Two Higgs doublets with 4th generation fermions - models for TeV-scale compositeness
aa r X i v : . [ h e p - ph ] M a y Udem-GPP-TH-11-199
Two Higgs doublets with 4th generation fermions - models for TeV-scalecompositeness
Shaouly Bar-Shalom ∗ Physics Department, Technion-Institute of Technology, Haifa 32000, Israel
Soumitra Nandi † Physique des Particules, Universit´e de Montr´eal, C.P. 6128,succ. centre-ville, Montr´eal, QC, Canada H3C 3J7
Amarjit Soni ‡ Theory Group, Brookhaven National Laboratory, Upton, NY 11973, USA (Dated: November 11, 2018)We construct a class of two Higgs doublets models with a 4th sequential generation of fermionsthat may effectively accommodate the low energy characteristics and phenomenology of a dynamicalelectroweak symmetry breaking scenario which is triggered by the condensates of the 4th familyfermions. In particular, we single out the heavy quarks by coupling the “heavier” Higgs doublet(Φ h ) which possesses a much larger VEV only to them while the “lighter” doublet (Φ ℓ ) couples onlyto the light fermions. We study the constraints on these models from precision electroweak data aswell as from flavor data. We also discuss some distinct new features that have direct consequenceson the production and decays of the 4th family quarks and leptons in high energy colliders; inparticular the conventional search strategies for t ′ and b ′ may need to be significantly revised. PACS numbers:
I. INTRODUCTION
One of the most studied, yet unresolved theoretical puzzles in modern particle physics is the origin of ElectroWeaksymmetry breaking (EWSB). Indeed, it is widely anticipated that the LHC will provide us with crucial answersregarding the underlying nature of EWSB: is the Higgs a fundamental scalar needing protection from SUSY or is it acomposite object. In the Standard Model (SM), EWSB is triggered by the Higgs mechanism, which assumes a singlefundamental scalar, the Higgs, with a mass at the EW-scale. This leads to the long standing difficulty known as thehierarchy problem: the presence of a fundamental EW-scale seems unnatural since there is a problem of stabilizingthe Higgs mass against radiative corrections without introducing a cutoff to the theory at the nearby TeV scale. Thehierarchy problem, which is usually being interpreted as evidence for new TeV-scale physics, has fueled much scientificeffort in the past several decades, both in theory and in experiment.Furthermore, recent flavor physics studies have revealed some degree of tension in the CKM fits for the SM with3 generations [1–5]. For example, there are indications that the “predicted” value of sin 2 β is larger than the valuemeasured directly via the “gold-plated” ψK s mode by as much as ∼ . σ [6]. On the other hand, the announcedCDF and DO results on the CP asymmetry S ψφ in B s → ψφ (at a higher luminosity around 6/fb) are larger than theSM prediction by about 1 σ [7], and at the same time, they find a surprisingly large CP-asymmetry in the same-signdimuons signal, which they attribute primarily to a ssl - the semileptonic asymmetry in B s → X s µν [8, 9].Interestingly, perhaps the simplest variant of the SM, known as the SM4, in which only a 4th sequential generationof fermion doublets is added to the theory (for reviews see [10–12]) can address some of the theoretical challengesassociated with the hierarchy problem [13–15] and can readily account for the CKM anomalies mentioned above[16–24]. In particular, as was suggested over two decades ago, a heavy 4th generation fermion may trigger dynamicalEWSB [13]. The picture that arises in this scenario is of new heavy fermions which have large Yukawa couplingsthat are driven to a Landau pole or a fixed point (which acts as a cutoff), possibly at the nearby TeV scale [14, 15].Consequently, some form of strong dynamics and/or compositeness may occur and the Higgs particles are viewed ascomposites primarily of the 4th generation fermions (see e.g., [25–27]), with condensates < Q ′ L t ′ R > = 0, < Q ′ L b ′ R > = 0 ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] (and possibly also < L ′ L ν ′ R > = 0, < L ′ L τ ′ R > = 0), which induce EWSB and generate a dynamical mass for thecondensing fermions. As for the CKM anomalies, the two extra phases that the SM4 possesses can give rise to a hostof non-standard CP asymmetries [28, 29] and, in addition, can significantly ameliorate the difficulties with regard tobaryogenesis that the SM has [28, 30, 31].Indeed, recent searches for 4th generation heavy quarks by the CDF Collaboration have found that m t ′ , m b ′ > ∼ II. TWO HIGGS DOUBLET MODELS FOR THE 4TH GENERATION FERMIONS - 4G2HDMS
Recall that in a type II 2HDM (see [38]) one Higgs doublet couples only to the the up-quarks while the 2nd Higgsdoublet couples to the down-quarks. It is straight forward to extend such a setup to the case of a 4th generationfermion doublet - this was considered in [36, 39, 40] and within a SUSY framework in [30, 41–43].Our aim here is to construct a new class of two Higgs doublet models (2HDMs) that can serve as a viable low energyeffective framework for models of 4th generation condensation. Thus, in analogy with the 2HDM setup proposed in[44], we construct our 4G2HDMs using different Yukawa textures than the “standard” 2HDM of type II. In particular,in our 4G2HDMs one of the Higgs fields (call it the “heavier” field) couples only to heavy fermionic states, while thesecond Higgs field (the “lighter” field) is responsible for the mass generation of all other (lighter) fermions. In thisway, the heavier field may be viewed as a ¯ q ′ q ′ composite with a condensate < q ′ q ′ > = 0.The Higgs potential is a general 2HDM one [38] and the Yukawa interaction Lagrangian of the quark sector isdefined as: L Y = − ¯ Q L (cid:16) Φ ℓ F · (cid:16) I − I α d β d d (cid:17) + Φ h F · I α d β d d (cid:17) d R − ¯ Q L (cid:16) ˜Φ ℓ G · (cid:0) I − I α u β u u (cid:1) + Φ h G · I α u β u u (cid:17) u R + h.c. , (1)where f L ( R ) are left(right)-handed fermion fields, Q L is the left-handed SU (2) quark doublet and F, G are general4 × ℓ,h are the Higgs doublets:Φ i = φ + iv i + φ i √ ! , ˜Φ i = v ∗ i + φ ∗ i √ − φ − i ! ,I is the identity matrix and I α q β q q ( q = d, u ) are diagonal 4 × I α q β q q ≡ diag (0 , , α q , β q ).The Yukawa texture of (1) can be realized in terms of a Z -symmetry under which the fields transform as follows:Φ ℓ → − Φ ℓ , Φ h → +Φ h , Q L → + Q L ,d R → − d R ( d = d, s ) , u R → − u R ( u = u, c ) ,b R → ( − α d b R , b ′ R → ( − β d b ′ R ,t R → ( − α u t R , t ′ R → ( − β u t ′ R . (2)One can thus construct several models in which the Yukawa interactions of the heavy fermionic states have a non-trivial structure, possibly associated with the compositeness scenario. Three particularly interesting models which wewill study in this paper are: • : ( α d , β d , α u , β u ) = (0 , , , h gives masses only to t ′ and b ′ , while Φ ℓ generatesmasses for all other quarks (including the top-quark). • : ( α d , β d , α u , β u ) = (1 , , , h is responsible for the massgeneration of the heavy quarks states of both the 3rd and 4th generation quarks, whereas Φ ℓ generates massesfor the light quarks of the 1st and 2nd generations. • : ( α d , β d , α u , β u ) = (0 , , , m t , m b ′ , m t ′ ∝ v h , so that only quarks with masses atthe EW-scale are coupled to the heavy doublet Φ h .The above 3 models represent, in our view, the minimal set of multi-Higgs frameworks that capture the compos-iteness scenarios associated with the heavy 4th generation fermions. Defining tan β ≡ v h /v ℓ , in the 4G2HDM-I weexpect tan β ∼ m q ′ /m t ∼ O (1) ( q ′ = t ′ or b ′ ), while for the 4G2HDM-II and 4G2HDM-III models, tan β ≫ v h ≫ v ℓ . However, there is a fundamental difference between our 4G2HDMs andthe Das and Kao 2HDM: the Das and Kao model which was constructed with three fermion generations has nonew heavy fermions (the heavier Higgs doublet, Φ h , couples only to the top-quark). Thus, without the new heavyfermionic degrees of freedom, the top-quark Yukawa coupling remains perturbative up to the Planck scale, so thattheir 2HDM does not have a natural low-energy cutoff as one would expect for the condensation picture. On the otherhand, in our 4G2HDMs the strong Yukawa couplings of the heavier Higgs field to the new heavy 4th family fermionsreaches a Landau pole at the near by TeV-scale, thus signaling new physics - possibly in the form of compositeness.Alternatively, our framework might be more naturally embedded into weakly coupled theories in 5 dimensions, seee.g., [25, 45].From the point of view of the leptonic sector, the type-I 4G2HDM is the more natural underlying setup that caneffectively accommodate the heavy masses of the 4th generation neutrino ν ′ . In particular, recall that the currentbounds on m ν ′ [49] indicate that ν ′ should have a mass at least at the EW-scale. The main glaring problem for theSM4 is the fact that it does not address the origin of such a heavy mass for ν ′ [48]. On the other hand, within our4G2HDM-I the heaviness of the 4th generation leptons (with respect to the lighter three generations) is effectivelyaccommodated by coupling them to the heavy Higgs doublet. This setup for the leptonic sector might also be aneffective underlying description of more elaborate construction in models of warped extra dimensions, see e.g., [45].The physical Higgs fields H ± and h, H, A ( h and H are the lighter and heavier CP-even neutral states, respectively,and A is the neutral CP-odd state) are obtained by diagonalizing the neutral and charged Higgs mass matrices:Φ + ℓ = c β G + − s β H + , Φ − h = s β G + + c β H + , Φ ℓ = c α H − s α h + i (cid:0) c β G − s β A (cid:1) , Φ h = s α H + c α h + i (cid:0) s β G + c β A (cid:1) , (3)where G + , G are the goldstone bosons, c β , s β ≡ cos β, sin β , c α , s α ≡ cos α, sin α and α is the mixing angle in theCP-even neutral Higgs sector.The Yukawa interactions between the physical Higgs bosons and quark states are then given by: L ( hq i q j ) = g m W ¯ q i (cid:26) m q i s α c β δ ij − (cid:18) c α s β + s α c β (cid:19) · (cid:2) m q i Σ qij R + m q j Σ q⋆ji L (cid:3)(cid:27) q j h , (4) L ( Hq i q j ) = g m W ¯ q i (cid:26) − m q i c α c β δ ij + (cid:18) c α c β − s α s β (cid:19) · (cid:2) m q i Σ qij R + m q j Σ q⋆ji L (cid:3)(cid:27) q j H , (5) L ( Aq i q j ) = − iI q gm W ¯ q i (cid:8) m q i tan βγ δ ij − (tan β + cot β ) · (cid:2) m q i Σ qij R − m q j Σ q⋆ji L (cid:3)(cid:9) q j A , (6) L ( H + u i d j ) = g √ m W ¯ u i (cid:8)(cid:2) m