Heavy Higgs of the Twin Higgs Models

Twin Higgs models are the prime illustration of neutral naturalness, where the new particles of the twin sector, gauge singlets of the Standard Model (SM), ameliorate the little hierarchy problem. In this work, we analyse phenomenological implications of the heavy Higgs of the Mirror Twin Higgs and Fraternal Twin Higgs models, when electroweak symmetry breaking is linearly realized. The most general structure of twin Higgs symmetry breaking, including explicit soft and hard breaking terms in the scalar potential, is employed. The direct and indirect searches at the LHC are used to probe the parameter space of Twin Higgs models through mixing of the heavy Higgs with the SM Higgs and decays of the heavy Higgs to the SM states. Moreover, for the Fraternal Twin Higgs, we study the production and decays of twin glueball and bottomonium states to the SM light fermions, which have interesting signatures involving displaced vertices and are potentially observable at the colliders.


Introduction
The discovery of a scalar boson with mass 125 GeV at the Large Hadron Collider (LHC) and its properties show that it is consistent with the Higgs boson of the Standard Model (SM). The SM of particle physics is the most accurate theory of elementary particles and their interactions, however, it is not a complete theory of nature since there are many unanswered puzzles which go beyond the Standard Model (BSM). In particular, the discovery of a relatively light SM Higgs has highlighted the need to address naturalness/hierarchy problem more than ever before, i.e. there should be some mechanism which cancels the SM radiative corrections to the Higgs mass. Motivated by naturalness, enormous work has been done in the last few decades constructing BSM scenarios. Most of the BSM models including supersymmetry, composite Higgs, and extra dimensions predict some sort of new physics (top partners) just above the electroweak scale to cancel the quantum corrections to the Higgs mass. However, the absence of such new physics signatures at the LHC pose a "little hierarchy" (fine-tuning) problem. Hence, there is a need to construct and analyse new BSM scenarios which can explain the elusiveness of the top-partners and solve the SM puzzles.
Neutral Naturalness is one such class of models, where the top-partners are not charged under the SM gauge symmetry, so they are very hard to produce at the colliders. Twin Higgs (TH) model [1][2][3][4] is the prime and first example of neutral naturalness, which is designed to resolve the little hierarchy problem and make the top-partners elusive. In a nutshell, TH models have two core ingredients: (i) the SM-like Higgs is a Goldstone boson which emerges due to spontaneous breaking of a global symmetry, hence its mass is protected, and (ii) the twin top partners are neutral ("colorless") under the SM gauge group, which explains their elusiveness at the LHC. In twin Higgs mechanism the cancellations of radiative corrections to the SM Higgs mass are due to the exchange of new twin states which are related to the SM by a discrete Z 2 symmetry.
The simplest realization of twin Higgs mechanism [1] employs an SU (4) global symmetry which is spontaneously broken to SU (3) at a high energy scale Λ and gives 7 Goldstone bosons and a radial mode. A subgroup SU (2)× SU (2) of SU (4) global symmetry is gauged, where SU (2) and SU (2) represent the SM and twin weak gauge groups, respectively (hat " " denotes twin sector). After TH electroweak symmetry breaking (EWSB), 6 of the Goldstone bosons of SU (4)/SU (3) coset are "eaten" by the weak gauge bosons of the SM (W ± ,Z) and twin (Ŵ ± ,Ẑ) sectors. The remaining one Goldstone boson serves as the SM Higgs. The original incarnation of this idea, where the twin sector has an exact mirror copy of the SM particle content, is referred as 'Mirror Twin Higgs' (MTH) model [1].
However, in order to manifest the features of the twin Higgs mechanism, one does not need to have an exact copy of the SM particle content in the twin sector, but rather a minimal number of states are necessary to ensure the cancellation of radiative corrections to the SM Higgs mass. This possibility was first discussed in the so-called 'Fraternal Twin Higgs' (FTH) model [5]. In order to solve the little hierarchy problem, the FTH model requires only the third generation of quarks and leptons, and the weak gauge bosons in the twin sector. Another striking difference of the FTH model to the MTH case is twin hadron spectroscopy in the two models. In particular, the FTH model has two interesting features: (i) unlike the MTH model, there are no fraternal twin hadrons with light quark flavors (û,d,ŝ,ĉ), but the fraternal twin hadrons are glueball [ĝĝ] and bottomonium [bb] states [5], and (ii) the fraternal twin QCD confinement scale,Λ qcd , can be much larger than that of the MTH model, which is due to fewer twin quark flavors and faster running of the twin QCD couplingsα qcd . Hence, the fraternal twin hadrons (glueballs/bottomoniums) can be relatively heavier [5]. Note that the twin hadrons are colorless under the SM QCD. The lightest stable twin hadrons in the MTH model are the mirror baryons which could be dark matter candidates [2], however, in the FTH case the stable twin hadronic states are the lightest glueball/bottomonium states [5]. Moreover, among the fraternal twin hadrons, the 0 ++ glueball and bottomonium states have the correct quantum numbers to mix with the scalar sector of the model 1 . This mixing leads to exotic decays of the fraternal 0 ++ twin hadrons back into the SM light quarks and leptons at the colliders [5]. These features of the FTH model are similar to those of the 'hidden valley' models [6][7][8][9][10][11].
There are many variants of TH models which offer UV complete description of the TH mechanism; including supersymmetric TH [12][13][14][15][16][17][18], composite/holographic TH [19][20][21][22][23][24], and some more recent TH constructions [25][26][27][28][29][30][31][32][33][34][35]. In most of the TH models, it is assumed that the radial mode of TH EWSB is of the order of symmetry breaking scale or heavier. Hence the radial mode is integrated out and low-energy effective theory contains only the Goldstone bosons. However, if the TH spontaneous symmetry breaking is linearly realized then, in principle, the radial mode can be much lighter than the symmetry breaking scale and can be present in the low-energy theory 2 , in fact, is favored by the electroweak precisions tests (EWPT) (see below, Sec. 2.2). Note that the TH symmetry breaking can be linearly realized in a perturbative UV complete TH model, e.g. supersymmetric TH [16]. However, we work in an effective field theory (EFT) framework of the TH models without considering any specific perturbative UV complete scenario. In this work, we explore the scalar sector of the TH mechanism with a focus on the phenomenological implications of the radial mode. The twin radial mode is heavier than the SM-like Higgs of mass 125 GeV, hence we refer it as 'heavy' twin Higgs.
