aa r X i v : . [ h e p - ph ] A ug Clockwork origin of neutrino mixings
Teruyuki Kitabayashi ∗ Department of Physics, Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan
The clockwork mechanism provides a natural way to obtain hierarchical masses and couplings ina theory. We propose a clockwork model that has nine clockwork generations. In this model, thecandidates of the origin of the neutrino mixings are nine Yukawa mass matrix elements Y aβ thatconnect neutrinos and clockwork fermions, nine clockwork mass ratios q aβ , and nine numbers ofclockwork fermions n aβ , where a, β = 1 , ,
3. Assuming | Y aβ | = 1, the neutrino mixings originatefrom the pure clockwork sector. We show that the observed neutrino mixings are exactly obtainedfrom a clockwork model in the case of the q aβ origin scenario. In the n aβ origin scenario, the correctorder of magnitude of the observed neutrino mixings is obtained from a clockwork model. PACS numbers: 12.60.-i, 12.90.+b, 14.60.Pq, 14.60.St
I. INTRODUCTION
One of the outstanding problems in particle physicsis the origin of the neutrino masses and mixings [1].There are theoretical mechanisms to generate tiny neu-trino masses, such as seesaw mechanisms [2–5], radiativemechanisms [6–13], and the scotogenic model [14]. Onthe other hand, the neutrino mixings have been studiedunder assumptions of the existence of underlying flavorsymmetries in the theories [15–17].Recently, a new mechanism, the clockwork mechanism[18], has attracted attention. The clockwork mechanismprovides a new natural way to obtain hierarchical massesand couplings in a theory. In a series of the gears in aclock, large (small) movement of the gear in one side ofthe series can generate a small (large) movement of thegear in the opposite side. In the theories based on theclockwork mechanisms, a large number of fields, so-calledclockwork gears, is introduced. The zero-mode state ofthe clockwork gears ψ (0) R on the one side of the series ofthe clockwork gears connects to the gear on the oppositeside ψ R via intermediate gears. We obtain the relation ψ R ∼ q n ψ (0) R , (1)where q ( q >
1) denotes the mass ratio of the gears and n denotes the number of gears [19, 20]. Even if the massratio q is not so hierarchical, e.g., q = 1 . , q = 2 .
0, etc, alarge suppression factor 1 /q n for large n may provide asmall coupling or mass for ψ R in the model. The applica-tions of the clockwork mechanism have been extensivelystudied in the literature, e.g., for the axion [21–30], forinflation [31, 32], for dark matter [33–37], for the muon g − ∗ Electronic address: [email protected] masses [20, 46, 47] and for their masses and mixings [48].Up to now, there have only been discussions of neutrinomixings with the clockwork mechanisms in Ref. [48] byIbarra, et.al . In this model, the neutrino mass m aβν isobtained as m aβν = f ( Y aβ , q β , n β ) , (2)where a = 1 , , β ≥ Y aβ denotes the Yukawa coupling (whichconnects the standard model sector to the clockwork sec-tor), q β is the clockwork mass ratio, and n β is numberof clockwork fields in the β th clockwork generation. Themain role of the clockwork sector, e.g., q β and n β , isthe genesis of the tiny neutrino masses. On the otherhand, the mixings of the neutrinos are originated fromthe Yukawa couplings.In this paper, we extend the clockwork model pro-posed by Ibarra et.al. , [48] to propose a clockwork modelthat has nine clockwork generations. In the extendedmodel, only three clockwork generations can couple withone generation of the standard model lepton doublet, theother three clockwork generations can only couple withanother one generation of the lepton doublet, and theremaining three clockwork generations can only couplewith the remaining one generation of the lepton doublet.The final expression of neutrino mass is obtained as afunction of the Y aβ , q aβ , and n aβ , m aβν = f ( Y aβ , q aβ , n aβ ) , (3)where a, β = 1 , ,
3. In this model, not only the Yukawacoupling Y aβ but also q aβ and n aβ can be the origin ofthe neutrino mixings. Indeed, we will show that a modelwith the democratic Yukawa matrix | Y aβ | = 1 is consis-tent with the observed neutrino masses and mixings. Inthis case, the mixings of the neutrinos originate from theclockwork fields instead of the Yukawa couplings.The paper is organized as follows. In Sec.II, we presenta brief review of the neutrinos masses and mixings andthe fermion clockwork mechanisms. In Sec.III, we pro-pose a clockwork model, which has the origin of the neu-trino mixings in the pure clockwork sector. Section IV isdevoted to a summary. II. BRIEF REVIEWS
In this section, we just review the basic picture of theneutrino physics and clockwork mechanisms. This reviewdoes not include any new findings. We also show ournotations and some assumptions in this paper.