d j tan β · V u i d j − m d k (tan β + cot β ) · V ik Σ dkj (cid:3) R + (cid:2) − m u i tan β · V u i d j + m u k (tan β + cot β ) · Σ u⋆ki V kj (cid:3) L (cid:9) d j H + , (7)where q = d or u for down or up-quarks with weak Isospin I d = − and I u = + , respectively, and R ( L ) = (1 + ( − ) γ ). Also, V is the 4 × d (Σ u ) are new mixing matrices in the down(up)-quark sectors,obtained after diagonalizing the quarks mass matrices:Σ dij = Σ dij ( α d , β d , D R ) = α d D ⋆R, i D R, j + β d D ⋆R, i D R, j , Σ uij = Σ uij ( α u , β u , U R ) = α u U ⋆R, i U R, j + β u U ⋆R, i U R, j , (8)where D R , U R are the rotation (unitary) matrices of the right-handed down and up-quarks, respectively. Notice thatΣ u and Σ d depend only on the elements of the 3rd and 4th rows of U R and D R , respectively, and on whether α q and/or β q are “turned on”. For example, in model 4G2HDM-I, for which ( α d , β d , α u , β u ) = (0 , , , U R and D R are relevant.Recall that in standard frameworks such as the single-Higgs SM4 or 2HDMs of types I and II [38], the right-handed mixing matrices U R and D R are non-physical in the sense that they are “rotated away” in the diagonalizationprocedure of the quark masses. On the other hand, in our 4G2HDMs some elements of these matrices can, in principle,be measured in Higgs-fermion systems, as we will later show. One can, thus, treat these matrices as unknowns, byexpressing physical observables in terms of the elements of the 3rd and 4th rows of U R and D R , or study thereproperties under some theoretically motivated parameterization. In particular, inspired by the working assumptionof our 4G2HDMs and by the observed flavor pattern in the up and down-quark sectors, we may assume the followingstructure (see also [44] for the 3 × D R = cos θ ds − sin θ ds sin θ ds cos θ bb ′ ǫ ⋆s − cos θ ds cos θ bb ′ ǫ ⋆s sin θ ds cos θ ds − sin θ ds sin θ bb ′ ǫ ⋆s e − iδ b cos θ ds sin θ bb ′ ǫ ⋆s e − iδ b ǫ s cos θ bb ′ − sin θ bb ′ e − iδ b θ bb ′ e iδ b cos θ bb ′ , (9) U R = cos θ uc − sin θ uc sin θ uc cos θ tt ′ ǫ ⋆c − cos θ uc cos θ tt ′ ǫ ⋆c sin θ uc cos θ uc − sin θ uc sin θ tt ′ ǫ ⋆c e − iδ t cos θ uc sin θ tt ′ ǫ ⋆c e − iδ t ǫ c cos θ tt ′ − sin θ tt ′ e − iδ t θ tt ′ e iδ t cos θ tt ′ , (10)where ǫ s = m s m b e iδ s and ǫ c = m c m t e iδ c , so that unitarity of D R and U R is restored at 1st order in ǫ s and ǫ c , respectively.In the limit sin θ uc ∼ m u /m c << θ ds ∼ m d /m s << [1] U R and D R simplify to (similar textures can befound in Randall-Sundrum warped models of flavor [46, 47]): D R = − ǫ ⋆s (cid:16) − | ǫ b | (cid:17) ǫ ⋆s ǫ ⋆b ǫ s (cid:16) − | ǫ b | (cid:17) − ǫ ⋆b ǫ b (cid:16) − | ǫ b | (cid:17) , U R = − ǫ ⋆c (cid:16) − | ǫ t | (cid:17) ǫ ⋆c ǫ ⋆t ǫ c (cid:16) − | ǫ t | (cid:17) − ǫ ⋆t ǫ t (cid:16) − | ǫ t | (cid:17) , (11)where we have further defined ǫ b = sin θ bb ′ e iδ b , ǫ t = sin θ tt ′ e iδ t . (12)We thus obtain for the Σ mixing matrices in Eq. 8 (in each element keeping only the leading terms in ǫ q , q = s, c, b, t ):Σ d = α d | ǫ s | α d ǫ ⋆s (cid:16) − | ǫ b | (cid:17) − α d ǫ ⋆s ǫ ⋆b α d ǫ s (cid:16) − | ǫ b | (cid:17) α d (cid:16) − | ǫ b | (cid:17) + β d | ǫ b | ( β d − α d ) ǫ ⋆b (cid:16) − | ǫ b | (cid:17) − α d ǫ s ǫ b ( β d − α d ) ǫ b (cid:16) − | ǫ b | (cid:17) α d | ǫ b | + β d (cid:16) − | ǫ b | (cid:17) , (13)and similarly for Σ u by replacing α d , β d → α u , β u and ǫ s , ǫ b → ǫ c , ǫ t .A natural choice which we will adopt in some instances below is: | ǫ t | = sin θ tt ′ ∼ m t /m t ′ and | ǫ b | = sin θ bb ′ ∼ m b /m b ′ . [1] The mixing angles θ uc and θ ds have no effect in our models as they enter only in the 1st and 2nd rows of U R and D R which have nophysical outcome. III. CONSTRAINTS ON THE 4G2HDMS
We now consider constraints from PEWD and from flavor physics in b-quark systems; namely ¯ B → X s γ and B q − ¯ B q ( q = d, s ) mixing. The PEWD constraints can be divided into the effects of the heavy new physics which does anddoes not couple directly to the SM ordinary fermions. For the former we consider constraints from Z → b ¯ b , which ismainly sensitive to the H + t ′ b and W + t ′ b couplings in our models. The effects which do not involve direct couplingsto the ordinary fermions, are analyzed by the quantum oblique corrections to the gauge-bosons 2-point functions,which can be parameterized in terms of the oblique parameters S,T and U [50]. It should be noted that, as far as theoblique parameters are concerned, the contribution from our 4G2HDMs is identical at the 1-loop level to that of any2HDM, since the new Hf f
Yukawa interactions in our models do not contribute at 1-loop to the gauge-bosons selfenergies. A. ¯ B → X s γ and B q - ¯ B q mixing
1. ¯ B → X s γ The inclusive radiative decays of the B meson are known to be a very sensitive probe of new physics. Strongconstraints on new physics from ¯ B → X s γ [51–53] crucially depend on theoretical uncertainties in the SM predictionfor this decay. At the parton level, the decay process B → X s γ is induced by the flavor changing (FC) decay of theb-quark into a strange quark.The current experimental world average is given by [7],BR[ ¯ B → X s γ ] = (3 . ± . ± . × − . (14)In the SM, the calculation of the decay rate is most conveniently performed after decoupling the electroweak bosonsand the top quark. In the resulting effective theory, the relevant FC weak interactions are given by a linear combinationof dimension-five and -six operators [54] O , = (¯ s Γ i c )(¯ c Γ ′ i b ) , (current-currentoperators) O , , , = (¯ s Γ i b ) P q (¯ q Γ ′ i q ) , (four-quarkpenguin operators) O = em b π ¯ s L σ µν b R F µν , (photonic dipoleoperator) O = gm b π ¯ s L σ µν T a b R G aµν . (gluonic dipoleoperator) . (15)The Wilson coefficients, C i , of these operators are perturbatively calculable at the renormalization scale µ ∼ ( m W , m t ) and the Renormalization Group Equations (RGE) can be used to evaluate C i at the scale µ b ∼ m b / µ b . At present, all the relevant Wilson coefficients C i ( µ b ) are known at the Next-to-Next-to-Leading-Order (NNLO) [55–62]. However, the matrix elements of the opera-tors O i consists of perturbative and non-perturbative corrections. As far as the perturbative corrections are concerned,they are reduced dramatically after the completion of Next-to-Leading-Order (NLO) and NNLO QCD calculations.A further improvement comes from electroweak corrections [63–66]. On the other hand, no satisfactory quantitativeestimates of all the non-perturbative effects are available, but they are believed to be ≈
5% [67].In the SM within the leading log approximation, the ¯ B → X s γ amplitude is proportional to the (effective) Wilsoncoefficient of the operator O . The well-known [68] expression for this coefficient reads C (0)eff7 ( µ b ) = η C (0)7 ( µ ) + 83 (cid:16) η − η (cid:17) C (0)8 ( µ ) + X i =1 h i η a i , (16)where η = α s ( µ ) /α s ( µ b ) and h i = (cid:0) − − − − . − . − . − . (cid:1) . (17)Separating the charm and top contributions, and neglecting the CKM-suppressed u -quark contribution, eq. (16)can be written as [69] C (0)eff7 ( µ b ) = X c + X t , (18)where the charm contribution, given by X c , is obtained from eq. (16) by the replacement: C (0)7 ( µ ) → − and C (0)8 ( µ ) → − , X c = − η − (cid:16) η − η (cid:17) + X i =1 h i η a i , (19)which is equivalent to including only charm contributions to the matching conditions for the corresponding operators.Analogously, only the top-loop contributes to X t and the expression is given by X t = − A t ( x t ) η − F t ( x t ) (cid:16) η − η (cid:17) , (20)with x t ≡ ( m t ( µ ) /m W ) and A t ( x ) = − x +2 x x − ln x + − x +153 x − x +4636( x − ,F t ( x ) = x x − ln x + − x +9 x − x +812( x − . Including the perturbative, electroweak and the available non-perturbative corrections, the branching ratio of¯ B → X s γ , with an energy cut–off E in the ¯ B -meson rest frame, can be written as follows [69]:BR[ ¯ B → X s γ ] subtracted ψ, ψ ′ E γ >E = BR[ ¯ B → X c e ¯ ν ] exp (cid:12)(cid:12)(cid:12)(cid:12) V ∗ ts V tb V cb (cid:12)(cid:12)(cid:12)(cid:12) α em π C [ P ( E ) + N ( E )] , (21)where α em = α on shellem [64], N ( E ) denotes the non-perturbative correction and P ( E ) is given by the perturbativeratio Γ[ b → X s γ ] E γ >E | V cb /V ub | Γ[ b → X u e ¯ ν ] = (cid:12)(cid:12)(cid:12)(cid:12) V ∗ ts V tb V cb (cid:12)(cid:12)(cid:12)(cid:12) α em π P ( E ) . (22)In their approach (see [69]) the charmless semileptonic rate has been chosen as the normalization factor in eq. (22),whereas C in eq. (21) is given by C = (cid:12)(cid:12)(cid:12)(cid:12) V ub V cb (cid:12)(cid:12)(cid:12)(cid:12) Γ[ ¯ B → X c e ¯ ν ]Γ[ ¯ B → X u e ¯ ν ] . (23)Furthermore, the perturbative quantity P ( E ) can be written as [69]: P ( E ) = (cid:12)(cid:12)(cid:12)(cid:12) K c + (cid:18) α s ( µ ) π ln µ m t (cid:19) r ( µ ) K t + ε ew (cid:12)(cid:12)(cid:12)(cid:12) + B ( E ) , (24)where K t contains the top-quark contribution to the b → sγ amplitude and K c contains the remaining contributions,among which the charm loops are by far the dominant one. Also, the electroweak correction to the b → sγ amplitude isdenoted in Eq. 24 by ε ew and B ( E ) is the bremsstrahlung function which contains the effects of b → sγg and b → sγq ¯ q ( q = u, d, s ) transitions and which is the only E -dependent part in P ( E ).The NLO expression for K t is given by [69] γ γ γ γu i u i W ± W ± u i u i H ± H ± b W ± s b u i s b H ± s b u i s FIG. 1:
Examples of one-loop 1PI diagrams that contribute to b → sγ in the 4G2HDM, with W -bosons, charged Higgs and 4thgeneration quarks exchanges ( u i = u, c, t, t ′ ). K t = (cid:20) − α s ( m b ) + α s ( µ ) π ln µ m t x ∂∂x (cid:21) (cid:20) − η A ( x t ) + 43 (cid:16) η − η (cid:17) F ( x t ) (cid:21) + α s ( µ b )4 π ( E ( x t ) X k =1 e k η ( a k + )+ η (cid:20) − ηA ( x t ) + (cid:18) − η − π − (cid:18) ln m b µ b + η ln µ m t (cid:19)(cid:19) A ( x t )+ 43 ηF ( x t ) + (cid:18) − η + 1627 π + 329 (cid:18) ln m b µ b + η ln µ m t (cid:19)(cid:19) F ( x t ) (cid:21) + η (cid:20) − ηF ( x t ) + (cid:18) − η − π ( π + i ) − (cid:18) ln m b µ b + η ln µ m t (cid:19)(cid:19) F ( x t ) (cid:21)(cid:27) , (25)where the functions A t ( x ) and F t ( x ) and the expression for K c are given in Ref. [69].For the electroweak ( ε ew ) and non-perturbative ( N ( E )) corrections in eq. (21) we consider the following values[69], ε ew ≈ . . . . N ( E ) = 0 . ± . . (26)Other required inputs which we take from [69] are, r ( µ = m t ) = 0 . ± . µ b ± (parametric errors) (27) C = 0 .