Twin Higgs scalar sector provides a portal between the visible (SM) and invisible (twin) sectors. Hence, the discovery potential of twin Higgs mechanism needs a detailed phenomenological analysis of the scalar sector of TH models. Recently, there has been some work done on different aspects of the TH scalar sector involving only the SM Higgs, see e.g. [5,[25][26][27][28][29][30][31][32][33][39][40][41][42]. In this work, we aim to investigate the phenomenological aspects of the scalar sector of MTH and FTH models focusing on heavy twin Higgs. We parametrize the fundamental scalar of SU (4), which breaks the global symmetry, by two Higgs doublets H and H. We consider the most general form of TH scalar potential with soft and hard Z 2 discrete and SU (4) global symmetry breaking terms. The effective TH scalar potential has five parameters, however, after fixing the SM-like Higgs mass m h = 125 GeV and its vacuum expectation value (vev) v = 246 GeV, we are left with three free parameters; the heavy twin Higgs mass, its vev, and the hard Z 2 breaking parameter. In this work, we explore the parameter space spanned by these three free parameters of the TH models at the LHC. Note that relatively small number of free parameters offers a unique opportunity to discover the twin Higgs mechanism at the current and/or future colliders.
To analyse phenomenological aspects of the scalar sector of the MTH and FTH models, we calculate the branching fractions of the heavy Higgs to the SM and dark (twin) states. It turns out that dominant decay channels of heavy twin Higgs are pairs of the SM Higgs, SM weak gauge bosons, and twin weak gauge bosons (if kinematically allowed), roughly with equal branching fractions. This makes the heavy Higgs decays to the SM massive gauge bosons and the SM Higgs as the prime channels to discover the TH models at the colliders. We perform a detailed analysis of these channels at the LHC with 14 TeV centerof-mass energy. Moreover, indirect searches at the LHC, via the SM Higgs signal-strength measurements and its invisible decay width, provide a complementary probe to the scalar sector of the TH models.
In the case of FTH, the phenomenological study of the twin hadrons (twin glueball and bottomonium states) is much more involved, mainly due to the lack of proper knowledge of twin QCD confinement/hadronization. However, for our purposes, we make conservative estimates of the twin hadron production through the heavy twin Higgs. The production rates for the twin 0 ++ hadrons through heavy twin Higgs are about the same order as those of the SM Higgs and massive gauge bosons. These twin 0 ++ hadrons (glueball and/or bottomonium states) decay back to the SM light quarks and leptons through the Higgs mixing, see a prototype process in Fig. 1. The 0 ++ twin hadrons decays are either prompt or displaced depending on their kinematics. Especially, the displaced signals offer a unique discovery potential for the FTH model at the colliders. Figure 1: An illustration of a benchmark process of 0 ++ twin glueballĜ 0 + production via heavy twin Higgsĥ at the LHC and decays to the SM bottom quarks through an off-shell SM Higgs.
The paper is organized as follows. Twin Higgs symmetry breaking details, the choice of physical basis, and EWPT constraints on TH models are presented in Sec. 2. The phenomenological analysis of the twin Higgs of the MTH and FTH models are presented in Secs. 3 and 4, respectively. Section 5 contains the summary and conclusions. We collect supplementary material in Appendix A, which includes Feynman rules and partial decay widths of the heavy twin Higgs to the SM and twin sector states.

Electroweak Symmetry Breaking in Twin Higgs Models
We consider the TH models where an SU (4) global symmetry is spontaneously broken to SU (3) providing 7 Goldstone bosons. For the TH symmetry breaking we employ a scalar H which belongs to fundamental representation of SU (4) and parametrize as, where Π is a matrix containing the 7 Goldstone bosons and H is a radial Higgs which acquires a vev H ≡ f 0 . Twin Higgs potential for the EWSB is where using (2.1), we get a quartic potential for the radial Higgs H, In order to have a global minimum of the above potential we require, The above conditions insure that mass squared of the radial fluctuation, m 2 h 0 = 2λf 2 0 , is positive for f 2 0 > 0 and λ > 0. Note that mass of the radial mode can be much lower than the TH symmetry breaking scale f 0 for λ < 1. Whereas for λ 1, we recover a non-linear sigma model version of the twin Higgs models where the radial mode is decoupled.
In this work, we parametrize the scalar H in a linear sigma model framework, i.e.
where H and H are doublets under the SM and twin SU (2) weak groups, respectively, whereas v andv are their respective vevs. The most general TH effective scalar potential including explicit soft and hard Z 2 discrete and SU (4) global symmetry breaking terms can be written as [5,16], Note that one can rewrite all other possible Z 2 /SU (4) explicit symmetry breaking terms in the form of above potential parameters. As mentioned above, we do not aim to consider any specific UV complete description of TH models, rather we assume that the above form of effective potential can arise from a particular UV complete model, e.g. [16]. Before moving forward, we would like to discuss the role/significance of each term in the above effective potential.
1. The λ-term is invariant under SU (4) global symmetry as well as respects Z 2 discrete symmetry under which H ↔ H. As noted above, for f 2 0 > 0, this terms is the source of spontaneous breaking of SU (4) global symmetry to SU (3) giving 7 Goldstone bosons plus a radial modeĥ 0 .
2. The κ-term is invariant under the discrete Z 2 symmetry, however, breaks SU (4) global symmetry explicitly. For κ > 0, the vevs v andv are degenerate in the absence of Z 2 breaking term, i.e. v =v = f 0 / (2λ + κ), which is phenomenologically not a viable option.
3. The σ-term 'softly' breaks both SU (4) global and Z 2 discrete symmetries, as a result a misalignment of the two vevs is generated.
4. The ρ-term captures 'hard' breaking effects of Z 2 /SU (4) symmetries and it helps to reduce the fine-tuning in TH models [16].
Note that the Z 2 /SU (4) breaking terms are important in the above potential to generate mass of the SM-like Higgs boson, which would have been a pure Goldstone boson in the absence of such terms. Moreover, these explicit symmetry breaking parameters are expected to be smaller than the symmetry preserving parameter, i.e. (κ, σ, ρ) ≤ λ, in order to not produce large radiative corrections to the SM-like Higgs mass. We impose unitary gauge where 6 Goldstone bosons are "eaten" by the weak gauge bosons of the SM (W ± ,Z) and twin (Ŵ ± ,Ẑ) sectors. Hence, in the unitary gauge, we write the SM Higgs H and twin Higgs H doublets as, Minimization of the scalar potential (2.5) results in the following values of vevs, v 2 and , .
