Clockwork for neutrino masses and lepton flavor violation

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Physics Letters B

www.elsevier.com/locate/physletb

Clockwork for neutrino masses and lepton flavor violation

Alejandro Ibarra

^a,b

, Ashwani Kushwaha

^c

, Sudhir K. Vempati

^c,∗

aPhysik-DepartmentT30d,TechnischeUniversitätMünchen,James-Franck-Straße,85748Garching,Germany bSchoolofPhysics,KoreaInstituteforAdvancedStudy,Seoul02455,SouthKorea

cCentreforHighEnergyPhysics,IndianInstituteofScience,C.V.RamanAvenue,Bangalore560012,India

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

Articlehistory:

Received18November2017

Receivedinrevisedform14February2018 Accepted20February2018

Availableonline26February2018 Editor:G.F.Giudice

WeinvestigatethegenerationofsmallneutrinomassesinaclockworkframeworkwhichincludesDirac masstermsaswellasMajoranamasstermsforthenewfermions.Wederiveanalyticformulasforthe massesofthenewparticlesandfortheirYukawacouplingstotheleptondoublets,inthescenariowhere theclockworkparametersareuniversal.WhentheuniversalMajoranamassvanishes,thezeromodeof theclockworksectorformsaDiracpairwiththeactiveneutrino,withamasswhichisinagreementwith oscillationsexperimentsforasufficientlylargenumberofclockworkgears.Ontheotherhand,whenit doesnotvanish,neutrinomassesaregeneratedviatheseesawmechanism.Inthiscase,andduetothe factthattheeffectiveYukawacouplingsofthehighermodescanbesizable,neutrinomassescanonlybe suppressedbypostulatingalargeMajoranamassscale.Finally,wediscusstheconstraintsonthemass scaleoftheclockworkfermionsfromthenon-observationoftherareleptonicdecayμ_→eγ.

©^{2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

The smallness of neutrino masses stands as one of the most puzzlingopenquestionsinFundamentalPhysics.Aplausiblesolu- tiontothispuzzleisprovidedbytheseesawmechanism,inwhich thesmallnessof neutrinomassesis explainedby thebreaking of theleptonnumberataveryhighenergyscale[1–5].Modelswith conserved leptonnumber, onthe other hand,can alsoreproduce theobservations, atthe expenseofpostulating tiny Yukawa cou- plings of the neutrino to the Standard Model Higgs. Such small parameters are usually regarded as unnatural, however the ex- istence of tiny Yukawa couplings is a phenomenologically viable possibility, andcan be accomplishedin furtherextensions ofthe model(forreviewsandrecentmodels,seee.g.in [6–19]).

Recently, a new mechanism of generating small couplings in theoriescoupledto theStandardModelhas beenintroduced [20, 21].Themechanism,reminiscentofdeconstructionmodels[22,23], canbesummarizedasalinearquivermodelwithnolargehierar- chies in the theory parameters, that gives rise to site-dependent suppressedcouplingstothezero-mode[24].Originally,introduced foraquiverofAbelianGoldstonebosons(axions),ithasbeengen- eralized to fermions, vectors and other fields [24,25] (See also

*

Correspondingauthor.

E-mailaddresses:ibarra@tum.de(A. Ibarra),ashwani@chep.iisc.ernet.in (A. Kushwaha),vempati@chep.iisc.ernet.in(S.K. Vempati).

[26]). Applications and generalizations of this mechanism have beendiscussedin[27–43],andspecificallyframeworkstoexplain theobservedpatteroffermionmassesin[44–46].

