Physics Letters B

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Physics Letters B

www.elsevier.com/locate/physletb

Leptonic CP violation in flipped SU(5) GUT from Z 12 − I orbifold compactification

Junu Jeong

^a,b

, Jihn E. Kim

^a,c,d,∗

, Soonkeon Nam

^c

aCenterforAxionandPrecisionPhysicsResearch(InstituteofBasicScience),KAISTMunjiCampus,193Munjiro,Daejeon34051,RepublicofKorea bDepartmentofPhysics,KAIST,291Daehakro,Yuseong-Gu,Daejeon,14141,RepublicofKorea

cDepartmentofPhysics,KyungHeeUniversity,26Gyungheedaero,Dongdaemun-Gu,Seoul02447,RepublicofKorea dDepartmentofPhysicsandAstronomy,SeoulNationalUniversity,1Gwanakro,Gwanak-Gu,Seoul08826,RepublicofKorea

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received9January2019

Receivedinrevisedform4February2019 Accepted21February2019

Availableonline26February2019 Editor:M.Cvetiˇc

Keywords:

Stringcompactification FlippedSU(5)GUT PMNSmatrix Kim-Seoform Jarlskogdeterminant

We obtain a phenomenologically acceptable PMNS matrix in a flipped SU(5) model, possessing the Z4R discretesymmetry, fromthe compactification ofheterotic string E8×^E8.To analyze the Jarlskog determinantefficiently,weincludethesimpleKim-SeoformforthePontecorbo-Maki-Nakagawa-Sakata matrix.Wealsonotedthat|δPMNS|^{63oforthenormalhierarchyofneutrinomasseswiththePDGbook} parametrization.

©^{2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

The most urgent theoretical issue in the standard model(SM) is probing the symmetry structure from which the observed flavor phenomenacan beunderstood.It isdesirableifsuchsymmetry resultsfromanultra-violet completedtheory.Atpresent, stringtheory isconsidered tobethemostattractiveoneamongvariousultra-violetcompletedtheories,chieflybecauseitunifiesgravityonthesame groundasgaugetheories.TheSMisobtainedfromcompactificationofsixextradimensions[1–6],andthesymmetrystructureofflavors istheonerealizedbelowthecompactificationscale.

Previously,moststudiesalongthisdirectionwerecenteredonobtainingthreefamiliesinthestandard-likemodels[7,8].¹ Sincethere aretoomanyYukawacouplingsinstandard-likemodels,hereweattempttoworkina grandunification(GUT) models.TheGUTgroup weusehereisthe rank5flippedSU(5)GUT[11,12],SU(5)_flip≡^SU(5)×^U(1)X,which wasobtainedfromcompactifications viafermionic string[13] andZ12−^I orbifold[14].The simplestGUTSU(5)fromstringcompactification isnotaccompanying an adjointrepresentation atthelevel1constructionwheretheHiggsmultipletneededforbreakingSU(5)downtotheSMislacking.Therank5SU(5)flip requires 10⊕^{10forbreakingitdowntotherank4SMgaugegroup,andthemodelof}[10] containsthem.

Timeis ripe enough to study the details of the flavor structure fromstring compactification to see whether they convergeto the observeddata.The studyisnowpossibleintheZ_4R model[9] basedon[10] wheretheneededZ_4R quantumnumbersofalllightchiral fields are presented.In the quark sector, the Cabibbo-Kobayashi-Maskawa matrix [15,16] has been studied in ourprevious paper[17].

Inthispaper,wepresentanumericalstudyonthePontecorbo-Maki-Nakagawa-Sakata (PMNS)matrix[18,19] viamanyU(1)’sarising in stringcompactification.Theheterotic stringE₈×^E₈ ^{hasrank16gaugesector.Themodelpresentedin[10] hastheSU}(5)flipfromE₈ and

*

Correspondingauthor.

E-mailaddress:jihnekim@gmail.com(J.E. Kim).

1 Formorereferences,see[9].

https://doi.org/10.1016/j.physletb.2019.02.035

0370-2693/©^{2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

Fig. 1.A neutrino interaction with the SM fields only.

Table 1

U(1)chargesofmatterfieldsintheflippedSU(5).ξiandη¯icontaintheleft-handedquarkandleptondoublets,respectively,inthei-thfamily.