We define a parameter ξ ≡ v/f . Roughly speaking, 2ξ 2 measures the amount of fine-tuning in our model 3 . In a limit κ λ, the ξ 2 can be written as, In order to have small fine-tuning in the model, i.e. ξ 2 1 or v 2 f 2 , one needs a tuning in the parameters σ and ρ, such that σ ρ for fixed λ and κ. This result implies that large hard breaking effects could potentially lower the amount of fine-tuning in TH models, see also a similar discussion in [16].
Moving on, the mass-squared matrix for the scalar fluctuations (h 0 ,ĥ 0 ) is, We diagonalize M 2 by an orthogonal rotational matrix R as where (h,ĥ) are the two scalar mass-eigenstates (without the subscript '0') with masses (m 2 h , m 2 h ). The two physical scalars, SM Higgs h and heavy twin Higgsĥ, are admixture of h 0 andĥ 0 , and are defined as h = h 0 cosα −ĥ 0 sinα,ĥ = h 0 sinα +ĥ 0 cosα . (2.12) The corresponding mass eigenvalues are, where (2.14) The mixing angle is given by For more intuition of the above results, we expand in the limit ξ 2 = v 2 /f 2 1. At the leading order in this approximation, we get the mass eigenvalues as, The above expressions manifestly show that in the limit ξ 2 1 and fixed v 2 , the SM Higgs mass is directly related to hard breaking parameter ρ. Therefore, the hard breaking parameter ρ would be severely constrained by the SM Higgs mass. However, it is possible to have sizable ρ, in order to reduce the fine-tuning as mentioned above in Eq. (2.9), but at the price of reducing κ to get the SM Higgs mass correct, see also [16]. Furthermore, in the same approximation ξ 2 1, the mixing angle can be written as, which shows a dependence of sin 2 α v 2 /f 2 similar to composite Higgs model. Before going ahead, we would like to comment on the most relevant scalar sector couplings, the triple Higgs couplings g hhh and gĥ hh , which are given as Trilinear Higgs coupling of the SM-like Higgs g hhh is a very crucial indirect probe of new physics at the current and future colliders. Any deviation from its corresponding SM value, would be a signal of new physics. At the high-luminosity LHC (HL-LHC), i.e. √ s = 14 TeV with 3000 fb −1 , it is expected to achieve sensitivity of |1− g hhh /g sm hhh | ∼ 0.5, [44]. However, as we see in the following sections, in most of TH parameter space trilinear Higgs coupling is very mildly constraining. On the other hand, the triple Higgs coupling gĥ hh is crucial for the decay of heavy twin Higgs to a pair of SM Higgs.

Physical Basis for Twin Higgs models
Twin Higgs effective scalar potential (2.6) contains five real parameters, f 0 , λ, κ, σ and ρ. For phenomenological analysis, it is more convenient to choose a basis where these parameters can be expressed in more physical observables. Physical observables for linear TH models are the SM Higgs mass m h , heavy twin Higgs mass mĥ, SM vev v, and twin vevv (or equivalently f ). We fix the SM Higgs mass m h = 125 GeV and its vev v = 246 GeV. This leaves us with three free parameters mĥ, f , and hard breaking parameter ρ. Since we require ρ ≤ λ, therefore, it is instructive to defineρ as In the following sections we work in (mĥ −f/v) plane and constraint the parameter space through direct searches of the heavy twin Higgs and indirect limits from the SM Higgs data at the LHC. The trade of potential parameters in the 'physical TH basis' is as, Advantage of the above parametrization is that it leaves the hard breaking parameterρ as a free parameter, which gives us a handle on hard breaking effects in the model. Potential parameters have the following values in the physical basis: In our phenomenological analysis in Secs. 3 and 4, we require the following conditions on potential parameters in order to satisfy unitarity and perturbativity constraints, along with the SM Higgs mass m h = 125 GeV and its vev v = 246 GeV. If any (or more than one) of the above conditions is not met then the parameter space would be called "unphysical".

Electroweak Precision Constraints in Twin Higgs Models
In this subsection, we briefly comment on the constraints in the TH models from the electroweak precision data when the TH symmetry breaking is linearly realized and a relatively light radial mode (twin Higgs) is present in the effective theory. In particular, the electroweak oblique parameters S and T receive infrared contributions -due to the mixing of the SM-like Higgs with the heavy states and the reduced coupling of the SM-like Higgs to the SM gauge bosons -which are proportional to and Λ is the scale associated with the heavy states [45]. In the linear TH models, a = g h = cos α (see Fig. 16) and the cutoff scale Λ is the heavy twin Higgs mass, i.e. Λ = mĥ. Hence, the shifts of the oblique parameters S and T due to the SM-like Higgs and heavy twin Higgs in the TH models are [5,45], (2.28) In Fig. 2, we employ Gfitter values for S = 0.05±0.11 and T = 0.09±0.13 with free U for the The left (right) plot considers hard breaking contributionsρ = 0(0.5) in the TH effective potential. Note that the presence of hard breaking contributionsρ > 0 (ρ < 0) slightly worsens (improves) the electroweak precision constraints as compared to theρ = 0 case. This is mainly due to the fact that the hard breaking effects are proportional to the mixing angle α, see also below. There are two interesting features we observe from Fig. 2: (i) for a fixed value of f /v, the TH contributions to the oblique parameters S, T are smaller for lighter twin Higgs, and (ii) for a fixed twin Higgs mass, the oblique parameters S, T receive contribution proportional to v 2 /f 2 , this feature is similar to composite Higgs models. In the following sections for the phenomenological analysis, we implement the electroweak precision constraints at 95% C.L on the TH parameter space.

Heavy Higgs Phenomenology in Mirror Twin Higgs
The most general effective renormalizable Lagrangian for the MTH model is 4 excluding the SM Higgs potential andL321 is the corresponding twin sector Lagrangian. 4 We have not included a renormalizable Z2 soft breaking term involving mixing of the SM hypercharge U (1)Y and twin hypercharge U (1)Ŷ , i.e. εBµνB µν . It provides a vector portal between the SM and twin sectors, see [47].
The effective scalar potential V eff (H, H) for the MTH model is given in Eq. 2.6. Notable features of the MTH model are: Gauge structure of twin LagrangianL321 is a mirror copy of the SM Lagrangian L 321 , and the twin sector particle content is the same as that of the SM.
