SUSY Meets Her Twin

We investigate the general structure of mirror symmetry breaking in the Twin Higgs scenario. We show, using the IR effective theory, that a significant gain in fine tuning can be achieved if the symmetry is broken hardly. We emphasize that weakly coupled UV completions can naturally accommodate this scenario. We analyze SUSY UV completions and present a simple Twin SUSY model with a tuning of around 10% and colored superpartners as heavy as 2 TeV. The collider signatures of general Twin SUSY models are discussed with a focus on the extended Higgs sectors.


Introduction
Supersymmetry (SUSY) and Compositeness are the leading new physics (NP) candidates to solve the hierarchy problem of the Standard Model (SM). However, the null results of the LHC searches for new colored particles already put these ideas under pressure. While it is plausible that one of these theories indeed solves the big hierarchy problem, it is not easy to see how one can avoid the little hierarchy problem, i.e. the mismatch between the electroweak (EW) scale and the scale of top-partners. It is certainly possible that this residual hierarchy is not resolved in nature, and naturalness might not be the only criterion for physics beyond the SM (BSM). Nevertheless, it is interesting to explore alternatives that can solve the little hierarchy problem and circumvent the LHC bounds.
hard and soft breaking of the mirror symmetry can be easily achieved, and study their LHC phenomenology. We show that the FT gain of hard Z 2 -breaking models is substantial and only limited by the Higgs mass constraint.
Our paper is structured as follows. We review the Twin Higgs mechanism with an emphasis on its symmetries and the FT gain in Sec. 2, where we work exclusively in the Twin Higgs IR effective theory and do not ask any questions about possible UV completions. We reproduce the bound on the FT for soft Z 2 -breaking and show how this bound can be relaxed by adding hard Z 2 -breaking. We analyze in detail the parameter space of the IR effective theory, emphasize the role of the SM-like Higgs mass constraint, and discuss how this requirement determines the final gain in FT of hard Z 2 -breaking vs soft Z 2 -breaking.
In Sec. 3 we discuss SUSY UV completions of the Twin Higgs. First, we review the SUSY Twin Higgs with soft Z 2 -breaking focusing on the simplest scenario of Ref. [20]. We explore the parameter space of this model showing that its FT gain scales as λ/λ SM and the effective SU (4)-invariant quartic λ is far from being maximized. Second, we construct SUSY Twin Higgs models with hard Z 2 -breaking. The minimal model fails to decouple the colored states due the Higgs mass constraint. We show that in next to minimal realizations we can comply with such a constraint, decouple the colored states and achieve a FT of around 10%, a factor of about 5 better than in the soft Z 2 -breaking scenario.
In Sec. 4 we present a phenomenological analysis restricted to the extended Higgs sectors of Twin SUSY theories. We leave for future investigations an analysis including the full matter sector, which can lead to many other interesting signatures both at the LHC and in cosmological observables (see e.g. Ref. [26][27][28][29][30] for further LHC studies and [31][32][33][34][35][36] for the exploration of viable cosmological scenarios). The best probes of Twin SUSY constructions are certainly direct searches of the Twin Higgs, which looks very much like a singlet mixing with the SM Higgs [37][38][39][40][41] and for this reason can be hunted for in direct searches of a resonance decaying into di-bosons. Such searches are always better suited for weakly coupled UV completions of the Twin Higgs rather than indirect searches from Higgs coupling measurements. We also find that if the twin Higgs becomes too heavy to be abundantly produced at the LHC, the MSSM-like Higgses become sensibly lighter than their mirror states. In this case we can hunt for them in direct searches for MSSM Higgses in the low tan β region [42], via their indirect effects on SM Higgs couplings and on the b → sγ transition rate [43].
Sec. 5 is devoted to our conclusions and future directions. The technical details, such as RGE formulae, detailed calculations of the Higgs mass spectrum and branching ratios are relegated to appendices.

EWSB and fine tuning in Twin Higgs Models
In this section we discuss the general structure of the Twin Higgs potential, and study the fine tuning associated to soft and hard Z 2 -breaking. We suggest that the hard breaking can lead to a substantial improvement in FT.

Setup
The basic idea of the Twin Higgs mechanism is that the SM-like Higgs is light compared to the new physics scale, because it is a pseudo-Goldstone boson (PGB) of an approximate global SU (4) symmetry, spontaneously broken down to SU (3). This SU (4) is merely an accidental symmetry that holds at one-loop at the level of quadratic terms, as a result of a Z 2 mirror symmetry between the SM and the "twin" sector. As we will see later, this Z 2 has to be broken explicitly in the Higgs potential, in order to allow for a realistic Higgs sector. By construction the Twin Higgs model is an effective theory that can resolve the hierarchy problem only up to a scale of around ∼ 5 TeV. Above this scale it should be UV-completed by a theory which solves the big hierarchy problem.
