Weinberg dimension-5 operator by vector-like lepton doublets
aa r X i v : . [ h e p - ph ] J un Weinberg dimension-5 operator by vector-like lepton doublets
Pei-Hong Gu ∗ School of Physics, Jiulonghu Campus, Southeast University, Nanjing 211189, China
It is well known that a Weinberg dimension-5 operator for small neutrino masses can be realizedat tree level in three types of renormalizable models: (i) the type-I seesaw mediated by fermionsinglets, (ii) the type-II seesaw mediated by Higgs triplets, (iii) the type-III seesaw mediated byfermion triplets. We here point out such operator can be also induced at tree level by vector-likelepton doublets in association with unusual fermion singlets, Higgs triplets or fermion triplets. Ifthese unusual fermion singlets, Higgs triplets or fermion triplets are heavy enough, their decays cangenerate a lepton asymmetry to explain the cosmic baryon asymmetry, meanwhile, the vector-likelepton doublets can lead to a novel inverse or linear seesaw with rich observable phenomena. Wefurther specify our scenario can be naturally embedded into a grand unification theory without theconventional type-I, type-II or type-III seesaw.
PACS numbers: 14.60.Pq, 14.60.Hi, 98.80.Cq, 12.60.CN, 12.60.Fr
The discovery of neutrino oscillations indicates thatthree flavors of neutrinos should be massive and mixing[1]. Within the context of the standard model (SM), wecan consider a Weinberg dimension-5 operator [2],
L ⊃ −
12Λ ¯ l L φφ T l cL + H.c. , (1)to generate the required neutrino masses, L ⊃ − m ν ¯ ν L ν cL + H.c. with m ν = h φ i Λ . (2)In Eq. (1), l L and φ are the lepton and Higgs doublets, l L (1 , , − ) , φ (1 , , − ) . (3)Here and thereafter the brackets following the fields de-scribe the transformations under the SU (3) c × SU (2) L × U (1) Y gauge groups.It has been well known that the Weinberg dimension-5operator (1) can be realized at tree level in three typesof renormalizable models: (i) the type-I seesaw mediatedby fermion singlets [3–6], (ii) the type-II seesaw mediatedby Higgs triplets [7–11], (iii) the type-III seesaw mediatedby fermion triplets [12, 13], i.e. L I ⊃ − M N (cid:0) ¯ N cR N R + H.c. (cid:1) − y N ¯ l L φN R + H.c.with N R (1 , , , (4) L II ⊃ − M Tr (cid:0) ∆ † ∆ (cid:1) − µ ∆ (cid:0) φ T iτ ∆ φ + H.c. (cid:1) − f ∆ ¯ l cL iτ ∆ l L + H.c. with ∆(1 , , +1) , (5) L III ⊃ − M T (cid:2) Tr (cid:0) ¯ T cL iτ T L iτ (cid:1) + H.c. (cid:3) − y T ¯ l cL iτ T L ˜ φ +H.c. with T L (1 , , . (6) ∗ Electronic address: [email protected]
As shown in Fig. 1, we can integrate out the right-handedneutrino singlets N R , the Higgs triplets ∆ and the leptontriplets T L from the type-I, II and III seesaw models.Then the effective cutoff in Eq. (1) should be1Λ I = − y N M N y TN , II = − f † ∆ µ ∆ M , III = − y ∗ T M T y † T . (7)In the following we shall show the other renormalizableways to realize the Weinberg dimension-5 operator (1) attree level. Specifically, we introduce two types of fermiondoublets with opposite hypercharges, ψ L (1 , , + ) , ψ ′ L (1 , , − ) . (8)We then construct the models in association with fermionsinglets, L IVa ⊃ − M S (cid:0) ¯ S cR S R + H.c. (cid:1) − M ψ (cid:0) ¯ ψ cL iτ ψ ′ L + H.c. (cid:1) − m D ¯ l cL iτ ψ L − f ′ S ¯ ψ ′ L φS R + H.c.with S R (1 , , , (9a) L IVb ⊃ − M S (cid:0) ¯ S cR S ′ R + H.c. (cid:1) − M ψ (cid:0) ¯ ψ cL iτ ψ ′ L + H.c. (cid:1) − m D ¯ l cL iτ ψ L − f S ¯ l L φS R − f ′ S ¯ ψ ′ L φS ′ R + H.c.with S R (1 , , , S ′ R (1 , , , (9b)the models in association with Higgs triplets, L Va ⊃ − M Tr (cid:0) Σ † Σ (cid:1) − µ Σ (cid:0) φ T iτ Σ φ + H.c. (cid:1) − M ψ (cid:0) ¯ ψ cL iτ ψ ′ L + H.c. (cid:1) − m D ¯ l cL iτ ψ L − f Σ ¯ ψ ′ cL iτ Σ ψ ′ L + H.c. with Σ(1 , , +1) , (10a) L Vb ⊃ − M Tr (cid:0) Σ † Σ (cid:1) − µ Σ (cid:0) φ T iτ Σ φ + H.c. (cid:1) − M ψ (cid:0) ¯ ψ cL iτ ψ ′ L + H.c. (cid:1) − m D ¯ l cL iτ ψ L − f Σ ¯ ψ ′ cL iτ Σ l L + H.c. with Σ(1 , , +1) , (10b) l L N R N R l L φ φ l L ∆ l L φ φ l L T L T L l L φ φ FIG. 1: The Weinberg dimension-5 operator induced by the tyep-I, type-II and type-III seesaw at tree level. l L ψ L ψ ′ L S ′ R S R l L φ φl L l L φ φψ L ψ ′ L S R S R ψ ′ L ψ L l L S R S ′ R ψ ′ L ψ L l L φ φ + FIG. 2: The Weinberg dimension-5 operator induced by the vector-like lepton doublets and the fermion singlets at tree level. l L ψ L ψ ′ L l L Σ φ φl L Σ l L φ φψ L ψ ′ L ψ ′ L ψ L l L ψ ′ L ψ L l L Σ φ φ + FIG. 3: The Weinberg dimension-5 operator induced by the vector-like lepton doublets and the Higgs triplets at tree level. l L ψ L ψ ′ L X ′ L X L l L φ φl L l L φ φψ L ψ ′ L X L X L ψ ′ L ψ L l L X L X ′ L ψ ′ L ψ L l L φ φ + FIG. 4: The Weinberg dimension-5 operator induced by the vector-like lepton doublets and the fermion triplets at tree level. F F ′ F ′ F F F ′ H ′ H H ′ H H H H F F ′ F ′ F F F H H H ′ H ′ H ′ H H + FIG. 5: The Weinberg dimension-5 operator from an SO (10) model where the vector-like lepton doublets are from the (10 F , ′ F )representations while the fermion singlets are from the (1 ′ F , F ) representations. The 1 ′ H Higgs scalar can obtain an inducedVEV at the TeV scale after a PQ symmetry is spontaneously broken. The 16 ′ H Higgs scalar then can obtain its VEV belowthe TeV scale after the 16 H Higgs scalar spontaneously breaks the left-right symmetry. as well as the models in association with fermion triplets, L VIa ⊃ − M X (cid:2) Tr (cid:0) ¯ X cL iτ X L iτ (cid:1) + H.c. (cid:3) − M ψ (cid:0) ¯ ψ cL iτ ψ ′ L + H.c. (cid:1) − m D ¯ l cL iτ ψ L − f ′ X ¯ ψ ′ cL ı τ X L φ + H.c. with X L (1 , , , (11a) L VIb ⊃ − M X (cid:2) Tr (cid:0) ¯ X cL iτ X ′ L iτ (cid:1) + H.c. (cid:3) − M ψ (cid:0) ¯ ψ cL iτ ψ ′ L + H.c. (cid:1) − m D ¯ l cL iτ ψ L − f X ¯ l cL iτ X L φ − f ′ X ¯ ψ ′ cL iτ X ′ L φ + H.c.with X L (1 , , , X ′ L (1 , , . (11b)Here we have imposed a global symmetry of lepton num-ber, which is softly broken only by the m D terms, inorder to forbid the other mass and Yukawa terms involv-ing the non-SM fields. For this purpose, ( ψ L , ψ ′ L , S R )carry the lepton numbers L = (0 , ,
0) for the model(9a), ( ψ L , ψ ′ L , S R , S ′ R ) carry L = (+1 , − , +1 , −
1) for the model (9b), ( ψ L , ψ ′ L , Σ) carry L = (0 , ,
0) for themodel (10a) or L = (+1 , − ,
0) for the model (10b),( ψ L , ψ ′ L , X L ) carry L = (0 , ,