Twin sector gauge and Yukawa couplings are almost equal to that of the SM, i.e.
where g s , g and g denote the SU (3) c , SU (2) L and U (1) Y gauge couplings, respectively, whereas y f are the fermion Yukawa couplings. This is required to cancel the quantum corrections to the SM Higgs mass.
Masses of the twin sector gauge bosons mV and fermions mf scale universally w.r.t. the SM counterparts as, where V = (W ± , Z) and f denotes the SM fermions (twin sector are with a hat).
We assume a mechanism for twin neutrino mass generation exits, see e.g. [32,48]. However, neutrinos play almost no role in the phenomenology of TH scalar sector so they are irrelevant to our discussion.
We collect the most relevant Feynman rules for the scalar sector of the MTH model in Fig. 16 (Appendix A), where apart from tree-level vertices for the SM and twin-sector massive gauge bosons and fermions, we also include loop induced vertices for massless gauge bosons of the two sectors. Note that the SM-like Higgs couplings to the SM sector particles are suppressed by g h = cos α to their respective SM values, which is a common feature of BSM scenarios where the SM Higgs mixes with another scalar with a mixing angle α. The SM Higgs coupling measurements at the LHC put severe constraints on the mixing angle and exclude some of the parameter space of the MTH model. We collect formulae for partial decay widths of heavy twin Higgs to the SM and twin states in Appendix A. For computing heavy twin Higgs cross-sections, we adopt a strategy similar to the one outlined in [49]. For different initial (production) and final (decay) states, the cross-section of heavy twin Higgs is parametrized as: is the SM-like Higgs production cross-section calculated at the heavy twin Higgs mass for Y Y = (gg, V V ). The SM Higgs cross-sections σ SM (h → Y Y )| m h =mĥ are obtained from Higgs Cross-section Working Group [50]. Above B(ĥ → XX) is the twin Higgs branching fraction to XX final state, whereas, effective coupling Cĥ Y Y is defined as, is the SM Higgs partial decay width evaluated at the heavy Higgs mass and Γ(ĥ → Y Y ) is the twin Higgs decay width, see Appendix A. For Y Y = gg and V V , one finds the effective twin Higgs coupling, which is a universal rescaling of heavy twin Higgs production at the LHC and future colliders. Note that the main production channel for the heavy twin Higgs is gluon-gluon fusion (ggf), whereas, the vector-boson fusion (vbf) is subdominant in most of the parameter space of TH models. Similarly, for the SM Higgs with m h = 125 GeV, the cross-section times branching fractions are given as, where for Y Y = (gg, V V ) the effective SM Higgs coupling is The SM Higgs signal-strength measurements in many production and decay channels have been measured with unforeseeable precision at the LHC run-1 [51] and they provide stringent constraints on BSM models. For numerical analysis of the twin Higgs phenomenology, we employ a modified version of HDECAY code [52] incorporating the twin sector, where all the appropriate QCD and electroweak corrections are included in the SM as well as in the twin sector. Note that in the MTH running of twin gauge couplings is similar to those of the SM. We also assume that at the cutoff scale Λ, the SM and twin gauge coupling as well the Yukawa couplings are exactly equal. Moreover, the heavy twin Higgs (and the SM-like Higgs) decays to the twin states are invisible at the colliders.
In Fig. 3 we show branching fractions (larger than a percent) of heavy twin Higgs for different final states of the SM and twin (invisible) sector as a function of the heavy Higgs mass for f/v = 3 (left-panel) and 6 (right-panel), andρ = 0. Notice the lower mass of the heavy Higgs is shifted to higher values as f/v increases. This is due to the fact that with the increase in f/v the lower twin Higgs masses do not reconcile with the SM Higgs mass 125 GeV. For a given production mode (e.g. gluon fusion), the relative magnitudes of these branching ratios determine the relative rates for the various final states. In the absence of the Z 2 /SU (4) hard breaking term, i.e.ρ = 0, it is clear from Fig. 3 the heavy Higgs branching fractions to the SM vector bosons W ± , Z, Higgs h, and the twin vector bosonŝ W ± ,Ẑ, are the most dominant channels. Moreover, forρ = 0 we find,  Figure 3: These plots show heavy twin Higgs branching ratios in the MTH to different final states as a function of its mass. We take f/v = 3 (left-panel) and 6 (right-panel) forρ = 0 and Λ = 5 TeV.  which is a clear manifestation of the Goldstone boson equivalence theorem. Since the longitudinal modes of massive vector bosons and the SM Higgs are Goldstone modes of the TH spontaneous symmetry breaking. Since the radial mode couples universally to all the Goldstone modes, therefore, one would expect that the branching ratio of heavy twin Higgs in the visible (W ± , Z, h) sector would be ∼ 4/7 and that in the invisible (Ŵ ± ,Ẑ) sector would be ∼ 3/7, see also [36]. This gives a profound prediction for the TH models that the decay rates of heavy Higgs to the SM Higgs and vector bosons (say ZZ) are equal. Such an observation at the LHC and/or future colliders would provide a clear distinction between TH and other BSM models.
On the other hand, in the presence of explicit Z 2 /SU (4) hard breaking ρ-term in the TH scalar potential (2.6), the above-mentioned universality among the interactions of heavy twin Higgs to the SM Higgs/vector bosons and twin vector bosons would break. To see this explicit symmetry breaking effects, in Fig 4 we plot the twin Higgs branching fractions for ρ = 0.5 and −0.5, whereρ ≡ ρ/λ. As we are working in EFT framework where the sign and/or the size of explicit hard braking term is a model-dependent quantity so we consider both sign possibilities, see for instance [16]. Notice for the case of positiveρ (Fig 4 left-panel), there is a sharp increase of SM branching fractions, whereas this behavior is reversed for the case of negativeρ (right-panel), as compared to Fig 3 (left-panel) whereρ = 0. The reason for this is the effects of explicit breaking on mixing angle α, which increases for positiveρ and vice versa (see also (2.16)). Hence,ρ > 0 enhances the rates of heavy twin Higgs to the SM sector, whereasρ < 0 suppresses those rates. As a consequence, the LHC (and/or future colliders) would be able to probe more region of TH parameter space withρ > 0 than for ρ < 0 (see below).
In • Gray region of Figs. 5 and 6 represents unphysical parameter space excluded by the theoretical constraints as outlined in the previous section.
• Orange area (Figs. 5 and 6) captures parameter spaces where heavy twin Higgs width is greater than its mass, Γĥ > mĥ.  • Blue region (Fig. 5) presents projected 95% C.L. exclusion limit for cross-section times branching fraction of heavy twin Higgs to SM vector bosons σ ggf h→ZZ for the HL-LHC at 14 TeV with an integrated luminosity of 3000 fb −1 . This estimate is adopted from [36].