The gauge group of the Twin Higgs is extended to two copies of the SM, which we denote by G A and G B , where here and in the following A refers to the visible sector and B to the twin sector. The Higgs sector of the original Twin Higgs model [1] consists of two copies of the SM Higgs potential, with two Higgs doublets H A and H B transforming under G A and G B respectively. The full matter content of the SM is also doubled in the A and B sector. However, for the purpose of analyzing the FT, we focus on the top quarks and their twin partners that have the largest couplings to the Higgs sector, in the spirit of Ref. [26]. Although the contribution of the gauge sector to the little hierarchy problem is not negligible numerically, we do not address it explicitly, since the discussion closely follows the one of the top sector. We also assume for simplicity a perfect mirror Z 2 symmetry between the visible and the twin sector, which we break explicitly only in the Higgs sector. In agreement with the original model we imagine that the low energy effective theory has no light states that carry both visible and mirror quantum numbers. 3 Let us now write down the most general renormalizable Higgs potential for the visible and the twin Higgs: Other terms one might be tempted to write down lead to an equivalent potential up to redefinitions of the existing couplings. We now analyze this potential dividing all the terms into three different categories: SU (4)-preserving, SU (4)-breaking but Z 2 -preserving, and Z 2 -breaking. 1. SU (4)-preserving terms. These are the first two in the potential, to λ and m 2 . If m 2 < 0 and λ > 0, the global SU (4) symmetry is spontaneously broken down to SU (3). We denote the scale of the SU (4) breaking by f , and consequently f 2 ≡ v 2 A + v 2 B . If we disregard the rest of the potential, after the breaking of the SU (4) global symmetry down to SU (3) we get one real scalar (the so-called radial mode or the "twin Higgs") with mass m h T = 2 √ λf and seven Goldstone bosons. After we gauge the SU (2) × SU (2) subgroup of the SU (4), and identify it with the visible and 3 States carrying both GA and GB quantum numbers can be present in other implementations of the Twin Higgs mechanism, for example if SU (3)A × SU (3)B is embedded in SU (6) [1,4]. See Ref. [30] for a discussion of the phenomenology of these states. dark 4 EW gauge groups, 6 of the Goldstone bosons are "eaten" by the visible and the mirror W and Z-bosons, while the remaining one is identified with the SM Higgs boson.
2. SU (4)-breaking but Z 2 -preserving. This is the term proportional to κ. If this coefficient is smaller than zero, the mirror symmetry is spontaneously broken in the vacuum, and only one of the Higgses, either H A or H B , gets a VEV. This possibility does not lead to a viable phenomenology in our setup and we will not discuss it further (see however Ref. [22]). Instead, if κ is larger than zero, the Z 2 symmetry is preserved by the vacuum, and H A and H B get equal VEVs v A = v B = f / √ 2. We further consider 0 < κ λ, such that the SU (4)-breaking term gives a sub-leading contribution to the mass of the radial mode, but generates a small SM-like Higgs mass m h = √ 2κf . The hierarchy between the two quartics in the potential is technically natural and so is the hierarchy between the radial mode and the SM Higgs, because the latter is a PGB in the limit κ λ. The main problem with the model at the present stage is the unbroken Z 2 symmetry which implies that the SM Higgs is an equal superposition of the visible and the mirror Higgs and it couples with equal strength to the mirror and the visible gauge bosons and matter. The couplings of the SM-like Higgs to the visible gauge bosons and fermions would be then reduced by a factor 1/ √ 2. This scenario is excluded both by LEP EWPM and the LHC Higgs coupling measurements. Therefore we must include also explicit Z 2 -breaking terms.
3. Z 2 -breaking terms. There are two ways to break the mirror symmetry within a renormalizable theory: via a relevant operator proportional toμ in Eq. (2.1), or via a marginal one -the ρ term. We defineμ 2 ≡ σf 2 and work with the dimensionless parameters σ and ρ, which are responsible for the Higgs VEV misalignment and the mixing angle between the SM-like Higgs and the twin Higgs. Maximal mixing is already excluded and the largest possible misalignment allowed by data translates into the bound f /v 2.3 or f 400 GeV (see for example Ref. [37]).

Electroweak Symmetry Breaking and Radiative Corrections
In order to analyze EWSB and the Higgs mass in the Twin Higgs scenario it is instructive to integrate out the heavy radial mode and switch to an effective Higgs theory, where we can write the Higgs fields in a non-linear realization as Hereafter we identify φ with the SM-like Higgs. It is straightforward to plug these expressions into Eq. (2.1) and obtain the effective SM Higgs potential in the low energy effective theory. Minimization of this potential with respect to φ yields the following expressions for the VEV and the mass of the SM-like Higgs: 5 Most of previous work on Twin Higgs has concentrated on soft Z 2 -breaking, i.e σ ρ. In this paper we go further and consider hard Z 2 -breaking, i.e. the presence of a tree-level ρ, such that ρ σ. A hard breaking of the mirror symmetry was already introduced in some specific Twin Higgs models (see for example Refs. [19,25]). Note that, in models where Z 2 -breaking effects are generated by loops of the matter/gauge sector, ρ is unavoidably generated. In that case, however, it is typically smaller than σ. 6 Here for the first time we attempt to provide a systematic understanding of the breaking of Z 2 mirror symmetry in the Higgs sector from the EFT point of view. 5 The same expressions can alternatively be obtained at the level of the linear sigma model (2.1) by solving the EWSB conditions and expanding them at leading order in κ, σ λ. We refer to Appendix B.1 for a discussion of the sub-leading corrections in this expansion. 6 A notable example are models where the "twin" hypercharge U (1)B is ungauged [26] and Z2-breaking effects are generated by U (1)A gauge loops. In this case σ ∼ g 2 1 /16π 2 and ρ ∼ g 4 1 /16π 2 and the latter can safely be neglected.
