A tale of twin Higgs: natural twin two Higgs doublet models

In original twin Higgs model, vacuum misalignment between electroweak and new physics scales is realized by adding explicit ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_2 $$\end{document} breaking term. Introducing additional twin Higgs could accommodate spontaneous ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_2 $$\end{document} breaking, which explains origin of this misalignment. We introduce a class of twin two Higgs doublet models with most general scalar potential, and discuss general conditions which trigger electroweak and ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_2 $$\end{document} symmetry breaking. Various scenarios on realising the vacuum misalignment are systematically discussed in a natural composite two Higgs double model framework: explicit ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_2 $$\end{document} breaking, radiative ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_2 $$\end{document} breaking, tadpole-induced ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_2 $$\end{document} breaking, and quartic-induced ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_2 $$\end{document} breaking. We investigate the Higgs mass spectra and Higgs phenomenology in these scenarios.


Introduction
The discovery of a 125 GeV Higgs boson at the LHC [1,2] is a great triumph of the Standard Model (SM) of particle physics. Although it confirms the Higgs mechanism, it sharpens existing naturalness problem. Naturalness tells us that the weak scale should be insensitive to quantum effects from physics at very higher scale. However, in SM, the large, quadratically divergent radiative corrections to the Higgs mass parameter destabilize the electroweak scale. From theoretical point of view, the SM should be well-behaved up to Planck scale. The existing hierarchy between the Planck and weak scales requires that the quantum corrections to the Higgs mass parameter should cancel against the Higgs bare mass to obtain the observed 125 GeV Higgs boson mass. The large cancellation indicates -1 -

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existence of fine-tuning between the tree-level Higgs mass parameter and loop-level Higgs mass corrections. This is the well-known hierarchy problem [3].
The dynamical solution to the naturalness problem is to introduce a new symmetry which protects the Higgs mass against large radiative corrections. Under this direction are weak scale supersymmetry [4], and composite Higgs [5][6][7], etc. These new physics (NP) models introduce symmetry partners of the SM fields that cancel the quadratically divergent corrections to the Higgs boson mass. Because the dominant quantum correction to the Higgs mass involves in the SM top quark in the self-energy loop, the top quark partner is typically most relevant new particle to the quadratic cancellation. The new symmetry not only relates the top partner with the SM top quark, but also relates the Higgs coupling of the top partner to the one of the top quark. This enforces quadratic cancellation between the top quark and top partner contributions. Since the top partners typically carry SM color charge, the search limits of these top partners at the LHC have reached 700∼ 800 GeV. This already leads to around 10% level of tuning between the weak scale and NP scale. This is known as the little hierarchy problem.
One way to avoid the little hierarchy problem is the neutral naturalness [8][9][10][11], that symmetry partners are not charged under the SM gauge groups. This lowers the NP cutoff scale, and thus softens the little hierarchy problem. The twin Higgs model [8,9] [see also refs. [12][13][14][15][16][17] and [23][24][25]] introduces the mirror copy of the SM, the twin sector, which is neutral under the SM gauge group. The Higgs sector respects the approximate global U(4) symmetry, which is broken spontaneously to U (3) at NP scale f . The U (4) symmetry is broken at the loop level via radiative corrections from the gauge and Yukawa interactions. Thus the Higgs boson is the pseudo Goldstone Boson (PGB) of the symmetry breaking. Imposing a discrete Z 2 symmetry between SM and twin sectors ensures that radiative corrections to the Higgs mass squared are still U(4) symmetric. Thus there is no quadratically divergent radiative corrections to the Higgs mass terms. At the same time, the Z 2 symmetry needs to be broken at electroweak scale. Otherwise, the Z 2 symmetry induces symmetric VEVs at NP scale. It is necessary to realize vacuum misalignment v f (and thus some level of little hierarchy) to separate the electroweak and NP scales. This implies a moderate amount of tuning (approximately 2v 2 f 2 ). If the Higgs boson is the PGB, the Higgs field should respect the shift symmetry. The shift symmetry is approximately broken by radiative corrections. Considering radiative corrections only, the typical Higgs potential [18] could be parametrized as Here a and b denote radiative corrections, with the form where g * denotes the typical SM couplings, such as top Yukawa coupling, and m * represents the top partner mass. If there is no other contribution than the radiative corrections, the Higgs VEV can be obtained as h = a/bf f.

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Depending on the breaking pattern, the scalars in twin 2HDM could be partially Goldstone Bosons or completely Goldstone bosons. Through the twin 2HDM framework, physics behind the spontaneous Z 2 breaking scenarios could be explained. The above radiative, tadpole induced, and quartic induced symmetry breaking mechanisms are also classified and considered in a unified framework with composite two twin Higgses. The collider phenomenology of the twin two Higgs models is quite similar to the one of elementary 2HDM, except that twin 2HDM also involves in the twin hadron phenomenology. Only when we identify the signatures of the twin hadrons from the twin Higgs decays, we will be able to distinguish the twin 2HDM from the elementary 2HDM.
The paper is organized as follows. In section 2 we briefly review the original twin Higgs and the vacuum misalignment in this model. In section 3 we introduce the most general scalar potential and its radiative corrections in the twin two Higgs model. Then we investigate the conditions for symmetry breaking and vacuum misalignment in section 4. Subsequently in section 5 we classify various Z 2 symmetry breaking scenarios in a natural composite two Higgs doublet framework. Section 6 discuss Higgs phenomenology in each scenario. Finally we conclude this paper. In appendix A and B, we list the calculation details of the two twin Higgs models.

