Two-Higgs-Doublet Model

Transcription

1 Two-Higgs-Doublet Model Logan A. Morrison University of California, Santa Cruz March 18, 016 Logan A. Morrison (UCSC) HDM March 18, / 7

2 Overview 1 Review of SM HDM Formalism HDM Potential HDM VEVs Characters in the theory 3 Flavor Changing Neutral Currents Higgs Basis Yukawa Coupling Removing FCNCs: Type 1 and Type II HDM 4 Application Rare B meson decays Logan A. Morrison (UCSC) HDM March 18, 016 / 7

3 Motivation Super symmetry needs two Higgs doublets SM cannot explain the observed baryon asymmetry, possible in HDM CP Violations Candidates for dark matter Many more.. Logan A. Morrison (UCSC) HDM March 18, / 7

4 Review of SM Part I: Review of SM Logan A. Morrison (UCSC) HDM March 18, / 7

5 Review of SM SM Higgs In the SM, we have one complex scalar doublet ( ) φ1 Φ = The scalar potential is given by φ V = µ Φ Φ + 1 λ (Φ Φ ) (1) where µ and λ are real. Logan A. Morrison (UCSC) HDM March 18, / 7

6 Review of SM SM Higgs (Continued) We assume that the scalar potential has a non-zero vev: φ = 1 ( ) 0 v Expanding about the vev ( G + ) Φ = (v + H + ig 0 )/ () (3) where G is a complex field and H, G 0 are real fields The potential becomes V = µ4 λ + µ H + L Int Only H gets a mass and it s mass matrix, V H H = µ is trivial Logan A. Morrison (UCSC) HDM March 18, / 7

7 Review of SM SM Higgs (Continued) We thus have a charged goldstone boson: G +, a nuetral goldstone boson: G 0 and a Higgs: H This scalar field couples to the gauge fields via the covariant derivative D µ = µ + igt a A a µ + ig 1 YB µ (4) This gives us the W s, the Z and the photon field W ± = 1 ( A 1 µ ia ) µ ; Z 0 µ = ga3 µ g B µ ; (5) g + g A µ = g A 3 µ + gb µ g + g (6) We then couple the scalar field to the fermions to give them mass via Yukawa couplings L Yukawa = λ e ψ L Φψ E λ u ψ Q1 Φψ u λ d ψ Q Φψ d (7) Logan A. Morrison (UCSC) HDM March 18, / 7

8 HDM Formalism Part : HDM Formalism Logan A. Morrison (UCSC) HDM March 18, / 7

9 HDM Formalism HDM Potential Two-Higgs Doublet Model (HDM) Now we add a second Higgs doublet The most general, renormalizable two-doublet scalar potential is [1] ( ) V = m11φ 1 Φ 1 + mφ Φ m1φ 1 Φ + H.C. (8) + 1 ( ) λ 1 Φ 1 Φ 1 ( ) ( ) ( ) 1 + λ Φ Φ + λ3 Φ 1 Φ 1 Φ Φ ) ( ) [ + λ 4 (Φ 1 Φ Φ 1 ( ) Φ 1 + λ 5 Φ 1 Φ ) ( ) ( ) ( ) ] + λ 6 (Φ 1 Φ 1 Φ 1 Φ + λ 7 Φ Φ Φ 1 Φ + H.C. where m 11, m and λ 1,,3,4 are real and m 1 and λ 5,6,7 are in general complex Logan A. Morrison (UCSC) HDM March 18, / 7

10 HDM Formalism HDM Potential Two-Higgs Doublet Model (HDM) The potential can be written more compactly as V = a,b=1 µ ab ( Φ aφ b ) + 1 a,b,c,d=1 λ ab,cd ( Φ aφ b ) ( Φ cφ d ) (9) where µ 11 = m11 µ 1 = m1 µ 1 = ( m1 µ = m ) λ 11,11 = λ 1 λ 11, = λ,11 = λ 3 λ 1,1 = λ 5 λ 11,1 = λ 1,11 = λ 6 λ,1 = λ 1, = λ 7 λ, = λ λ 1, = λ,11 = λ 4 λ 1,1 = λ 5 λ 11,1 = λ 1,11 = λ 6 λ,1 = λ 1, = λ 7 Logan A. Morrison (UCSC) HDM March 18, / 7

11 HDM Formalism HDM VEVs HDM VEVs There are three types of vevs in the HDM Normal vacua (neutral, CP conserving, v = ( ) 0v1 Φ 1 = v1 + v 46GeV ) ( ) 0v and Φ = (10) CP breaking vacua: where the vevs have a relative phase 0 Φ 1 = v 1 e iθ and Φ = 0 v (11) Charged vacua: Φ 1 = 1 ( ) α v 1 and Φ = 1 ( ) 0 v (1) (These vevs are bad: they give the photon a mass) Logan A. Morrison (UCSC) HDM March 18, / 7

