We review the possible role that multi-Higgs models may play in our understanding of the dynamics of a heavy 4th sequential generation of fermions. We describe the underlying ingredients of such models, focusing on two Higgs doublets, and discuss how they may effectively accommodate the low-energy phenomenology of such new heavy fermionic degrees of freedom. We also discuss the constraints on these models from precision electroweak data as well as from flavor physics and the implications for collider searches of the Higgs particles and of the 4th generation fermions, bearing in mind the recent observation of a light Higgs with a mass of ~125 GeV.
1. Introduction: The “Need” of a Multi-Higgs Setup for the 4th Generation
The minimal and perhaps the simplest framework for incorporating 4th generation fermions can be constructed by adding to the standard model (SM) a 4th sequential generation of fermion (quarks and leptons) doublets (for reviews see [1–3]). This framework, which is widely known as the SM4, can already address some of the leading theoretical challenges in particle physics:
the hierarchy problem [4–11],
the origin of matter/antimatter asymmetry in the universe [12–16],
flavor physics and CKM anomalies [17–30].
Unfortunately, the current bounds on the masses of the 4th generation quarks within the SM4 are rather high, reaching up to ~600 GeV [31–34], that is, around the unitarity bounds on quark masses [35–37]. The implications of such a “superheavy” 4th generation spectrum are far reaching. In fact, the SM4 as such is also strongly disfavored from searches at the LHC [38–41] and Tevatron [42] of the single Higgs particle of this model, essentially excluding the SM4 Higgs with masses up to 600 GeV [43, 44] and, thus, making it incompatible with the recent observation/evidence of a light Higgs with a mass of ~125 GeV [45, 46] (for a recent comprehensive analysis of the SM4 status in light of the latest Higgs results and electroweak precision data (EWPD), we refer the reader to [47]). These rather stringent limits on the SM4 raise several questions at the fundamental level:
Are superheavy fermionic degrees of freedom a surprise or is that expected once new physics (NP), beyond the SM4 (BSM4), is assumed to enter at the TeV scale?
Are such heavy fermions linked to strong dynamics and/or to compositeness at the nearby TeV scale?
What sub-TeV degrees of freedom should we expect if indeed such heavy fermions are found? And what is the proper framework/effective theory required to describe the corresponding low energy dynamics?
How do such heavy fermions affect Higgs physics?
Can one construct a natural framework for 4th generation heavy fermions with a mass in the range 400–600 GeV that is consistent with EWPD and that is not excluded by the recent direct measurements from present high-energy colliders?
What type of indirect hints for BSM4 dynamics can we expect in low energy flavor physics?
In this paper we will try to address these questions by considering a class of BSM4 low energy effective theories which are based on multi-Higgs models.
Let us start by studying the hints for BSM4 and strong dynamics from the evolution of the 4th generation Yukawa coupling y4, under some simplifying assumptions. In particular, one can write the RGE of y4 assuming SM4 dynamics and neglecting the gauge and the top-Yukawa couplings and taking all 4th generation Yukawa couplings equal [48]
(1)(16π2)μ∂∂μy4≃(2y4)3.
This yields a Landau Pole (defined by 1/y42(μ=Λy)→0) at Λy~m4eπ2v2/2m42, giving Λy~8,3,2 TeV for m4~300,400,500 GeV. Therefore, within the SM4, the 4th generation Yukawa couplings are expected to “run into” a Landau Pole at the near by TeV scale.
In fact, there are additional strong indications from the Higgs sector that a heavy 4th generation of fermions is tied with new strong dynamics at the near by TeV scale and that the SM4 is not the adequate framework to describe the new TeV-scale physics
(1)The Higgs Mass Correction Due to Such Heavy Fermions Is Pushed to the Cutoff Scale. To see that, one can calculate the self-energy 1-loop correction to the Higgs mass with the exchange of a heavy fermion q′ and set the cutoff to Λ>mq′, obtaining
(2)δmH2~(mq′400GeV)2·Λ2,
indicating that a heavy 4th family fermion with a mass around 400 GeV cannot coexist with the recently observed single light Higgs, since in the absence of fine tuning, the Higgs mass should be pushed up to the cutoff scale where the NP enters (in which case the definition of the Higgs particle becomes meaningless).
(2)The SM4 Higgs Quartic Coupling (λ)and a Heavy Higgs. One can again study the RGE for λ, assuming SM4 dynamics and neglecting the gauge and the top-Yukawa couplings and taking all 4th generation Yukawa couplings equal. One then obtains [48]
(3)(16π2)μ∂∂μλ≃24λ2+16y42(2λ-y42)θ(μ-m4),
giving a Landau Pole (i.e., λ(μ=Λλ)→∞) at Λλ~4.3,2.5,2.1 TeV for mH~500,600,700 GeV and, thus, indicating that a light Higgs is not consistent with the SM4 if the NP scale is at the few-TeV range. Indeed, solving the full RGE for the SM4 one finds that mH≳mq′ when the cutoff of the theory is set to the TeV scale, that is, to the proper cutoff of the SM4 when mq′~𝒪(500) GeV [48]. The implications of a heavy Higgs in this mass range was considered, for example, in [49–52], claiming that the heavy SM4 Higgs case can relax the currently reported exclusion on the SM4. However, the heavy SM4 Higgs scenario is now in contradiction with the recent measurements of the two experiments at the LHC, which observe a light Higgs boson with a mass of ~125 GeV [38–41]. On the other hand, as will be shown in this paper (and was also demonstrated before in [48] for the case of the popular 2HDM of type II with a 4th generation of doublets), a multi-Higgs setup for the 4th generation theory can relax the constraint mH≳mq′.
Thus, under the assumption that heavy 4th generation quarks exist, if one assumes a light Higgs with a mass around 125 GeV and seriously takes into account the fact that low energy 4th generation theories possess a new threshold/cutoff (or a fixed point; see, e.g., [53, 54]) at the TeV scale, then one is forced to consider extensions of the naive SM4 with more than one Higgs doublet which, in turn, leads to the possibility that the Higgs particles (or some of the Higgs particles) may be composites primarily of the 4th generation fermions (see, e.g., [55–60]), with condensates 〈QL′tR′〉≠0, 〈QL′bR′〉≠0 (and possibly also 〈LL′νR′〉≠0, 〈LL′τR′〉≠0). These condensates then induce EWSB and generate a dynamical mass for the condensing fermions. This viewpoint in fact dates back to an “old” idea suggested more than two decades ago [4]; that a heavy top quark may be used to form a tt- condensate which could trigger dynamical EWSB. Although, this top-condensate mechanism led to the prediction of a too large mt, this idea ignited further thoughts and studies towards the possibility that 4th generation fermions may play an important role in dynamical EWSB [4–9]. In particular, due to the presence of such heavy fermionic degrees of freedom, some form of strong dynamics and/or compositeness may occur at the near by TeV scale.
In this article, we will review the above viewpoint which was also adopted in [61]: that theories which contain such heavy fermionic states are inevitably cutoff at the near by TeV scale and are, therefore, more naturally embedded at low energies in multi-Higgs models, which are the proper low-energy effective frameworks for describing the sub-TeV dynamics of 4th generation fermions. As mentioned above, in this picture, the Higgs particles are viewed as the composite scalars that emerge as manifestations of the different possible bound states of the fundamental heavy fermions. This approach was considered already 20 years ago by Luty [62] and more recently in [60], where an attempt to put 4th degeneration heavy fermions into an effective multi-(composite) Higgs doublets model was made, using a Nambu-Jona-Lasinio (NJL) type approach.
The phenomenology of multi-Higgs models with a 4th family of fermions was studied to some extent recently in [48, 63–69] and within a SUSY framework in [14, 16, 70–72]. In this article, we will further study the phenomenology of 2HDM frameworks with a 4th family of fermions, focusing on a new class of 2HDM’s “for the 4th generation” (named hereafter 4G2HDM) that can effectively address the low-energy phenomenology of a TeV-scale dynamical EWSB scenario, which is possibly triggered by the condensates of the 4th generation fermions.
We will first describe a few viable manifestations of a 2HDM framework with a 4th generation of fermions, focusing on the 4G2HDM framework of [61]. We will then discuss the constraints on such 4th generation 2HDM models from PEWD as well as from flavor physics. We will end by studying the expected implication of such 2HDM frameworks on direct searches for the 4th generation fermions and for the Higgs particle(s), assuming the existence of a light Higgs with a mass of 125 GeV.
2. 2HDM’s and 4th Generation Fermions
Assuming a common generic 2HDM potential, the phenomenology of 2HDM’s is generically encoded in the texture of the Yukawa interaction Lagrangian. The simplest variant of a 2HDM with 4th generations of fermions can be constructed based on the so-called type II 2HDM (which we denote hereafter by 2HDMII), in which one of the Higgs doublets couples only to up-type fermions and the other to down-type ones. This setup ensures the absence of tree-level flavor changing neutral currents (FCNC) and is, therefore, widely favored when confronted with low energy flavor data. The Yukawa terms of the 2HDMII, extended to include the extra 4th generation quark doublet, are (and similarly in the leptonic sector)
(4)ℒY=-Q-LΦdFddR-Q-LΦ~uFuuR+h.c. ,
where fL(R) are left-(right) handed fermion fields, QL is the left-handed SU(2) quark doublet, and Fd, Fu are general 4×4 Yukawa matrices in flavor space. Also, Φd,u are the Higgs doublets:
(5)Φi=(ϕi+vi+ϕi02),Φ~i=(vi*+ϕi0*2-ϕi-).
Motivated by the idea that the low energy scalar degrees of freedom may be the composites of the heavy 4th generation fermions, it is possible to construct a new class of 2HDM’s that effectively parameterize 4th generation condensation by giving a special status to the 4th family fermions. This was done in [61], where (in the spirit of the Das and Kao 2HDM that was based on the SM’s three families of fermions [73]) one of the Higgs fields (ϕh—call it the “heavier” field) was assumed to couple only to heavy fermionic states, while the second Higgs field (ϕℓ—the “lighter” field) is responsible for the mass generation of all other (lighter) fermions. The possible viable variants of this approach can be parameterized as [61] (and similarly in the leptonic sector)
(6)ℒY=-Q-L(ΦℓFd·(I-ℐdαdβd)+ΦhFd·ℐdαdβd)dR-Q-L(Φ~ℓFu·(I-ℐuαuβu)+ΦhFu·ℐuαuβu)uR+h.c.,
where Φℓ,h are the two Higgs doublets, I is the identity matrix, and ℐqαqβq (q=d,u) are diagonal 4×4 matrices defined by ℐqαqβq≡diag(0,0,αq,βq).