Original twin Higgs and vacuum misalignment
We first briefly review the twin Higgs model [8,9,[13][14][15][16] and how the vacuum misalignment is realized in this model. The original twin Higgs model consists of a mirror copy of the SM content, called the twin sector. We use the labels A and B to denote the SM and twin sector respectively. The twin sector is related to the SM sector by a Z 2 exchange symmetry: A ↔ B. The Higgs sector consists of the SM Higgs doublet H A and the twin Higgs doublet H B . Due to the Z 2 symmetry, the Higgs potential preserves an approximate global symmetry U(4): to have correct field normalization. Expanding out the exponential and taking the unitary gauge we obtain the explicit form where the field h denotes the SM Higgs doublet h = h + h 0 . The global symmetry U(4) is explicitly broken once the SM and its mirror gauge group SM A × SM B are gauged, and the Yukawa interactions are introduced. Both the gauge and Yukawa interactions give rise to radiative corrections to the quadratic part of the scalar potential. The leading correction to the potential induced by gauging the where g A and g B are the gauge couplings of the SM A × SM B gauge group. Here if the Z 2 symmetry is imposed, the leading corrections to the quadratic part of the scalar potential accidentally respect the original U(4) symmetry. Thus corrections from the gauge sector cannot contribute to the masses of the Goldstone bosons. Similarly, consider the Yukawa sector by focusing on the top Yukawa couplings, which takes the form where q A,B and t A,B are the left-handed SU(2) A,B doublet quark and right-handed SU(2) A,B singlet top quark in the SM and twin sectors. The leading corrections take the form 1 Different notations on field definition and VEVs are used in literatures [8,9]. Here we define the field and take notation on field VEVs HB = f and HA = v = 174 GeV. Using the same field definition, another notation on field VEVs HB = f and HA = v/ √ 2 = 174 GeV are also used in literature [15]. Finally some literature [29] uses the following field definition Note that the normalization of the hi is different, with Rehi + iImhi to have correct field normalization. Under this notation, the VEVs are HB = f / √ 2 and HA = v/ √ 2 = 174 GeV.

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Similarly the Z 2 symmetry ensures that the leading corrections respect the U(4) symmetry. Therefore, there is no quadratically divergent contribution to the Higgs boson mass at one loop order. Although the Z 2 symmetry ensures the quadratically divergent corrections respect the U(4) symmetry, the gauge and Yukawa interactions still break the U(4) symmetry via the logarithmically divergent corrections. The leading logarithmically divergent corrections take the form (2.8) The sub-leading corrections proportional to g 4 take the similar form with opposite sign. However, since both the squared mass and quartic coupling come from the same loopsuppressed corrections, the VEV is obtained to be at the scale f as mentioned in introduction. This fact can be seen if we write the scalar potential including the radiative corrections, to good approximation, as Here δ denotes the small U(4)-violating but Z 2 -preserving loop corrections on the quartic potential, with δ λ. According to eq. (2.8), the Yukawa interactions lead to δ It is interesting to note that the symmetry breaking structure is controlled by the sign of the δ: • if δ < 0 (such as, only including loop corrections from the gauge interactions), the potential induces which breaks the Z 2 symmetry spontaneously.
• if δ > 0 (such as, adding loop corrections from the Yukawa interactions), the potential induces which preserves the Z 2 symmetry.
The original twin Higgs belongs to the second case: the vacuum is equally aligned with the two sectors. In order to realize the symmetry breaking at the electroweak scale, the VEV must be misaligned to be asymmetric with H A = v f . This requires explicit Z 2 symmetry breaking by adding • either a soft Z 2 -breaking mass term • or a hard Z 2 -breaking quartic term (2.14) we obtain the dominant Higgs potential (2.15) If the m 2 A is smaller than δf 2 term, the mass term could be negative, which induces electroweak symmetry breaking, and the Higgs boson h obtains its mass. We minimize the potential and obtain the electroweak VEV from the tadpole condition To realize electroweak VEV, m 2 A should be comparable to the δf 2 term. This implies a moderate tuning between δf 2 and m 2 A . We estimate the tuning using the following approximation: which respect the twin parity Z 2 .

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The scalar pontential of the fields H 1 and H 2 is similar to the two Higgs doublet model. In the generalized two Higgs doublet framework, we write the general twin Higgs potential Here all the parameters are taken to be real for simplicity. Note that refs. [19,20] only contains λ 1,2,4 terms in the potential. The symmetries of the potential are recognised as follows: • First, of course, all the terms in the potential preserve the twin parity Z 2 symmetry: • The first line of the potential eq. (3.2) has the global U(4) 1 × U(4) 2 symmetry.
• While the second and the third lines of the eq. (3.2) explicitly break the global symmetry U(4) 1 × U(4) 2 → U(4) V . If λ 5 is zero but λ 4 is non-zero, an additional global U(1) symmetry exists.
To avoid tree-level Higgs mediated flavor changing neutral current, similar to 2HDM, a softly-broken discrete symmetry Z 2 : H 1 → H 1 H 2 → −H 2 is imposed on the quartic terms, which implies that λ 6 = λ 7 = 0, whereas m 2 12 = 0 is still allowed. The two Higgs sector is weakly gauged under the mirror SM gauge group. The gauge symmetry is applied 2 under The covariant kinetic terms of the Higgs fields are written as The global symmetry is weakly broken by the loop effects from the gauge interactions. If the mass terms µ 2 1,2 are positive, the fields H 1 and H 2 vacua take the form

