The Two-Higgs-Doublet Model: Past, Present and Future

Transcription

1 The Two-Higgs-Doublet Model: Past, Present and Future Augusto Barroso Fest Howard E. Haber 24 October 2008

2 Outline A (biased) history of the two-higgs-doublet model (2HDM) The paradox of tan β The general Two-Higgs-Doublet Model Basis-independent techniques Basis-independent (invariant) form of the Higgs couplings The Higgs-fermion interaction The significance of tan β? Many tan β-like parameters? CP symmetry and custodial symmetry Basis-independent conditions for a CP-conserving neutral Higgs sector Basis-independent conditions for custodial symmetry Implications for the T -paparmeter Lessons for future work

3 Highlights of the history of the 2HDM T.D. Lee, A Theory of Spontaneous T Violation, Phys. Rev. D8, 1226 (1973). The first motivated 2HDM: an attempt to find a new source of CP-violation. S.L. Glashow and S. Weinberg, Natural Conservation Laws For Neutral Currents, Phys. Rev. D15, 1958 (1977). To avoid neutral-higgs-mediated tree-level flavor changing neutral currents (FCNCs), all fermions of a given electric charge can couple to at most one Higgs doublet (in a model with multiple scalar doublets). N.G. Deshpande and E. Ma, Pattern Of Symmetry Breaking With Two Higgs Doublets, Phys. Rev. D18, 2574 (1978). Parameters of the Higgs potential had to lie in an appropriate region of parameter space to ensure that U(1) EM is not broken.

4 J.F. Donoghue and L. F. Li, Properties Of Charged Higgs Bosons, Phys. Rev. D19, 945 (1979). The inventors of the 2HDM with Type-II Higgs-fermion interactions: one Higgs doublet couples to up-type fermions and the other Higgs doublet couples to down-type fermions. H.E. Haber, G.L. Kane and T. Sterling, The Fermion Mass Scale And Possible Effects Of Higgs Bosons On Experimental Observables, Nucl. Phys. B161, 493 (1979). The inventors of the 2HDM with Type-I Higgs-fermion interactions: one Higgs doublet couples to both up-type and down-type fermions, and the other Higgs doublet does not couple at all to the fermions. L.J. Hall and M.B. Wise, Flavor Changing Higgs Boson Couplings, Nucl. Phys. B187, 397 (1981). The inventors of the Type-I and Type-II nomenclature.

5 T.P. Cheng and M. Sher, Mass Matrix Ansatz and Flavor Nonconservation in Models with Multiple Higgs Doublets, Phys. Rev. D35, 3484 (1987). The first realistic Type-III 2HDM (defined as a 2HDM with all possible Higgs-fermion couplings allowed). Other important 2HDM milestones the axion as the CP-odd scalar of a 2HDM [the Peccei-Quinn mechanism]. the requirement of a second Higgs doublet in the minimal supersymmetric extension of the Standard Model (MSSM). In a supersymmetric extension of a one-doublet Standard Model, the corresponding higgsinos are anomalous. Anomalies are canceled if the higgsino doublets come in pairs with opposite sign hypercharges. Influential early papers: Fayet; Inoue et al.; Flores and Sher; and Gunion and Haber.

6 Contributions to 2HDM Physics by A. Barroso and collaborators R. Santos and A. Barroso, Renormalization of two-higgs-doublet models, Phys.Rev. D56, 5366 (1997). Renormalization of the CP-conserving, FCNC preserving 2HDM [with Model I and II Yukawa couplings generalized to allow for different patterns of Higgs couplings to quarks and leptons]. J. Velhinho, R. Santos and A. Barroso, Tree level stability in two-higgs-doublet models, Phys. Lett. B322, 213 (1994). P.M. Ferreira, R. Santos and A. Barroso, Stablility of the tree-level vacuun in two- Higgs doublet models against charge of CP spontaneous violation, Phys. Lett. B603, 219 (2004) [Erratum: B629, 219 (2004)]. A. Barroso, P.M. Ferreira and R. Santos, Charge and CP symmetry breaking in two-higgs doublet models, Phys. Lett. B632, 684 (2006). A. Barroso, P.M. Ferreira and R. Santos, Neutral minima in two-higgs doublet models, Phys. Lett. B652, 181 (2007). A series of seminal papers on the vacuum structure of the 2HDM.