State(P+kV0) i RX(Sect.) QR Q1 Q2 Q3 Q4 Q5 Q6 Qanom Q18 Q20 Q22

ξ3 (+ + + − −; − − +)(⁰⁸) 0 10₋1(U3) +¹ −⁶ −⁶ +^{6 0 0 0} −¹³ +¹ −¹ +¹

¯

η3 (+ − − − −; + − −)(0⁸) 0 5+3(U3) +1 +6 −6 −6 0 0 0 −1 +1 −1 +1

τ^c (+ + + + +; − + −)(0⁸) 0 1₋5(U3) +¹ −⁶ +⁶ −^{6 0 0 0} +⁵ +¹ −¹ +¹

ξ2 (+ + + − −; −¹6,−¹6,−¹6)(0⁸) ⁺₄¹ 10₋1(T₄⁰) −¹ −² −² −^{2 0 0 0} −³ −¹ −¹ −¹

¯

η2 (+ − − − −; −¹6,−¹6,−¹6)(0⁸) ⁺₄¹ 5₊3(T⁰₄) −¹ −² −² −^{2 0 0 0} −³ −¹ −¹ −¹

μ^c (+ + + + +; −¹6,−¹6,−¹6)(0⁸) ⁺₄¹ 1₋5(T⁰₄) −¹ −² −² −^{2 0 0 0} −³ −¹ −¹ −¹

ξ1 (+ + + − −; −¹6,−¹6,−¹6)(0⁸) ⁺₄¹ 10−1(T₄⁰) −1 −2 −2 −2 0 0 0 −3 −1 −1 −1

¯

η1 (+ − − − −; −¹6,−¹6,−¹6)(0⁸) ⁺₄¹ 5₊3(T⁰₄) −¹ −² −² −^{2 0 0 0} −³ −¹ −¹ −¹

e^c (+ + + + +; −¹6,−¹6,−¹6)(0⁸) ⁺₄¹ 1−5(T⁰₄) −1 −2 −2 −2 0 0 0 −3 −1 −1 −1

HuL (+^{1 0 0 0 0};^{0 0 0})(0⁵;⁻2¹ +1

2 0) ⁺₃¹ 2·⁵−2(T6) −^{2 0 0 0} −^{12 0 0 0} −¹ −¹ −¹

HdL (−1 0 0 0 0;0 0 0)(0⁵;⁺2¹ −1

2 0) ⁺₃¹ 2·5+2(T6) +2 0 0 0 +12 0 0 0 −1 −1 −1

SU(5)×^{SU(2) fromE}₈^{,andwe can consider 6extra U(1) gauge groups. Thesemany U(1)’s make itpossible to haveflavor dependent} Yukawacouplings.²

StringcompactificationinourexampleallowsalltheneededYukawacouplingsintheSMasnon-renormalizableforms.Therehasbeen an ambitious attempt[21] to relate the originofthe

μ

termwith themagnitudes ofneutrino massesby introducing justone singlet chiral field beyondthe minimal supersymmetricSM. Thistry allowed only renormalizable couplings. Therefore,withso many singlets participatingthroughnon-renormalizableYukawacouplingsinourmodel,thisdesignisnotapplicable.Therelationsinourmodelmight beintertwinedinanelaboratewaysincetheZ4R discretesymmetryautomaticallygivesthe

μ

termattheelectroweakscale[22].Since ourZ4R isasubgroupofanAbeliangaugegroupU(1)⁶,itcannotbeanon-Abeliandiscretesymmetrysuchastheinteresting A4 symmetry [23].

The massmatrices leading to the Kim-Seo(KS) parametrization [24] ofthe CKM and PMNSmixing matriceshave a smallnumber ofcomplexelements,bymaking onerowconsistofrealnumbers.Attheplaces wherecomplexentriesareallowed,we allocatetheCP phase.TheKSformhasanotheradvantagethattheJarlskogdeterminantis J= −^ImV₃₁^KSV₂₂^KSV₁₃^{KS [25].Towardamodelbuilding,thenext} stepistoobtainphenomenologicallyacceptablemassmatrices.IntheflippedSU(5),theneutrinomassmatrixissymmetric,canbemade real,andhenceweproposeinthispapertoputtheCPphaseinthechargedleptonmassmatrix.