As a guideline for model building, it is very helpful to visualize the parameter space of the Twin Higgs model when both soft and hard Z 2 -breaking terms are present. In order to do this we solve the equations for the measured values of the Higgs mass (2.4) and the VEV (2.3) and remain with only two free parameters, 7 which we choose to be the Z 2 -breaking ones, namely σ and ρ. The values of the Z 2 -preserving quartic κ and of the SU (4)-breaking scale f are then fixed at each point in the (σ, ρ) plane. We show the contours of κ and the region that satisfy the constraint f /v > 2.3 in Fig. 1. The first striking message of this figure is that, given the values of the Higgs boson mass and the electroweak VEV, the acceptable values of the Z 2 -odd parameters are confined to a narrow band in the Z 2 -breaking parameter space. It is also instructive to consider the two limiting cases of dominant soft and dominant hard Z 2 -breaking. The corresponding regions in the parameter space are marked in Fig. 1 with red and green stars, respectively. The values of κ in both regions can also be inferred from the figure: in the hard Z 2 -breaking region κ is forced to be roughly a factor of 5 smaller than in the soft Z 2 -breaking region as a consequence of the Higgs mass constraint. Now we are ready to include the radiative corrections to the various parameters. These are coming from top loops and EW gauge boson loops (we discuss only the former in detail) but also from Higgs quartic loops, because the hard Z 2 -breaking quartic ρ reintroduces for σ a quadratic sensitivity to the mass threshold at which Z 2 -breaking is generated.
For the radiative top corrections, we should a priori consider the contributions of both visible and mirror tops and assume that the numerical value of the top-Higgs coupling in the visible and the twin sector can be different from one another. However, as was already pointed out in Ref. [26], the top Yukawas in the visible and the hidden sector should agree to a level of better than 1% in order to avoid an unacceptable FT. To simplify our discussion we enforce an exact Z 2 symmetry between the top sectors (and the gauge sectors also).
The dominant radiative corrections to the dimensionless parameters read where we kept frozen the Higgs dependence in the logarithms and expanded to first order in κ and v/f . The running top Yukawa coupling, y t , is evaluated at the UV cutoff scale Λ t , fixing the scheme dependence of our one-loop computation. Several clarifications are in order. First, we introduced in these expressions two different mass thresholds: Λ t and Λ ρ , cutting-off the top loops and the Higgs loops respectively. Loosely speaking, in a UV complete natural theory Λ t will be identified with the mass scale of new colored states, while the states associated with Λ ρ can be complete SM singlets. 7 We use m h = 125 GeV and v = 174 GeV in all our calculations. Here we allow for a ±5 GeV Higgs mass shift because we do not take into account higher order corrections. While in strongly coupled UV completions it is hard to imagine a wide separation between Λ t and Λ ρ , in weakly coupled UV completions ( e.g. SUSY) there could be appreciable differences between the two. Therefore, we keep track of these scales separately and we will see in the concrete SUSY models of Sec. 3 that Λ t and Λ ρ correspond to different SUSY mass thresholds. Of course we should keep in mind that these scales cannot be arbitrarily separated from one another, because of higher-order quantum corrections. Second, in Eq. (2.7) we introduced a new parameter . This parameter stand for the sign of the UV mass threshold corrections and a priori = ±1. Since one cannot calculate the sign of these radiative corrections within the IR effective theory, we remain agnostic and consider both positive and negative threshold corrections. As we will see in the next section, within a full UV complete theory this sign is determined unambiguously.
Having at hand all the radiative contributions to ρ, κ and σ, we can estimate how big are the radiative contributions to the Higgs mass. We concentrate on two extreme cases: pure soft breaking, defined as ρ 0 = 0, and pure hard breaking, defined as σ 0 = 0. From Eq. (2.4) we can infer that the radiative contributions to the Higgs mass squared, up to O(v 2 /f 2 ) corrections, are given by ∆m 2 h = 4v 2 (2∆κ + ∆ρ). We show the size of these corrections in Fig. 2. The first important conclusion that we draw from this figure is that while radiative corrections contribute at least ∼ 60% of the Higgs mass for pure soft Z 2 -breaking, the radiative corrections typically overshoot the Higgs mass for pure hard Z 2 -breaking , though not by orders of magnitude. In what follows we will see that in the hard breaking case the FT ∆ v/f can be ameliorated compared to the soft case at the price of adjusting the Higgs mass (with either a large Λ ρ or a negative tree-level contribution to κ).

Fine Tuning in the Low Energy Effective Theory and Beyond
We are now ready to discuss the FT quantitatively. For illustration purposes let us start from the pure soft case, where the IR FT is well-known to scale as ∼ f 2 /v 2 . More precisely, quantifying the fine-tuningà la Barbieri-Giudice [44], one gets This is only a part of the total fine-tuning that one can estimate in the IR effective theory.
On top of that one should also consider the fine-tuning of the scale f , which one can think of as the inverse Fermi constant of the spontaneously broken SU (4), with respect to the top cutoff scale Λ t . Strictly speaking this fine-tuning should be computed in the full UV theory, but one can get a reasonable estimate by analyzing the threshold corrections to the scale f in the effective theory. These radiative corrections are given by (2.9) We will see in the next section how this expression reproduces the dominant RGE effects of the UV theory, once the "cut-off" scales are identified with physical mass thresholds. A priori the sensitivities to the thresholds Λ t and Λ λ are equally dangerous and, unlike in the SM, it is not clear that the dominant sensitivity comes from the tops rather than the Higgses themselves. However, at the threshold Λ λ we expect to find colorless particles (singlet scalars and singlinos), which are weakly constrained by the LHC. Instead, one expects the colored top partners at the scale Λ t , which are more tightly bounded from direct searches. Therefore we will assume that Λ t > Λ λ and will mostly worry about the top threshold as main source of the fine-tuning. From Eq. (2.9) we can estimate the fine-tuning ∆ f /Λ of the scale f 2 with respect to the scale Λ t as: 8 What is now the total fine-tuning of the soft Twin Higgs model? It is tempting to say that From the point of view of the effective field theory this factorization is definitely correct and we can estimate the total fine-tuning as  We will see later that this result is an excellent approximation also well beyond the effective IR theory. Interestingly, Eq. (2.11) reproduces the familiar expression for the fine-tuning in common SUSY models up to the crucial change λ ↔ λ SM , where the effective SM quartic is defined via m 2 h = 4λ SM v 2 , such that λ SM ≈ 0.13 (as we will see later we can identify Λ t with the average stop soft mass M s times a log encoding the dependence on the high SUSY scale). We conclude that the fine-tuning in the Twin Higgs with a softly broken Z 2 can be reduced with respect to the garden-variety SUSY by a factor λ SM /λ at best, in agreement with the results of Ref. [20] and with later claims about the generic nature of the fine-tuning in Twin Higgs models (see e.g. Ref. [10]). This parametric structure of the fine-tuning measure is characteristic of so-called double-protection models [11][12][13][14][15][16][17], in which the Higgs is a PGB, and the scale of the spontaneous global symmetry breaking is protected by SUSY.