7 Basis-independent techniques for the 2HDM L. Lavoura and J.P. Silva, Fundamental CP violating quantities in a SU(2) U(1) model with many Higgs doublets, Phys.Rev.D50, 4619 (1994). F.J. Botella and J.P. Silva, Jarlskog-like invariants for theories with scalars and fermions, Phys.Rev.D51, 3870 (1995). Invariants that govern whether CP is violated (spontaneously or explicitly) in the 2HDM. S. Davidson and H.E. Haber, Basis-independent methods for the two-higgs-doublet model, Phys.Rev.D72, (2005) [Erratum: D72, (2005)]. J.F. Gunion and H.E. Haber, Conditions for CP-violation in the general two-higgsdoublet model, Phys.Rev.D72, (2005). H.E. Haber and D. O Neil, Basis-independent methods for the two-higgs-doublet model. II: The significance of tan β, Phys.Rev.D74, (2006) [Erratum: D74, (2006)]. A comprehensive basis-independent treatment of the 2HDM and an identification of the physical observables. Related work by Ivanov and by Nishi is especially notable.

8 The MSSM Higgs sector The Higgs sector of the MSSM (at tree-level) is a constrained Type-II 2HDM. One of the key parameters of the model is: tan β v u /v d, where v u [v d ] is the vacuum expectation value of the neutral Higgs boson that couples exclusively to up-type [down-type] fermions. But, one-loop radiative effects generate corrections to the tree-level structure of the model due to SUSY-breaking effects that enter in loops. In particular, for MSSM Higgs couplings to fermions, Yukawa vertex corrections modify the effective Lagrangian that describes the coupling of the Higgs bosons to the third generation quarks: L eff = ɛ ij [(h b + δh b ) b R H i dq j L +(h t + δh t ) t R Q i LH j u ] +Δh b br Q k LH k u +Δh t t R Q k LH k d +h.c. Thus, the MSSM Higgs-sector is actually a type-iii model.

9 For example, in some MSSM parameter regimes (corresponding to large tan β and large supersymmetry-breaking scale compared to v), Δh b h b [ 2αs 3π μm g I(M 2 b1,m 2 b2,m 2 g )+ ] h2 t 16π 2μA t I(M 2 t 1,M 2 t 2,μ 2 ) The tree-level relation between m b and h b is modified (first pointed out by Hempfling and later emphasized strongly by Carena, Olechowski, Pokorski and Wagner): 2mb h b = v cos β(1 + Δ b ), where Δ b (Δh b /h b )tanβ. That is, Δ b is tan β-enhanced, and governs the leading one-loop correction to the physical Higgs couplings to third generation quarks. In typical models at large tan β, Δ b can be of order 0.1 or larger and of either sign.. I(a, b, c) =[ab ln(a/b) +bc ln(b/c) +ca ln(c/a)]/(a b)(b c)(a c).

10 The paradox of tan β If the 2HDM is realized in nature, it is likely that its effective Lagrangian will consist of all possible dimension-four terms or less, consistent with the electroweak gauge invariance that is a general type-iii model. The general 2HDM consists of two identical (hypercharge-one) scalar doublets Φ 1 and Φ 2. One can always redefine the basis, so the parameter tan β v 2 /v 1 is not meaningful! Nevertheless, the literature is filled with 2HDM Feynman rules that depend on tan β and many phenomenological proposals to measure it! Hence, the paradox.

11 The parameter tan β makes sense only if there is a physical principle that distinguishes between Φ 1 and Φ 2. Such a principle is model-dependent. Any experimental study of 2HDM physics should avoid theoretical bias in defining their measurements. The theoretical interpretation should be a consequence of the observations. To determine the relevant physical quantities for measurements, one must develop basis-independent techniques. Inspired by a beautifully written chapter on the 2HDM by G. Branco, L. Lavoura and J.P. Silva, in CP Violation (Oxford University Press, Oxford, UK, 1999), my collaborators (S. Davidson, J.F. Gunion and D. O Neil) and I set out to develop the basis independent formalism of the 2HDM in order to identify the relevant invariant (basis-independent) quantities. In particular, O Neil and I were able to write down a complete set of Feynman rules that completely avoid the parameter tan β, while describing all the CP-violating and flavor-violating phenomena in an elegant form.