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Similar to the 2HDM model, let us define the mixing angle β and scale f Depending on the global symmetry before the symmetry breaking, there could be seven or fourteen Goldstone bosons. In the following, we discuss the nonlinear parametrization of the fields H i in U (4) (1) U(4)/U(3) symmetry breaking. The most general scalar potential in eq. (3.2) exhibits the global U(4) symmetry. The VEVs will break the symmetries of the Lagrangian spontaneously: global symmetry: gauge symmetry: The SUSY twin Higgs model [21,22] belongs to this breaking pattern. Similar to the original twin Higgs, there are seven Goldstone bosons. To isolate the Goldstone bosons in the fields, similar to 2HDM, it is convenient to work in the Higgs basis by rotating the fields After rotation, only the field H obtain VEV. Similar to the original twin Higgs, the field H can be parametrized non-linearly. After rotation, the two fields becomes (3.10) where the field h denotes the SM Higgs doublet h = h + h 0 , and C ± and N are Goldstone bosons in the B sector, which are absorbed by the twin gauge bosons. Therefore, similar to the original twin Higgs, taking the expansion, the field H takes the form Here the field H plays the role of the twin Higgs as the original twin Higgs model. Another field H does not obtain VEV, and thus it is just another scalar quadruplet in this model. quartic λ 4 and λ 5 terms are taken to be small, and thus the U(4) ×U(4) symmetry becomes approximate. The VEVs will break the symmetries of the Lagrangian spontaneously: global symmetry: gauge symmetry: In this case, the approximate global symmetry breaking is Let us parametrize the fields H 1 and H 2 nonlinearly in terms of the nonlinear sigma fields. Assuming the radial models ρ 1 and ρ 2 in H 1 and H 2 are heavy, the fields H 1 and H 2 are parametrized nonlinearly as (3.13) Expanding out the exponential we obtain the explicit form where are Goldstone bosons in the sector A, and C i , N i are Goldstone bosons in the sector B. When U(4) × U(4)-breaking terms exist, one combination of the h ± i , and one combination of the C ± i becomes pseudo Goldstone bosons.

Fermion assignments
In the twin Higgs model, the SM fermions are extended to include mirror fermions: where the quantum number assignments are ( If there are two twin Higgses, the general Yukawa interactions could be written as Similar to 2HDM, it is possible to induce Higgs mediated FCNC processes in visible sector.
To avoid such problem, the discrete Z 2 symmetry H 1 → H 1 , H 2 → −H 2 can also be applied to the fermion contents, which are identified as the Type-I, II, X, Y 2HDMs [30]. Here for simplicity, we adopt the type-I Yukawa structure: all fermions only couple with H 1 . Similar to 2HDM, it is straightforward to extend type-I Yukawa structure to other Yukawa structures.

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(1) Fermion assignment: mirror fermions. In this setup, similar to the original twin Higgs model, the 2HDM top Yukawa interactions are In the above Lagrangian the U(4) symmetry is explicitly broken by the Yukawa terms. Similar to the SM fermions, the mirror fermions are treated as the chiral fermions. The fermion masses are y 2 f 2 indicates the quadratically divergent cancellation. Of course, it is also possible to treat the mirror fermions vector-like [31] with Here additional fermion degree of freedoms are introduced to make the mirror fermions vector-like. This will lift the mirror fermion masses but not affect the quadratically divergent cancellation in the Higgs potential. Here we only take chiral fermion case.
(2) Fermion assignment: SU(6)×SU(4) fermions. To keep the U(4) invariant form, the following fermions [8,9] are introduced: In the SU(6) × SU(4) invariant form, the fermions are assembled as Similar for the leptons. The U(4) × U(4) invariant top Yukawa interactions are written as To lift the non-SM fermions masses, additional vector-like fermion mass terms are introduced as The vector-like mass terms exhibit U(4) × U(4) breaking effects in the Yukawa sector. Expanding the Yukawa interactions, we obtain Thus the mass matrices are

Radiative corrections
The gauge and Yukawa interactions break the global symmetry explicitly, which generate the scalar potential for the pseudo-Goldstone bosons. The one-loop Coleman-Weinberg potential in Landau gauge is where the super-trace STr is taken among all the dynamical fields that have the Higgs dependent masses. The Higgs dependent gauge boson masses are for the SU(2) × SU(2) gauge bosons, and similarly for the U(1) × U(1) gauge boson masses. The Higgs dependent top sector masses in the fermion assignment I are The field dependent top sector masses in the fermion assignment II are Let us examine how the quadratic divergence cancels at the one-loop again due to the Z 2 symmetry. The leading corrections to the quadratic part of the scalar potential are from the gauge sector, and due to the Yukawa interactions in the top sector. Note that both quadratic contributions respect the original U(4) symmetry, and thus there is no quadratically divergent contribution to the Higgs boson masses. Therefore the leading corrections are the quartic terms in the effective potential. The radiative corrections to the gauge sector is .
Similarly for the U(1) sector. The radiative corrections to the top sector in the mirror fermion model are (3.33) -12 -

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In most general case, the dominant contributions of the radiative corrections could be parametrized as Note that there could have δ 6,7 terms in the scalar potential (just like the λ 6,7 terms in 2HDM). However, since we have taken the λ 6,7 terms to be zero, and we adopt the Type-I Yukawa structure, the radiative corrections could not generate δ 6,7 terms. We list the coefficients in eq. (3.34): from gauge interactions [19]. In the Type-I Yukawa structure, the Yukawa interactions induce for the fermion assignment I and (3.37) for the fermion assignment II [8,9]. In other Yukawa structures, the Yukawa radiative corrections could be different. Here other non-logarithm contributions and small radiative contributions from scalar self-interactions are neglected.
The overall radiative corrections are the sum over gauge boson and fermion contributions. Note that the above radiative corrections are independent of the breaking patterns.
Given the gauge and fermion assignments, the radiative corrections is completely determined by gauge and Yukawa couplings. In the following, we take general form of δ 1−5 . In the numerical calculation, we take the values from the fermion assignments I: (3.38) This serves as our benchmark point in the following discussions.