575 (1 ± . ± . ± .
02) (28) a ( z ) = (0 . ± .
25) + i (1 . ± .
15) (29) b ( z ) = ( − . ± .
01) + i (0 . ± . , (30)where a ( z ) and b ( z ) are the z -dependent terms in K c ( z = ( m c /m b ) , see Eq. 3.7 in [69]).With these inputs the NLO prediction for the branching fraction of B → X s γ is [69]BR[ ¯ B → X s γ ] E γ > . = (3 . ± . × − . (31)In the SM4 there are no new operators other than the ones present in the SM. However, there are extra contributionsto the Wilson coefficients corresponding to the operators O and O from t ′ -loop [16–19]. In our 4G2HDMs the newingredient with respect to the SM4 is the presence of the charged Higgs which gives new contributions to the Wilsoncoefficients of the effective theory. Examples of the 1-loop diagrams that contribute to b → sγ in our 4G2HDMs aregiven in Fig. 1. [2] In order to include the charged-Higgs effect we need to compute the new Wilson coefficients at the matching scale µ (The new H + u i d j Yukawa interactions in our models are given in Eq. 7). At the LO, the charged-Higgs contributions,with the t -quark in the loops are given by (see also [70]), δC (0) effi ( µ
0) = 0 i = 1 , ..., [2] We are considering only the charged Higgs contributions to b → sγ and neglecting the flavor changing neutral Higgs 1-loop exchanges,which are much smaller in our models due to the very small b − s and b ′ − s transitions as embedded in Σ d (see Eq. 13). δC (0 t ) eff , ( µ
0) = A U t F (1)7 , ( y t ) + A D t F (2)7 , ( y t ) , (33)and that of t ′ in the loops are given by δC (0) ′ eff , ( µ
0) = A U t ′ F (1)7 , ( y t ′ ) + A D t ′ F (2)7 , ( y t ′ ) , (34)where y i = ¯ m i ( µ ) m H + , and the functions F (1 , , ( y i ) are given by [51, 53, 70] F (1)7 ( y i ) = y i (7 − y i − y i )24( y i − + y i (3 y i − y i − ln y i ,F (1)8 ( y i ) = y i (2 + 5 y i − y i )8( y i − − y i y i − ln y i ,F (2)7 ( y i ) = y i (3 − y i )12( y i − + y i (3 y i − y i − ln y i ,F (2)8 ( y i ) = y i (3 − y i )4( y i − − y i y i − ln y i . (35)Dropping terms proportional to m s (the strange-quark mass) and also neglecting the terms proportional to Σ bb ∝| ǫ b | (which is expected to be small compared to the leading terms), the factors A U t/t ′ and A D t/t ′ in Eqs. 33 and 34are given by A U t = ( A u − A u Σ tt ) + r y t ′ y t ( V ∗ t ′ s V ∗ ts + V t ′ b V tb )Σ t ′ t ( A u Σ tt − A u A u ) + y t ′ y t λ t ′ sb λ tsb A u Σ t ′ t ,A D t = − A d A u + A d A u Σ tt + m b ′ m b V tb ′ V tb ( A d A u − A d A u Σ tt )Σ b ′ b − r y t ′ y t m b ′ λ t ′ bs m b λ tbs A u A d Σ t ′ t Σ b ′ b + r y t ′ y t V ∗ t ′ s V ∗ ts A d A u Σ t ′ t ,A U t ′ = ( A u − A u Σ t ′ t ′ ) + r y t y t ′ ( V ∗ ts V ∗ t ′ s + V tb V t ′ b )Σ tt ′ ( A u Σ t ′ t ′ − A u A u ) + y t y t ′ λ tsb λ t ′ sb A u Σ tt ′ ,A D t ′ = − A d A u + A d A u Σ t ′ t ′ + m b ′ m b V t ′ b ′ V t ′ b ( A d A u − A d A u Σ t ′ t ′ )Σ b ′ b − r y t y t ′ m b ′ λ tbs m b λ t ′ bs V t ′ b ′ V tb A u A d Σ t ′ t Σ b ′ b + r y t y t ′ V ∗ ts V ∗ t ′ s A d A u Σ tt ′ . (36)where for later convenience we have defined A u = A d = tan β , A u = A d = tan β + cot β . (37)In all the cases where the new physics contributions do not involve new operators (and in which C new k ( µ ) = 0 for k = 1 , , , , K t given in Eq. 24, which in our4G2HDMs should be replaced by K t → K Wt + V t ′ b V ∗ t ′ s V tb V ∗ ts K Wt ′ + K Ht + V t ′ b V ∗ t ′ s V tb V ∗ ts K Ht ′ , (38)where K Wt , K Wt ′ , K Ht and K Ht ′ represent the W and charged-Higgs contributions to the b → sγ amplitudes from t and t ′ loops (see Fig. 1). In particular, K Wt ′ can be obtained simply by replacing (neglecting ln ( µ m t )) E ( x t ) → E ( x t ′ ) ,A ( x t ) → A ( x t ′ ) ,A ( x t ) → A ( x t ′ ) ,F ( x t ) → F ( x t ′ ) ,F ( x t ) → F ( x t ′ ) , (39)in eq. 25. On the other hand, K Ht and K Ht ′ , which represent the net contributions to the b → sγ amplitude fromcharged-Higgs exchanges (with t and t ′ as the internal quark, respectively), can be obtained from Eq. 25 by calculatingthe functions E ( y i ) , A ( y i ) , A ( y i ) , F ( y i ) and F ( y i ) ( i = t or i = t ′ ). The LO functions A ( y t ) and A ( y t ′ ) aregiven by A ( y t ) = − δC (0) eff ( µ ) , F ( y t ) = − δC (0) eff ( µ ) ,A ( y t ′ ) = − δC (0) ′ eff ( µ ) , F ( y t ′ ) = − δC (0) ′ eff ( µ ) . (40)and the NLO functions A ( y t ) and A ( y t ′ ) by A ( y i ) = − δC (1)7 ( µ ) , F ( y i ) = − δC (1)8 ( µ ) . (41)The NLO contributions to the Wilson coefficients (in our 4G2HDMs) are given by [3] δC (1) effi ( µ ) = 0 i = 1 , , , , E ( y i ) = δC (1)4 ( µ ) = A U i (cid:20) y i − y i − y i ) ln y i + − y i + 29 y i − y i − y i ) (cid:21) , (43)and δC (1)7 ( µ ) = A U i (cid:26) y i − y i + 36 y i − y i ) Li (cid:18) − y i (cid:19) + − y i + 807 y i − y i + 7 y i − y i ) ln y i + − y i + 7569 y i − y i + 797 y i − y i ) + (cid:20) y i + 46 y i − y i − y i ) ln y i + − y i + 135 y i − y i − y i − y i ) (cid:21) ln µ m i (cid:27) + A D i (cid:26) − y i + 112 y i − y i − y i ) Li (cid:18) − y i (cid:19) + 14 y i − y i + 66 y i − y i ) ln y i + 8 y i − y i + 28 y i − y i ) + (cid:20) − y i − y i + 32 y i − y i ) ln y i + 16 y i − y i + 42 y i − y i ) (cid:21) ln µ m i (cid:27) , (44) δC (1)8 ( µ ) = A U i (cid:26) y i − y i + 30 y i − y i ) Li (cid:18) − y i (cid:19) + − y i + 321 y i − y i − y i − y i ) ln y i + − y i + 7650 y i − y i + 1130 y i − y i ) + (cid:20) − y i − y i − y i ) ln y i + − y i + 18 y i − y i − y i − y i ) (cid:21) ln µ m i (cid:27) + A D i (cid:26) − y i + 25 y i − y i − y i ) Li (cid:18) − y i (cid:19) + 34 y i − y i + 165 y i − y i ) ln y i + 29 y i − y i + 143 y i − y i ) + (cid:20) y i + 19 y i − y i ) ln y i + 7 y i − y i + 81 y i − y i ) (cid:21) ln µ m i (cid:27) . (45)The electroweak and non-perturbative corrections are retained to their SM predictions as given in [69] (see alsoeq. 26), i.e., we do not take into account the effect of our 4G2HDM on these corrections. [3] The NLO results for the Wilson coefficients in a 2HDM can be found in [51, 71]. B q − ¯ B q mixingIn the SM, B q − ¯ B q mixing ( q = d, s ) proceeds to an excellent approximation only through the box diagrams withinternal top quark exchanges. On the other hand, in our 4G2HDMs there are additional contributions to B q − ¯ B q mixing coming from the loop exchanges of the t ′ and charged-Higgs.In the 4G2HDM, the mass difference ∆ M q = 2 | M q | is given at LO by [4] M q = G F π M W f B q B q M B q [ M W W + M HH + M HW ] , (46)where we have used h B q | (¯ sb ) ( V − A ) (¯ sb ) ( V − A ) | B q i = 83 f B q B q M B q , (47) h B q | (¯ sb ) ( S + P ) (¯ sb ) ( S + P ) | B q i = − f B q B q M B q . (48)and M W W = λ tbq η tt S W W ( x t ) + λ t ′ bq η t ′ t ′ S W W ( x t ′ ) + 2 λ tbq λ t ′ bq η tt ′ S W W ( x t , x t ′ ) ,M HH = λ tbq S HH ( y t ) + λ t ′ bq S HH ( y t ′ ) + 2 λ tbq λ t ′ bq S HH ( y t , y t ′ ) ,M HW = λ tbq S HW ( x t , z ) + λ t ′ bq S HW ( x t ′ , z ) + 2 λ tbq λ t ′ bq S HW ( x t , x t ′ , z ) , (49)with z = m H + m W , x i = m i m W , y i = m i m H + ( i = t or t ′ ) and λ ud i d j ≡ V ⋆ud i V ud j .The contributions from W -exchange diagrams with q i and q j ( i, j are generation indices) as the internal quarks aregiven by, S W W ( x i , x j ) = x i x j (cid:26)(cid:20)
14 + 32 1(1 − x j ) −
34 1(1 − x j ) (cid:21) ln x j ( x j − x i ) + ( x j → x i ) −