12 The General Two-Higgs-Doublet Model Consider the 2HDM potential in a generic basis: V = m 2 11 Φ 1 Φ 1 + m 2 22 Φ 2 Φ 2 [m 2 12 Φ 1 Φ 2 +h.c.] λ 1(Φ 1 Φ 1) λ 2(Φ 2 Φ 2) 2 + λ 3 (Φ 1 Φ 1)(Φ 2 Φ 2)+λ 4 (Φ 1 Φ 2)(Φ 2 Φ 1) { λ 5(Φ 1 Φ 2) 2 + [ λ 6 (Φ 1 Φ 1)+λ 7 (Φ 2 Φ 2) ] } Φ 1 Φ 2 +h.c. A basis change consists of a U(2) transformation Φ a U a bφ b (and Φ ā =Φ bu bā ). Rewrite V in a U(2)-covariant notation: V = Y a bφ āφ b Z a bc d(φ āφ b )(Φ cφ d ) where Z a bc d = Z c da b and hermiticity implies Y a b =(Y bā ) and Z a bc d =(Z bād c ). The barred indices help keep track of which indices transform with U and which transform with U. For example, Y a b U a c Y c du and Z d b a bc d U aē U U f b cḡu h d Z e fg h.

13 The most general U(1) EM -conserving vacuum expectation value (vev) is: Φ a = v 2 ( 0 v a ), with v a e iη ( cβ s β e iξ ), where v 2m W /g = 246 GeV. The overall phase η is arbitrary (and can be removed with a U(1) Y hypercharge transformation). If we define the hermitian matrix V a b ˆv aˆv b, then the scalar potential minimum condition is given by the invariant condition: Tr (VY)+ 1 2 v2 Z a bc dv bā V d c =0. The orthonormal eigenvectors of V a b are ˆv b and ŵ b v c ɛ cb (with ɛ 12 = ɛ 21 =1, ɛ 11 = ɛ 22 =0). Note that ˆv b ŵ b =0. Under a U(2) transformation, ˆv a U a bˆv b, but: ŵ a (det U) 1 U a b ŵ b, where det U e iχ is a pure phase. That is, ŵ a is a pseudo-vector with respect to U(2). One can use ŵ a to construct a proper second-rank tensor: W a b ŵ a ŵ b δ a b V a b. Remark: U(2) = SU(2) U(1) Y /Z 2. The parameters m 2 11, m2 22, m2 12,andλ 1,...,λ 7 are invariant under U(1) Y transformations, but change under a flavor -SU(2) transformation; whereas ˆv transforms under the full U(2) group.

14 A list of invariant and pseudo-invariant quantities Y 1 Tr (YV), Y 2 Tr (YW), Z 1 Z a bc d V bā V d c, Z 2 Z a bc d W bā W d c, Z 3 Z a bc d V bā W d c, Z 4 Z a bc d V b c W dā are invariants, whereas the following (potentially complex) pseudo-invariants Y 3 Y a b v ā ŵ b, Z 5 Z a bc d v ā ŵ b v c ŵ d, Z 6 Z a bc d v ā v b v c ŵ d, Z 7 Z a bc d v ā ŵ b ŵ c ŵ d. transform as [Y 3,Z 6,Z 7 ] (det U) 1 [Y 3,Z 6,Z 7 ] and Z 5 (det U) 2 Z 5. Physical quantities must be invariants. For example, the charged Higgs boson mass is m 2 H ± = Y Z 3v 2. Pseudo-invariants are useful because one can always combine two such quantities to create an invariant.

15 The invariants and pseudo-invariants in the generic basis are given by: Y 1 = m 2 11 c2 β + m2 22 s2 β Re(m2 12 eiξ )s 2β, Y 2 = m 2 11 s2 β + m2 22 c2 β +Re(m2 12 eiξ )s 2β, Y 3 e iξ = 1 2 (m2 22 m2 11 )s 2β Re(m2 12 eiξ )c 2β i Im(m 2 12 eiξ ), Z 1 = λ 1 c 4 β + λ 2 s4 β λ 345 s2 2β +2s 2β Z 2 = λ 1 s 4 β + λ 2 c4 β λ 345 s2 2β 2s 2β [ c 2 β Re(λ 6 eiξ )+s 2 ] β Re(λ 7 eiξ ) [ s 2 β Re(λ 6 eiξ )+c 2 ] β Re(λ 7 eiξ ) Z 3 = 1 4 s2 2β [λ 1 + λ 2 2λ 345 ]+λ 3 s 2β c 2β Re[(λ 6 λ 7 )eiξ ],,, Z 4 = 1 4 s2 2β [λ 1 + λ 2 2λ 345 ]+λ 4 s 2β c 2β Re[(λ 6 λ 7 )eiξ ], Z 5 e 2iξ = 1 4 s2 2β [λ 1 + λ 2 2λ 345 ]+Re(λ 5 e2iξ )+ic 2β Im(λ 5 e 2iξ ), Z 6 e iξ = 1 2 s 2β Z 7 e iξ = 1 2 s 2β s 2β c 2β Re[(λ 6 λ 7 )e iξ ] is 2β Im[(λ 6 λ 7 )e iξ )], [ λ 1 c 2 ] β λ 2 s2 β λ 345 c 2β iim(λ 5 e2iξ ) + c β c 3β Re(λ 6 e iξ ), +s β s 3β Re(λ 7 e iξ )+ic 2 β Im(λ 6 eiξ )+is 2 β Im(λ 7 eiξ ), [ λ 1 s 2 ] β λ 2 c2 β + λ 345 c 2β + iim(λ 5 e2iξ ) + s β s 3β Re(λ 6 e iξ ) +c β c 3β Re(λ 7 e iξ )+is 2 β Im(λ 6 eiξ )+ic 2 β Im(λ 7 eiξ ). where λ 345 λ 3 + λ 4 +Re(λ 5 e 2iξ ).