12 HDM Formalism Characters in the theory Higgs Masses For illustration and simplicity, we will work with a CP conserving vacua and a scalar potential which is CP conserving, in which the quartic terms obeys a Z symmetry: Z : Φ 1 Φ 1 and Φ Φ (13) Under these assumptions, the scalar potential is ( ) V = m11φ 1 Φ 1 + mφ Φ m1 Φ 1 Φ + Φ Φ 1 (14) + 1 ( ) λ 1 Φ 1 Φ 1 ( ) ( ) ( ) 1 + λ Φ Φ + λ3 Φ 1 Φ 1 Φ Φ ) ( ) + λ 4 (Φ 1 Φ Φ Φ 1 + λ ( ( ) 5 ( ) ) Φ 1 Φ + Φ Φ 1 Expanding about these vev s the fields are ( φ + ) a Φ a = (v a + ρ a + iη a )/, a {1, } (15) Logan A. Morrison (UCSC) HDM March 18, / 7

13 HDM Formalism Characters in the theory Higgs Masses (continued) There are 8 fields. Three fields get eaten up by the W ± µ and the Z 0 µ The remaining content is: two neutral scalars, a pseudo scalar and a charged scalar. To determine mass matricies, one needs to enfore extrema of the V potential: v 1 = V v = 0, which occurs only when 0 = m 11v 1 Re ( m 1) v + v = m v Re ( m 1) v1 + v 3 λ 1 λ v 1v λ + v v1 λ 345 (16) (17) where λ 345 = λ 3 + λ 4 + λ 5 Logan A. Morrison (UCSC) HDM March 18, / 7

14 HDM Formalism Characters in the theory Charged Higgs For the charged scalars, the mass matrix is ) ] v (Mφ = V ± ij φ i φ + = [m 1 (λ 4 + λ 5 ) v 1 v v 1 j 1 1 v 1 v This matrix has one zero eigenvalue corresponding to the charged Goldstone boson (which gets eaten by the W ). The mass of the charged Higgs is H + = v φ + 1 v 1φ + v1 + v M H + = v v 1 v [ m 1 v 1 v (λ 4 + λ 5 ) ] Logan A. Morrison (UCSC) HDM March 18, / 7

15 HDM Formalism Characters in the theory Pseudo Scalar Higgs The mass matrix for the pseudo scalar Higgs is ( ) ( ) ( M A )ij = V m = 1 v λ v 1 v 5 η i η j v 1 v v 1 v v1 Diagonalizing this, we find one massless pseudo scalar (which gets eaten by the Z boson) and a massive pseudo scalar with mass M A : A = v η 1 v 1 η v1 + v M A = (v 1 + v ) ( m 1 v 1 v λ 5 ) Logan A. Morrison (UCSC) HDM March 18, / 7

16 HDM Formalism Characters in the theory Neutral Scalar Higgs Lastly we have the mass matrix for the neutral scalar Higgs ( ) ( M h )ij = V A C = ρ i ρ j C B where A = m λ 1 v 1 + λ 345 v (18) B = m + 3λ v + λ 345 v 1 (19) C = m1 + λ 345 v 1 v (0) This can be diagonalized by and angle α, resulting in the physical fields h and H: H = ( ρ 1 cos α ρ sin α) h = (ρ 1 sin α ρ cos α) (1) Logan A. Morrison (UCSC) HDM March 18, / 7

17 HDM Formalism Characters in the theory The masses of the two neutral scalar Higges are [] [ M h = v λ ˆλ ] cos(β α) sin(β α) [ M H = v λ + ˆλ ] sin(β α) cos(β α) () (3) where β is defined by tan β v /v 1 and λ = λ 1 cos 4 (β) + λ sin (β) + 1 λ 345 sin (β) (4) ˆλ = 1 ] [λ sin(β) 1 cos (β) λ sin (β) λ 345 cos(β) (5) Logan A. Morrison (UCSC) HDM March 18, / 7

18 Flavor Changing Neutral Currents Part 3: Flavor Changing Neutral Currents Logan A. Morrison (UCSC) HDM March 18, / 7

19 Flavor Changing Neutral Currents Higgs Basis Higgs Basis To examine FCNCs it is convenient to work in the Higgs Basis To get to the Higgs Basis, we rotate our basis {Φ 1, Φ } such that Φ 1 = 0 and Φ = 0. Assuming the general vevs of the form Φ 1 = ( 0 v 1 / ) (, Φ = 0 v e iδ / The Higgs basis is achieved by rotating via a unitary matrix U ( H a = U ab Φ b = e iδ/ v1 e iδ/ v e iδ/ ) ( ) Φ1 v v e iδ/ v 1 e iδ/ Φ b=1 where v = v1 + v. This is the Higgs Basis. Expanding about the vev the fields are ( G + ) ( H 1 = (v + H + ig 0 )/ H + ), H = (R + ii )/ ) (6) (7) (8) where R, L, H are neutral, G 0, G + and are Goldstones Logan A. Morrison (UCSC) HDM March 18, / 7