34 1(1 − x i )(1 − x j ) (cid:27) , (50)and S W W ( x i ) ≡ S W W ( x i , x i ) can be obtained from Eq. 50 by taking the limit x j → x i .The contributions from the H + -exchange diagrams are given by S HH ( y t , y t ′ ) = zS L S L (cid:20) S k HH ( y t , y t ′ )4 B L B L − x b S mHH ( y t , y t ′ ) B R B R (cid:21) ,S HH ( y t ) = zS L (cid:20) S k HH ( y t )4 B L − x b S mHH ( y t ) B R (cid:21) ,S HH ( y t ′ ) = zS L (cid:20) S k HH ( y t ′ )4 B L − x b S mHH ( y t ′ ) B R (cid:21) , (51)where x b = m b m H + M Bq m b ( m b ) , S k HH ( y i , y j ) = y i y j ( y i − y j ) y i ln y i (1 − y i ) − y j ln y j (1 − y j ) ! + 1(1 − y i )(1 − y j ) ) , (52) S k HH ( y i ) = S k HH ( y i , y j ) y j → y i , (53) S mHH ( y i , y j ) = y i y j (cid:26) y i − y j ) (cid:18) y i ln y i (1 − y i ) − y j ln y j (1 − y j ) (cid:19) + 1(1 − y i )(1 − y j ) (cid:27) , (54) S mHH ( y i ) = S mHH ( y i , y j ) y j → y i . (55) [4] The LO results for B q − ¯ B q mixing in a “standard” 2HDM of type II with three generations of fermion doublets are given in [72]. B L = − A u + A u Σ t ′ t ′ + A u m t m t ′ V tb V t ′ b Σ tt ′ ,B L = − A u + A u Σ tt + A u m t ′ m t V t ′ b V tb Σ t ′ t ,S L = − A u + A u Σ t ′ t ′ + A u m t m t ′ V ∗ ts V ∗ t ′ s Σ tt ′ ,S L = − A u + A u Σ tt + A u m t ′ m t V ∗ t ′ s V ∗ ts Σ t ′ t ,B R = A d − A d Σ bb − A d m b ′ m b V t ′ b ′ V t ′ b Σ b ′ b ,B R = A d − A d Σ bb − A d m b ′ m b V tb ′ V tb Σ b ′ b , (56)are obtained from the b → t, t ′ and t, t ′ → s vertices in the box diagrams.The functions S HW ( x i , x j , z ) obtained from diagrams with both W and H + -exchanges are given by S HW ( x t , x t ′ , z ) = 2 x t x t ′ ( S L B L + S L B L ) (cid:20) S ( x t , x t ′ , z )4 + S ( x t , x t ′ , z ) (cid:21) , (57) S HW ( x t , z ) = 2 x t S L B L (cid:20) S ( x t , z )4 + S ( x t , z ) (cid:21) , (58) S HW ( x t ′ , z ) = 2 x t ′ S L B L (cid:20) S ( x t ′ , z )4 + S ( x t ′ , z ) (cid:21) , (59)where S ( x i , x j , z ) = z ln z (1 − z )( z − x i )( z − x j ) + x i ln x i (1 − x i )( x i − z )( x i − x j ) + x j ln x j (1 − x j )( x j − z )( x j − x i ) ,S ( x i , x j , z ) = − z ln z (1 − z )( z − x i )( z − x j ) − x i ln x i (1 − x i )( x i − z )( x i − x j ) − x j ln x j (1 − x j )( x j − z )( x j − x i ) , (60)and the functions S ( x i , z ) and S ( x i , z ) can be derived from the expressions for S ( x i , x j , z ) and S ( x i , x j , z ), respec-tively, by taking the limit x j → x i .3. Combined constraintsUsing the analysis above, we derive below the constraints on our 4G2HDMs that come from Br ( B → X s γ ) and∆ M q ( q = d, s ). For the B-physics parameters we use the inputs given in Table I. As an illustration, the 4thgeneration quark masses are fixed to m t ′ = 500 GeV and m b ′ = 450 GeV, consistent with the direct limits from theTevatron [32] and the perturbative unitarity upper bounds [34, 73]. [5] We vary the charged Higgs mass in the range200 GeV < m H + < ǫ t in the range 0 < ǫ t <
1, while fixing ǫ b = m b /m b ′ ( ∼ . × V t ′ b in the range 0 < | V t ′ b | < . | λ t ′ sb | ≤ . β in the range, 1 < tan β <
30. We made a scan over the entire parameter space by a flat randomnumber generator and obtained bounds and correlations among the various parameters mentioned above.Let us first consider the case V t ′ b →
0, corresponding to the “3+1” scenario, in which the 4th generation quarksdo not mix with the quarks of the 1st three generations (we assume that | V t ′ b | >> | V t ′ s | , | V t ′ d | ). In this case, thetop-quark loops become dominant, since contributions to the amplitudes of B → X s γ and B q - ¯ B q mixing from t ′ -loopsare mostly suppressed apart from the terms which are proportional to ( m b ′ /m b ) · λ tbs (see Eqs. 36 and 56).In Figs. 2, 3 and 4 we plot the allowed ranges in the m H + − tan β (left plots) and the tan β − ǫ t (right plots) planes,in the 4G2HDM of types I, II and III, respectively, using | V t ′ b | = 0 .
001 (with | λ t ′ sb | = 10 − correspondingly). [5] There is a very weak dependence of B → X s γ and B − ¯ B -mixing on the b ′ -mass, since it enters only in the the H + ud Yukawa couplingswith no dynamical and/or kinematical dependence. f bd √ B bd = 0 . ± .
015 GeV [74, 75] | V ub | = (32 . ± . × − a ξ = 1 . ± .
042 [74, 75] | V cb | = (40 . ± . × − η t = 0 . ± . γ = (73 . ± . ◦ ∆ M s = (17 . ± . ps − BR ( B → X s γ ) = (3 . ± . × − ∆ M d = (0 . ± . ps − m b ( m b ) = 4 . GeVf B = (0 . ± . α s ( M Z ) = 0 . m polet = (170 ±
4) GeV τ B + = 1 . psm τ = 1 .
77 GeV a It is the weighted average of V inlub = (40 . ± . ± . × − and V exlub = (29 . ± . × − . In our numerical analysis, we increasethe error on V ub by 50% and take the total error to be around 12% due to the appreciable disagreement between the two determinations. TABLE I: Inputs used in order to constrain the 4G2HDM parameter space. When not explicitly stated, we take the inputsfrom Particle Data Group [49].
200 300 400 500 600 700 800 900 1000 1 1.2 1.4 1.6 1.8 2 2.2 2.4 M H + ( G e V ) tan β | λ t’bs | = 1x10 -5 |V t’b | = 0.001 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t an β ε t M H + = 750M H + = 400 FIG. 2:
The allowed parameter space in the m H + − tan β and tan β − ǫ t planes, following constraints from B → X s γ and B q - ¯ B q mixing, in the 4G2HDM-I, for V t ′ b = 0 . , m t ′ = 500 GeV, m b ′ = 450 GeV and ǫ b = m b /m b ′ . We see that in the type-I 4G2HDM, the “3+1” scenario typically imposes tan β ∼ ǫ t typically larger thanabout 0.4 when m H + < ∼
500 GeV. In particular, for a fixed ǫ t the upper bound on tan β is reduced with the chargedHiggs mass, allowing m H + > ∼
200 GeV for tan β ∼ m H + > ∼
500 GeV for tan β > ∼ .
5. In the type IIand type III 4G2HDMs we observe a similar correlation between tan β and m H + , however, larger tan β are allowedfor ǫ t < ∼ m t /m t ′ and a charged Higgs mass typically heavier than 400 GeV.Let us now turn to the case of a Cabbibo size mixing between the 4th and 3rd generation quarks, setting | V t ′ b | = | V tb ′ | = 0 .
2. In Fig. 5 we show the allowed parameter space in the tan β − ǫ t plane in the 4G2HDM-I, II and IIIwith | V t ′ b | = 0 . m t ′ = 500 GeV, m b ′ = 450 GeV and ǫ b = m b /m b ′ . In addition, we take | λ t ′ sb | = 0 .
004 for Type-Iand 0.001 for Type-II and III models and depict these correlations for two different values of the charged Higgs mass: M H + = 400 and 750 GeV. In the type II and type III 4G2HDMs we see a similar behavior as in the no mixing case( V t ′ b → V t ′ b allows for a slightly larger tan β , i.e., up to tan β ∼
100 200 300 400 500 600 700 800 900 1000 5 10 15 20 25 30 M H + ( G e V ) tan β | λ t’bs | = 1x10 -5 |V t’b | = 0.001 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t an β ε t M H + =750M H + =400 FIG. 3:
Same as Fig. 2 for the 4G2HDM-II.
100 200 300 400 500 600 700 800 900 1000 5 10 15 20 25 30 M H + ( G e V ) tan β | λ t’bs | = 1x10 -5 |V t’b | = 0.001 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t an β ε t M H + = 750M H + = 400 FIG. 4:
Same as Fig. 2 for the 4G2HDM-III. for ǫ t > ∼ . ǫ t ∼ m t /m t ′ , in Figs. 6 and 7 we plot tan β as a function of M H + (where | λ t ′ sb | is kept free) and of λ t ′ bs (where M H + is kept free), respectively, in the three different types of our 4G2HDMs.We note that, similar to the no mixing case, larger values of tan β are allowed in the 4G2HDM of types II and III.Furthermore, m H + ∼
300 GeV and tan β ∼ | λ t ′ bs | up to0.01 is allowed in the case of the 4G2HDM-I and II, while in 4G2HDM-III | λ t ′ sb | < ∼ .
