JHEP02(2020)162 X U(1) × Springer L January 14, 2020 -Breaking Scale

JHEP02(2020)162 X U(1) × Springer L January 14, 2020 -breaking scale. February 12, 2020 February 26, 2020 SU(3) : September 12, 2019 : : X : × c , U(1) Revised × L Accepted Published Received Published for SISSA by https://doi.org/10.1007/JHEP02(2020)162 [email protected] , . 3 1908.09384 The Authors. c Beyond Standard Model, Global Symmetries, Neutrino Physics, Spontaneous

The extensions of the Standard Model based on the SU(3) , [email protected] Department of Physics, andP.O. Helsinki Box Institute 64, of FI-00014 Physics, UniversityE-mail: of Helsinki, Finland [email protected] Open Access Article funded by SCOAP Keywords: Symmetry Breaking ArXiv ePrint: three right-handed neutrino singlets.this model. We The show scale that ofThe the the model seesaw seesaw we mechanism mechanism present is here is nearincluding simultaneously the realized neutrinos, explains SU(3) and in the the mass number hierarchy of of all families. the fermions, Abstract: gauge group (331-models) have been advocatednature. to explain It the has number been of recently fermionexplain shown families the that mass in hierarchy the of Froggatt-Nielsen the mechanism,in charged a fermions, popular an can way be economical to incorporated fashion into the (FN331). 331-setting In this work we extend the FN331-model to include Froggatt-Nielsen mechanism Katri Huitu, Niko Koivunen and Timo J. Kärkkäinen Natural neutrino sector in a 331-model with JHEP02(2020)162 14 9 17 16 – 1 – 11 6 18 22 3 12 14 9 22 22 22 22 3 6 4 1 20 7 2.3.1 Charged gauge bosons A.1 CP-even scalars A.2 CP-odd scalars A.3 Charged scalars 8.1 The FN-charges8.2 for the numerical Numerical example values for leptons 6.1 Neutrino mass6.2 matrices Neutrino eigenstates framework 2.1 Fermion representations 2.2 Scalar sector 2.3 Gauge sector 1 Introduction After the discovery ofparticle the predicted Higgs by the boson Standard at Modelhas the (SM) moved has Large been on Hadron confirmed. to Collider, Today, a particle the physics new last era, elementary where we attempt to answer the problems plagueing the SM B Neutral gauge boson masses A Scalar mass matrices 8 Constraints and numerical examples 9 Conclusion 6 Neutrino masses and eigenstates 7 Neutrino coupling to charged gauge bosons and PMNS-matrix 4 Charged lepton Yukawa couplings and5 masses Neutrino mass matrix 3 The Yukawa sector and the Froggatt-Nielsen mechanism in the 331- Contents 1 Introduction 2 Particle content JHEP02(2020)162 = × L β (1.1) SU(2) × gauge group c 3 on the other X √ ]. / 1 22 U(1) ]–[ × gauge group contains -triplets. The models = L L 14 X β U(1) SU(3) × × c L ]. The Froggatt-Nielsen mechanism X, 26 SU(3) 1 eV and the sum of their masses is less , . + ] the gauge anomalies cancel only if the × 25 8 c 22 βT ]–[ 8 + are extensively studied in the literature. The – 2 – 3 ] can be incorporated into the 331-models with 1 T ] 24 = 22 ], which is a statistical fit to SM using LEP data. This 7 Q ]–[ ]–[ ]. 13 1 23 -sextet. The models based on the 3 [ L √ 008 [ . / ] does not exhibit the cancellation of gauge anomalies and does not explain 0 1 13 defines the particle content of the model. The models with β = 984 3 contain particles with exotic electric charges such as doubly charged . 3 do not contain particles with exotic electric charges. Also the models √ β √ / 1 = β ] and = 12 3 without extending the scalar sector [ β ]–[ √ = 0 have been studied [ ]. This neutrino sector is identical to the one presented in [ -triplets and an SU(3) 12 eV from cosmological constraints by the PLANCK experiment. Neutrino masses 8 . 1 . This means that the electric charge can be defined in multiple different ways in the L β 15 Y Even though the 331-models can shed light on the number of fermion familes, the Extensions of the Standard Model based on the SU(3) In Nature three generations of quarks and leptons have been observed. Number of 3 [ The model presented in [ 1 = √ neutrinos is massless andare the needed two to other lifttwo mass the [ degenerate one eigenvalue at from tree-level. zero and Loop to corrections break thethe number degeneracy of of fermion families. the other (FN) is a well established methodmodel to with explain the incorporated massboth hierarchy FN-mechanism of the the (FN331) fermions, number and canThe a of therefore 331- neutrino fermion masses simultaneously families and explain The mixings and neutrino are mass the not matrix mass naturally is hierarchy explained antisymmetric in in of the FN331-model, the FN331-model however. charged and therefore fermions. one of the with fermion mass hierarchy is left unexplainedthat in the the traditional Froggatt-Nielsen models. mechanismβ Recently [ it was shown models with scalars and gauge bosons.SU(3) They also contain ahand very large have scalar simpler sector, scalarbased composed sector, on of composed three from only three SU(3) U(1) 331-models: where the parameter (331-models) have been proposed in thein literature Nature. to explain the In numbernumber of the fermion of traditional families fermion 331-models familesone [ is additional three. diagonal The generator SU(3) compared to the SM-gauge group SU(3) SM neutrinos are massive, withthan masses 0 less than 0 are not included in thethe Standard next Model, lightest and massive theythe particle, are six heaviest electron, orders particle, and of twelve top magnitudeflavour orders lighter problem. quark. of than magnitudes This lighter huge than range of different masses gives birth to the mass, neutrino mixing and fermion mass hierarchies. neutrino flavours is 2 is a strong indication for exactly threeby generations SM of itself. matter, which, We however, know is from not imposed neutrino oscillation experiments that at least two of the three with economical extensions. The problems include number of generations, nonzero neutrino JHEP02(2020)162 - - × ]– X C FN 29 (2.2) (2.1) U(1) . × 0) , L 1 , breaking scale (1 X ∼ -breaking scale as -singlets: L X R,i U(1) ]. We aim to generate × U(1) 27 L ,N × 1) L − mass however suggest that the , 0 1 , Z (1 ∼ X, + 2 R,i 8 T -breaking scale as possible. The suppression 3 1 X generators. We also introduce global U(1) √ , e L – 3 – 3 − , U(1) 3 2 T , × L = (FCNC) allows for SU(3) = 1 Q we study the lepton mass matrices and mixings and finally , i 7 ]. The collider bounds on the 1 3 to − 4 26 , 3 we review the Froggatt-Nielsen mechanism in the 331-setting, the , would result in essentially a same model. 