16 The Higgs basis and parameter counting Define new Higgs-doublet fields: H 1 ˆv â Φ a and H 2 ŵ â Φ a Then, H 0 1 = v 2, H 0 2 =0, where v = 246 GeV. Note that H 0 1 is an invariant field, where H0 2 is pseudo-invariant (corresponding to a possible rephasing of H 2 ). The Higgs potential in this basis is: V = Y 1 H 1 H 1 + Y 2 H 2 H 2 +[Y 3 H 1 H 2 +h.c.] Z 1(H 1 H 1) Z 2(H 2 H 2) 2 + Z 3 (H 1 H 1)(H 2 H 2)+Z 4 (H 1 H 2)(H 2 H 1) { Z 5(H 1 H 2) 2 + [ Z 6 (H 1 H 1)+Z 7 (H 2 H 2) ] } H 1 H 2 +h.c., where the coefficients of V correspond to the (pseudo-)invariants introduced previously. The potential minimum conditions are: Y 1 = 1 2 Z 1v 2 and Y 3 = 1 2 Z 6v 2. Thus, the independent degrees of freedom of the model comprise nine real parameters, Y 1 (or equivalently v), Y 2, Z 1, Z 2, Z 3,andZ 4, Z 5, Z 6 and Z 7, and two relative phases, arg(z 5 Z2 6 ) and arg(z 5 Z2 7 ). This yields 11 real parameters that are required to specify the most general 2HDM.

17 The Higgs mass-eigenstate basis The three physical neutral Higgs boson mass-eigenstates are determined by diagonalizing a 3 3 squared-mass matrix that is defined in a basis in which only one of the neutral Higgs bosons has a vacuum expectation value (the so-called Higgs basis ). The diagonalizing matrix is a 3 3 real orthogonal matrix that depends on three angles: θ 12, θ 13 and θ 23. Under a U(2) transformation, θ 12,θ 13 are invariant, and e iθ 23 (det U) 1 e iθ 23. One can express the mass eigenstate neutral Higgs directly in terms of the original shifted neutral fields, Φ 0 a Φ 0 a v v a / 2: h k = 1 [ ] Φ 0 ā (q k1 v a + q k2 ŵ a e iθ 23 )+(qk1 v ā + qk2ŵ āe iθ 23 )Φ 0 a, 2 for k =1,...,4, whereh 4 = G 0. The invariant quantities q kj are given by:

18 k q k1 q k2 1 c 12 c 13 s 12 ic 12 s 13 2 s 12 c 13 c 12 is 12 s 13 3 s 13 ic 13 4 i 0 Since ŵ a e iθ 23 is a proper U(2)-vector, we see that the mass-eigenstate fields are indeed U(2)-invariant fields. Inverting the previous result yields: Φ a = v a v 2 G + v a + H + ŵ a k=1 ( qk1 v a + q k2 e iθ 23ŵ a ) hk. If Im (Z5Z 6)=0, 2 then the neutral scalar squared-mass matrix can be transformed into block diagonal form, containing the squared-mass of a CP-odd neutral Higgs mass-eigenstate and a 2 2 sub-matrix that yields the squared-masses of two CP-even neutral Higgs mass-eigenstates.

