Multiple seesaw mechanisms of neutrino masses at the TeV scale
aa r X i v : . [ h e p - ph ] J u l Multiple seesaw mechanisms of neutrino masses at the TeV scale
Zhi-zhong Xing ∗ and Shun Zhou † Institute of High Energy Physics and Theoretical Physics Center for Science Facilities,Chinese Academy of Sciences, P.O. Box 918, Beijing 100049, China
Abstract
In pursuit of a balance between theoretical naturalness and experimental testa-bility, we propose two classes of multiple seesaw mechanisms at the TeV scale tounderstand the origin of tiny neutrino masses. They are novel extensions of thecanonical and double seesaw mechanisms, respectively, by introducing even andodd numbers of gauge-singlet fermions and scalars. It is thanks to a proper im-plementation of the global U(1) × Z N symmetry that the overall neutrino massmatrix in either class has a suggestive nearest-neighbor-interaction pattern. Webriefly discuss possible consequences of these TeV-scale seesaw scenarios, which canhopefully be explored in the upcoming Large Hadron Collider and precision neutrinoexperiments, and present a simple but instructive example of model building. PACS numbers: 14.60.Pq, 12.15.Ff, 13.66.-a, 98.80.Cq ∗ E-mail: [email protected] † E-mail: [email protected] seesaw ideas [2, 3, 4] are most brilliant and might even leadus to a true theory of neutrino masses.The canonical (type-I) seesaw mechanism [2] can naturally work at a superhigh energyscale Λ SS ∼ GeV to generate tiny neutrino masses of order Λ / Λ SS ∼ . EW ∼ GeV being the electroweak scale. To be more specific, the effective Majoranamass matrix of three light neutrinos is given by M ν = − M D M − M T D in the leading-orderapproximation, where M D ∼ O (Λ EW ) originates from the Yukawa interactions betweenthe SM lepton doublet ℓ L and the right-handed neutrinos N i R (for i = 1 , , M R ∼O (Λ SS ) is a symmetric matrix coming from the lepton-number-violating Majorana massterm of N i R . This seesaw picture is technically natural because it allows the relevantYukawa couplings to be O (1) and requires little fine-tuning of the textures of M D and M R , but it loses the direct testability on the experimental side and causes a hierarchyproblem on the theoretical side (as long as Λ SS > GeV [5]). A possible way outof the impasse is to lower the seesaw scale down to Λ SS ∼ M D /M R so as to make it possibleto produce and detect heavy Majorana neutrinos at the LHC via their charged-currentinteractions. This prerequisite unavoidably requires a terrible fine-tuning of M D and M R ,because one has to impose M R ∼ M D /M R ∼ − · · · − and M ν ∼ . unnaturalness problem built in the TeV seesaw mechanism.We stress that a multiple seesaw mechanism at the TeV scale may satisfy both natu-ralness and testability requirements. To illustrate, we assume that the small mass scale ofthree light neutrinos arises from a naive seesaw relation m ∼ ( λ Λ EW ) n +1 / Λ n SS , where λ isa dimensionless Yukawa coupling coefficient and n is an arbitrary integer larger than one.Without any terrible fine-tuning, the seesaw scale can be estimated fromΛ SS ∼ λ n +1 n (cid:20) Λ EW
100 GeV (cid:21) n +1 n (cid:20) . m (cid:21) n n +6) n GeV . (1)A numerical change of Λ SS with n and λ is shown in Fig. 1, where Λ SS ∼ n ≥ λ ≥ − . Hence the multiple seesaw idea is expected towork at the TeV scale and provide us with a novel approach to bridge the gap betweentheoretical naturalness and experimental testability of the canonical seesaw mechanism.The simplest way to build a multiple seesaw model at the TeV scale is to extend thecanonical seesaw mechanism by introducing a number of gauge-singlet fermions S in R and2calars Φ n (for i = 1 , , n = 1 , , · · · ). We find that a proper implementation of theglobal U(1) × Z N symmetry leads us to two classes of multiple seesaw mechanisms withthe nearest-neighbor-interaction pattern — an interesting form of the overall 3 ( n + 2) × n + 2) neutrino mass matrix in which every 3 × S in R and