A. Observed neutrino masses and mixings
The simple clockwork model of fermions yields theDirac neutrinos [18]. The models of the Majorana neu-trinos with the clockwork mechanisms are also discussed[20, 48]. The minimal setup of the model is enough tobuild a possible model that can explain both the observedneutrino masses and mixing by only pure clockwork pa-rameters ( q aβ and n aβ ) in the next section. Thus, theDirac neutrinos are employed in this model.The neutrino mass matrix m ν = m m m m m m m m m , (4)satisfies the relation m ν m † ν = U PMNS m m
00 0 m U † PMNS , (5)where m , m and m denote the neutrino mass eigen-states and U PMNS = (6) c c s c s − s c − c s s c c − s s s s c s s − c c s − c s − s c s c c , denotes the mixing matrix [49]. We use the abbreviations c ij = cos θ ij and s ij = sin θ ij ( i, j =1,2,3) and ignore the CP -violating phase. The relation between CP and theclockwork sector is one of the important problems forthe clockwork mechanism. It seems that the CP vio-lation can be achieved if the Yukawa couplings are notdemocratic but have different phases (and magnitude);however, in this paper, we will employ the democraticYukawa coupling and we would like to omit the study of CP structure in the clockwork mechanism.The neutrino mass ordering (either the normal massordering m < m < m or the inverted mass ordering m < m < m ) is unsolved problems. The best-fit val-ues of the squared mass differences ∆ m ij = m i − m j andthe mixing angles for normal mass ordering are estimatedas [50]∆ m / (10 − eV ) = 7 .
50 (7 . − . , ∆ m / (10 − eV ) = 2 .
524 (2 . − . ,θ / ◦ = 33 .
56 (31 . − . ,θ / ◦ = 41 . . − . ,θ / ◦ = 8 .
46 (7 . − . , (7) where the parentheses denote the 3 σ region. The neu-trino mass matrix m ν = . m . m . m − . m . m . m . m − . m . m , (8)with m = q . × − + m eV ,m = q . × − + m eV , (9)is consistent with the best-fit values of neutrino oscil-lation parameters in the case of normal mass ordering.On the other hand, in the inverted mass ordering, thesquared mass differences and the mixing angles are esti-mated as [50]∆ m / (10 − eV ) = 7 .
50 (7 . − . , − ∆ m / (10 − eV ) = 2 .
514 (2 . − . ,θ / ◦ = 33 .
56 (31 . − . ,θ / ◦ = 50 . . − . ,θ / ◦ = 8 .
49 (8 . − . , (10)and the neutrino mass matrix m ν = . m . m . m − . m . m . m . m − . m . m , (11)with m = q m − . × − eV ,m = q . × − + m eV , (12)is consistent with the best-fit values of neutrino oscilla-tion parameters. B. Fermionic clockwork mechanism
In the clockwork sector, there are n left-handed chiralfermions, ψ Li ( i = 0 , , · · · , n − n +1 right-handedchiral fermions, ψ Ri ( i = 0 , , · · · , n ). The clockworkLagrangian is [18, 36, 48] L cw = L kin + L nearest + L mass , (13)where L kin denotes the kinetic term for clockworkfermions, L nearest = − n − X i =0 (cid:0) m i ¯ ψ Li ψ Ri − m ′ i ¯ ψ Li ψ Ri +1 + h . c . (cid:1) , (14)denotes the nearest-neighbor interaction term, and L mass = − n − X i =0 M Li ψ cLi ψ Li − n X i =0 M Ri ψ cRi ψ Ri , (15)denotes the Majorana mass term. We take the universalDirac mass assumption, m i = m , m ′ i = mq , and theuniversal Majorana mass assumption, M Li = M Ri = m ˜ q ,for all i [48]. The universal Dirac mass assumption andthe universal Majorana mass assumption are enough toconstruct a model that can generate both the observedtiny neutrino masses and mixing pattern by only pureclockwork sector ( q aβ and n aβ ) in the next section.The nearest-neighbor interaction term can be writtenin the simple form L nearest = − (cid:0) Ψ c M Ψ + H . c . (cid:1) , (16)whereΨ = ( ψ L , ψ L , · · · , ψ Ln − , ψ cR , ψ cR , · · · ψ cRn ) , (17)and M = m ˜ q · · · − q · · ·