Inthisworkweexploretheapplicationofthefermionicclock- work to the generation of smallneutrino masses. Concretely, we identify the right-handed neutrinos with the zero modes of a clockwork sector [24], such thatsmallcouplings canbe naturally generatedandthereforesmallneutrinomasses.Weanalyzeinde- tailtheframeworkwheretheDiracmasses,Majoranamassesand nearest neighbor interactionsareuniversal, complementingprevi- ous studiesin[45,46] wheretheMajoranamasstermislocalized on just one of the modes. We derive analytical formulas for the massesofthenewparticlesandfortheircouplingstotheStandard Modelfermions,forthecaseswhentheMajoranamasstermsare includedintheLagrangianandwhentheyarevanishing.Weshow thattheclockworkmechanism,i.e.,thesuppressionoftheYukawa couplings bysitedependent powerfactors, isnot affectedby the presenceoftheMajoranamassterms.Infact,whilethezeromode contribution is a combination of the clockwork suppression and the Majorana seesaw,the scale is however set by the dominant contribution by the gears, which have O(1) Yukawa couplings, through the standardseesaw mechanism. Furthermore,while the clockwork mechanismsuppressesthecouplingsofthezeromode, the couplingsofthehighermodescanbe sizableandinduce,via loops,potentiallylargeratesfortheleptonicraredecays.

The restofthepaperisorganized asfollows.Insection 2,we presentthemostgeneralframeworkforclockwork neutrinoswith https://doi.org/10.1016/j.physletb.2018.02.047

0370-2693/©^{2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

DiracandMajoranamassterms,andwediscusstheirphenomenol- ogyinsubsections2.1and2.2,respectively.Insection3,wediscuss leptonflavor violationintheclockworkscenarioandcalculatelim- itsonthegearmasses.Weclosewithasummary.

2. Neutrinosinclockwork

We extendthe Standard Modelwith n left-handedandn+¹ right-handed chiral fermions, singlets under the Standard Model gaugegroup,whichwedenoteasψLi(i=⁰,...,n−¹)andψRi(i= 0,...,n)respectively.TheLagrangianofthemodelreads:

L

=

LSM

+

LClockwork

+

Lint

,

(1)

whereLSMistheStandardModelLagrangian,LClockworkisthepart oftheLagrangianinvolvingonlythenewfermionsinglets,andLint istheinteractiontermofthenewfieldswiththeStandardModel fields. Following [24], we assume that the Standard Model only couplestothelastsiteofthefermionicclockwork,therefore,

Lint

= −

^Y

H LL

ψ

Rn

,

(2) with H=ⁱ

τ

2H^∗, H the Standard Model Higgs doublet and LL theleft handed lepton fields (weassume only one generation of fermions;the generalizationto morethanone generation willbe discussedbelow).

Infullgenerality,theclockworkLagrangiancanbecastas:

LClockwork

=

Lkin

−

n−¹

i=0

m_i

ψ

_Li

ψ

Ri

−

^m_i

ψ

_Li

ψ

Ri+¹

+

^h

.

c

.

−

n−1

i=⁰ 1

2M_Li

ψ

_Li^c

ψ

Li

−

n i=⁰ 1

2M_Ri

ψ

^c_Ri

ψ

Ri

,

(3)

whereLkin denotes the kinetic term forall fermions, andm,m andML,R are massparameters. Denoting =(ψL0,ψL1,...ψLn−¹, ψ^c_R0,ψ^c_R1,...,ψ_Rn^c ),theclockworkLagrangiancanbewritteninthe compactform:

LClockwork

=

LKin

−

¹

2

(

^cM

+

^h

.

c

.)

(4)

with M a (2n+¹)×(2n+¹) mass matrix. We note that Lkin isinvariant underthe globalgroup U(n)L×^U(n+¹)R. Themass termsmibreaktheglobalgroupU(n)L×^U(n+¹)R→n−¹

i=⁰U(1)i, where U(1)i acts as ψL,i→^eiαiψL,i, ψRi →^eiαiψRi, and com- binedwiththemasstermsm_i,breaktheglobalsymmetryU(n)L× U(n+¹)R→^U(1)CW,whereU(1)CWactsasψL,i→^eiαψL,i,ψR,i→ eⁱαψR,i for all i. Finally, MLi and MRi are Majorana masses for the left and right handed singlet fields. It is sufficient that M_Li or MRi is non-vanishing forone i to break the symmetry group U(n)L×^U(n+¹)R→^nothing.