Φ n and correspondsto an appealing extension of the canonical seesaw mechanism, while the second class hasan odd number of S in R and Φ n and is actually a straightforward extension of the doubleseesaw mechanism [7]. Their possible collider signatures and low-energy consequences,together with a simple example of model building, will be briefly discussed.The spirit of multiple seesaw mechanisms is to make a harmless extension of the SMby adding three right-handed neutrinos N i R together with some gauge-singlet fermions S in R and scalars Φ n (for i = 1 , , n = 1 , , · · · ). Allowing for lepton number violationto a certain extent, we can write the gauge-invariant Lagrangian for neutrino masses as − L ν = ℓ L Y ν ˜ HN R + N c R Y S S Φ + n X i =2 S c ( i − Y S i S i R Φ i + 12 S cn R M µ S n R + h . c . , (2)where ℓ L and ˜ H ≡ iσ H ∗ stand respectively for the SU(2) L lepton and Higgs doublets, Y ν and Y S i (for i = 1 , , · · · , n ) are the 3 × M µ isa symmetric Majorana mass matrix. After spontaneous gauge symmetry breaking, wearrive at the overall 3 ( n + 2) × n + 2) neutrino mass matrix M in the flavor basisdefined by ( ν L , N c R , S c , · · · , S cn R ) and their charge-conjugate states: M = M D · · · M T D M S · · · M TS M S · · ·
00 0 M TS . . . . . . ... . . . . . . M S n − ... ... ... . . . M TS n − M S n · · · M TS n M µ , (3)where M D ≡ Y ν h H i and M S i = Y S i h Φ i i (for i = 1 , , · · · , n ) are 3 × N R = S for simplicity, one can observe that the Yukawa interactions between S i R and S j R exist if and only if their subscripts satisfy the selection rule | i − j | = 1 (for i, j = 0 , , , · · · , n ). Note that M manifests a very suggestive nearest-neighbor-interactionpattern, which has attracted a lot of attention in the quark sector to understand theobserved hierarchies of quark masses and flavor mixing angles [8]. Such a special structureof M , or equivalently that of L ν in Eq. (2), may arise from a proper implementationof the global U(1) × Z N symmetry. The unique generator of the cyclic group Z N is ̟ = e iπ/N , which produces all the group elements Z N = { , ̟, ̟ , ̟ , · · · , ̟ N − } .By definition, a field Ψ with the charge q transforms as Ψ → e iπq/N Ψ under Z N (for q = 0 , , , · · · , N − Z N charges of therelevant fields in Eq. (2) in the following way:3. The global U(1) symmetry can be identified with the lepton number L , namely L ( ℓ L ) = L ( E R ) = +1, where E R represents the charged-lepton singlets in the SM.We arrange the lepton numbers of gauge-singlet fermions and scalars to be L ( N R ) =+1, L ( S k R ) = ( − k and L (Φ k ) = 0 (for k = 1 , , · · · , n ). It turns out that only theMajorana mass term S cn R M µ S n R in L ν explicitly violates the U(1) symmetry. Afterthis assignment, other lepton-number-violating mass terms (e.g., N c R M R N R in thecanonical seesaw mechanism) may also appear in the Lagrangian, but they can beeliminated by invoking the discrete Z N symmetry.2. We assign the Z N charge of S n R as q ( S n R ) = N . Then it is easy to verify thatthe Majorana mass term S cn R M µ S n R is invariant under the Z N transformation. Ifall the other gauge-singlet fermions S k R (for k = 1 , , · · · , n −
1) take any chargesin { , , · · · , N − } other than N , their corresponding Majorana mass terms areaccordingly forbidden. Given q ( ℓ L ) = q ( E R ) = q ( N R ) = 1, both the charges of S k R (for k = 1 , , · · · , n −
1) and those of Φ i (for i = 1 , , · · · , n ) can be properly chosenso as to achieve the nearest-neighbor-interaction form of L ν as shown in Eq. (2).But the solution to this kind of charge assignment may not be unique, because fora given value of n one can always take N ≫ n to fulfill all the above-mentionedrequirements [9]. Simple examples (with n = 1 , ,