1 3 are the diagonal SU(3) 3 1 8 √ ∼ T -breaking scale has to be at least around 7 TeV [ . We define the electric charge as: L we present numerical example for the neutrino masses and mixings. = + X X    8 β and 0 i i i 3 e ν ν -triplets and the right-handed leptons are assigned to SU(3) T U(1) U(1)    . In section L × × 2 = L L The paper is structured as follows. We present the particle content of the model in The hierachical structure of the neutrino Yukawa couplings is determined by the FN- Our aim is to extend the neutrino sector to make it natural and explain the neutrino The choice L,i ] for the neutrino sector. Here the seesaw is combined with the FN-mechanism, which 2 flavour changing neutral currents L 34 to SU(3) where the symmetry, under which fermions and some of2.1 the scalars are charged. Fermion representations Let us now write down the fermion representations. The left-handed leptons are assigned 2 Particle content We propose a model where theSU(3) gauge group of the Standard Model is extended to SU(3) section FN331-model. In sections in the section neutrino mixings are generatedexperimentally without known fine-tuning. to be Allachieved order-one-numbers. in the FN-setting PMNS-matrix This elements byequal. kind are assigning of all We pattern the showmixing of FN-charges that angles mixing of are the can the produced correct be within left-handed sub-eV experimental leptons limits. neutrino to masses, be mass square differences and mechanism. By medium-energyand mass we of refer electron. toconsistent with a We experimental will mass data, present which scale the turns between lowest out possible to the be SU(3) mechanism. active approximately The neutrinos 7 TeV. hierarchy of the neutrino Yukawa couplings can be arranged so that the due to the FN-mechanism. Thebreaking seesaw scale. scale We is study essentially as the lowof SU(3) same as the SU(3) low as 5 TeV as shown inSU(3) [ singlet neutrinos at medium-energy scale and sterile neutrinos at TeV scale, utilizing seesaw masses and mixingsFN331-model without where fine-tuning we add at threethe right-handed tree-level. neutrino tree-level singlets masses to We for the propose[ all model. the an This neutrinos allows extension andallows of implementation the of the seesaw seesaw scale mechanism to [ be low thanks to suppression in the neutrino Yukawa couplings JHEP02(2020)162 3 / . 1 with (2.7) (2.3) (2.4) (2.5) (2.6) − 3 1 U = . − 3 , , ] and they X 3 2 , , 1 26 ]. This is in , 3 and / antitriplets and 26 = 1 25 1 ∼ L −    , i 0 0 − 0 χ χ χ 1 3 3 triplets with    − / , 1 1 = , − 3 = , χ ∼ , X 2 0) 3 1 , R, ∗ − 3 with electric charge , , 3 2 , (3 1 D are new leptons with electric charges 0. The ∼ ,D symmetry. This will be discussed in detail in , L ∼ ]. It was however recently shown that tree-level R,i – 4 – 1 3 and -anomaly. All the gauge anomalies will cancel    L N    2 3 22 − 1 3 FN 2 0 , , 3 0 − u 0 D 1 1 ]–[ d ρ ρ D ρ , , − 8 and 3 3       are not present in the model studied in [ 0 L,i = = ν ∼ ∼ R,i , contain only two types of scalar triplets with neutral entries: 3 1 R 3 N L, 1 R, 3 in order to generate the masses for all the charged fermions 1 3 √ , ρ / , − . The 3 , 2 3 X = 2 3 = , . 3. One must include at least two 3 / β 3 ,D , X ,Q 1 1 U(1) ,U ∼ L − 1 3 3, which will mix with the SM quarks. All the fermions are also charged L ×    = / 2 3 − ∼ 2 L    1 , , 2 u 1 1 1 1 X d D    , , U d u − 3 3 + 0 + 0       η η SU(3) η = = ∼ ∼ 3 and ×    / 2 1 c = R,i R,i When we take into account the colour, there are six fermion triplets and six antriplets, We have introduced new quarks The cancellation of anomalies requires the number of fermion triplets to be the same L, L, d u = 2 η Q Q at tree level. We choose to have this minimal scalar sector: 2.2 Scalar sector The 331-models with X and one triplet with generations inevitably leads toplaguing the scalar traditional mediated 331-models [ FCNCsscalar at mediated tree-level, FCNCs whichcontrast of is to quarks the a are traditional feature the suppressed 331-models, tree-level which in scalar offer mediated the no FCNCs. FN331-model natural suppression [ mechanism for under the global Froggatt-Nielsenthe U(1) dedicated section ensuring the cancellation ofwith pure this particle SU(3) content.placed in The a anomaly different cancellation representation than forces the one other quark two. generation The to unequal be treatment of quark electric charge 2 one family to asecond triplet. and the We third choose into an to antitriplet: assign first quark generations into triplet and the SU(3) three right-handed neutrinos allow the tree-level masses for all the neutrinos. as antitriplets. This is achieved by assigning two quark families to SU(3) The numbers in the parantheses label the transformation propeties under the gauge group JHEP02(2020)162 χ & with (2.9) (2.8) , u and (2.10) 2 v ρ FN -symmetry. 2 FN ) . χ †    χ ( 0 0 u 3 to be comparable to λ    ) ) f ρ χ + 2 ]. The Froggatt-Nielsen † † 1 2 χ √ χ ) 26 ρ )( )( , † carries a non-zero FN-charge = ρ χ ρ † † i 25 ( χ rotation. We choose to rotate ρ ρ † 2 χ ( ( h ρ λ L 23 23 + 0 e λ λ χ . , 2 c is gauge invariant. Also according to . )    η 1 ρ ) + ) + χ 1 2 † † 0 η χ v v η symmetry and we take them to be of the † ρ † ( ) + h 1 χ 1 χ    L η . Note that since the scalar triplets χ λ − )( )( † 2 1 . η χ 1 ρ -combination can play the role of the flavon in + ) † † ( √ . – 5 – b χ η η c χ † ( ( . † = ρ -symmetry, and we assume them to be: = χ 13 13 i L 2 3 e λ λ ρ + h µ h soft k Particle V + χ ) are charged under the global symmery U(1) ) + ) + FN-charge j , ρ η ρ ρ † † † i . The FN U(1) charges of the scalars. 2.7    ρ ρ ρ break the SU(2) η 0 2 2 0 0 )( )( v 1 µ ijk η ρ -symmetry is spontaneously broken by the scalar field VEVs. v † †    + ( η η -symmetric scalar potential is, ( ( f 2 FN η 1 Table 1 2 † , the gauge invariant combination and 12 12 √ FN η break the SU(3) 0 e λ λ √ 1 2 1 -breaking scale. The soft-breaking term is also chosen to be large, v = 3 triplet VEVs so that the upper component VEV goes to zero. This µ + + + u X / i 1 η = h − and U(1) , in order to decouple the pseudo-Goldstone boson in the low energies. FN = 2 2 V ) × v X L heavy ) and the table v ( 2.8 All the complex phases in the scalar potential can be absorbed into the fields and The scalar potential is greatly simplified due to inclusion of global U(1) The scalar triplets in eq. ( All the neutral fields can in general develop a nonzero vacuum expectation value (TeV). The VEVs ∼ − therefore all the parameters inof the the scalar scalars potential are willthe real. therefore SU(3) The not real mix. andb imaginary parts We choose the parameter However the global U(1) This leaves one Golstoneadd boson the to following soft the FN-breaking physical term spectrum. to the In potential: order to give it a mass