005 is typically required. t an β ε t M H + = 750M H + = 400 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t an β ε t M H + =750M H + =400 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t an β ε t M H + = 750M H + = 400 FIG. 5:
Allowed parameter space in the tan β − ǫ t plane in the 4G2HDM of type-I (left), type-II (middle) and type-III (right),for | V t ′ b | = 0 . , m t ′ = 500 GeV, m b ′ = 450 GeV, ǫ b = m b /m b ′ , | λ t ′ sb | = 0 . and with m H + = 400 and 750 GeV.
200 300 400 500 600 700 800 900 1000 1 1.2 1.4 1.6 1.8 2 2.2 M H + ( G e V ) tan β ε t =0.34 400 500 600 700 800 900 1000 1 2 3 4 5 6 7 8 9 10 M H + ( G e V ) tan β ε t = 0.34 400 500 600 700 800 900 1000 1 2 3 4 5 6 7 8 9 10 M H + ( G e V ) tan β ε t =0.34 FIG. 6:
Allowed parameter space in the m H + − tan β plane in the 4G2HDM of type-I (left), type-II (middle) and type-III (right),for | V t ′ b | = 0 . , m t ′ = 500 GeV, m b ′ = 450 GeV, ǫ b = m b /m b ′ , | λ t ′ sb | = 0 . and ǫ t = 0 . ∼ m t /m t ′ ) . t an β | λ t’bs | ε t =0.34 1 2 3 4 5 6 7 8 9 0 0.002 0.004 0.006 0.008 0.01 t an β | λ t’bs | ε t = 0.34 1 2 3 4 5 6 7 8 9 0 0.002 0.004 0.006 0.008 t an β | λ t’bs | ε t = 0.34 FIG. 7:
Allowed parameter space in the tan β − | λ t ′ sb | plane in the 4G2HDM of type-I (left), type-II (middle) and type-III (right),for a fixed | V t ′ b | = 0 . , ǫ t ∼ m t /m t ′ and for m t ′ = 500 GeV, m b ′ = 450 GeV and ǫ b = m b /m b ′ . To summarize this section, we find that the parameter space of our 4G2HDMs, when subject to constraints from Br ( B → X s γ ) and B q − ¯ B q mixing, can be characterized by the following features: • In the type II and III 4G2HDMs large tan β > ∼
20 are allowed for ǫ t < ∼ . • In the 4G2HDM-I tan β is typically restricted to be tan β ∼ O (1) with ǫ t ∼ m t /m t ′ , reaching at most tan β ∼ ǫ t ∼ V t ′ b ∼ O (0 . • The charged Higgs mass is typically heavier than about 400 GeV in the type II and III 4G2HDM and is allowedto be as light as 200-300 GeV (depending on V t ′ b ) in the 4G2HDM-I. In all three models the lower bound on m H + increases (typically linearly) with tan β ; reaching m H + ∼ β ∼ β ∼ ǫ t ∼ m t /m t ′ . • In the 4G2HDM-III, | λ t ′ sb | < ∼ .
005 is required if ǫ t ∼ m t /m t ′ , but values up to | λ t ′ sb | ∼ .
01 are still allowed inthe 4G2HDMs of types I and II.
B. Constraints from Z → b ¯ b It has been long known that the decay Z → b ¯ b is very sensitive to effects of new heavy particles, in particular, tothe dynamics of multi-Higgs models through loop exchanges of both neutral and charged Higgs particles (see e.g.,[77, 78]). The Zb ¯ b vertex can be parameterized as follows: V qqZ ≡ − i gc W ¯ qγ µ (¯ g qL L + ¯ g qR R ) qZ µ , (61)where s W ( c W ) = sin θ W (cos θ W ), L ( R ) = (1 − (+) γ ) / g qL,R = g SMqL,R + g newqL,R , (62)so that g SMqL,R are the SM (1-loop) quantities and g newqL,R are the new physics 1-loop corrections.The effects of the new physics, g newqL,R , is best studied via the well measured quantity R b : R b ≡ Γ( Z → b ¯ b )Γ( Z → hadrons) , (63)which is a rather clean test of the SM. In particular, being a ratio between two hadronic rates, most of the electroweak,oblique and QCD corrections cancel between numerator and denumerator.Following the analysis in [77], we parameterize the effects of new physics in R b in terms of the corrections δ b and δ c to the decays Z → b ¯ b and Z → c ¯ c , respectively: R b = R SMb δ b R SMb δ b + R SMc δ c , (64)5where R SMb = 0 . ± . R SMc = 0 . ± . δ q are the new physics corrections defined in terms of the Zq ¯ q couplings as: δ q = 2 g SMqL g newqL + g SMqR g newqR (cid:16) g SMqL (cid:17) + (cid:16) g SMqR (cid:17) , (65) H+H+ H+H+ H+ µ Z µ Z µ Z µ Z (1) (2) i dd IJ (3) (4)d IJ d J i d dd IJ d I i dd IJ u uu ii uu FIG. 8:
One-loop diagrams for corrections to Z → d I ¯ d J from charged Higgs loops. With the new scalar-fermion interactions in Eqs. 4-7, the corrections to R b from a 4th generation quarks in our4G2HDMs are of three types: (i) the SM4-like corrections due to the 1-loop W − t ′ exchanges (see also [17, 80, 81]),(ii) the 1-loop diagrams in Fig. 8 involving the H + − t ′ exchanges and (iii) the 1-loop corrections involving the FC H bb ′ interactions (coming from the non-diagonal 34 and 43 elements in Σ d ), where H = h, H or A .Let us first consider the SM4-like (non-decoupling) correction to R b , i.e., g SM qL from the 1-loop diagrams involvingthe W − t ′ exchanges (which are also present in our 4G2HDMs). It is given by [17, 80]: g SM qL = g π c W (cid:18) m t ′ m Z − m t m Z (cid:19) sin θ , (66)where θ is the mixing angle between the 3rd and 4th generation quarks, i.e., defining | V t ′ b | = | V tb ′ | ≡ sin θ . ThisSM4-like effect on R b is plotted in Fig. 9. We see that R b puts rather stringent constraints on the m t ′ − θ plane whichis the SM4 subspace of the parameter space of our 4G2HDMs. In particular, increasing the t ′ mass would tightenthe constraints on θ ; e.g., for m t ′ ∼
500 GeV the t ′ − b mixing angle is restricted to θ < ∼ .
2. The upper boundon θ stays roughly the same in our 4G2HDMs where the effects from the charged Higgs loops are included. Forconcreteness, for the rest of this section we will fix θ to either θ = 0 or θ = 0 . , .
2, representing the no-mixingor mixing cases.Using the generic formula given in [82], we calculated the 1-loop corrections to R b from the charged-Higgs and fromthe FC neutral-Higgs exchanges and found that: • In all three models, i.e., 4G2HDM-I,II,III, δ c ≪ δ b , so that we can safely neglect the new effects in Z → c ¯ c . • The 1-loop FC neutral-Higgs contributions are much smaller than the 1-loop charged-Higgs contributions shownin Fig. 8, in particular for ǫ b ≪
1. We, therefore, focus below only on the leading effects coming from thecharged-Higgs sector.6 θ [rad]0.2130.2140.2150.2160.2170.2180.219 R b SM4 m t’ =400 GeVm t’ =500 GeVm t’ =600 GeVexp−2 σ
300 350 400 450 500 550 600 650 700 m t’ [GeV] R b SM4 θ =0.1 θ =0.2 exp−2 σ
200 300 400 500 600 700 800 900 1000 M H+ [GeV] R b m t’ =500 GeV , ε t = m t / m t’ tan β =1 , θ =0tan β =1 , θ =0.2tan β =5 , θ =0tan β =5 , θ =0.2exp−2 σ
300 350 400 450 500 550 600 650 700 m t’ [GeV] R b θ =0.2, ε t =m t / m t’ m H + =400 GeV, tan β =1m H + =400 GeV, tan β =5m H + =750 GeV, tan β =1m H + =750 GeV, tan β =5exp−2 σ FIG. 9:
Upper plots: R b in the SM4, as a function of θ for several values of the t ′ mass (left) and as a function of m t ′ for θ = 0 . and . (right). Lower plots: R b in the 4G2HDM-I, as a function of the charged Higgs mass for m t ′ = 500 GeV, ǫ t = m t /m t ′ and for (tan β, θ ) = (1 , , (1 , . , (5 , , (5 , . (left), and as a function of m t ′ for θ = 0 . , ǫ t = m t /m t ′ andfor ( m H + [GeV] , tan β ) = (400 , , (400 , , (750 , , (750 , (right). • The charged-Higgs interactions in models 4G2HDM-II and 4G2HDM-III have negligible effects on R b and are,therefore, not constrained by this quantity. On the other hand, R b is rather sensitive to the charged Higgs loopexchanges within our type I 4G2HDM.In light of the above findings, we plot in Fig. 9 the quantity R b in the 4G2HDM-I (calculated from Eq. 64), as afunction of the charged Higgs and t ′ masses, fixing ǫ t = m t /m t ′ and focusing on the values tan β = 1 , θ = 0 , . m H + = 400 ,
750 GeV. We see that, while there are no constraints from R b on the charged Higgs and t ′ massesif tan β = 1, for higher values of tan β a more restricted region of the charged Higgs mass is allowed which againdepends on θ , e.g., for tan β = 5, 550 GeV < ∼ m H + < ∼
800 GeV, and m t ′ < ∼
500 GeV is required in order for R b to bewithin its 2 σ measured value ( R expb = 0 . ± . m H + − tan β plane in the 4G2HDM-I, subject to the R b constraint7 tan β m H + [ G e V ] m t’ =500 GeV , m b’ =450 GeV , θ =0.2 , ε b =m b /m b’ ε t = ε b =m b / m b’ tan β m H + [ G e V ] m t’ =500 GeV , m b’ =450 GeV , θ =0.2 , ε b =m b /m b’ ε t =m t / m t’ tan β m H + [ G e V ] m t’ =500 GeV , m b’ =450 GeV , θ =0.2 , ε b =m b /m b’ ε t =1 FIG. 10:
Allowed area in the m H + − tan β in the 4G2HDM-I, subject to the R b measurement (within σ ), for m t ′ = 500 GeV, m b ′ = 450 GeV, θ = 0 . , ǫ b = m b /m b ′ and for three values of the t − t ′ mixing parameter: ǫ t = ǫ b ∼ . (left plot), ǫ t = m t /m t ′ ∼ . (middle plot) and ǫ t = 1 (right plot). (2 σ ), for tan β in the range 1-15, fixing m t ′ = 500 GeV, m b ′ = 450 GeV, θ = 0 . ǫ b = m b /m b ′ (which also enters the t ′ bH + vertex) and for three representative values of the t − t ′ mixing parameter: ǫ t = ǫ b ∼ . ǫ t = m t /m t ′ ∼ . ǫ t = 1. As expected, when tan β is lowered, the constraints on the charged Higgs mass are weakened. We seee.g., that for ǫ t = m t /m t ′ ∼ .