3) will be presented below.We remark that our multiple seesaw picture should be the simplest extension of thecanonical seesaw mechanism, since it does not invoke the help of either additional SU(2) L fermion doublets [10] or a new isospin 3/2 Higgs multiplet [11]. We also stress that thedouble seesaw scenario [7] is only the simplest example in one class of our multiple seesawmechanisms (with an odd number of S in R or Φ n ) and cannot reflect any salient featuresof the other class of multiple seesaw mechanisms (with an even number of S in R or Φ n ).Now let us diagonalize M in Eq. (3) to achieve the effective mass matrix of three lightneutrinos M ν in the multiple seesaw mechanisms. Note that M can be rewritten as M = (cid:18) ˜ M D ˜ M T D ˜ M µ (cid:19) , (4)where ˜ M D = ( M D ) denotes a 3 × n + 1) matrix and ˜ M µ is a symmetric 3 ( n + 1) × n + 1) matrix. Taking the mass scale of ˜ M µ to be much higher than that of ˜ M D , one caneasily obtain M ν = − ˜ M D ˜ M − µ ˜ M T D for three light Majorana neutrinos in the leading-orderapproximation. Because the elements in the fourth to 3 n -th columns of ˜ M D are exactlyzero, only the 3 × M − µ is relevant to the calculation of M ν . Withoutloss of generality, the inverse of ˜ M µ can be figured out by assuming all the non-zero 3 × M to be of rank three. We find two types of solutions [9], depending onwhether n is even or odd, and thus arrive at two classes of multiple seesaw mechanisms: Class A of multiple seesaw mechanisms — they contain an even number of gauge-singlet fermions S in R and scalars Φ n (i.e., n = 2 k with k = 0 , , , · · · ) and correspond to anovel extension of the canonical seesaw picture. The effective mass matrix of three light4able 1: The charges of relevant fermion and scalar fields under the U(1) × Z symmetryin the multiple seesaw mechanism with n = 2. ℓ L H E R N R S S Φ Φ L +1 0 +1 +1 − q +1 0 +1 +1 +2 +3 +3 +1Majorana neutrinos is given by M ν = − M D " k Y i =1 (cid:16) M TS i − (cid:17) − M S i M − µ " k Y i =1 (cid:16) M TS i − (cid:17) − M S i T M T D (5)in the leading-order approximation. The k = 0 case is obviously equivalent to the canon-ical seesaw mechanism (i.e., M ν = − M D M − M T D by setting S = N R and M µ = M R ).If M S i ∼ M D ∼ O (Λ EW ) and M S i − ∼ M µ ∼ O (Λ SS ) hold (for i = 1 , , · · · , k ), Eq. (5)leads to M ν ∼ Λ k +1)EW / Λ k +1SS , which can effectively lower the conventional seesaw scaleΛ SS ∼ GeV down to the TeV scale as illustrated in Fig. 1.Taking n = 2 (i.e., k = 1) for example [12], we arrive at the minimal extension of thecanonical seesaw mechanism: M ν = − M D (cid:16) M TS (cid:17) − M S M − µ M TS (cid:16) M S (cid:17) − M T D . (6)This effective multiple seesaw mass term is illustrated in Fig. 2(a). The nearest-neighbor-interaction pattern of M with n = 2 can be obtained by imposing the global U(1) × Z symmetry on L ν , in which the proper charge assignment is listed in Table 1. Class B of multiple seesaw mechanisms — they contain an odd number of gauge-singletfermions S in R and scalars Φ n (i.e., n = 2 k + 1 with k = 0 , , , · · · ) and correspond to aninteresting extension of the double seesaw picture. The effective mass matrix of threelight Majorana neutrinos reads M ν = M D " k Y i =1 (cid:16) M TS i − (cid:17) − M S i M TS k +1 (cid:17) − M µ (cid:16) M S k +1 (cid:17) − " k Y i =1 (cid:16) M TS i − (cid:17) − M S i T M T D (7) in the leading-order approximation. The k = 0 case just corresponds to the double seesawscenario with a very low mass scale of M µ [7]: M ν = M D (cid:16) M TS (cid:17) − M µ (cid:16) M S (cid:17) − M T D . Notethat the nearest-neighbor-interaction pattern of M in the double seesaw mechanism isguaranteed by an implementation of the global U(1) × Z symmetry with the followingcharge assignment: L ( ℓ L ) = L ( E R ) = L ( N R ) = +1, L ( S ) = − L ( H ) = L (Φ ) = 0, q ( ℓ L ) = q ( E R ) = q ( N R ) = q (Φ ) = +1, q ( H ) = 0 and q ( S ) = +2.If M S i ∼ M D ∼ O (Λ EW ) and M S i − ∼ O (Λ SS ) hold (for i = 1 , , · · · , k ), the massscale of M µ is in general unnecessary to be as small as that given by the double seesawmechanism. To be more specific, let us consider the