What multiple Higgs doublets can do for you

Transcription

1 What multiple Higgs doublets can do for you Igor Ivanov CFTP, Instituto Superior Técnico, Universidade de Lisboa University of Lund, November 14, 2016 based on: I. P. Ivanov, J. P. Silva, PRD 93, (2016) A. Aranda, I. P. Ivanov, E. Jiménez, arxiv:

2 1 Multi-Higgs-doublet models Overstretching the SM Higgs Fermion sector from NHDM symmetries Astroparticle consequences 2 A hidden beauty: 4-3HDM 3 Conclusions Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 2/29

3 The usual picture of SM particle content: Does not really represent which forces act on which particles... Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 3/29

4 left fermions right fermions u c t u c t d s b d s b g... 1 g 8 SU(3) ν e ν μ ν τ e μ τ W... W 1 3 H e μ τ B SU(2) U(1) Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 4/29

5 Brout-Englert-Higgs mechanism in SM Standard Model: 3-in-1 Start with matter fields, organize them into left doublets Q L = (u L, d L ) and right singlets u R, d R, require local gauge symmetry get gauge interactions for free all fields are massless. add Higgs doublet with mexican hat potential get massive W and Z, as well as fermions, without spoiling high-energy features. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 5/29

6 Electroweak symmetry breaking Brout-Englert-Higgs mechanism of electroweak symmetry breaking: Bosons partially mix: W 1, W 2, W 3, B, (H +, H 0 ) W ±, Z, γ, h. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 6/29

7 Electroweak symmetry breaking Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 7/29

8 Overstretching the Higgs mechanism The scalar sector of SM is overly minimalistic. We ask the single Higgs doublet to accomplish three simultaneous tasks: give masses of W and Z via D µ φ 2, give masses of down-quarks and leptons via Q L φd R. give masses of up-quarks via Q L φu R, where φ iσ 2 φ. The gauge/fermion structure of SM does not require the Higgs sector to be minimal! It can well be rich and it can extend the idea of generations to scalars. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 8/29

9 Overstretching the Higgs mechanism With so many tasks to do, the Higgs sector of the SM is completely exhausted : does not explain the hierarchical fermion masses and the mixing patterns nor offers any relation among them (flavour sector = the ugly part of the SM). boring flavor properties of the Higgs boson exchange: no FCNC, no LFV, -violation must be inserted by hand, unable to generate astroparticle and cosmological phenomena we observe: no DM, insufficient cosmological EWPT no sizable baryon asymmetry. All these features can be successfully reproduced with the SM fermions and gauge interactions but with an extended scalar sector, in particular, in N-Higgs-doublet models (NHDM). Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 9/29

10 Quark masses and mixing For three generations d i = (d, s, b), u i = (u, c, t), Yukawa interactions are written as Q Li Γ ij φd Rj + q Li ij φurj + h.c. d Li (M d ) ij d Rj + ū Li (M u ) ij u Rj + h.c. where the 3 3 mass matrices are M d = Γ v 2, M u = v 2. They can be diagonalized: V dl M d V dr = D d, V ul M u V ur = D u, but then the charged current matrix can become non-trivial: ū Li γ µ W + µ d Li ū Li γ µ W + µ V ij d Lj, where V ij = V ul V dl δ ij. Diagonalizing M d, M u, we also diagonalize hf f -interactions no FCNC or LFV in Higgs exchanges. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 10/29

11 Quark masses and mixing With N Higgs doublets φ a, the procedure is the same: ( ) QLi Γ (a) ij φ a d Rj + q Li (a) ij φ a u Rj + h.c. a Vacuum coresponds to a vev alignment φ 0 a = v a / 2, which gives M d = 1 Γ (a) v a, M u = 1 (a) va. 2 2 a Dangerous tree-level FCNC can be avoided with natural flavour conservation via discrete symmetries [Weinberg, Glashow, 1977; Pachos, 1977]. Individual Γ (a) and (a) can be very simple, symmetry-constrained. In M d and M u, this symmetry is ruined but can leave traces in masses/mixing. If complex vevs v a get a relative phase, then -violation can appear spontaneously for real Γ and [T.D.Lee, 1973, Branco, 1979]. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 11/29 a