we mechanism can thus beinto implemented the without model. introducing new scalar degrees ofThe freedom most general U(1) the charge assignment presented inare the in table the same representation,eq. ( the combination and has a non-zero VEV.the Therefore Froggatt-Nielsen the mechanism, as was demonstrated in [ The VEVs O order of the electroweak scale. one of the rotation will leave the rest of the VEVs general. So we have vacuum structure: (VEV). The minima are related to each other by SU(3) JHEP02(2020)162 2 . / a µ λ V gauge (2.11) = L a T , -boson and Z      µ 8 ). Details of the 0 µ − µ W 0 µ 0 6 5 2 X V √ of the SM, new heavy iW − − µ 8 µ W . These fields are identified 4 and a non-hermitian heavy , W , µ µ 6 0 4 f W 1 Z ( √ Z f 2 W + µ 1 . The off-diagonal gauge bosons + µ XB 0 + √ 0 . . x , , ) µ V ) ) B W 3 ig µ A ≡ µ µ does not mix with the other neutral 5 2 7 W 0 − µ Y, µ 2 and the heavy new gauge bosons 5 iW 1 iW iW X √ aµ W µ − ∓ ∓ − W µ a µ µ W 2 charged µ 4 1 6 T 8 gauge couplings respectively. The M W W W W ( 8 – 6 – =1 T ( ( X 6 a X 2 2 2 1 Y 1 √ 3 1 1 ∗ are the Gell-Mann matrices. The SU(3) √ − µ √ √ 0 0 0 µ ig + a . The field = = = X W µ λ µ L ⊃ − -breaking scale and they are very heavy. The details of 4 3 0 µ 0 µ ∓ µ µ will form neutral mass eigenstates: photon, 0 0 X and U(1) f W ∂ W X µ V 2 4 W 1 L = √ triplet is: W U(1) µ and L      0 D × 2 Z , new heavy neutral gauge boson L 1 and √ . Thus there are four CP-even, two CP-odd and two charged scalars µ 0 V B = X , µ aµ 8 are the SU(3) generators, where W W a x , will form the SM gauge bosons L g T µ 3 µ 0 8 =1 W a X V and 3 g and physical neutral non-hermitian gauge boson The scalar sector has five CP-even, five CP-odd and four charged scalars. One CP- µ 0 W 2.3.1 Charged gaugeThe bosons mass term for the charged gauge bosons is given by, The fields new heavy gauge bosons gauge bosons and is aas mass a eigenstate, with sameneutral mass gauge as boson masses are given in the appendix where we have denoted, where are the SU(3) bosons are, variant derivative for SU(3) masses around the SU(3) the scalar mass matrices are provided in the2.3 appendix Gauge sector As previously mentioned theThe gauge 331-models sector will of 331-model contain five is additional enlarged gauge compared to bosons the compared SM. to the SM. The co- even, three CP-odd and twoare charged absorbed scalars by are the massless gaugecharged would-be-Goldstone bosons gauge bosons of boson that the model,neutral gauge namely boson the left in the physical spectrum. All the physical scalars, except the 125 GeV scalar, have their JHEP02(2020)162 is ). µ 8 (2.13) (2.15) (2.16) (2.12) (2.14) W . The SM fermions, Y , , ) the SM Higgs VEV is   U(1) ) ! , 2 × 2 u , ) 2.13 L 2 light + 2 heavy sm + µ v , 0 2 v v 2 2 + µ , due to large difference between v 0 v . 1 ! 4

v 2 1 SU(2) θ V 2 = ( 3 + v g O light × θ V heavy 2 v 0 v ! 2 light 2 − C 2 heavy + v v v 2 v ( 1 ∼ 2 3 u + cos v 4

g 2 2 light θ + sin 2 2 heavy + µ + v u 2 0 v ) O + µ 2 2 u 2 1 0 + 2

1 v v . The experimental bound for the mixing is v 2 2 2 -breaking VEVs. The SM gauge boson µ – 7 – ) + v − θ W O 0 + v 2 1 Y 4 θ W is tiny, v 2 V u . According to eq. ( 0 + = 2 3 0 + sin g v 2 θ + 2 µ ( , v 0 − u 2 2 U(1) 3 V 2 1 2 4 v g u v v = cos + = 2 ( 1 × tan 2 v 2 2   3 3 2 + 2 µ = 4 4 + L and g g µ v = W V is mostly µ + = = ( light 2 W 0 µ v is defined as: v V 2 V 2 W θ 2 charged ) and, m m and + µ and SU(2) M 0 and X , u ,V µ 0 2 + µ v ], and has been taken into account in our numerical analysis (section 0 W 1 U(1) W = )[ × 2 = 246 GeV. − = ( L T heavy sm (10 v v Y O The mass eigenstates are defined as framework symmetry. The FN-framework introduces a new complex scalar field, the flavon, which . | N θ Z is a singlet under standard modelthe gauge SM group Higgs SU(3) and theFN-symmetry flavon is are to charged forbid under theYukawa the couplings SM FN are Yukawa couplings, symmery. generated save The as perhaps key effective the property couplings top of instead. quark. the The SM Next we study themechanism Yukawa to sector generate of the theoriginal hierarchical model. Froggatt-Nielsen model structure extends We of the are thesymmetry), Standard employing whose Model fermion symmetry the with Yukawa group a couplings. Froggatt-Nielsen in flavour the symmetry The simplest (FN case is global or local U(1) or a discrete the SU(3) almost totally | 3 The Yukawa sector and the Froggatt-Nielsen mechanism in the 331- where the mixing angle The mixing angle between where where related to the triplet VEVs through the relation is the charged gauge boson mass matrix. The eigenvalues of the matrix are, where JHEP02(2020)162 3 S ). (3.2) (3.3) (3.4) (3.1) ). 5.1 R,j . 1. The c 2.8 f . i , will act < ) gives the S χ , obtains a ) h + h † 2 χ . The power ρ 3.3 † f L,i R,j ρ ¯ ψ (2Λ 2 Sf 2.1 / ) √ represent here the f L,i u ¯ ψ # 2 v 0 ij R,j u ) χ 2 f s f 1 n v = ( . v ( − ij ., ) + . c . and s f . ρ c 2 and therefore the amount of . n † 0 i Λ 0 ( χ ) u S S χ f ij q L,i S + h ) ij ¯ + h ( ψ ) ij f s + q ) f s ) n f s R,j c ∗ c R,j ( 2 0 0 f R,j v ) + R,i Sf ) ρ f ≡ ]. The addition of complex scalar field f i ( 2, as can be seen from the eq. ( " . The f q L,i R,j S / ij ¯ h f χ ) ) ij ψ 26 ( , s f ) u q ) n ij + c f s 2 ( ) or ) 25 y v s f S f ( L,i n ρ ( f L,i ¯ ( ) + 2 ψ ij – 8 – u ( ψ , ) = ( 2 f ( L,i q s f η f L,i v i 2Λ ¯ χ 2 ψ ¯ n ψ χ † ( Λ † ij ρ q ) ρ ij contains the neccesary incredients to house Froggatt- + ( s f h and determined by the FN charge conservation (see h ) n s f ( 4 ij c = 2.2 R,j ) f f s ) ij ) c i ) = ( Particle i ( s f χ FN charge S h ij n h ) ( f s + . The FN charges of fermions and the scalar fields. + L ⊃ y , ( χ 2 S ( ( † Λ ) ··· i f L,i ρ ¯ + ψ h Table 2 . ij acquire VEVs: c + ) . f s ρ χ y ( is a dimensionless order-one number, Λ is the scale of the new physics, +h were negative, we would have to include operator ( ij = ( ) and ij ij ) f s ) ρ c f s s f is positive integer number ): c n ( 2 ij The usual 331-model Yukawa terms are generated as effective couplings when the scalar The effective operator that generates the Yukawa couplings of the charged fermions is: The FN mechanism can be economically incorporated into a 331-model. The scalar sec- The effective operator that generates theIf neutrino ( Yukawa couplings is presented later in eq. ( ) 3 4 s f n FN-charges of the SMsuppression fermions each determine Yukawa the couplinghierarchy power obtains. by ( assigning One larger FN can charges obtain to the the observed lighter fermions fermion compared mass to the heavier usual Yukawa terms of thenow defined model, as: as in the original FN model. The Yukawa coupling is Hierarchical Yukawa couplings are produced by assuming that where only the renormalizable contributions are kept. The first term in eq. ( triplets denotes any of thefermion three triplets, scalar antitriplets triplets and( singlets that were introducedtable in section where ( Nielsen mechanism, as was demonstratedto in act [ as aas flavon the is flavon thus insteadnonzero unneccesary. of vacuum expectation single Here value, complex a scalar gauge field. singlet combination, The effective flavon, tor we have introduced in section JHEP02(2020)162 × L (5.1) (4.2) (4.3) (4.4) (4.5) (4.1) 7 TeV. & ., c . and we will is + h scale heavy v 0 v χ . 