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Fermion Mass Hierarchy in Lifshitz Type Gauge Theory 0 1 0 2 Kaneta Kawamura n Kunio ∗and Yoshiharu † a Department of Physics, Shinshu University, J Matsumoto 390-8621, Japan 8 2 January 27, 2010 ] h p - p e h Abstract [ We study the origin of fermion mass hierarchy and flavor mix- 2 v ing in a Lifshitz type extension of the standard model including an 0 extra scalar field. We show that the hierarchical structure can orig- 2 inate from renormalizable interactions. In contrast to the ordinary 9 2 Froggatt-Nielsen mechanism, the higher the dimension of associated . 9 operators,theheavierthefermionmasses. Tinymassesforleft-handed 0 neutrinos are obtained without introducing right-handed neutrinos. 9 0 : The origin of fermion mass hierarchy and flavor mixing is one of the v biggest problem in particle physics. In the standard model (SM), the hier- i X archical structure originates from the texture of Yukawa couplings. Because r a the Yukawa couplings are free parameters, their values should be determined by a theory beyond the SM. Hence the structure of Yukawa couplings can give us valuable clues for exploring an underlying theory. Recently, an exotic theory beyond the SM and/or the minimal supersym- metric SM (MSSM) has been proposed.[1] The candidate theory is a Lifshitz type extension of the SM and/or the MSSM.1 This type of theory is assumed ∗E-mail: [email protected] †E-mail: [email protected] 1ALifshitztypeextensionofgravitytheorywasproposedbyHoˇrava.[2,3,4]Properties of Lifshitz type field theory have been investigated in Ref. [5, 6, 7, 8, 9, 10]. 1 to have a fixed point with anisotropic scaling characterized by a dynamical critical exponent z(> 1) above a high-energy scale M . The system does ℓ not possess the relativistic invariance for z = 1. The Lorentz invariance is 6 expected to emerge after the transition from z = 1 to z = 1 around M .2 ℓ 6 In this letter, we study the origin of fermion mass hierarchy and flavor mixing in a Lifshitz type extension of the SM including an extra scalar field. We show that the hierarchical structure can originate from renormalizable interactions. The basic idea is as follows. The Lifshitz type theory can be renormaliz- ablebypowercounting, eventhoughitcontainshigher-dimensional operators which make the theory with z = 1 non-renormalizable. Operators of dimen- sionality 4 + r (r > 0), O(4+r), become irrelevant ones O(4+r)/Mr after the ℓ transition from z = 1 to z = 1 (and the dimensional reduction if extra di- 6 mensions exist) around M . The contributions from these operators are, in ℓ general, negligibly small if M is sufficiently large. For example, M should ℓ ℓ be larger than O(1015∼16) GeV in order to suppress proton decay processes. Suppose that a symmetry is broken down spontaneously at a high-energy scale M larger than M and O(4+r) change into M4+r−qO(q). Then these SB ℓ SB operators can be relevant ones such as (M /M )4+r−qM4−qO(q) for q 4 SB ℓ ℓ ≤ below M . Here, we assume that renormalizable terms including parameters ℓ with positive mass dimensions originate from a specific dynamics characterized by a scale M and every parameter with mass dimension is given by a ℓ power of M . In this case, the hierarchy among couplings related to O(q) can ℓ originate from the difference of exponents in (M /M )4+r−q. If we apply it SB ℓ to the Yukawa couplings, we find an interesting feature that the higher the dimension of associated original operators, the heavier the fermion masses.3 For the sake of completeness, we explain the ordinary Froggatt-Nielsen mechanism[16] for our framework. When we suppose that the Lifshitz type theoryisaneffectiveonederivedfromanunderlyingtheory,non-renormalizable terms can appear after integrating out superheavy fields. That is, operators ofdimensionalityD+z+p, O(D+z+p), canbederived withthesuppression factor Λp in the Lifshitz type theory on D+1-dimensional space-time, where Λ is a cut-off scale or a mass scale related to superheavy fields. After a symme- 2 In Ref. [11, 12, 13], properties and renormalizability for quantum field theories with Lorentz symmetry breaking terms have been studied intensively on the basis of “weighted power counting”. Furthermore, extensions of the SM have been proposed for this framework.