Therefore, the fine-tuning gain is at best moderate in weakly coupled theories where perturbativity implies that λ cannot exceed ∼ 1 − 1.5. In practice, the gain is even more modest, because getting such high values of λ is not easy in full SUSY UV complete models, where λ is a function of other parameters of the underlying theory. This difficulty has motivated model building in the direction of strongly-coupled Twin Higgs UV completions [5,6], with the hope to saturate the parametric gain by taking λ ≈ 4π. This kind of constructions have their own problems, most notably the large number of states below the cut-off scale and the tensions with EW precision measurements, which again give an upper bound on the value of λ [10,21].
What happens if we consider pure hard Z 2 breaking, i.e. σ 0 = 0? In this case σ is entirely given by the radiative corrections in Eq. (2.7). In this case the fine-tuning in the IR effective theory is given by the logarithmic variation of v 2 with respect to the parameter ρ and reads where we eliminated ρ using the EWSB condition in Eq. (2.3) and we dropped the log Λ ρ contributions to σ because they are largely subdominant to the quadratic contributions in most portions of the parameter space (but not all). Here we have introduced for further convenience (2.13) Interestingly the first piece in (2.12) is precisely the fine-tuning in the pure soft breaking scenario, but in the hard breaking case it is multiplied by a function F (v, f ; Λ ρ ). We illustrate the behavior of this function in Fig. 3. 9 This function is always smaller than 1 and in the limit 3Λ 2 ρ 32π 2 v 2 reduces to 2v 2 /f 2 , while in the limit 3Λ 2 ρ 16π 2 f 2 it saturates the upper bound of 1. If we assume that the fine-tuning of the scale f 2 with respect to the top cutoff scale is not different from the soft case and the factorization works, this tells us (at least naively) that the reduction of the fine-tuning with respect to common SUSY scenarios should be (λ SM /λ) × F (v, f ; Λ ρ ), which can be a significant improvement compared to the Twin Higgs with softly broken Z 2 .
Alternatively one can understand the parametric dependence of the fine-tuning measure in the hard breaking scenario by fixing the cut-off scale Λ ρ using the EWSB condition in Eq. (2.3), so that the fine-tuning in Eq. (2.12) can be rewritten as (2.14) This expression makes manifest that the gain in fine-tuning of the hard-breaking scenario with respect to the soft breaking one is roughly proportional to the relative size of quartic couplings κ and ρ. In the second equality we have used that the combination of the two quartics in the denominator is fixed by the Higgs mass constraint (2.4), so that the gain in fine-tuning depends just on the value of κ, which has to be a small (see Fig. 1), in agreement with our requirement to minimize the fine-tuning. Clearly κ cannot be arbitrarily small and its natural value is indicated by Eq. (2.5), numerically O(10 −2 . . . 10 −3 ). As we see in Fig. 2 this slightly overshoots the Higgs mass, but the overshooting is O(1) rather than order of magnitude, therefore the fine-tuning associated with adjusting the value of κ is also merely of order O(1).
In Fig. 4 we show the improvement in the fine-tuning of two different hard-breaking scenarios. Both these scenarios will have direct analogs in the SUSY UV completions to be discussed in Sec  In the white region Λ t < λf or Λ ρ < λf with λ = 1. Left: we take κ 0 = 10 −3 and the black dashed contours show the × Λ ρ (in TeV units) adjusted to obtain m h = 125 GeV. The light red/blue region is the allowed parameters space with = ±1. Right: we fix = −1 and Λ ρ = 1 TeV and fix κ 0 (black dashed contours) to get m h = 125 GeV. The light red/blue region is the allowed parameters space with κ 0 ≶ 0. and a small κ 0 that we take 10 −3 having in mind its typical value in SUSY UV completions at small tan β. The value of × Λ ρ is indicated by the black dashed isolines on the left panel of Fig. 4. Since κ 0 is fixed, this quantity is determined in order to get the measured Higgs mass value m h = 125 GeV. Interestingly, in the light blue region of Fig. 4 where the threshold correction proportional to Λ ρ is negative (i.e. for = −1), the Higgs mass induces a strong upper bound on Λ t . In the opposite case where the threshold correction is positive, one can always adjust the Higgs mass in Eq. (2.4) via a large radiatively induced σ, increasing Λ ρ . Of course this comes at the price of reducing the gain in fine-tuning of the hard-breaking model, as it is shown by the blue isolines on the left panel of Fig. 4.