0) for the model (11a),( ψ L , ψ ′ L , X L , X ′ L ) carry L = (+1 , − , − , +1) for themodel (11b). By integrating out the non-SM fields fromEqs. (9a-11b), the effective cutoff in Eq. (1) can be re-spectively given by1Λ IVa = m ∗ D M ψ f ′ S M S f ′ TS M ψ m † D , (12a)1Λ IVb = − f S M S f ′ TS M ψ m † D − m ∗ D M ψ f ′ S M S f TS , (12b)1Λ Va = m ∗ D M ψ f † Σ M ψ m † D µ Σ M , (13a)1Λ Vb = − f † Σ M ψ m † D + m ∗ D M ψ f ∗ Σ ! µ Σ M , (13b)1Λ VIa = m ∗ D M ψ f ′∗ X M X f ′† X M ψ m † D , (14a)1Λ VIb = − f ∗ X M X f ′† X M ψ m † D − m ∗ D M ψ f ′∗ X M X f † X . (14b)The relevant diagrams are shown in Fig. 2, Fig. 3 andFig. 4.If the vector-like lepton doublets ( ψ L , ψ ′ L ) are heavyenough, they can be integrated out before the fermionsinglets ( S R , S ′ R ), the Higgs triplets Σ or the fermiontriplets ( X L , X ′ L ). This means the models (9a-11b) cangive nothing but the type-I, type-II or type-III see-saw. Alternatively, the fermion singlets ( S R , S ′ R ), theHiggs triplets Σ and the fermion triplets ( X L , X ′ L ) couldbe much heavier than the vector-like lepton doublets( ψ L , ψ ′ L ). In consequence, we could first integrate outthese fermion singlets, Higgs triplets or fermion tripletsand then study the phenomena from the vector-like lep-ton doublets. For example, the models (9a), (10a) and(11a) indeed can lead to an inverse seesaw [14]. In thisinverse seesaw scenario, the charged and neutral com-ponents of the vector-like lepton doublets ( ψ L , ψ ′ L ) canrespectively mix with the SM charged leptons and neutri-nos up to the bounds allowed by the precision measure-ments [15]. Such significant mixings can be also inducedby the m D term in the models (9b), (10b) and (11b),which now have a few features of the linear seesaw [16–18]. We hence could expect the models (9a-11b) to beverified at the LHC and other colliders, similar to theinverse seesaw with vector-like lepton triplets [19].It is easy to see our models can be embedded into an SU (5) grand unification theory (GUT) by placing thevector-like lepton doublets in the 5-dimensional repre-sentations. We further consider the SO (10) GUT. InFig. 5, we indicate the model (9b) can originate froman SO (10) GUT where the vector-like lepton doubletsare from the (10 F , ′ F ) representations [20] while thefermion singlets are from the (1 ′ F , F ) representations.In this model we have imposed a global U (1) P Q sym-metry under which the 16 H field carries a charge − ′ H , 1 ′ F and 10 ′ F fields carry a charge +1, while the16 ′ H and 10 F carry a charge +2. Consequently, the trilin-ear coupling 10 H H H should be absent and hence the[ SU (2) L ]-singlet components of the 16 H Higgs scalar can-not obtain a vacuum expectation value (VEV). The 1 ′ H Higgs scalar has a quartic coupling with another 1 H Higgsscalar, i.e. ¯1 ′ H H . After the 1 H Higgs scalar develops aVEV around 10 − GeV, the 1 ′ H Higgs scalar can beexpected to obtain a VEV around 10 − GeV if its massterm is below the GUT scale. Since the (10 F , ′ F ) fieldscontains the color-triplet fermions, the U (1) P Q symme-try should be the Peccei-Quinn (PQ) symmetry [21] for aKSVZ invisible axion [22, 23]. We would like to empha-size