[14, 15] 3 This feature changes into an opposite one if M is smaller than M . SB ℓ 2 try breaking at M smaller than Λ, O(D+z+p) change into (M /Λ)pO(D+z) SB SB and the hierarchy among couplings related to O(D+z) can originate from the difference of exponents in (M /Λ)p.4 In this case, it is known that the SB higher the dimension of associated original operators, the lighter the fermion masses. First, let us explain the fermion mass hierarchy and the flavor mixing. In the SM, the Yukawa interactions for quarks and charged leptons are given by (u) (d) (e)¯ = f q¯ h u +f q¯ hd +f l he +h.c. , (1) LY ij Li u Rj ij Li Rj ij Li Rj (X) where f (X = u,d,e) are the Yukawa couplings, i,j are family indices ij (i,j = 1,2,3), q¯ are Hermitian conjugates of left-handed quark doublets, Li u are right-handed up type quark singlets, d are right-handed down type Ri Ri ¯ quark singlets, l are Hermitian conjugates of left-handed lepton doublets, Li e are right-handed electron type lepton singlets, h (or h iτ h∗) is a weak Ri u 2 ≡ Higgs doublet and h.c. represents Hermitian conjugates of former terms. Quark masses and charged lepton masses are the eigenvalues of fermion mass matrices M given by X v v v (u) (d) (e) (M ) = f , (M ) = f , (M ) = f , (2) u ij ij √2 d ij ij √2 e ij ij √2 where v(= 246GeV) is the vacuum expectation value (VEV) of neutral component (h0) of h.5 Using unitary matrices S and T , M are diagonalized X X X as S†M T = diag(m ,m ,m ) , S†M T = diag(m ,m ,m ) , u u u u c t d d d d s b S†M T = diag(m ,m ,m ) . (3) e e e e µ τ The quark flavor mixing is given by the Kobayashi-Maskawa matrix V = KM S†S .[17]Experimental data[18]showtheexistence offermionmasshierarchy u d and flavor mixing. Because the flavor structure originates from the texture of Yukawa couplings and the Yukawa couplings are free parameters in the 4 Strictly speaking, an extra factor such as (M /Λ)γp appears after the dimensional ℓ reduction and the redefinition of fields where γ is a constant related to z and the dimen- sionality of extra space. See (19). 5 IntheMSSM,v/√2isreplacedbythecorrespondingone,i.e.,eitherVEVforneutral components (h0,h0) of two Higgs doublets. u d 3 SM, we need a theory beyond the SM to strip the structure of its aura of mystery. Suppose that an underlying theory holds above O(1015∼16) GeV. Considering renormalization effects in the SM or the MSSM, the magnitude of each fermion mass and each entry in V at O(1016) GeV can be roughly KM represented as v v (m ,m ,m ) (λ7,λ4,1) , (m ,m ,m ) (λ6,λ4,λ2) , u c t d s b ∼ √2 ∼ √2 v (m ,m ,m ) (λ7,λ4,λ2) (4) e µ τ ∼ √2 and 1 λ λ3 (V ) λ 1 λ2 , (5) KM ij   ∼ λ3 λ2 1     where we use the Cabibbo angle λ sinθ 0.23.[19] For V , recall the C KM ≡ ∼ Wolfenstein parameterization.