12 Quark masses and mixing Starting from late 1970: intenstive NHDM model-building, aiming at deriving properties of quark and lepton masses and mixing from symmetry considerations and relating quark mixing angles with masses. simplest texture-based approaches: 2 generations [Weinberg, 1977; Wilczek, Zee, 1977] with sin θ C m d /m s, 3-generation Ansatz [Fritzsch, 1978] with full expression of V ij via mass ratios and phases. global permutation symmetry groups S 3 or S 4 : [Pakvasa, Sugawara, 1978, 1979, + Yamanaka, 1982] in early 80 s, perfectly agreed on the CKM matrix. rephasing + permutation global symmetry groups (54) which makes Γ (a) and (a) very simple [Segre, Weldon, Wyers, 1979]: mass hierarchy may come from vev hierarchy v 1 v 2 v 3, though it is difficult to arrange that in the scalar potential. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 12/29

13 Scalar dark matter More Higgses more fun! some Higgs fields participate in EWSB massive W and Z, massive fermions, other can be inert: no coupling to fermions, and no role in W and Z masses. If inert scalars are protected by a new parity (do not get vev) the lightest parity-odd scalar is stable scalar DM. This does not require any fine-tuning! Example: Inert doublet model = 2HDM with exact Z 2 -symmetry [Deshpande, Ma, 1978; Barbieri et al, 2006, Lopez Honorez et al, 2006]. Two Higgs doublets φ and φ D, with φ D = 0. Extra Higgses from φ D : H ±, H, A, with the lightest one being the DM + interesting collider phenomenology. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 13/29

14 Cosmological phase transition Baryon asymmetry requires a strong first-order thermal electroweak phase transition in early Universe. v(t c ) T c > 1. The SM could satisfy it only for light Higgs (m h < 50 GeV) [Kajantie et al, 1996; Csikor, 1999]. Extra scalars modify the Higgs potential and can produce strong EWPT, from [Turok, Zadrozny, 1992] on, recent detailed work [Dorsch, Huber, No, 2013]. Can be probed in principle with future GW observatories! Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 14/29

15 4-3HDM: a three-higgs-doublet model with order-4 -symmetry Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 14/29

16 Freedom of defining In QFT, the discrete transformations such as are not uniquely defined a priori [e.g. Feinberg, Weinberg, 1959]. For example, in NHDM with doublets φ i, i = 1,..., N, the transformation φ i X ij φ j, X U(N), with any X can play the role of the transformation [e.g. Branco, Lavoura, Silva, 1999]. The standard convention φ i φ i is basis-dependent. I will show that in models with several gauge-blind scalars the freedom of defining is even larger. In particular, it can lead to scalars which are -half-odd: J : Φ(x, t) i Φ( x, t). Notice: (1) no conjugation, (2) 4: order-4 transformation, J 2 I, J 4 = I. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 15/29

17 Freedom of defining In QFT, the discrete transformations such as are not uniquely defined a priori [e.g. Feinberg, Weinberg, 1959]. For example, in NHDM with doublets φ i, i = 1,..., N, the transformation φ i X ij φ j, X U(N), with any X can play the role of the transformation [e.g. Branco, Lavoura, Silva, 1999]. The standard convention φ i φ i is basis-dependent. I will show that in models with several gauge-blind scalars the freedom of defining is even larger. In particular, it can lead to scalars which are -half-odd: J : Φ(x, t) i Φ( x, t). Notice: (1) no conjugation, (2) 4: order-4 transformation, J 2 I, J 4 = I. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 15/29

18 4-3HDM Consider 3HDM with the following potential V = V 0 + V 1 (notation: i φ i ): [ V 0 = m11 2 (1 1) m22 2 ( ) + λ 1 (1 1) 2 + λ 2 (2 2) 2 + (3 3) 2] ] +λ 3 (1 1)( ) + λ 3 (2 2)(3 3) + λ 4 [(1 2)(2 1) + (1 3)(3 1) + λ 4 (2 3)(3 2), with all parameters real, and V 1 = λ 5 (3 1)(2 1) + λ 6 2 [ (2 1) 2 (3 1) 2] [ ] + λ 8 (2 3) 2 + λ 9 (2 3) (2 2) (3 3) + h.c. with real λ 5,6 and complex λ 8,9. It is invariant under order-4 : J : φ i X ij φ j, X = 0 0 i. 0 i 0 Its square, J 2 = XX = diag(1, 1, 1) and J 4 = I. This model has no other symmetries [Ivanov, Keus, Vdovin, 2012]. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 16/29

