasummary.AppendixAshowstherelevantmultiplicationrulesofthe

A4 group.

2. ModularA4groupandmodularformsoflevel3

Themodulargroup



isthegroupoflinearfractionaltransformations

γ

actingonthecomplexvariable

τ

belongingtotheupper-half complexplaneasfollows:

γ τ

=

a

τ

+

b

c

τ

+

d

,

where a

,

b

,

c

,

d

∈ Z

and ad

−

bc

=

1

,

Im

τ

>

0

.

(2.1)

Thegroup



isgeneratedbytwotransformations S andT satisfying

S2

= (

S T

)

3

=

I

,

(2.2)

whereI istheidentityelement.Representing S andT as

S

=



0 1

−

1 0



,

T

=



1 1 0 1



,

(2.3) onefinds

τ

−

→ −

S 1

τ

,

τ

T

−→

τ

+

1

.

(2.4)

The modulargroup



is isomorphic to the projectivespecial linear group P S L

(2,

Z)

=

S L

(2,

Z)/Z

2,where S L

(2,

Z)

isthe special

lineargroup of2

×

2 matriceswithintegerelementsandunitdeterminant,and

Z

2

= {

I

,

−

I

}

isitscentre(I beingtheidentityelement).

Thespeciallineargroup S L

(2,

Z)

= (

1)

≡ 

containsaseriesofinfinitenormalsubgroups

(

N

),

N

=

1,2,3,

. . .

:

(

N

)

=



a b c d



∈

S L

(

2

,

Z) ,



a b c d



=



1 0 0 1



(

mod N

)



,

(2.5)

called the principal congruence subgroups. For N

=

1 and 2, we define the groups

(

N

)

≡ (

N

)/

{

I

,

−

I

}

with

(1)

≡ 

). For N

>

2,

(

N

)

≡ (

N

)

since

(

N

)

doesnotcontainthesubgroup

{

I

,

−

I

}

.ForeachN,theassociatedlinearfractionaltransformationsoftheformin eq. (2.1) areinaone-to-onecorrespondencewiththeelementsof

(

N

).

Thequotientgroups



N

≡ /(

N

)

arecalledfinite modulargroups. ForN

≤

5, thesegroupsareisomorphic tonon-Abelian discrete

groupswidelyusedinflavourmodelbuilding(see,e.g.,[17]):



2



S3,



3



A4,



4



S4 and



5



A5.Wewillbeinterestedinthefinite

modulargroup



3



A4.

Modularformsofweightk andlevel N areholomorphicfunctions f

(

τ

)

transformingundertheactionof

(

N

)

inthefollowingway:

Herek isevenandnon-negative, andN isnatural.Modularforms ofweightk and level N spana linearspaceoffinitedimension.The dimensionofthelinearspaceofmodularformsofweightk andlevel3,

M

k

(

3



A4

),

isk

+

1.There existsabasis inthisspacesuch

thatamultipletofmodularforms fi

(

τ

)

transformsaccordingtoaunitaryrepresentation

ρ

ofthefinitegroup



N:

fi

(

γ τ

)

= (

c

τ

+

d

)

k

ρ

(

γ

)

i j fj(

τ

) ,

γ

∈  .

(2.7)

InthecaseofN

=

3 ofinterest,thethreelinearindependentweight2modularformsformatriplet of A4 [16].Theseformshavebeen

explicitlyobtained[16] intermsoftheDedekindeta-function

η

(

τ

):

η

(

τ

)

=

q1/24 ∞



n=1

(

1

−

qn

) ,

(2.8)

whereq

=

e2πi_{τ .Inwhatfollowswewillusethefollowingbasisofthe A}

4 generators S andT inthetripletrepresentation:

S

=

1 3

⎛

⎝

−

21

−1

2 22 2 2

−

1

⎞

⎠ ,

T

=

⎛

⎝

10

ω

0 00 0 0

ω

2

⎞

⎠ ,

(2.9)

where

ω

=

ei23π . _{Themodularforms}

₍

_Y(2) 1

,

Y

(2) 2

,

Y

(2)

3

)

transformingasatriplet of A4 can bewritteninterms of

η

(

τ

)

anditsderivative

[16]: Y₁(2)

(

τ

)

=

i 2

π



η

(

τ

/

3

)

η

(

τ

/

3

)

+

η

((

τ

+

1

)/

3

)

η

((

τ

+

1

)/

3

)

+

η

((

τ

+

2

)/

3

)

η

((

τ

+

2

)/

3

)

−

27

η

(

3

τ

)

η

(

3

τ

)



,

Y₂(2)

(

τ

)

=

−

i

π



η

(

τ

/

3

)

η

(

τ

/

3

)

+

ω

2

η

((

τ

+

1

)/

3

)

η

((

τ

+

1

)/

3

)

+

ω

η

((

τ

+

2

)/

3

)

η

((

τ

+

2

)/

3

)



,

(2.10) Y₃(2)

(

τ

)

=

−

i

π



η

(

τ

/

3

)

η

(

τ

/

3

)

+

ω

η

((

τ

+

1

)/

3

)

η

((

τ

+

1

)/

3

)

+

ω

2

η

((

τ

+

2

)/

3

)

η

((

τ

+

2

)/

3

)



.

Theoverall coefficientineq. (2.10) isone possiblechoice; itcannot be uniquelydetermined.The tripletmodularforms Y₁(2_,2)_,3 havethe followingq-expansions: Y(₃2)

=

⎛

⎜

⎝

Y₁(2)

(

τ

)

Y₂(2)

(

τ

)

Y₃(2)

(

τ

)

⎞

⎟

⎠ =

⎛

⎝

1

+

12q

+

36q 2

₊

_12q3

_{+ . . .}

−

6q1/3

(

1

+

7q

+

8q2

+ . . . )

−

18q2/3

₍

₁

₊

_2q

₊

_5q2

_{+ . . . )}

⎞

⎠ .