35, tan β ∼ m H + values ranging from 200 GeV up to the TeVscale, while for tan β ∼ < ∼ m H + < ∼
750 GeV.Note however, that in the 4G2HDM-I, tan β = 5 with ǫ t = m t /m t ′ is not allowed by constraints from B-physics flavordata (see previous section). C. Constraints from the Oblique parameters
The sensitivity of 4th generation fermions to PEWD within the minimal SM4 framework was extensively analyzedin the past decade [49, 80, 83–87]. One of the immediate interesting consequences of the presence of the 4th gen-eration fermion doublet (with respect to the PEWD constraints) is that it allows for a considerably heavier Higgs,thus removing the slight tension between the LEPII bound on the mass of the SM Higgs m H > ∼
115 GeV and thecorresponding theoretical best fitted value (to PEWD) m H = 87 +35 − GeV [49]. In fact, a Higgs with m H > ∼
300 GeVbecomes favored in the SM4 when m t ′ − m b ′ ∼
50 GeV and θ is of the size of the Cabbibo angle, see e.g., [80, 85].On the other hand, if, as in our case, the 4th generation fermions are embedded in a 2HDM framework, then there isa wider range of parameter space for which a lighter Higgs with a mass of O (100) GeV is allowed (see [36] and ouranalysis below). In addition, in the 2HDM case, the LEPII lower bound m H > ∼
115 GeV can be relaxed, dependingon the value of sin( α − β ) [sin( α − β ) = 1 corresponds to the current SM bound] which controls the ZZH couplingresponsible for the Higgs production mechanism at LEP.In general, the contributions to the oblique parameters ( S , T , U ) of 4th generation fermions (∆ S f , ∆ T f , ∆ U f ) andof extra scalars (∆ S s , ∆ T s , ∆ U s ) are calculated with respect to the SM values and are bounded by a fit to PEWD[88]: ∆ S = S − S SM = 0 . ± . T = T − T SM = 0 . ± . U = U − U SM = 0 . ± . , (67)where, following the fit made in [88], the SM values are defined for a Higgs mass reference value of M refh = 120 GeVand for m t = 173 . S − T plane which,for a given confidence level (CL), is defined by (see e.g., [23]): (cid:18) S − S exp T − T exp (cid:19) T (cid:18) σ S σ S σ T ρσ S σ T ρ σ T (cid:19) (cid:18) S − S exp T − T exp (cid:19) = − − CL ) , (68)where S exp = 0 .
02 and T exp = 0 .
05 are the best fitted (central) values in Eq. 67, σ S = 0 . , σ T = 0 .
12 are thecorresponding standard deviations and ρ = 0 .
879 [88] is the (strong) correlation factor between S and T.8Note that the contribution of the Higgs spectrum of our 4G2HDMs to S and T are identical to that of any general2HDM. We thus use the analytical expressions given in [83], where we also include in ∆ T f the new contributions fromthe W t ′ b and W tb ′ off-diagonal CKM mixing angles (see e.g., [80]):∆ T f = 38 πs W c W (cid:18) | V t ′ b ′ | F t ′ b ′ + | V t ′ b | F t ′ b + | V tb ′ | F tb ′ − | V tb | F tb + 13 F ℓ ν (cid:19) , (69)with F ij = x i + x j − x i x j x i − x j log x i x j , (70)and x k ≡ ( m k /m Z ) .We first “blindly” (randomly) scan our parameter space, varying them in the ranges: tan β ≤ θ ≤ . ≤ m h ≤ m h ≤ m H ≤ . ≤ m A ≤ ≤ m t ′ , m b ′ ≤
600 GeV,100 GeV ≤ m ν ′ , m τ ′ ≤ . [6] and the CP-even neutral Higgs mixing angle in the range 0 ≤ α ≤ π .We use a sample of 100000 models (i.e., points in parameter space varied in the above specified ranges) and plotthe result in Fig. 11. We find that out of the 100000 models about 3000 are within the 99%CL contour, 1500 withinthe 95%CL contour and 100 within the 68%CL contour. We compare these results to the SM4 case also shown inFig. 11 (again using a sample of 100000 models), where the 4th generation quark and lepton masses as well as the(single) neutral Higgs mass are varied in the same ranges as specified above. We find that in the SM4 case onlya few points (out of the 100000) are within the 68%CL S-T contour, while the number of SM4 points within the95%CL and 99%CL allowed contours are comparable to the 2HDM case. This quantifies the slight preferability ofthe 2HDM (with respect to the amount of fine tuning required for compatibility with the available precision data) asan underlying framework for a 4th generation model.We also examined the correlation in the m H + − tan β plane, when subject to the PEWD S-T constraint. This isshown in Fig. 12, where the data points are taken from the same 100000 sample used in Fig. 11 (i.e., the rest of theparameter space was varied in the ranges specified above). We see that compatibility with PEWD mostly requirestan β ∼ O (1) with a small number of points in parameter space having tan β > ∼ −0.4 −0.2 0 0.2 0.4−0.4−0.200.20.40.6 S T
99% CL95% CL68% CL −0.4 −0.2 0 0.2 0.4−0.4−0.200.20.40.6 S T
99% CL95% CL68% CL
FIG. 11:
The allowed points in parameter space projected onto the 68%, 95% and 99% allowed contours in the S-T plane, inthe 4G2HDMs (left) and in the SM4 (right). The data points are varied in the ranges: tan β ≤ , θ ≤ . ,
100 GeV ≤ m h ≤ , m h ≤ m H ≤ . ,
100 GeV ≤ m A ≤ ,
400 GeV ≤ m t ′ , m b ′ ≤
600 GeV ,
100 GeV ≤ m ν ′ , m τ ′ ≤ . andthe CP-even neutral Higgs mixing angle in the range < ∼ α < ∼ π . [6] Note that the perturbative unitarity upper bounds on the lepton masses are about twice larger than those on the quark masses [73];thus allowing 4th generation lepton masses around 1 TeV. m H + [ G e V ] tan β FIG. 12:
95% CL allowed range in the m H + − tan β plane. The data points are varied as specified in Fig. 11. Next we consider the correlations between the mass splitting among the 4th generation quark masses, ∆ m q ′ ≡ m t ′ − m b ′ , and the lepton masses ∆ m ℓ ′ ≡ m ν − m ℓ . In Fig. 13 we plot the 95%CL allowed regions (i.e., subjectto the measured 95%CL contour in the S-T plane) for both the 4G2HDMs and the SM4 in the ∆ m q ′ − ∆ m ℓ ′ plane,again using the same data set of 100000 models used in Fig. 11. We see that, while in the SM4 case the allowed masssplittings are restricted to −
100 GeV < ∆ m q ′ <
100 GeV and −
200 GeV < ∆ m ℓ ′ <
200 GeV, in the 4G2HDMs thesemass splitting ranges are significantly extended to: −
200 GeV < ∆ m q ′ <
200 GeV and −
500 GeV < ∆ m ℓ ′ <
400 GeV. −300 −200 −100 0 100 200 300−600−400−2000200400600 m ν ′ − m τ ′ m t ′ − m b ′ −300 −200 −100 0 100 200 300−600−400−2000200400600 m ν ′ − m τ ′ m t ′ − m b ′ FIG. 13:
Allowed regions in the ∆ m q ′ − ∆ m ℓ ′ plane within the CL contour in the S-T plane, for the 4G2HDMs (left) andfor the SM4 (right). The data points are varied as in Fig. 11.
In Figs. 14 we again plot the 95%CL allowed regions in the ∆ m q ′ − ∆ m ℓ ′ plane, for both the 4G2HDMs and theSM4, considering now the “3+1” scenario, i.e., with a vanishing mixing between the 4th generation quarks and thelighter three generations; θ = 0. The rest of the parameter space is varied as in Fig. 13. We see that in the SM40 −300 −200 −100 0 100 200 300−600−400−2000200400600 m ν ′ − m τ ′ m t ′ − m b ′ −300 −200 −100 0 100 200 300−600−400−2000200400600 m ν ′ − m τ ′ m t ′ − m b ′ FIG. 14:
Same as Fig. 13 but for θ = 0 ; the rest of the parameter space is varied as in Fig. 13. with θ → | ∆ m q ′ | < ∼
50 GeV and | ∆ m ℓ ′ | < ∼
100 GeV. On the other hand, the implications ofthe no-mixing case on the 4G2HDMs are mild as there are still points/models for which the 4th generation quarksand leptons are both almost degenerate. For such small (or no) 4th generation fermion mass splitting the amount ofisospin breaking required to compensate for the effect of the extra fermions and Higgs particles on S and T is providedby a mass splitting among the Higgs particles, as is shown below.In order to demonstrate the interplay between the mass splittings in the Higgs and fermion sectors, we choose a morespecific framework - partly motivated by our theoretical prejudice towards the possibility of dynamical EWSB, drivenby the condensation(s) of the 4th generation fermions. In particular, we set α ∼ π/
2, for which case H ∼ Re(Φ h )and h ∼ Re(Φ ℓ ); the heavier Higgs may be thus identified as a possible ¯ Q ′ Q ′ ( Q ′ = t ′ , b ′ ) condensate, with a typicalmass of m H < ∼ m Q ′ [89]. We thus set m H = 1 TeV and take a nearly degenerate 4th generation quark doublet with m t ′ = 500 GeV and m b ′ = 490 GeV. We further study two representative values for tan β : tan β = 1 and tan β = 5,recalling that for tan β ∼ O (1), H + and A are roughly equal admixtures of Φ ℓ and Φ h , while if tan β >>
1, one has H + ∼ Φ + ℓ and A ∼ Im(Φ h ). The charged Higgs mass is set to m H + = 600 GeV, so that it is within the R b constraintsfor both tan β = 1 and tan β = 5 when m b m ′ b < ∼ ǫ t < ∼ m t m ′ t (see previous section). For simplicity we furthermore set θ = 0and vary the 4th generation lepton masses in the range 100 GeV < ∼ m ν ′ , m τ ′ < ∼ . h and A , in the range 100 GeV < ∼ m h , m A < ∼ m ℓ ′ − m h , the m h − m A and the ∆ m ℓ ′ − ( m h − m A ) planes, using again a sample of 100000 models withtan β = 1 and tan β = 5, respectively. Under the above set of inputs, we find the following noticeable features: • There are allowed sets of points in parameter space (i.e., models) where both the 4th generation quarks andleptons are nearly degenerate with a mass splitting smaller than 50 GeV. These solutions require m h and m A to have a mass splitting smaller than about 400 GeV and to be within the narrow black bands in the m h − m A plane, as seen in Figs. 15 and 16. • There are allowed sets of points with a large splitting between the 4th generation leptons, | m ν ′ − m τ ′ | > m h and m A ; 400 GeV < ∼ | m h − m A | < ∼
800 GeV if tan β = 1and 600 GeV < ∼ | m h − m A | < ∼
800 GeV if tan β = 5. • For tan β = 1, a splitting in the 4th generation lepton sector of | m ν ′ − m τ ′ | >
200 GeV requires m h to be largerthan about 400 GeV.Finally, in Table II we give a list of interesting points (models) in parameter space (of our 4G2HDM of types I, IIand III) that pass all the constraints considered in this chapter, i.e., from the S and T parameters, from R b and from1
200 400 600 800 1000−400−300−200−1000100200300400 m h m ν ′ − m τ ′ tan β =1
200 400 600 800 1000−400−300−200−1000100200300400 m h −m A m ν ′ − m τ ′ tan β =1
200 400 600 800 10001002003004005006007008009001000 m A m h tan β =1 FIG. 15: 95%
CL allowed regions in the ∆ m ℓ ′ − m h plane (left), in the ∆ m ℓ ′ − ( m h − m A ) plane (middle) and in the m h − m A plane (right), for tan β = 1 , m H + = 600 GeV, θ = 0 , α ∼ π/ , m t ′ = 500 GeV and m b ′ = 490 GeV. The lepton massesand Higgs masses are varied in the ranges:
100 GeV < ∼ m ν ′ , m τ ′ < ∼ . and
100 GeV < ∼ m h , m A < ∼ . The blackdots correspond to solutions with | ∆ m ℓ ′ | < GeV, the red dot to solutions with m τ ′ − m ν ′ > GeV and the blue dots tosolutions with m ν ′ − m τ ′ > GeV.
