Higgs data does not rule out a sequential fourth generation with an extended scalar sector

Contrary to the common perception, we show that the current Higgs data does not eliminate the possibility of a sequential fourth generation that get their masses through the same Higgs mechanism as the first three generations. The inability to fix the sign of the bottom-quark Yukawa coupling from the available data plays a crucial role in accommodating a chiral fourth generation which is consistent with the bounds on the Higgs signal strengths. We show that effects of such a fourth generation can remain completely hidden not only in the production of the Higgs boson through gluon fusion but also to its subsequent decay to $\gamma\gamma$ and $ Z\gamma$. This, however, is feasible only if the scalar sector of the Standard Model is extended. We also provide a practical example illustrating how our general prescription can be embedded in a realistic model.

Introduction: Since the first observation of a new resonance at the Large Hadron Collider (LHC) [1,2] in 2012, the particle physics community converged on the view that this is a Standard Model (SM)like Higgs boson if not the SM Higgs boson itself. The fact that the LHC Higgs data is gradually drifting towards the SM expectations [3] has pushed many scenarios that go beyond the SM (BSM) to some contrived corner of the parameter space. For some BSM, the situation is even worse as they have nowhere to hide, because quantum effects coming from some of the heavy degrees of freedom do not decouple and hence leave observable imprints in the Higgs signal strengths. The SM extended by a chiral fourth generation (SM4) [4] constitutes such an example. The fourth generation quark masses, so heavy as to avoid the direct detection bound, are proportional to the corresponding Yukawa couplings and thus, their contributions to the gg → h production amplitude saturate to a constant value just as in the case of the top quark loop. Consequently, the gluon-gluon fusion (ggF) amplitude for the Higgs production increase roughly by a factor of 3 compared to the SM, enhancing the cross-section by a factor of 9. This should have been reflected as a huge enhancement in the Higgs signal strengths, nothing like which has been observed, leading us to believe that the possibility of a sequential fourth generation is strongly disfavored from the existing data [5][6][7][8][9].
Efforts have been made to counter the enhanced production in the ggF channel by reducing the branching ratios (BRs) of the Higgs boson into different visible channels. This can be achieved, e.g., by adjusting the mass of the fourth generation neutrino (m ν ) so that the Higgs boson mainly decays invisibly into a pair of fourth generation neutrinos thereby increasing the total decay width of the Higgs boson [10][11][12][13][14][15][16][17][18]. However this possibility also fell out of favor after the arrival of the Higgs data [8].
Another way to avoid the enhancement in the ggF production channel is to assume that the heavy chiral fourth generation receives its masses from a different scalar other than the SM Higgs doublet 1 . This option has been considered in the framework of a two Higgs doublet model (2HDM) [21][22][23][24].
In this article, we ask a more ambitious question: Can we still accommodate an extra generation of fermions coupling to the SM Higgs in an identical way as the first three generations, without altering appreciably the Higgs signal strengths from their corresponding SM expectations? We will answer in the affirmative and provide a practical example of such a scenario. As we will explain, this is feasible because the Yukawa couplings of up-and down-type quarks can be so arranged as to make the extra contributions cancel out. Such a cancellation, quite remarkably, is also effective for loop-induced Higgs decays. However, a realization of this is not possible with only one scalar doublet, 2 so one needs to extend the scalar sector of the SM.
Making the fourth generation survive: The way to make a fourth generation survive is to note the subtle fact that the sign of the bottom quark Yukawa coupling is not determined from the present LHC data. The sign of the top quark Yukawa coupling has to be positive with respect to the W W h coupling (where h is the SM, or SM-like, Higgs boson with a mass of 125 GeV), which can be inferred from the interference in the h → γγ amplitude. The mass being small, effect of the bottom quark loop is negligible as far as current precision is concerned.
Let us now define the Higgs coupling modification factors relative to the SM as follows: where, x = W, Z and other massive fermions, and g xxh is the generic coupling. For the SM, all κ x = 1, which is of course completely consistent with the LHC data. However, as the tree-level two-body decays of h cannot shed any light on the phase of the coupling, one may also entertain an alternative scenario, namely, (for down type quarks and charged leptons) .
In what follows, we will refer to this possibility as the wrong sign limit. The modification factor for the gg → h production cross section in the presence of fourth generation quarks is given by where, using τ x ≡ (2m x /m h ) 2 , the expression for F 1/2 is given by [25] Since we are concerned with heavy fermions, we can take τ x > 1 for x = (t, t , b ), and then For chiral fermions much heavier than m h = 125 GeV, the loop function F 1/2 saturates to a constant value [25] and the new physics (NP) contribution becomes proportional to (κ t + κ b ). Clearly, in the SM-like limit (κ t = κ b = 1), R gg = 9, which dealt a killer blow to the sequential fourth generation within the SM. But, more importantly, in the wrong sign limit, the NP contributions coming from the t and b loops are of opposite sign and hence there exists the possibility that they may cancel each other. Strictly speaking, the cancellation is perfect in the limit m t = m b , but even if there is a mass splitting, it is still exact for all intents and purposes as long as t and b are much heavier than h. Thus, there is no enhancement in Higgs production through gluon-gluon fusion.