00 ˜ q · · · · · · · · · ˜ q − q · · · q · · · − q · · · q · · · · · · − q q . (18)The eigenvalues of the (2 n + 1) × (2 n + 1) matrix M areobtained as [36] m = m ˜ q,m k = m (cid:16) ˜ q − p λ k (cid:17) , k = 1 , · · · , n,m n + k = m (cid:16) ˜ q + p λ k (cid:17) , k = 1 , · · · , n, (19)where λ k = q + 1 − q cos kπn + 1 . (20)The interaction eigenstates Ψ i and mass eigenstates,denoted by χ i , are related to each other by the uni-tary transformation Ψ i = P j U ij χ j , where U is the(2 n + 1) × (2 n + 1) unitary matrix U = ~ √ U L − √ U L ~u R √ U R √ U R ! , (21)with ~ i = 0 , i = 1 , · · · , n, ( ~u R ) i = 1 q i s q − q − q − n , i = 0 , · · · , n, ( U L ) ij = r n + 1 sin ijπn + 1 , i, j = 1 , · · · , n, ( U R ) ij = s n + 1) λ j (cid:18) q sin ijπn + 1 − sin ( i + 1) jπn + 1 (cid:19) ,i = 0 , · · · , n j = 1 , · · · , n. (22) The total Lagrangian of the standard model with theclockwork sector reads L = L SM + L cw + L int , (23)where L SM is the standard model Lagrangian and L int describes the interactions between the standard modelsector and the clockwork sector. We assume that thelast site of the clockwork fields only couples to the left-handed neutrinos in the standard model [18] L int = − Y ˜ H ¯ Lψ Rn , (24)where L denotes the left-handed lepton doublet, ˜ H = iτ H ∗ ( H denotes the standard model Higgs doublet),and Y denotes the Yukawa matrix. In general, the mix-ing matrix U PMNS is related to the neutrino-diagonalizingmatrix U ν and the charged-lepton diagonalizing matrix U ℓ as U PMNS = U † ℓ U ν [51]. In this paper, to discussthe possible origin in the clockwork sector with a sim-ple setup, we assume that the charged leptons are flavordiagonal. In terms of the mass eigenstates, we have L int = − Y ˜ H ¯ LU nk χ k ≡ − n X k =0 Y k ¯ L ˜ Hχ k , (25)where Y = Y ( U R ) n = Yq n s q − q − q − n , (26) Y k = Y n + k = 1 √ Y ( U R ) nk = Y s n + 1) λ k (cid:18) q sin nkπn + 1 (cid:19) , k = 1 , · · · , n. Now we generalize the above setup to three leptonicgenerations and N clockwork generations. The nearest-neighbor interaction term for N clockwork generationsis L nearest = − n − X i =0 (cid:16) m iαβ ¯ ψ αLi ψ βRi − m ′ iαβ ¯ ψ αLi ψ βRi +1 + H . c . (cid:17) , (27)where α, β = 1 , · · · N . For simplicity, we assume m iαβ = mδ αβ , m ′ iαβ = mq α δ αβ and M αβLi = M αβRi = m ˜ qδ αβ = 0[48]. The nearest-neighbor interaction term can be L nearest = − (cid:0) Ψ αc M αβ Ψ β + H . c . (cid:1) , (28)whereΨ α = (cid:0) ψ αL , ψ αL , · · · , ψ αLn − , ψ αcR , ψ αcR , · · · ψ αcRn (cid:1) . (29)In terms of the mass eigenstates χ βk (Ψ αi = P j U αβij χ βj ),the interactions between the left-handed neutrinos andclockwork fields can be written as L int = − n X k =0 Y aβk ¯ L a ˜ Hχ βk , (30)where a = 1 , ,
3. We define new fields N αL =( ν αL , N αL , · · · , N αLn ) and N αR = ( N αR , N αR , · · · , N αRn )where N αLk = 1 √ (cid:0) − χ αk + χ αk + n (cid:1) , k = 1 , · · · , n,N αRk = 1 √ (cid:0) χ αk + χ αk + n (cid:1) , k = 0 , · · · , n, (31)for α = 1 , · · · , N . The nearest-neighbor interaction termcan be cast as: L nearest = − N αL m αν N αR + H . c ., (32)and we have the interaction Lagrangian L int = − n X k =0 Y aβk ¯ L a ˜ H N βRk , (33)with Y aβk = Y aα U αβnk for the Dirac neutrinos. After elec-troweak symmetry breaking, the neutrino mass matrix isto be m ν = N βR N βR N βR · · · N βRn ν aL vY aβ vY aβ vY aβ · · · vY aβn N βL M β · · · N βL M β · · · N βLn · · · M βn , (34)where v = 246 / √ M βk denotesthe mass of the k th clockwork fields for the Dirac pair( N βL , N