U(4)/U(3) breaking pattern
In this breaking pattern, due to existence of the m 2 12 term and λ 4−5 terms in the potential, the global symmetry breaking pattern is U(4) → U(3), with seven Goldstone bosons generated. The δ 1−5 terms further trigger spontaneous symmetry breaking, and some Goldstone bosons become PGBs.
The radiative corrections from the gauge and Yukawa interactions trigger symmetry breaking on A sector. According to eq. (3.10), only one combination of the two twin Higgses H 1,2 obtains VEV. Denoting the VEV θ ≡ h 0 f we obtain the field VEVs in the Higgs basis, or the H 1,2 basis: (4.1) Let us calculate the VEV θ = h 0 /f using the tadpole conditions. The tadpole conditions determine only only the mass-squared parameters µ 2 1,2 , but also the VEV θ. The full tadpole conditions are listed in the appendix A. Here we only list the tadpole conditions which determine the VEV: If f 1 = f 2 and δ 1 = δ 2 , the two conditions give rise to Note that the VEVs are equally aligned because of the Z 2 symmetry. Similar to the original twin Higgs model, adding soft or hard breaking terms will realize vacuum misalignment.
Here we add the soft mass breaking terms in the scalar potential Taking the soft breaking terms into account, we obtain new relevant tadpole conditions  where . From the above relations we see that θ could be less than π/4 only if there are the following relations Since both F 1A and F 2A are free parameters, These relations could be easily satisfied.
The left panel of the figure 1 shows the correlations between F 1A and F 2A given the t β and benchmark parameters in eq. (3.38). Given the soft mass term F 1A or F 2A , we could determine the VEV θ using tadpole conditions in eq. (4.5). Figure 1 (middle) shows the F 1A or F 2A vesus the θ value for different tan β. It shows that the solution θ < π/4 does exist, and thus the vacuum misalignment is realized. Although the tadpole conditions in eq. (4.5) determine θ, we need to know the scale f to further determine v. To obtain the VEV v at electroweak scale, the following condition should be imposed Given tan β, there are relations between θ and f 1 , which are shown in figure 1 (right). In summary, given appropriate values of m 2 1A (or m 2 2A ) and t β , θ and f 1 are totally determined, and the vacuum misalignment could be relized.
(4.12) Figure 3 shows the (θ 1 , θ 2 ) contours imposed by two conditions for different (Ω 1 , Ω 2 ). We note that the two conditions are symmetric under θ 1 ↔ −θ 1 if |Ω 1 + Ω 2 | > 1, while they are symmetric under θ 2 ↔ −θ 2 if |Ω 1 + Ω 2 | < 1. This symmetric behaviour can be seen from the upper left and middle panels of figure 3. Therefore we can determine the solution for (θ 1 , θ 2 ): θ 2 = 0, θ 1 ≤ π/4, for |Ω 1 + Ω 2 | > 1 θ 1 = 0, θ 2 ≥ π/4, for |Ω 1 + Ω 2 | < 1.  figure 3, if Ω 2 = 0, we have either θ 1 = π/4 (if |Ω 1 + Ω 2 | > 1) or θ 2 = π/4 (if |Ω 1 + Ω 2 | < 1). According to the middle panel, when Ω 2 < 0, we have either θ 1 < π/4 (if |Ω 1 + Ω 2 | > 1) or θ 2 > π/4 (if |Ω 1 + Ω 2 | < 1). On the right panel, it shows as Ω 2 decreases, the value of θ 1 decreases. Thus we could obtain appropriate asymmetric vacuum θ 1 when we vary Ω 2 . When we take |Ω 1 + Ω 2 | > 1, θ 1 could be smaller than π/4 as we vary Ω 2 . Thus even without tree-level breaking parameters, the vacuum misalignment could still happen. This is the scenario of radiative Z 2 symmetry breaking [28]. Turning on tree-level breaking terms (F λ , F m ) will change the contours between θ 1 and θ 2 imposed by the second tadpole condition. For simplicity, let us turn on single tree-level breaking term: F λ or F m . Figure 3 (lower panel) shows the (θ 1 , θ 2 ) contours imposed by two conditions for different (Ω 1 , Ω 2 , F m ). For comparison, we use the same values of the (Ω 1 , Ω 2 ) in both the upper and lower panels of figure 3. We find that turning on F m shifts the intersection point between the contour and the x-axis to lower θ 1 , and also change the convex behavior of the contour. Thus F m plays a similar role as Ω 2 . Figure 3 (left) show that even Ω 2 is zero, turning on F m will obtain the following solution: (4.14) The vacuum misalignment could be realized via the bilinear term m 2 12 . This is the scenario of tadpole induced Z 2 symmetry breaking [26,27]. The middle and right panels of figure 3 show that turning on Ω 2 will also obtain viable solutions. And the larger Ω 2 , the smaller θ 1 . Our discussion on the tree-level breaking term F m could also be applied to the case  with only F λ . The results are quite similar to the one in figure 3. In this case, the λ 45 plays the role to obtain vacuum misalignment. This is a new scenario: quartic induced Z 2 symmetry breaking. The set of parameters (Ω 1 , Ω 2 , F λ , F m ) could only determine (θ 1 , θ 2 ), but not the VEVs (v 1 , v 2 ). To obtain the electroweak VEVs, additional condition (the VEV condition) needs to be imposed: where v = 174 GeV. Given t β and θ 1,2 , we could determine f 1 and f 2 . Figure 4 (left) shows the VEV contour curves on the (θ 1 , θ 2 ) plane for different f 1 and t β . f 1 determines the intersection point between the curve and the x-axis, while t β determines the curvature behaviour of the curve. The middle and right panels of figure 4 show that once t β is fixed, f 1 could be determined and thus the VEV contour is fixed. There is one special case. In figure 4 (middle panel), when the tree-level breaking term is off, f 1 keeps the same for different t β . So in radiative Z 2 case, f 1 can be determined by two parameters (Ω 1 , Ω 2 ). Given the vacuum misalignment condition θ 1 < π/4, we could estimate the parameter region for the tree-level breaking parameters F m , F λ and global symmetry breaking scales f 1,2 . Figure 5 (left) shows the values of F m or F λ as functions of θ 1 , which determines θ 1 for different t β . Interestingly, even when F m or F λ is absent, we could still obtain θ 1 < π/4, which corresponds to the radiative breaking scenario. Figure 5 (right) shows that once θ 1 (and t β ) is known, f 1 is totally determined. And the larger t β , the larger f 1 . This relation is quite general and does not depend on scenarios. Thus in tadpole or quartic induced symmetry breaking, only two independent parameters are needed, which are typically taken to be F m (F λ ) and t β . If there is no tree-level breaking term, only one parameter t β could determine the VEVs.
Finally let us summarize what we have obtained so far from the tadpole conditions. The tadpole conditions determine (θ 1 , θ 2 ), which depends on (Ω 1 , Ω 2 ) and/or (F m , F λ ).
• m 2 12 -induced Z 2 breaking [26,27], when Ω 1 = 0, m 2 12 = 0. The tree-level potential is The dominant radiative corrections to the scalar potential are the same as the radiative Z 2 breaking case. Similarly Ω 1 determines whether the electroweak symmetry breaking could happen. However, in the parameter region that t β is small, Ω 2 alone can not obtain small enough θ 1 . And in certain case Ω 2 is also small. In those cases, the parameter m 2 12 could play the role to obtain appropriate θ 1 . It is the m 2 12 which determines whether vacuum misalignment could happen. As explained in ref. [26] -20 -
It is also possible that both m 2 12 and λ 4,5 terms exist in the potential. In this case, it is the m 2 12 and λ 4,5 which determine whether vacuum misalignment could happen. This scenario is mixture of the tadpole and quartic induced Z 2 breaking scenarios.