The Yukawa interaction Lagrangian of (6) can lead to several interesting textures that can be realized in terms of a Z2-symmetry under which the fields transform as follows:
(7)Φℓ⟶-Φℓ,Φh⟶+Φh,QL⟶+QL,dR⟶-dR(d=d,s),uR⟶-uR(u=u,c),bR⟶(-1)1+αdbR,bR′⟶(-1)1+βdbR′,tR⟶(-1)1+αutR,tR′⟶(-1)1+βutR′,
which allows us to construct several models that have a non-trivial Yukawa structure and that are potentially associated with the following compositeness scenario
Type I 4G2HDM: denoted hereafter by 4G2HDMI and defined by (αd,βd,αu,βu)=(0,1,0,1), in which case Φh gives masses only to t′ and b′, while Φℓ generates masses for all other quarks (including the top quark). For this model, which seems to be the natural choice for the leptonic sector, we expect
(8)tanβ≡vhvℓ≈mq′mt~𝒪(1).
Type II 4G2HDM: denoted hereafter by 4G2HDMII and defined by (αd,βd,αu,βu)=(1,1,1,1), in which case the heavy condensate Φh couples to the heavy quarks states of both the 3rd and 4th generations t and b quarks, whereas Φℓ couples to the light quarks of the 1st and 2nd generations. For this model one expects tanβ≫1.
Type III 4G2HDM: denoted hereafter by 4G2HDMIII and defined by (αd,βd,αu,βu)=(0,1,1,1), in which case mt, mb′, and mt′∝vh, so that only quarks with masses at the EW-scale are coupled to the heavy doublet Φh. Here also one expects tanβ≫1.
The Yukawa interactions for these models are given by [61]
(9)ℒ(hqiqj)=g2mWq-i{mqisαcβδij-(cαsβ+sαcβ)·[mqiΣijqR+mqjΣjiq⋆L](cαsβ+sαcβ)}qjh,ℒ(Hqiqj)=g2mWq-i{-mqicαcβδij+(cαcβ-sαsβ)·[mqiΣijqR+mqjΣjiq⋆L](cαcβ-sαsβ)}qjH,ℒ(Aqiqj)=-iIqgmWq-i{[mqiΣijqR-mqjΣjiq⋆L]mqitanβγ5δij-(tanβ+cotβ)·[mqiΣijqR-mqjΣjiq⋆L]}qjA,ℒ(H+uidj)=g2mWu-i{[mdjtanβ·Vuidj-mdk(tanβ+cotβ)·VikΣkjdmdjtanβ]R+[-muitanβ·Vuidj+muk(tanβ+cotβ)·Σkiu⋆Vkj-muitanβ·Vuidj]L-muitanβ·Vuidj}djH+,
where V is the 4×4 CKM matrix, q=d or u for down or up quarks with weak isospin Id=-1/2 and Iu=+1/2, respectively, and R(L)=(1/2)(1+(-)γ5). Also, the 4G2HDM type, that is, the 4G2HDMI, 4G2HDMII, and 4G2HDMIII, as well as FCNC effects are all encoded in Σd and Σu, which are new mixing matrices in the down- (up-) quark sectors, obtained after diagonalizing the quarks mass matrices:
(10)Σijd=Σijd(αd,βd,DR)=αdDR,3i⋆DR,3j+βdDR,4i⋆DR,4j,Σiju=Σiju(αu,βu,UR)=αuUR,3i⋆UR,3j+βuUR,4i⋆UR,4j,
depending on DR, UR which are the rotation (unitary) matrices of the right-handed down and up quarks, respectively, and on whether αq and/or βq are “turned on.” This is in contrast to “standard” frameworks such as the SM4 and the 2HDM’s of types I and II, where the right-handed mixing matrices UR and DR are nonphysical being “rotated away” in the diagonalization procedure of the quark masses. Indeed, in the 4G2HDM’s described above some elements of DR and UR can, in principle, be measured in Higgs-fermion systems, as we will later show.
In particular, inspired by the working assumption of the 4G2HDM’s and by the observed flavor pattern in the up-and down-quark sectors, it was shown in [61] that the new mixing matrices Σd and Σu are expected to have the following form: (11)Σu=(00000αu|ϵc|2αuϵc⋆(1-|ϵt|22)-αuϵc⋆ϵt⋆0αuϵc(1-|ϵt|22)αu(1-|ϵt|22)+βu|ϵt|2(βt-αt)ϵt⋆(1-|ϵt|22)0-αuϵcϵt(βu-αu)ϵt(1-|ϵt|22)αu|ϵt|2+βu(1-|ϵt|22)),
and similarly for Σd by replacing αu,βu→αd,βd and ϵc,ϵt→ϵs,ϵb. The new parameters ϵc, ϵt are free parameters that effectively control the mixing between the 4th generation t′ and the 2nd and 3rd generation quarks c and t, respectively. Thus, a natural choice which will be adopted here in some instances is |ϵt|=~mt/mt′, |ϵb|=~mb/mb′ and ϵs,ϵc→0.
3. Constraints on 2HDM’s with a 4th Generation of Fermions3.1. Constraints from Electroweak Precision Data: Oblique Parameters
The sensitivity of EWPD to 4th generation fermions within the minimal SM4 framework was extensively analyzed in the past decade [74–80]. Here we are interested instead in the constraints that EWPD impose on 2HDM’s with a 4th generation family. As usual, the effects of the NP can be divided into the effects of the heavy NP which does and which does not couple directly to the ordinary SM fermions. For the former, the leading effect comes from the decay Z→bb-, which is mainly sensitive to the H+t′b and W+t′b couplings through one-loop exchanges of H+ and W+ shown in Figure 1, and which was analyzed in detail in [61].
One-loop diagrams for corrections to Z→dId-J from charged Higgs loops. Similar diagrams with W-t′ loops contribute as well.
On the other hand, the effects which do not involve direct couplings to the ordinary fermions can be 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 [81, 82]. For the oblique parameters the effects of a 2HDM with a 4th generation are common to any variant of a 2HDM framework (including the 2HDMII, and the 4G2HDMI, 4G2HDMII and 4G2HDMIII described in the preivous section), since the Hff Yukawa interactions of any 2HDM do not contribute at 1 loop to the gauge-bosons self-energies.
In particular, apart from the pure 1-loop Higgs exchanges, one also has to include the new contributions from t′ and b′ exchanges which shift the T parameter (ΔTf) and which involve the new SM4-like diagonal coupling Wt′b′ as well as the Wt′b and Wtb′ off-diagonal vertices (see, e.g., [79]):
(12)ΔTf=38πsW2cW2(13|Vt′b′|2Ft′b′+|Vt′b|2Ft′b+|Vtb′|2Ftb′-|Vtb|2Ftb+13Fℓ4ν413|Vt′b′|2),
with
(13)Fij=xi+xj2-xixjxi-xjlogxixj,
and xk≡(mk/mZ)2.
The complete set of corrections to the S and T parameters within a 2HDM with a 4th generation of fermions was considered in [61, 75, 83]. Following the recent analysis in [61], we show in Figure 2 the results of “blindly” (randomly) scanning the relevant parameter space with 100000 models, where we set the light Higgs mass to be mh=125GeV and vary the rest of the relevant parameters within the ranges: tanβ≤30, θ34≤0.3, 150GeV≤mH≤1TeV, 150GeV≤mA≤1TeV, 200GeV≤mH+≤1TeV, 400GeV≤mt′, mb′≤600GeV, and 100GeV≤mν′ and mτ′≤1.2TeV, and the CP-even neutral Higgs mixing angle in the range 0≤α≤2π. In particular, we plot in Figure 2 the allowed points in parameter space projected onto the 68%, 95%, and 99% allowed contours in the S-T plane, and the 95% CL allowed range in the mH+-tanβ and the Δmq′-Δmℓ′ planes, corresponding to the 95% CL contour in the S-T plane.
(a) The allowed points in parameter space projected onto the 68%, 95%, and 99% CL contours in the S-T plane. (b) 95% CL allowed range in the mH+-tanβ plane. (c) Allowed region in the Δmq′-Δmℓ′ plane within the 95% CL contour in the S-T plane. All plots are for any 2HDM setup (such as the 2HDMII and the three types of the 4G2HDM; see text) and with 100000 data points setting the light Higgs mass to mh=125 GeV and varying the rest of the parameters in the ranges tanβ≤30, θ34≤0.3, 150GeV≤mH≤1TeV, 150GeV≤mA≤1TeV, 200GeV≤mH+≤1TeV, 400GeV≤mt′, mb′≤600GeV, 100GeV≤mν′, and mτ′≤1.2TeV and the CP-even neutral Higgs mixing angle in the range 0≲α≲2π.
We find that compatibility with PEWD mostly requires tanβ~𝒪(1) with a small number of points in parameter space having tanβ≳5. We also find that the 2HDM frameworks allow 4th generation quarks and leptons mass splittings extended to -200GeV≲Δmq′≲200GeV and -400GeV≲Δmℓ′≲400GeV, and “solutions” where both the quarks and the leptons of the 4th generation doublets are degenerate. For the cases of a small (or no) 4th generation fermion mass splitting, the amount of isospin breaking required to compensate for the effect of the extra fermions and Higgs particles on S and T is provided by a mass splitting among the Higgs particles; see [61].
The effects of the NP in Z→bb- are best studied via the well-measured quantity Rb:
(14)Rb≡Γ(Z→bb-)Γ(Z→hadrons),
which is a rather clean test of the SM, since being a ratio between two hadronic rates, most of the electroweak, oblique, and QCD corrections cancel between numerator and denumerator.
Following [61], the effects of NP in Rb can be parameterized in terms of the corrections δb and δc to the decays Z→bb- and Z→cc-, respectively:
(15)Rb=RbSM1+δb1+RbSMδb+RcSMδc,
where RbSM and RcSM are the corresponding 1-loop quantities calculated in the SM and δq are the NP corrections defined in terms of the Zqq- couplings as
(16)δq=2gqLSMgqLnew+gqRSMgqRnew(gqLSM)2+(gqRSM)2,
where
(17)VqqZ≡-igcWq-γμ(g-qLL+g-qRR)qZμ,
with sW(cW)=sinθW(cosθW), L(R)=(1-(+)γ5)/2 and g-qL,R=gqL,RSM+gqL,Rnew, so that gqL,RSM are the SM (1-loop) quantities and gqL,Rnew are the NP 1-loop corrections.