(2.11)

Theysatisfyalsotheconstraint[16]:

(

Y₂(2)

)

2

+

2Y₁(2)Y₃(2)

=

0

.

(2.12)

3. ResidualsymmetriesofA4andmodularforms

Residualsymmetriesarise whenevertheVEVofthemodulus

τ

breaksthe modulargroup



only partially,i.e., thelittlegroup (sta-biliser)of

τ



isnon-trivial.ResidualsymmetrieshavebeeninvestigatedinthecaseofmodularS4invariancein[22],andof A5 invariance

in[24],whereviablemodelsofleptonmassesandmixinghavealsobeenconstructed.Inthepresentworkweconsidermodelsoflepton flavourbasedontheresidualsymmetriesofthemodular A4 invariance.

There are only 2 inequivalent finite points with non-trivial little groups of

,

namely,

τ



= −

1/2

+

i

√

3/2

≡

τL

and

τ



=

i

≡

τC

[22].The firstpoint istheleft cusp inthe fundamentaldomain ofthemodulargroup, whichisinvariant underthe S T transformation

τ

= −

1/(

τ

+

1).Indeed,

Z

S T

3

= {

I

,

S T

,

(

S T

)

2

}

isoneofsubgroupsof A4group.2Therightcuspat

τ



=

1/2

+

i

√

3/2

≡

τR

isrelatedto

τ

L

bytheT transformation.The

τ



=

i pointisinvariantundertheS transformation

τ

= −

1/

τ

.Thesubgroup

Z

S

2

= {

I

,

S

}

of A4ispreserved

at

τ



=

τC

.Thereisalsoinfinitepoint

τ



=

i

∞

≡

τT

,inwhichthesubgroup

Z

T

3

= {

I

,

T

,

T2

}

ofA4 ispreserved.

Itispossibleto calculatethevaluesofthe A4 triplet modularformsofweight2atthe symmetrypoints

τ

L,

τ

C and

τ

T.The results

arereportedinTable1inwhichthevaluesofthemodularformsat

τ



=

τR

arealsogiven, tobecomparedwiththoseattheothertwo points.

Aswehavenoted, thedimension ofthelinearspace

M

k

(

3



A4

)

ofmodularforms ofweightk andlevel 3isk

+

1.The modular

formsof weights higherthan 2can be obtainedfrom themodular formsof weight2. Theytransformaccordingto certain irreducible representationsof the A4 group.Indeed, forweight 4 we have5 independent modularforms, whichare constructed by the weight 2

modularforms throughthe tensor productof 3

⊗

3 (see AppendixA). We obtain one triplet 3 and two singlets1, 1 ,while thethird singlet1 vanishes:

2 _{Intherecentpublication[38] theauthorsobtainτ= −1}

Table 1

Modularformsofweight2 and4 andthemagnitudeofY1(2)atrelevantτ. weight 2 weight 4 τ 3 3 {1, 1 } Y1(2) τL Y1(2)(1,ω,− 1 2ω2) 3(Y (2) 1 )2(1,− 1 2ω,ω2), {0, 9 4(Y (2) 1 )2ω} 0.9486... τR Y1(2)(1,ω2,− 1 2ω) 3(Y (2) 1 )2(1,− 1 2ω2,ω), {0, 9 4(Y (2) 1 )2ω2} 0.9486... τC Y1(2)(1,1− √ 3,−2+√3) (Y1(2))2(1,1,1), (Y (2) 1 )2{6 √ 3−9, 9−6√3} 1.0225... τT Y1(2)(1,0,0) (Y (2) 1 )2(1,0,0), {(Y (2) 1 )2, 0} 1 Table 2

ThechargeassignmentofSU(2), A4,andmodular weights (kIforfieldsandk forcouplingY ).Theright-handedcharged leptonsareassignedthreeA4singlets,respectively.

L (eR,μR,τR) Hu Hd Y SU(2) 2 1 2 2 1 A4 3 (1,1 ,1 ) 1 1 3,1,1 kI kL (keR,kμR,kτR) 0 0 k Y(₃4)

≡

⎛

⎜

⎝

Y₁(4) Y₂(4) Y₃(4)

⎞

⎟

⎠ =

2₃

⎛

⎜

⎝

(

Y₁(2)

)

2

−

Y₂(2)Y₃(2)

(

Y₃(2)

)

2

−

Y₁(2)Y₂(2)

(

Y₂(2)

)

2

−

Y₁(2)Y₃(2)

⎞

⎟

⎠ ,

(3.1) Y₁(4)

= (

Y₁(2)

)

2

+

2Y₂(2)Y₃(2)

,

Y₁(4 )

= (

Y3(2)

)

2

₊

_2Y(2) 1 Y (2) 2

,

Y (4) 1

= (

Y (2) 2

)

2

₊

_2Y(2) 1 Y (2) 3

≡

0 (3.2)

wherethevanishingY(₁4 )isduetotheconditioninEq. (2.12).UsingEq. (3.2) wecancalculatethevaluesofthemodularformsofweight 4,transformingas3 and

{

1

,

1

}

,atthesymmetrypoints

τ

L,

τ

C and

τ

T.WeshowtheresultsalsoinTable1.

4. Leptonmassmatriceswithresidualsymmetry

Wewillconsidernextmodularinvariantlepton flavourmodelswiththe A4 symmetry, assumingthatthemassiveneutrinosare

Ma-jorana particles and that the neutrino masses originate from the Weinberg dimension 5 operator. There is a certain freedom for the assignmentsofirreduciblerepresentationsandmodularweightstoleptons.Wesupposethatthreeleft-handed(LH)leptondoubletsform atripletofthe A4 group.The Higgsdoublets aresupposedtobezeroweightsingletsof A4.Thegeneric assignmentsofrepresentations

andmodularweightskI totheMSSMfields3arepresentedinTable2.Inordertoconstructmodelswithminimalnumberofparameters,

weintroducenoflavons.Forthechargedleptons,weassignthethreeright-handed(RH)chargedleptonfieldsforthreedifferentsinglet representationsof A4,

(

1

,

1

,

1

).