1 c u v . 2 0 v v ., c + h . + γ obtains a VEV: . 0 ) 0 ]. This also ensures that the ∗ ij + h η l χ η 0 ( 26 u v ij e β R,j c ) e -breaking scale + ≡ c L,j X ∗ L,i 0 , when we study numerical examples. ) . ¯ e 0 L η ( 8 ρ ( e ij q . 0 2 e diag U(1) v m α L,i 2 v 0 )+ ¯ v L m √ × = R,j . e ij e = L c ]. We will safely ignore the flavour violating ( αβγ y . R,j q † e e R 27 – 9 – = )+ ij U ) + h L,i e ∗ e ij L,i ¯ e η N ¯ L m n m ( ) and the operator of the second kind is: ( R,j e e ij L q e y U i 3.1 ij e ij η χ 2 l c h † Λ ρ + ) is not proportional to the Yukawa matrix and is therefore = L,i ¯ L e ij R,j 3.3 ij e ij y ) y ∗ ηe N η c L,i ( ¯ L ⊃ L e ij y L ⊃ 3. The first term is the traditional 331 Yukawa term for the charged = , 2 , 3 TeV, satisfying bounds in [ lepton & = 1 L i, j -breaking. We assume that the SU(3) X The charged leptons acquire masses as the scalar triplet The second term in eq. ( mass is 0 The neutrino Yukawa couplings originatetype from was already effective presented operators in of eq. two ( types. The first simultaneously with the Yukawacouplings coupling. in the Therefore standardleptons there is Yukawa will couplings. coming be from The the no Froggatt-Nielsen only flavour mechanism flavour which changing is violation5 however to suppressed. the charged Neutrino mass matrix The charged lepton mass matrix is diagonalized as: The charged lepton mass matrix proportional to the Yukawa matrix will be diagonalized where the charged lepton mass matrix is, mechanism. As stated earlier,ignore this it additional in term the is following. suppressed by The the charged lepton Yukawa matrix is given as follows: We will specify the FN-charges we use later in section where leptons whereas the second term is the additional Yukawa interaction term due to the FN- Z contributions as they are heavily suppressed. 4 Charged lepton Yukawa couplingsCharged lepton and Yukawa masses couplings are: fine-tuning the couplings themselves. flavour violating. ThisU(1) flavour violating part isThis suppressed is by enough the to scale suppress the of quark the FCNCs SU(3) as shown in [ ones. This is in contrast to the Standard Model where the hierarchy is obtained only by JHEP02(2020)162 ) 5.2 (5.8) (5.4) (5.5) (5.6) (5.7) (5.2) (5.3) ., c . + h . 0 c . χ R,j 0 u R,j 1 R,j 2 ., v + h . N v c will also have a Majo- N i . χN ij i ) ρ + ρ R,j ∗ h L,i h η N R,i 0 + h N ¯ n 0 L L,i ( u c N 0 L,i χ ) ., N ij ¯ 0 ¯ L L c R,j ij . y ) + R,j N , ij N ∗ N ij c y ∗ , N η + 0 ij . N y ) 2 0 c ) + h n ( 0 ij v ij + n ρ ij n R,j R,j R,j γ N ij -breaking VEVs by, = ( 0 R,j ) M N N ij , due to presence of the antisymmetric N 2 c i X + ( ) ( c N 1 2 ρN ∗ q ∗ ji √ c γ ) η η e = . ) i ( = )+ h o L,i + 2 ∗ q ) ( − R,j ) ¯ U(1) η L χ β R,j 2 R,i ρ ( N 0 h )+ R,j uv ) ( ( q = N ij R,j N β × q N q ( y N ) ( c L,j q )+ N r ij i L,j c L )+ )+ L i ij e + ( χ L c L,j – 10 – χ q R,j h ( M ij R,j R,i γ h M L c N ) ( N )+ N ( Λ = 1 2 L,i ∗ L,i α ( ( q 0 L,i q q η ¯ ¯ L c L,i α L,i ¯ L ( L = )+ = Λ L )+ ¯ L β ( N ij ) produce the following Yukawa couplings for neutrinos: ) χ q 2 0 N ij L,i ij αβγ 0 L,i † ¯ y L Λ ¯ y L ρ ( αβγ c L,j ij 5.1 ( M q ij ) 0 ij + q L e ∗ ij n ( N η N ij e Majorana N ij 0 = c M ij + c c ij α L,i c L ) 0 ). ¯ is in principle a free parameter. We choose the mass scale to be ) contains the standard Yukawa interactions for the neutrinos L ) and ( ∗ is antisymmetric: = = = ( N η 0 M 5.2 n 5.2 ij 3.1 ij αβγ N N ij ( ij M e 0 e y n y ij L ⊃ + e 3 and the Yukawa couplings are, neutrino mass = , L 2 , in the eq. ( = 1 neutrino αβγ L i, j The full contribution to the neutrino masses is finally given by the following terms: The neutrino masses will be generated by the Yukawa terms in the first line of eq. ( The first line in ( The messenger scale is related to the SU(3) Majorana mass term becomes, where the Majorana mass matrix is, where the mass scale same as the FN-messengers, in order not to introduce new mass scales into the model. The and the two lastFroggatt-Nielsen mechanism, lines which we contain will the ignore due additional to Yukawa them interactionsas being originating the suppressed. scalars from obtain VEVs. the rana The mass right-handed term neutrino which singlet is generated by the following operator, The Yukawa coupling tensor where The operators in eqs. ( JHEP02(2020)162 5 as: . (6.2) (6.1) (5.9) c ν . (5.10) U + h . 3    , 2 c ) , 0 L L R ν ν N , = 1 ( i       ]. ij    e . ∗ ∗ 2 ) 0 36 ) , ∗ v R,i N √ N 0 35 M N m m = ( D ij † † ) ) m ). D N 0 ( 0 ) has been studied in the literature m 7.4 . m ( ν 5.9 and U ∗ † e ) L ) D U N 0 2( N ij 0 9 matrix form as: m confidence level [ ∼ m y ( × 2( 2 σ and the neutrino diagonalization matrix – 11 – u √ e    L (1) numbers in contrast to CKM-matrix where dis- U PMNS + ., R c O U . N ij N y c 2 2 + h ) v 0 √ L 0.287–0.532 0.486–0.706 0.604–0.754 0.797–0.840 0.518–0.585 0.143–0.156 0.233–0.495 0.448–0.679 0.639–0.783 0 N ν ], where it was studied in great detail. We will therefore not = ( ν    c N ij M 26 ) 0 , c = L ) | 0 ν 25 ( N ( , m 1 2 1 2 ] only by the additional neutrino singlets PMNS N ij y U ≡ = | 26 2 1 , v 3 sub-matrices are: √ 25 × = N ij m ]. neutrino mass The PMNS-matrix elements are The determining factor in our choice of leptonic FN-charges is the form of the PMNS We next determine the pattern of FN-charges for the leptons using the experimental The exact form of the PMNS matrix is given later in eq. ( L 28 5 tinct hierarchy is present.charged lepton The diagonalization matrix PMNS-matrix is given schematically by the left-handed where the value of each entry is given at 3 consider it here.presented We in instead [ concentrate on lepton sector whichmatrix. differs from The thecollaboration current model are: experimental values of the PMNS-matrix elements by the NuFit is important as theobservables: left-handed the fermion CKM-matrix diagonalizationfermion and matrices FN-charges the can enter PMNS-matrix. ensure theare that two Proper produced also physical the choice correctly, and hierarchy