We assume for simplicity universal Dirac masses, Majorana massesandnearestneighborinteractions,namelymi=^m,m_i=^mq M_Ri=^MLi=^{mq forall i.Underthis}assumption,themassmatrix reads:

M

=

^m

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎝

q 0

· · ·

^{0 1}

−

^q

· · ·

⁰ 0

^q

· · ·

^{0 0 1}

· · ·

⁰

.. . .. . .. . .. . .. . .. . .. . .. .

0 0

· · ·

^{q 0 0 0}

−

^q 1 0

· · ·

⁰

q 0

· · ·

⁰

−

q 1

· · ·

0 0

q

· · ·

0

.. . .. . .. . .. . .. . .. . .. . .. .

0 0

· · · −

q 0 0 0

q

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎠

,

(5)

whichhaseigenvaluesM_k givenby:

M0

=

^m

q

,

M_k

=

^m

^q

−

^m

λ

k

,

k

=

¹

, . . . ,

n

,

M_n+k

=

^m

q

+

^m

λ

_k

,

k

=

¹

, . . . ,

n

,

(6)

withλk definedas

λ

_k

≡

^q2

+

¹

−

^{2qcos k}

π

n

+

1

.

(7)

Withourconventions,theeigenvaluescanbepositiveornegative;

thephysicalmassescorrespondtothemodulioftheeigenvalues.

Themasseigenstates,whichwedenoteas

χ

k,arerelatedtothe interactioneigenstatesjbytheunitarytransformationU^,namely

j=

jUjk

χ

k.ThematrixU ^{canbeexplicitly}calculated,there- sultbeing:

U

=

⎛

⎝

0 √¹

2U_L

−

^√1₂^UL

u_R √¹

2U_R √¹ 2U_R

⎞

⎠ .

(8)

where0 anduR aren-dimensionalvectors,withentries:

0

_j

=

⁰

,

j

=

¹

, ...,

n

,

(9)

(

u_R

)

j

=

¹ q^j

q²

−

¹

q²

−

^q−2n

,

j

=

¹

, ...,

n

,

(10)

whileUL andUR are,respectively,n×^nand(n+¹)×^nmatrices withelements

(

UL

)

jk

=

2

n

+

1sin jk

π

n

+

1

,

j

,

k

=

1

, ...,

n

, (

U_R

)

_jk

=

2

(

n

+

¹

)λ

_k

qsin jk

π

n

+

¹

−

^sin

(

j

+

¹

)

k

π

n

+

¹

,

j

=

⁰

, ..,

n

,

k

=

¹

, ...,

n

,

(11)

WenotethatthemixingmatrixUdoesnotdependontheparam- eter^q,whichisaconsequenceofourassumptionofuniversalityof theMajoranamassesMRi=^MLi=^mqforalli.

TheinteractionLagrangian oftheclockwork fieldstotheStan- dard Model fields, Eq. (4), can now be recast in terms of mass eigenstates:

Lint

= −

^{Y L}L

HUnk

χ

_k

≡ −

2n k=⁰

Y_kLL

H

χ

_k

,

(12) where

Y0

≡

^Y

(

uR

)

n

=

^Y qⁿ

q²

−

¹

q²

−

^q−2n

,

(13)

Y_k

=

^Yk+ⁿ

≡ √

¹

2Y

(

U_R

)

nk

=

^Y

1

(

n

+

¹

)λ

k

qsin nk

π

n

+

¹

,

k

=

¹

, ...,

n

.

(14)

Thecomponents(uR)n and(UR)np,whichdescribethefractionof thenth “gear”inthezeromode,willplayamajorroleinthephe- nomenology,astheyparametrizetheportalstrengthbetweenthe StandardModelsectorandtheclockworksector.