On the right panel of Fig. 4 we consider a purely hard Z 2 -breaking case with σ 0 = 0, Λ ρ = 1 TeV and we fix = −1 (i.e. the sign of the threshold correction in Eq. (2.7)). If we require κ 0 to compensate for the overshooting of the Higgs mass, than a negative κ 0 is needed in order to avoid an upper bound on the scale of the colored states Λ t 1 TeV. This leads to a new contribution to the fine-tuning from the adjustment of κ 0 against ∆κ that we sum up in quadrature with Eq. (2.12), and the final gain in fine-tuning with respect to the soft breaking scenario is indicated by the blue isolines of Fig. 4 (right).
We now comment on the caveats of our effective analysis. First, the factorization of the fine-tuning measure very often fails in UV complete models. This failure is related to RGE effects that introduce a dependence of f on the Z 2 -breaking parameters. These effects could be important in UV complete hard breaking models as we will see in Sec. 3.
Second, there is the log-dependence of both κ and σ on Λ ρ , which we have neglected in order to have a simple analytical understanding. Third there might be extra threshold corrections coming from the UV theory which sensibly affect the Higgs mass estimate. All these effects can slightly weaken our naive estimates of the reduction in fine-tuning.
In the next section we will show how all these caveats are taken under control in explicit SUSY UV completions, which indeed follow the parametrical intuition developed here in most of the parameter space. In particular we will show examples of hard breaking models that sensibly ameliorate the fine-tuning of the Twin Higgs.

SUSY UV completions
In this section we discuss simple SUSY UV completions of the Twin Higgs. In the minimal setup the Twin SUSY UV completion consists of two copies of the MSSM (MSSM A × MSSM B ) that are symmetric under the action of the Z 2 exchange symmetry A ↔ B. The Higgs sector of the model consists of two replicas of the two Higgs doublet model charged under the A and the B EW group respectively.
Even if the most general SUSY potential is quite involved, we can develop an analytical understanding by going in the decoupling limit of the SUSY Higgses and matching it to the non-SUSY Twin Higgs potential in Eq. (2.1) (with 2 doublets H A,B ) by replacing Of course this matching gives a good description of the SUSY Twin Higgs EWSB as long as the SUSY Higgses are heavier than the Twin Higgs. We give a treatment of the full four Higgs doublet model beyond this approximation in Appendix B. We first review Twin SUSY constructions with only soft Z 2 -breaking, which exemplifies several generic features of the SUSY UV completions of the Twin Higgs. We pay particular attention to the simplest soft Twin SUSY model proposed in Ref. [20] and validate our analytical understanding against a full numerical treatment. We show how the parameter space of this model can be completely solved once the Higgs mass and EWSB constraints are imposed and comment on possible directions to ameliorate the fine tuning.
Then we move to Twin SUSY models with hard Z 2 -breaking, which UV-complete some previously described setups and feature similar improvement in the FT. We first discuss the prominent role of the Higgs mass constraint in these models showing that the simplest implementation of hard Z 2 -breaking in SUSY has a strong upper bound on the stop mass scale M s 10 once the constraint m h = 125 GeV is imposed. Finally, we present an explicit model where the measured Higgs mass is obtained via a negative contribution to κ, very much in the spirit of the effective theory presented in the right panel of Fig. 4. In this model we get ∼ 10% FT with M s ∼ 2 TeV and λ S = 1, ameliorating the fine-tuning of soft Twin-SUSY by a factor of about ∼ 5.

Soft Twin SUSY
In this subsection we analyze SUSY UV completions of the Twin Higgs with softly broken mirror symmetry. Even though this kind of UV completions have already been discussed to some extent in the literature, reviewing them carefully will clarify the basic building blocks of any SUSY UV completion of the Twin Higgs. Analogously to Sec. 2, we organize our discussion according to the global symmetries that each term of the scalar potential preserves.
Generating the SU (4)-invariant part of the Twin Higgs potential in Eq. (2.1) already introduces some degree of model dependence in SUSY UV completions. Indeed, if we consider the matter content of just two copies of the MSSM, the only SU (4)-preserving operators at the renormalizable level are the mass terms. Therefore the SU (4)-invariant quartics require some additional dynamics. The simplest possibility is to use a non-decoupling Fterm from a heavy singlet S that has NMSSM-like couplings with the A and B Higgses, as in Refs. [18,20]. The superpotential and soft masses of this setup are To make our equations more compact, we have switched here to manifestly SU (4) invariant notations. We will further use , wherever the SU (4) conventions are appropriate, and with a slight abuse of notation, we will use H u,d both for the Higgs superfields and their lowest components.
We also assume that the singlet soft mass is much larger than the SUSY one and integrate out S in this limit. The potential we get is: where we have defined m 2 u,d = µ 2 + m 2 H u,d . We also kept just the renormalizable operators and neglected the extra quartics of order O(M 2 S /m 2 S ), O(µM S /m 2 S ) and O(µ 2 /m 2 S ). By construction these sub-leading quartic terms are also SU (4)-invariant.