ThestringmodelgivestheYukawacouplingsforthechargedleptonmassesandneutrinomasses.Thus,inSec.2wepresentthelepton massmatricesfromtheflippedSU(5)model,i.e.allowedbythequantumnumbersofRef. [9],intheformsrelatedtotheKSmixingmatrix.

Then,we locatepossiblephasesinthecomplexvacuumexpectationvalues(VEVs)ofthe SMsingletfields

σ

i.NextinSec.3,we relate theseleptonicmassmatriceswiththePMNSmatrixandcompareitwiththedatapresentedintheParticleDataBook[26].InSec.4,we presentnumerical analysesandobtain|δPMNS|<62.8^o forthe normalhierarchyofneutrinomasseswiththePDGbookparametrization [26].Section5isaconclusion.ThemagnitudeontheweakCPviolation J [27] isbrieflydiscussedinAppendixA.

2. Suggestionfromtheflipped SU(5)model

IfweconsideronlytheSMparticles,neutrinomassesarisefromthediagramshowninFig.1.Anyfurtherattachmentstothisdiagram areSM singletscalars.Ifwe considerthequantumnumbers underSU(2)W×^U(1)Y,two neutrinoshave1₋1⊕^3↑₋1 where ↑^{means that} the 3rd component of the weak isospin is +^{1. Possible additional scalar} attachments to Fig. 1 must carry quantum number 1₊₁ or 3^↓₊₁ together withtwo H_u⁰’s, and 1₊1 is ruled out because ¹+¹ breaks U(1)em. 3^↓₊₁ allows the scalar attachments, shown as Hu⊕ Hu in Fig. 1. Depending ondetails of highenergyfields, implied by thequestion markinthe gray, two typesof neutrinomassesare named,TypeIseesaw[28] andTypeIIseesaw[29].TypeIIIseesaw[30] requiresmorelightparticlesattheelectroweakscale.Fromthe SU(5)flipspectrashowninRef. [20],wenote thatthereisnoSU(2)tripletrepresentation;henceonlyTypeIseesawisallowed fromour stringcompactification.

2 Amongthese,theanomalousU(1)canworkasaflavorsymmetry[20].NotethatinadditiontotheseweuseU(1)’sfromtheextradimensionstowardZ4Rdiscrete symmetryinRef. [9].

Table 2

U(1)chargesofL-handedneutralscalars(but σ7,8forR-handed).Wekeptuptooneoscillatorrepresented asNumberofresultingfields(number of oscillating mode).For example,n(11¯)meansthatthereresultsnmultiplicitieswithoneoscillator ⁻₁₂⁵.For Q18,20,22charges,herewelistedonlythoseofL-handedfields,participatinginthe Yukawacouplings.σ2,3,4,11,15,21,22,23,24havephasei=0,whichcanbeusedtobreakZ4RdowntoZ2R.