At this point, one might suspect that such a perfect cancellation might not occur for the h → γγ decay. But we should remember that, for anomaly cancellation [26,27], one needs to include an extra chiral generation of leptons too. The fourth generation charged lepton, τ , also contributes to the diphoton decay. Consequently, the NP contribution to the h → γγ amplitude, in the heavy mass limit, is proportional to where, Q f is the electric charge of the fermion f , and N c = 3 for quarks and 1 for leptons. One can easily check that κ γγ = 0 in the wrong sign limit. Thus a heavy chiral extra generation can remain perfectly hidden from the LHC Higgs data in this limit. In passing, we note that, the quantity where T f 3 denotes the isospin projection of f L , also vanishes in the wrong sign limit leaving no trace of extra generations in the h → Zγ decay as well. The mechanism, obviously, works even for two or more such heavy generations as well.
A practical example: The reader, at this point, might wonder whether the conspiracy of couplings given in Eq. (2) can be realized in a gauge theoretic model. The wrong sign limit is not achievable in the SM itself with only one scalar doublet, because of the uniform proportionality between the fermion masses and their Yukawa couplings. Also, one may note that the cancellation of the bad high-energy behaviour for the f f → W L W L scattering amplitude is sensitive to the relative sign between the W W h and f f h couplings [28]. Consequently, in the wrong sign limit, the amplitude at high energies for dd → W L W L (where d represents a generic down-type quark or a charged lepton) will grow and eventually violate the partial-wave unitarity, if only one Higgs doublet is present in the theory. Hence, we must extend the scalar sector.
As an example, let us consider the Type-II two Higgs doublet model (2HDM) [29] where an additional Z 2 symmetry is employed so that one of the doublets (φ 2 ) couples only to the T 3 = + 1 2 fermions and the other doublet (φ 1 ) couples only to the T 3 = − 1 2 fermions, thereby ensuring the absence of any flavor changing neutral current (FCNC) mediated by neutral scalars. We follow the convention of Ref. [29] for the parameters of the potential as well as the couplings of the scalars to the fermions and gauge bosons, but extend this popular framework by adding one complete extra chiral generation of leptons and quarks. Note that we have introduced a right-handed neutrino for the fourth generation, and made it heavy enough to have no significant impact in collider as well as astrophysical experiments. We assume that (B − L) symmetry remains exact in the Lagrangian and thus consider the neutrinos as Dirac particles. The lightest CP-even scalar (h) in the spectrum of the physical particles is then identified with the 125 GeV resonance observed at the LHC. Denoting the ratio of the vacuum expectation values (VEVs) of the two scalar doublets by tan β = v 2 /v 1 , we can write the Higgs coupling modification factors as [29] , (for down type quarks and charged leptons) (8c) where α is the mixing angle that takes us from the Lagrangian basis to the physical basis in the CP-even scalar sector. Evidently, the condition cos(β − α) = 2 tan β , with, tan β 2 (9) leads to the desired wrong sign limit of Eq. (2) [30][31][32][33][34]. We also note that by demanding only κ u = −κ d = 1, one obtains from Eqs. (8b) and (8c) the condition, cos(β − α) = sin 2β (10) which reduces to Eq. (9) in the large tan β limit. To demonstrate our proposition explicitly, we try to find the allowed region in the tan β -cos(β−α) plane so that the Higgs signal strengths into different production and decay channels remain within the experimental limits [3]. 3 We display our result in Fig. 1. This plot shows the allowed parameter space in terms of the two free parameters of the model, namely, cos(β − α) and tan β, for the benchmark point For the parameter space displayed in Fig. 1, the BSM effects to the oblique parameters S and T , taking into account both fermionic [4,36] and scalar [37,38] contributions, are within the experimental limits given by [39] 4 ∆S = 0.05 ± 0.10 , ∆T = 0.08 ± 0.12 .
In the limit tan β 2 or equivalently cos(β − α) → 0, one gets (see Fig. 2) We have checked that all theoretical constraints on the potential, e.g., perturbative unitarity and stability, are satisfied for such a benchmark point. One may wonder whether such a benchmark point would not be in conflict with the direct search limits from the LHC. We discuss later how such bounds can possibly be evaded. At the same time higher mass values for the fourth generation fermions satisfying the ∆S and ∆T constraints would have been equally good for us.