19 If Im (Z5Z 6) 2 0,wecanwriteZ 6 Z 6 e iθ 6. Then the neutral scalar mass-eigenstates do not possess definite CP quantum numbers, and the three invariant mixing angles θ 12, θ 13 and φ 6 θ 6 θ 23 are non-trivial. The angles θ 13 and φ 6 are determined modulo π from tan θ 13 = Im(Z 5 e 2iθ 23) 2Re(Z 6 e iθ 23 ), tan 2θ 13 = 2Im(Z 6 e iθ23) Z 1 A 2 /v 2, where A 2 Y [Z 3 + Z 4 Re(Z 5 e 2iθ 23)]v 2. These equations exhibit multiple solutions (modulo π) corresponding to different orderings of the h k masses. Finally, tan 2θ 12 = 2cos2θ 13 Re(Z 6 e iθ 23) c 13 [c 2 13 (A2 /v 2 Z 1 )+cos2θ 13 Re(Z 5 e 2iθ 23 )]. For a given solution of θ 13 and φ 6, the two solutions for θ 12 (modulo π) correspond to the two possible relative mass orderings of h 1 and h 2.

20 The gauge boson Higgs boson interactions ( L VVH = gm W W μ + W μ + gm Z s 2 W Zμ (W + μ G + W μ G+ ), g ) m 2c Z ZμZ μ Re(q k1 )h k + em W A μ (W μ + G + Wμ G+ ) W L VVHH = 1 4 g 2 W μ + W μ + g2 8c 2 ZμZ μ Re(qj1 q k1 + q j2 q k2 ) h j h k W g 2 W μ + W μ + e 2 AμA μ + g2 ( 12 c 2 s 2 ) 2 W Z μz μ + 2ge ( 12 s 2 ) c W AμZ μ (G + G + H + H ) W W { ega μ W μ + g2 s 2 W Z μ W + 2c μ (q k1 G + q k2 e iθ 23H } )h k +h.c., W L VHH = g Im(q 4c j1 q { k1 + q j2 q k2 )Zμ h j μ h k 1 2 g iw μ + [q k1 G μ hk + q k2 e iθ 23H ] μ hk W [ + iea μ + ig ] ( 12 s 2 ) c W Z μ (G + μ G + H + μ H ). W } +h.c.

21 The cubic and quartic Higgs couplings [ L 3h = 1 2 vh j h k h l q j1 qk1 Re(q l1 )Z 1 + q j2 q k2 Re(q l1 )(Z 3 + Z 4 )+Re(q j1 q k2 q l2 Z 5 e 2iθ 23) ( +Re [2q j1 + qj1 ]q k1 q l2 Z 6 e iθ 23 ) +Re(qj2 ] q k2 q l2 Z 7 e iθ 23) vh k G + G [ Re(q k1 )Z 1 +Re(q k2 e iθ ] 23Z 6 ) + vh k H + H [ Re(q k1 )Z 3 +Re(q k2 e iθ ] 23Z 7 ) 1 2 vh k { G H + e iθ 23 [ qk2 Z 4 + q k2 e 2iθ 23Z 5 +2Re(q k1 )Z 6 e iθ 23 ] } +h.c., [ L 4h = 1 8 h j h k h l h m q j1 q k1 ql1 q m1 Z 1 + q j2 q k2 q l2 q m2 Z 2 +2q j1 q k1 q l2 q m2 (Z 3 + Z 4 ) +2Re(qj1 q k1 q l2 q m2 Z 5 e 2iθ 23) +4Re(q j1 qk1 q l1 q m2 Z 6 e iθ 23) +4Re(qj1 ] q k2 q l2 q m2 Z 7 e iθ 23) 2 1h j h k G+ G [ q j1 qk1 ] Z 1 + q j2 q k2 Z 3 +2Re(q j1 q k2 Z 6 e iθ 23) 2 1h j h k H+ H [ q j2 qk2 ] Z 2 + q j1 q k1 Z 3 +2Re(q j1 q k2 Z 7 e iθ 23) 1 2 h j h k { G H + e iθ 23 [ q j1 qk2 Z 4 + q j1 q k2 Z 5 e 2iθ 23 + q j1 qk1 Z 6 e iθ 23 + q j2 qk2 Z 7 e iθ 23 ] } +h.c. 1 2 Z 1 G+ G G + G 1 2 Z 2 H+ H H + H (Z 3 + Z 4 )G + G H + H 1 2 (Z 5 H+ H + G G + Z 5 H H G + G + ) G + G (Z 6 H + G + Z 6 H G + ) H + H (Z 7 H + G + Z 7 H G + ).