State(P+^kV0) i (N^L)j P·^RX(Sect.) QR Q1 Q2 Q3 Q4 Q5 Q6 Qanom Q18 Q20 Q22

^∗₁ (+ + + − −;⁰³)(0⁵;⁻4¹−¹ 4 +²

4) 0 2(11) 210₋1(T3)L +^{4 0 0 0 0} +⁹ +^{3 −}7³³ −¹ +¹ −¹

^∗₁ (+ + + − −;⁰³)(0⁵;⁻4¹−1 4 +2

4) ⁺₃² 1(13) 110₋1(T3)L +^{4 0 0 0 0} +⁹ +^{3 −}7³³ −¹ +¹ −¹

2 (+ + − − −;0³)(0⁵;⁺4¹+¹ 4 −²

4) 0 2(1¯1) 210+1(T3)L −4 0 0 0 0 −9 −3 ⁺₇³³ −1 −1 −1

2 (+ + − − −;⁰³)(0⁵;⁺4¹+1 4 −2

4) ⁺₃¹ 1(1_¯3) 110₊1(T3)L −^{4 0 0 0 0} −⁹ −^{3 +}7³³ −¹ −¹ −¹

σ1 (0⁵;⁻3²−2 3 −2

3)(0⁸) ⁺₄¹ 0 2·10(T⁰₄) −4 −8 −8 −8 0 0 0 −12 −1 −1 −1

σ2 (0⁵;⁻3²+1 3 +1

3)(0⁸) 0 3(1¯1) 3·¹⁰(T⁰₄) 0 −⁸ +⁴ +^{4 0 0 0} −² −¹ −¹ −¹

σ3 (0⁵;¹3 −2 3

1

3)(0⁸) 0 3(1_¯1) 3·¹0(T⁰₄) 0 +⁴ −⁸ +^{4 0 0 0} −⁸ −¹ −¹ −¹

σ4 (0⁵;¹3 1 3−²

3)(0⁸) 0 3(1¯1) 3·10(T⁰₄) 0 +4 +4 −8 0 0 0 +10 −1 −1 −1

σ5 (0⁵;⁰¹⁰)(0⁵;¹2−1

2 0) ⁺₂¹ 0 2·¹0(T6) +^{4 0} +^{12 0} +^{12 0 0} +¹⁴ −¹ −¹ −¹

σ6 (0⁵;001)(0⁵;⁻2¹ 1

20) ⁺₂¹ 0 2·10(T6) 0 0 0 +12 −12 0 0 −4 −1 −1 −1

σ7 (0⁵;⁰−¹⁰)(0⁵;⁻2¹ 1

20) ⁺₂¹ 0 2·¹0(T6)R +^{4 0} +^{12 0} +^{12 0 0} +¹⁴ −¹ +¹ −¹

σ8 (0⁵;00−1)(0⁵;¹2−1

2 0) ⁺₂¹ 0 2·10(T6)R −2 0 0 +12 −12 0 0 −4 −1 +1 −1

σ11 (0⁵;⁻2¹−1 2 −1

2)(0⁵;³4−1 4 −1

2) ⁺₃² 2(11+¹³,11¯+¹3¯) 2·¹⁰(T3) −⁶ −⁶ −⁶ −⁶ +¹² −⁹ −^{3 −}7³⁰ +¹ +¹ −¹ σ₁₁ (0⁵;⁻2¹−1

2 −1 2)(0⁵;³4−1

4 −1

2) 0 4(11+¹3,1_1¯+¹3¯) 4·¹0(T3) −⁶ −⁶ −⁶ −⁶ +¹² −⁹ −^{3 −}7³⁰ −¹ +¹ +¹ σ12 (0⁵;⁻2¹

1 2 1 2)(0⁵;³4−¹

4 −¹

2) ⁺₃¹ 2(11+13,11¯+13¯) 2·10(T3) −2 −6 +6 +6 +12 −9 −3 ⁺₇⁴⁰ +1 +1 −1 σ₁₂ (0⁵;⁻2¹

1 2 1 2)(0⁵;³4−1

4 −1

2) ⁺₃² 2(11+¹3,1_1¯+¹3¯) 2·¹0(T3) −² −⁶ +⁶ +⁶ +¹² −⁹ −^{3 +}7⁴⁰ −¹ +¹ +¹ σ13 (0⁵;¹2

1 2−1

2 )(0⁵;⁻4¹ 3 4−1

2) ⁺₃¹ 2(11+13,11¯+13¯) 2·10(T3) −6 +6 +6 −6 −12 −9 −3 ⁺¹²⁴₇ +1 +1 −1 σ13 (0⁵;¹2

1 2−1

2 )(0⁵;⁻4¹ 3 4−1

2) ⁺₃² 2(11+¹3,1_1¯+¹3¯) 2·¹0(T3) −⁶ +⁶ +⁶ −⁶ −¹² −⁹ −^{3 +124}7 −¹ +¹ +¹ σ14 (0⁵;¹2

1 2−1

2 )(0⁵;⁻4¹ −1 4

1

2) ⁺₃² 2(1¯1)+1(13¯) 3·10(T3) +4 +6 +6 −6 0 +9 +3 ⁺₇⁵⁸ −1 +1 +1

σ15 (0⁵;⁻2¹−¹ 2 −¹

2)(0⁵;⁺4³ −¹ 4 −¹

2) ⁺₃² 2(11+¹³,11¯+¹3¯) 2·¹⁰(T3) −⁶ −⁶ −⁶ −⁶ +¹² −⁹ −^{3 −}7³⁰ +¹ +¹ −¹ σ₁₅ (0⁵;⁻2¹−1