ofidentical of no the to the fine-tuning the left-handed CKM- is one and required. in PMNS-matrices [ The quark sector of our model is 6 Neutrino masses andThe eigenstates Froggatt-Nielsen charges determine thefermion FN-charges hierarchy should of be the chosen fermionbecomes so mass that right, the matrices. thus order the The ofalso mass magnitude determine hierarchy of the is the structure fermion explained of the masses without matrices fine-tuning. that diagonalize The the fermion FN mass charges matrix. This values of thehierarchy PMNS of matrix the neutrino asdiagonalization mass guidance. of matrix the becomes neutrino Once clear, mass and the matrix. we FN-charges can are proceed with known, the the block exact where the 3 The mass matrix within [ same structure as in eq. ( The neutrino masses can be written in a 9 JHEP02(2020)162 . Y R M (6.5) (6.6) (6.7) (6.3) (6.4) U(1) × and , L D M ) do not have heavy . v breaking VEVs, 6.7 ! ∼ X ∗ ) ∗ ij N 0 M M will have in total three U(1) m ν × ”. Our neutrino sector is are much larger than the † depend on the FN-charges L M ) R ν and N . 0 ( 0 M U L. heavy m L ( ≡ , also have this anarchical texture. ) = 0 and

) ν 3 3 e , L U are proportional to SU(2) R, = c L, heavy U ! ” and “ N D v R L can be written in the following notation: ( T D R ( q ν M M q ∼ M M . The eigenvalues of the heavier block will M L N 0 ij D ) = ) = 0 2 2 M medium and also have diﬀerent orders of magnitude: matrix by R, c L, – 12 –

R L N , m ) will have nine eigenvalues corresponding to nine ( ( M . Note that sub-matrices in eq. ( = q q M +1 2 5.9 ∗ ν L ) M N ) = ) = u, v are proportional to the SU(3) 1 1 m light c L, ( = R R, v in eq. ( L N M ( ( ∼ † ν q q ) N ij M heavy D v . Therefore the eigenvalues of the 1 m v 2( , m and +1 1 and = L 0 and neutrino diagonalization matrix . This hierarchy is reﬂected in the eigenvalues of the matrix: it has three 2 , v 0 T D v D e L v M U M light = v ∼ , whereas the entries in the sub-matrix light 2 v v D ij ” and six heavier eigenvalues. is heavily suppressed compared to m The elements in the heavier block N and 0 light m therefore be in twosubject distinct to scales kind we of call “double-seesaw”.“light” “ eigenvalues, The three neutrino “medium” mass eigenvalues matrix and three “heavy” eigenvalues. an internal hierarchy, but distinctThe hierarchy entries is in present the between sub-matrix u sub-matrices breaking VEVs entries in the “ The order of magnitude of the sub-matrices are given by where where 6.1 Neutrino mass matrices The neutrino mass matrix Majorana neutrinos. The neutrino mass matrix We can now see thewith order the of block magnitude in diagonalization the of neutrino the mass neutrino matrix mass elements matrix. and proceed The FN-charges of thelight-neutrinos. right-handed We will neutrino choose singlets thesimplicity: do FN-charge of not the aﬀect right-handed neutrinos the to hierarchy be of zero the for tation matrix This is the method weof adopt the here. left-handed leptons. The The hierarchyare anarchical of hierarchy treated is equally achieved when under all thelepton the FN-symmetry. triplets lepton We families to will have therefore equal choose FN-charges: from now on all the The observed PMNS-hierarchy is naturally obtained, if the left-handed charged lepton ro- JHEP02(2020)162 6 × 3 (6.8) (6.9) (6.14) (6.11) (6.12) (6.13) (6.10) 1 − ) T are to the . ) N Z 0 m heavy v (( ] and is the electroweak † T ∼ ) N W D , m light , 1 m v ,     − 2( )     3 † ! heavy,ij × 6 N 3 3 3 3 0 m 1 ) × × × ∗ × 3 1 − 6 3 3 3 heavy m ) ) D × 3 ( 0 0 C 6 , where 1 3 † † 1 m N × ) m 0 × 3 B D 1 3 C 2 and † 1 0 ) 1 m 1 2 C m 1 3 3 ( B heavy − )( 3 ) 1 2 B × × , − . ∗ ( 3 3 N /v × L 1 + 2 3 medium 0 0 N 2 − (1 − 0 m ∗ m 3 2 D light m (1 M v +( ( × 3 , 3 3 m 3 3 ∗ 1 N † × 3 0 ) 2 × × heavy × − 3 light † N 3 3 1 1 v × 3 ) 0 M m 0 0 3 3 ) − 3 1 C m – 13 – ) † N 1 ) m × 1 ∼ 0 × − † 1 ∗ = 6 1 3 ) B     C ) ): † m C N 0 − 1 1 † 1 1 2 0 ( ( ) , is however present, due to the Froggatt-Nielsen N = C B ∗ 0 B − m 6.7 1 2 − ( M can be block-diagonalized into three blocks, each cor- ( +2 m 1 M ∗ L − ( ν ( − , − (1 2 † N ∗ ) WZ medium,ij . D M 3 3 3 ν T N ∗ (1 m 3 0 [ ) × × × m × 3 3 3 M

3 N − m M 1 0 0 0 T ) , m 2 1 = m = 2 = = separates the three “light”-neutrinos from the six heavier ones,     W ) with eq. ( B +2 (( ( T = W L T further block diagonalizes the block of heavier neutrinos into block W 2 Z ) light 6.13 Z 3 heavy D Z m × medium m 3 m ) 2 light m 1 heavy 1 2( v is the scale of new physics, which is characteristic to the seesaw-mechanism. v B − ( ∼ heavy v = = 6 light,ij × 3 ) m The order of magnitude of light-, medium- and heavy-neutrino masses can now be The light, medium and heavy blocks can be written at lowest order as: The neutrino mass matrix 1 B ( The light-neutrino masses arescale proportional and to Additional suppression factor, mechanism. The masses of the “medium”-neutrinos are heavily suppressed compared to estimated using eq. ( and, with, and, where unitary matrix and unitary matrix of “medium”-mass neutrinos andleading “heavy” order: neutrinos. The matrices responding to these eigenvalue-types according to: JHEP02(2020)162 Z boson (6.17) (6.18) (6.19) (6.16) (6.15) µ W , . ,L ,L 2. They are there- heavy / ν , heavy Z · ν ,L ) is fully diagonalized i · m +1 5.9 L (1) heavy . ν 0 O · , N + ) † light heavy L v . v in eq. ( ,L W , that mixes with the , h †    ν µ O O 3    Z 3 3 WZU V † M × ( × × 3 + + N ν 3 3 U medium 0 0 U ν ,L ,L M = · 1 1 1 1 1 1 1 1 1 ) 3 3 3 T × × ×     -breaking scale and will therefore decouple.    L 3 n 3 3 W 0 0 X U ,L ∼ medium medium T ,L ,L 3 ν ν 3 3 O – 14 – 1 Z have no internal hierarchy. N 1 · · × × × 1 × T 3 U(1) ν 3 3 × + i 3 light × ,U 3 heavy 0 0 1 U U ν 3 medium n (1) ν × diagonalize light, medium and heavy neutrino blocks ,L ν ] on the number of light neutrinos. However, suppres- heavy    O L 7 ,U = ( N     m ν light light are anarchical in nature, + = U heavy v ]–[ ν v U ≡ 1 · N ,L h U diag ν i U and . The mixing between the charged gauge bosons is however and O M boson make their contribution to the invisible decay with of light +1 ), the mass eigenstate neutrinos are: mass n + ν L ν Z U · and ,L , i 2.3.1 6.15 n is given by, ν 1 medium U U light heavy light v U m , v ν ν , h · light U heavy v v O -breaking scale, making them typically lighter than light h (1) = X m O O c ) = = R U(1) 0 L L N ν ν ( × L              The mixing between the neutrinos can be estimated with the use of FN-textures of the According the eq. ( 7 Neutrino coupling toOur charged model gauge includes bosons additional andas charged PMNS-matrix shown gauge in boson the section neutrino Yukawa couplings as: as the blocks The unitary matrices respectively. The with unitary matrix The neutrino mass eigenstatesblocks are are obtained diagonalized. onceaccording The the to: neutrino light, mass medium matrix and heavy neutrino fore subject to the LEPsion bound on [ their couplings to boson tiny, as will behave their demonstrated masses later around for the our SU(3) benchmark points.6.2 The heavy neutrinos Neutrino eigenstates SU(3) JHEP02(2020)162 2 ), is − ν 10 (7.4) (7.5) (7.6) (7.1) (7.2) (7.3) U 2 heavy ≥ /v ., ). We note c 2 light . v 6.1 ( very small and . + h O c . + θ µ . ν W ) and is significantly + h U L e + µ † µ 1 1 V γ 2 heavy i † + e θ L µ θB /v U W i cos 2 light θ θ v + sin 0 L,i ( 6 † e 1 to light neutrinos can be writen as, sin sin O , the left-handed diagonalization matrix , µ . 