• Green shaded area (Fig. 6) shows projected 95% C.L. exclusion limit for cross-section times branching fraction of heavy Higgs to a pair of SM Higgs bosons σ ggf h→hh at the HL-LHC, as adopted from [36].
From Figs. 5 and 6 one can notice a clear enhancement in the heavy twin Higgs crosssections to the SM vector bosons and the Higgs boson forρ = 0.5 as compared toρ = 0 case. As we discussed above, by turning on the hard breaking effects (ρ > 0) not only cross-sections of heavy twin Higgs to the visible sector would increase due to increase in the mixing angle but also the SM Higgs signal-strength would increase. As a result, on one hand, the observability potential of heavy twin Higgs through direct search would increase at colliders, on the other hand, the indirect constraints from the SM Higgs signal-strength measurements would become stringent. However, we note that forρ < 0 the mixing angle decreases, hence making it harder to exclude the parameter space via direct or indirect searches at colliders. As discussed above, a generic prediction of the TH models is that heavy Higgs branching fractions to hh is comparable to that of the V V , hence the signal of heavy Higgs to a pair of SM Higgs is very important for the discovery potential of TH mechanism. However notice that the HL-LHC reach for σ ggf h→hh is about an order of magnitude less than that of the σ ggf h→ZZ . We summarize results for the MTH parameter scan in Fig. 5 and 6 as: (i) The LHC run-1 and early run-2 data only constraints a small region of parameters (iii) Direct searches are more sensitive to the low mass region, whereas the indirect constraints are more effective in the high mass regions of the MTH model. Current indirect constraint implies f/v < ∼ 3, however the reach for the signal-strength measurement at the HL-LHC is about 5% [44] which implies constraints up to f/v < ∼ 6.
For completeness in Fig. 7 we show the contours of the ratio of heavy twin Higgs width over its mass Γĥ/mĥ (solid orange), the ratio of the SM-like Higgs trilinear coupling over its SM value g hhh /g sm hhh (dashed purple), and the SM Higgs invisible (twin) branching fraction B(h → inv.) (dotted), in mĥ −f/v plane forρ = 0 (left) andρ = 0.5 (right). Moreover, in Fig. 7 we also show 95% C.L. constraints on the TH parameter space due to the EWPT (gray hatched region), see Sec. 2.2 for details. Note the indirect probes via the SM-like Higgs trilinear coupling g hhh /g sm hhh and/or the SM Higgs invisible (twin) branching fraction B(h → inv) are far less constraining at the LHC 5 than the Higgs coupling measurements. However, at the future colliders they can be complimentary probe channels.

Heavy Higgs Phenomenology in Fraternal Twin Higgs
The FTH model has the minimal twin sector matter content required to cancel the radiative corrections to the SM Higgs mass [5]. The Lagrangian for the FTH model is, where L 321 is the SM Lagrangian and V eff (H, H) is the scalar potential (same as the MTH model) given in Eq. 2.6. However, the FTH model differs from the MTH case in the twin sector Lagrangian,L32, especially in the matter content. The most salient features of the FTH model as compared to the MTH case are as follows [5]: Gauge symmetry of the FTH LagrangianL32 is SU (3) c × SU (2) L , i.e. there is no twin hypercharge U (1)Ŷ gauge group and hence, no twin photon in the FTH model.
Twin fermionic part of the FTH model contains only third generation of twin quarks (t,b) and leptons (τ ,ν τ ), which are necessary to cancel the SM radiative corrections to the SM Higgs mass.
Twin sector gauge couplings are required to be very close (better than 1%) to that of the SM at the cutoff scale Λ, i.e.
Similarly, twin top Yukawa couplingŷt is required to be very close to the SM top Yukawa coupling, i.e.ŷt y t . These requirements are motivated by the naturalness arguments in cancellation of the radiative corrections.
Contrary to the top Yukawa coupling, the light twin fermions Yukawa couplings (ŷb,ŷτ ) could be substantially different from their SM counterparts.
Twin sector masses of the weak gauge bosons mV and fermions mf scale w.r.t. their SM counterparts as, where the light twin fermions receive an extraŷf /y f enhancement in their masses (and also in their couplings) as compared to the MTH model.
The light twin fermions Yukawa couplings (ŷb,ŷτ ) are essentially free parameters. Which implies an enhancement of the heavy Higgs (and the SM Higgs if mf < m h /2) decay rates to light twin fermions by a factor y 2 f /y 2 f as compared to the MTH case. The most remarkable feature of the FTH model is the relative large QCD confinement scale (due to faster running of the twin QCD coupling constantα qcd ) and subsequent hadronization of twin gluons/bottom quarks to twin glueball/bottomonium states, respectively. In particular, the 0 ++ twin hadrons mix with the Higgs bosons and lead to prompt and/or displaced exotic decays to the SM light fermions.  In the FTH model, the heavy twin Higgs decay rates and cross-sections to the SM sector would be almost same as for the MTH case (Sec. 3). Moreover, the twin weak gauge bosonsŴ ,Ẑ and twin topt of the FTH have the same masses and coupling to heavy twin Higgs as those for the MTH case. Therefore, their the branching fractions/cross-sections would not change substantially as compared to the MTH case. However, light twin fermion Yukawa couplings can be enhanced, as a result their production cross-sections would get enhancement. To be precise, heavy twin Higgs decay rates to the light twin fermions get an extra (ŷf /y f ) 2 enhancement factor as compared to that of the MTH model. The on-shell decays of heavy Higgs to a pair of twin massive vectors bosons (ĥ →VV ) remain same as in the MTH scenario. However, the off-shell decaysĥ →V * V * → 4f involve total widths of the vector bosons ΓV , which change as compared to the MTH case due to fewer light twin fermions. In the FTH model, total decay widths of the twin bosonsŴ andẐ are (assuming mf mV ), whereĝ V andĝ A are the vector and axial couplings of the twin fermionsf = (b,τ ,ν τ ).