22 Example: Higgs self-couplings Lightest neutral Higgs boson cubic self-coupling: g(h 1 h 1 h 1 )= 3v { Z 1 c 3 12 c3 13 +(Z 3 + Z 4 )c 12 c 13 s c 12 c 13 Re(s Z 5e 2iθ 23 ) 3c 2 12 c2 13 Re(s 123Z 6 e iθ 23 ) s Re(s 123 Z 7 e iθ 23 ) } Lightest neutral Higgs boson quartic self-coupling: g(h 1 h 1 h 1 h 1 )= 3 { Z 1 c 4 12 c Z 2 s (Z 3 + Z 4 )c 2 12 c2 13 s c 2 12 c2 13 Re(s2 123 Z 5e 2iθ 23 ) 4c 3 12 c3 13 Re(s 123Z 6 e iθ 23 ) 4c 12 c 13 s Re(s 123 Z 7 e iθ 23 ) } where s 123 s 12 + ic 12 s 13. Note that these quantities depend on U(2)-invariants. In particular Z 5 e 2iθ 23, Z 6 e iθ 23 and Z 7 e iθ 23 are U(2)-invariants!

23 The Higgs-fermion Yukawa couplings The Yukawa Lagrangian can be written in terms of the quark mass-eigenstate fields as: L Y = U L Φ0 ā ηu a U R + D L K Φ ā ηu a U R + U L KΦ + a ηd ā D R + D L Φ 0 a ηd ā D R +h.c., where Φā ( Φ 0, Φ )=iσ 2 Φ ā and K is the CKM mixing matrix. The ηu,d are 3 3 Yukawa coupling matrices. We can construct invariant and pseudo-invariant matrix Yukawa couplings: κ Q v ā ηq a, ρq ŵ ā ηq a, where Q = U or D. Inverting these equations yields: η Q a = κq v a + ρ Q ŵ a. Under a U(2) transformation, κ Q is invariant, whereas ρ Q (det U)ρ Q. By construction, κ U and κ D are proportional to the (real non-negative) diagonal quark mass matrices M U and M D, respectively. In particular, M U = v 2 κ U =diag(m u,m c,m t ), M D = v 2 κ D =diag(m d,m s,m b ). The matrices ρ U and ρ D are independent complex 3 3 matrices.

24 The final form for the Yukawa couplings of the mass-eigenstate Higgs bosons and the Goldstone bosons to the quarks is: L Y = 1 { v D M D (q k1 P R + q k1 P L)+ v ] [q } k2 [e iθ 23 ρ D ] P R + q k2 eiθ 23 ρ D P L Dh k { v U M U (q k1 P L + q k1 P R)+ v ] [q } k2 eiθ 23 ρ U P R + q k2 [e iθ 23 ρ U ] P L Uh k 2 { + U [K[ρ D ] P R [ρ U ] KP L ] DH v } U [KM DP R M U KP L ] DG + +h.c.. By writing [ρ Q ] H + =[ρ Q e iθ 23] [e iθ 23H + ], we see that the Higgs-fermion Yukawa couplings depend only on invariant quantities: the diagonal quark mass matrices, ρ Q e iθ 23, and the invariant angles θ 12 and θ 13. The couplings of the neutral Higgs bosons to quark pairs are generically CP-violating as a result of the complexity of the q k2 and the fact that the matrices e iθ 23ρ Q are not generally hermitian or anti-hermitian. L Y also exhibits Higgs-mediated flavor-changing neutral currents (FCNCs) at tree-level by virtue of the fact that the ρ Q are not flavor-diagonal. Thus, for a phenomenologically acceptable theory, the off-diagonal elements of ρ Q must be small.

25 The significance of tan β So far, tan β has been completely absent from the Higgs couplings. This must be so, since tan β is basis-dependent in a general 2HDM. However, a particular 2HDM may single out a preferred basis, in which case tan β would be promoted to an observable. To simplify the discussion, we focus on a one-generation model, where the Yukawa coupling matrices are simply numbers. As an example, the MSSM Higgs sector is a type-ii 2HDM, i.e., η U 1 = ηd 2 =0. A basis-independent condition for type-ii is: = 0. In the preferred basis, η D ā ηu a ˆv =(cosβ, sin βe iξ ) and ŵ =( sin βe iξ, cos β). Evaluating κ Q =ˆv η Q and ρ Q = ŵ η Q in the preferred basis, it follows that: e iξ tan β = ρd κ D = κu ρ U, where κ Q = 2m Q /v. These two definitions are consistent if κ D κ U + ρ D ρ U =0is satisfied. But this is equivalent to the type-ii condition, η D ā ηu a =0.