βR ).In this model, the neutrino masses are to be small viazero-mode interactions of clockwork fermions; however,in general, unsuppressed effects in low-energy phenom-ena, such as an unobserved lepton flavor-violating decay µ → eγ , are allowed. The upper bound of the leptonflavor-violating processes yields constraints on the massscale of the clockwork fermions. The clockwork fermionsmust be larger than approximately 40 TeV in order toevade the experimental constraints, and the condition of M βk ≫ vY aβ is required [48].With the relation of M βk ≫ vY aβ , the active neutrinomasses are obtained as m aβν = vY aβ = vY aβ q n β β vuut q β − q β − q − n β β , (35)where q β and n β denote the clockwork mass ratio andthe number of clockwork fermions in the β th clockworkgeneration, respectively.In the next section, we propose a clockwork model thatis based on the model in this section. Since the basicstructures of these two models are the same, the phe-nomenological consistency of the model in the next sec-tion is guaranteed with the requirement of M βk ≫ vY aβ [48]. III. ORIGIN OF NEUTRINO MIXINGS
First of all, we would like to show briefly the main newresult in this paper.In the previously proposed model [48], the neutrinomasses are obtained as m aβν ∝ q n β β vuut q β − q β − q − n β β , a = 1 , , , (36)for democratic Yukawa couplings. In the matrix form,we have m ν = m m m m m m m m m = m m m m m m m m m , (37)for β = 1 , ,
3. The number of parameters in this equa-tion is not enough to generate the observed neutrino mix-ing patterns. We cannot obtain both the tiny neutrinomasses and neutrino mixings by the tuning of only q β and n β .In this section, we propose a new nine-generationclockwork model with clockwork lepton numbers. In thisnew model, the neutrino masses are finally obtained as m aβν ∝ q n aβ aβ vuut q aβ − q aβ − q − n aβ aβ ,a = 1 , , , β = 1 , , , (38)for democratic Yukawa couplings; then, we can obtainboth the correct tiny neutrino masses and mixings bythe tuning of only q aβ and n aβ .The Yukawa couplings connect the standard model sec-tor and clockwork sector. On the other hand, q aβ and n aβ are pure clockwork parameters. The origin of neutrinomixing can be in the pure clockwork sector in the newmodel. This is the main new finding in this paper. A. Model
We extend the clockwork model proposed by Ibarra et.al. [48] to propose a new clockwork model that hasnine generations in the clockwork sector. We assumethat only three clockwork generations can couple withone generation of the standard model lepton doublet, an-other three clockwork generations can only couple withanother one generation of the lepton doublet and theremaining three clockwork generations can only couplewith the remaining one generation of the lepton doublet.Under these assumptions, the interaction Lagrangian,in terms of N R , for three leptonic generations and nineclockwork generations is L int = − n X k =0 Y aβk ¯ L a ˜ H N βRk ,a = 1 , , , β = 1 , · · · , (39)where Y aβk = ∗ , β = , , a = 14 , , a = 27 , , a = 3 , , others , (40)or equivalently, Y aβk = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , (41)where ∗ denotes nonzero values. If we assign the leptonnumber to the clockwork sector as shown in Table I andassume the lepton number is conserved in the interactionsof Y aβk ¯ L a ˜ H N βRk , we obtain the configuration in Eq.