200 400 600 800 1000−400−300−200−1000100200300400 m h m ν ′ − m τ ′ tan β =5
200 400 600 800 1000−400−300−200−1000100200300400 m h −m A m ν ′ − m τ ′ tan β =5
200 400 600 800 10001002003004005006007008009001000 m A m h tan β =5 FIG. 16:
Same as Fig. 15 for tan β = 5 . B-physics flavor data. In particular, the list includes models with mass splittings between the up and down partnersof both the 4th family quarks and leptons larger than 150 GeV, models with a light 100 −
200 GeV neutral Higgs,models with degenerate 4th generation doublets, models with a large inverted mass hierarchy in the quark doublet,i.e., m b ′ − m t ′ >
150 GeV, models with a light charged Higgs with a mass smaller than 500 GeV, models with aCabbibo size as well as an O (0 .
01) size t ′ − b / t − b ′ mixing angle (i.e., θ ). IV. PHENOMENOLOGY OF THE YUKAWA SECTOR IN THE 4G2HDM-I
Although this paper is not aimed to explore in detail the phenomenological consequences of the modifications tothe Higgs Yukawa interactions involving the 4th generation quarks in our 4G2HDMs, in order to give a feel for theirimportance for collider searches of the 4th generation fermions, we consider below some phenomenological aspectsof the 4G2HDM-I which is defined by ( α d , β d , α u , β u ) = (0 , , , O ( ǫ q ), q = b, t ):Σ d ≃ | ǫ b | ǫ ⋆b ǫ b (cid:16) − | ǫ b | (cid:17) , Σ u ≃ | ǫ t | ǫ ⋆t ǫ t (cid:16) − | ǫ t | (cid:17) , (71)so that Σ u,d = 0 if i or j = 3 ,
4. This leads to new interesting patterns (in flavor space) of the H q i q j Yukawa2 tan β = 1, ǫ t = m t /m t ′ Point m t ′ m b ′ m ν ′ m τ ′ m h m A m H m H + sin θ α . π . π . π . π . π . π a . π . π . π
10 4G2HDM-I 500 450 302 414 220 793 1001 750 0.05 π/
211 4G2HDM-I 500 450 424 410 120 597 1479 750 0.2 π/
212 4G2HDM-I 500 450 147 127 350 716 506 400 0.05 π/ b ) 4G2HDM-I 450 500 225 235 220 782 303 300 0.2 π/ β = 5, ǫ t = m t /m t ′
14 4G2HDM-II,III 542 386 938 740 126 458 1141 738 0.094 0 . π
15 4G2HDM-II,III 544 367 305 310 179 417 1255 706 0.117 1 . π
16 4G2HDM-II,III 517 366 393 211 295 130 1347 801 0.188 0 . π
17 4G2HDM-II,III 430 412 193 175 246 568 904 617 0.18 0 . π
18 4G2HDM-II,III 463 451 398 418 170 593 1218 715 0.026 0 . π
19 4G2HDM-II,III 381 465 545 622 135 145 1084 803 0.051 0 . π
20 4G2HDM-II,III 514 371 106 610 122 295 1495 819 0.031 1 . π
21 4G2HDM-II,III 496 399 541 617 148 343 1054 780 0.03 1 . π
22 4G2HDM-II,III 463 481 959 784 105 319 918 760 0.188 0 . π
23 4G2HDM-II,III 504 508 497 545 140 118 1175 748 0.166 0 . π tan β = 20, ǫ t < .
124 4G2HDM-II,III 521 362 178 191 177 231 775 525 0.03 1 . π
25 4G2HDM-II,III 535 381 568 399 435 573 1500 954 0.073 0 . π
26 4G2HDM-II,III 542 372 106 314 510 268 1382 450 0.158 0 . π
27 4G2HDM-II,III 369 527 212 565 571 233 1335 669 0.175 1 . π
28 4G2HDM-II,III 459 440 684 702 142 455 631 400 0.101 0 . π
29 4G2HDM-II,III 546 517 260 661 111 216 1347 940 0.186 0 . π
30 4G2HDM-II,III 411 456 126 423 140 163 1261 940 0.1843 0 . π a point requires | λ t ′ sb | < ∼ − . b point requires ǫ b ∼ m b /m b ′ in order to have BR( b ′ → tH + ) ∼ O (1) (see Fig. 19. TABLE II: List of points (models) in parameter space for our 4G2HDMs of types I, II and III, allowed at 95%CL by PEWDand B-physics flavor data. The 2nd column denotes the model(s) for which the point is applicable. Points 1-3,14-16 and 24 have m t ′ − m b ′ >
150 GeV with a light CP-even Higgs of mass m h < ∼
300 GeV, while points 4,5,27 have a large inverted splitting m b ′ − m t ′ >
150 GeV with a heavier h. Points 6,7 and 17,18,28 have nearly degenerate 4th generation quark and leptondoublets, while points 22,23 have a nearly degenerate 4th generation quark doublet with a lepton doublet heavier than thequark doublet. Points 8,19 have m b ′ − m t ′ > m W and a light charged Higgs, while points 9,16 have m t ′ − m b ′ ∼
150 GeV witha light Higgs mass of m h ∼
100 GeV. Points 1,8,16,17,22,23,26,27,29,30 all have a large t ′ − b/t − b ′ mixing angle: θ > ∼ . t ′ → th ) ∼ O (1) (see Fig. 17 in the next section), point 12 gives BR( t ′ → bH + ) ∼ O (1) (seeFig. 18 in the next section) and point 13 gives BR( b ′ → tH + ) ∼ O (1) (see Fig. 19 in the next section). interactions in Eqs. 4-7 ( H = h, H, A ). In particular, the most notable new features of the 4G2HDM-I are:1. There are no tree-level FC neutral currents (FCNC) among the quarks of the 1st, 2nd and 3rd generations.That is, no tree-level c → u , s → d transitions, as well as no t → u , t → c , b → d and b → s ones.2. There are no tree-level FCNC effects involving transitions between the quarks of the 4th generation and the1st and 2nd generations, i.e, no t ′ → u , t ′ → c , b ′ → d and b ′ → s transitions. This, makes the 4G2HDM-Icompatible with all FCNC constraints coming from light meson mixings and decays, i.e., in the K and D systems.3. There are new potentially large tree-level FCNC effects in the H q i q j couplings involving the 3rd and 4thgeneration quarks (i.e., i, j = 3 , H ′ t ′ t and H ′ b ′ b interactions are (taking α → π/
300 350 400 450 500 m b’ [GeV] B r a n c h i n g R a t i o (cid:24) m h =120 GeV , θ =0.05 m t’ =500 GeV , tan β =1 , ε t =m t / m t’ h =120 GeV , θ =0.2 t’ −> t ht’ −> b’ Wt’ −> b W300 350 400 450 500 m b’ [GeV] m h =220 GeV , θ =0.05m h =220 GeV , θ =0.2 FIG. 17:
The branching ratio for the t ′ decay channels t ′ → th , t ′ → bW and t ′ → b ′ W ( ⋆ ) ( W ( ⋆ ) is either on-shell or off-shell depending on the b ′ mass), as a function of m b ′ for m t ′ = 500 GeV, ǫ t = m t /m t ′ , tan β = 1 and ( m h [GeV] , θ ) =(120 , . , (120 , . , (220 , . , (220 , . , as indicated. Also, α = π/ and m H + > m t ′ , m A > m t ′ is assumed. L ( ht ′ t ) = − g m t ′ m W ǫ t q t β ¯ t ′ (cid:18) R + m t m t ′ L (cid:19) th , (72) L ( Ht ′ t ) = − g m t ′ m W ǫ t q t β t β ¯ t ′ (cid:18) R + m t m t ′ L (cid:19) tH , (73) L ( At ′ t ) = i g m t ′ m W ǫ t t β t β ¯ t ′ (cid:18) R − m t m t ′ L (cid:19) tA , (74)and similarly for the H b ′ b interactions by changing ǫ t → ǫ b (and an extra minus sign in the Ab ′ b coupling).We thus see that, if ǫ t ∼ m t /m t ′ , then the above couplings can become sizable, e.g., to the level that it mightdominate the decay pattern of the t ′ (see below). In fact, we also expect large FC effects in b ′ → b transitionssince, even for a very small ǫ b ∼ m b /m b ′ , the FC hb ′ b and Ab ′ b Yukawa couplings can become sizable if e.g.,tan β ∼
5, i.e., in which case they are ∝ m b m W .4. The flavor diagonal interactions of the Higgs species with the up-quarks, H uu , are proportional to tan β ,thus being a factor of tan β larger than the corresponding “conventional” 2HDMs couplings, for which thesecouplings are ∝ cot β (e.g., as in the 2HDM of type II which also underlies the supersymmetric Higgs sector).In particular, the H tt couplings in our 4G2HDM-I are given by:4 ε t B R (t’ −> b H + ) (cid:24) m b’ =400 GeV , θ =0.05 m t’ =500 GeV , m H+ =400 GeV , tan β =1 , ε b =m b /m b’ b’ =400 GeV , θ =0.2 m h =200 GeVm h =350 GeV ε t m b’ =450 GeV , θ =0.05m b’ =450 GeV , θ =0.2 FIG. 18:
The branching ratios for the decay t ′ → bH + as a function of ǫ t for m t ′ = 500 GeV, m H + = 400 GeV, tan β = 1 , ǫ b = m b /m b ′ , m h = 220 and 350 GeV and ( m b ′ [GeV] , θ ) = (400 , . , (400 , . , (450 , . , (450 , . , as indicated. Also, α = π/ and m A > m t ′ is assumed. L ( htt ) ≈ g m t m W q t β (cid:0) − | ǫ t | (cid:1) ¯ tth | ǫ t | ≪ −→ g m t m W q t β ¯ tth , (75) L ( Htt ) ≈ − g m t m W q t β t β | ǫ t | ¯ ttH , (76) L ( Att ) ≈ − i g m t m W t β h − (cid:16) t − β (cid:17) | ǫ t | i ¯ tγ tA | ǫ t | ≪ −→ − i g m t m W t β ¯ tγ tA . (77)We see that the htt and Att