After electroweak symmetry breaking new mass terms arise which mix the Standard Model neutrino with the clockwork

fermions. The mass matrix of the 2n+2 electrically neutral fermionfieldsofthemodelreads:

m_ν

=

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎝

ν

L

χ

0

χ

1

χ

2

· · · χ

2n

ν

_L 0 v Y₀ v Y₁ v Y₂

· · ·

^{v Y}2n

χ

₀ v Y₀ M₀ 0 0

· · ·

⁰

χ

1 v Y1 0 M1 0

χ

2 v Y2 0 0 M2

· · ·

⁰

.. . .. . .. . .. . .. . .. . .. .

χ

2n v Y_2n 0 0 0

· · ·

^M2n

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎠

,

(15)

where v=²⁴⁶/√

2 GeV is the Higgs vacuumexpectation value.

Upondiagonalizingthismassmatrix,one findsamassfortheac- tive neutrino. Furthermore, the off-diagonal entries in the mass matrix translate into charged current interactions between the chargedleptonandthek-thmode,aswellasneutral-currentand Higgsinteractionsofthelightneutrino,proportionalto∼^{v Y}k/M_k, andwhichcanbesizable.

Inorderto accommodatetheleptonic mixingobserved inNa- tureitis necessarytointroduce threegenerations ofleptondou- blets, as well as N generation of clockwork fermions, each con- sisting of nα left-handed and nα +1 right-handed gears, where

α

₌1,. . . ,N (phenomenologically, N≥^{2, inorderto account for} thetwoobservedoscillationfrequencies).Furthermore,theYukawa couplinginEq. (2) andallthemassparametersinEq. (3) mustbe promotedtomatricesinflavorspace.Inthisworkwewillassume forsimplicitymαβ

i =^mδ^αβ,mαβ

i =^mqαδ^αβ Mαβ

Ri =^Mα_Li^β=^mqαδ^αβ forall i.Namely, the massparameterm is universal forallgears and all generations, while the mass parameters m, M_R and M_L are commonforall gearswithin one generation, butinprinciple differentamonggenerations.

Denoting ^α=(ψ_L0^α,ψ_L1^α,...,ψ_Ln^α₋₁,ψ_R0^αc,ψ^α_R1^c,...,ψ_Rn^αc) as the fermion field which has as component all the clockwork fields withinthegeneration

α

,theclockworkandinteractionLagrangian canbewrittenas:

LClockwork

=

LKin

−

¹

2

(

^αcM^αβ

^β

+

^h

.

c

.) ,

(16) Lint

= −

^YaαLa_L

H

ψ

_R^α_,n

,

(17)

where a=¹,2,3 and

α

,β=¹,...,N. As for the one generation case, we assumedthat the Standard Model lepton doublets only coupletothen-thsitesoftheN clockworkgenerations.

The Lagrangian expressed in the mass eigenstate basis, ^α_k = U_kj^αβ

χ

^β_j,read:

LClockwork

=

LKin

−

¹

2

( χ

_k^αcM_k^α

χ

_k^α

+

h

.

c

.) ,

(18) Lint

= −

2n k=0

Y_k^aβL^a_L

χ

_k^β

,

(19)

whereY_k^aβ≡^YaαU_nk^{αβ with}U_nk^{αβ thematrixthatmixesfermionsof} different clockwork gears anddifferent generations. Finally,after electroweak symmetry breaking, the mass matrix of the N(2n+ 1)+3 electricallyneutralfermionsofthemodelreads:

m_ν

=

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎝

ν

^a_L

χ

₀^β

χ

₁^β

χ

₂^β

· · · χ

_2n^β

ν

^a_L 0 v Y₀^aβ v Y₁^aβ v Y₂^aβ

· · ·

v Y^a_2n^β

χ

₀^β v Y₀^βa M^β₀ 0 0

· · ·

0

χ

₁^β v Y₁^βa 0 M^β₁ 0

χ

₂^β v Y₂^βa 0 0 M^β₂

· · ·

⁰

.. . .. . .. . .. . .. . .. . .. .

χ

_2n^β v Y_2n^βa 0 0 0

· · ·

^Mβ_2n

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎠

.

(20)

Thismatrixhasingeneralanon-trivialflavorstructure andleads not only tomixingamongthe threeactive neutrinos,butalso to potentially large lepton flavor violating charged current, neutral current and Higgs interactions, thus providing a possible test of thisframework,aswillbediscussedinSection3.