The corrections to Rb from the 4th generation quarks in the 4G2HDMI, 4G2HDMII, and 4G2HDMIII are of three types (see [61]), where in all cases one finds that δc≪δb, so that one can safely neglect the effects from Z→cc-.
3.2.1. SM4-Like Corrections
These are the corrections to gqL due to the 1-loop W-t′ exchanges (denoted here as gqLSM4), which are given by [18, 79, 84, 85]
(18)gqLSM4=g264π2cW2(mt′2mZ2-mt2mZ2)sin2θ34,
where θ34 is the mixing angle between the 3rd and 4th generation quarks, that is, defining |Vt′b|=|Vtb′|≡sinθ34, and the 2nd term ∝-sin2θ34mt2/mZ2 is the decrease from the SM’s tb correction to the W-boson vacuum polarization, which in the 4th generation case is ∝|Vtb|2=cos2θ34=1-sin2θ34.
The SM4-like effect on Rb is plotted in Figure 3, from which we can see that Rb puts rather stringent constraints on the mt′-θ34 plane which is the SM4 subspace of the parameter space of any 2HDM containing a 4th generation of fermions. For example, for mt′~500 GeV the t′-b mixing angle is restricted to θ34≲0.2.
Rb in the SM4, as a function of θ34 for several values of the t′ mass (a) and as a function of mt′ for θ34=0.1 and 0.2 (b). Figure taken from [61].
The corrections from the 1-loop H+-t′ exchanges are plotted in Figure 1. In the 4G2HDM of types II and III, these charged Higgs exchange diagrams are found to have negligible effects on Rb and are, therefore, not constrained by this quantity. On the other hand, Rb is rather sensitive to the charged Higgs 1-loop exchanges within the 4G2HDMI. This can be seen in Figure 4, where Rb is plotted (for the 4G2HDMI case) as a function of the charged Higgs and t′ masses, fixing ϵt=mt/mt′ and focusing on the values tanβ=1,5, θ34=0,0.2, and mH+=400,750 GeV.
Rb in the 4G2HDMI (upper plots, figure taken from [61]) and in the 2HDMII (lower plots), as a function of the charged Higgs mass (left plots) for mt′=500 GeV, and (tanβ,θ34)=(1,0),(1,0.2),(5,0),(5,0.2), and as a function of mt′ (right plots), for θ34=0.2 and (mH+[GeV],tanβ)=(400,1),(400,5),(750,1),(750,5) (right). In the 4G2HDMI case we use ϵt=mt/mt′. The long-dashed horizontal lines represent the upper and lower 2σ (measured) bounds on Rb.
In Figure 5 we further plot the allowed ranges in the mH+-tanβ plane in the 4G2HDMI, subject to the Rb constraint (at 2σ), for tanβ in the range 1–15, fixing mt′=500 GeV, mb′=450 GeV, θ34=0.2, ϵb=mb/mb′ (which also enters the t′bH+ vertex) and for three representative values of the t-t′ mixing parameter: ϵt=ϵb~0.01, ϵt=mt/mt′~0.35, and ϵt=1. We see, that, as expected, when tanβ is lowered, the constraints on the charged Higgs mass are weakened. In particular, while there are no constraints from Rb on the charged Higgs and t′ masses if tanβ~𝒪(1), for higher values of tanβ a more restricted region of the charged Higgs mass is imposed which again depends on θ34. We see for example, that for ϵt=mt/mt′~0.35, tanβ~1 is compatible with mH+ values ranging from 200 GeV up to the TeV scale, while for tanβ~5, the charged Higgs mass is restricted to be within the range 450GeV≲mH+≲750 GeV.
Allowed area in the mH+-tanβ in the 4G2HDMI, subject to the Rb measurement (within 2σ), for mt′=500 GeV, mb′=450 GeV, θ34=0.2, and ϵb=mb/mb′ and for three values of the t-t′ mixing parameter: ϵt=ϵb~0.01 (a), ϵt=mt/mt′~0.35 (b), and ϵt=1 (c). Figure taken from [61].
For the case of the 2HDMII (i.e., extended with a 4th family of fermions), which is also plotted in Figure 4, we find that there is essentially no constraint in the mH+-tanβ plane for mt′≲500 GeV.
3.2.3. The Flavor Changing <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M264"><mml:msup><mml:mrow><mml:mi>ℋ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mi>b</mml:mi><mml:msup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> Interactions
The 1-loop corrections to Rb which involve the flavor changing (FC) ℋ0bb′ interactions emanate from the nondiagonal 34 and 43 elements in Σd, with ℋ0=h,H, or A. These corrections are found to be much smaller than 1-loop H+ exchanges, so that they can be safely neglected, in particular for ϵb≪1.
3.3. Constraints from Flavor in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M272"><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:math></inline-formula> Physics3.3.1. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M273"><mml:mover accent="true"><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mo>-</mml:mo></mml:mover><mml:mo mathvariant="bold">→</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mi>γ</mml:mi></mml:math></inline-formula>
Flavor physics plays an important role in discriminating between the various NP models. In this regard, FCNC decays can provide key information about the SM and its various extensions.
The inclusive radiative decay B-→Xsγ is indeed known to be a very sensitive probe of NP. The underlying process is induced by the FC decay of the b-quark into a strange quark and a photon. The Br(B-→Xsγ) has already carved out large regions of the parameter space of most of the NP models [89–102]. On the other hand, model independent analysis in the effective field theory approach without [103] and with [104] the assumption of minimal flavor violation also show the strong constraining power of the decay B-→Xsγ. Once more precise data from super-B factories are available, this decay will undoubtedly be more efficient in selecting the viable regions of the parameter space in the various classes of NP models.
The calculation of the decay rate of the B-→Xsγ transition is most conveniently performed after integrating out the heavy degrees of freedom. The resulting effective theory contains various FC dimension-five and -six local interactions and the inclusive decay rate is given by
(19)Γ(b→Xsγ)Eγ>E0=GF2mb5αem32π4|Vts*Vtb|2∑i,j=18Ci(μb)Cj(μb)Gij(E0,μb),
where the Wilson coefficients, Ci, of the effective operators (see below) are perturbatively calculable at the relevant renormalization scale and the Renormalization Group Equations (RGE) can be used to evaluate Ci at the scale μb~mb/2. At present, all the relevant Wilson coefficients Ci(μb) are known at the Next-to-Next-to-Leading Order (NNLO) [105–116] and Gij(E0,μb) is determined by the matrix elements of the operators O1,…,O8 [107, 108]:
(20)O1,2=(s-Γic)(c-Γi′b)(current-current operators)O3,4,5,6=(s-Γib)∑q(q-Γi′q)(four-quark penguin operators)O7=emb16π2s-LσμνbRFμν(photonic dipole operator)O8=gmb16π2s-LσμνTabRGμνa(gluonic dipole operator),
which consists of perturbative and nonperturbative corrections. The perturbative corrections are well under control and are fully known at NLO QCD [117]. However, quantitative estimates of all the non-perturbative effects are not available, although they are believed to be ≈5% [117].
The inclusive branching ratio in the SM is given by [89]
(21)ℬ(B-⟶Xsγ)Eγ>1.6GeVNNLO=(3.15±0.23)×10-4,
whereas the current experimental data gives [118]
(22)ℬ(B-⟶Xsγ)Eγ>1.6GeVexp=(3.55±0.24±0.09)×10-4.
The SM prediction is, thus, consistent with the experiment (both having a 7% error) and is therefore useful for constraining many extensions of the SM.
In the SM4, there are no new operators other than the ones present in the SM. However, there are extra contributions to the Wilson coefficients corresponding to the operators O7 and O8 from t′-loops [17–20]. In a 2HDM framework with a 4th generation family, the new ingredient with respect to the SM4 is the presence of the charged Higgs 1-loop exchanges which contribute to the Wilson coefficients of the effective theory. In particular, at the parton level within a 2HDM, B-→Xsγ proceeds via the penguin diagrams depicted in Figure 6. As was shown in [61], in the 4G2HDMI, 4G2HDMII, and 4G2HDMIII frameworks, the leading effects enter in C7 and C8 from the 1-loop exchanges of t′-W, t-H+, and t′-H+.
Examples of one-loop 1PI diagrams that contribute to b→sγ in a 2HDM framework, with W bosons, charged Higgs, and 4th generation quarks exchanges (ui=u,c,t,t′).
An important role for constraining NP in the b-quark system is also played by Bq-B-q (q=d,s) mixing, the phenomenon of which is described by the dispersive part M12q of the Bq mixing amplitude. The current theory precision is limited by lattice results; the SM prediction still allows NP contributions to |M12s| of order 20% [119].
Within a 2HDM setup, the leading contribution to Bq-B-q (q=d,s) mixing comes from the box diagrams shown in Figure 7, where the G boson is replaced by the charged Higgs H+, and the fermions ui,j are replaced by (t,t′). Thus, the net contribution to the mass difference ΔMq=2|M12q| is given by [61]
(23)M12q=GF212π2MW2fBq2BqMBq[MWW+MHH+MHW],
where
(24)MWW=λbqt2ηttSWW(xt)+λbqt′2ηt′t′SWW(xt′)MWW=+2λbqtλbqt′ηtt′SWW(xt,xt′),MHH=λbqt2SHH(yt)+λbqt′2SHH(yt′)MHH=+2λbqtλbqt′SHH(yt,yt′),MHW=λbqt2SHW(xt,z)+λbqt′2SHW(xt′,z)MHW=+2λbqtλbqt′SHW(xt,xt′,z),
and z=mH+2/mW2, xi=mi2/mW2, yi=mi2/mH+2 (i=t or t′), and λdidju≡Vudi⋆Vudj. Here, MWW, MHH, and MHW are the contributions from the box diagrams with the combination of the gauge bosons (W,W), (W,H), and (H,H) in the internal lines (H stands for the charged Higgs), respectively. The detailed expression for the various Inami-Lim functions Si,j is given in [61].
The B0-B-0 (representative) box diagrams with different combinations of the gauge bosons (W,G) and the fermions (ui,uj) in the internal lines. The same diagrams contribute to Bs-B-s mixing with the d-quark in the external lines replaced by the s-quark.
For the B-physics parameters, we use the inputs given in Table 1, and for the 4th generation quark masses, we take mt′=500 GeV and mb′=450 GeV.