Therefore,there arethree independentcouplingconstants inthe superpotentialofthe chargedlepton

sector.Thesecouplingconstants canbeadjusted totheobservedchargedleptonmasses.Since therearethreesingletirreducible repre-sentationsinthe A4 group, thereare sixcasesfortheassignment ofthethreeRH chargedlepton fields.However, thisambiguity does

notaffectthematrixwhichactsontheLHchargedleptonfieldsandentersintotheexpressionforthePMNSmatrix.Thus,effectivelywe havethefollowinguniqueformforthesuperpotential:

we

=

α

eRHd

(

LY

)

1

+ β

μ

RHd

(

LY

)

1

+

γ τ

RHd

(

LY

)

1

,

(4.1)

wν

= −

1



(

HuHuLLY

)

1

,

(4.2)

wherethesumsofthemodularweightsshouldvanish.Theparameters

α

,

β,

γ

and



areconstantcoefficients.

4.1. Chargedleptonmassmatrixwithresidualsymmetry

Byusingthedecompositionofthe A4 tensorproducts giveninAppendixA,thesuperpotentialinEq. (4.1) leadstoamassmatrixof

chargedleptons,whichiswrittenintermsofmodularformsofA4 tripletwithweightk:

ME

=

vd

⎛

⎝

α

0 0

β

00 0 0

γ

⎞

⎠

⎛

⎜

⎝

Y₁(k) Y₃(k) Y₂(k) Y₂(k) Y₁(k) Y₃(k) Y₃(k) Y₂(k) Y₁(k)

⎞

⎟

⎠

R L

,

(4.3)

wherevd

≡

H0d



.Withoutlossofgeneralitythecoefficients

α

,

β,

and

γ

canbemaderealpositivebyrephasing theRHchargedlepton

fields.

3 _{Forthemodularweightsofchiralsuperfieldswefollowthesignconventionwhichisoppositetothatofthemodularforms,i.e. afield}

φ(I)_{transformsasφ}(I)_→

Wewilldiscussnextthechargedleptonmassmatrixatthespecificpointsof

τ

=

τL

,

τ

R

τC

,

τ

T inthecaseofweightk

=

2.At

τ

=

τL

,

thematrixM†_EME,whichisrelevantfortheleft-handedmixing,isgivenas:

M†_EME

=

9 4v 2 d

(

Y (2) 1

)

2

⎛

⎝

α

2

_{+ β}

2

₊

_γ

2

_/

₄

₋

_ω

2

_/

₂

_α

2

₊

_ω

2

_β

2

₋

_ω

2

_/

₂

_γ

2

_ωα

2

₋

_ω

_/

₂

_β

2

₋

_ω

_/

₂

_γ

2

−

ω

/

2

α

2

₊

_ω

_β

2

₋

_ω

_/

₂

_γ

2

_α

2

_/

₄

_{+ β}

2

₊

_γ

2

₋

_ω

2

_/

₂

_α

2

₋

_ω

2

_/

₂

_β

2

₊

_ω

2

_γ

2

ω

2

_α

2

₋

_ω

2

_/

₂

_β

2

₋

_ω

2

_/

₂

_γ

2

₋

_ω

_/

₂

_α

2

₋

_ω

_/

₂

_β

2

₊

_ωγ

2

_α

2

_{+ β}

2

_/

₄

₊

_γ

2

⎞

⎠ .

(4.4) Itiseasilynoticedthatthismatrixcommuteswith S T ,whichisguaranteedbytheresidualsymmetry

Z

S T₃ at

τ

=

τL

,where

S T

=

1 3

⎛

⎝

−1

2

ω

2

ω

2 2

−

ω

2

ω

2 2 2

ω

−

ω

2

⎞

⎠ .

(4.5)

BothmatricesM†_EME andS T arediagonalizedbytheunitarymatrixUE:

UE

≡

T S

=

1 3

⎛

⎝

−

2

ω

1

−

2

ω

22

ω

2

ω

2 ₂

_ω

2

₋

_ω

2

⎞

⎠ ,

U†_ES T UE

=

T

=

diag

(

1

,

ω

,

ω

2

),

U†_EM†_EMEUE

=

9 4v 2 d

(

Y (2) 1

)

2_diag

₍

_γ

2

_,

_α

2

_{, β}

2

_),

(4.6)

whereUE isindependentofparameters

α

,

β,

γ

.

Ontheotherhand,at

τ

=

τR

,wehave: M†_EME

=

9 4v 2 d

(

Y (2) 1

)

2

⎛

⎝

α

2

_{+ β}

2

₊

_γ

2

_/

₄

₋

_ω

_/

₂

_α

2

₊

_ω

_β

2

₋

_ω

_/

₂

_γ

2

_ω

2

_α

2

₋

_ω

2

_/

₂

_β

2

₋

_ω

2

_/

₂

_γ

2

−

ω

2

_/

₂

_α

2

₊

_ω

2

_β

2

₋

_ω

2

_/

₂

_γ

2

_α

2

_/

₄

_{+ β}

2

₊

_γ

2

₋

_ω

_/

₂

_α

2

₋

_ω

_/

₂

_β

2

₊

_ωγ

2

ωα

2

−

ω

/

2

β

2

−

ω

/

2

γ

2

−

ω

2

/

2

α

2

−

ω

2

/

2

β

2

+

ω

2

γ

2

α

2

+ β

2

/

4

+

γ

2

⎞

⎠ .

(4.7) ThematrixM†_EME inEq. (4.7) commuteswith

T S

=

1 3

⎛

⎝

−

2

ω

1

−

2

ω

22

ω

2

ω

2 ₂

_ω

2

₋

_ω

2

⎞

⎠ .