26 Since ρ Q is a pseudo-invariant, we can eliminate ξ by rephasing Φ 2. Hence, tan β = ρd κ D = κu ρ U, with 0 β π/2. Indeed, tan β is now a physical parameter, and the ρ Q are no longer independent: ρ D = 2md tan β v, ρ U = 2mu cot β v. In the more general (type-iii) 2HDM, tan β is not a meaningful parameter. Nevertheless, one can introduce three tan β-like parameters: tan β d ρd κ D, tan β u κu ρ U, tan β e ρe κ E, the last one corresponding to the Higgs-lepton interaction. In a type-iii 2HDM, there is no reason for the three parameters above to coincide. Interpretation: In the Higgs basis, up and down-type quarks interact with both Higgs doublets. But, clearly there exists some basis (i.e., a rotation by angle βu from the Higgs basis) for which only one of the two up-type quark Yukawa couplings is non-vanishing. This defines the physical angle βu.

27 The MSSM Higgs sector is a type-iii 2HDM Recall the effective one-loop Higgs-fermion Yukawa couplings in the MSSM are of the form: L eff = ɛ ij [(h b + δh b ) b R Hd i Qj L +(h t + δh t ) t R Q i ] L Hj u +Δh b br Q k L Hk u +Δh t t R Q k L Hk d +h.c. For illustrative purposes, we neglect CP violation in the following simplified discussion. Keeping only the leading tan β-enhanced terms, Δ b (Δh b /h b )tanβ, tan β b vρd tan β, tan β t 2 mb 1+Δ b 2 mt vρ U tan β 1 tan β (Δh t /h t ). Thus, supersymmetry-breaking loop-effects can yield observable differences between tan β-like parameters that are defined in terms of basis-independent quantities. In particular, the leading one-loop tan β-enhanced corrections are automatically incorporated into: g Ab b = m b v tan β b, g At t = m t v cot β t.

28 Conditions for neutral Higgs CP-conservation Im (Z5Z 6)=Im(Z 2 5Z 7)=Im(Z 2 5(Z 6 + Z 7 ) 2 )=0. In this case a real basis exists in which all potentially complex coefficients of the scalar potential in the Higgs basis are real (as the scalar potential minimum condition fixes Y 3 = 1 2 Z 6v 2 ). Z 5 (ρ Q ) 2, Z 6 ρ Q and Z 7 ρ Q are hermitian (Q = U, D and E). This guarantees that the couplings of the neutral Higgs boson to fermion pairs are CP-invariant. If the two conditions above are satisfied, then the neutral Higgs bosons are eigenstates of CP, and the only source of CP-violation is the unremovable phase in the CKM matrix that enters via the charged current interactions mediated by either W ±, H ± or G ± exchange.

29 A singular point in the parameter space of CP-conserving 2HDMs: Y 3 = Z 6 = Z 7 =0. One neutral Higgs boson (call it h 0 1) is CP-even, with couplings identical to the SM Higgs boson. The other two neutral Higgs bosons (h 0 2 and h 0 3) have opposite CP quantum numbers, but the Higgs self-interactions and Higgs boson-vector boson interactions do not determine which of these two neutral Higgs bosons is the CP-odd state. To identify the CP-odd state, we must examine the Higgs-fermion Yukawa couplings. CP-invariance requires that Im(Z 5 e 2iθ 23) =0and Z 5 (ρ Q ) 2 is hermitian. It then follows that: (e iθ 23 ρ Q ) = ± Z 5 Z 5 e iθ 23 ρ Q = ±e iθ 23 ρ Q. For one choice of sign, h 0 3 is the CP-odd state, whereas for the other choice of sign, h 0 2 is the CP-odd state.

30 Conditions for custodial symmetry In the Standard model, the scalar sector exhibits a global SU(2) L SU(2) R symmetry that is violated only by hypercharge gauge interactions and the Higgs-fermion Yukawa couplings. This global symmetry would be exact in the limit of g =0and h t = h b. In the custodial symmetric limit the electroweak ρ-parameter, ρ m 2 W m 2 Z cos θ W =1, to all orders in perturbation theory. In models with only Higgs doublets, with g 0and h t h b, radiative corrections generate corrections to the tree-level relation, ρ =1. Pomarol and Vega studied the implications of custodial symmetry for the 2HDM. They identified to separate realizations, but failed to realize that their two cases were actually related by a change of Higgs basis! Clearly, basis-independent methods can be valuable here!