Inputs used for the B-physics parameters in the analysis below. When not explicitly stated, we take the inputs from Particle Data Group [74].
fbdBbd=0.224±0.015 GeV [86, 87]
|Vub|=(32.8±2.6)×10-4a
ξ=1.232±0.042 [86, 87]
|Vcb|=(40.86±1.0)×10-3
ηt=0.5765±0.0065 [88]
γ=(73.0±13.0)°
ΔMs=(17.77±0.12)ps-1
ℬℜ(B→Xsγ)=(3.55±0.25)×10-4
ΔMd=(0.507±0.005)ps-1
mb(mb)=4.23GeV
fB=(0.208±0.008) GeV
αs(MZ)=0.11
mtpole=(170±4) GeV
τB+=1.63ps
mτ=1.77 GeV
aIt is the weighted average of Vubinc=(40.1±2.7±4.0)×10-4 and Vubexc=(29.7±3.1)×10-4 for the inclusive and exclusive values, respectively. In our numerical analysis, we increase the error on Vub by 50% and take the total error to be around 12% due to the appreciable disagreement between the two determinations.
3.3.3. Constraints from <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M358"><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:math></inline-formula> Physics: Results
For the “standard” 2HDMII with four generations we find that the constraints from Br(B→Xsγ) and ΔMq(q=d,s) have a simple pattern in the mH+-tanβ plane. In particular, with mt′~500 GeV we find that MH+≳600 GeV for tanβ=1, while MH+≳500 GeV for tanβ=5.
For the 4G2HDM’s of types I, II, and III, the combined constraints on their parameter space from both Br(B→Xsγ) and ΔMq(q=d,s) are summarized below. In Figures 8 and 9 we show a sample of the results obtained in [61], where the allowed ranges are shown in the mH+-tanβ and the tanβ-ϵt planes, respectively. In these plots we use |Vt′b|=0.001—corresponding to the “3+1” scenario with a negligible 4th-3rd generation mixing, that is, with |λsbt′|=10-5 correspondingly. We see, for example, that in the 4G2HDMI, the “3+1” scenario typically imposes tanβ~1 with ϵt typically larger than about 0.4 when mH+≲500 GeV. In the 4G2HDMII and the 4G2HDMIII one observes a similar correlation between tanβ and mH+; however, larger tanβ values are allowed for ϵt≲mt/mt′ and a charged Higgs mass is typically heavier than 400 GeV.
The “3+1” scenario, Vt′b=0.001 (|λsbt′|=10-5): the allowed parameter space in the mH+-tanβ plane, following constraints from B→Xsγ and Bq-B-q mixing, in the 4G2HDMI (a), the 4G2HDMII (b), and the 4G2HDMIII (c), for mt′=500 GeV, mb′=450 GeV, ϵb=mb/mb′, and ϵt=0.34(~mt/mt′). Figure taken from [61].
The “3+1” scenario, Vt′b=0.001 (|λsbt′|=10-5): the allowed parameter space in the tanβ-ϵt plane, following constraints from B→Xsγ and Bq-B-q mixing, in the 4G2HDMI (a), the 4G2HDMII (b), and the 4G2HDMIII (c), for mt′=500 GeV, mb′=450 GeV, ϵb=mb/mb′ and with mH+=400 and 750 GeV. Figure taken from [61].
For the case of a Cabibbo size mixing between the 4th and 3rd generation quarks, we set |Vt′b|=|Vtb′|=0.2 and show in Figures 10 and 11 the allowed parameter space in the mH+-tanβ and tanβ-ϵt planes, in the 4G2HDM’s of types I, II, and III, with mt′=500 GeV, mb′=450 GeV, and ϵb=mb/mb′. In the 4G2HDMII and the 4G2HDMIII we see a similar behavior as in the no-mixing case (i.e., as in the case Vt′b→0), while in the 4G2HDMI we see that “turning on” Vt′b allows for a slightly larger tanβ, that is, up to tanβ~5 for ϵt≳0.9. Also, similar to the no-mixing case, larger values of tanβ are allowed in the 4G2HDMII and 4G2HDMIII. Furthermore, mH+~300 GeV and tanβ~1 are allowed in the 4G2HDMI.
The Cabibbo size mixing case, Vt′b=0.2 (|λsbt′|=0.004): the allowed parameter space in the mH+-tanβ plane, following constraints from B→Xsγ and Bq-B-q mixing, in the 4G2HDMI (a), 4G2HDMII (b), and 4G2HDMIII (c), for mt′=500 GeV, mb′=450 GeV, ϵb=mb/mb′, and ϵt=0.34(~mt/mt′). Figure taken from [61].
The Cabibbo size mixing case, Vt′b=0.2 (|λsbt′|=0.004): the allowed parameter space in the tanβ-ϵt plane, following constraints from B→Xsγ and Bq-B-q mixing, in the 4G2HDMI (a), 4G2HDMII (b), and 4G2HDMIII (c), for mt′=500 GeV, mb′=450 GeV, and ϵb=mb/mb′ and with mH+=400 and 750 GeV. Figure taken from [61].
3.4. Combined Constraints and Points of Interest
In Table 2 we give a sample list of interesting points (models) in parameter space of the 4G2HDMI that “survive” all constraints from EWPD and flavor physics in the 4G2HDMI, for mh=125 GeV, tanβ=1, and ϵt=mt/mt′. The list includes (see also caption of Table 2) models with a 4th generation mass splitting (between the up and down partners of both the 4th family quarks and leptons) larger than 150 GeV; models where both the 4th generation quarks and leptons are nearly degenerate; models with a light to intermediate neutral Higgs spectrum, that is, mh=125 GeV and mA or mH in the range 150–300 GeV; models with a large inverted mass hierarchy in the quark doublet, that is, mb′-mt′>150 GeV; models with a light charged Higgs with a mass smaller than 400 GeV and models with a Cabibbo size as well as an 𝒪(0.01) size t′-b/t-b′ mixing angle.
List of points (models) in parameter space for the 4G2HDM’s of type I with mh=125 GeV, tanβ=1, and ϵt=mt/mt′, allowed at 95% CL by EWPD and B-physics flavor data. Points 1-2 have mt′-mb′>150 GeV, while points 3–5 have a large inverted splitting mb′-mt′>150 GeV. Points 6 and 7 have nearly degenerate 4th generation quark and lepton doublets. Points 8–11 give BR(t′→th)~𝒪(1) (see Figure 16 in Section 5), while points 12 and 13 give BR(b′→tH+)~𝒪(1) (see Figure 17 in Section 5). Point 4 has a light 180 GeV pseudoscalar Higgs (A) and points 12 and 13 have a light 300 GeV charged Higgs. Points 1, 8, and 9 have a small t′-b/t-b′ mixing angle (θ34≤0.05), while points 4 and 10–13 have a Cabibbo size t′-b/t-b′ mixing angle (θ34~0.2).
4G2HDMI: mh=125 GeV, tan β=1, ϵt=mt/mt′
Point no.
mt′
mb′
mν′
mτ′
mA
mH
mH+
sinθ34
α
|λsbt′|
1
570
403
118
184
319
993
806
0.02
0.46π
<0.002
2
596
435
124
277
840
172
595
0.09
0.32π
<0.0005
3
425
591
1151
1085
817
203
646
0.08
0.46π
<0.001
4
441
595
385
556
180
998
661
0.21
0.69π
<0.001
5
429
580
587
759
978
304
454
0.13
0.95π
<0.0005
6
555
564
1185
1180
501
674
661
0.06
0.62π
<0.0007
7
409
401
424
429
509
837
472
0.1
0.68π
<0.0006
8
500
450
1079
1005
745
439
750
0.05
π/2
<0.0006
9
500
450
160
176
733
414
750
0.05
π/2
<0.0006
10
500
450
786
652
833
308
750
0.2
π/2
<0.0006
11
500
450
211
268
798
289
750
0.2
π/2
<0.0006
12
450
500
711
618
500
215
300
0.2
π/2
<0.004
13^{a}
450
500
108
253
872
295
300
0.2
π/2
<0.004
aPoints 12 and 13 require ϵb≲mb/mb′ in order to have BR(b′→tH+)~𝒪(1) (see Figure 17).
4. Other Useful Effects in Flavor Physics
We discuss below some important low energy observables, which are potentially sensitive to the 4th generation dynamics within the multi-Higgs framework, and have shown some degree of discrepancy between their measured values and the SM predictions.
The muon anomalous magnetic moment (μAMM), aμ=(gμ-2)/2, is well known to play an important role in the search for NP. In the SM, the total contributions to the μAMM, aμSM, can be divided into three parts: the QED, the electroweak (EW), and the hadronic contributions. While the QED [120–125] and EW [126–129] contributions are well understood, the main theoretical uncertainty lies with the hadronic part which is difficult to control [130, 131].
Since the first precision measurement of aμ, there has been a discrepancy between its experimental value and the SM prediction. This discrepancy has been slowly growing due to recent impressive theoretical and experimental progress. Comparing theory and experiment, the deviation amounts to [132]
(25)aμexp-aμSM=(255±80)·10-11
which corresponds to a~3σ effect. In order to confirm this result, the uncertainties have to be further reduced.
It is interesting to interpret the difference as a contribution from loop exchanges of new particles. A number of groups have studied the contribution to aμ in various extensions of the SM to constrain their parameters space (for reviews see [133, 134]). In most extensions of the SM, new charged or neutral states can contribute to the μAMM at the one-loop (lowest) level. In [135], we have shown that the ~3σ excess in aμ (with respect to the SM prediction) can be accounted for by one-loop exchanges of the heavy 4th generation neutrino (ν′) in the 4G2HDMI setup when applied to the leptonic sector (i.e., where the “heavy” Higgs doublet couples only to the 4th generation lepton doublet and the “light” Higgs doublet couples to leptons of the lighter 1st–3rd generations; see [135]).
The effective vertex of a photon with a charged fermion can in general be written as
(26)u-(p′)eΓμu(p)=u-(p′)e[γμF1(q2)+iσμνqν2mfF2(q2)]u(p),
where, to lowest order, F1(0)=1 and F2(0)=0. While F1(0) remains unity at all orders due to charge conservation, quantum corrections yield F2(0)≠0. Thus, since gμ≡2·(F1(0)+F2(0)), it follows that aμ≡(gμ-2)/2=F2(0).