1 B 1 ,j ν γ µ 1 . 0 L,i    B U +1 e W 0 e i L,i e L µ    θB − ν m γ θ θU + ¯ 1 1 1 1 1 1 1 1 1 0 ≤ cos L,i θ ν 1 2 cos 1 1 1 1 1 1 1 1 1 cos    e i,j + ¯ sin    m ≈ e induces nonunitarity effects to neutrino oscilla- ∼ † – 15 – 1 L θ ) the light-neutrino diagonalization matrix U † B 1 ∼ 0 L,i 1 e is similarly suppressed by a factor ]. In any case, nonunitary mixing strength of B e cos L − µ B 1 † 1 U 6.17 γ ν PMNS 39 PMNS 1 2 0 L,i – U U ) the coupling of U θB e L,i is without a hierarchy: θB e L − µ ¯ ν 37 e 7 TeV, which makes the mixing angle L 1 γ θU − 6.18 U L,i . h & | ¯ ν ) † 2 ν 3 h g c L,j U 2 √ 3 = cos L g ( ,L √ ) and ( + q heavy − v ) = 4.5 light ¯ c L,i ν PMNS L 2 ( 3 U q | g gCC √ 1. As stated in eq. ( L ∼ ⊃ is not Hermitian, the anti-Hermitian part of it induces neutrino decay. Since ≈ ij 1 ) e 9 L B . gCC U 0 L ≈ The PMNS matrix therefore is: The term proportional to sin The term proportional to cos When charged lepton mass matrix satisfies θ 6 which is compatible withhere the experimental that measurements this presented is in eq. the ( extent whichsatisfies: Froggatt-Nielsen ( setting can predict the structure of The texture for the PMNS is therefore anarchical as well, anarchical. Since the leptonlepton triplet diagonalization FN-charges matrix are identical also the left-handed charged We have chosen the cos is ruled out. but since the nonunitarity and unstability effects areder both of small, this we paper. shall ignore them in the remain- tions, which is anviation expected from effect the due unitarity tosmaller is than inclusion suppressed the current of by bounds sterile the [ neutrinos factor in the model. De- from which we can identify the PMNS matrix: With the use of eqs. ( tiny. The neutrino gauge eigenstates couple to the physical charged gauge bosons as: JHEP02(2020)162 2 21 m -bosons Z -breaking X ] all constrain U(1) 45 × L ], cosmic microwave 44 Re. Searches for these 50 TeV the medium-scale 187 ] and data from supernova ∼ 43 - and SU(3) heavy Y Ni and v 63 decay [ U(1) S, β × 35 L F, 20 H, – 16 – 3 12 eV. . 0 ]. As of now, the PMNS matrix is consistent with ]. The coupling of medium mass neutrinos to < 42 7 – ν – . The heavy TeV-scale sterile neutrinos in principle can 1 m 40 1 2 [ P ]. We will show the constraints from kink searches to one of have discarded large mixings of electron neutrino to medium- / scales. The masses of the medium-scale neutrinos is determined Z 49 2 corresponding to inverted and normal mass orderings, respec- m , X , . 48 -breaking scale and in the case of 8 = 1 X U(1) j × L U(1) kink searches × ]. From these, a lower bound for two heavier light neutrinos can be deduced, (with L 2 ]. Although it is not possible to distinguish the models experimentally from | 36 , j 3 51 35 ) from electron energy spectrum of tritium , m e ν ∆ 50 ( | Seesaw mechanisms have been successfully applied to the 331 models also previously Our model predicts neutrinos in three different mass scales: the three sub-eV neutrinos, The existence of medium-mass sterile neutrinos at eV and keV scale would distort the m is heavily suppressed by thescales ratio and between SU(2) they willevident pass in the our LEP benchmark limits points. on the number ofe.g. light [ neutrinos. This becomes sub-eV and SU(3) by the SU(3) neutrino masses are around keVsquared scale. differences The and mixings. sub-eV neutrinos The are keVby neutrinos constrained the that by our LEP their model bound mass predicts are ontheir constrained masses the smaller number than of light neutrinos, “light” here meaning neutrinos with be probed with a next-generationis collider large. experiments, if However, theirneutrino our mixing component with benchmark to active points active flavours neutrinos, correspond rendering to them extremely completely tiny inaccessible. three TeV-scale heavy, sterile mostly right-handed neutrinos, and three neutrinos with masses between the electron energy spectrum, andbe different sterile detected neutrino via mass unstable rangesdistortions, nuclei, i.e. of such this as distortion can mass sterile neutrinos [ our benchmark points in figure and tively) [ being approximately 9 meVCosmological and constraints are 50 meV. The lightest neutrino state may be massless. SN1987a burst. Also, neutrinolessbackground double and beta growth decay of experimentsthe [ large upper limits scale of structuresstricter flavour in by neutrino one the masses, order early anddition, of their universe neutrino magnitude, sum. [ but oscillation are experiments Cosmological dependent provide constraints on neutrino are the mass cosmological squared model. differences, ∆ In ad- 8 Constraints and numerical examples There are many importantwhen experimental considering constraints the that neutrinothe have sector most to well-known of and be any restrictive.of taken model. Least into model-dependent Constraints account is for the active direct detection neutrinos bound are metries in the neutrinoanarchical sector mixing. [ It isduce up the correct to lepton the masses numerics andis the to the PMNS-matrix acquire within focus the the of experimental order-one limits, section coefficients which that pro- PMNS-matrix. This is in contrast to many models involving more elaborate flavour sym- JHEP02(2020)162 (8.3) (8.1) (8.2) ]. to be around ). The CLFV 55 heavy 6.19 v . ) 0 l L , mediates the CC-NSI ]. We choose the values P µ will be: µ 26 V i , lγ ¯ m 25 )( -breaking scale is fixed. is determined by the light-neutrino -breaking scale -breaking scale. The non-standard X X light,L,β X , L,i ν µ L , ], in contrast to our model. +2 γ U(1) L U(1) 2 U(1) 55 2 V 2 sm × , × v × m L L 54 L 2 light ∼ heavy light,L,α v v – 17 – ν ,L (¯ 0 ∼ ll αβ ,L 0 i ll αβ ). All the new gauge bosons and scalars of our model m F 13 − G 7 TeV. Indeed, for our numerical benchmarks, the NSI 2 ] contain exotic charged