Clockwork origin of neutrino mixings

The clockwork mechanism provides a natural way to obtain hierarchical masses and couplings in a theory. We propose a clockwork model which has nine clockwork generations. In this model, the candidates of the origin of the neutrino mixings is nine Yukawa mass matrix elements $Y^{a\beta}$ which connect neutrinos and clockwork fermions, nine clockwork mass ratios $q_{a\beta}$ and nine numbers of clockwork fermions $n_{a\beta}$, where $a, \beta=1,2,3$. Assuming $|Y^{a\beta}|=1$, the neutrino mixings are originate from pure clockwork sector. We show that the observed neutrino mixings are exactly obtained from a clockwork model in the case of $q_{a\beta}$ origin scenario. In the $n_{a\beta}$ origin scenario, the correct order of magnitude of the observed neutrino mixings are obtained from a clockwork model.

Recently, a new mechanism, the clockwork mechanism [18], attracts attention. The clockwork mechanism provides a new natural way to obtain hierarchical masses and couplings in a theory. In a series of the gears in a clock, large (small) movement of the gear in one side of the series can generate a small (large) movement of the gear in the opposite side. In the theories based on the clockwork mechanisms, large number of fields, so-called clockwork gears, are introduced. The zero mode state of the clockwork gears ψ (0) R in the one side of the series of the clockwork gears connect to the gear in the opposite side ψ R0 via intermediate gears. We obtain the following relation where q (q > 1) denotes the mass ratio of the gears and n denotes the number of gears [19,20]. Even if the mass ratio q is not so hierarchical, e.g. q = 1.5, q = 2.0, etc, a large suppression factor 1/q n for large n may provide a small coupling or mass for ψ R0 in the model. The applications of the clockwork mechanism have been extensively studied in the literature, e.g., for axion [21][22][23][24][25][26][27][28][29][30], for inflation [31,32], for dark matter [33][34][35][36][37], for muon g−2 [38], for string theory [39][40][41], for gravity [42,43], for charged fermion masses and mixings [19] and for quark masses and mixings [44]. The applications of the clockwork mechanism for the neutrino sector have been studied for tiny neutrino * Electronic address: teruyuki@tokai-u.jp masses [20,45,46] and for their masses and mixings [47]. Up to now, we can see the discussions of neutrino mixings with the clockwork mechanisms only in Ref. [47] by Ibarra, et.al. In this model, the neutrino mass m aβ ν is obtained as where a = 1, 2, 3 and β ≥ 2 for observed three neutrino generations, Y aβ denotes the Yukawa coupling (which connects the standard model sector to the clockwork sector), q β is the clockwork mass ratio and n β is number of clockwork fields in the β-th clockwork generation. The main role of the clockwork sector, e.g., q β and n β , is genesis of the tine neutrino masses. On the other hand, the mixings of the neutrinos are originated from the Yukawa couplings.
In this paper, we extend the clockwork model proposed by Ibarra et.al., [47] to propose a clockwork model which has nine clockwork generations. In the extended model, only three clockwork generations can couple with one generation of the standard model lepton doublet, other three clockwork generations can only couple with other one generation of the lepton doublet and the remaining three clockwork generations can only couple with the remaining one generation of the lepton doublet. The final expression of neutrino mass is obtained as a function of the Y aβ , q aβ and n aβ : where a, β = 1, 2, 3. In this model, not only the Yukawa coupling Y aβ but also q aβ and n aβ can be origin of the neutrino mixings. Indeed, we will show that a model with the democratic Yukawa matrix |Y aβ | = 1 is consistent with the observed neutrino masses and mixings. In this case, the mixings of the neutrinos are originated from the clockwork fields instead of the Yukawa couplings. The paper is organized as follows. In Sec.II, we present brief review of the neutrinos masses and mixings, and the fermion clockwork mechanisms. In Sec.III, we propose a clockwork model which has the origin of the neutrino mixings in the pure clockwork sector. Sec.IV is devoted to a summary.