20 Flavor Changing Neutral Currents Yukawa Coupling Coupling to Fermions: Yukawa Couplings The most general gauge invariant coupling we can have between spinors and the Higges are L Y = [ j=1 Q L (Φ j Yj D n R + Φ ) ] j Yj U p R + L L Φ j Yj e l R + H.C. (9) where Q L, L L, n R, p R and l R are flavor three vectors and Φ j = iσ Φ j and Yj D s and Yj U s are 3 3 coupling matrices. Rotating into the Higgs basis and defining mass matrices M n and N n Φ 1 = 1 v (ṽ 1H 1 + ṽ H ) Φ = 1 v (ṽ 1 H 1 ṽ H ) M n = 1 (ṽ1 V1 d + ṽ Y d ) Nn = 1 (ṽ Y1 d ṽ 1 Y d ) Logan A. Morrison (UCSC) HDM March 18, / 7

21 Flavor Changing Neutral Currents Yukawa Coupling Yukawa Couplings (Continued) In terms of the mass matrices, M n and N n, we have j=1 Q L Φ j Yj d n R = v Q L (M n H 1 + N n H )n R (30) In general, we can bi-diagonalize M n by rotating the flavor basis of Q L and n R Q L = U L Q L We when have new mass matrices n R = U n R n R M d = U L M nu n R and N d = U L N nu n R (31) where M d is diagonal: M d = diag(m d, m s, m b ) In general, N d will not be diagonal. If it is not, there will be flavor changing neutral currents Logan A. Morrison (UCSC) HDM March 18, / 7

22 Flavor Changing Neutral Currents Yukawa Coupling Yukawa Couplings (Continued) If FCNCs were allowed, we could have, for example, H d s which would give rise to K- K oscillations at tree level [1] d d H K 0 K 0 s s Figure: FCNC lead to effects such a Kaon osc. at tree level This affects the rate of Kaon oscillation and thus require the neutral flavor changing mediator to have a mass of 10 TeV Theories with FCNC can still are still viable Logan A. Morrison (UCSC) HDM March 18, 016 / 7

23 Flavor Changing Neutral Currents Removing FCNCs: Type 1 and Type II HDM Removing FCNCs: Type I and II HDMs It is possible to rid our theory of FCNC by requiring that all particles with same quantum numbers couple to the one Higgs (for example M n N n ) There are two ways to do this, called: Type I and Type II models Type I: all quarks couple to the same Higgs doublet. This can be achieved by imposing a Z symmetry: Z : Φ 1 Φ 1 and Φ Φ (3) Type II: all Q = /3 couple one doublet (say Φ 1 ) and all Q = 1/3 quarks couple the other double Φ. This type is enforced using Φ 1 Φ 1 and d i R d i R (33) The type II model uses the same Yukawa couplings as MSSM Logan A. Morrison (UCSC) HDM March 18, / 7

24 Flavor Changing Neutral Currents Removing FCNCs: Type 1 and Type II HDM Yukawa Couplings in Type I and II models Recall that the neutral scalars and pseudo scalar are given by H = ( ρ 1 cos α ρ sin α) h = (ρ 1 sin α ρ cos α) (34) 1 A = (v η 1 v 1 η ) (35) v1 + v In the Type I model (tan β v /v 1 ) h A H up-type quarks cos α/ sin β cot β sin α/ sin β down-type quarks and leptons cos α/ sin β cot β sin α/ sin β In the Type II model h A H up-type quarks cos α/ sin β cot β sin α/ sin β down-type quarks and leptons sin α/ cos β tan β cos α/ cos β Logan A. Morrison (UCSC) HDM March 18, / 7

25 Flavor Changing Neutral Currents Removing FCNCs: Type 1 and Type II HDM Charged Higgs Couplings in Type I and II models The charges Higgs Largrangian is [ ] ( ) L H ± = v H+ V ij ū i Xu m ui P L + X d m dj P R dj + m l X l v l P L l + H.C. where V ij is the CMK matrix The couplings X u, X d and X l are Type I Type II X u cot β cot β X d cot β tan β X l cot β tan β (36) Logan A. Morrison (UCSC) HDM March 18, / 7

26 Application Rare B meson decays Application The HDM makes testable predictions such as the decay width of B mesons ( B D τ ν or B Dτ ν) In the SM, theses decay are mediated via W s SM doesn t predict correct decay rate of the B In the HDM, this process can also occur through a charge Higgs boson It is still unclear if the HDM can explain the discrepancy in the decay rate of the B [3] τ B 0 b d H c d ν τ D + Logan A. Morrison (UCSC) HDM March 18, / 7

27 Application Rare B meson decays References G. Branco, P. Ferreira, L. Lavoura, M. Rebelo, M. Sher, and J.P. Silva, Physics Reports, 516(1-), (01) H.E. Haber, (013, March ). Toward a set of HDM benchmarks. Retrieved from haber/webpage/thdmbench.pdf A. Celis, M. Jung, X. Li, A. Pitch, B D τν τ decays in two-higgs-doublet models, arxiv: D.M. Asner, T. Barklow, C. Calancha, K. Fujii, N. Graf, H.E. Haber, A. Ishikawa, S. Kanemura, S. Kawada, M. Kurata, et al., ILC Higgs White Paper, arxiv: H.E. Haber, O. Stal, New LHC Benchmarks for the CP-conserving Two-Higgs-Doublet Model, arxiv: Logan A. Morrison (UCSC) HDM March 18, / 7