In what follows, we will consider separately the case when the universal Majorana mass is vanishing and when it is non- vanishing.

2.1. VanishinguniversalMajoranamass

WeconsiderfirstthecasewheretheuniversalMajoranamassis equaltozero.Inthiscase,theglobalsymmetryoftheLagrangian isbrokenasU(n)L×^U(n+¹)R→^U(1)CW,whichwillbeidentified withtotal lepton number.The eigenstates andeigenvaluesof the massmatrixcan bedeterminedusingtheresultsofSection 2,by settingq=^0.

ItisusefultorecasttheclockworkLagrangianas

Lclockwork

=

Lkin

−

^NLm_ν^DN_R

+

^h

.

c

.

(21)

where we have defined new fields N_L =(

ν

L,N_L1,...,N_Ln) and N_R=(N_R0,N_R1,...,N_Rn),with

NRk

= √

¹

2

( χ

k

+ χ

k+ⁿ

) ,

k

=

0

, ...

n

,

(22)

N_Lk

= √

¹

2

( − χ

_k

+ χ

_k+n

) ,

k

=

¹

, ...

n

.

(23)

Inthisbasis,themassmatrixhastheform:

m_ν^D

=

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎝

NR0 NR1 NR2

· · ·

^NRn

ν

L 0 0 0

· · ·

⁰

NL1 0 M1 0

· · ·

⁰ NL2 0 0 M2

· · ·

0

.. . .. . .. . .. . . . . .. .

N_Ln 0 0 0

· · ·

^Mn

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎠

.

(24)

where M_k=^m√

λk,withλk definedinEq. (7).Namely, thefields

ν

L andNR0 formamassless Dirac pair,whilethe fields N_Rk and NLk form, for k=¹,...,n, Diracpairs withmass Mk.The overall scaleofthemassivepairsisdeterminedbytheparameterm,and themassdifferencebetweenpairsdependsonq andn.Assuming q>1, one obtains that the massesof the modeswithk>0 in- creasemonotonicallywithn,fromM1≈^m(q−¹)toMn≈^m(q+¹). In Fig. 1, left panel, we show forillustration the mass spectrum of the particles ofthe clockwork sector, labeled by k, takingfor concretenessn=^{10 andq}=^{2.Themassspectrumhasbeennor-} malizedtom.

The mass spectrum is modified after electroweak symmetry breaking by the interactions with the Higgs field. Expressed in termsofN_Rk,theinteractionLagrangianreads:

Fig. 1.Diracmasses(leftpanel)andYukawacouplings(rightpanel)ofthesingletfermionsoftheclockworksector,normalizedrespectivelytomandY,forthespecificcase n=^{10 andq}=^2.

Lint

=

n k=⁰

Y_kLL

H N_Rk

+

^h

.

c

.

(25) with

Y₀

≡

^Y

(

u_R

)

_n

=

^Y qⁿ

q²

−

¹

q²

−

^q−2n

,

(26)

Y_k

≡

^Y

(

U_R

)

nk

=

^Y

2

(

n

+

¹

)λ

k

qsin nk

π

n

+

¹

,

k

=

¹

, ...,

n

.

(27) TheYukawacouplingofthemasslessmodeY0 issuppressedbyqⁿ, providedq>1,whereasthecouplingsofthekth-modeare ofthe same orderas Y. This is illustrated inFig. 1, right panel, which shows the Yukawa couplings of the clockwork fermions to the StandardModelleptondoublets,normalizedtoY,forthesameval- uesofnandq asintheleftpanel(inthiscase,|^Y0|/Y≈⁸×¹⁰⁻⁴ andisnotvisiblefromthefigure.)

Themassmatrixofthe electricallyneutralfermionfieldsnow reads:

m_ν^D

=

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎝

N_R0 N_R1 N_R2

· · ·

^NRn

ν

L v Y₀ v Y₁ v Y₂

· · ·

^{v Y}n

N_L1 0 M₁ 0

· · ·

⁰ N_L2 0 0 M₂

· · ·

⁰

.. . .. . .. . .. . . . . .. .