In the 4G2HDMI [61, 135] the one-loop contribution to the muon anomaly can be subdivided as
(27)aμ=[aμ]WSM4+[aμ]ℋ4G2HDMI,
where [aμ]ℋ4G2HDMI contains the charged and neutral Higgs contributions coming from the one-loop diagrams in Figure 12, where the diagrams with τ′ and ν′ in the loop dominate. The SM4-like contribution, [aμ]WSM4, comes from the one-loop diagram with W±-ν′ in the loop and is given by [136]
(28)[aμ]WSM4|U24|2=GFmμ242π2A(xν′),
where U24 is the 24 element of the CKM-like PMNS leptonic matrix, xi=mi2/mW2. For values of mν′ in the range 100GeV≲mν′≲1000GeV, one finds 1.5×10-9≲[aμ]WSM4/|U24|2≲3.0×10-9, so that for |U24|2≪1 (as expected) the simple SM4 cannot accommodate the observed discrepancy in aμ. The detail expression for [aμ]ℋ4G2HDMI has been given in [135]. It is interesting to note that the dominant contribution to [aμ]ℋ4G2HDMI, or for that matter to aμ, comes from the charged Higgs loops and the contribution from diagrams with the neutral Higgs exchanges is subleading [135]. In addition, aμ was found to be sensitive only to the product δΣ2·δU2, where
(29)δUi≡Ui4*U44*,δΣi≡Σ4ie*Σ44ν,
and Σe(Σν) are the new mixing matrices (i.e., in the 4G2HDMI) in the charged (neutral) leptonic sectors. That is, similar to the quark sector (see (10)), these matrices are obtained after diagonalizing the lepton mass matrices
(30)Σije=LR,4i⋆LR,4j,Σijν=NR,4i⋆NR,4j,
where LR and NR are the rotation (unitary) matrices of the right-handed charged and neutral leptons, respectively.^{1}
One-loop diagrams for li→ljγ with charged and neutral scalar exchanges.
In Figure 13 we plot aμ as a function of the product δΣ2·δU2 (assuming its real) for several values of mν′ and mH+ and fixing mτ′=mν′. Depending on the mass mν′, we find that δU2·δΣ2~10-3-10-2 is typically required to accommodate the measured value of aμ.
The muon g-2 as a function of the product δΣ2·δU2, for mν′=100,200,400 GeV, mτ′=mν′ and with mH+=500 GeV (a) and mH+=700 GeV (b). The horizontal lines are the measured 1-σ bounds on aμ (see (25)). Figure taken from [135].
The constraints on the 4G2HDMI parameters and in particular on the quantities δΣ2 and δU2 which control the μAMM were studied in [135], by analyzing the lepton flavor violating (LFV) decays τ→μγ and μ→eγ. These decays are absent in the SM and are useful for constraining NP models that can potentially contribute to the muon anomaly.
The current experimental 90% CL upper bounds on these LFV decays are [74, 137]
(31)Br(τ⟶μγ)<4.4×10-8,Br(μ⟶eγ)<2.4×10-12.
The amplitude for the transition ℓi→ℓjγ can be defined as
(32)ℳ(ℓi⟶ℓjγ)=u-ℓj(p′)[iσμνqν(A+Bγ5)]uℓi(p)ϵμ*,
where ϵμ* is the photon polarization. The decay width is then given by
(33)Γ(ℓi⟶ℓjγ)=mℓi38π(1-mℓj2mℓi2)×[(1+mℓj2mℓi2)(|A|2+|B|2)+4mℓjmℓi(|A|2-|B|2)].
Here also, the new 4G2HDMI contribution to the amplitude, ℳ(ℓi→ℓjγ)4G2HDMI, can be divided as
(34)ℳ(ℓi⟶ℓjγ)4G2HDMI≡ℳWSM4(ℓi⟶ℓjγ)+ℳH+4G2HDMI(ℓi⟶ℓjγ)+ℳℋ04G2HDMI(ℓi⟶ℓjγ),
where ℳWSM4(ℓi→ℓjγ) is the SM4-like W-exchange contribution which is much smaller than the charged and neutral Higgs amplitudes, ℳH+4G2HDMI(ℓi→ℓjγ) and ℳℋ04G2HDMI(ℓi→ℓjγ) (calculated from the diagrams in Figure 12). As in the μAMM case, the dominant contribution to LFV decays was found to be from the charged Higgs exchange diagrams [135]. In addition, the decays μ→eγ and τ→μγ are sensitive to δU2 and δΣ2 through the products (δU2δΣ1,δU1δΣ2) and (δU3δΣ2,δU2δΣ3), respectively, so that, in principle, one can avoid constraints on the quantities δU2 and δΣ2 if δU1, δU3, δΣ1, and δΣ3 are sufficiently small.
In [135], we have shown that it is possible to address both the BR(μ→eγ) and the muon anomaly aμ within the 4G2HDMI framework, if δU1≪δU2 and δΣ1≪δΣ2, which is indeed expected if we consider the observed hierarchical pattern of the quark’s CKM matrix as a guide. However, in order to account also for the measured upper limit on BR(τ→μγ) (see (31)), one requires that δU3<δU2 and δΣ3<δΣ2. Therefore, the typical benchmark texture for the 4th generation elements of the matrices Ui4Σ4ie that can account for the observed muon anomaly and still be consistent with the current constraints from the LFV decays τ→μγ and μ→eγ is
(35)Ui4~(Σ4ie)T≃(ϵ5ϵϵ21),
Where, for example, ϵ~0.1 for mν′=100 GeV.
The above texture implies a hierarchical pattern which is different from what one would expect from the observed hierarchical pattern of the quark’s CKM matrix. Nonetheless, without a fundamental theory of flavor, our insight for flavor should be data driven also in the leptonic sector. Besides, the above texture is sensitive to the current precision in the measurement of the muon g-2 which can change for example, if more accurate calculations end up showing that part of the hadronic contributions cannot be ignored.
Among the various Bq rare decays, the purely leptonic Bd/s→μ+μ- decays are highly sensitive to indirect effects of NP, since the quark level decays are based on the FCNC b→d,s transitions which are severely (loop) suppressed in the SM. In particular, the decay Bs→μ+μ- has received special attention in the past decade, since its branching fraction, Br(Bs→μ+μ-), can be significantly enhanced by loop exchanges of new particles predicted by various NP scenarios. For example, Br(Bs→μ+μ-) imposes restrictive constraints on the SUSY parameter space (see, e.g., [138–140]), where in some scenarios better limits than those obtained from direct searches have been claimed. However, the excluded SUSY parameter space depends strongly on the choice of tanβ since the Bs→μ+μ- rate typically varies as (tanβ)6.
In the LHC era the current limit on Br(Bs→μ+μ-) has been improved. The two different experiments LHCb and CMS, using 1fb-1 and 5fb-1 data sample, respectively, yield [141, 142]
(36)Br(Bs⟶μ+μ-)<4.5×10-9,LHCb@95% CL<7.7×10-9,CMS@95%C,
whereas the SM prediction for this decay is [19]
(37)Br(Bs⟶μ+μ-)=(3.2±0.2)·10-9.
In fact, LHCb has the sensitivity to measure the Br(Bs→μ+μ-) down to ~2×10-9, which is about 5σ smaller than the SM prediction.
In general, the matrix element for the decay B-s→ℓ+ℓ- can be written as [143]
(38)ℳ=GFα22πsinθW2[FSℓ-ℓ+FPℓ-γ5ℓ+FAPμℓ-γμγ5ℓ],
where Pμ is the four-momentum of the initial Bs meson and Fi’s are functions of Lorentz invariant quantities. Squaring the matrix and summing over the lepton spins, we obtain the branching fraction
(39)Br(B-s⟶ℓ+ℓ-)=GF2α2MBsτBs64π31-4mℓ2MBs2×[(1-4mℓ2MBs2)|FS|2+|FP+2mℓFA|2].
In the SM, the dominant effect in B-s→ℓ+ℓ- arises from the diagrams shown in Figure 14, which contribute only to FA in (38).
Dominant SM diagrams for the decay Bd′→ℓ+ℓ-, d′=d or s.
As in other NP models, in the 4G2HDMI there will be contributions to FS, FP, and FA coming from the charged Higgs exchange penguin and box diagrams (replacing W+→H+ in Figure 14). In [61], constraints on the 4G2HDMI parameter spaces were estimated, using the recent data on Br(Bs→μ+μ-). This was done in the context of the muon (g-2), in the sense that only those interactions (in the leptonic vertex) which are associated with aμ have been considered. In particular, considering only the ℓ±ν′H± vertex, the only diagrams that contribute to B-s→ℓ+ℓ- are the Higgs exchange box diagrams in Figure 14, where one or two W-bosons are replaced by H+ and (t,νℓ) are being replaced by both (t,ν′) and (t′,ν′). It was then found that the contribution from the new box diagrams in the 4G2HDMI that involve the heavy 4th generation neutrino is consistent with the current experimental bound on BR(Bs→μμ) for values of δU2 and δΣ2 that reproduce the observed muon g-2 see Figure 15.
BR(Bs→μμ) as a function of λbst′≡Vt′bVt′s*, from box diagrams with H+ and (t,ν′), (t′,ν′) exchanges in the 4G2HDMI. The parameters δU2 and δΣ2 are varied within the constraints imposed by aμ (see the previous section), keeping both of them ≲ 0.2. Also shown are the experimental 95% CL upper bounds from LHCb (red horizontal line) and from CMS (green horizontal line). The SM predicted range of values (at 1σ) is shown within the black horizontal lines. Figure taken from [135].
It is also interesting to note that the Br(Bs→μ+μ-), in both the SM4 and the 4G2HDMI, can differ from the SM value by at-most a factor of 𝒪(3) in either direction (for a detail discussions see [135]).
Other purely leptonic and semileptonic decays of the B meson, such as B→τ decays, can also provide useful tests of the SM and its extensions. Of particular interest are the purely leptonic B→τν and the semileptonic B+→D(*)τν decays. The SM contribution to the branching ratios of these decays arises at the tree-level from the charged weak interactions. An important NP contribution to these decays is the tree level exchange of a charged Higgs in multi-Higgs models, so that these decays offer interesting probes of the Higgs sector and, particularly, of its Yukawa interactions.
The SM expression for the decay rate of B→τν is given by
(40)Br(B⟶τν)SM=GF2mτ2mB8π(1-mτ2mB2)2fB2|Vub|2τB,
where fB is the decay constant and τB is the B+ life time. The SM prediction for Br(B+→τ+ν) is, therefore, sensitive to the decay constant fB and to the CKM element |Vub| and is thus limited by the uncertainty in the determination of these quantities. Using the available constraints on fB and the inclusive determination of Vub: fB=200±20MeV and Vub=(39.9±1.5±4.0)·10-4 [144], the SM prediction for the decay rate is
(41)Br(B⟶τν)SM=(0.86±0.12)·10-4.
Furthermore, the SM prediction on Br(B→τν), obtained directly from a fit to various other observables (i.e., without using Vub and the lattice results for fB) is [144]
(42)Br(B⟶τν)SM=(0.73±0.12)·10-4.
Both results show some degree of discrepancy with the current world average on BR(B→τν) which is [118]
(43)Br(B+⟶τ+ντ)=(1.67±0.3)·10-4.
We want to indicate here how the 4G2HDM can address this if the discrepancy is confirmed.
From the theoretical point of view, several models of NP predict large deviations from the SM for processes involving third generation fermions. For instance, in a “standard” 2HDM where the two Higgs doublets are coupled separately to up- and down-type quarks (i.e., the 2HDMII setup described in Section 2), the B→τν amplitude receives an additional tree-level contribution from the heavy charged-Higgs exchange, leading to
(44)Br(B⟶τν)2HDMIIBr(B⟶τν)SM=[1-mB2tan2βMH2],
so that for large tanβ, the r.h.s. of (44) can be significantly different from “1.” However, in this particular case (of the 2HDMII), the charged-Higgs contribution reduces the SM value for the branching ratio, thus further worsening the situation with respect to the experimentally measured value.
In the 4G2HDMI, the effective tree-level interactions that will contribute to B→τν can be written as
(45)ℋeff=GFVub2[u-γμ(1-γ5)bτ-γμ(1-γ5)ν-mτmbAbuMH2×{Auℓu-(1+γ5)bτ-(1-γ5)ν+Adℓu-(1+γ5)bτ-(1+γ5)ν}mτmbAbuMH2],
where the second term represents the tree-level charged-Higgs exchange and the first term results from the diagram with W boson exchange. Also, Abu, Auℓ, and Adℓ are factors coming from the b→uH and τ→ντH vertices, respectively, given by
(46)Abu=tanβ-(tanβ+cotβ)(Σbb+mb′mbVub′VubΣb′b),Auℓ=-tanβ+(tanβ+cotβ){Σ33ℓU33+mτ′mτΣ43ℓU43},Adℓ=-mντ′mτ(tanβ+cotβ)Σ43νU34.
A simple calculation, using (45) and (46), yields
(47)Br(B⟶τν)=Br(B⟶τν)SM[|1-mB2MH2AbuAuℓ|2+|mB2MH2AbuAdℓ|2].
Thus, taking, for example, Σij≈mj/mi, only a moderate enhancement to BR(B→τν) is possible at large tanβ. If, on the other hand, Σij≫mj/mi, then the BR(B→τν) can be significantly enhanced compared to the SM prediction. Of course, the experimental deviations at the moment are only a few sigmas, but, if they get confirmed, then we have indicated here how we may be able to address them.
Semileptonic B decays such as B→D(*)τν are more complicated to handle than the pure leptonic ones, since the theoretical predictions for these decays to exclusive final states require knowledge of the form factors involved. There are, however, several other observables (besides the branching fraction), such as the decay distributions and the τ polarization, which can be useful in this cases for probing NP.
As in the case of B→τν, the semileptonic decay B→D(*)τν is also known to be a sensitive mode to the tree-level charged-Higgs exchange. Furthermore, the precise measurement of B→D(*)ℓν at the B-factories and the theoretical developments of heavy-quark effective theory (HQET) has improved our understanding of exclusive semileptonic decays [74, 145].
In particular, the ratios R(D(*))≡BR(B→D(*)τν)/BR(B→D(*)ℓν) reduce considerably the main theoretical uncertainties and, hence, turn out to be a more useful observable [146]. The updated SM predictions of these rates, averaged over electron and muons, are given by [147, 148]
(48)R(D)SM=0.297±0.017,R(D*)SM=0.252±0.003,
so that at this level of precision the experimental uncertainties are expected to dominate.
The most recently measured values of these observables are given by [147, 148]
(49)R(D)exp=0.440±0.058±0.042,R(D*)exp=0.332±0.024±0.018.
The measured values, therefore, exceed the SM predictions for R(D)SM and R(D*)SM by 2.0σ and 2.7σ, respectively, so it is argued that the possibility of both the measured values agreeing with the SM is excluded at the 3.4σ level. In addition, the combined analysis of R(D) and R(D*) rules out the 2HDMII charged Higgs boson with 99.8% confidence level for any value of tanβ/MH when combined with Br(B→Xsγ); see [147, 148]. Once again, it is not clear to us how serious to take the indications of the deviations in (49). Nonetheless, we briefly indicate here how this discrepancy (if experimentally confirmed) can be addressed in the 4G2HDMI, for which the effective tree-level interactions that contribute to B→D(*)τν are given in (45) with the u-quark replaced by the c-quark. Thus, similar to the case of B→τν, we expect a moderate enhancement to both R(D) and R(D*) in the 4G2HDMI if Σij≈mj/mi and a larger effect for larger values of Σij.
5. New Aspects of the Phenomenology of the 4G2HDMI
In the 4G2HDMI (i.e., the 4G2HDM with βd=βu=0 see (6)), one obtains (see (11))
(50)Σd≃(0000000000|ϵb|2ϵb⋆00ϵb(1-|ϵb|22)),Σu≃(0000000000|ϵt|2ϵt⋆00ϵt(1-|ϵt|22)),
which leads to new interesting patterns (in flavor space) in both the neutral and charged Higgs sectors. For example, the ℋ0qiqj Yukawa interactions of (9) (ℋ0=h,H,A) give rise to potentially enhanced tree-level t′→t and b′→b FC transitions and absence of “dangerous” tree-level FCNC transitions between the 4th and the 1st and 2nd generations quarks as well as among the 1st-2nd and 3rd generation quarks. In particular, the FC ℋ′t′t interactions in this case are (taking α→π/2)
(51)ℒ(ht′t)=-g2mt′mWϵt1+tβ2t-′(R+mtmt′L)th,ℒ(Ht′t)=-g2mt′mWϵt1+tβ2tβt-′(R+mtmt′L)tH,ℒ(At′t)=ig2mt′mWϵt1+tβ2tβt-′(R-mtmt′L)tA,
and similarly for the ℋ0b′b vertices by changing ϵt→ϵb (and an extra minus sign in the Ab′b coupling).
If ϵt~mt/mt′, then the above ℋ′t′t couplings can become sizable, to the level that it might dominate the decay pattern of the t′ (see below). In fact, large FC effects are also expected in b′→b transitions since, even for a very small ϵb~mb/mb′, the FC hb′b and Ab′b Yukawa couplings can become sizable if, for example, tanβ~5, for which case they are ∝5mb/mW. Therefore, such new FCNC t′→t and b′→b transitions can have drastic phenomenological consequences for high-energy collider searches of the 4th generation fermions, as we be discussed below.
Furthermore, the flavor diagonal interactions of the Higgs species with the up quarks of the 1st, 2^{nd}, and 3rd generations are proportional to tanβ in this model, thus being a factor of tan2β larger than the corresponding “conventional” 2HDMII (i.e., the type II 2HDM) couplings (which are ∝cotβ). For example, this gives rise to an enhanced flavor diagonal htt- interactions, while suppressing the ht′t-′ one,
(52)ℒ(htt)≈g2mtmW1+tβ2(1-|ϵt|2)t-th→|ϵt|2≪1g2mtmW1+tβ2t-th,ℒ(ht′t′)≈g4mt′mW1+tβ2|ϵt|2t-′t′h,
when |ϵt|2→0.
Another important new feature of this model occurs in the charged Higgs couplings involving the 3rd and 4th generation quarks, which are completely altered by the presence of the Σd and Σu matrices and can thus lead to interesting new effects in both leptonic (see, previous section) and quark sectors. For example, taking Vt′b,Vtb′≪Vtb,Vt′b′, and the H+t′b and H+tb′ Yukawa couplings are given in the 4G2HDMI by
(53)ℒ(H+t′b)≈g2mWtβ(1+tβ-2)t-′ℒ(H+t′b)≈×(mtϵtVtbL-mb′ϵbVt′b′R)bH+,ℒ(H+tb′)≈g2mWtβ(1+tβ-2)t-ℒ(H+tb′)≈×(mt′ϵt⋆Vt′b′L-mbϵb⋆VtbR)b′H+.
Recalling that in the “standard” 2HDMII (which would underies a supersymmetric four-generation model) the t-RbL′H+ would be ∝mtVtb′/tβ, we find that in the 4G2HDMI the t-RbL′H+ coupling is potentially enhanced by a factor of
(54)t-RbL′H+(4G2HDMI)t-RbL′H+(2HDMII)~ϵt·tβ2·mt′mt·VtbVtb′,
so that if, for example, tβ~1 and ϵt~mt/mt′, there is a factor of Vtb/Vt′b enhancement to the t-RbL′H+ interaction.
These new aspects of phenomenology in the Yukawa interactions sector can have far reaching implications for collider searches of the heavy 4th generation quarks and leptons, as will be discussed in more detail in the next sections. To see that, one can study the new decay patterns of t′ and b′ that follow from the above new Yukawa terms. In particular, in Figure 16 we plot the branching ratios of the leading t′ decay channels (assuming mH+,mA>mt′): t′→th,bW,b′W(⋆) (W(⋆) stands for either on-shell or off-shell W depending on mb′), as a function of the b′ mass. We use mh=125 GeV, mt′=500 GeV, tanβ=1, ϵt=mt/mt′, and θ34=0.05 and 0.2. We see that the BR(t′→th) can easily reach 𝒪(1) (even for a rather large θ34~0.2 for which t′→bW becomes sizable), in particular when mt′-mb′<mW; see for example, points 8–11 in Table 2 for which BR(t′→th)~𝒪(1).
The branching ratios for the t′ decay channels t′→th, t′→bW, and t′→b′W(⋆) (W(⋆) are either on shell or off shell depending on the b′ mass) in the 4G2HDMI, as a function of mb′, for mh=125 GeV, mt′=500 GeV, ϵt=mt/mt′, tanβ=1, θ34=0.05 (a), and θ34=0.2 (b). Also, α=π/2, and mH+>mt′, mA>mt′ are assumed.
In Figure 17 we plot the branching ratios of the leading b′ decay channels b′→tH-,bh,tW,t′W, as a function of ϵb for mb′=500 GeV, mh=125 GeV, tanβ=1, mH+=300 GeV, ϵt=mt/mt′, and θ34=0.05 and 0.2. We see that in the b′ case the dominance of b′→tH- (if kinematically allowed) should be much more pronounced due to the expected smallness of the b-b′ mixing parameter, ϵb, which controls the FC decay b′→bh; see, for example, points 12 and 13 in Table 2 for which BR(b′→bH-)~𝒪(1). On the other hand, if ϵb is larger than about 0.4, then b′→bh dominates.