In Fig. 8, we show the heavy twin Higgs branching fractions (≥ 1%) to the SM and twin states, where the left-and right-panel are for yb/y b 1.2 and 6, respectively, for fix values of f/v = 3,ρ = 0, andΛ qcd = 7.35 GeV (which corresponds to the lightest glueball mass m0 + = 50 GeV). For simplicity, in this work we take twin tau Yukawa as yτ/y τ = yb/y b , which fixes the twin sector fermion masses up to only one free parameter, the twin bottom Yukawa couplingŷb (or mb). Notice the heavy twin Higgs branching fractions do not substantially change for the twin massive gauge bosons and twin top for yb/y b 1.2 and 6 cases. However, the heavy twin Higgs branching fraction for twin bottom quark, B(ĥ →bb) (shown as dotted curve in Fig. 8), increases about an order of magnitude for a given mĥ value as we increase yb/y b from ∼ 1 to 6. The effect of explicit hard breaking parameterρ on the heavy Higgs branching fractions is similar to the MTH case, i.e. there is an enhancement in the SM branching fractions and subsequently suppression to twin sector branching ratios forρ > 0, and vice versa forρ < 0.
The most notable difference of the FTH model as compared to the MTH case, is the running of twin QCD coupling constantα qcd from the cutoff scale Λ to the confinement scaleΛ qcd . Since the number of quark flavors is less in the FTH (so faster running) as a result the confinement scaleΛ qcd would be larger than the SM QCD confinement scale. In the following numerical analysis we employ the NNLO twin QCD correctedα qcd , however, we require at the cutoff scale Λ,ˆΛ  The above variation in the twin strong coupling constant induce a rather mild finetuning ∼ 30% in the SM Higgs mass. A small variation ofα qcd as compared to α qcd at the high scale Λ significantly change the confinement scale which has very crucial phenomenological consequences. In Fig. 9 we plot the twin confinement scaleΛ qcd as function ofα qcd /α qcd at the cutoff scale Λ = 5 TeV, where we kept fix f/v = 3,ρ = 0, and yb/y b = 5.

Fraternal Twin Hadron Production and Decays
In the FTH model due to fewer twin quark flavors and potentially large confinement scalê Λ qcd , the twin QCD confines to form twin glueball and bottomonium [bb] states. The lightest of the twin glueballs is a 0 ++ stateĜ 0 + with mass m0 + 6.8Λ qcd . Notice that the lightest glueballĜ 0 + has the correct quantum numbers to mix with the SM Higgs (and also twin Higgs) which allows its decay to the SM light fermions. In analogy to the SM bottomonium spectra, the lightest twin bottomonium state isη(0 −+ ), and next to lightest isΥ(1 −− ) state. However, the 0 ++ p-wave stateχ 0 with mass mχ 2mb has the correct quantum numbers to mix with the TH scalar sector such thatχ 0 decays in the visible sector are also possible. Phenomenological aspects of production and decays of the twin glueball and twin bottomonium states due to the SM-like Higgs has been discussed in [5,[39][40][41][42], see also [6][7][8][9][10][11][57][58][59][60]. In this work, we follow these analyses closely but focus on the production of twin hadrons due to heavy twin Higgs. Since twin hadronization is poorly understood, especially, without the proper knowledge of the non-perturbative twin QCD effects, its hard to make a precise estimate of twin hadron production. However, the perturbative estimates can be made quite accurately and the ignorance of twin hadronization can be controllably parametrized, see for instance [5,11,39,41,42]. The twin QCD hadronization of twin gluons and bottom quarks lead to a dozen of meta-stable states in their spectrums. However, we are interested in the production of 0 ++ states, i.e. twin glueballĜ 0 + and twin bottomoniumχ 0 , via the scalar sector of the FTH model. Such that the 0 ++ twin hadrons mix with the SM Higgs and lead to exotic decays to the SM light fermions.
We parametrize heavy twin Higgs partial decay rate to twin glueballĜ 0 + and twin bottomoniumχ 0 states as, where NĜ and N [bb] are the hadronic multiplicities of glueball and bottomonium states at the heavy Higgs mass mĥ, respectively. Whereas,κ 0 is the fraction of glueball or bottomonium states in the 0 ++ state. Twin QCD hadronic average multiplicity N had is defined as [61], is the twin QCD beta function,α qcd (s) is the twin QCD coupling evaluated at the center-of-mass energy √ s, and nf are the number of twin quark flavors below the confinement scaleΛ qcd . The normalization constant N 0 is a free parameter which we fix by setting the number of twin hadrons at a fixed low energy. For example, for a glueball of mass m0 + = 50 GeV, a scalar at mass 125 GeV can produce roughly two glueballs, so we fix N 0 by the condition NĜ 125 = 2. The above hadron multiplicity for the dark QCD has been tested by [62] with showering/hadronization in PYTHIA [63] and a very good agreement has been found. In Fig. 10 (left-panel) we plot the average multiplicity of twin glueballs as a function of the twin Higgs mass for different choices of the lightest glueball mass m0 + . Note that we normalize the glueball multiplicity w.r.t. to the SM Higgs, i.e. we take NĜ 125 = (6, 4, 3, 2) corresponding to m0 + = (10, 25, 40, 50) GeV, respectively, which are rather conservative numbers. Moreover, when the twin bottomonium states are heavier than the lightest glueball, then through the relaxation/annihilation these bottomonium states decay to twin gluons which then hadronize to form the glueballs. In the following we assume if m [bb] > 2m0 + then all the bottomonium states decay to the glueballs, i.e.
[bb] → glueballs. The rightpanel of Fig. 10 show the production cross-section of the lightest glueball state with mass m0 + = 10 GeV via the heavy twin Higgs. The heavy Higgs production is considered by the ggf (solid curve) and ggf+vbf (dashed curve) at the LHC with center-of-mass energy 14 TeV. As mentioned above, around a dozen (depending on their masses) meta-stable glueball states are formed and the probability to produce the lightest glueballĜ 0 + is parameterized bŷ κ 0 , we take a conservative choice ofκ 0 = 10% (blue curves) andκ 0 = 25% (red curves). In Fig. 10 (right-panel) we consider mb = 15 GeV (nf = 0), such that only stable twin hadrons are the glueballs, whereas, other parameters are f /v = 3 andρ = 0. The bumps in the curves of glueball cross-section indicate thresholds for the twinẐ and top quarkt, as the decays of twinẐ boson andt quark to twin bottomb quarks lead to hadronize and decay to the Figure 10: The left-graph shows average glueball multiplicity NĜ as a function of the heavy Higgs mass for different choices of the lightest glueball mass and the corresponding normalized number of glueball states at the SM Higgs mass. In the right-panel we plot cross-section times branching fraction of the heavy twin Higgs decaying to the lightest glueballsĜ 0 + with mass m0+ = 10 GeV as a function of the twin Higgs mass. The twin Higgs at the 14 TeV LHC is produced via ggf (solid) and ggf+vbf (dashed). We takeκ 0 = 10% (blue) andκ 0 = 25% (red) for the lightest glueball multiplicity.
glueball states. Note that there could be non-perturbative effects in the production of the twin glueballs as large as 10% as argued in [5], however, we assume they are within the uncertainty of our approximation for the average hadron multiplicity and theκ 0 parameter. Nevertheless, the above parameterization provides a reasonable estimate for the production of fraternal twin hadrons.