Spontaneous Z 2 breaking in composite 2HDM
In above section, we discussed how the tadpole conditions determine the electroweak vacua (θ 1 , θ 2 ). Three different mechanisms could lead to the vacuum misalignment θ 2 < θ 1 < π/4, and realize the spontaneous Z 2 breaking. Let us understand the physics behind these Z 2 breaking scenarios.
Since the electroweak symmetry breaking only involves in the PGBs in visible A sector, we will simplify the original scalar potential in eq. (3.2) and eq. (3.34) by setting Expanding the potential to the quartic order, we obtain the approximated potential of the visible sector in the 2HDM framework: Unlike the elementary 2HDM potential, in this composite 2HDM potential, all the coefficients in the potential are proportional to the tree-level and loop-induced breaking terms: Note that there is no dependence on the tree-level parameters λ 1−3 . Since the H 1A and H 2A are pseudo-Goldstone bosons, we identify this scenario as composite 2HDM, to distinguish it from the elementary 2HDM.

Radiative Z 2 breaking
In this scenario, the tree-level breaking terms m 2 12 and λ 4,5 do not exist. From the above potential, the Higgs mass squared terms reduce to 4) and the quartic terms reduce to 6) Note that the δ 4,5 terms are absent. The form of the potential is the same as the inert Higgs doublet potential [32], although all the terms are generated radiatively. The first two terms in the potential, determine the electroweak vacuum: To obtain the electroweak VEV v = 174 GeV, the two terms in the equation should cancel with each other. We know although contributions δ 1 (from Yukawa corrections) and δ 345 (from gauge corrections) have opposite sign, the adequate cancellation will not happen if δ 1 δ 345 . To have adequate cancellation, the second term in the mass-squared µ 2 H 2A needs to be enhanced by assigning large t β . On the other hand, because λ 1A keeps large compared to the mass-squared µ 2 H 2A , the electroweak VEV is obtained. Finally, let us write the masses of the PGBs. The Higgs boson mass reads If the Z2 symmetry between H2A ↔ H2B is exact, the terms |H1B| 2 |H2B| 2 and |H1A| 2 |H2A| 2 generate opposite but equal mass terms for H1A. Thus the Z2 symmetry between H1A ↔ H1B is still unbroken.
In elementary 2HDM model, the masses of the charged and CP-odd neutral scalar are only proportional to δ 4,5 , which are very small. In this scenario, the inert scalar masses also have δ 2,3 dependences, which induce large masses for the inert scalars. Therefore, the radiative Z 2 breaking scenario can be viewed as a natural UV completion of the inert Higgs doublet model.