Throughout this paper we will often trade the b-term for the mass of the heavy CPodd Higgs m A T = 2b/ sin(2β). Note, that unlike in the MSSM, where 2b/ sin(2β) is the mass-squared of the CP-odd Higgs, here, in the SUSY Twin Higgs, it controls the masssquared of the mirror CP-odd Higgs. The mass of the "visible" CP-odd Higgs turns out to be always lighter: . We further match the SU (4)-invariant parameters of the SUSY potential in Eq. (3.3) to the parameters of the Twin Higgs potential in Eq. (2.1): In these expressions we have disregarded the terms that depend on the difference between the β angles in the different sectors. We will later analyze in detail the role of the β-misalignment. So far we are taking the SUSY decoupling limit, which is defined by Eq. (3.1), while we are keeping the twin Higgs in the spectrum treating the Twin Higgs model as a full linear sigma model. The effective PGB theory discussed in Sec. 2 can be obtained integrating out the twin Higgs and expanding at the first non-trivial order in λ κ, σ (see Appendix B for details on this point). Eq. (3.4) immediately explains the origin of the problem already mentioned in the previous section: SUSY UV completions have a hard time to maximize λ and therefore also the fine-tuning gain, which is λ SM /λ. In SUSY UV completions λ S , rather than λ, should be perturbative. Moreover, as we will shortly see, in order to get the right Higgs mass we will have to stick to moderate values of tan β, thus further suppressing the effective λ. As a result, we will generally get λ < 1, such that the gain in fine tuning will never be large. In passing, we notice that the fine tuning in the soft models can be slightly ameliorated by changing the functional dependence of λ on tan β and other fundamental parameters, as we comment later on.
We now proceed to discus the leading Z 2 -even but SU (4)-breaking operators in the Twin SUSY potential. At leading order the SU (4)-breaking originates at tree-level from the electroweak D-terms and at one-loop from the top Yukawa sector: where we defined g 2 ew = g 2 + g 2 , and y t is the SUSY superpotential coupling, related to the top mass as m t = y t v sin β. In order to get Eq. (3.6) we compute the CW potential and and set the dynamical Higgses to their VEVs in the non-polynomial terms. The RH and LH stops mass are assumed to be equal, as well as the soft masses in the A and B sectors. 11 Matching again the effective SU (4)-breaking potential to Eq. (2.1) we identify This expression indicates that the SU (4)-breaking Z 2 -even quartic κ gets an unavoidable positive contribution from the EW D-terms at tree level in any SUSY UV completion. The ballpark of this contribution is O(10 −2 . . . 10 −3 ) depending on tan β. The contribution from the top-stop sector reproduces the result obtained in Eq. (2.5). Given that top loops alone already set the Higgs mass in the right ballpark (see the blue contours in Fig. 2), we expect that for a fixed M s a Twin SUSY model would have an upper bound on tan β. We will later see that because of this bound only models with small or moderate tan β are viable.
In those Twin SUSY theories, where the mirror symmetry is broken only softly, the Z 2 -breaking terms can be introduced as soft masses in the potential: These can easily be matched to the respective term in the IR effective potential in Eq. (2.1) where we defined σ u,d = ∆m 2 u,d /f 2 and assume ∆b = 0. Using the matching conditions described so far, we can rephrase in SUSY language the two EWSB conditions, which were already captured by the low energy Twin Higgs potential in Eq. (2.1). The first EWSB condition is the one we discussed in the PGB limit in Eq. (2.3) which fixes the EW scale v in terms of f . This is achieved in soft Z 2 -breaking models by balancing σ against κ. These two parameters are determined in terms of four parameters of the SUSY theory: β and M s determine κ through Eq. (3.7), while σ u , σ d and β give σ through Eq. (3.9).
Solving the EWSB condition in the IR effective theory we can trade σ for f /v. The second combination of σ u and σ d is unimportant in the δt β = 0 limit. Imposing the Higgs mass constraint in the PGB limit then fixes κ and allows us to predict the value of β in the (f /v, M s ) plane by solving Eq. (3.7) for β. This procedure holds only as far as κ/λ 1 and we will later discuss how the Higgs mass formula gets modified beyond the PGB approximation.
The second condition is essentially the SU (4) breaking condition, namely f 2 ≈ m 2 /2λ, rephrased in terms of the parameters of the underlining SUSY theory (note that here m stands for the mass parameter in Eq. (2.1)) using the matching conditions in Eq. (3.4). The dependence of the scale f on the stop threshold M s can then be estimated via the RGE of m Hu , which largely dominates the fine-tuning measure. In this approximation we × δm 2 Hu . The fine tuning measure derived in Eq. (2.11) in terms of the SUSY parameters is In this estimate we also neglect the contribution of the singlet soft mass m S in the RGE for m Hu . This contribution turns out to be sub-leading with respect to the stops one as long as m S ∼ 1 . . . 2 TeV. Looking back at the low energy fine-tuning (2.11) we see that in the SUSY theory Λ 2 . The UV cut-off Λ SU SY can be interpreted as a scale where the soft masses are formed, corresponding for example to the messenger scale in low energy SUSY-breaking scenarios. For the purpose of numerical calculations we will further use Λ SU SY = 100 M s .
All in all the gain in fine tuning of soft Twin SUSY with respect to standard SUSY models like the NMSSM is , (3.11) in agreement with the analysis performed in Ref. [20] as well as with our expectations from the IR effective theory analysis in Sec. 2. It is important to remember that the Higgs mass We summarize the behavior of the soft Twin SUSY model in Fig. 5. We work directly in the SUSY theory, solve numerically the EWSB conditions and the Higgs mass constraint, and present the allowed parameter space in the (f /v, M s ) plane. In this plane we show the behavior of the fine tuning measure, the minimal value of tan β that satisfies the Higgs mass constraint 12 and the mass of the next-to-lightest CP-even Higgs. We show our results for two different choices of the mirror CP-odd Higgs masses m A T = 1.5, 3 TeV. In the rest of the section we give an analytical understanding of the results of Fig. 5.
The fine tuning we show in Fig. 5 is computedà la Giudice-Barbieri by evaluating numerically the logarithmic derivatives of the EW scale v with respect to all the UV parameters of the full soft Twin SUSY model: (3.12) In this expression i runs over the parameters of the Higgs sector All these parameters are considered at the "messenger scale" Λ SU SY for the purpose of the fine-tuning computation and we use full one-loop RGE (see Appendix A) to obtain the final fine tuning.