The lightest twin glueballĜ 0 + mixes with the TH scalar sector, which leads to its decay to the SM light quarks and leptons, see Fig. 11 (left). In the following, we calculate decay width of the twin glueballĜ 0 + to the SM light fermions. The first step is to calculate the twin gluon coupling to the SM Higgs 6 , which is given by, where i =t,b (dominant contribution is by twin top loop) and in the limit τt = 4m 2 t /m 2 h 1 the triangle loop functionF 1/2 (τt) → −4/3, whereasĝ h = − sinα (see also Fig. 16). Hence in the low-energy effective theory, twin gluon coupling to the SM Higgs is sinα Tr G a µν G a,µν h . (4.10) Now following [11] we can write decay rate of the glueball to the SM light states Y as, where F0 + ≡ 0|Tr[Ĝ a µνĜ a,µν ]|0 ++ is the decay constant of the lightest glueballĜ 0 + and its value by lattice is 4πα qcd F0 + = 3.06 m 3 0 + [64]. Above, Γ SM (h → Y Y ) is the SM Higgs decay rate to the SM light states Y evaluated at the twin glueball mass m0 + . Assuming the twin leptons (ˆ =τ ,ν) are heavy enough such that the invisible twin weak decays ofχ 0 →Υˆ ˆ are kinematically forbidden. Then, as mentioned above, for mχ > 2m0 + , bottomonium stateχ 0 decays to the glueballs, otherwise for mχ < 2m0 + ,χ 0 decays to the visible sector fermions. The decays ofχ 0 state to the SM sector would meditate through the SM Higgs, as shown in Fig. 11 (right), and the interaction is given by, An estimate of the bottomonium stateχ 0 decay to the SM light fermions via an off-shell Higgs using non-relativistic quantum mechanism is [5], where R (0) is derivative of the radial wave-function of p-wave stateχ 0 w.r.t. its radius. We use the estimate of |R (0)| 2 given by [5], which employs linear confining potential, but neglecting the relativistic corrections which could be potentially important. The estimate of |R (0)| 2 forχ 0 state is [5], (4.14) In the following subsection we calculate the production cross-section for the twin hadrons via heavy twin Higgs in mĥ −f/v plane. The decay length of the twin 0 ++ hadron is, Depending on kinematics of the produced twin hadrons (0 ++ ) their decay-length cτ 0 ++ can be from very short < ∼ 10 −3 [m] to very long > ∼ 10 2 [m]. For very short decay-length the daughter decays to the SM light fermions would be prompt, so the signature at the colliders could be high multiplicity jets, see e.g. [65,66]. On the contrary, for very long decay-length the twin-hadrons would escape the detectors and signatures would be missing energy. However, the most promising signals would appear for the case when the twin hadrons decay-length is of the size of detectors, i.e. 10 −3 m < ∼ cτ 0 ++ < ∼ 10 2 m, such that the signatures would appear as displaced vertices at the colliders. There are ongoing searches at the LHC for long-lived particles at the ATLAS [67][68][69], the CMS [70,71] and the LHCb [72] experiments.

Fraternal Twin Hadron Phenomenology
In this subsection, we calculate the production cross-section of twin glueballĜ 0 + and twin bottomoniumχ 0 states via heavy twin Higgs at the 14 TeV LHC 7 . For this purpose, we choose several benchmark cases where either the lightest twin hadron is a glueball or a bottomonium state. In Fig. 12 we plot the contours of production cross-section [fb] (solid blue) and proper decay-length [m] (dashed green) of glueballsĜ 0 + with mass m0 + = 10 GeV in mĥ−f/v plane, and the twin bottom quark mass mb = 15 GeV. The left and right panels of Fig. 12 are for ρ = 0 GeV andρ = 0.5 GeV, respectively. For the chosen masses of twin hadrons we normalize average hadron multiplicity at the SM Higgs mass as NĜ 125 = 6 and N [bb] 125 = 3 for the twin glueball and bottomonium states, respectively. Note that for mb = 15 GeV the twin bottomonium states with mass m [bb] 30 GeV are unstable and we assume they decay 100% to the twin glueballs. Moreover, we setκ 0 = 0.25, the probability to produce the lightest glueball stateĜ 0 + . The shaded regions correspond to the unphysical parameters (gray), exclusion by the SM Higgs signal-strength measurements (red), and exclusion by the SM invisible branching fraction (hatched light-gray). Notice that heavy twin Higgs cross-section to twin glueballsĜ 0 + is comparable to its cross-section to the SM Higgs/vector bosons. The decay length of twin glueballĜ 0 + with mass m0 + = 10 GeV is rather large > ∼ 10 m in most of the FTH parameter space, so it decays outside the detector and hence would appear as a missing energy signal.
Current LHC searches [67][68][69][70][71][72] for long-lived particles are not sensitive for the crosssections obtained here for the twin hadron production via heavy twin Higgs so we do not get any constraints, however in future some of the regions of the FTH parameter space can be probed by these searches. The studies of twin hadron production via the SM Higgs for the FTH model seems to offer more feasibility for the observation of twin hadrons at the LHC, see [39][40][41][42]. Projected reach for the HL-LHC is to explore the TH parameter space up to f/v < ∼ 6 in the twin glueball mass window m0 + ≈ [10,60], [39]. However, the twin hadron production through the SM Higgs is limited only for the low mass ( < ∼ m h /2), whereas, their production via heavy Higgs decay offers chances to probe heavier twin hadrons as well.

Heavy Twin Higgs Phenomenology
In this subsection, we briefly discuss the direct and indirect phenomenological prospects of the heavy twin Higgs in the weak sector. As we have learned from Fig. 8, the dominant decay modes of the twin Higgs in the FTH model are the SM and twin massive gauge bosons and the SM Higgs. Roughly speaking, phenomenology of heavy twin Higgs in the FTH model is very similar to that of the MTH case. The only difference, which can be crucial for indirect constraints, is the flexibility of twin light fermion masses which may enhance the SM Higgs invisible decay width.