leptons not present in the FN331 ) all the light-neutrino masses ], which contain additional sources for charged lepton flavour (10 & √ 2 O 53 51 , − 6.14 V 50 = m ] and [ CC NSI 52 . The neutrinos will still mediate charged lepton decays at loop-level. L 4 mediated contribution to the NSI will be heavily suppressed by its mass: ) and due to small mixing between neutrinos presented in eq. ( µ V 5.3 The FN-charge of the left-handed lepton triplet Our model possesses the extended particle content of the 331-model. The additional The sterile neutrinos of our model come in two distinct mass ranges “medium” and The tree-level CLFV decays of the charged leptons are heavily suppressed as we have where the only free parameter is the FN-charge of the lepton-triplet. masses. According to eq. ( 8.1 The FN-chargesFor for the the numerical numerical example we example take7 the to SU(3) 50 TeV, as forfor this the scale leptonic the FN-charges quark so sector that was studied the in SU(3) [ making it negligible as parameters have magnitude have their masses proportionalinteractions to mediated the SU(3) by chargedheavy masses. scalars will therefore also be suppressed due to their The do not pose any bounds on our model.gauge This bosons is and in scalars of contrast ourinteractions. to model the For could model example potentially mediate in the thegiven [ additional non-standard by, charged neutrino gauge boson, decays of charged leptons can be significant [ “heavy”. In our benchmark points theleptons medium and sterile neutrinos cannot are decay lighter thantheir into the masses them. charged at 10–100 The TeV scale heavy and are sterile not neutrinos produced on in collider the experiments other and therefore hand have stated in section These are also heavilyeq. suppressed ( due todecays of small charged neutrino leptons do Yukawa-couplings331-models presented not such pose in as [ constraints to ourviolation model, such in as constrast bileptons. to Also some in other 331-models with inverse seesaw mechanism the CLFV otherwise, for example, [ model. If theexperiments, non-Standard in Model the particles case of ofother the the in FN331 model a model, calculable turn the way masses out and and to thus couplings be the are measurement within related can to reach hint each of towards the model. each other solely by the light neutrino sector, the particle spectra of the models may differ JHEP02(2020)162 - 5] X , light 5 . (8.4) (8.5) v . The [0 , U(1) 2 3          × | ∈ eV c 3 L | − 9485 5412 1934 5618 6293 2469 4777 4131 ...... 3761 . 4 2 10 − − × . The FN-charges ]: ) . By setting 3 2 j 0852 2 5597 4 6004 2 8619 2 1831 4 , and the FN charges . . . . . 36 5776 2 033 032 1 , . . . 1 4 0 0 4 1174 4777 m . . v 0 3618 1 . − − − − − 1 4 35 +0 − − − − 2 i and , presented in table 525 0 . m 7567 3890 v . . 7356 0 7189 8685 6590 . . . . 2 3 0703 2 3618 3761 = . . . 3 0 3 0 BP3 ) are in the interval = (2 − − 4 2 1 2 ij .          2 32 5.2 m and m    = = = 4 4 4 ∆ N N M c c c 6 6 6 , BP2 2 , . 9 9 9 -breaking VEVs 1 eV ) and eq. ( 5 Y    – 18 – − 0 BP1    4.2 v 10          ) of the right-handed charged leptons as is evident U(1) ∼ × ) are the sole source of charged lepton mass hierarchy, R,i × e 0577 0976 9093 ) . . . e L R,i ( 3 2 0 0397 1615 9481 5320 0345 m 21 20 . . . . . e 5651 6579 . . q . . ( − − − 2 0 q +0 − 8 or 9, one obtains light-neutrino masses from the correct − 39 . ∼ 6284 1689 1615 0 5320 0 . . . . 1251 5925 4 4094 4 L . . . 3 3 2 1 2832 2 3937 2 7926 . . . , the SU(2) = (7 − − − − 2 v 2 21 m 5178 9467 3 59250397 0 2 40949481 0 1 9815 2 0777 4 0 0 0 1 and ...... ∆ 9737 5272 3 . . 0 0 0 4 1 4 1 4 u 1 2 − − − − − − − − , ). The FN-charges             = = = = 4.2 03eV . e ∗ ∗ N 0 c N η N η 0 c c c < As a summary the chosen lepton FN-charges are presented in table When the FN-charge of the lepton triplet is fixed, the charged lepton mass hierarchy Experimentally the light-neutrino masses are constrained by [ 1 m Benchmark point 2. to retain naturalness ofbreaking the VEVs parameters. Wefor choose charged different leptons. values for Below we SU(3) list theBenchmark explicit point values for 1. the coupling matrices we used. of the scalar triplets were presented in table 8.2 Numerical values forWe have leptons chosen three benchmarkorder-one points coefficients introduced in eq. ( depends only on thefrom FN-charges eq. ( as all the left-handed leptoncharged lepton triplet charges FN-charges so are that identical. their We mass choose matrix the texture right-handed becomes: where the neutrinoto mass electroweak squared scale differences and ballpark. are: We will ∆ choose these values for our numerical example. JHEP02(2020)162          66249 88535 57821 . . . 3984 2496 0 3 1 mixing angle. . . ) experiments 4849 . 4 2 − − − 6042 8805 6798 µ . . . − − νββ W the disappearance - µ 67661 88535 4909 3984 . . V 41124 5694 2 9699 2 . . 1954 1 . . . 2 3 2035 1 . 1 4 . BP2 1 4 − − − − − − and similarly the expected 2 4 7 9 1 and 3 7.5 − − 10.7 BP3 − 5467 2 1954 4849 , since the present experimental 2421 66249 203.205 . . . . . 1270 1 3340 72993 1 41124 3 2 1 10 . . . . 2 0 4 3 3 1 BP1 − −    ∼ BP3       2 = | 21 19 29.5 ej = = M BP2 . For c U 1 3 197.5999 | 4 N M 8 2 c − − c − =4 j 6 10 (10), respectively. The medium-mass neutrinos P – 19 – 48 55    ∼ O 100    75.8 BP1 237.05 2    |    µj    5841 0283 9120 U 68750 . . . | . 77676 22936 6207 3 2 1 . . 2 . 5080 2478 7095 8557 4851 . . ) ) ) ) . . . 1 − − − =4 − c α 3 1 1 6215 3745 R (100) and 6 j R R . . − τ e L µ − − − (TeV) ( O ( (GeV) ( ( (GeV) (TeV) P q q it is apparent that next-generation neutrino oscillation exper- q q 0 2 1 u 9615 3998 v v Λ (TeV ) v 2 . . 7798 . 69692 6081 2 2310 2 9354 2 2 . . . . Benchmarks 54563 2 64608 2 3745 0 0524 9733 2219 4 . . . 3 . . . − − 9513 . 1 3 1 1 and − disappearance or neutrinoless double beta decay (0 − − − − 1 µ ν 2359 2 8536 4 0302 7233 3 . . . . 9602 2 0244 . . 1 3 1 0 for the resulting neutrino masses, mass squared differences and effective 1 2 6215 42031 2578 5279 3 0 . . . . − − − − 95332 1 58451 3 5625 2219 0 1 . . . . has a disapprearance effect has the greatest prospect of being detected in future, since the sterile com- 4 3 1 1 4 2 0 1 1       e − − − − ν = =          e BP3 . The numerical values of vacuum expectation values and FN charge assignments of leptons N c = = = 0 c e ∗ N c See table From the figures N η 0 c disappearance should be c µ limits are only approximatelypoints, one degree of magnitudeponent weaker. of Of ourν three benchmark effect is smaller by a factor of strengths of nonunitary and nonstandard interactions, as well as the iments measuring have a moderate possibility of supporting our model at Benchmark point 3. Table 3 for our benchmarks. JHEP02(2020)162 ) it 4 m shows the 1 , (mass shows the con- 0 1 ν 04 2 . 