Higgs alignment and the top quark

There is a surprising connection between the top quark and Higgs alignment in Gildener-Weinberg multi-Higgs doublet models. Were it not for the top quark and its large mass, the coupling of the 125 GeV Higgs boson $H$ to gauge bosons and fermions would be indistinguishable from those of the Standard Model Higgs. The top quark's coupling to a single Higgs doublet breaks this perfect alignment in higher orders of the Coleman-Weinberg loop expansion of the effective potential. But the effect is still small, $<{\cal{O}}(1\%)$, and probably experimentally inaccessible.


I. Introduction
The 125 GeV Higgs boson H discovered at the LHC in 2012 [1,2] is consistent in all measurements with the single Higgs boson of the Standard Model (SM). This is dramatically illustrated in Fig. 1 where the couplings of fermions and weak gauge bosons to H as measured by ATLAS and CMS are plotted. This degree of agreement is puzzling. Many well-motivated attempts to cure the problems of the SM -most famously, naturalness -require two or more Higgs multiplets. Why, then, does H have SM couplings? The usual answer is "Higgs alignment" [3,4,5,6]. However, with a few exceptions that rely on elaborate global symmetries or supersymmetry [7,8,9,10], implementations of alignment suffer large radiative corrections.
In Gildener-Weinberg (GW) multi-Higgs models of electroweak (EW) symmetry breaking [11], the classical Lagrangian is scale-invariant -so that the Higgs potential is purely quartic and fermions acquire mass only from Higgs boson vacuum expectation values. At tree level, H is a Goldstone boson of spontaneously broken scale symmetry. And, in this approximation, H naturally has the same structure as the Goldstone bosons eaten by W ± and Z 0 . In N -Higgs-doublet models (NHDMs), these Goldstone bosons are where v i is the VEV of the CP -even scalar Thus, H has exactly the same couplings to EW gauge bosons and to fermions (and, hence, to the gluon and the photon) as the single Higgs boson of the Standard Model; i.e., H is aligned. In Sec. II of this paper, we show that, but for the top quark, this alignment would be perfect through second order in the Coleman-Weinberg loop expansion of the effective potential [12]. The top quark's presence upsets perfect alignment, but only by a small amount, at most O(1%). In Sec. III we discuss the experimental consequences of this alignment. In short, experimental searches for new Higgs bosons, such as H , A and H ± , via weak vector boson fusion or decay and Drell-Yan production in association with H, will remain fruitless. We also update the promising paths to discovery of these new Higgses at the LHC. They rely on the fact that these new bosons must lie below 400-500 GeV.