(4.8)

The fact that M†_EME and T S commute is a consequence of the residual symmetry

Z

3T S at

τ

=

τR

. The matrices M †

EME and S T are

diagonalisedbytheunitarymatrix:

UE

≡

S T

=

1 3

⎛

⎝

−

1 2

ω

2

ω

2 2

−

ω

2

ω

2 2 2

ω

−

ω

2

⎞

⎠ .

(4.9)

At

τ

=

τC

,thedeterminantofME vanishes.Indeed,thismassmatrixleadstoamasslesschargedlepton,andthuscannotbeusedfor

modelbuilding.

Finally,at

τ

=

τT

weobtaintherealdiagonalmatrix: ME

=

vdY1(2)

⎛

⎝

α

0

β

0 00 0 0

γ

⎞

⎠ .

(4.10)

Inthecaseofmodularformsofweight 4wecanobtain achargedleptonmassmatrixinwhichthemodularforms transformingas

1 and1 do notcontribute. As seenin Table1,the weight4 triplet modularformscoincide withweight2 ones at

τ

=

τL

,

τ

R.Indeed,

M†_EME isobtainedby replacingparameters

(

α

,

β,

γ

)

ofthemassmatricesinEqs. (4.4) and (4.7) with

(

γ

,

α

,

β),

respectively.Therefore,

themixingmatricesinEqs. (4.6) and(4.9) arethesame.

At

τ

=

τC

,the chargedleptonmassmatrixisofrankone, i.e.,two masslesschargedleptons appearsincethe tripletmodularforms areproportionalto

(1,

1,1).At

τ

=

τT

,thechargedleptonmassmatrixisequaltothe diagonalonegiveninEq. (4.10) sincethetriplet weight4modularformscoincidewiththeweight2 modularforms.

4.2.Neutrinomassmatrix(Weinbergoperator)

TheneutrinomassmatrixiswrittenintermsofA4 tripletmodularformsofweightk byusingthesuperpotentialinEq. (4.2):

Mν

=

v2_u



⎛

⎜

⎝

2Y₁(k)

−Y

₃(k)

−Y

(₂k)

−

Y₃(k) 2Y₂(k)

−

Y(₁k)

−

Y₂(k)

−

Y₁(k) 2Y₃(k)

⎞

⎟

⎠

LL

,

(4.11) wherevu

≡

Hu0



.

Inthecaseofweight 2 modularformsitiseasilycheckedthattwo lightestneutrinomassesare degenerateat

τ

=

τL

,

τ

R,whilethe

determinantof Mν vanishes at

τ

=

τC

. In the lattercase one neutrinois massless and two neutrinomassesare degenerate.The two lightestneutrinomassesaredegeneratealsoat

τ

=

τT

.Itmaybe helpfultoadd acomment:thesedegeneraciesofneutrinomassesstill holdevenifweusetheseesawmechanismbyintroducingthethreeright-handedneutrinofieldsasA4 triplet.Thus,therealisticneutrino

massmatrixisnotobtainedasfaraswetakeweight2 modularformsat

τ

=

τL

,

τ

R

,

τ

C

,

τ

T.

Inthecaseofweight4 modularforms,thereisonecandidatethatcanbeconsistentwiththeobservedneutrinomasses.At

τ

=

τL

,

τ

R,

theneutrinomassterm3L3LY(₃4)issimilartothecaseofweight 2,wheretwoneutrinomassesaredegenerate.Inthecaseofweight 4, thesinglet1 alsocontributestotheneutrinomassmatrixthroughthecoupling3L3LY(₁4 ).However, thisadditionaltermcannot resolve thedegeneracy.

Itiseasily noticedthattwoneutrinomassesaredegeneratealsoat

τ

=

τT

sinceY₃(4)

∼ (

1, 0, 0).AnadditionalY(₁4) doesnotchange thissituation.

At

τ

=

τC

, thetriplet modularform, as seenin Table1,is Y₃(4)

∼ (

1, 1, 1),whichallows to getlarge mixingangles. Moreover, we

have1 and 1 modularfunctions.Therefore,weexpectnearlytri-bimaximalmixingpatternofPMNSmatrixwiththreedifferentmassive

neutrinos.TheLHweak-eigenstateneutrinofieldscoupletoY₃(4).ThiscouplingleadstothefollowingneutrinoMajoranamassmatrix:

Mν

=

v2_u



(

Y (2) 1

)

2

⎛

⎝

−1

2

−

21

−

−1

1

−

1

−

1 2

⎞

⎠ .

(4.12)

Moreover,theLHneutrinofieldscouplealsotoY(₁4)andY₁(4 ),whichgivesthefollowingadditionalcontributionstotheneutrinoMajorana massmatrixMν : 3

(

2

√

3

−

3

)

v 2 u



(

Y (2) 1

)

2

⎛

⎝

1 0 00 0 1 0 1 0

⎞

⎠ ,

−

3

(

2

√

3

−

3

)

v 2 u



(

Y (2) 1

)

2

⎛

⎝

0 0 10 1 0 1 0 0

⎞

⎠ ,

(4.13)

whereeachofthesetwotermsenters Mν withitsownarbitraryconstant.

Tosummarise,thechargedleptonmassmatrixcouldbeconsistentwithobservedmassesat

τ

=

τL

,

τ

R

,

τ

=

τT

forbothcasesofweight

2 and4 modular forms.Onthe other hand,the neutrinoMajoranamassmatrixis consistentwithobserved massesonly at

τ

=

τC

for weight4 modularforms.Thereisnocommonsymmetryvalueof

τ

,whichleadstochargedleptonandneutrinomassesthatareconsistent withthedata.