19 Physical scalars -conserving minimum: v i = (v, 0, 0). Physical scalars: h SM, degenerate H ± 2,3, and two pairs of degenerate neutrals: the heavier H and A and the ligher h and a, with masses M 2, m 2 = m v 2 ( ) λ 3 + λ 4 ± λ λ2 6. However, these real neutrals are not -eigenstates: H A, A H, h a, a h. Can we combine them into neutral complex -eigenstate fields? Φ = 1 2 (H ia), ϕ = 1 2 (h + ia), Φ iφ, ϕ iϕ. Conserved quantum number: not -parity but -charge q defined mod 4. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 17/29

20 Physical scalars -conserving minimum: v i = (v, 0, 0). Physical scalars: h SM, degenerate H ± 2,3, and two pairs of degenerate neutrals: the heavier H and A and the ligher h and a, with masses M 2, m 2 = m v 2 ( ) λ 3 + λ 4 ± λ λ2 6. However, these real neutrals are not -eigenstates: H A, A H, h a, a h. Can we combine them into neutral complex -eigenstate fields? Φ = 1 2 (H ia), ϕ = 1 2 (h + ia), Φ iφ, ϕ iϕ. Conserved quantum number: not -parity but -charge q defined mod 4. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 17/29

21 To C, or not to C that is the question Φ(x, t) iφ( x, t) looks like P transformation rather than. Where did we lose the C in? Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 18/29

22 Back to basics Single complex scalar field: φ(x, t) = dp [ a(p)e ipx + b (p)e ipx]. If we define φ φ a(p) b( p), b(p) a( p), which means b 0 is antiparticle of a 0. Two complex mass-degenerate scalars φ i (x), i = 1, 2. If φ 1 φ 2, φ 2 φ 1, then a 1 (p) b 2 ( p), b 1 (p) a 2 ( p) under, which means By choosing X in φ i b 2 0 is antiparticle of a 1 0, b 1 0 is antiparticle of a 2 0, X ij φ j, we assign who is antiparticle of whom. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 19/29

23 Back to basics Single complex scalar field: φ(x, t) = dp [ a(p)e ipx + b (p)e ipx]. If we define φ φ a(p) b( p), b(p) a( p), which means b 0 is antiparticle of a 0. Two complex mass-degenerate scalars φ i (x), i = 1, 2. If φ 1 φ 2, φ 2 φ 1, then a 1 (p) b 2 ( p), b 1 (p) a 2 ( p) under, which means By choosing X in φ i b 2 0 is antiparticle of a 1 0, b 1 0 is antiparticle of a 2 0, X ij φ j, we assign who is antiparticle of whom. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 19/29

24 Basis change Within the same φ 1 ( ) η ξ φ 2, φ 2 = 1 2 ( φ 1 example, define η and ξ as ) ( φ1, η η, ξ ξ. φ 2 ) Remember: η and ξ are mass-degenerate. Combine two -even fields Reη and Imξ Φ = Reη i Imξ. Combine two -odd fields Reξ and Imη Φ = Reξ i Imη. Φ Φ, Φ Φ. Conjugation disappeared. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 20/29

25 Basis change One can now describe passage from (φ 1, φ 2 ) (η, ξ) (Φ, Φ) via ( ) = Φ Φ 1 ( ) ( ) 1 1 φ φ. 2 This basis change undoes the conjugation. This is a norm-preserving non-holomorphic map on C 2, not the usual basis change. It is a valid O(4)-rotation in R 4 spanned by (Reη, Imη, Reξ, Imξ). If complex fields are gauge-blind (do not carry any non-zero gauge quantum number), the group of basis changes increases from U(2) to O(4). Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 21/29

26 For mass-degenerate gauge-blind scalars, conjugating or not under is a matter of basis choice. In all transformations, we never redefined the transformation itself. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 22/29