N_Ln 0 0 0

· · ·

^Mn

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎠

.

(28)

Concretely,a masstermfortheactive neutrinosisgenerated.As- suming that M_k^Y0v, which as we will see below is justified fromthecurrentlimitsonrareleptonic decays,one canapproxi- matetheactiveneutrinomassby

m_ν

≈

v Y0 (29)

and can be made small by choosing appropriate values of Y, q andn.Forinstance,assuming Y=O(1),q=^{2,oneobtains mν}= O(⁰.1)eV forn≈^40.

Thegeneralization ofthe abovesetup tothree leptonic gener- ationsand N clockwork generationsisstraightforward.The clock- workLagrangianis:

Lclockwork

=

Lkin

−

^Nα_L^mα_ν^Nα_R

+

^h

.

c

.

(30) withNα

L =(

ν

_L^α,Nα L1,...,Nα

Ln)andNα R=(Nα

R0,Nα R1,...,Nα

Rn),where

N^α_Rk

= √

¹

2

( χ

_k^α

+ χ

_k^α_+n

) ,

k

=

⁰

, ...,

n

α =

¹

...,

N

,

(31) N^α_Lk

= √

¹

2

(− χ

_k^α

+ χ

_k^α_+n

) ,

k

=

1

, ...,

n

, α =

1

...,

N

,

(32) andtheinteractionLagrangian,

Lint

= −

n k=⁰

Y_k^aβL^a_L

H₀N^β_Rk

,

(33)

withY_k^aβ=^YaαU_nk^αβ.

Afterelectroweaksymmetrybreakingtheneutrinomassmatrix reads:

m^D_ν

=

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎝

N^β_R0 N^β_R1 N^β_R2

· · ·

N^β_Rn

ν

_L^a v Y^a₀^β v Y^a₁^β v Y^a₂^β

· · ·

v Y_n^aβ N^β_L1 0 M₁^β 0

· · ·

0 N^β_L2 0 0 M₂^β

· · ·

⁰

.. . .. . .. . .. . . . . .. .

N^β_Ln 0 0 0

· · ·

^Mn^β

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎠

.

(34)

where M_k^β is themass ofk-thclockwork gearforthe Dirac pair N^β_L,N^β_R.

Weanalyzeindetailthe casewheretheclockwork consistsof two generations withn₁ andn₂ gears,respectively. Wescan Y^aα within the ranges ¹₄ <|^Yaα|<4, qα between1.5 and 6 and nα between15and55, andweselect thepoints that reproducethe observed values ofthe solarand atmosphericmass splitting and mixingangleswithin1

σ

,asdeterminedinRef. [47].InFig.2(left panel)weshowasgreencircles(yellowtriangles)thevaluesofn1 (n2) as a function of q1 (q2) that satisfy the experimental con- straints. As apparent from the plot, larger qα require a smaller numberofgearstoreproducethesmallneutrinoYukawacoupling.

Furthermore,theallowed valuesforn1 andn2 haveabigoverlap, whichisaconsequenceofourassumptionofcomparableelements inthecouplingY^aα andthenecessityofproducingamildhierar- chybetweenthesolarandtheatmosphericneutrinomassscales.

Inparticular,we findthatthe scenariowithq1=^q2 andn1=ⁿ2, namelythescenariowheretheclockworkparametersareuniversal also amonggenerations, is allowed by observations.This isillus- trated inFig. 2 (rightpanel), whichshows theallowed values of q1−^q2asafunctionofn1−ⁿ2,anddemonstratestheexistenceof viablepointsatthepointq1=^q2 andn1=ⁿ2.Oneconcretepoint whichleadstothecorrectneutrinoparametersis:

Fig. 2.Valuesofq1 andq2asafunctionofn1andn2(leftpanel),anddifferencesbetweenthem(rightpanel),compatiblewiththemeasuredvaluesoftheneutrinomass splittingsandmixingangleswithin1σ forascenariowithtwoclockworkgenerations.