A. Observed neutrino masses and mixings
The simple clockwork model of fermions yields the Dirac neutrinos [18]. Although the models of the Majorana neutrinos with the clockwork mechanisms are discussed [20,47], we assume that the neutrinos are Dirac particles for simplicity.
The neutrino mass matrix satisfies the following relation [48] where m 1 , m 2 and m 3 denote the neutrino mass eigenstates and denotes the mixing matrix [49]. We use abbreviation c ij = cos θ ij and s ij = sin θ ij (i, j=1,2,3) and ignore the CP violating phase.
Although the neutrino mass ordering (either the normal mass ordering m 1 < m 2 < m 3 or the inverted mass ordering m 3 < m 1 < m 2 ) is unsolved problems, a global analysis shows that the preference for the normal mass ordering is mostly due to neutrino oscillation measurements [50,51]. We assume the normal mass ordering. The best-fit values of the squared mass differences ∆m 2 ij = m 2 i − m 2 j and the mixing angles for normal mass ordering are estimated as [52] where the parentheses denotes 3σ region. With the bestfit values, the neutrino mass matrix is to be where m 2 = 7.50 × 10 −5 + m 2 1 eV, B. Fermionic clockwork mechanism In the clockwork sector, there are n left-handed chiral fermions: ψ Li , (i = 0, 1, · · · , n−1) and n+1 right-handed chiral fermions: ψ Ri , (i = 0, 1, · · · , n). The clockwork Lagrangian is [18,36,47] where L kin denotes the kinetic term for clockwork fermions, denotes the nearest neighbor interaction term and denotes the Majorana mass term. For simplicity, we take the universal Dirac mass assumption: m i = m, m ′ i = mq and the universal Majorana mass assumption: M Li = M Ri = mq for all i [47]. The nearest neighbor interaction term can be written in the following simple form where and The eigenvalues of the (2n + 1) × (2n + 1) matrix M are obtained as [36] m 0 = mq, where The interaction eigenstates Ψ i and mass eigenstates, denoted by χ i , are related each other by the unitary trans- with The total Lagrangian of the standard model with clockwork sector reads where L SM is the standard model Lagrangian and L int describes the interactions between the standard model sector and the clockwork sector. We assume that the last site of the clockwork fields only couples to the left-handed neutrinos (left-handed lepton doublets) in the standard model [18] where L denotes the left-handed lepton doublet,H = iτ 2 H * (H denotes the standard model Higgs doublet) and Y denotes Yukawa matrix. In the terms of the mass eigenstates, we have where Now we generalize the above setup to three leptonic generations and N clockwork generations. The nearest neighbor interaction term for N clockwork generations is where α, β = 1, · · · N . For simplicity, we assume m iαβ = mδ αβ , m ′ iαβ = mq α δ αβ and M αβ Li = M αβ Ri = mqδ αβ = 0 [47]. The nearest neighbor interaction term can be where In the terms of the mass eigenstates χ β k (Ψ α i = j U αβ ij χ β j ), the interactions between the left-handed neutrinos and clockwork fields can be written as where a = 1, 2, 3.
for α = 1, · · · , N . The nearest neighbor interaction term can be cast as: and we have the following interaction Lagrangian with Y aβ k = Y aα U αβ nk for the Dirac neutrinos. After electroweak symmetry breaking, the neutrino mass matrix is to be where v = 246/ √ 2 GeV denotes the vacuum expectation value of the standard model Higgs field and M β k denotes the mass of the k-th clockwork fields for the Dirac pair (N β L , N β R ). Assuming M β k ≫ vY aβ 0 , the active neutrino masses are obtained as where q β and n β denote the clockwork mass ratio and the number of clockwork fermions in the β-th clockwork generation, respectively.