01 . 55 60 120 0 01–0 . . . < . 0 6.79–8.01 Unknown 2.412–2.625 . . We have also included the ] and the MiniBooNe anomaly Experimental values 49 BP3 gauge symmetry and economically 3 − X ]. We calculated the active-medium 10 2.98 9.06 50.8 62.8 7.33 2.50 11.2 48.2 65.1 ] for BP3 0.014 46 0.0109 0.0206 0.0735 ∼ 48 U(1) × 4 − L 10 , the lightest sterile neutrino 129 204 ] and kink searches in single beta decay energy 1.85 8.93 51.1 61.9 7.64 2.61 2.03 5.42 69.3 BP2 0.387 – 20 – 0.0047 ∼ 48 SU(3) , BP3 5 × 47 − c 10 184 380 523 8.59 51.2 59.8 7.39 2.62 1.36 4.99 12.8 BP1 0.0234 0.0010 ∼ | ) ) θ 2 | 2 eV (meV) experiments [ eV 3 3 5 − − m mixing (keV) (keV) (keV) (TeV) (TeV) (TeV) νββ (meV) (meV) (meV) (10 + (10 4 5 6 | 7 8 9 1 2 3 2 µ 2 21 2 32 m m m m m m m m m m W - NSI strength m m Benchmarks + µ ∆ ∆ 1 | V Nonunitary strength m . mixing for our benchmarks. . The computed values of neutrino masses, effective neutrino interaction strength and µ W BP3 - µ PMNS-matrix, is also explained without fine-tuning since the FN mechanism can enforce combination of the seesawis and a FN viable mechanisms. candidateacquire for additional Lightest the suppression sterile MiniBooNescale due neutrino oscillation to to of anomaly. be the quite our The low,heavy FN model around light-neutrino neutrinos mechanism, few masses in TeV. allowing the This the allows future for Majorana colliders. the mass possible The collider mixing searches of of the the neutrinos, represented by the The FN331-model is based onincorporates SU(3) the Froggatt-Nielsen mechanismnumber into of it, fermion thus families simultaneouslyextended and explaining the the the mass FN331-model hierarchytree-level with of masses three charged right-handed for fermions. neutrino allThe In singlets. of neutrino this work the masses This we neutrinos, and allowed which mixings for the in original this FN331-model model was are lacking. naturally explained by utilizing a straints from muon neutrino disappearancefor experiments [ 9 Conclusion will be able toneutrino mixing account matrices the and MiniBooNe have illustratedconstraints anomaly them from at [ 0 constraint plots.spectra Figure of various unstableexpected radioactive sensitivity isotopes of [ KATRIN experiment after three-year run. Figure Table 4 V lie on the eV-scale. For the case on JHEP02(2020)162 . Mass is in GeV units. Black dots ej U – 21 – , but for mixing of muon neutrinos. 1 . BP3 . Same as figure Figure 2 . Constraints for the matrix element absolute values squared describing the strength of Acknowledgments The authors acknowledge theHiggs). H2020-MSCA-RICE-2014 NK grant is no. supported by 645722 Vilho, (NonMinimal- Yrjöand Kalle VäisäläFoundation. the correct texturehere for offers the an lepton explanation forfamilies mass the and matrices. whole the fermion mass sector:while As hierarchy fulfilling it a all explains of the summary the all experimental number the of of constraints. model the fermion fermions, presented thus solving the flavour problem Figure 1 mixing of electron neutrino anddenote medium-massive the neutrinos, corresponding values for JHEP02(2020)162 . .         2            u v 2 (B.1) b u b v b − + 0 1 2 − 0 u 2 fv 2 v fv 1 v 1 v v u 1 0 b 13 ) − − v v fv e λ 23 u + f u 1 2 23 0 1 2 e λ + u e λ v v + 2 + . 1 2 u + fv 0 13 23 v 2 2 2 v            λ λ − u v 23 23 3 2 13 e u λ λ 23 v λ 2 ( e λ 1 2 b v 2 e λ 1 2 1 1 2 0 1 b v − b − 1 23 fv + v u . The imaginary part of 2 0 λ 1 u u v v v 1 2 u µ 0 2 2 b 2 v 5 f v v v 1 2 ) 0 1 v v − ) v v W 2 2 b 23 2 2 2 2 v 0 u e λ 12 v − b f 2 λ + v fu 2 b fv λ v 2 1 2 2 + v λ + 12 2 ( v + + 2 0 v and e λ . b u 2 23 1 1 v 1 2 ) u 2 1 2 v v λ 2 µ 0 v v 2 v − 12 ( fv C, v 4 v + b 23 e λ 2 u 2 23 ) − H, λ 1 2 23 v u 2 12 A, 2 1 e λ 0 W 1 2 v v e λ u e λ 1 2 v + 23 2 1 2 + , 1 2 b e v λ 2 1 2 2 odd µ even v 1 2 2 ( 2 v 1 + v − − B b v − 1 2 1 fu f u 2 v v 2 2 v 2 + cp , 12 2 cp 2 − b u v + v 1 2 v e µ λ 2 23 2 1 1 2 2 v v b 0 0 b − 1 2 1 M M e v λ 23 v v 2 v charged scalar fv 1 2 W − 0 ( T e λ T 12 23 + v 1 2 – 22 – − , 2 b b 3 e λ e λ A M 4 u 0 H 1 2 µ 12 g 1 2 1 v 1 2 + T 3 v e λ 1 2 − 1 2 0 u u f 1 0 C = 2 v W v 0 u 1 2 v 0 fu b fv 5 v v u fu fv 23 fv L ⊃ 2 2 + f W L ⊃ 13 e λ v − − u L ⊃ 1 e 1 2 λ 1 + 1 2 M v 1 2 u v v 2 2 2 0 1 u 0 fv 1 1 2 2 λ 1 + v v v 23 v v 0 0 v ) and v 2 fv 0 fu f 2 1 e λ 2 + 2 fv 5 23 12 ) and f v v 1 2 − v λ ) and λ λ u 2 0 + 12 5 12 12 , h            v e λ e λ e λ u 4 0 1 1 2 1 , ξ 1 2 13 1 2 v , χ v fu = 4 e λ fv 2 , h + f 1 2 + v − 3 , ξ 0 2 − u 0 + 1 , ρ v 3 1 v odd u v v 2 , h fv 0 0 + 0 12 − 2 , ξ v v f v λ 2 1 2 , η 13 12 cp         λ , h λ λ , ξ + 2 1 0 M 1 = h η ξ            = = ( = ( = ( T T T even C A H − decouples from the other neutral gauge bosons and acquires a mass: 2 cp 0 µ 2 charged scalar 0 M M B Neutral gauge bosonThere masses are five neutral gaugeX bosons: The charged scalar mass term is where A.3 Charged scalars where A.2 CP-odd scalars The CP-odd scalar mass term is where A.1 CP-even scalars The CP-even scalar mass term is A Scalar mass matrices JHEP02(2020)162 . , (B.2)             2 2 ) (2018) B 292 u 2 v 2 + 1 3 2 v v 2 1 v 2 √ ( v 1 v D 98 3 x v annihilation g g − 3 + 4 − 2 1 , e − v Phys. Lett. collisions near the + ! e , )) ) − ) 2 2 e 0 2 0 0 v ) + v 2 v light Phys. Rev. 2 2 heavy e + v 2 2 v v , +4 ) 1 u 3 2 2 + 2 g v

2 2 3 x x + u 3 g g g ( 2 2 2 1 9 2 4 √ + 3 v x O v ( 3 2 2 g g ( v 3 4 3 x + +2( + g g . − 2 2 1 1 3 2 v v ) (B.3) ( ! 2 µ − − ( 5 u 2 X, u )) + 2 1 2 iW light )) 2 . heavy 2 v 0 v 2 2 2 2 ]. v v − u v + + 2

µ neutral 2 2 ) 2 2 4 + 2 2 u v 3 0 3 v O M g ) and v 2 3 x + 1 W 0 3 √ SPIRE 3 g µ T 2 2 v − ( √ 2 ]. v √ 4 +4( v 2 1 – 23 – IN 3 boson resonance parameters in 2 2 1 Review of particle physics v X ) + − , ( [ v 1 2 ) 1 2 √ +2( ,W Z + 2 1 u 2 1 2 µ v 0 2 v x 2 = x v SPIRE + g ( − g + ( ,B 0 IN µ 2 2 L ⊃ µ 2 2 [ v + ) 8 X ( + 4 v 2 ), which permits any use, distribution and reproduction in 2 (1995) 47 3 0 2 x 2 3 g v + g g ,W 3 2 1 3 2 µ 2 + v collaboration, 3 9 0 u 2 + 2 v 3 ( 2 3 v C 65 + W 2 1 2 2 − g 1 3 3 √ , 4 4 Measurement of single photon production in 2 1 v 2 v g g 3 0 v ( (1989) 2173 Measurements of ]. v = ( 3 x g g = 0 = = = Determination of the number of light neutrino species CC-BY 4.0 3 2 T 63 0 ]. 4 2 γ 2 Z 2 Z − X f This article is distributed under the terms of the Creative Commons 2 W Z. Phys. m SPIRE m , m             m IN 2 3 [ 4 SPIRE g IN collaboration, [ = collaboration, physical neutral non-hermitean gauge boson 2 Phys. Rev. Lett. 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