[20] Our present goal is to derive the structure (4) and (5) using a specific theory beyond the SM. An exotic candidate beyond the SM is a Lifshitz type extension of the SM or the MSSM. Next let us explain a framework of Lifshitz type field theory [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] briefly. Space-time is assumed to be factorized into a product of 3-dimensional Euclidean space R3, extra n-dimensional compact space and time R, whose coordinates are denoted by xi (i = 1,2,3), yk (k = 1, ,n) and t. The notation xI (I = 1, ,n+3) is also used for the ··· ··· (n + 3)-dimensional space coordinates. The dimensions of xi, yk and t are defined as [xi] = [yk] = 1 , [t] = z , (6) − − where z is the dynamical critical exponent, which characterizes anisotropic scaling xi bxi, yk byk and t bzt at the fixed point. The system → → → does not possess the relativistic invariance for z = 1 but it possesses spatial 6 rotational invariance and translational invariance. The kinetic terms for a complex scalar field Φ and a spinor field Ψ are given by 2 ∂Φ ∂ dtd3xdny +Ψ¯iΓ0 Ψ+ , (7) (cid:12) ∂t (cid:12) ∂t ··· Z (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)  (cid:12) (cid:12) 4 where Γ0 corresponds to the time component of the gamma matrices and the ellipsis stands for terms including spatial derivatives. Ψ is a spinor defined on Rn+3 and it transforms as Ψ(x) Ψ′ Ψ′(x′) = S(O)Ψ(x) , (8) → → i S(O) e−4iωIJΣIJ , ΣIJ ΓIΓJ ΓJΓI , (9) ≡ ≡ 2 − (cid:16) (cid:17) under the spatial rotation xI x′I = OIxJ. Here, the ΓI are gamma → J matrices, the ω are parameters related to the rotation angles θI with IJ ω = ε θK and OI is the orthogonal matrix given by OI = (eω)I. S(O) IJ − IJK J J J satisfies the following relations: S†(O)ΓIS(O) = OIΓJ , S†(O)S(O) = , (10) J I where is the unit matrix. Chiral fermions on R3 are assumed to appear af- I terthedimensional reduction, e.g., throughtheorbifoldbreakingmechanism. The engineering dimensions of Φ and Ψ are given by 3+n z 3+n [Φ] = − , [Ψ] = , (11) 2 2 respectively. Then the dimension of operator Ψ†ΦNΨ is given by 3+n z [Ψ¯ΦNΨ] = − N +3+n . (12) 2 The operator Ψ¯ΦNΨ becomes relevant or a renormalizable term if its dimension is less than or equals to 3 + n + z. The operator including higher spatial derivatives such as Φ† 2zΦ can also become relevant. The theory ∇ can be renormalizable by power counting, even though it contains higher- dimensional operatorswhich make thetheorywithz = 1 non-renormalizable. The Lorentz invariance is expected to emerge as an accidental symmetry after the transition from a high-energy theory with z = 1 to that with z = 1 6 around M .6 The magnitude of Lorentz symmetry breaking terms is esti- ℓ mated and it gives constraints on parameters.[23, 24, 25, 26],[1] Now let us explore the origin of texture of Yukawa couplings in a Lifshitz type extension of the SM including an extra scalar field Φ.7 We introduce an 6 TherehasbeenaproposalthattheLorentzinvarianceappearsatanattractiveinfra- red fixed point.[21, 22] 7 The study based on the Lifshitz type extension of the MSSM can be carried out by the introduction of two Higgs doublets and similar results can be obtained. 5 extra U(1) symmetry denoted by U(1) and assume that Φ is a singlet under A the SM gauge group SU(3) SU(2) U(1) but has a non-zero U(1) C L Y A × × charge and U(1) is spontaneously broken down by the non-vanishing VEV A of Φ above Mℓ. The interactions such as Φm(ijX)Ψ¯iH(u)Ψj are determined by (X) U(1) . Here, m are zero or positive integers, and Ψ and H are fermions A ij i (u) and a boson which contain SM fermions ψ and a weak Higgs boson h as i (u) zero modes, respectively. The origin of extra U(1) symmetry is not specified in our analysis for simplicity.8 The action is given by 1 dtd3xdny D Φ 2 Φ†(D†D )zΦ C D Φ 2 | t | − κ2 I I − Φ| I | Z (cid:20) 1 + D H 2 H†(D†D )zH C D H 2 +Ψ¯ iΓ0D Ψ | t | − κ2 I I − H | I | i t i h −ξ1i2Ψ¯i(iΓIDI)zΨi −CΨiΨ¯iiΓIDIΨi +γi(jX)Φm(ijX)Ψ¯iH(u)Ψj +···# , (13) where κ2, κ2 and ξ2 are dimensionless parameters concerning Lorentz sym- h i metry violating terms, D and D are covariant derivatives and the ellipsis t I stands for other terms. The engineering dimensions of fields (Φ,H ,Ψ ) and (u) i (X) parameters (C ,C ,C ,γ ) are given by Φ H Ψi ij 3+n