5. Modelswithresidualsymmetry

Asseenintheprevioussection,wecouldnotfindmodelswithonemodulus

τ

andwithresidualsymmetry,whichare phenomenolog-icallyviable.Therefore,weconsiderthecaseofhavingtwomoduliinthetheory: one

τ

_{,responsibleviaitsVEVforthebreakingofthe} modular A4symmetryinthechargedleptonsector,andtheanotherone

τ

ν ,breakingthemodularsymmetryintheneutrinosector.Our

approachhereispurelyphenomenological.Constructingamodelwithtwodifferentmoduliinthechargedleptonandneutrinosectorsis outofthe scopeofourstudy,itisa subjectofongoingresearch andworkinprogress.However,there arehintsfromtherecentstudy [39] thatthismightbepossible.4 Amodelwithtwodifferentmoduliinthequarkandleptonsectors,associatedrespectivelywithS3 and A4 modularsymmetries,hasbeenpresentedrecentlyin[40].

Wepresentnext oursetup. Forthechargedleptonmass matrix,wetake weight 2 modularformsat

τ



₌

_τT

_{(Case I)orat}

_τ



₌

_τL

(CaseII).5 Atthesametimewe useweight 4 modularformsat

τ

ν

₌

_τC

_{forconstructingtheneutrinoMajoranamassterm.Inorderfor}

themodularweightinthesuperpotentialtovanish,weassignthefollowingweightstotheLHleptonandRHchargedleptonfields:

kL

=

2

,

keR

=

kμR

=

kτR

=

0

,

(5.1)

wherethenotationsareself-explanatory.WenotethatkL

=

2 iscommoninboth

τ

and

τ

ν modularspaces.

Then,thechargedleptonmassmatrixisobtainedbyusingasinputtheexpressionsfortheweight2 modularformsgiveninTable1. At

τ

T,itisadiagonalmatrix: ME

=

vd

⎛

⎝

α

0 0

β

00 0 0

γ

⎞

⎠ :

Case I

.

(5.2)

At

τ

=

τL

,thechargedleptonmassmatrixhastheform: ME

=

vd

⎛

⎝

α

0 0

β

00 0 0

γ

⎞

⎠

⎛

⎝

1

ω

2

₋

1 2

ω

−

1 2

ω

1

ω

2

ω

2

₋

1 2

ω

1

⎞

⎠

R L

:

Case II

.

(5.3)

ThematrixM†_EME,whichisrelevantforthecalculationoftheleft-handedmixing,isgiveninEq. (4.4).

4 _{Theauthorsof[39] writeintheConclusions:“Aswefinddifferentflavorsymmetriesatdifferentpointsinmodulispace(inparticularinsixcompactdimensions),fields} thatliveatdifferentlocationsinmodulispacefeeladifferentamountofflavorsymmetry.(...)Thiscouldleadtoadifferentflavor- andCP-structureforthevarioussectors ofthestandardmodellikeup- ordown-quarks,chargedleptonsorneutrinos.”

5 _{Thesamenumericalresultsareobtainedatτ}

Theneutrinomassmatrixrepresentsa sumofthecontributions ofmodularformsof3,1 and 1 , withthe termsinvolvingthetwo singletmodularformsenteringthesumwitharbitrarycomplexcoefficients A and B:

Mν

=

v2 u



(

Y (2) 1

)

2

⎧

⎨

⎩

⎛

⎝

₋

21

−

21

−

−

11

−1

−1

2

⎞

⎠ +

⎡

⎣

A

⎛

⎝

1 0 00 0 1 0 1 0

⎞

⎠ −

B

⎛

⎝

0 0 10 1 0 1 0 0

⎞

⎠

⎤

⎦

⎫

⎬

⎭

,

(5.4)

wheretheconstantsofthetwotermsinEq. (4.13) areabsorbedintheparameters A and B.

ThetwomodelswithchargedleptonmassmatrixME specifiedinEqs. (5.2) and(5.3) andneutrinomassmatrixMν giveninEq. (5.4),

aswewillshow,leadtothesamephenomenology.

Asanalternativetothemodelswithtwomoduli

τ

 _and

_τ

_{ν ,wepresentnextamodelwithonemodulus}

_τ

_{ν andoneflavon,breaking} themodular symmetry to

Z

S

2 and

Z

T3 in theneutrino andcharged leptonsectors respectively andleading to the chargedlepton and

neutrinomassmatricesgivenin Eqs. (5.2) and(5.4). Weintroduce an A4 triplet flavon

φ

withmodularweight kφ

= −

3.Incontrastto Eq. (5.1),themodularweightsoftheLHleptondoubletandRHchargedleptonfieldsarechosenasfollows:

kL

=

2

,

keR

=

kμR

=

kτR

=

1

.

(5.5)

As a consequence, the modular functions Y(i) do not couple to the charged lepton sector, butcouple to the neutrinosector because

Y(i) _{havepositive evenmodularweights. Ontheother hand,theflavon}

_φ

_{couplesonlyto thechargedleptonsector becauseofits odd} weight.6 ThecorrespondingtermsofthesuperpotentialarethesameasgiveninEq. (4.1) withthemodularformY replacedbytheflavon

φ.

Moreover, wecan easilyobtain therequisiteVEV

φ

=

vE

(1,

0,0)T preserving

Z

T3, vE beingaconstant parameter,fromthepotential

analysisasseeninRefs. [12,13].Finally,wegetthechargedleptonandneutrinomassmatricesgiveninEqs. (5.2) and(5.4).Thisflavon modelwithonemodulus

τ

ν leadstothesamephenomenologyasthemodelsconsideredearlierwithtwodifferentmoduli

τ

 _and

_τ

_{ν .}

5.1.Theneutrinomixing

IncaseI,onlytheneutrinomassmatrixcontributestothePMNSmatrixsincethechargedleptonmassmatrixisdiagonal.Theneutrino massmatrixinthiscaseleads totheso calledTM2 mixingformofPMNSmatrixUPMNS [35,36] wherethesecond columnofUPMNS is

trimaximal: U_PMNSI

=

⎛

⎜

⎝

2 √ 6 1 √ 3 0

−

_√1 6 1 √ 3

−

1 √ 2

−

_√1 6 1 √ 3 1 √ 2

⎞

⎟

⎠

⎛

⎝

cos

θ

0 e iφ_sin

_θ

0 1 0

−e

−iφ_sin

_θ

_{0 cos}

_θ

⎞

⎠

P

.