One more comment is appropriate here regarding the total fine tuning measure in Eq. (3.12) that we are using. This is slightly different from the measure which has been used by Barbieri and Giudice in [44], which assumed the total fine-tuning to be the maximum of the one-parameter fine tunings, rather than their sum in quadrature, as we do. We choose this particular measure to give a numerical expression to an intuition, that if a model has a similar fine-tuning with respect to two independent parameters, it is more fine-tuned, than a model that is one-tuned only with respect to a single parameter. We checked it explicitly, and in vast majority of the parameter space our measure is not very different numerically from the nominal Barbieri-Giudice measure. Note however, that in case of multiple free parameters and small fine-tuning our measure overestimates the fine-tuning with respect to the Barbieri-Giudice measure.
As expected, our tuning measure can be well approximated by formula (3.10), meaning that the gross features of the Twin SUSY model are captured by the PGB intuition we have developed so far. In particular the factorized formula for the tuning measure works pretty accurately and we can rewrite Eq. (3.12) as where the last expression exactly reproduces the approximated formula in Eq. We now analyze the role of the Higgs mass in the soft Twin SUSY parameter space. In particular, we would like to understand the absolute lower bound on M s which is present in both plots and the upper bound on f /v at fixed M s in the left panel. These two bounds are responsible for the white region in Fig. 5, where there is no value of tan β for which the Higgs mass constraint can be satisfied. Assuming that there is an upper bound on tan β, the lower bound on M s can be easily understood from the PGB mass formula: which shows that for most of the moderate tan β we undershoot m h = 125 GeV unless we enhance the radiative corrections, yielding a lower bound on the M s . To understand the upper bound on tan β we have to go beyond the PGB approximation, which breaks down for large tan β, because the PGB approximation requires Beyond the PGB approximation we should take into account all the κ/λ corrections, which is not easy to do analytically (see however Appendix B). Nonetheless, we can arrive to the right conclusions analyzing the first non-vanishing correction to the PGB approximation: As tan β grows, the term in the brackets becomes important, and drives the Higgs mass down, signalling that in the full Higgs mass we have a "sweet spot" for the β angle at which we can maximize the Higgs mass, and therefore the minimal allowed value for M s . We show the value of the Higgs mass as a function of tan β and f /v using the full numerical calculation at all orders in κ/λ on the left panel Fig. 6. This confirms that the full function of the Higgs mass has a sweet spot for tan β, in qualitative agreement with the simplified formula in Eq. (3.16).
The upper bound on f /v at fixed M s instead strongly depends on the value of m A T which controls the masses of the SUSY Higgses (see Table 1 and Appendix B for details). In particular we see in Fig. 5 that the upper bound on f /v disappears for m A T = 3 TeV. This feature is also illustrated on the right panel of Fig. 6 and can be understood as a consequence of level splitting between the SM Higgs and the CP-even SUSY Higgs whose mass squared is proportional to ∝ (m 2 A T − λ 2 S f 2 ). For fixed m A T increasing f /v reduces the SUSY Higgs, thus enhancing the level splitting and decreasing the SM Higgs mass. This effect can always be compensated by taking a larger m A T (see Fig. 6), however this comes at the price of an O(1) increasing of the overall fine-tuning for M s 3 TeV.
Interestingly, an indirect bound on f /v at fixed M s implies an upper bound on the twin Higgs mass. Since the twin Higgs is the radial mode of the SU (4)-charged Higgs, its mass is roughly ∼ λ S f up to the κ/λ corrections. For example, for M s = 2 TeV and m A T = 1.5 TeV the twin Higgs cannot exceed 500 GeV and can be probed at the LHC as we will discuss in more detail in Sec. 4.
Up to now we completely ignored the role of δt β which we set to zero in Fig. 5 for simplicity. However, in the full SUSY theory there are two more relations which determine tan β and the misalignment between the β angles in terms of the fundamental parameters (at the leading order in δt β ): where m 2 A = m 2 A T − λ 2 S f 2 is the mass of the MSSM-like CP-odd Higgs. From these expressions one can estimate what would be the natural value for the β angle misalignment between the different sectors. If we require no unnatural cancellations and the "sweet spot" values for tan β (namely of order 2...3), we expect that |δt β | 1, but usually not smaller than 0.1. Because these natural values are not necessarily small, we show in Fig. 7 how the non-trivial β-angle misalignment changes the allowed parameter space and the fine-tuning. Negative values of δt β are allowed, because σ d can be either positive or negative, allowing to extend the parameter space at larger f /v. A positive δt β makes instead the parameter space shrink. The qualitative features of the plots are however left unchanged with respect to the δt β = 0 case.
We conclude that this particular soft Twin SUSY model has a number of features which crucially depend on the fact that λ is generated via the non-decoupling F-term of a NMSSM-like singlet and hence suppressed even for moderate tan β. In particular the tan β-dependence of λ renders the behavior of the tuning measure sub-optimal and certainly motivates further exploration in the theory space of soft Z 2 -breaking models. For example, one can consider models along the lines of [45] where an extra U (1) x is gauged and both A and B Higgses carry the same U (1) x charge. If the U (1) x is spontaneously broken by fields with large soft masses, one would obtain a non-decoupling SU (4)-symmetric quartic from the D-terms of the U (1) x , getting a very different dependence of the effective λ on the β-angle. We leave an exploration of soft Twin SUSY models for future work.