31 In the 2HDM, custodial symmetry implies that the Higgs sector is CPconserving. In addition, it imposes one extra basis-independent condition: where Z 4 = ε 56 Z 5, if Z 6 0, ε 57 Z 5, if Z 7 0, ± Z 5, if Y 3 = Z 6 = Z 7 =0. ε 56 Z 5Z6 2 Z 5 Z 6 =+1 or 1, ε 2 57 Z 5Z 2 7 =+1 or 1, Z 5 Z 7 2 for the cases of Z 6 0and Z 7 0, respectively. Note that if either Z 6 or Z 7 is non-zero, then in the real basis, custodial symmetry implies that Z 4 = Z 5 (the sign of Z 5 is invariant in this case under O(2) transformations between any two real bases). In contrast, if Y 3 = Z 6 = Z 7 =0, then one can transform H 2 ih 2 and change the sign of Z 5 without leaving the real Higgs basis. Hence, in this case custodial symmetry implies Z 4 = ± Z 5.

32 ThechargedHiggsbosonmassisgivenby M 2 H ± = Y Z 3. If A 0 is the CP-odd Higgs boson, one finds: m 2 A = Y (Z 3 + Z 4 ε 56 Z 5 ), if Z 6 0, Y (Z 3 + Z 4 ε 57 Z 5 ), if Z 7 0. Hence custodial symmetry implies that m H ± = m A, if Z 6 0or Z 7 0. If Y 3 = Z 6 = Z 7 =0,then m 2 h 2,h 3 = Y v2 (Z 3 + Z 4 Z 5 ), in which case custodial symmetry implies that H ± is mass-degenerate with either h 2 or h 3. As previously noted, either h 2 or h 3 can be CP-even, depending on the sign choice in the relation (e iθ 23ρ Q ) = ±e iθ 23ρ Q. Thus,forthecaseofY 3 = Z 6 = Z 7 =0, imposing the custodial symmetry can yield m 2 H ± = m 2 H,whereH is a CP-even Higgs boson! This is the twisted scenario of Gerard and Herquet.

33 If the custodial symmetry is violated, then one-loop radiative corrections can shift the tree-level result of ρ =1. Denoting αt δρ = ρ 1, we find that the contribution of a general (possibly CP-violating) Higgs sector to the T parameter [Haber and O Neil, in preparation] is given by the basis independent result: T = g 2 [ 3 q k2 2 F (m 2 k,m2 H ± ) q 2 k1 F (m2 i,m2 j ) ]+O(g 2 ), i j k, 64π 2 m 2 W k=1 where m k m hk and F (x, y) 1 2 (x + y) xy x y ln(x/y), F(x, x) =0. This result is consistent with a recent computation of Grimus, Lavoura, Ogreid and Osland. One can check that in the custodial symmetric limit where g =0and m H ± = m A (or m H ± = m H in the special case of Y 3 = Z 6 = Z 7 =0), the Higgs contributions to T vanish exactly!

34 Lessons for future work If phenomena consistent with the 2HDM are found, we will not know a priori the underlying structure that governs the model. In this case, one needs a model-independent analysis of the data that allows for the most general CP-violating Model-III. Instead of claiming that you have measured tan β (unless you wish to test a specific theoretical framework), measure the physical parameters of the model. Examples include the tan β-like parameters introduced in the one-generation model. (For three generations, the formalism becomes more complicated. However, one has good reason to assume that the third generation quark Higgs Yukawa couplings dominate.) Which tan β-like parameters will be measured in precision Higgs studies at the ILC? How can one best treat the full three-generation model at one-loop order?

35 Even in the MSSM where tan β at tree-level is physically well-defined, the scheme presented here might be useful in achieving a more direct connection between model parameters and physical observables (when radiative corrections are incorporated). The basis-independent formalism is very powerful in identifying physical observables. It is also provides a valuable tool for studying additional underlying symmetries that can constrain the model. For example, it provides important insights into the nature of custodial symmetry, CPsymmetry, and other possible discrete symmetries of the 2HDM. Best wishes, Augusto, on your special day. May you have a fruitful and Higgsful retirement!