We consider one benchmark scenario for the choice of twin glueball mass m0 + = 25 GeV and twin bottom mass mb = 75. Moreover, we assume that twin lepton mass (Yukawa) scale as mτ = mb m b m τ . In the left-panel of Fig. 15 we plot contours of the twin Higgs cross-section (solid blue) to a pair of SM Z bosons and the SM Higgs signal-strength µ ggf h→ZZ (dashed red) forρ = 0 in mĥ − f/v plane. The right-panel of Fig. 15 shows the concours of the ratio of heavy twin Higgs width over its mass Γĥ/mĥ (solid orange), the SM Higgs trilinear coupling over its SM value g hhh /g sm hhh (shaded purple), and the SM Higgs invisible (twin sector) branching ratio B(h → inv.) (dotted gray). The shaded regions represent: unphysical parameters (gray), exclusion by the SM Higgs signal-strength measurements (red), exclusion by the SM invisible branching fraction (hatched light-gray), exclusion by the EWPT at 95% C.L (below the gray line, hatched area), exclusion by the direct heavy scalar searches at the LHC (purple), and the HL-LHC projected reach for direct searches (light blue). Note that for our choice of twin bottom mass mb = 75, we have mτ 30 GeV, which implies that the SM Higgs decays tobb are forbidden but toττ are allowed. The invisible decays of the SM Higgs are only in twin gluonic and twin leptonic channels.

Conclusions
In this work, we have explored phenomenological implications of the radial mode of Twin Higgs models, when the TH symmetry breaking is linearly realized. We consider an SU (4) global symmetry for TH models which is spontaneously broken to SU (3) and gives 7 Goldstone bosons. Out of these 7 Goldstone bosons, 6 are "eaten" by the SM and twin weak gauge bosons, and hence leaving only one Goldstone boson which plays the role of the SM Higgs boson. We consider an EFT approach and employ the most general symmetry breaking structure for the TH scalar potential including explicit soft and hard breaking Z 2 discrete and SU (4) global symmetries. These explicit breaking terms are the source of generating the SM Higgs mass. These terms are model dependent quantities and are calculable in a given perturbative UV complete TH model. The mixing of the SM-like Higgs and the twin Higgs (radial mode) provide a portal to the SM and twin sectors, which could potentially lead to the discovery of TH mechanism.
We have considered two Twin Higgs scenarios; the MTH, which has an exact mirror of the SM field content in the twin sector, and the FTH, which has the minimal field content needed to solve the little hierarchy problem. The symmetry breaking pattern of the TH mechanism is such that the SM Higgs and weak gauge bosons, and twin weak gauge bosons are the Goldstone bosons of same broken symmetry, therefore their interactions with the radial mode (heavy twin Higgs) are fixed. As a result, the heavy twin Higgs branching fractions to the SM Higgs and weak gauge bosons W ± , Z are equal. Furthermore, the twin Higgs branching fractions to the SM sector as compared to the twin sector are approximately ∼ 4/3. However, as shown in Secs. 3 and 4, this universal behavior changes considerably when explicit hard breaking is turned on. Employing the current experimental direct and indirect searches at the LHC for a heavy scalar we explore the parameter space of the TH models. Current LHC direct searches for a heavy scalar to a pair of weak gauge boson constraint a small region of the MTH parameter space, in particular excluding the twin Higgs < ∼ 500 GeV mass. Whereas, the searches in the other decay channels especially to a pair of SM Higgs are not constraining with the available data. However, it is shown that the HL-LHC reach, at 14 TeV with 3000 fb −1 , can discover/exclude the heavy twin Higgs up to 1 TeV mass. On the other hand, indirect searches at the LHC by the SM Higgs signal-strength measurements are quite constraining. In particular, the current LHC data excludes f/v < ∼ 3 without explicit hard breaking contributions. In the presence of explicit hard breaking contributions, the f/v ratio could be less or more constrained depending on the sign of these terms. Moreover, the HL-LHC reach for the SM Higgs signal-strength measurements could potentially exclude the MTH parameter space up to f/v < ∼ 6.
In Sec. 4, we have explored phenomenology of the twin Higgs in the FTH model. In particular, we have discussed the consequences of twin QCD confinement. In the absence of light quark flavors in the FTH model, twin QCD confinement/hadronization leads to the formation of twin glueball/bottomonium states, which have interesting phenomenological signatures. We have discussed the production of these twin hadrons via heavy twin Higgs decays at the LHC. More importantly, the 0 ++ glueballĜ 0 + and bottomoniumχ 0 states have the correct quantum number to mix with the scalar sector of the model, as a result, decays of these twin hadrons back to the SM light fermions are possible. The lessons we have learned from the twin hadron phenomenology are: (i) The heavy Higgs cross-section to the 0 ++ twin hadrons at the LHC is comparable to that of the SM vector bosons, and (ii) Depending on the masses of 0 ++ twin hadrons, their decays can be prompt, displaced or even can escape the detectors. These signatures offer a unique experimental opportunity to search for multi-jet prompt or displaced objects and/or missing energy signals at the colliders. Hence, any signals of such events at the LHC could lead to the discovery of twin Higgs framework.  Heavy Higgs decays into gauge bosons: The heavy twin Higgs partial decay width to on-shell SM massive gauge bosons is, where, λ(x 1 , x 2 ) ≡ (1 − x 1 − x 2 ) 2 − 4x 1 x 2 with x 1,2 = m 2 V 1,2 /m 2 h and S V is a symmetry factor, for identical gauge bosons it is 2 otherwise 1. Moreover, we also include the heavy Higgs decays with off-shell massive gauge bosons, which allows four-body decays of the heavy Higgs, as: The partial decay widths of the twin Higgs to massless gauge bosons of the SM are, where the triangle loop functions F 1/2 , F 1 , A F and A W are adopted from the Higgs Hunter's Guide [74] and for completeness we collect them below. It is straightforward to obtain the heavy twin Higgs partial decay widths to the twin gauge bosons from the above relations by interchanging V →V , gĥ →ĝĥ, G f →Ĝ f , α em → α em , α qcd →α qcd F 1/2 →F 1/2 , F 1 →F 1 , A F →Â F and A W →Â W .
Heavy Higgs decays into SM di-Higgs: The heavy twin Higgs partial decay width to a pair of the SM Higgs is, where the trilinear coupling gĥ hh is given in (2.19).