this GUT does not result in the conventional type-Iand type-II seesaw. Actually the usual right-handed neu-trinos from the 16 F representations now construct the fermion singlets with the 1 ′ F representations. Similarly,the other models (9a) and (10a-11b) can be also embed-ded into an SO (10) context such as the above GUT forthe model (9b).We then simply explain how to produce the cosmicbaryon asymmetry in our models. For example, we con-sider the models (10a) and (10b). If and only if the kine-matics is allowed, the Higgs triplets Σ can have two decaymodes as below,Σ → ψ ′ cL + ψ ′ cL or ψ cL + l cL , Σ → φ ∗ + φ ∗ . (15)As long as the CP is not conserved, we can expect a CPasymmetry in the above decays, ε a = 2 Γ(Σ a → ψ ′ cL + ψ ′ cL ) − Γ(Σ ∗ a → ψ ′ L + ψ ′ L )Γ(Σ a → ψ ′ cL + ψ ′ cL ) + Γ(Σ a → φ ∗ + φ ∗ )= 2 Γ(Σ ∗ a → φ + φ ) − Γ(Σ a → φ ∗ + φ ∗ )Γ(Σ a → ψ ′ cL + ψ ′ cL ) + Γ(Σ a → φ ∗ + φ ∗ ) = 0 or ε a = 2 Γ(Σ a → ψ cL + l cL ) − Γ(Σ ∗ a → ψ L + l L )Γ(Σ a → ψ cL + l cL ) + Γ(Σ a → φ ∗ + φ ∗ )= 2 Γ(Σ ∗ a → φ + φ ) − Γ(Σ a → φ ∗ + φ ∗ )Γ(Σ a → ψ cL + l cL ) + Γ(Σ a → φ ∗ + φ ∗ ) = 0 . (16)This CP asymmetry can arrive at a nonzero value if themodels (10a) and (10b) contain at least two Higgs tripletsΣ , ,... , see Fig. 6. After the Higgs triplets Σ a go out ofequilibrium, their CP-violating decays (16) can generatean asymmetry stored in the vector-like lepton doublets( ψ L , ψ ′ L ) and the SM lepton doublets l L . The ψ ′ L or ψ L asymmetry eventually can contribute to the l L asymme-try because of the M ψ and m D terms in Eqs. (10a) and(10b). The l L asymmetry can be partially converted to abaryon asymmetry through the sphaleron [24] processes.This leptogenesis [25] scenario is very similar to the lep-togenesis in type-II seesaw [26, 27]. The details can befound in literatures such as [27–29]. Similarly, the othermodels (9a-9b) and (11a-11b) can accommodate a lepto-genesis through the decays of the heavy fermion singletsor triplets.In this work we have clarified the Weinberg dimension-5 operator can be induced at tree level by the vector-like lepton doublets in association with the unusualfermion singlets, Higgs triplets or fermion triplets, be-sides the well-known type-I, type-II and type-III see-saw. The vector-like lepton doublets can lead to richobservable phenomena if they are at the TeV scale.Meanwhile, when the fermion singlets, Higgs triplets orfermion triplets are very heavy, their out-of-equilibriumand CP-violating decays can accommodate a leptogene-sis to explain the baryon asymmetry in the universe. Ourscenario can be naturally embedded into some SU (5)or SO (10) GUTs without the conventional seesaw. Inthese SU (5) or SO (10) GUTs, the 5-dimensional or10-dimensional representations for the vector-like leptondoublets can also result in the KSVZ invisible axion bytheir color-triplet components if a PQ symmetry is intro-duced properly. Σ a ψ ′ cL ( ψ cL ) ψ ′ cL ( l cL ) +Σ a φ ∗ φ ∗ + Σ a φφ Σ b ψ ′ cL ( ψ cL ) ψ ′ cL ( l cL )Σ a ψ ′ L ( ψ L ) ψ ′ L ( l L ) Σ b φ ∗ φ ∗ FIG. 6: The heavy Higgs triplet decays at one-loop order.
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