Tadpole induced Z 2 breaking
The radiative Z 2 breaking scenario can only realize the electroweak symmetry breaking when δ 345 is non-zero, and if the enhancement from t β 1 exists. Otherwise, vacuum misalignment cannot be obtained by purely radiative Z 2 breaking. Thus when δ 345 is zero, or t β ∼ 1, the tadpole induced Z 2 breaking scenario should play the role of electroweak symmetry breaking. However, the price to pay is introducing additional m 2 12 term. Let us turn on m 2 12 gradually to see how the VEVs θ 1,2 vary. When m 2 12 term is off, from the radiative breaking scenario, the VEVs have Thus m h 1 is much heavier than m h 2 . When gradually turning on m 2 12 term, h 2 starts to obtain small VEV. This can be seen from the potential by assuming h 1 is too heavy and decoupled. After integrating out h 1 , the potential generates an effective tadpole term. The h 2 potential is dominated by the tadpole and quadratic terms which gradually becomes large as we increase m 2 12 . At the same time, the VEV of h 1 decreases. This can be seen from the H 1A potential. Assuming the VEV h 2 is small, the relevant H 1A potential is Here the tadpole contribution is negligible due to h 1 > h 2 . From the potential, we see that as the m 2 12 becomes larger, there is large cancellation in the quadratic term, and thus the VEV h 1 becomes smaller. Therefore, the bilinear term m 2 12 plays the role of the effective tadpole. As the effective tadpole term increases, the VEV θ 1 decreases from π/4, while the VEV θ 2 increases from 0. The vacuum misalignment θ 2 < θ 1 < π/4 could be realized when an appropriate m 2 12 term is taken.

Quartic induced Z 2 breaking
In this scenario, only quartic breaking terms λ 4,5 are kept in the tree level potential. Unlike the m 2 12 case, the quartic breaking scenario works for both small t β and large t β regions. The λ 45 terms appear in both quadratic term µ 2 1A and the bilinear term in the potential. The quadratic term has We see that both δ 345 and λ 45 contribute to the quadratic term to have the opposite δ 1 corrections. Furthermore, it also generates the effective tadpole term, which transits θ 1 to θ 2 . Therefore, the quartic induced breaking scenario has the ingredients of the radiative and tadpole scenarios to break the Z 2 symmetry. In this scenario, there should be much larger viable parameter regions which generate the appropriate Z 2 breaking.

Higgs mass spectra
In this natural composite 2HDM framework, the Higgs sector contains two Higgs doublets H 1A , H 2A in A sector, and another two Higgs doublets H 1B , H 2B (with two neutral radial mode decoupled) in B sector. Six exact GBs: three (z ±,0 ) from H iA and three (C ± , N 0 ) from H iB are generated. All of them are eaten by gauge bosons in A and B sectors. Depending on the breaking pattern, other particles than the exact GBs in the scalar multiplets could be PGBs or just scalar particles. We present details of the mass spectra in two breaking pattern in appendix A and B. Here we summarize main features of the mass spectra based on results in appendix A and B.
• Explicit soft Z 2 breaking In the Higgs basis, the field H plays the role of twin Higgs, while another field H is just additional scalar U(4) multiplet. Thus among seven GBs, six are eaten by gauge bosons, and one PGB is the Higgs boson. For the additional scalars in H , the masses are which only depends on U(4) breaking parameters m 2 12 , m 2 12A and λ 4,5 in the potential. If the tree-level terms λ 4,5 do not exist, then all the new scalars have degenerate masses. In SUSY extension of the twin Higgs model [22], the mass spectra are much simplified due to the global symmetries.
Similar to the radiative breaking scenario, all the scalar components except the radial modes in twin two Higgs doublets are PGBs. The difference between two scenarios is that in this tadpole scenario there are mixing between two Higgs doublet in A sector, with mixing angle β A , defined in appendix B. All the masses of the -25 -

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charged Higgses and CP odd Higgses depend on m 2 12 and are nearly degenerate when θ 1,2 are much smaller than π/4. The mass spectra read Similar to the tadpole induced breaking scenario, all the masses of the charged Higgses and CP odd Higgses depend on λ 4,5 . The difference between quartic and tadpole scenarios is that in this scenario there are mass splittings between charged and neutral CP odd Higgses unless λ 4 = λ 5 . The charged scalar masses are and the CP-odd scalar masses are presented in appendix B.
In all scenarios, the SM Higgs boson originates from the mixing between h 1 and h 2 in visible sector. 6 We present the mass matrices of the Higgs boson in appendix A and B. Figure 6 shows the mass spectra in above four scenarios. The independent parameters in four scenarios are taken to be (θ 1 , t β , m 12 = 500, m 12A = 200) (explicit Z 2 breaking), θ 1 (radiative breaking), θ 1 , t β = 3 (tadpole breaking), and θ 1 , t β = 2.8 (quartic breaking). Figure 6 shows that typically charge and neutral CP odd Higgses (H ± , A 0 ) in visible sector have degenerate masses, and similarly for charged and neutral CP odd Higgses (H ± , A 0 ) in twin sector. In the spontaneous Z 2 breaking scenarios, there are only two free parameters (t β , θ 1 ). Imposing the condition of the 125 GeV Higgs boson mass provides additional constraint on the model parameters. In figure 6, if we identify the Higgs boson mass to be 125 GeV (dashed line), the θ 1 totally fixed once we fix t β .