(40).The interaction Lagrangian reads L int = (42) − ˜ H ¯ L n X k =0 Y k N Rk + n X k =0 Y k N Rk + n X k =0 Y k N Rk ! − ˜ H ¯ L n X k =0 Y k N Rk + n X k =0 Y k N Rk + n X k =0 Y k N Rk ! − ˜ H ¯ L n X k =0 Y k N Rk + n X k =0 Y k N Rk + n X k =0 Y k N Rk ! . Assuming M βk ≫ vY aβ , after electroweak symmetrybreaking, the neutrino masses are m aβν = vY aβ = vY aβ q n β β vuut q β − q β − q − n β β , (43)where( a, β ) = (1 , , (1 , , (1 , , (2 , , (2 , , (2 , , (3 , , (3 , , (3 , . (44)We arrange these nine neutrino masses m ν , · · · , m ν intothe neutrino mass matrix as m ν = m ν m ν m ν m ν m ν m ν m ν m ν m ν = vY vY vY vY vY vY vY vY vY = f ( Y , q , n ) f ( Y , q , n ) f ( Y , q , n ) f ( Y , q , n ) f ( Y , q , n ) f ( Y , q , n ) f ( Y , q , n ) f ( Y , q , n ) f ( Y , q , n ) , (45)and rename the model parameters as follows: m ν m ν m ν m ν m ν m ν m ν m ν m ν → m ν m ν m ν m ν m ν m ν m ν m ν m ν , (46) TABLE I: Lepton numbers of the β th clockwork generations. β L L L Y Y Y Y Y Y Y Y Y → Y Y Y Y Y Y Y Y Y , (47) Y Y Y Y Y Y Y Y Y → Y Y Y Y Y Y Y Y Y , (48) q q q q q q q q q → q q q q q q q q q , (49)and n n n n n n n n n → n n n n n n n n n . (50)After the renaming, the neutrino mass matrix is de-noted by m ν = m ν m ν m ν m ν m ν m ν m ν m ν m ν = vY vY vY vY vY vY vY vY vY , (51)where m aβν = vY aβ q n aβ aβ vuut q aβ − q aβ − q − n aβ aβ ,a = 1 , , , β = 1 , , . (52)Now, we obtain the similar relation of Eq.(35), but q β and n β are replaced with q aβ and n aβ . This replacementallows us to obtain the correct tiny neutrino masses andmixings by the tuning of only pure clockwork parameters( q aβ and n aβ ). These replacements are possible from theassignment of the lepton number in the Table I as well asfrom the lepton-number conservations in the interaction Y aβk ¯ L a ˜ H N βRk .There are the following four possible origins of the neu-trino mixings in this model: (a) the Yukawa matrix Y aβ [48], (b) the clockwork mass ratio in the aβ th generations q aβ , (c) the number of clockwork fermions in the aβ th gen-erations n aβ , (d) the others (both of Y aβ , q aβ , etc).We assume a democratic form of the Yukawa matrix[52], | Y aβ | = 1 , (53)more concretely, Y aβ = sign( m ν ) sign( m ν ) sign( m ν )sign( m ν ) sign( m ν ) sign( m ν )sign( m ν ) sign( m ν ) sign( m ν ) . (54)The democratic Yukawa couplings could be regarded asan extreme limit of the random matrices models of flavors[19, 44, 52]. In the random matrices scheme of flavors, theproducts of random O (1) matrices possess a hierarchicalspectrum. In the democratic Yukawa couplings case, theneutrino mass m aβν only depends on the clockwork massratio q aβ and number of clockwork fermions n aβ , (cid:12)(cid:12) m aβν (cid:12)(cid:12) = vq n aβ aβ vuut q aβ − q aβ − q − n aβ aβ , (55)and the cases b, c and d are relevant for possible originsof the neutrino mixing.In what follows, we study the cases b, c and d. Al-though the neutrino mass ordering is an unsolved prob-lem, a global analysis shows that the preference for thenormal mass ordering is mostly due to neutrino oscilla-tion measurements [53, 54]. From this experimental fact,we use the relations of neutrino masses with normal massordering in the main part of the remainder of this paper. B. q aβ with universal n First, to see the relation between neutrino mixings and q aβ , we assume that the number of clockwork fermionsis common for all clockwork generations. According toRefs. [44], this “universal n ” limit of the clockwork modelis reminiscent of the Randall-Sundrum flavor models [55,56]. In this case, the origin of the neutrino mixings isthe clockwork mass ratios q aβ (case b in Sec.III A). Forexample, if we take the universal number of clockworkfermions n = n aβ = 50 , (56)for all a and β , the clockwork mass ratios q q q q q q q q q = .