Fig. 3.Majoranamasses(leftpanel)andYukawacouplings(rightpanel)ofthesingletfermionsoftheclockworksector,normalizedrespectivelytomandY,forthespecific casen=10,q=2 andq=0.1 (darkblue)orq=10 (lightblue).(Forinterpretationofthecolorsinthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

Y

=

⎛

⎝

⁰

.

49 0

.

89 3

.

62 1

.

27 3

.

61 2

.

54

⎞

⎠

(35)

withq₁=^q2=¹.79 andn₁=ⁿ2=^52.

2.2. Non-vanishinguniversalMajoranamass

InthiscasethemassmatrixofthemodelisgivenbyEq. (15) and the Yukawa couplings by Eq. (27). Identifyingq as the or- derparameteroftheU(1)CWsymmetrybreaking,onecanconsider twolimitsofinterest:^qq,1 and^qq,1.

Fig.3showsthemassesofthesingletfermions(leftpanel)and theircorrespondingYukawacouplings(rightpanel)forthespecific casen=^10,q=^2,andq=⁰.1 (darkblue)or^q=^{10 (lightblue);}

the former case corresponds to a mild breaking of the U(1)CW symmetry and the latter to a strong breaking. For^q=⁰.1 one notices that the mode k and the mode n+^{k have very similar} massesandsuggest a pseudo-Diracstructure, whichresults from themildU(1)CWbreaking;inthelimit^q→^{0,theywouldforman} exact Diracpair andhave identicalmasses. Forq=^{10, however,} themassesofallthemodesaremarkedlydifferent.

On the other hand, the Yukawa couplings of the singlet fermionstotheleft-handedleptons,shownintherightpanel,do notdepend onthe valueofq,asdemonstratedin subsection2.1.

The phenomenology ofthescenarioq^q,1 is then verysimilar to the one already discussed in subsection 2.1, while the phe- nomenologyof the scenarioq^q,1 can be rather distinct from the one in the (pseudo-)Dirac case. Indeed, in this scenario one obtainsamassfortheactiveneutrinothroughtheseesawmecha- nismgivenby:

m_ν

≈

k

Y_k²v²

M_k

.

(36)

Then, since the couplings for the higher modes are expected to be O(^Y),theresultingneutrinomasscanbeorders ofmagnitude largerthanthevalueinferredfromoscillationexperiments,unless Y^{1 and/or the gearmasses are very large, inthe same spirit} asinthestandardseesawmechanism. Arelatedanalysiswas also presentedin[46].

3. Leptonflavorviolation

TheclockworkmechanismsuppressestheYukawacouplingsfor thezeromode,henceexplainingthesmallnessofneutrinomasses.

However the Yukawacouplingsforthe highermodesare ingen- eral unsuppressed andcan lead to observable effects at low en- ergies.Inparticular,theleptonflavorviolationgenericallypresent intheYukawacouplingsofthehighermodescontributes,through quantum effectsinduced by clockwork fermions,to generaterare leptonicdecays(suchasli→^lj

γ

)or

μ

-econversioninnuclei,with ratesthatcouldbeatthereachofcurrentorfutureexperimentsif thegearmassesaresufficientlylow.

We calculate the rate for l_i→^lj

γ

following [48–50]. For N clockworkgenerations,weobtain:

B

( μ →

^e

γ )

³

α

emv⁴ 8

π

N α=¹

nα k=¹

Y_k^eαY_k^μα M_k^α2 F

(

x^α_k

)

2

,

where

α

emisthefinestructureconstant,nαisthenumberofgears in the

α

-thgeneration, Mα

k is themass ofthe k-thmode in the

Fig. 4.PredictedvalueofBr(μ→^eγ)forpointsoftheparameterspacereproducing theobservedneutrinooscillationparameters,asafunctionofthemassofthefirst clockworkgear.TheblacksolidlineshowsthecurrentupperlimitfromtheMEG experiment.