Collider constraints
Let us first consider the visible sector. The visible sector contains the same particle contents as the ones in the 2HDM. The phenomenology in visible sector should be very similar to the one in 2HDM, except that there could be additional decay channels to the twin particles. For simplicity, we take the Type-I Yukawa structure in this work, although other Yukawa structure, such as Type-II, X, Y, are possible. Let us setup the notation similar to 2HDM. 7 According to appendix A and B, the 2HDM mixing angles and electroweak VEV 6 The exception is that in radiative breaking scenario, there is no mixing between h1 and h2. The Higgs mass is proportional to the breaking parameters δ1−5 and/or m 2 12 (λ4,5). 7 Note that definition of the mixing angle α is opposite from the typical notation, such as ref. [30].
mixing angle between charged/CP-odd scalars, v = √ f 1 sin θ 1 +f 2 sin θ 2 , electroweak vacuum, α, mixing angle between CP even scalars. (6.6) The normalized Higgs couplings to the SM gauge bosons and fermions are Here the SM couplings are taken to be g SM These Higgs couplings are constrained by the Higgs coupling measurements at the LHC [37,38]. The charged and CP-odd neutral scalars in visible sector have the same constraints as the one in 2HDM. On the other hand, the CP-even neutral scalars have additional decay channels to the twin particles, and thus need additional care.
The twin sector includes another two Higgs doublet H 1B,2B , the mirror gauge bosons, and mirror fermions. The mirror gauge bosons are mirror photon, and mirror W B , Z B , which absorb three GBs in two Higgs doublets. For simplicity, two radial modes in H 1B,2B are assumed to be decoupled. The physical scalars in twin sector are charged and neutral CP odd scalars H ± , A 0 . The mirror fermions have very rich twin hadron phenomenology [33] because they are charged under the mirror QCD. Since the twin fermions are mirror copy of the SM particles, the mirror fermion phenomenology should be similar to the original twin Higgs. For simplicity, we take the fermion setup in the fraternal twin Higgs model [33], and leave more general discussion for future. The fermionic ingredients of the fraternal twin Higgs setup are summarized as follows: • To avoid the twin SU(3) and twin SU(2) anomalies, the whole third generation twin fermions are introduced: twin top, bottom, tau, and twin tau neutrino, but not the first two generations; • The fermion Yukawa interactions are taken to be the fermion assignment I in our discussion; • The twin SU(3) has confinement, which indicates the existence of the twin glue-balls, and twin bottomonium and hadrons below confinement scalar Λ 3 ∼ O(10)Λ QCD .

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To be specific, we take the twin bottom Yukawa coupling the same as the bottom Yukawa coupling, which indicates m b B m b f v . Thus the Higgs boson could decay into b B : h → b BbB . Because twin fermions are SM charge neutral, some of them could be dark matter candidate. This has been discussed in refs. [33][34][35][36].
The Higgs boson and the heavier CP even neutral scalar provide connection between visible and twin sector. The Higgs boson also couples to the twin particles due to its PGB feature. Here we denote the VEV in twin sector v ≡ f cos θ (v ≡ f 2 1 cos θ 2 1 + f 2 2 cos θ 2 2 ), and mixing angle β (β B ) in explicit (spontaneous) breaking pattern. The normalized Higgs couplings to the twin gauge bosons and fermions are sin θ 1 c α c β B + sin θ 2 s α s β B , Tadpole and Quartic Z 2 Breaking, (6.9) , Tadpole and Quartic Z 2 Breaking. (6.10) Here the SM-like couplings are taken to be g SM Since in general the normalized couplings of the Higgs boson to the twin gauge bosons and fermions are different, the calculation of the signal strength is not just a simple scaling. We calculate the Higgs invisible decay widths based on the above couplings.
We take the latest LHC results on the Higgs coupling measurements [37,38] and Higgs invisible decays [39], and perform a global fit on the model parameters. In the tadpole and quartic Z 2 breaking scenarios, if we fix the parameter t β , there is only one free parameter. Thus in all the spontaneous Z 2 breaking scenarios, we will vary θ 1 and fix t β . Furthermore, the explicit breaking scenario is not considered here, since it should be less constrained than the other three scenarios. In the following, we perform a global fitting on the Higgs signal strength. In the case where the Higgs coupling measurements are well within the Gaussian statistical regime, the likelihood function is defined Based on Higgs signal strengths at the 8 TeV LHC with 20.7 fb −1 data [37], a statistical analysis is performed by the Lilith package [40]. Figure 7 (left panel) shows the loglikelihood profile ∆(−2 log L) as the function of θ 1 , in three scenarios. Here in tadpole and quartic scenarios we fix the parameter t β = 3 and t β = 2.5. Up to the 2σ level, the exclusion limits in three scenarios are that θ 1 should typically be less than 0.2. This put very strong constraints on the model parameter. Looking back to figure 6, we note that both this Higgs coupling constraints and the requirement on 125 GeV Higgs mass should be satisfied. The tadpole and quartic breaking scenarios are viable, but there is tension  Figure 7. On the left, it shows the log-likelihood profile ∆(−2 log L) as the function of θ 1 in three scenarios. Here 1σ, 2σ, 3σ errors are also shown. On the middle, it shows the signal strength in gluon fusion production and subsequent V V decays, and Higgs invisible branching ratio as function of θ 1 in three scenarios. The bound on the invisible decay branching ratio is Br inv < 0.23. On the right, the S − T oblique parameter contours at the 1σ, 2σ levels are shown. The dotted points are the parameter points in three scenarios: radiative (orange color), tadpole (blue), and quartic (green) Z 2 breaking scenarios. between the Higgs coupling constraints and the 125 GeV Higgs boson mass requirement in the radiative breaking scenario. If the U(4) fermion assignment is taken in the radiative breaking scenario, such tension does not exist, and there are viable θ 1 parameter regions which could satisfy both conditions. This viable fermion assignment has been discussed in ref. [28]. Although the invisible decay width has been taken into account indirectly in the above global fitting, we would like to consider constraints from the direct searches on the Higgs invisible decays. The updated upper limits on the invisible decay branching ratio is Br inv < 0.23 [39]. Figure 7 (middle panel) shows the invisible decay branching ratio as the function of θ 1 . As a comparison, we also plot the signal strength in the gluon fusion process gg → h 1 → V V . From figure 7, we see that the direct searches on the invisible decays put much weaker constraints than the Higgs coupling measurements. The high luminosity LHC will improve sensitivity of signal strengths to around 5% assuming current uncertainty with 3 ab −1 luminosity [41]. Thus we should be able to explore more parameter regions at the high luminosity LHC.