z 3+n [Φ] = [H ] = − , [Ψ ] = (14) (u) i 2 2 and [C ] = [C ] = 2(z 1) , [C ] = z 1 , Φ H − Ψi − (X) (3+n z)(m +1) [γ(X)] = z − ij , (15) ij − 2 respectively. We assume that renormalizable terms including parameters with positive mass dimensions originate from a specific dynamics characterized by a scale M and the parameters are given by a power of M , which ℓ ℓ is similar to the soft supersymmetry (SUSY) breaking parameters in SUSY models.9 The magnitude of parameters is not necessarily O(1) in the unit 8 If U(1) is an anomalousgaugesymmetry, we need to introduce extra fields in order A for the theory to be harmless. 9 For example, the soft supersymmetry breaking terms are given by a power of the gravitino mass m3/2 in the gravity mediation. 6 of M but can be much smaller like most A terms in SUSY models. On the ℓ other hand, we assume that parameters in non-renormalizable terms are sup- pressed by a power of cutoff scale Λ as usual. To become relativistic below M , finetuning among parameters is required such as C = C = C2 for all ℓ Φ H Ψi species. In this setup, parameters are expressed as C = C = M2(z−1) , C = Mz−1 , Φ H ℓ Ψi ℓ γ0(X)Mz−(3+n−z)(2m(ijX)+1) z (3+n−z)(m(ijX)+1) , γi(jX) =  γi0j(X)Λzℓ−(3+n−z)(2m(ijX)+1) z ≥< (3+n−z)(2m(ijX)+1)! , (16) ij 2 ! after a suitable rescaling of fields. Here, γ0(X) is a dimensionless parameter. ij We assume that the volume of extra n-dimensional space is 1/Mn. After the ℓ redefinition of time variable and fields as z−n−1 x Mz−1t , Φ˜ M 2 Φ = φ+ , 0 ≡ ℓ ≡ ℓ ··· ˜ z−n−1 ˜ −n H M 2 H = h + , Ψ M 2Ψ = ψ + (17) (u) ≡ ℓ (u) (u) ··· i ≡ ℓ i i ··· and the dimensional reduction of extra dimensions, the following action for zero modes is derived from (13), m(X) φ ij d4x D h 2 +ψ¯iγµD ψ +γ0(X) h i ψ¯h ψ + , (18) Z | µ | i µ i ij Mℓ! i (u) j ···     wheretheellipsis standsfortheYukawa interactionsfromnon-renormalizable terms and so on. The dimensions of φ, h and ψ are [φ] = [h ] = 1 and (u) i (u) [ψ ] = 3/2. The Yukawa couplings are given by10 i m(X) γ0(X) hφi ij z (3+n−z)(m(ijX)+1) , fi(jX) =  γi0j(X) MMℓℓ!3+n2−3z Mℓ 1+n2−z hφi m(ijX) z ≥< (3+n−z)(2m(ijX)+1)! . (19) 10 Thecontijribut(cid:18)ionΛs f(cid:19)rom volum(cid:18)e sΛup(cid:19)pression fΛactor can al so appear after2the dim!en- sional reduction.[27] In this case, the difference of field configurations related to interactions can be important to study the origin of mass hierarchy. Here, we do not consider them for simplicity. 7 (X) The exponents m are determined from the U(1) charge conservation: ij A (u) m Q (φ)+Q (q¯ )+Q (u )+Q (h ) = 0 , (20) ij A A Li A Rj A u (d) m Q (φ)+Q (q¯ )+Q (d )+Q (h) = 0 , (21) ij A A Li A Rj A (e) ¯ m Q (φ)+Q (l )+Q (e )+Q (h) = 0 , (22) ij A A Li A Rj A where Q represents the charge of U(1) . The first one in (19) comes from A A renormalizable terms and the ratio M / φ can play a role of λ, in the case ℓ h i 0(X) that there is no hierarchy among each entry in γ and φ is larger than ij h i M . In other words, there is a build-in mechanism to generate the hierarchy ℓ using the ratio M / φ on the basis of relevant operators. On the other ℓ h i hand, the second one in (19) comes from non-renormalizable terms which might originate from some renormalizable interactions after integrating out superheavy fields. The mechanism to generate the hierarchy using the ratio 1+n−z (Mℓ/Λ) 2 φ /Λ is regarded as the Lifshitz type extended version of the h i Froggatt-Nielsen mechanism.11 There is a possibility that the hierarchy of Yukawa couplings stems from the mixture of first and second ones. Next we consider the case with z = 4 and n = 2 as an example. The Yukawa couplings are given by m(X) φ ij 0(X) (X) γ h i (0 m 7) , fi(jX) =  i0j(X) MΛℓ!72 φ m(ijX) ≤(X) ij ≤ (23) γ h i (m > 7) . If M / φ λ,theifjerm(cid:18)ioMnℓm(cid:19)as s √hieMraℓΛrc!hy (4) canibje derived from the U(1) ℓ A h i ∼ charge assignment:12 Q (q¯ ) = (0,1,3) , Q (u ) = (0,2,4) , Q (d ) = (1,2,2) , A Li A Ri A Ri ¯ Q (l ) = (0,1,1) , Q (e ) = (0,2,4) , Q (h ) = 0 , A Li A Ri A (u) Q (φ) = 1 , (24) A − 11 We need some selection rule related to interactions in order for the Froggatt-Nielsen mechanism to work. In most cases, one uses an anomalous U(1) gauge symmetry,[28, 29] whose anomalies are canceled via the Green-Schwarz mechanism,[30] motivated by superstring theories. A discrete horizontal symmetry is also used.[31] 12 The U(1) charge assignment is not unique. A 8 wherewe assume γ0(X) = O((M / φ )7). If φ /√M Λ λ, thefermionmass ij ℓ h i h i ℓ ∼ hierarchy (4) can be derived from the U(1) charge assignment: A Q (q¯ ) = (11,10,8) , Q (u ) = (4,2,0) , Q (d ) = (3,2,2) , A Li A Ri A Ri ¯ Q (l ) = (9,8,8) , Q (e ) = (6,4,2) , Q (h ) = 0 , A Li A Ri A (u) Q (φ) = 1 , (25) A − where we assume γ0(X) = O(Λ1/2M15/2/ φ 8). In either case, the quark flavor ij ℓ h i mixing due to the Kobayashi-Maskawa matrix (5) can be obtained by 1 λ λ3 (V ) λ|QA(q¯Li)−QA(q¯Lj)| λ 1 λ2 . (26) KM ij   ∼ ∼ λ3 λ2 1     Finally, we discuss the neutrino sector. The Majorana mass matrix M ν of left-handed neutrinos is usually obtained throught the see-saw mechanism such that[32, 33, 34] v v (M ) = f(ν) M−1 f(ν) , (27) ν ij ia √2 R ab bj √2 (cid:16) (cid:17) (ν) where f is the Yukawa coupling among l , right-handed neutrinos ν ia Li Ra and h and (M ) is the superheavy Majorana mass matrix of right-handed u R ab neutrinos. In our Lifshitz type extension of the SM including Φ, M can ν be obtained without introducing right-handed neutrinos from the following relevant interactions: γ(ν)Φmi(jν)L¯ iτ τLc Htiτ τH , (28) ij i 2 j · u 2 u where L are fermions whose zero modes are l , and superscripts c and t j Li stand for the charge conjugation and transpose of the relevant field, respec- (ν) tively. The exponent m is determined by ij (ν) ¯ ¯ m Q (φ)+Q (l )+Q (l )+2Q (h ) = 0 . (29) ij A A Li A Lj A u In our model, M is given by ν γ0(ν) hφi m(ijν)v2 z (3+n−z)(m(ijν)+2) , (Mν)ij = γi0j(ν) MMℓℓ!2+n2−M2zℓ Mℓ 1+n2−z hφi m(ijνv)2 z <≥ (3+n−z)2(m(ijν)+2)! , (30)  ij (cid:18) Λ (cid:19) (cid:18) Λ (cid:19) Λ  2Λ 2 ! 9 0(ν) where γ is a dimensionless parameter. Using a unitary matrix S , M is ij ν ν diagonarized as S†M S = diag(m ,m ,m ) , (31) ν ν ν ν1 ν2 ν3 where m < m < m . The lepton flavor mixing is given by the Maki- ν1 ν2 ν3 Nakagawa-Sakata matrix V = S†S .[35] Using the experimental data for MNS e ν neutrinos,[18] we find that there are two large mixings: sin22θ 0.88 , sin22θ > 0.92 (32) 12 23 ∼ and the hierarchy between mass-squared differences for solar neutrinos ∆m2 ⊙ and for the atmospheric neutrinos ∆m2: ⊕ ∆m2 m2 m2 ⊙ = | ν2 − ν1| λ2 . (33) ∆m2 m2 m2 ∼ ⊕ | ν3 − ν2| If three neutrino masses do not degenerate, the hierarchy (33) suggests the relation: (m ,m ) (λ,1) 0.05eV . (34) ν2 ν3 ∼ × Using our mechanism with the U(1) charge assignment (24) or the ordinary Froggatt-Nielsen mechanism with the U(1) charge assignment (25), M and ν V are estimated as MNS λ2 λ λ (M ) λ∓(QA(¯lLi)+QA(¯lLj)) λ 1 1 , (35) ν ij   ∝ ∝ λ 1 1     1 λ λ V λ|QA(¯lLi)−QA(¯lLj)|S λ 1 1 S , (36) MNS ν   ν ∼ ∼ λ 1 1     wheretheminusandtheplussignin(35)forourmechanismandtheFroggatt- Nielsen mechanism, respectively. This type of neutrino mass matrix has been proposed and studied in Refs. [36, 37, 38]. The above matrices (35) and (36) haveaninteresting propertythatthebi-largemixing canbenaturallyderived if we obtain the mass relation (34) after the diagonalization of M . ν Finally, we discuss the lepton number and/or baryon number violating process through four fermi interactions. The four fermi interactions originate 10