The branching ratios for the b′ decay channels b′→tH+, b′→bh, b′→tW, and b′→t′W in the 4G2HDMI, as a function of ϵb for mh=125 GeV, mb′=500 GeV, mt′=400 GeV, mH+=300 GeV, tanβ=1, ϵt=mt/mt′, and θ34=0.05 (a) and θ34=0.2 (b). Also, α=π/2 and mA>mb′ are assumed.
6. Implications of the 4G2HDMI for Direct Searches of 4th Generation Quarks
The direct searches of the 4th generation quarks at the LHC currently provide the most stringent limits on their masses. In particular, CMS reported a 450 GeV lower limit [31] on the t′ mass in the semileptonic channel (pp→t′t-′→[W+]hadronicb[W-]leptonicb¯→ℓνbqq¯b¯) and a 557 GeV lower limit [32] in the dilepton channel (pp→t′t-′→[W+]leptonicb[W-]leptonicb¯→ℓ+ℓ-νν¯bb¯). The most recent lower bounds on the b′ mass are 480 GeV [33] (ATLAS) and 611 GeV [34] (CMS).
These searches assumed Br(t′→bW+)~𝒪(1), as expected within the SM4 framework. As was argued above, this is quite unlikely to be the case in models with more than one Higgs doublet, for which new decay patterns can emerge from the interaction of the heavy quarks with the extended Higgs sector, for example, t′→ht(b′→hb), t′→H+b(t′→H+b). In addition, the SM4 forbidden channels t′→b′W and b′→t′W, depending on the mass hierarchy in the fourth generation doublet, may no longer be in contradiction with the EWPD if there are more Higgs doublets (see [61] and Section 3) and may be kinematically open as well. Taking into account such possible new decay modes to the neutral and charged scalars, one can define the generic signature [149]: t′t-′/b′b-′→nWW+nbb, with nW and nb being the number of W and b and b- jets in the event, respectively.
Focusing on the t′ case, [150, 151] have reinterpreted the ATLAS b′ search (reported in [33]) to extract limits on t′ if it decays via non-SM4 channels such as t′→th and t′→tZ, whereas [149] have considered, more specifically, the decay channels t′t-′→6W+2b and t′t-′→2W+6b as representatives of such new signatures beyond the SM4. As was indeed demonstrated in both [149–151], when t′→bW and b′→tW are no longer the leading decay channels, the attempts to impose the SM4-motivated dynamics on processes with a completely different topology result in a relaxed limit on the fourth generation quarks with respect to the SM4 case. Specifically, for the t′, the CMS analysis in the semileptonic channel was based on the complete reconstruction of each ℓνbqq¯b¯ event (including the reconstruction of the hadronic W). The total distribution of Mfit (the reconstructed mass of the t′) and HT (the scalar sum of all transverse momenta in the event) was used to set a bound on the t′ mass. On the other hand, for the new signatures (e.g., t′→th), the number of jets in each event is higher (e.g., in the 2W+6b signature, there are 8 jets in the semileptonic channel) and the reconstruction will miss a large part of them, resulting in HT and Mfit being substantially lower—peaking around the main tt¯ background. An example of this effect is plotted in Figure 18.
Mfit distribution for the SM4 2W+2b→lνb¯bq¯q signature (blue) and for the 4G2HDMI 6W+2b signature (red), for a set of 7 TeV LHC events with ∫Ldt=1fb-1. For both signatures, mt′=450 GeV is assumed. The peak of the distribution of Mfit for the SM4 signature is around mt′, while for the new signature the peak is shifted to a significantly lower value coinciding with the peak of the t¯t background. Figure taken from [149].
The analysis in the dilepton channel relies on the fact that Mlb, which is the invariant mass of a pair of any lepton and a b jet in the event, is much higher in the underlying t′t-′ signal with respect to the leading tt¯ background. In particular, in the case of tt¯, Mlb has an upper bound that corresponds to the mass of the top quark, and therefore in the region above ~170 GeV (the “signal region”) Mlb is a clean signal of the SM4-like t′t′ production. However, this dilepton search strategy will fail for signatures with more than 2 leptons or b jets, as in the case of the 4G2HDMI 2W+6b and 6W+2b signatures, since the combinatorial background will lower Mlb, resulting in much less events in the signal region. An example for this effect is plotted in Figure 19.
Mlbmin for the SM4-like pp→t′t¯′→2W+2b signature (red) and for the 4G2HDMI pp→t′t¯′→2W+6b signature (blue) with mt′=350 GeV for a set of 7 TeV LHC events with ∫Ldt=5fb-1 in the dilepton channel. The black line is plotted at the top mass and the region to the right of this line is the “signal region.” Figure taken from [149].
Assuming now that the physics which underlies the 4th generation dynamics goes beyond the SM4, one can estimate the extent to which the new signatures are already excluded by the current LHC searches [149–151]. Here we will briefly recapitulate the analysis performed in [149] for both the semileptonic and dilepton channels mentioned above. For the semileptonic channel, [149] demonstrated, using a naive simulation of the new beyond SM4 signals in question, what the exclusion plot would be (using the CMS search strategy which is based on the SM4 t′→bW decay topology) if the data contains the 4G2HDMI signals. This was done by “injecting” t′t′→6b+2W events with mt′=350 GeV and t′t′→2b+6W events with mt′=450 GeV. The results are shown in Figure 20, which shows that the expected exclusion curves for the background + t′t′→6b+2W and background + t′t′→2b+6W cases are less than 2σ apart from the background-only curve. The curves for the 4G2HDMI signatures with mt′=350–450 GeV lie between the two signal curves shown in the figure. Thus, using the CMS analysis one would not be able to differentiate between the no-signal and the 4G2HDMI signal scenarios within 2σ, so that we expect the bound on the t′ mass within the 4G2HDMI framework to be no larger than about 400 GeV in the semileptonic channel. This result is consistent with the most stringent existing limit, mt′>423 GeV, calculated in [150, 151] by using templates from the b′ search at ATLAS [33] and assuming that BR(t′→th)~1.
The 95% CL exclusion plot distribution on the t′ mass assuming the SM4 signature in the semileptonic channel (1ℓ+nj+ET). For the case of background only, the red dotted line is the median and the yellow and green bands are the ±1 and ±2 standard deviations accordingly. The black line is the median for background + t′t′→2b+6W with mt′=450 GeV and the blue line is the median for background + t′t′→6b+2W with mt′=350 GeV. The curves for the 4G2HDMI signatures with mt′=350–450 GeV lie between these two lines. Figure taken from [149].
For the dilepton channel, the number of events with Mlbmin in the signal region is negligible for mt′=350 GeV (the lowest mass considered in the CMS analysis) and even less than that for higher mt′ (see Figure 19). One can, therefore, conclude that the CMS dilepton analysis is completely irrelevant for the 4G2HDMI signatures.
As was suggested in [149], an analysis that uses a more general reconstruction method could avoid the kinematic misrepresentation of the beyond SM4 events in both the semileptonic and dilepton channels and thus yield a higher sensitivity to NP (beyond the SM4) events containing the 4th generation fermions. An example of that is plotted in Figure 21 for the semileptonic channel, which shows how the misconstruction of the t′ mass can be surmounted.
Comparison between Mfit=m(lνb)=m(qq¯b)—the reconstructed t′ mass using the CMS method—(in blue) and Mgen=m(left side)=m(right side)—the reconstructed t′ mass using the method suggested in [149]—(in red), for the pp→t′t¯′→2W+6b signature with mt′=450 GeV at the LHC with a c.m. of 7 TEV and ∫Ldt=1fb-1, in the semileptonic channel (1ℓ+nj+ET). See also text. Figure taken from [149].
7. Implications for Direct Searches of the Higgs
The recently observed new Higgs-like particle with a mass of ~125 GeV (at the level of ~5σ see [38–41]) is the first potential evidence for a Higgs boson which can be consistent with the SM picture. Furthermore, a study of the combined Tevatron data has revealed a smaller broad excess in the bb¯W channel, which can be related to the production of hW with a Higgs mass between 115 GeV and 135 GeV [42]. These searches further exclude an SM Higgs with masses between ~130 and 600 GeV.
The quantity that is usually being used for comparison between the LHC and Tevatron results and the expected signals in various models is the ratio
(55)RXXModel(Obs)=σ(pp/pp¯⟶h⟶XX)Model(Obs)σ(pp/pp¯⟶h⟶XX)SM,
which is the observed ratio of cross-sections, that is, the signal strengths RXXObs, and the errors in the different channels are [38–42].^{2}
VV→h→γγ: 2.2±1.4 (taken from γγ+2j),
gg→h→γγ: 1.68±0.42,
gg→h→WW*: 0.78±0.3,
gg→h→ZZ*: 0.83±0.3,
gg→h→ττ: 0.2±0.85,
pp/pp¯→hW→bb¯W: 1.8±1.5.
One can easily notice that the channels which have the highest sensitivity to the Higgs signals and contributed the most to the recent 125 GeV Higgs discovery are h→γγ and h→ZZ*,WW*. In all other channels the results are not conclusive, and at this time, they are consistent with the background-only hypothesis at the level of less than 2σ.
As was shown recently in [47], the above reported measurements are not compatible with the SM4 at the level of 5σ. In particular, light Higgs production through gluon fusion is enhanced by a factor of ~10 in the SM4 due to the contribution of diagrams with t′ and b′ in the loops, which in general leads to larger signals (than what was observed at the LHC) in the h→ZZ/WW/ττ channels. For a light Higgs with a mass mh<150 GeV and 4th generation masses of 𝒪(600) GeV, h→ZZ/WW is in fact suppressed by a factor of ~0.2 due to NLO corrections [152, 153], and the exclusion is based mainly on the ττ channel. In the h→γγ channel there is also a substantial suppression of 𝒪(0.1) due to (accidental) destructive interference in the loop [77, 154] and another 𝒪(0.1) factor due to NLO corrections [152, 153]. If ν4 is taken to be light enough, then Br(h→ν4ν4) becomes 𝒪(1), suppressing all the other channels, and the exclusion gets eased. This, however, further suppresses the γγ channel to the level that the observed excess can no longer be accounted for [45]. Therefore, as was also noted in [45, 46, 155], the SM4 is strongly disfavored for any mν4, even without considering the ττ channel.
The comparison to any given model can be performed using a χ2 fit defined as
(56)χ2=∑X(RXXModel-RXXObs)2σXX2,
where σXX are the errors on the observed cross-sections and RXXModel is calculated using the program Hdecay [156] with recent NLO contributions (which also include the heavy 4th generation fermions for the 4th generation scenarios). One can take advantage of the fact that σ(YY→h)Model/σ(YY→h)SM=Γ(h→YY)Model/Γ(h→YY)SM and calculate RXXModel using
(57)RXXModel=Γ(h⟶YY)ModelΓ(h⟶YY)SM·Br(h⟶XX)ModelBr(h⟶XX)SM,
where YY→h is the Higgs production mechanism, that is, either by gluon fusion gg→h, vector boson fusion WW/ZZ→h, or associated Higgs-W production, W*→hW at Tevatron.