A. Model
We extend the clockwork model proposed by Ibarra et.al., [47] to propose a new clockwork model which has nine generations in the clockwork sector. We assume that only three clockwork generations can couple with one generation of the standard model lepton doublet, other three clockwork generations can only couple with other one generation of the lepton doublet and the remaining three clockwork generations can only couple with the remaining one generation of the lepton doublet.
Under these assumptions, the interaction Lagrangian, in terms of N R , for three leptonic generations and nine clockwork generations is where where * denotes nonzero values. If we assign the lepton number to the clockwork sector as shown in TABLE.I and assume the lepton number is conserved in the interactions of Y aβ kL aH 0 N β Rk , we obtain the configuration in Eq.(34). The interaction Lagrangian reads: Assuming M β k ≫ vY aβ 0 , after electroweak symmetry breaking, the neutrino masses are to be  , (3,8), (3,9).
We arrange these nine neutrino masses m 11 ν , · · · , m 39 ν into the neutrino mass matrix as and rename the model parameters as follows,   We assume a democratic form of the Yukawa matrix more concretely, In this case, the neutrino mass m aβ ν only depends on the clockwork mass ratio q aβ and number of clockwork fermions n aβ : B. q aβ origin with universal n First, we assume that the number of clockwork fermions is common for all clockwork generations. In this case, the origin of the neutrino mixings is the clockwork mass ratios q aβ (case (b) in Sec.III A). For example, if we take the following universal number of clockwork fermions for all a and β, the following clockwork mass ratios  Second, we assume that the clockwork mass ratio is common for all clockwork generations. In this case, the origin of the neutrino mixings is the number of the clockwork fermions n aβ (case (c) in Sec.III A). For example, if we take the following universal clockwork mass ratio for all a and β, the following numbers of clockwork fermions Although these predicted values (excepted with θ 23 ) are out of range of the 3σ region in Eq. (7), the order of the magnitude of these values are consistent with the observed data. We should perform more general parameter search with various set of the universal clockwork mass ratio and number of clockwork fermions {q, n aβ }; however, this is a numerical challenging task. For example, there are ∼ 10 21 loops in the code to perform a numerical search for q = 1.01, 1.02, · · · , 4.00, n aβ = 10, 11, · · · , 100 and m 1 = 0.001, 0.002, · · · , 0.03 eV for only the best-fit values of the neutrino parameters. In this paper, we abort such a full parameter search and only show some examples of the parameter set which are consistent with neutrino observations. D. n aβ origin with quasi-universal q If we relax the universal q requirement and allow the existence of the small perturbations of the clockwork mass ratios, ∆q (∆q ≪ q), we can obtain the collect neutrino mass parameters within the n aβ origin scenario of the neutrino mixings (case (c) with small correction of the universal clockwork mass ratio in Sec.III A). For example, if we take the following quasi-universal clockwork mass ratios, These predicted values are consistent with the observed data in Eq. (7).
E. effective n aβ origin with universal q Finally, we show an alternative way to obtain the correct neutrino mixings with the n aβ origin scenario for a universal clockwork mass ratio q.
Although the number of the clockwork fermions in the aβ-th clockwork generation n aβ should be a real integer number, we relax this requirement (small correction of case (c) in Sec.III A). In this case, for example, if we take the following universal clockwork mass ratio for all a and β, the following effective numbers of clock- yield the best-fit values of the squared mass differences and the mixing angles in Eq. (7) for m 1 = 0.01eV. Figure.2 shows the magnitude of the effective number of clockwork fermions n aβ for the best-fit values of the squared mass differences and the mixing angles under the normal mass ordering condition, where q denotes universal clockwork mass ratio (q = 2.0 in the upper panel and q = 2.5 in the lower panel).

IV. SUMMARY
We have proposed a clockwork model which has nine clockwork generations. Only three clockwork generations can couple with one generation of the standard model lepton doublet, other three clockwork generations can only couple with other one generation of the lepton doublet and the remaining three clockwork generations can only couple with the remaining one generation of the lepton doublet. Under these assumptions, the neutrino masses depend on the nine Yukawa matrix elements Y aβ , nine clockwork mass ratios q aβ and nine numbers of clockwork fermions n aβ . In this model, the candidates of the origins of the neutrino mixings are Y aβ , q aβ and n aβ . We have assumed |Y aβ | = 1, thus the Yukawa coupling is not main origin of the neutrino mixings. The main origin of the neutrino mixing is in the clockwork sector, q aβ and n aβ , in this model.
We have shown that the observed neutrino mixings are exactly obtained with a clockwork model in the case of q aβ origin scenario. In the n aβ origin scenario, although the predicted values (excepted with θ 23 ) are out of range of the 3σ region, the correct order of magnitude of the observed neutrino mixings are obtained from a clockwork model. To obtain the neutrino parameters within 3σ region in the n aβ origin scenario, it is suggested that some modification schemes should be employed, such as the quasi-universal q or the effective n aβ .
Finally, we would like comment on the collider phenomenology. In the model proposed in this paper, the neutrino masses are to be small via zero mode interactions of clockwork fermions: however, in general, unsuppressed effects at low energy phenomena, such as an unobserved lepton flavor violating decay µ → eγ, are allowed. The upper bound of the lepton flavor violating processes yield constraints on the mass scale of the clockwork fermions. Ibarra et.al. shown that the clockwork fermions must be larger than ∼ 40 TeV in order to evade the experimental constraints [47]. In this paper, we have assumed that the clockwork fermions are heavy enough to consistent with collider phenomenology.