841 1 .
845 1 . .
860 1 .
844 1 . .
882 1 .
841 1 . (57)yield the best-fit values of the squared mass differencesand the mixing angles in Eq.(7) for m = 0 . a β m =0.001 eVm =0.005 eVm =0.01 eVm =0.03 eV 1.5 2 2.5 3 3.5 4 4.5q11 q12 q13 q21 q22 q23 q31 q32 q33n = 50 q a β m =0.001 eVm =0.005 eVm =0.01 eVm =0.03 eV FIG. 1: The magnitude of the clockwork mass ratio q aβ for thebest-fit values of the squared mass differences and the mixingangles under the normal mass ordering condition, where n denotes the universal number of fermions for all clockworkgenerations ( n = 30 in the upper panel and n = 50 in thelower panel). Figure 1 shows the magnitude of the clockwork massratio q aβ for the best-fit values of the squared mass dif-ferences and the mixing angles under the normal massordering condition, where n denotes the universal num-ber of fermions for all clockwork generations ( n = 30 inthe upper panel and n = 50 in the lower panel). Theupper limit of m ≤ .
03 is obtained from the observeddata m ν < .
120 by the Planck Collaboration [57].Because q a ( a = 1 , ,
3) depends on m a ν ∝ m and m = p . × − + m ∼ √ . × − eV for m ≤ .
03, the magnitudes of q , q , and q are al-most independent of the lightest neutrino mass m , aswe see in Fig.1. C. n aβ with universal q Second, to see the relation between neutrino mixingsand n aβ , we assume that the clockwork mass ratio is com-mon for all clockwork generations. According to Refs.[44, 52], the clockwork model of flavor in this “univer-sal q ” limit is equivalent to the Froggatt-Nielsen modelswith a U (1) symmetry [58–60]. In this case, the originof the neutrino mixings is the number of the clockworkfermions n aβ (case c in Sec.III A). For example, if we takethe universal clockwork mass ratio q = q aβ = 2 . , (58)for all a and β , the numbers of clockwork fermions n n n n n n n n n =
44 44 4445 44 4246 44 42 (59)yield ∆ m = 8 . × − eV , ∆ m = 1 . × − eV ,θ = 40 . ◦ ,θ = 45 . ◦ ,θ = 10 . ◦ . (60)Although these predicted values (except θ ) are out ofrange of the 3 σ region in Eq.(7), the order of magnitudeof these values is consistent with the observed data.We should perform more general parameter searchwith various sets of the universal clockwork mass ra-tio and numbers of clockwork fermions { q, n aβ } ; how-ever, this is a numerically challenging task. For exam-ple, there are approximately 10 loops in the code toperform a numerical search for q = 1 . , . , · · · , . n aβ = 10 , , · · · ,
100 and m = 0 . , . , · · · , .
03 eVfor only the best-fit values of the neutrino parameters.In this paper, we abort such a full parameter search andonly show some examples of the parameter set that areconsistent with neutrino observations. D. n aβ with quasiuniversal q If we relax the universal q requirement and allow theexistence of the small perturbations of the clockworkmass ratios, ∆ q (∆ q ≪ q ), we can obtain the correct neu-trino mass parameters within the n aβ origin scenario ofthe neutrino mixings (case c with small correction of theuniversal clockwork mass ratio in Sec.III A). For example,if we take the quasiuniversal clockwork mass ratios q q q q q q q q q = q + ∆ q (61)= 2 . + .
01 0 − . − .
02 0 . − . − .
