Phenomenology of the new light Higgs bosons in Gildener-Weinberg model

Gildener-Weinberg (GW) models of electroweak symmetry breaking are especially interesting because the low mass and nearly Standard Model couplings of the $125\,{\rm GeV}$ Higgs boson, $H$, are protected by approximate scale symmetry. Another important but so far under-appreciated feature of these models is that a sum rule bounds the masses of the new charged and neutral Higgs bosons appearing in {\em all} these models to be below about $500\,{\rm GeV}$. Therefore, they are within reach of LHC data currently or soon to be in hand. Also so far unnoticed of these models, certain cubic and quartic Higgs scalar couplings vanish at the classical level. This is due to spontaneous breaking of the scale symmetry. These couplings become nonzero from explicit scale breaking in the Coleman-Weinberg loop expansion of the effective potential. In a two-Higgs doublet GW model, we calculate $\lambda_{HHH} \simeq 2(\lambda_{HHH})_{\rm SM} = 64\,{\rm GeV}$. This corresponds to $\sigma(pp \to HH) \cong 15$--$20\,{\rm fb}$, its {\em minimum} value for $\sqrt{s} = 13$--$14\,{\rm TeV}$ at the LHC. It will require at least the $27\,{\rm TeV}$ HE-LHC to observe this cross section. We also find $\lambda_{HHHH} \simeq 4(\lambda_{HHHH})_{\rm SM} = 0.129$, whose observation in $pp \to HHH$ requires a $100\,{\rm TeV}$ collider. Because of the above-mentioned sum rule, these results apply to {\em all} GW models. In view of this unpromising forecast, we stress that LHC searches for the new relatively light Higgs bosons of GW models are by far the surest way to test them in this decade.


I. SYNOPSIS
Section II of this paper reviews the Gildener-Weinberg (GW) mechanism for producing a model of a naturally light and aligned Higgs boson H in multi-Higgs-scalar models of electroweak symmetry breaking [1]. This is done in the context of a two-Higgs doublet model (2HDM) due to Lee and Pilaftsis [2]. The tree-level Higgs potential in GW models is scale invariant, but that symmetry can be spontaneously broken, resulting in H as a massless dilaton with exactly Standard Model (SM) couplings to gauge bosons and fermions. This scale symmetry is explicitly broken in one-loop order of the Coleman-Weinberg effective potential [3], resulting in M 2 H > 0 but only small deviations from its exact SM couplings. An important corollary of the formula for M 2 H is a sum rule for the masses of the additional Higgs scalars, generically H. In any GW model of electroweak breaking in which the only weak bosons are W AE and Z 0 and the only heavy fermion is the top quark, the sum rule in the first-order loop-perturbation theory is [2,4,5] In the GW-2HDM model, the additional Higgs bosons are a charged pair, H AE , and one CP-even and one CP-odd scalar, which we call H 2 and A. This sum rule has profound consequences for the phenomenology of GW models that this paper emphasizes. For example, in a search for these new Higgses, care must be taken in using the sum rule to estimate the light scalar's mass when the other scalar masses are assumed to exceed 400-500 GeV. In Sec. III, we discuss features of the cubic and quartic Higgs boson self-couplings peculiar to GW models. As a consequence of unbroken scale invariance in the classical Higgs potential, certain of them vanish. These couplings do become nonzero once the scale symmetry is explicitly broken. We calculate the most important of these, finding that the experimentally most relevant ones, λ HHH and λ HHHH , imply σðpp → HHÞ and σðpp → HHHÞ too small to detect at even the High-Luminosity (HL) LHC [6]. Again, because of the sum rule (1), this conclusion is true in all GW models of electroweak symmetry breaking, regardless of their Higgs sector.
This leads to Sec. IV, where we refocus on direct searches at the LHC for the new light Higgs bosons of GW models. We briefly summarize these Higgses' main search channels and the status of these searches. Substantial progress is in reach of data in hand or to be collected in the near future. There is nothing exotic about these searches; what is required for discovery or exclusion is greater sensitivity at relatively low masses.