As explained earlier, the mass splitting between the quarks does not affect the cancellation as the production amplitude in the ggF channel becomes insensitive to the precise mass values at the heavy mass limit; the same applies for the leptons while computing the h → γγ amplitude.
Note that a large splitting between m H and m A contributes to the electroweak T -parameter with a negative sign [29,[40][41][42]. This fact can be used to allow for much larger mass splittings between the components of fourth generation fermion doublets compared to that in the case of SM4 [4]. For the benchmark of Eq. (11) we find from Fig. 1 that the model automatically tends to the wrong sign limit, vindicating our assertion. We have also checked that the nature of the plot does not crucially depend on our choice of the benchmark as long as all the nonstandard masses are considerably heavier than 125 GeV. Figure 1: The gray shaded region, based on the benchmark given in Eq. (11), is allowed at 95% CL from the current data for Higgs signal strengths [3], and ∆S, ∆T for the benchmark of Eq. (11). The dashed and dotted lines represent the contours for cos(β − α) = 2/ tan β and cos(β − α) = sin 2β respectively. Caveat emptor: Lower bounds on the fourth generation quark masses (m q 700 GeV) have been placed from the direct searches at the LHC [43,44]. These bounds, however, crucially depend on the assumption that the lighter of the fourth generation quarks (q ) decays into a light quark from the first three generations accompanied by a W -boson. But these decay modes can become subdominant if V i4 , V 4i ≈ 0, (i = 1, 2, 3) in the 4 × 4 CKM matrix. In this case, q becomes quasi-stable and, in our toy model in the framework of a Type II 2HDM, it mainly decays through the loop-induced neutral currents. In BSM scenarios with tree level FCNC, where t → th is the dominant decay mode for t , the bound on the fourth generation mass can be relaxed up to m t > 350 GeV [45]. The bound of m t ∼ 850 GeV for t decaying through neutral-current channels, as recently found by the ATLAS collaboration [46], can be evaded if there are some exotic channels present. In addition to the direct searches, there are also constraints arising from tree-level unitarity of the scattering amplitudes for 2 → 2 processes of the type f 1 f 2 → f 3 f 4 (f i denotes a generic fermion). The unitarity bound for a fourth generation quark is m q 550 GeV [36,47] whereas that for a leptonic fourth generation is m 1.2 TeV [47]. All these considerations together justifies our choice of the benchmark in Eq. (11).
In the moderate tan β region as shown in Fig. 1, one should take into account the bound coming from the decay H → τ + τ − [48]. For tan β ∼ 10, BR(H → τ + τ − ) ∼ O(10 −2 ) for a type II 2HDM. In fact, the ATLAS bound, when translated to the MSSM case, gives m H/A 250 GeV for tan β ∼ 10 [48]. This constraint can be diluted further if we allow H to decay invisibly into a pair of fourth generation neutrinos.
At this point, let us note that there should be some new dynamics not much above the scale of the new fermion masses. Such new dynamics will, in all probability, alter the bounds arising from tree-unitarity. The justification for such new dynamics comes from the fact that the presence of extra heavy chiral fermions will usually give large negative contributions to the evolution of the scalar quartic couplings [4]. This can potentially render the vacuum unstable very quickly after the effects of heavy fourth generation set in. This problem can possibly be diluted by considering a metastable vacuum instead of an absolutely stable one [4] and/or adding extra singlets to the scalar potential, which can give positive contribution to the concerned renormalization group equations.
In passing, we also note that, while such a fourth generation can remain hidden in single production of the Higgs boson, it should show up if one considers the double Higgs production, gg → hh, because the new amplitudes will then add up instead of cancelling each other. Thus, a significant enhancement of the gg → hh rate may be taken as a possible signature of such an extension of the SM. Non-observation of such an enhancement will similarly help to rule the model out.
Summary: To summarize, we have presented a general recipe for resurrecting one, or more, sequential fermion generations in the precision Higgs era ushered by the LHC. Admittedly, we did not provide a complete model which solves all the problems faced by a heavy chiral generation. Nevertheless, we have demonstrated how, in a toy scenario based on a type II 2HDM, the sequential fourth generation can overcome its biggest threat, viz, the consistency of LHC Higgs data with the SM expectations. Thus our toy model can be taken as a constituent part of a more elaborate framework which can address all the other issues related to such extra chiral generations like the loss of unitarity or the stability of the electroweak vacuum way above the TeV scale. In anticipation that it will be long before the LHC starts probing the sign of the bottom quark Yukawa coupling [49], a sequential fourth generation can remain hidden in the wrong sign limit for many years to come. Hopefully this unconventional possibility can rekindle the interest in the study of extra fermionic generations.