minimal extension of the double5able 2: The charges of relevant fermion and scalar fields under the U(1) × Z symmetryin the multiple seesaw mechanism with n = 3. ℓ L H E R N R S S S Φ Φ Φ L +1 0 +1 +1 − − q +1 0 +1 +1 +2 +3 +5 +7 +5 +2seesaw picture by taking n = 3. In this case, we impose the U(1) × Z symmetry on L ν with a proper charge assignment listed in Table 2 to assure the nearest-neighbor-interaction form of M . The corresponding formula of M ν is M ν = M D (cid:16) M TS (cid:17) − M S (cid:16) M TS (cid:17) − M µ (cid:16) M S (cid:17) − M TS (cid:16) M S (cid:17) − M T D . (8)This effective multiple seesaw mass term is illustrated in Fig. 2(b). It becomes obviousthat the proportionality of M ν to M µ in Eq. (8) is doubly suppressed not only by the ratio M D /M S ∼ Λ EW / Λ SS but also by the ratio M S /M S ∼ Λ EW / Λ SS , and thus M ν ∼ . Y ν ∼ Y S ∼ Y S ∼ Y S ∼ O (1) and M µ ∼ SS ∼ Charged-current interactions of neutrinos — they are important for both productionand detection of light and heavy Majorana neutrinos in a realistic experiment. To definethe neutrino mass eigenstates, we diagonalize the overall mass matrix M in Eq. (4) bymeans of the following unitary transformation: (cid:18) V ˜ R ˜ S ˜ U (cid:19) † (cid:18) ˜ M D ˜ M T D ˜ M µ (cid:19) (cid:18) V ˜ R ˜ S ˜ U (cid:19) ∗ = (cid:18) c M ν c M N + S (cid:19) , (9)where c M ν ≡ Diag { m , m , m } contains the masses of three light Majorana neutrinos(ˆ ν , ˆ ν , ˆ ν ), and c M N + S denotes a diagonal matrix whose eigenvalues are the masses of3 ( n + 1) heavy Majorana neutrinos ( ˆ N , ˆ S , · · · , ˆ S n ; and each of them consists of threecomponents). The SM charged-current interactions of ν e , ν µ and ν τ can therefore beexpressed, in terms of the mass eigenstates of light and heavy Majorana neutrinos, as − L cc = g √ e µ τ ) L γ µ V ˆ ν ˆ ν ˆ ν L + ˜ R ˆ N ˆ S ...ˆ S n L W − µ + h . c . (10)in the basis where the mass eigenstates of three charged leptons are identified with theirflavor eigenstates. Note that V is the 3 × × n + 1) matrix ˜ R governs the strength of charged-currentinteractions of heavy Majorana neutrinos. Note also that both V V † + ˜ R ˜ R † = and V c M ν V T + ˜ R c M N + S ˜ R T = hold, and thus V must be non-unitary. It is ˜ R that measuresthe deviation of V from unitarity in neutrino oscillations and determines the collidersignatures of heavy Majorana neutrinos at the LHC.6e expect that our multiple seesaw idea can lead to rich phenomenology at both theTeV scale and lower energies. For simplicity, here we only mention a few aspects of thephenomenological consequences of multiple seesaw mechanisms. • Non-unitary neutrino mixing and CP violation . Since V is non-unitary, it generallyinvolves a number of new flavor mixing parameters and new CP-violating phases [13].Novel CP-violating effects in the medium-baseline ν µ → ν τ and ν µ → ν τ oscillationsmay therefore show up and provide a promising signature of the unitarity violationof V , which could be measured at a neutrino factory [14]. • Signatures of heavy Majorana neutrinos at the LHC . Given ˜ M D ∼ O (Λ EW ) and˜ M µ ∼ O (Λ SS ) ∼ O (1) TeV, it is straightforward to obtain ˜ R ≈ ˜ M D ˜ M − µ ˜ U ∼ O (0 . | ˜ R | [15]. For Class A of multiple seesaw mechanisms, their clear LHC signatures are expected tobe the like-sign dilepton events arising from the lepton-number-violating processes pp → l ± α l ± β X (for α, β = e, µ, τ ) mediated by heavy Majorana neutrinos [16]. For Class B of multiple seesaw mechanisms with M µ ≪ Λ EW , the mass spectrum ofheavy Majorana neutrinos generally exhibits a pairing phenomenon in which thenearest-neighbor Majorana neutrinos have nearly degenerate masses and can becombined to form pseudo-Dirac particles. This feature has already been observedin the double seesaw model [7]. Therefore, the discriminating collider signatures atthe LHC are expected to be the pp → l ± α l ± β l ∓ γ X processes (for α, β, γ = e, µ, τ ) [17]. • Possible candidates for