II. THE TWO-HIGGS DOUBLET MODEL
In 1976, Gildener and Weinberg proposed a scheme, based on broken scale symmetry, to generate a light Higgs boson in multiscalar models of electroweak symmetry breaking. In essence, their motivation was to generalize the work of Coleman and Weinberg [3] to completely general electroweak models, with arbitrary gauge groups and representations of the fermions and scalars. What GW did not appreciate then-there was no reason for them towas that their Higgs boson was also aligned [7]. That is, of all the scalars, its couplings to gauge bosons and fermions were exactly those of the single Higgs boson of the SM [8]. Like the Higgs boson's mass, its alignment is protected by the approximate scale symmetry [5].
GW assumed an electroweak Lagrangian whose Higgs potential V 0 has only quartic interactions. With no quadratic nor cubic Higgs couplings and, assuming that gauge boson and fermion masses arise entirely from their couplings to Higgs scalars, the GW theory is scale invariant at the classical level. This Lagrangian may, however, have a nontrivial extremum. If it does, it is along a ray in scalarfield space and it is a flat minimum if the quartic couplings satisfy certain positivity conditions. Thus, scale symmetry is spontaneously broken at tree level, and there is a massless (Goldstone) dilaton, H, which GW called the "scalon." Higgs alignment is a simple consequence of the linear combination of fields composing H having the same form as the Goldstone bosons w AE and z that become the longitudinal components of the W AE and Z bosons; see Eqs. (8) below.
Importantly, scale symmetry is explicitly broken by the first-order term V 1 in the Coleman-Weinberg loop expansion of the effective scalar potential [3]: V 0 þ V 1 can have a deeper minimum than the trivial one at zero fields. If it does, it occurs at a specific vacuum expectation value (VEV) hHi ¼ v, explicitly breaking scale invariance. Then M H and all other masses in the theory are proportional to v. The GW scheme is the only one we know in which the entire breaking of scale and electroweak symmetries is caused by the same electroweak operator, namely, hHi. Hence, the dilaton decay constant f ¼ v [9], which we take to be 246 GeV.
In 2012, Lee and Pilaftsis (LP) proposed a simple 2HDM model of the GW mechanism employing the Higgs doublets [2]: Here, ρ i and a i are neutral CP-even and -odd fields. Their potential is All five quartic couplings are real so that V 0 is CP invariant as well. This potential is consistent with a Z 2 symmetry that prevents tree-level flavor-changing interactions among fermions, ψ, induced by neutral scalar exchange [10]: This is the usual type-I 2HDM [11], but with Φ 1 and Φ 2 interchanged; we refer henceforth to this version of the model as the GW-2HDM. This choice of Higgs couplings differs from LP's choice of type II [2]. It was made to remain consistent with limits from CMS [12] and ATLAS [13] on charged Higgs decay into tb. The limits from these papers are consistent with tan β ≲ 0.5 for M H AE ≲ 500 GeV. This range of tan β also suppresses gg → AðH 0 Þ →bb;tt, where AðH 0 Þ is a CP-odd (-even) Higgs, relative to a heavy Higgs boson H with SM couplings. See the discussion and references in Ref. [5]. The potential V 0 can have a flat minimum along the ray Here ϕ > 0 is any real mass scale, c β ¼ cos β and s β ¼ sin β. The nontrivial tree-level extremal conditions are (for β ≠ 0, π=2) As explained in Sec. III, V 0β ¼ 0 is a consequence of the scale-invariance of the classical potential V 0 . The squared "mass" matrices of the CP-odd, charged, and