(5.6)

Here

θ

and

φ

are arbitrary mixing angle and phase, respectively, and P is a diagonal phase matrix containing contributions to the MajoranaphasesofUPMNS.Employingthe standardparametrisation ofUPMNS (see, e.g.,[1]),itispossible toshow that thetrimaximal

mixingpatternleadstothefollowingrelationbetweenthereactorangle

θ

13 and

θ

,betweentheatmosphericneutrinomixingangle

θ

23

and

θ

13and

θ,

andsumrulesforthesolarneutrinomixingangle

θ

12andfortheDiracphase

δ

[35,36] (seealso[9,29]):

sin2

θ

13

=

2 3sin 2

_{θ ,}

_(5.7) sin2

θ

12

=

1 3 cos2

_θ

13

,

(5.8) sin2

θ

23

=

1 2

+

s13 2



2

−

3s2₁₃ 1

−

s2₁₃ cos

φ ,

(5.9)

cos

δ

=

cos 2

θ

23cos 2

θ

13 sin 2

θ

23sin

θ

13

(

2

−

3 sin2

θ

13

)

1 2

.

(5.10)

Usingthe3

σ

allowedrangeofsin2

θ

13from[2] andEq. (5.7) wegetthefollowingconstraintsonsin

θ:

0

.

17

 |

sin

θ

| 

0

.

19

.

(5.11)

Toleadingorderins13weobtainfromEq. (5.9): 1 2

−

s13

√

2

∼

<

_sin2

_θ

23

∼

<

1 2

+

s13

√

2

,

or 0

.

391

(

0

.

390

)

∼

<

_sin2

_θ

23

∼

<

0

.

609

(

0

.

611

) ,

(5.12)

wherethenumericalvaluescorrespondtothemaximal allowedvalueofsin2

θ

13at3

σ

C.L.forNO(IO)neutrinomassspectrum[2].The

intervalofpossiblevaluesofsin2

θ

23ineq. (5.12) issomewhatwiderthan the3

σ

rangesofexperimentallyallowedvaluesofsin2

θ

23for

NOandIOspectragivenin[2].Usingthe3

σ

allowedrangesofsin2

θ

23andsin2

θ

13forNO(IO)spectrafrom[2] andEq. (5.9) wealsoget:

−

0

.

640

(

−

0

.

508

)



cos

φ

≤

1

.

(5.13)

Thephase

φ

isrelatedtotheDiracphase

δ

[9]:

sin 2

θ

23sin

δ

=

sin

φ .

(5.14)

TheMajoranaphase

α

31

/2 of

thestandardparametrisationofUPMNS[1] receivescontributionsfromthephase

φ

via[9]

α

31 2

=

ξ

31

2

+

α

2

+

α

3

,

(5.15)

wherethephase

ξ

31willbespecifiedlater,

α

2

=

arg



−

√

c 2

−

s

√

6e iφ



_,

_α

3

=

arg



c

√

2

−

s

√

6e iφ



_,

_(5.16) sin

α

2

= −

s

√

6 sin

φ

s23c13

= −

tan

θ

13cos

θ

23sin

δ ,

(5.17)

sin

α

3

= −

s

√

6 sin

φ

c23c13

= −

tan

θ

13sin

θ

23sin

δ .

(5.18)

Wealsohave[9]:

sin

(φ

−

α

2

−

α

3

)

= −

sin

δ .

(5.19)

Forfurtherdiscussionofphenomenologyoftheneutrinotrimaximalmixing(5.6),see,e.g.,[9,14,30,37].

IncaseII,thecontributionoftherotationofthechargedleptonsectorisaddedtothetrimaximalmixing,whichisderivedfromthe neutrinomassmatrixinEq. (5.4).ThemixingmatrixinthechargedleptonsectoristhematrixUE inEq. (4.6).ThePMNSmatrixisgiven

by: U_PMNSII

=

1 3

⎛

⎝

−

2

ω

1

−

2

ω

22

ω

2

ω

2 ₂

_ω

2

₋

_ω

2

⎞

⎠

†

⎛

⎜

⎝

2 √ 6 1 √ 3 0

−

_√1 6 1 √ 3

−

1 √ 2

−

_√1 6 1 √ 3 1 √ 2

⎞

⎟

⎠

⎛

⎝

cos

θ

0 e iφ_sin

_θ

0 1 0

−

e−iφ_sin

_θ

_{0 cos}

_θ

⎞

⎠

P

.

(5.20)

Itisstraightforwardtocheckthatafterasubstitution

θ

→ θ −

π

/2,

φ

→ −φ

,thePMNSmatrix (5.20) canberewrittenas U_PMNSII

=

⎛

⎝

−

01 eiπ0/3 0₀ 0 0 e−iπ/3

⎞

⎠

U_PMNSI

⎛

⎝

e i(φ−π/2) _{0 0} 0 1 0 0 0 e−i(φ+π/2)

⎞

⎠ .

(5.21)

Theleftmostphase matrixdoesnot contributeto themixing,since itseffectcan beabsorbedintothechargedleptonfield phases.The rightmostphasematrixcontributesonlytotheMajoranaphases,thereforethenumericalpredictionsinthiscasearethesameasinCase I, apart possiblyfrom thecorresponding shift ofthe Majoranaphases. However, ascan be shownanalytically, andwe have confirmed numerically,alsothepredictionsfortheMajoranaphasesinCaseIIcoincidewiththepredictionsincaseI.