011 0 − . , the number of clockwork fermions, the same as Eq.(59), n n n n n n n n n =
44 44 4445 44 4246 44 42 (62)
30 35 40 45 50 55 60 65 70n11 n12 n13 n21 n22 n23 n31 n32 n33q = 2.0 n a β m =0.001 eVm =0.005 eVm =0.01 eVm =0.03 eV 30 35 40 45 50 55 60 65 70n11 n12 n13 n21 n22 n23 n31 n32 n33q = 2.5 n a β m =0.001 eVm =0.005 eVm =0.01 eVm =0.03 eV FIG. 2: The magnitude of the effective number of clockworkfermions n aβ for the best-fit values of the squared mass dif-ferences and the mixing angles under the normal mass order-ing condition, where q denotes universal clockwork mass ratio( q = 2 . q = 2 . yields ∆ m = 7 . × − eV , ∆ m = 2 . × − eV ,θ = 31 . ◦ ,θ = 41 . ◦ ,θ = 8 . ◦ . (63)These predicted values are consistent with the observeddata in Eq.(7). E. Effective n aβ with universal q We show an alternative way to obtain the correct neu-trino mixings with the n aβ origin scenario for a universalclockwork mass ratio q .Although the number of the clockwork fermions in the aβ th clockwork generation n aβ should be a real integernumber, we relax this requirement (small correction ofcase c in Sec.III A). In this case, for example, if we takethe universal clockwork mass ratio q = q aβ = 2 , (64)for all a and β , the effective numbers of clockworkfermions n n n n n n n n n = .
06 44 .
24 44 . .
79 44 .
19 42 . .
64 44 .
08 41 . (65)yield the best-fit values of the squared mass differencesand the mixing angles in Eq.(7) for m = 0 . n aβ for the best-fit values of thesquared mass differences and the mixing angles underthe normal mass ordering condition, where q denotes theuniversal clockwork mass ratio ( q = 2 . q = 2 . F. Inverted mass ordering
We would like to address briefly some subjects for theinverted mass ordering of the neutrinos.In the case of the universal n , the universal number ofclockwork fermions n = n aβ = 50 , (66)with the clockwork mass ratios q q q q q q q q q = .
782 1 .
796 1 . .
803 1 .
801 1 . .
813 1 .
787 1 . , (67)yields the best-fit values of the squared mass differencesand the mixing angles in Eq.(10) for m = 0 . n setup can work for the case of the invertedmass ordering as well as for the case of the normal massordering.Moreover, in the case of inverted mass ordering withuniversal q , the order of the magnitude of the predictedvalues of neutrino oscillation parameters can be consis-tent with the observed data; however, these predictedvalues are out of range of the 3 σ region in Eq.(10). Wehave encountered the same situation in Sec.III C. IV. SUMMARY
We have proposed a clockwork model that has nineclockwork generations. Only three clockwork generationscan couple with one generation of the standard modellepton doublet; another three clockwork generations canonly couple with another one generation of the leptondoublet and the remaining three clockwork generationscan only couple with the remaining one generation ofthe lepton doublet. Under these assumptions, the neu-trino masses depend on the nine Yukawa matrix elements Y aβ , nine clockwork mass ratios q aβ , and nine numbersof clockwork fermions n aβ . In this model, the candidatesof the origins of the neutrino mixings are Y aβ , q aβ , and n aβ . We have assumed | Y aβ | = 1; thus, the Yukawa cou-pling is not the main origin of the neutrino mixings. Themain origin of the neutrino mixing is in the clockworksector, q aβ and n aβ , in this model.We have shown that the observed neutrino mixingsare exactly obtained with a clockwork model in the caseof the q aβ origin scenario. In the n aβ origin scenario,although the predicted values (except θ ) are out ofthe range of the 3 σ region, the correct order of mag-nitude of the observed neutrino mixings is obtained froma clockwork model. To obtain the neutrino parameterswithin the 3 σ region in the n aβ origin scenario, it is sug-gested that some modification schemes should be em-ployed, such as the quasiuniversal q or the effective n aβ .Finally, we would like to comment that there is nolepton number, or some symmetry, in the clockwork sec-tor in almost all of the clockwork models. On the con-trary, in this paper, we obtain both the correct neutrinomasses and mixings by assigning the lepton numbers tothe clockwork sector. We can expect that the symmet-ric argument is getting more important for further modelbuilding in the context of the clockwork mechanism. [1] S. F. King, J. Phys. G , 123001 (2015).[2] P. Minkowski, Phys. Lett. B , 421 (1977).[3] T. Yanagida, in Proceedings of the Workshop on UnifiedTheories and Baryon Number in the Universe , edited byA. Sawada and A. Sugamoto KEK Report No. 79-18,(KEK, Tsukuba,1979), p. 95.[4] M. Gell-Mann, P. Ramond, and R. Slansky, in
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