2 −1

2)(0⁵;⁺4³ −1 4 −1

2) 0 2(11+¹3,1_1¯+¹3¯) 4·¹0(T3) −⁶ −⁶ −⁶ −⁶ +¹² −⁹ −^{3 −}7³⁰ −¹ +¹ +¹ σ16 (0⁵;⁻2¹+¹

2 +¹

2)(0⁵;⁺4³ −¹ 4 −¹

2) ⁺₃¹ 2(11+13,11¯+13¯) 2·10(T3) −2 −6 +6 +6 +12 −9 −3 ⁺₇⁴⁰ +1 +1 −1 σ₁₆ (0⁵;⁻2¹+1

2 +1

2)(0⁵;⁺4³ −1 4 −1

2) ⁺₃² 2(11+¹3,1_1¯+¹3¯) 2·¹0(T3) −² −⁶ +⁶ +⁶ +¹² −⁹ −^{3 +}7⁴⁰ −¹ +¹ +¹ σ17 (0⁵;⁺2¹+1

2 −1

2)(0⁵;⁻4¹ +3 4 −1

2) ⁺₃¹ 2(11+13,11¯+13¯) 2·10(T3) −6 +6 +6 −6 −12 −9 −3 ⁺¹²⁴₇ +1 +1 −1 σ17 (0⁵;⁺2¹+1

2 −1

2)(0⁵;⁻4¹ +3 4 −1

2) ⁺₃² 2(11+¹³,1_1¯+¹3¯) 2·¹⁰(T3) −⁶ +⁶ +⁶ −⁶ −¹² −⁹ −^{3 +124}7 −¹ +¹ +¹ σ18 (0⁵;¹2 +1

2 −1

2)(0⁵;⁺4³−1 4 −1

2) ⁺₃² 2(1_¯1)+¹(1_3¯) 2·¹0(T3) +⁴ +⁶ +⁶ −^{6 0} +⁹ +^{3 +}7⁵⁸ −¹ +¹ +¹

σ21 (0⁵;⁻6¹−¹ 6 −¹

6)(0⁵;¹4 1 4 −¹

2) 0 1(1¯1) 10(T₁⁰) +2 −2 −2 −2 0 +9 +3 ⁺₇¹² −1 −1 −1

σ22 (0⁵;⁻6⁵ 1 6 1 6)(0⁵;¹4

1 4−1

2) 0 1(1_¯1+¹3) 10(T₅⁰) +² −¹⁰ +² +^{2 0} +⁹ +^{3 −}7² −¹ +¹ +¹

σ23 (0⁵;¹6 −5 6

1 6)(0⁵;¹4

1 4−1

2) 0 1(1¯1+13) 10(T₅⁰) +2 −10 +2 +2 0 +9 +3 ⁻₇⁴⁴ −1 +1 +1

σ24 (0⁵;¹6 1 6−5

6)(0⁵;¹4 1 4−1

2) 0 1(1_¯1+¹3) 10(T₅⁰) +² −¹⁰ +² +^{2 0} +⁹ +^{3 +}7⁸² −¹ +¹ +¹

TheSM fieldsfrom[9] areshowninTable1,andtheSM singletfields,includingthose in10₋₁ and10₊₁ ofSU(5)flip,areshownin Table2.ConsideringtheSM singletattachments to Fig.1,letusconsidertheneutrinomassoperatorsallowed bythequantumnumbers ofTables1and2.Firstly,thediagonalmassesare

M^ν₃₃

∝

M

˜

¹³₃

d²

ϑ

5₊₃

(

U₃

,

0

; +

¹

)

5₊₃

(

U₃

,

0

; +

¹

)

5₋₂

(

T₆

,

¹

3

; −

²

)

5₋₂

(

T₆

,

¹

3

; −

²

)

10₋₁

(

T₃

,

¹

3

; +

⁴

)

10₋₁

(

T₃

,

0

; +

⁴

)

M^ν₂₂

∝

M

˜

¹⁴₂

d²

ϑ

5₊₃

(

T₄⁰

,

¹

4

; −

¹

)

5₊₃

(

T₄⁰

,

¹

4

; −

¹

)