5.2. TheneutrinomassesandMajoranaphases

Itfollowsfrom(5.4) thattheneutrinomassmatrixMν isalinearcombinationofthreebasismatrices:

M1

=

⎛

⎝

₋

21

−

21

−

−

11

−1

−1

2

⎞

⎠ ,

M2

=

⎛

⎝

1 0 00 0 1 0 1 0

⎞

⎠ ,

M3

=

⎛

⎝

0 0 10 1 0 1 0 0

⎞

⎠ .

(5.22)

TodiagonalizeMν ,itisconvenienttorewriteitinadifferentbasis:

M ₁

=

√

1 3

(

M2

+

2M3

)

=

1

√

3

⎛

⎝

1 0 20 2 1 2 1 0

⎞

⎠ ,

M ₂

=

M2

+

1 3M1

=

1 3

⎛

⎝

−

51

−

21

−

21

−

1 2 2

⎞

⎠ ,

M ₃

=

M2

−

1 3M1

=

1 3

⎛

⎝

11

−

12 14 1 4

−2

⎞

⎠ ,

Mν

=

c



M₁

+

aM₂

+

bM ₃



,

(5.23)

wherea andb arearbitrarycomplexcoefficientsandc istheoverallscalefactorwhichcanberenderedrealpositive. Mν isdiagonalized byaunitarymatrixU◦_{ν of}thefollowingform:

U◦_ν

=

VTBMU13

(θ, φ) ,

(5.24)

sothat Mν

= (

U◦_ν

)

∗Mdiagν

(

Uν◦

)

†,withM diag ν

=

diag



m1e−i2φ1

,

m2e−i2φ2

,

m3e−i2φ3



,wheremie−i2φi arecomplexeigenvaluesandmi

≥

0 are

theneutrinomasses.7_{Extractingthephases}

_φ

i fromMdiagν ,wefind:

Mdiagν

=

e−i2φ1P∗diag

(

m1

,

m2

,

m3

)

P∗

,

P

=

diag



1

,

ei(φ2−φ1)

_,

_ei(φ3−φ1)



,

(5.25)

wherethephases

(φ

2

− φ

1

)

and

(φ

3

− φ

1

)

contribute tothe Majoranaphases

α

21

/2 and

α

31

/2 of

thestandard parametrisationofthe

PMNSmatrix[1].Thus,thePMNSmatrixhastheform:

UPMNS

=

U◦_νP

=

e−i2φ1VTBMU13

(θ, φ)

P

,

(5.26)

wherethecommonphasefactore−i2φ1 _{isunphysical.Thephase}

_ξ

31

/2 in

Eq. (5.15) canbeidentifiednowwith

(φ

3

− φ

1

):

ξ

31

/2

= φ

3

− φ

1.

Thus,theMajoranaphases

α

21

/2 and

α

31

/2 are

givenby:

α

21

2

= φ

2

− φ

1

,

α

31

2

= φ

3

− φ

1

+

α

2

+

α

3

.

(5.27)

Thecomplex rotation parameters

θ

and

φ

are fixed by a choice ofa and b,which we will now show explicitly.We find by direct calculationthat U◦_νTM ₁U◦_ν

=

⎛

⎝

−e

−iφ_{sin 2}

_θ

_{0 cos 2}

_θ

0

√

3 0 cos 2

θ

0 eiφ_{sin 2}

_θ

⎞

⎠ ,

U◦_νTM ₂U◦_ν

=

⎛

⎝

2 cos 2

_θ

_{0 e}iφ_{sin 2}

_θ

0 1 0 eiφ_{sin 2}

_θ

_{0 2e}2iφ_sin2

_θ

⎞

⎠ ,

U◦_νTM ₃U◦_ν

=

⎛

⎝

−

2e −2iφ_sin2

_θ

_{0 e}−iφ_{sin 2}

_θ

0 1 0 e−iφ_{sin 2}

_θ

₀

₋

_{2 cos}2

_θ

⎞

⎠ .

(5.28)

Thus,theneutrinomassmatrixMν isdiagonalizedwhenthecorrespondinglinearcombinationoftheoff-diagonalentriesvanishes,which leadsto

cos 2

θ

+

aeiφsin 2

θ

+

be−iφsin 2

θ

=

0

⇔

aeiφ

+

be−iφ

= −

cot 2

θ.

(5.29) Theaboveconditionisequivalentto:

eiφ

= ±

a ∗

₋

_b

|

a∗

−

b

|

,

cot 2

θ

= ∓

|

a

|

2

_{− |}

_b

_|

2

|

a∗

−

b

|

.

(5.30)

Itprovesconvenienttointroducethecomplexparameter

z

=

aeiφ

−

be−iφ

= ±

|

a

|

2

_{+ |}

_b

_|

2

₋

_2ab

|

a∗

−

b

|

.

(5.31)

Using

(θ,

φ,

z

)

isareparametrisationof

(

a

,

b

)

determinedby(5.30) and(5.31).Theinverseparametertransformationisgivenby

a

=

e −iφ 2

(

z

−

cot 2

θ ) ,

b

=

e iφ 2

(

−

z

−

cot 2

θ ) .

(5.32)

Theneutrinomassmatrixeigenvaluesarethecorrespondinglinearcombinationsofthediagonalentriesin(5.28):

m1e−i(2φ1−φ)

=

c



z

−

1 sin 2

θ



,

m2e−i2φ2

=

c

√

3

−

iz sin

φ

−

cot 2

θ

cos

φ



,

m3e−i(2φ3+φ)

=

c



z

+

1 sin 2

θ



.