In multi-Higgs 4th generation frameworks, the picture becomes more complicated, since there are new scalar states with new Yukawa couplings depending on tanβ and α (α is the mixing angle in the neutral Higgs sector), as well as couplings to the W and the Z bosons which are proportional to sin(α-β) and cos(α-β) (with the exception of the pseudoscalar A which does not couple at tree-level to the W and the Z). Furthermore, the specific Yukawa structure can vary depending on the type of the multi-Higgs model; for example, for the 4G2HDMI case considered below there is an additional parameter, ϵt, which parameterizes the tR-tR′ mixing (see Section 2 and [61]). In Figure 22 we plot the branching ratios of h as a function of α in the 4G2HDMI, for mh=125 GeV, tanβ=1, ϵt=0.5, and M4G=mt′=mb′=ml4=mν4=400 GeV.
The relevant branching ratios of h in the 4G2HDMI, as a function of α, with mh=125 GeV, M4G=400 GeV, ϵt=0.5, and tanβ=1. Figure taken from [157].
Let us now examine how well the 2HDM scenarios with a 4th generation of fermions fit the measured Higgs mediated cross-sections listed above with mh=125 GeV. The simplest case to study is the “standard” 2HDMII (i.e., the 2HDM of type II extended to include a fourth fermion family) with the pseudoscalar A being the lightest scalar, since its couplings do not depend on α [65, 66]. However, as was already noted in [66], for the “standard” 2HDMII the case of a light A decaying to the γγ mode is excluded when all 4th generation fermions are heavy. With the new results, in particular, the signals of the 125 GeV Higgs decaying into a pair of vector bosons, the case of the A being the lightest scalar is excluded irrespective of the 4th generation fermion masses.
Here we wish to extend the previous analysis made for the 2HDMII scenario by calculating the χ2 for the light Higgs with a mass mh=125 GeV, both for the 4G2HDMI of [61] and for the 2HDMII with a 4th generation of fermions, and to compare it to the SM. We follow the analysis in [157], which used the latest version of H decay [156], where all the relevant couplings for the 4G2HDMI and for the 2HDMII frameworks were inserted. For the treatment of the NLO corrections to h→VV, [157] used the approximation of a degenerate 4th generation spectrum, where two cases were studied: mt′=mb′=mℓ4=mν4≡M4G=400 and 600 GeV (while the first case, i.e., M4G=400, is excluded for the SM4, it is not necessarily excluded for the 2HDM setups, as discussed in the previous section). Note that the 4th generation neutrino is taken to be heavy enough, so that the decays of the light Higgs into a pair of ν′ are not considered, thus limiting the discussion to the effects of the altered Higgs couplings in the 2HDM frameworks with respect to the SM4.
Indeed, [157] found that the best fit is obtained for the light CP-even Higgs, h, whereas the other neutral Higgs particles of the 2HDM setups, that is, H and A, cannot account for the observed data.
The resulting χ2 and P values in the 4G2HDMI case (combining all the six reported Higgs decay channels above), with mh=125 GeV, M4G=400 and 600 GeV, ϵt=0.1 and 0.5, and for 0.7<tanβ<1.4 (this range is roughly the EWPD and flavor physics allowed range in these 2HDM setups; see Section 3), are shown in Figure 23. The value of the Higgs mixing angle α is the one which minimizes the χ2 for each value of tanβ. The SM best fit is also shown in the plot. In Figure 24 we further show the resulting χ2 and P-values as a function of tanβ, this time minimizing for each value of tanβ with respect to both α and ϵt (in the 4G2HDMI case). For comparison, we also show in Figure 24 the χ2 and P-values for a 125 GeV h in the 2HDMII with a 4th generation and in the SM.
χ2 (a) and P values (b), as a function of tanβ, for the lightest 4G2HDMI CP-even scalar h, with mh=125 GeV, ϵt=0.1 and 0.5, and M4G≡mt′=mb′=ml4=mν4=400 and 600 GeV. The value of the Higgs mixing angle α is the one which minimizes χ2 for each value of tanβ. The SM best fit is shown by the horizontal dashed line and the dash-dotted line in the right plot corresponds to P=0.05 and serves as a reference line. Figure taken from [157].
Same as Figure 23, while here we minimize with respect to both ϵt and α for each value of tanβ. Also shown are the χ2 and P values for a 125 GeV Higgs in the SM and in the type II 2HDM with a 4th generation of fermions (denoted by 2HDMII). Figure taken from [157].
Looking at the P-values in Figures 23 and 24 (which “measure” the extent to which a given model can be successfully used to interpret the Higgs data in all the measured decay channels) we see that h of the 4G2HDMI with tanβ~𝒪(1) and M4G=400–600 GeV is a good candidate for the recently observed 125 GeV Higgs, giving a fit comparable to the SM fit. This conclusion is not changed by explicitly adding the EWPD as an additional constraint to the above analysis (i.e., the P-values stay roughly the same; see [157]). The “standard” 2HDMII setup with M4G=400 GeV is also found to be consistent with the Higgs data in a narrower range of tanβ≲0.9. Also, the fit favors a large t-t′ mixing parameter ϵt, implying BR(t′→th)~𝒪(1) which completely changes the t′ decay pattern [61] and, therefore, significantly relaxing the current bounds on mt′ (see previous section).
However, more data is required to effectively distinguish between the 4G2HDMI scalars and the SM Higgs. In particular, in Figure 25 we show the individual pulls and the signal strengths for the best fitted h signals (i.e., with mh=125 GeV) in the 4G2HDMI with M4G=400 GeV. We can see that appreciable deviations from the SM are expected in the channels gg→h→ττ, VV→h→γγ, and hV→bbV. In particular, the most notable effects are about a 1.5σ deviation (from the observed value) in the VBF diphoton channel VV→h→γγ and a 2–2.5σ deviation in the gg→h→ττ channel. The deviations in these channels are in fact a prediction of the 4G2HDMI strictly based on the current Higgs data, which could play a crucial role as data with higher statistics becomes available. They can be understood as follows: the channels that dominate the fit (i.e., having a higher statistical significance due to their smaller errors) are gg→h→γγ,ZZ*,WW*. Thus, since the gg→h production vertex is generically enhanced by the t′ and b′ loops, the fit then searches for values of the relevant 4G2HDMI parameters which decrease the h→γγ,ZZ*,WW* decays in the appropriate amount. This in turn leads to an enhanced gg→h→ττ (i.e., due to the enhancement in the gg→h production vertex) and to a decrease in the VV→h→γγ and pp-/pp→W→hW→bbW, which are independent of the enhanced ggh vertex but are sensitive to the decreased VVh one. It is important to note that some of the characteristics of these “predictions" can change with more data collected.
The individual pulls (RXXModel-RXXObs)/σXX (a) and the signal strengths RXXModel (b), in the different channels, that correspond to the best fitted 4G2HDMI curve with mh=125 GeV and M4G=400 GeV, shown in Figure 24. Figure taken from [157].
Finally, [157] also finds that for the best fitted 4G2HDMI case, the heavier CP-even scalar, H, is excluded by the current data (in particular by the ZZ and WW searches) up to mH~500 GeV, whereas a CP-odd state, A, as light as 130 GeV is allowed by current data (for more details see [157]).
8. Summary
We have addressed several fundamental and challenging questions (that we have outlined in the introduction) regarding the nature and underlying dynamics of the physics and phenomenology of 4th generation fermions, if they exist. We have argued the following.
The current stringent bounds on the masses of the 4th generation quarks, that is, mq′≳400 GeV, are indicative of NP, possibly of a strongly coupled nature, since such new heavy fermionic degrees of freedom naturally lead to a Landau pole at the nearby TeV-scale, which may be viewed as the cutoff of 4th generation low-energy theories.
The fact that the 4th generation fermions must be so heavy is, therefore, of no surprise since their large mass stands out as a strong hint for the widely expected new TeV scale physics, where the new heavy fermionic states may be considered to be the agents of EWSB.
If indeed the 4th generation fermions are linked to strong dynamics and/or to compositeness at the nearby TeV scale, then one is forced to extend the minimally constructed SM4 framework which is not compatible with this viewpoint and neither with current data. In particular, in this case one should expect the sub-TeV particle spectrum to accommodate several new scalar composites of the 4th family fermions. The challenge in this scenario is to construct a viable theory that can adequately parameterize the physics of TeV-scale compositeness and that will guide us to the detection of these new states at the LHC.
We have, thus, suggested and reviewed a class of 2HDM’s—extended to include a 4th family of fermions—that can serve as low-energy effective models for the TeV-scale compositeness scenario and then analyzed/discussed
the constraints on these models from EWPD as well as from low-energy flavor physics,
the expected new phenomenology and the implications for collider searches of the 4th generation heavy fermions as well as of the multi-Higgs states of these models.
We have found that it is indeed possible to construct a natural 2HDM framework with heavy 4th generation fermions with a mass in the range 400–600 GeV, which is consistent with EWPD and which is not excluded by the recent direct measurements at the current-high energy colliders.
In particular, we found that, under the 2HDM frameworks for the 4th generation described in this paper, one can
relax the current mass bounds on the 4th generation quarks,
successfully fit the recently measured 125 GeV Higgs signals, to the parameters of the 2HDM with roughly similar quality of fit as the one achieved for the SM with 3 generations; this result is in sharp contrast to the poor fit obtained with the minimal SM4 setup which is, therefore, excluded.
Finally, we have shown that, if such an extended 4th generation 2HDM setup is realized in nature, then one should expect to observe further hints for the underlying TeV-scale dynamics in direct high-energy collider signals involving the 4th generation fermions and the associated new scalars as well as in low energy flavor physics.
Acknowledgments
SBS and MG acknowledge research support from the Technion. The work of AS was supported in part by the U.S. DOE Contract no. DE-AC02-98CH10886(BNL).
Endnotes
Note that since NR,4i and LR,4j parameterize mixings among the 4th generation and the 1st–3rd generations leptons, one expects Σijℓ≪Σ4kℓ for i,j,k=1,2,3; see (30).
We combine the results from the CMS and ATLAS experiments (for pp/pp¯→hW→bb¯W we combine the results from CMS and Tevatron), where in cases where the measured value was not explicitly given we estimate it from the published plots.
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