dark matter . One or more of the heavy Majorana neutrinosand gauge-singlet scalars in our multiple seesaw mechanisms could be arranged tohave a sufficiently long lifetime. Such weakly-interacting and massive particles mighttherefore be a plausible candidate for cold dark matter [18].One may explore more low-energy effects of multiple seesaw mechanisms, such as theircontributions to the lepton-flavor-violating processes µ → eγ and so on. It should alsobe interesting to explore possible baryogenesis via leptogenesis [19], based on a multipleseesaw picture, to interpret the cosmological matter-antimatter asymmetry.As a flexible and testable TeV seesaw scheme, the multiple seesaw mechanisms canalso provide us with plenty of room for model building. But the latter requires furtherinputs or assumptions. Here we present a simple but instructive example, in which allthe textures of 3 × M are symmetricand have the well-known Fritzsch pattern [8], M a = x a x a y a y a z a (11)with a = D , S , · · · , S n or µ , for illustration. Choosing the Fritzsch texture makes sensebecause it coincides with the nearest-neighbor-interaction form of M itself. We make an7dditional assumption that the ratio x a /y a is a constant independent of the subscript a .Then it is easy to show that the effective mass matrix of three light Majorana neutrinos M ν has the same Fritzsch texture in the leading-order approximation: M ν = − x x µ " k Y i =1 x S i x S i − x x µ " k Y i =1 x S i x S i − y y µ " k Y i =1 y S i y S i − y y µ " k Y i =1 y S i y S i − z z µ " k Y i =1 z S i z S i − (12)derived from Eq. (5) for Class A of multiple seesaw mechanisms (with n = 2 k for k =0 , , , · · · ); and M ν = x x S k +1 " k Y i =1 x S i x S i − x µ x x S k +1 " k Y i =1 x S i x S i − x µ y y S k +1 " k Y i =1 y S i y S i − y µ y y S k +1 " k Y i =1 y S i y S i − y µ z z S k +1 " k Y i =1 z S i z S i − z µ (13)obtained from Eq. (7) for Class B of multiple seesaw mechanisms (with n = 2 k + 1 for k = 0 , , , · · · ). This seesaw-invariant property of M ν is interesting since it exactly reflectshow two classes of multiple seesaw mechanisms work for every element of M ν . Note thatit is possible to interpret current experimental data on small neutrino masses and largeflavor mixing angles by taking both the texture of the light neutrino mass matrix M ν andthat of the charged-lepton mass matrix M l to be of the Fritzsch form [20]. Hence the aboveexamples are phenomenologically viable. Once the texture of M ν is fully reconstructedfrom more accurate neutrino oscillation data, one may then consider to quantitativelyexplore the textures of those 3 × M in such a multiple seesaw model.To conclude, new ideas are eagerly wanted in the upcoming LHC era to achieve aproper balance between theoretical naturalness and experimental testability of the elegantseesaw pictures, which ascribe the small masses of three known neutrinos to the existenceof some heavy degrees of freedom. In the present work we have extended the canonicaland double seesaw scenarios and proposed two classes of multiple seesaw mechanisms atthe TeV scale by introducing an arbitrary number of gauge-singlet fermions and scalarsinto the SM and by implementing the global U(1) × Z N symmetry in the neutrino sector.These new TeV-scale seesaw mechanisms are expected to lead to rich phenomenology atlow energies and open some new prospects for understanding the origin of tiny neutrinomasses and lepton number violation.This research was supported in part by the National Natural Science Foundation ofChina under grant No. 10425522 and No. 10875131.8 eferences [1] Particle Data Group, C. Amsler et al. , Phys. Lett. B , 1 (2008).[2] P. Minkowski, Phys. Lett. B , 421 (1977); T. Yanagida, in Proceedings of the Workshopon Unified Theory and the Baryon Number of the Universe , edited by O. Sawada and A.Sugamoto (KEK, Tsukuba, 1979), p. 95; M. Gell-Mann, P. Ramond, and R. Slansky, in
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