Yukawa interactions are indeed enhanced by a factor of t β relative to the conventional htt and Att couplings in multi-Higgs models (with no suppression from t − t ′ mixing parameter ǫ t ). On theother hand, the ht ′ t ′ and At ′ t ′ couplings are suppressed by the t − t ′ mixing parameter and by t β , respectively: L ( ht ′ t ′ ) ≈ g m t ′ m W q t β | ǫ t | ¯ t ′ t ′ h , (78) L ( Ht ′ t ′ ) ≈ − g m t ′ m W q t β t β (cid:18) − | ǫ t | (cid:19) ¯ t ′ t ′ H , (79) L ( At ′ t ′ ) ≈ − i g m t ′ m W t β (cid:20) − (cid:16) t − β (cid:17) (cid:18) − | ǫ t | (cid:19)(cid:21) ¯ t ′ γ t ′ A | ǫ t | ≪ −→ i g m t m W t β ¯ tγ tA . (80)5. The charged Higgs couplings involving the 3rd and 4th generation quarks are completely altered by the presenceof the Σ matrix in Eq. 7. For instance, the H + t ′ b and H + tb ′ couplings have new terms proportional to V tb and5 ε b B r a n c h i n g R a t i o (cid:24) m t’ =400 GeV , θ =0.05 m b’ =500 GeV , m h =220 GeV , m H+ =300 GeV , tan β =1 , ε t =m t /m t’ t’ =400 GeV , θ =0.2b’ −> b hb’ −> t H + b’ −> t Wb’ −> t’ W 0 0.1 0.2 0.3 0.4 0.5 ε b m t’ =450 GeV , θ =0.05m t’ =450 GeV , θ =0.2 FIG. 19:
The branching ratios for the decay channels b ′ → bh , b ′ → tH + , b ′ → tW and b ′ → t ′ W , as a func-tion of ǫ b for m b ′ = 500 GeV, m h = 220 GeV, m H + = 300 GeV, tan β = 1 , ǫ t = m t /m t ′ and ( m t ′ [GeV] , θ ) =(400 , . , (400 , . , (450 , . , (450 , . , as indicated. Also, α = π/ and m A > m b ′ is assumed. V t ′ b ′ . In particular, in the “3+1” scenario where V t ′ d i , V u i b ′ → i = 1 , ,
3, we have: L ( H + t ′ b ) ≈ g √ m W t β (cid:16) t − β (cid:17) ¯ t ′ ( m t ǫ t V tb L − m b ′ ǫ b V t ′ b ′ R ) bH + , (81) L ( H + tb ′ ) ≈ g √ m W t β (cid:16) t − β (cid:17) ¯ t ( m ′ t ǫ ⋆t V t ′ b ′ L − m b ǫ ⋆b V tb R ) b ′ H + . (82)Recall that in the standard 2HDM of type II that also underlies supersymmetry (assuming four generations offermions) the ¯ t ′ R b L H + would be ∝ m t ′ V t ′ b /t β . We thus see that in our 4G2HDM-I the ¯ t ′ R b L H + coupling ispotentially enhanced by a factor of t β · ǫ t · ( m t /m t ′ ) · ( V tb /V t ′ b ). For example, if t β = 3, m t ′ ∼
500 GeV and ǫ t ∼ m t /m t ′ we get a factor of V tb /V t ′ b enhancement to the ¯ t ′ R b L H + interaction.The implications of the above new Yukawa interactions can be far reaching with regard to the decay patterns of the t ′ and the b ′ and the search strategies for these heavy quarks. In particular, in Fig. 17 we plot the branching ratiosof the leading t ′ decay channels (assuming m H + , m A > m t ′ ): t ′ → th, bW, b ′ W ( ⋆ ) [ W ( ⋆ ) stands for either on-shell oroff-shell W depending on the m b ′ ], as a function of the b ′ mass. We use m t ′ = 500 GeV, tan β = 1, ǫ t = m t /m t ′ andthe following values for m h and θ : ( m h [GeV] , θ ) = (120 , . , (120 , . , (220 , . , (220 , . BR ( t ′ → th ) can easily reach O (1), in particular when m t ′ − m b ′ < m W and even for a rather large θ ∼ .
2; seee.g., points 10 and 11 in Table II for which BR ( t ′ → th ) ∼ O (1).In Fig. 18 we take m H + = 400 GeV (again assuming m A > m t ′ so that t ′ → tA is still kinematically closed) and plot BR ( t ′ → bH + ) as a function of ǫ t , for m t ′ = 500 GeV, tan β = 1, ǫ b = m b /m b ′ ∼ . m h = 200 and 350 GeV and thefollowing values for m b ′ and θ ( m b ′ [GeV] , θ ) = (400 , . , (400 , . , (450 , . , (450 , . t ′ → bH + can become important and even dominate if ǫ t > ∼ m t /m t ′ , in particular, when m t ′ − m b ′ < m W anda small mixing angle of θ ∼ O (0 . BR ( t ′ → bH + ) ∼ O (1).6In Fig. 19 we plot the branching ratios of the leading b ′ decay channels, as a function of ǫ b for m b ′ = 500 GeV,tan β = 1, m H + = 300 GeV, m h = 220 GeV, ǫ t = m t /m t ′ and the following values for m t ′ and θ ( m t ′ [GeV] , θ ) =(400 , . , (400 , . , (450 , . , (450 , . b ′ case the dominance of b ′ → tH − (if kinematicallyallowed) is much more pronounced due to the expected smallness of the b − b ′ mixing parameter, ǫ b , which controlsthe FC decay b ′ → bh ; see e.g., point 13 in Table II for which BR ( b ′ → bH − ) ∼ O (1).This change in the decay pattern of the 4th generation quarks can have important consequences for collider searchesof these heavy fermions. For example, as was already noticed in [34], if t ′ → th dominates then t ′ production at theLHC via gg → t ′ ¯ t ′ will lead to the dramatic signature of t ¯ thh . If m h < m W (so that h decays to b ¯ b ) this willgive a 6 b + 2 W signature (i.e., after the top decays via t → bW ), while if m h > m W , m Z the t ¯ thh final state canlead to either t ¯ thh → t ¯ tW + W − and/or t ¯ thh → t ¯ tZZ . In particular, notice that the former t ¯ tW + W − is the oneconventionally used for b ′ searches [32], while the latter will lead to e.g., a 2 b + 2 W + 4 ℓ signature which is expectedto have a rather small irreducible SM-like background (e.g., coming from gg → t ¯ th ) that can be further controlledusing the kinematic features of the process gg → t ′ ¯ t ′ → t ¯ thh → t ¯ tZZ . If, on the other hand, t ′ → bH + dominates,then the signature b ¯ bH + H − should be focused on. In this case the t ′ searches will depend on the H + decays, e.g., H + → tb or H + → τ ν , which will lead to gg → t ′ ¯ t ′ → b + 2 W or gg → t ′ ¯ t ′ → b + 2 τ + E T , respectively.For the b ′ the situation is similar, i.e, the new decays b ′ → bh and/or b ′ → tH − can also alter the search strategiesfor b ′ . For example, if b ′ → tH − dominates the b ′ decays, then gg → b ′ ¯ b ′ will lead to, e.g., a t ¯ tH − H + → t + 2 b signature as opposed to the “standard” 2 t + 2 W one when the b ′ decays via b ′ → tW [90].Clearly, these new 4th generation quark signatures deserve a detailed investigation which is beyond the scope ofthis paper and will be considered elsewhere [91]. V. SUMMARY
We have introduced a class of 2HDMs, which we named the 4G2HDM of types I, II and III. Our models are“designed” to give an effective low-energy description for the apparent heaviness of the 4th generation fermions andto address the possibility of dynamical EWSB which is driven by the condensates of these new heavy fermionic states.This is done by giving a special status to the 4th family fermions which are coupled to the scalar doublet that hasthe heavier VEV. Such setups give rise to very distinct Yukawa textures which can have drastic implications on thephenomenology of 4th generation fermions systems. We studied the constraints from PEWD and from flavor physicsin B-systems and outlined the allowed parameter space of our 4G2HDMs, which we find to have various differentfeatures than the simpler SM4 version with a single Higgs boson and a 4th family of fermions. For example, we findthat the mass splitting m t ′ − m b ′ and the inverted mass splitting m b ′ − m t ′ can be as large as 200 GeV, and that themass splitting in the 4th generation lepton doublet can be as large as 400 GeV.We focused on the 4G2HDM-I, where the Higgs doublet with the heavier VEV is coupled only to the 4th generationdoublet while the “lighter” Higgs doublet is coupled to all other quarks. This model is, in our view, somewhat bettermotivated as it provides a more natural setup in the leptonic sector, i.e., addressing the existence of a 4th generationEW-scale neutrino. In addition, it has very distinctive features in flavor space: there are no tree-level FCNC amongthe 1st three generation of fermions as well as no FCNC among the 4th generation fermions and the light fermions ofthe 1st and 2nd generations. On the other hand, the 4G2HDM-I does give rise to potentially large tree-level FCNC t ′ → t and b ′ → b transitions, which, as we briefly explored in the paper, can have significant implications on thesearch for the 4th generation quarks at high-energy colliders. For example, the FC decay t ′ → th can become thedominant t ′ decay channel and should therefore effect the search strategy for the t ′ .Finally, we note that the 4G2HDM setups can also alter the production and decay patterns of the Higgs particles athadron colliders. For example, the di-photon Higgs channel gg → h → γγ can be dramatically enhanced or suppressed(to the level of being unobservable at the LHC) compared to the SM4 case. The phenomenology of the productionand decay channels of the Higgs particles in the 4G2HDMs will be considered elsewhere. Acknowledgments:
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