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According to the updated results on oblique parameters via Gfitter package [42], the S, T parameters have S = 0.05 ± 0.09, T = 0.11 ± 0.13, with correlation coefficients of +0.90 between S and T . In this model, the S and T parameters contains two contributions: corrections from possible radial modes, and corrections from 2HDM scalars. The corrections from radial modes takes the form with radial modes ρ, while the 2HDM corrections [43] are roughly (6.13) In our numerical study, the complete form of the S, T parameters [30,43] are used. From the above, we see that if the radial modes decouple, or if the heavy scalars are degenerate, the oblique corrections are negligible. Figure 7 (right panel) plots the predicted S, T values in three scenarios, which we vary the parameter θ 1 while fixing t β = 3 in tadpole scenario, and t β = 2.5 in quartic scenario. According to the S − T oblique parameter contours at the 1σ, 2σ levels, we note that most of S, T parameter points are within the 2σ level contour. Thus the precision electroweak test usually provide weaker constraints on the model parameters than the one from the Higgs coupling measurements. Let us briefly discuss the distinct signatures of this model. First, like the original twin Higgs model, the twin hadron phenomenology [33] provides us very distinct signatures from other models. Furthermore, the additional charged and neutral scalars provide us a way to distinguish this model from the original twin Higgs. This has been explored in the 2HDM contents for the general case [30] and the inert case [44]. Finally, to distinguish it from the typical 2HDM, the signatures from the twin H ± and A 0 need to be explored. Furthermore, if the radial modes are not so heavy (thus not decoupled), exploring the radial mode decay channels could provide us different signatures from the typical 2HDM model. The detailed collider phenomenology would require studies of their own. Furthermore, the fine-tuning argument provides additional theoretical constraints on the models. We leave the detailed studies of these [45] in future.

Conclusions
In this work, we investigated a class of twin two Higgs models, in which the Higgs sector is extended to incorporate two twin Higgses and the global symmetry breaking pattern could be either U(4) → U(3) or [U(4) × U(4)] → [U(3) × U(3)]. The SM Higgs boson is identified as one of the pseudo Goldstone Bosons after symmetry breaking. The discrete Z 2 symmetry protects the Higgs mass term against the quadratically divergent radiative corrections. However, the Z 2 symmetry needs to be broken to generate electroweak scale, which should be separated from the new physics scale. Typically the soft or hard explicit Z 2 breaking terms were introduced to do so. We found that in the twin two Higgs setup, it is possible to realize spontaneous Z 2 breaking, without the need of explicit Z 2 breaking terms.

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We performed a systematical study on the general Z 2 breaking conditions in a natural composite two Higgs doublet framework, and discussed various possible scenarios which could realize the vacuum misalignment. In the radiative Z 2 breaking scenario, given the appropriate fermion assignments, the Z 2 symmetry could be spontaneously broken purely due to the radiative corrections to the Higgs potential. In this scenario, only one Higgs obtains the electroweak vacuum, and the other is just an inert Higgs. The tadpole-induced Z 2 breaking scenario can also be classified in this twin two Higgs doublet framework. In this scenario, the bilinear term in the scalar potential triggers the spontaneous Z 2 breaking. We also proposed a novel scenario: the quartic-induced Z 2 breaking scenario. In this scenario, the λ 4,5 terms instead of the bilinear term in the scalar potential trigger the spontaneous Z 2 breaking.
In the twin two Higgs models, we discussed phenomenology of the Higgs sector in the composite two Higgs doublet framework. Although the particle contents in the scalar sector are the same in each scenario, the Higgs mass spectra are quite distinct for each Z 2 breaking scenarios. The radiative Z 2 breaking scenario includes an inert Higgs doublet with degenerated masses. Both the tadpole-induced and quartic-induced Z 2 breaking scenarios contain additional scalars in two Higgs doublet model with not so degenerated masses. We calculated various Higgs couplings and utilized the the Higgs coupling measurements at the current LHC to constrain the model parameters. The additional scalars from the Higgs sector should be able to be probed at the Run-2 LHC and future colliders.