CP-even scalars are given by where g, g 0 are the electroweak SUð2Þ and Uð1Þ gauge couplings where hi means that the derivatives of V 0 þ V 1 are evaluated at the vacuum expectation values of the fields, and For nontrivial extrema with β ≠ 0, π=2, these conditions lead to a deeper minimum than the zeroth-order ones, This minimum occurs at a particular value v of the scale ϕ which, as we have said, is identified as the electroweak breaking scale The CP-odd and charged Higgs bosons' masses receive no contribution from V 1 and, so, they are given by Eqs. (8) with ϕ ¼ v. The CP-even masses, however, receive important corrections from V 1 . The eigenvectors H 1 and H 2 are The angle δ measures the departure of the Higgs boson H 1 from perfect alignment, and it should be small. Furthermore, the accuracy of first-order perturbation theory requires jδ=βj ≪ 1. Both these criteria are met in calculations with a wide range of input parameters; they are illustrated in Fig. 1. From now on we refer interchangeably to the 125 GeV Higgs boson as H 1 or H, as clarity requires. Its mass is given by [1,2,5] In accord with first-order perturbation theory, all the masses on the right side of this formula are obtained from zerothorder perturbation theory, i.e., from V 0 plus gauge and Yukawa interactions, with ϕ ¼ v. As we see in Fig. 1, the Higgs masses M H and M H 1 derived from Eq. (14) and from diagonalizing the one-loop mass matrix M H 0 þ , respectively, are extremely close, as they should be. This formula can be used in two related ways. First, assuming that there are no other heavy fermions and weak bosons, it implies a sum rule on all the new scalar masses in this GW-2HDM [2,4,5]: The sum rule is illustrated in Fig. 2 for The smallness of δ in Fig. 1 [14,15].) Due to the sum-rule constraint in Eq. (15), the mass of M H 0 is very sensitive to small changes in M H AE when it is large. Figure 2 suggests we can use the sum rule until M H 0 starts to dive to zero. To be quantitative about this, Fig. 3  The only model parameter that enters this calculation is tan β; Fig. 3 is practically independent of tan β. M H 0 ≅ M H 2 ¼ 412 GeV and, beyond it, M H 0 starts its dive. The most sensible thing to do, in our opinion, is to use the large CP-even mass eigenvalue, M H 2 , over the entire considered range of M H AE ¼ M A .
We will do this for our estimates of the scalars' production cross sections and decay branching ratios in Sec. IV. We recommend this approach for searches by ATLAS and CMS. For example, in a search involving the three GW Higgs bosons (say, pp → A → ZH 2 and pp → H AE → W AE H 2 , with H 2 → bb), one could use ellipsoidal search regions in (M H AE ; M A ; M H 2 ) space roughly consistent with M H AE ¼ M A and the sum rule and calculate the model's predicted σ · BR's accordingly. Therefore, as with H and H 1 , we refer henceforth to the heavier CP-even scalar as H 0 or H 2 , as clarity or the situation requires.

III. TRIPLE AND QUARTIC HIGGS COUPLINGS
In GW models of electroweak symmetry breaking, the tree-level triple-scalar couplings involving two or three of the Goldstone bosons H; z; w AE vanish, as do the quartic couplings involving three or four of them. This is unlike any other multi-Higgs model. The reason for this, of course, must be scale invariance of the tree-level Lagrangian, in particular, that the potential V 0 contains only quartic couplings. But how does it work? We show how in this section. Then we calculate at one-loop order the triple-scalar couplings involving at least one H ≅ H 1 and the quartic coupling λ H 1 H 1 H 1 H 1 .