5₋₂

(

T₆

,

¹

3

; −

²

)

5₋₂

(

T₆

,

¹

3

; −

²

)

10₋₁

(

T₃

,

¹

3

; +

⁴

)

10₋₁

(

T₃

,

0

; +

⁴

)

·

¹0

( σ

5

,

T6

,

¹ 2

; +

⁴

)

(1)

wherethelastnumberafter;isthe QR charge,andM˜3 andM˜2 aredeterminedby?inFig.1.Weneed QR=^{2 modulo4aboveford2}ϑ integration.Mν

11,Mν 12,Mν

21havethesamestructureasMν

22.Notethattheselectionruleismakingthephaseanintegermultiple,whichis satisfiedabovefori,viz.0+⁰+¹₃+¹₃+¹₃+⁰=^{1 and 1}₄+¹₄+¹₃+¹₃+¹₃+⁰+¹₂=^{2.Then,theabovemassesareestimatedas}

M^ν₃₃

∼

^{v2E W}M

˜

^M3₃²¹⁰

,

M^ν₂₂

∼

^{v2E WM102}

| σ

5

|

M

˜

⁴₂

,

(2)

whereM10= ¹⁰−¹ = ¹⁰+¹ .Then,neutrinomixingmassesaregenerallyoforder v²_{E W}/M˜ sincetheSMsingletVEV

σ

5 canbeatthe GUTscalewithoutbreakingZ4R.

Fortheoff-diagonalmassesbetweenU3 andT₄⁰neutrinos,weneed QR=^{0 modulo4ford2}ϑd²ϑ¯ integration.

M^ν₃₂

,

M^ν₃₁

∝

M

˜

¹⁶

d²

ϑ

d²

ϑ ¯

5₊₃

(

U₃

,

0

; +

¹

)

5₊₃

(

T₄⁰

,

¹

4

; −

¹

)

5₋₂

(

T₆

,

¹

3

; −

²

)

5₋₂

(

T₆

,

¹ 3

; −

²

)

·

10₋1

(

T3

,

0

; +

4

)

10₋1

(

T3

,

0

; +

4

) ·

10

( σ

1

,

T₄⁰

,

¹

4

; −

4

)

^∗10

( σ

14

,

T3

,

² 3

; +

4

)

^∗

.

(3)

Then,theabovemassmixingisestimatedas M^ν_13,23

∼

^{v2E WM210}

| σ

1

σ

14

|

M

˜

_×⁵

,

(4)

whereM˜_× issomemassscaledeterminedbytheaboveequations.Notethat^∗₂,1 and

σ

1 canhavetheGUTscaleVEVsbecauseallof themcarry Q_R=^{0 modulo4,andweobtainasimilarorderofmassforallofMν}_{11,12,22,33,13,23}^.

ComparingMν

11,22,31,32andMν 33, M^ν₁₁

,

M^ν₂₂

M^ν₃₃

≈ σ

5

M

˜ ,

^M

ν 31

,

M^ν₃₂

M₃₃^ν

≈ σ

1

σ

14

M

˜

²

,

(5)

wenotethattheneutrinomasshierarchycanbethenormalhierarchy(inthesensethat

ν

τ istheheaviest)iftheVEVsof

σ

singletsare comparablysmall,|

σ

1|,|

σ

5|<M.˜

Sinceweobtainedallentriesintheneutrinomassmatrix,hereweinvestigatehowtheCPphasecanbeinsertedinthemassmatrix ofthe Q_em= −^{1 leptonsandintheneutrinomassmatrix.}

2.1. NeutrinomassmatrixinspiredbyflippedSU(5)

InRef. [9] basedontheflippedSU(5)modelof[10],apossibleidentificationZ4R hasbeenachieved,forbiddingdimension-5Bviolating operatorsbutallowing theelectroweakscale

μ

termanddimension-5Lviolating Weinbergoperator.IntheflippedSU(5), theneutrino massesariseintheform

−

Lν^{I J}

=

^f_{I J}^(ν)

({ σ })

⁵₊^I,i₃⁵₊^J,₃^j5k₋₂

(

H_u

)

5^l₋₂

(

H_u

)[

¹⁰−1

(

H_GUT

)