(5.33)

7 _{Ingeneral,thestandardlabellingoftheneutrinomasses [1] correspondstosomepermutationoftheneutrinomassmatrixeigenvalues,whichaffectstheorderofthe} PMNSmatrixcolumns.However,theonlynon-trivialpermutationoftheTM2matrixcolumnsconsistentwiththeexperimentaldatais(321),whichisequivalenttoashift θ→ θ −π/2 uptoanunphysicaloverallcolumnsign.Hence,wecanassumethattheorderofneutrinomassmatrixeigenvaluescoincideswiththestandardlabelling withoutlossofgenerality.

Fig. 1. Correlationsbetweensin2_θ

23andthesumofneutrinomassesmi,betweensin2θ23andtheeffectiveMajoranamass|m|,andbetweentheMajoranaphasesα31 andα21inthecaseofNOneutrinomassspectrum.Seetextforfurtherdetails.

Fittingthe mass-squared differencesto experimentally observedvalues, wefind thefollowing constrainton z in terms of

θ

,

φ

and

r

≡ 

m2 21

/

m231:

|

z

−

z0

|

2

=

R2

,

sign

(

Re z

)

= ±

sign

(

sin 2

θ ) ,

(5.34)

wheretheplus(minus)signcorrespondstoNO(IO)spectrumofneutrinomasses,and

z0

(θ, φ,

r

)

=

1

−

2r

cos2

_φ

_{sin 2}

_θ

+

tan

φ

 √

3 cos

φ

−

cot 2

θ



i

,

R2

(θ, φ,

r

)

=

√

3

−

cot 2

θ

cos

φ



2

+

(

1

−

2r

)

2

−

cos2

φ

sin22

θ

!

cos4

φ.

(5.35)

Since

θ

andr aretightlyconstrainedbytheexperimentaldata,thesetofphenomenologicallyviablemodelsiseffectivelydescribedbytwo angles

φ

and

ψ,

withthelatterbeingtheangleparameteronthecircle (5.34),i.e.z

=

z0

+

Reiψ.Scanningthrough

φ

and

ψ

numerically,

wefindthattoeachsetoftheexperimentallyallowedvaluesofthemixinganglesandthemass-squareddifferencescorrespondsarange ofmodels(parameterisedby

ψ

)withdifferentvaluesoftheneutrinomassesandtheMajoranaphases.

Wereportthe numericalresultsinthecaseofNOspectruminFig.1.Theallowed rangeofthesumofneutrinomassesdependson thevalueofsin2

θ

23.Thelowerboundslightlydecreasesfrom0.097eVto0.074eVassin2

θ

23runsthroughits 3

σ

confidenceintervalof

[

0.46,0.58

]

.8Ontheotherhand,theupperboundishighlydependentonthevalueofsin2

θ

23,andtendstoinfinityassin2

θ

23approaches

0.5,whichcorresponds to

δ

= φ =

3

π

/2.

Thismeansthatatthispointthesumofneutrinomassesisallowed totakeanyvalue greater than its lower bound of 0.093 eV. The dependence of the effectiveMajorana mass

|

m

|

on sin2

θ

23 is qualitatively similar to that of

thesumofneutrinomasses.Themaximal valueof

|

m

| ∼

₌

0.059 eV ispracticallyindependentofsin2

θ

23for0.46

≤

sin2

θ

23

≤

0.55.The

lowerboundof

|

m

|

variesfrom0.0015 eVto0.0059eVforsin2

θ

23 inits3

σ

range.However, forvaluesofsin2

θ

23fromits 3

σ

range,

0.46

≤

sin2

θ

23

≤

0.58,

|

m

|

canhavevaluesintheinterval

[

0.0059,0.059

]

eV(seeFig.1).Most(ifnotall)ofthesevaluesmaybeprobed

inthefutureneutrinolessdoublebetadecayexperiments.

ThereisalsoastrongcorrelationbetweentheMajoranaphases.Thesetofbest-fitmodelscorrespondsto

φ

=

1.664

π

andleadstothe followingvaluesofobservables:

r

=

0

.

0299

,

δ

m2

=

7

.

34

·

10−5eV2

,



m2

=

2

.

455

·

10−3eV2

,

sin2

θ

12

=

0

.

3406

,

sin2

θ

13

=

0

.

02125

,

sin2

θ

23

=

0

.

5511

,

m1

=

0

.

0143

−

0

.

0612 eV

,

m2

=

0

.

0166

−

0

.

0618 eV

,

m3

=

0

.

0519

−

0

.

079 eV

,



imi

=

0

.

0828

−

0

.

2019 eV

,

|m| =

0

.

0029

−

0

.

0589 eV

,

δ/

π

=

1

.

339

,

(5.36)

consistentwiththeexperimentaldataat2.59

σ

C.L.

SimilaranalysiscanbeperformedinthecaseofIOneutrinomassspectrum.However,inthat casetheminimal valueofthesumof thethreeneutrinomassesis0.63eV,andwedonotanalysethiscasefurther.

6. Summary

Wehaveinvestigatedmodels ofleptonmassesandmixingbasedonmodular A4 flavour symmetrybrokentoresidualsymmetriesin

thechargedleptonandneutrinosectors. Thestandardcaseofthreeleptonfamilieswas considered.Inatheorybasedonfinitemodular flavour symmetry not only the matter fields, butalso the constants such as the Yukawa couplings transformnon-trivially under the modularsymmetry. These constants are writtenin termsofmodular formswhich are holomorphicfunctionsof acomplex scalarfield –the modulus

τ

.The modularforms havespecific transformationproperties underthemodular symmetrytransformations, whichare characterisedbyapositiveevennumberk called“weight”,anddependontheorderofthefinitemodulargroupviatheir“level”N.Inthe

8 _{Wedefinethenumberofstandarddeviationsfromtheχ}2_{minimumasNσ=}"_χ2_,whereχ2_{isasumofone-dimensionalprojectionsχ}2

j, j=1,2,3,4 from [2] fortheaccuratelyknowndimensionlessobservablessin2