The way to see simply why certain scalar couplings vanish is to write V 0 in the "aligned basis": On the ray Eq. (5) on which V 0 has nontrivial extrema, these fields are where ϕ ∈ ð0; ∞Þ is a constant mass scale. Then, in terms of the tree-level mass-eigenstate scalars, the fields Φ and Φ 0 are (after shifting H → H þ ϕ) Rewritten in terms of quartic polynomials in Φ and Φ 0 , By virtue of its scale invariance, V 0 is a homogeneous polynomial of degree four: Thus, V 0 vanishes at any extremum, in particular for Φ β ¼ ð0; ϕÞ= ffiffi ffi 2 p and Φ 0 β ¼ ð0; 0Þ, the flat direction associated with spontaneous scale symmetry breaking. We know that the conditions for the nontrivial extrema of V 0 are those in Eq. (6). It follows that the coefficients of ðΦ † ΦÞ 2 and ðΦ † ΦÞðΦ † Φ 0 þ Φ 0 † ΦÞ terms in V 0 vanish. It is easy to see why these coefficients, C 1 and C 2 , had to vanish. On the ray Neither operator vanishes; hence, their coefficients must. 2 This would not have happened had V 0 also contained polynomials of degree less than four. That is, spontaneously broken scale invariance is the reason for the vanishing Goldstone boson couplings at tree level. And it is obvious that this analysis using homogeneous polynomials of fourth degree generalizes to any GW model of the electroweak interactions. Using the tree-level extremal conditions, the nonzero coefficients in V 0 are simplified by using Then, From this, the masses in Eq. (7) may be read off from the first three terms. With foreknowledge, we now put ϕ ¼ v ¼ 246 GeV. Then the nonzero cubic terms in the tree-level potential, written in terms of mass eigenstate scalars of V 0 , are 3 The quartic terms are Recall from Eq.
We turn to the one-loop corrections, focusing on the triple-scalar couplings involving the 125 GeV Higgs boson, H ≅ H 1 , and the quartic coupling λ H 1 H 1 H 1 H 1 . For brevity, we include only those cubic couplings of H 1 with itself and with H 2 . The H 1 AA and H 1 H þ H − couplings are similar to H 1 H 2 H 2 , as may be inferred from the tree-level cubics in Eq. (27) and Table I below. There are two types of one-loop corrections: (i) those to V 0 obtained by writing the zerothorder CP-even fields in terms of H 1 and H 2 , Eqs. (13), and by using the one-loop extremal conditions, Eqs. (11); (ii) those obtained from V 1 in Eq. (9) by isolating the coefficients of H 3 , H 2 H 0 , etc.
(i) With ρ i shifted by v i , the cubic CP-even terms in V 0 are Our convention for the triple and quartic couplings of H 1 , for example, is that they are the coefficients of H 3 1 and H 4 1 in these two types of corrections. Then, the corrections to the triple-Higgs couplings from   Table II. A word of caution is in order here: These decay rates are dominated by the emission of longitudinally polarized weak bosons and are proportional to p 3 =M 2 W;Z , hence sensitive to the available phase space. At the LHC there are now 140 fb −1 of pp collision data at 13 TeV from run 2 and another 200 fb −1 at 14 TeV are expected from run 3 by the time it concludes at the end of 2024. With masses in the range 200-500 GeV, GW Higgs production rates are σðpp → H þ þ H − Þ ¼ ð0.1-1.0Þ pb × tan 2 β, σðpp→AÞ¼ð4.0-20Þpb×tan 2 β and σðpp → H 2 Þ ¼ ð2.0-7.0Þ pb × tan 2 β. Thus, unless tan β ≲ 0.2, there will be anywhere from 10 3 to several 10 6 of these GW Higgs bosons produced by the end of run 3. Given the large SM production of bb, direct detection of H 2 → bb via gluon fusion is the most difficult. There is no doubt that improved sensitivity in the low-mass region of H AE → tb is needed to access the expected cross sections. The decays Aðor H 2 Þ → ZH 2 ðor AÞ → l þ l − bb, H AE → W AE H 2 → l AE bb þ E T are helped by the narrow bb resonance and lepton kinematics. They may be easier than H AE → tb, but they cover a slimmer portion of (M H AE ; M A ; M H 2 ) space, the upper and lower ends of the allowed M H AE ¼ M A region.

ACKNOWLEDGMENTS
We are grateful for informative conversations with and advice from Tulika Bose, Kevin Black, Gustaaf Brooijmans, Jon Butterworth, Estia Eichten, Howard Georgi, Guoan Hu, FIG. 5. The cross sections for ffiffi ffi s p ¼ 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 → tH − . From Ref. [5].