10₋₁

(

H_GUT

)]

i jkl

+

^h

.

c

.,

(6) wherethecouplings f⁽ν)

I J arecomplexparameters, I and J areflavorindices,i,j,k,l,mareSU(5)indices,andthesubscriptistheU(1)_X quantumnumberofSU(5)_flip.5₋2 isusuallydenotedasHuL,and10₋1,togetherwith10₊1,istheten-pletneededforbreakingtherank 5gaugegroupSU(5)×^{U(1)ataGUTscaledowntotherank4SMgaugegroup.}

Consider 5₊^I,i₃5₊^J,₃^j inEq. (6) which is symmetricunder I and J.Thus, the neutrino massmatrix is symmetric.The Majorana phase factoredout in Eq. (20) is fromtheheavy neutrinos, whichwill not affectourstudyof CC interactions ofSec. 3.We assume that the neutrinomassmatrix,beingsymmetric,isreal.Thus, V⁽ν)canbeconsideredtobeanorthogonalmatrix O⁽ν).

2.2. MassmatrixofchargedleptonsinspiredbyflippedSU(5)

Ascommentedabove, wecanalways take V⁽ν) asa realmatrix O⁽ν).Thus,the PMNSmatrixgivenintheKSform, Eq. (25),can be representedas

V_KS^(l)†

=

^V(e)O(ν)T

=

⎛

⎝

^q11r11

+

^q12r₁₂

+

^q13r₁₃

,

q₁₁r₂₁

+

^q12r₂₂

+

^q13r₂₃

,

q₁₁r₃₁

+

^q12r₃₂

+

^q13r₃₃

,

q21r11

+

^q22r12

+

^q23r13

,

q21r21

+

^q22r22

+

^q23r23

,

q21r31

+

^q22r32

+

^q23r33

,

q₃₁r₁₁

+

^q32r₁₂

+

^q33r₁₃

,

q₃₁r₂₁

+

^q32r₂₂

+

^q33r₂₃

,

q₃₁r₃₁

+

^q32r₃₂

+

^q33r₃₃

,

⎞

⎠

(7)

wheretheelements V_{i j}^(l)=^qi j andO⁽ν)

i j =^ri j arecomplexandrealnumbers,respectively.Makingthe1strowrealfortheKSform,q11,q12 andq13arerequiredtobereal.

Theunitarymatricesrelatingtheweakeigenstateslamdmasseigenstatesi ofthechargedleptonsarenamedasV forL-handedfields andU forR-handedfields,

lL

=

3

j=¹

V_lj^(l)l⁽_jL^mass)

,

lR

=

3

j=¹

U_lj^(l)l⁽_{j R}^mass)

,

(8)

where¯l_L^mass=(1,2,3)L intermsofmasseigenstates1,2,3,andl^mass_R =(N₁,N₂,N₃)R.Themassmatrixbecomes

¯

l_L

(

V_e^(l)M^massU⁽_e^l)†

)

l_R (9)

intheweakeigenstatebasis.SinceR-handedleptonsarenotparticipatingintheCCinteractions,theleptonR-handedunitarymatrixU^(l) canbetakenastheidentitymatrix.Thus,themassmatrixintheweakbasisbecomes

M^(l)

=

V^(l)

⎛

⎝

m

˜

e

,

0

,

0 0

,

^m

˜

μ

,

0 0

,

0

,

1

⎞

⎠

_U^(l)†

=

⎛

⎝

^q11m

˜

e

,

q12m

˜

_μ

,

q13

q₂₁^m

˜

e

,

q₂₂^m

˜

μ

,

q₂₃ q₃₁m

˜

_e

,

q₃₂m

˜

_μ

,

q₃₃

⎞

⎠ =

⎛

⎝

^real

,

real

,

real complex

,

complex

,

complex complex

,

complex

,

complex

⎞

⎠

₍₁₀₎

whereV_{i j}^(l)=^qi j andweobtainedq₁₁,q₁₂andq₁₃arerealnumbers.

We show that the quantumnumbers of themodel presented in[9] allows an effectivemass matrix formEq. (10) forthe charged leptons.

−

L_l^{I J}

=

^f_{I J}^(l)

({ σ })

⁵₊^I,i₃¹₋^J₅⁵₊2,