II. Higgs alignment in the GW-2HDM
We discuss the top quark's role in Higgs alignment in the context of an N = 2 Higgs doublet model introduced by Lee and Pilaftsis in 2012 [13].
However, by the Glashow-Weinberg criterion that all quarks of a given electric charge must couple to a single Higgs doublet to avoid flavor-changing neutral current interactions mediated by neutral Higgs exchange [14], our conclusion is true in any GW-NHDM. This GW-2HDM was updated in 2018 [15] to make it consistent with LHC data at the time. The modification used the following Z 2 symmetry on the Higgs doublets and fermions: This is the usual type-I 2HDM [16], but with Φ 1 and Φ 2 interchanged. (The effect of this is that experimental lower limits on tan β = v 2 /v 1 in other type-I models are lower limits on cot β in this model.) The most important experimental constraint came from CMS [17] and ATLAS [18] searches for charged Higgs decay into tb. Consistency with those experiments required tan β < ∼ 0.5 for M H ± < ∼ 500 GeV. This is discussed in Sec. III. The GW tree-level potential for this model is purely quartic [11] so that, since all masses in the model arise from Higgs VEVs, the Lagrangian is scaleinvariant at this level: The five quartic couplings λ i in Eq. (5) are real and, so, V 0 is CP -invariant. The couplings λ 1,2 > 0 for positivity of the potential. The trivial minimum of V 0 occurs at Φ 1 = Φ 2 = 0. But a nontrivial flat minimum of V 0 can occur on the ray where c β = cos β, s β = sin β with β = 0, π/2 a fixed angle and 0 < φ < ∞ a real mass scale. 1 The nontrivial extremal conditions are 1 It is easily proved that any such purely quartic potential as well as its first derivatives vanish at any extremum so that V 0 (Φ iβ ) = 0 [19].
where λ 345 = λ 3 + λ 4 + λ 5 < 0 for positivity of V 0 . Eqs. (7) hold in all orders of the loop expansion of the effective potential [11]; this is important in our subsequent development (see Eqs. (12,13,14)). This extremum spontaneously (but not explicitly) breaks scale invariance, as well as the EW gauge symmetry, and H is the corresponding Goldstone boson.
Following Ref. [19], we use the "aligned basis" of the Higgs fields because the scalars' mass matrices will remain very nearly diagonal in that basis beyond the tree approximation (also see Ref. [6]). That is the essence of Higgs alignment in GW models and, in this and similar models, it is broken, but only slightly, by the top quark. This basis is: On the ray Eq. (6) on which V 0 has nontrivial extrema, these fields are Φ = (0, φ)/ √ 2 and Φ = 0. In this basis, H, z, w ± are massless and unmixed with the H , A, H ± whose "masses" are Thus, the flat potential is indeed a minimum on the ray (6) (albeit degenerate with the trivial one) if, like λ 345 , λ 5 and λ 45 = λ 4 + λ 5 are negative. To establish the top quark's role in Higgs alignment of GW-NHDMs, it suffices to consider this model in one-loop order of the effective potential, V 0 + V 1 . This potential provides a lower minimum than V 0 = 0 by picking out a particular value v of φ, explicitly breaking the scale symmetry of V 0 , and giving H a nonzero mass. The one-loop potential is [20] Only very massive particles contribute to V 1 . They are n = (W ± , Z, t, H , A, H ± ) in this model. The constants are α n = (6, 3, −12, 1, 1, 2); k n = 5/6 for the weak gauge bosons and 3/2 for scalars and the top-quark. 2 The background-field-dependent masses M 2 n in Eq. (10) are [21,13] n ∝ φ 2 is the actual squared "mass" of particle n. The form of M 2 t is dictated by the type-I coupling of fermions to the Φ 1 doublet in Eq.(4). This difference controls the breaking of Higgs alignment through second order in the loop expansion. The renormalization scale Λ GW will be fixed relative to the Higgs VEV v = 246 GeV in Eq. (15) below.
The one-loop extremal conditions are [11] ∂ Here, we follow GW's analysis by expanding around the tree-level VEVs H = φ, H = 0 while allowing for O(V 1 ) shifts δ 1 H and δ 1 H in those VEVs -and from perfect Higgs alignment. Recall that the tree-level extremal conditions (∂V 0 /∂H) = (∂V 0 /∂H ) = 0 remain in force. To O(V 1 ) this expansion results in where, by Eq. (11), the first derivative with respect to H of the n = t terms in V 1 vanish because they are quadratic in H . Eq. (13) provides a definition of the renormalization scale Λ GW in terms of the VEV φ = v at which the minimum of V 1 occurs. It can be rewritten as [11] where A = n α n M 4 n (ln(M 2 n /v 2 )−k n ) and B = n α n M 4 n , so that ln(Λ 2 GW /v 2 ) = A/B + 1 2 . 3 Note that M 2 n ∝ v 2 so that Λ GW /v is a function of coupling constants only.
From Eq. (14), the shift δ 1 H in H is given by the tadpole formula: As an example of its magnitude, we take M H = 400 GeV, Λ GW = 260 GeV and tan β = 0.5. Then δ 1 H = 1.57 GeV which, when added in quadrature with v = 246 GeV, amounts to an increase of 0.002 %. 4 Eq. (16) establishes the connection of the top quark to Higgs alignment: The large mass of the top quark ensures its appearance in the effective potential V 1 while the Glashow-Weinberg criterion [14] implies (∂M At this level, only the top quark prevents M 2 0 + being diagonal and the Higgs boson being completely aligned. To repeat: Because the Glashow-Weinberg criterion applies to any EW model in which quarks of a given charge acquire all their mass from the 3 As discussed in Ref. [11], Eq. (13) does not lead to a minimum of V 1 unless B > 0. With the known masses of W ± , Z, t and H, B > 0; see Eq. (17). 4 Because δ 1 H is not determined in O(V 1 ), we can set it to zero here. This is consistent with our expectation that δH = O(δ 2 ) where δ = O(V 1 ) is the H-H mixing angle. scalars, the top quark's role in Higgs alignment holds in any GW-NHDM. The additional complications of the two-loop effective potential do not alter this conclusion. 5