27 4 -half-odd scalars arise in a similar way. Starting point: φ 1 Operators: a 1 Basis change: iφ 2, φ 2 iφ 1. ib 2, b 2 ia 1 and a 2 ib 1, b 1 ( ) Φ ϕ = 1 ( ) ( ) 1 1 φ φ. 2 Result: Φ iφ, ϕ iϕ. For operators ia 2. a Φ = (a 1 + b 2 )/ 2 ia Φ, b Φ = (a 2 + b 1 )/ 2 ib Φ. An example of doubling of mass eigenstates beyond Kramers degeneracy (a possibility mentioned e.g. in Weinberg, vol. 1, app. 2C). Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 23/29

28 Yukawas for 4-3HDM Can -half-odd scalars have -conserving Yukawa interactions? Yes, provided the mixes the fermion families. In 4-3HDM, the charged lepton sector (here, φ a φ 0 a): Take φ a L Y = l Li Γ a ijl Rj φ a + l Ri (Γ a ij) l Lj φ a. X ab φ b, with the same X as before, and l i Yij lc j. Then, Y Γ 1Y = Γ 1, iy Γ 2Y = Γ 3, iy Γ 3Y = Γ 2. We switch to the fermion basis in which Y = 0 0 e iα, 0 e iα 0 -conservation requires that fermions be degenerate: m 2 = m 3. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 24/29

29 Yukawas for 4-3HDM case 1: α = ±π/4 + πk, order-8 transformation. Γ 1 = g g g 2, Γ 2 = case 2: α = ±π/2, order-4 transformation. Γ 1 = g g 2 g 3 0 g 3 g 2, Γ 2 = g 23 0 g g 12 g 13 g g , Γ 3 =, Γ 3 = ± ±g 32 0 g g 13 g 12 g g Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 25/29

30 Yukawas for 4-3HDM Explicitly, in case 1 [notation: (e, µ, τ) = (l 1, l 2, l 3 )]: where L Y = ( µτ)(gφ gϕ) + ( τγ 5 µ)( g Φ + g ϕ) + h.c., g = c γg 23 s γ g32, g = s γg 23 + c γ g32, tan 2γ = λ 6 /λ Notice that fermion biliears are -half-odd, and that insertion of γ 5 introduces an extra -oddness as in the usual case: µτ i µτ, τµ i τµ, µγ 5 τ i µγ 5 τ, τγ 5 µ i τγ 5 µ. -half-odd scalars do not have to be inert. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 26/29

31 Clash of two definitions Lagrangian contains: ( µγ µ µ + τγ µ τ)a µ and (ēµ + τe)φ + h.c. (case 2). Do Yukawa couplings of Φ generate particle-antiparticle asymmetry? No unambiguous answer exists! Depends on the particle-antiparticle assignment. Formal definition: just as above. Field µ creates µ and annihilates τ +, field τ creates τ + and annihilates µ. Then (ēµ + τe)φ is particle-antiparticle symmetric. But then γ τ + µ and τ µ +. Physical definition: we require, as usual, that γ l + l. Then, Φ decays produce only µ and τ + and strongly violate C-symmetry. Agnostic position: we refrain from assigning particle-antiparticle pairs. Then it is unclear what -conservation/violation should mean. It sounds pure semantics within the toy model. But it may have unusual consequences for realistic models with the broken 4. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 27/29

32 Remarks on phenomenology Does a model based on 4 lead to any phenomenological signal which cannot be mimicked by any usual -conserving model? ()a Φ a Φ () 1 = a Φ a Φ. We get a -odd pair of two identical bosons any pheno signal? A variation on inert doublet model based on 4 instead of + Z 2 : fully calculable, can be readily tested against collider data and DM searches. If 4 is spontaneous broken, can 4-3HDM with Yukawas reproduce the quark sector? How is the clash between two definitions resolved? Any specific predictions for the neutrino sector? Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 28/29

33 Conclusions Multi-Higgs-doublet models is a conceptually simple bsm framework with very rich phenomenology. 4-3HDM is the simplest model featuring -half-odd scalars: Φ iφ. Their origin is the extra freedom of basis change arising in models with mass-degenerate gauge-blind scalars. No fine-tuning is required: the gauge blindness, mass degeneracy, and -half-oddness appear naturally within 4-3HDM. The -half-odd scalars do not have to be inert: they can couple to fermions in -conserving way. The model is compact, analytically tractable, and can have interesting phenomenological consequences. Igor Ivanov (CFTP, IST) What NHDM can do for you Uni. Lund 29/29