α

-thgeneration(k=¹,...,nα),andxα

k ≡^Mα_k²/M²_W.Theloopfunc- tionF(x)isdefinedas

F

(

x

) ≡

¹

6

(

1

−

x

)

⁴

(

10

−

43x

+

78x²

−

49x³

+

4x⁴

−

18x³logx

) ,

(37) andhaslimitsF(0)=⁵/3 and F(∞)=²/3.

Thecurrentupperbound Br(

μ

→^e

γ

)≤⁴.2×^{10−13 fromthe} MEG experiment[51] (fora recentreview, please see[52])poses stringentconstraintsonthemassscaleoftheclockwork.InFig.4 weshow thebranchingratioexpectedforpoints reproducingthe measured neutrino parameters, assuming two clockwork genera- tions,asobtainedinthescanpresentedinsection2.1,asafunction ofthemassofthefirstclockwork gear.Itfollowsfromthefigure thattheclockworkgearsmustbelargerthan∼^{40TeVinorderto} evadetheexperimental constraints,unlessvery finecancellations amongall contributionstothisprocess exist.Foralargernumber ofclockworkgenerations weexpectevenstrongerlowerlimitson thelightestgearmass,duetothelargernumberofparticlesinthe loop.

4. Summary

Theoriginofsmallneutrinomassesremainsa mysterytothis day. The recently proposed clockwork mechanism provides new insights into this puzzle, as it naturally generates small param- eters in the effective Lagrangian. In the present work, we have scrutinizedthemechanismofneutrinomassgenerationwithinthe clockworkframework.Wehavegeneralizedtheclockwork formal- ismto include, inaddition toDirac massesandnearest neighbor interactions, also Majorana mass terms in the clockwork sector;

and we have derived analytical expressions for the masses and couplingsofthenewsingletfermionsforthespecificcasewhere theDiracmasses, Majoranamassesandnearest neighbor interac- tionsareuniversalamongallclockwork“gears”.

We have investigated in detail the impact of the Majorana masses in the clockwork sector in the generation of small neu- trinomasses.WhentheuniversalMajoranamassvanishes,thezero modeoftheclockworksectorisstrictlymasslessandformsaDirac pairwiththeactiveneutrino.Inthisframework,smallDiracneu- trinomasses can be generated fora sufficiently large numberof gears,dependingonthehierarchybetweenthemassscalesinthe clockworksector.Ontheotherhand,whentheuniversalneutrino massisnon-vanishing,thezeromodeisnolongermassless.How- ever, the corresponding Yukawa coupling still hasthe clockwork structure.Inthiscase,thecontributionfromthisparticularmode

istheresultoftheinterplaybetweenthe standardseesawmech- anism andthe“clockworked”Yukawacouplings.The contribution fromthegearsistypicallyproportionaltotheirO(¹)Yukawacou- plingsandtheyrequirea verylargeMajoranamassscaleinorder to reproduce the smallneutrino massesinferred from oscillation experiments.

TheStandardModelleptonscoupletothefermionsoftheclock- worksectorwithasitedependentstrength,givingriseto(possibly leptonflavor violating)chargedcurrent, neutralcurrentandHiggs boson interactions. We have investigated the constraints on this framework from the non-observation of the rare leptonic decay

μ

→^e

γ

.Ourresultsindicatethatthelightestparticleoftheclock- worksector musthavea mass^{40 TeV,iftheYukawacouplings} ofthefundamentaltheoryareO(1).

Acknowledgements

AIandSKVacknowledgepartialfinancialsupportfromtheDFG cluster of excellence EXC 153 “Origin and Structure of the Uni- verse”andAIfromtheCollaborativeResearchCenterSFB1258.SKV thanksthePhysicsDepartmentoftheTechnicalUniversity ofMu- nichforhospitality.SKVthanksthehospitalityofIPHT,CEA,Saclay duringthefinalstagesofthiswork.Healsoacknowledgessupport from Department of Science & Technology, Govt. of India, Grant No. EMR/2016/001097, ‘Nature of New Physics Beyond Standard Model’.

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