III. Experimental consequences
ATLAS and CMS discovered the 125 GeV Higgs H relatively easily because of its rather strong coupling to weak bosons: production via W W and ZZ fusion and decay to W W * and ZZ * . It's worth remembering that, despite its lower statistics, H → ZZ * → 4 leptons was much more convincing at first than H → γγ. Furthermore, gg fusion of H via the top-quark loop was important because thettH coupling has its full-strength value, M t /v. Because of this success, it seems, a lamp-post strategy has been adopted for many searches of Beyond-Standard-Model (BSM) Higgs bosons. This is especially true for heavier neutral Higgses such as H and A. The web abounds with searches for H and A production via weak-boson fusion or gluon fusion followed by their decay to weak boson pairs or to ZH, as well as W ± Z → H ± → W ± Z or W ± H; see, e.g., https://twiki.cern.ch/twiki/bin/view/AtlasPublic and https://cms-results.web.cern.ch/cms-results/public-results/ publications/HIG/SUS.html.
All of these processes are strongly hindered by Higgs alignment in GW models. 6 This fact is codified in the (non-Goldstone) Higgs bosons' interactions with electroweak bosons and fermions. They are taken from Ref. [15] but, because the H-H mixing angle δ < ∼ O(1%) [19], so that the rates for Higgs-alignment-violating processes are suppressed by at least a factor of 10 −4 , it is more illuminating to write them in terms of the aligned-basis fields H and H rather than the mass eigenstates H 1 = H cos δ − H sin δ and where V = U † L D L is the CKM matrix. Note that gluon fusion and two-photon decay of H and A via a top-quark loop are suppressed by tan 2 β < 0.25 [15]. So long as tan 2 β is not much smaller, this can, in principle, be overcome by the data expected in Run 3. And, so long as most BSM Higgs decays are to fermion pairs, the tan 2 β suppression does not affect decay branching ratios.
The prospects for testing the GW-2HDM (and similar models) are much brighter than these comments suggest. 7 They stem from the fact that the one-loop approximation for the Higgs boson's mass in Eq. (17) implies a sum rule for the BSM Higgs masses [13,22,15,23]: Eq. (22) tells us that the BSM scalars are lighter than 400-500 GeV. 8 In using this sum rule, we shall assume that M A = M H ± , an assumption justified by the fact that it makes the BSM scalars' contribution to the T -parameter vanish identically [24,25].) The principal search modes for the BSM scalars are via gluon fusion: gg → H +t b with H + → tb and W + H ; gg → A →bb,tt and ZH ; gg → H →bb,tt and ZA, W ± H ∓ .
Cross sections for these processes (with the tan β dependence factored out) are in Fig. 2. It is unlikely that an H or A →bb lighter than 2M t can be seen above the QCD background

1) gg → H +t b →ttbb
There have been four searches for this process relevant to the mass range of the GW-2HDM [17,18,27,28]. The first of these was a CMS search at 8 TeV; the other three are from 13-TeV data with the last an ATLAS search using the full Run 2 data set of 139 fb −1 . The 8-TeV search by CMS was used to set the limit tan β < ∼ 0.5 for 180 GeV < M H ± < 500 GeV. Subsequent searches have not improved on this limit despite the larger date-sets. For example, the limit for the 200-500 GeV mass range extracted from Ref.
[28] is tan β < 1.10 ± 0.14. The main reason for this disappointing outcome is the largett background at low masses and the fact that its rate increases with energy faster than the signal does. Given the payoff of a significant improvement in the limit on tan β, we urge ATLAS and CMS to find a way to overcome the problems of searches at low masses.  Figure 2: The cross sections for √ s = 13 TeV at the LHC for single Higgs production processes in the alignment limit (δ → 0) of the GW-2HDM with the dependence on tan β scaled out. Both charged Higgs states are included in pp → tbH − . From Ref. [15]. be included in a search for H + → tb. Even if that happened, the model expectation σ(gg → H +t b) = 0.075 pb for M H ± 400 GeV and tan β = 0.5 is well below the new 95% CL ATLAS limit of 0.42 pb.

2) gg → A/H →tt
A search by CMS with 35.9 fb −1 of data at 13 TeV for A/H →tt with low mass, 400 < M A/H < 750 GeV, is in Ref. [29]. CMS presented results in terms of allowed and excluded regions of the "coupling strength" g ϕ = λ ϕtt /(M t /v) and for fixed width-to-mass ratio Γ ϕ /M ϕ = 0.5-25%. In the GW-2HDM, g ϕ = tan β. For the CP -odd case, ϕ = A, with 400 GeV < M A < 500 GeV and all Γ A /M A consid-ered, the region tan β < 0.5 is not excluded. 9 This is possibly due to an excess at 400 GeV that corresponds to a global (local) significance of 1.9 (3.5 ± 0.3) σ for Γ A /M A 4%. The CMS paper notes that higherorder electroweak corrections to the SM tt threshold production may account for the excess and that further improvement in the theoretical description is needed.
There have been three published searches so far: [30, 31, 32]. The latter ATLAS search updates the former one with the full Run 2 data set. As noted above, the decay rates of these processes are proportional to p 3 and, therefore, they are sensitive to the available phase space. They quickly become dominant when M A = M H ± > ∼ 400 GeV or M H > ∼ 350 GeV [33,19]. Two examples of this are shown in Table 1 which has been updated from Ref. [19] to include the recent ATLAS analysis. The full Run 2 data set has significantly improved the ATLAS limits on σB which were 255 fb and 105 fb for a 36. A search for A(H ) → ZH (A) → + −b b using Contur [34] was carried out at the Les Houches 2019 "Physics at TeV colliders" workshop [33]. It showed no significant sensitivity to these processes for tan β < 0.5 except near M A ≥ 400 GeV where there was a ≥ 2σ exclusion requiring tan β < ∼ 0.3. This exclusion was based on ATLAS 8-TeV data for + − + jets. This study was recently extended using several full Run 2 ATLAS data sets (see https://hepcedar.gitlab.io/ contur-webpage/results/G-W/index.html.) The 1σ exclusion now covers tan β < 0.9 extending down to tan β 0.43 at M A = 150-200 GeV and tan β > ∼ 0.23 at M A > ∼ 400 GeV. The greatest sensitivity came from ATLAS jet and top measurements except near M A ≥ 400 GeV where ATLAS 4-lepton measurements dominated.