Let us finally mention some caveats of our analysis. First, the allowed parameter space presented in Fig. 5 depends crucially on our requirement m h = 125 ± 2 GeV. This range is probably too conservative since our one-loop computation is subject to large theoretical uncertainties dominated by two-loop QCD effects. It is known that these effects can sometimes shift the one-loop result by more than 2 GeV in SUSY theories where they have been already computed like in the MSSM. In particular a large positive shift from two loop effect might quantitatively change our conclusions on the allowed parameter space of the model. Nevertheless our analysis illustrates some important features of tree level effects to the Higgs mass that will certainly be present also after the inclusion of two loops corrections. Second, an interesting option to enlarge the parameter space of this particular soft Twin SUSY model and possibly ameliorate its fine tuning would be to introduce a sizeable negative δt β . This value does not come out naturally, but might be justified in a UV complete model where the origin of both Z 2 -even and Z 2 -odd soft masses is specified. We leave these questions open for future work.

Hard Twin SUSY
In this section we discuss models of Twin SUSY where a Z 2 -breaking quartic is generated at the tree-level. The easiest implementation of this class of models borrows the SU (4)invariant sector from the soft model described above and just extends the singlet sector adding a new singlet S A which couples to the A-sector Higgses only Integrating out the singlet in the limit m S A M S A , µ we generate a Z 2 -breaking quartic at treel level while the Z 2 -breaking soft masses in Eq. (3.8) are generated at 1-loop: with the same approximations that we have used before and neglecting the log-pieces in the CW potential. Matching Eq. (3.19) to the Z 2 -odd parameters of the Twin Higgs potential in Eq. (2.1) we get After comparing these matching conditions with the results of Sec. 2 we see that the required values of Z 2 -breaking quartic ρ 0.1 to get f /v 2.3 are easy to achieve with a moderate coupling λ A 0.6 as long as tan β 2.
The biggest problem of this model is that the one-loop threshold corrections to σ are negative and their size is controlled by the singlet mass. A negative σ is subject to a strong upper bound coming from the stability of the EWSB vacuum (c.f. Fig. 1 and the discussion in Sec. 2). In a SUSY UV completion we get a similar bound by solving the EWSB condition for f /v in terms of λ A as long as κ > 0. This poses an upper bound on the mass of the singlet m S A around 1 TeV. This bound was expected from Sec. 2 where for negative σ ( = −1) we got a strong constraint on Λ ρ which is here identified with the singlet mass up to numerical factors.
We illustrate the parameter space of the model in Fig. 8. The figure also shows that the upper bound on the singlet mass we derived in Eq. (3.21) gives indirectly a bound on the maximal allowed tan β, which is even stronger constrained than in the soft Twin SUSY. An important difference between the hard Z 2 -breaking scenario compared and the soft one is that the masses of the MSSM Higgses controlled by m A T = 2b/s 2β cannot be arbitrarily decoupled if δt β is fixed. The β-angle misalignment in this scenario is where δλ u = and it can only be positive. Completely neglecting the β-misalignment is often not a bad approximation in this scenario. Although light SUSY Higgses can help with the Higgs mass constraint because level splitting among CP-even Higgses can decrease the mass of the lightest eigenstate (i.e. the SM-Higgs), the masses of the SUSY Higgses are subject to bounds both from direct and indirect searches, e.g. b → sγ and the SM Higgs couplings fit. Even though these bounds are somewhat model dependent, we conservatively impose m A 400 GeV in our numerical scan, which correspond to m H ± 350 GeV using the formulas in Appendix B. We will further discuss these bounds in detail in Sec. 4. At the end of the day the simplest model of Twin SUSY has a fine tuning of around ∼ 10% but is so severely constrained by m h = 125 GeV that the SUSY colored states are always forced to be within the reach of the LHC. This traces back to the fact that with a negative σ and an irreducible tree level contribution to κ coming from the EW D-terms, we always overshoot the Higgs mass. The only way out of this is to take tan β ∼ 1 and render the log in Eq. (3.7) extremely small. The upper bound on M s was again expected from the discussion in Sec. 2 when for = −1 we got a strong upper bound on Λ t . Interestingly, the Higgs mass constraint is acting on the parameter space of hard Twin SUSY models in the opposite direction with respect to the standard MSSM-like SUSY constructions where it often gives a lower bound on M s .
A crucial question remains then to be addressed: Is it possible to find a model that decouples the colored states? Here we present an existence proof for such a model. Notice that in the IR effective theory for a given Λ ρ threshold one can access higher values of Λ t by introducing a negative contribution to κ at the tree level, compensating for the unavoidable positive contribution from the D-terms. Of course this is not the only way one can explore, but we focus on this option for concreteness and comment later on possible alternative solutions.
In order to get a negative contribution to κ we introduce a pair of bi-doublets B +B under SU (2) A × SU (2) B with hypercharges ±1/2 and the following potential Integrating out the bidoublet in the limit of m B M B , µ and matching it to the Twin Higgs potential we obtain: In the Twin Higgs parametrization in Eq. (2.1) this gives us a negative contribution to κ and a positive one to λ: Therefore we expect that for a given point in the (f /v , M s ) plane where the Z 2 -breaking quartic λ A is fixed to get the desired value of f /v, one can always choose an appropriate λ B in order to satisfy the Higgs mass constraint, hence rescuing the hard-breaking Z 2 -breaking model. An analytical understanding of the structure of the EWSB conditions can be easily derived from our previous discussion performing the shift: δλ u → δλ u − λ 2 BD 2 . Of course this solution for correcting the Higgs mass comes at a price. The main problem is the presence of states charged under both A and B gauge group that generically generate a mixing between the photon and the dark photon at one-loop, which is strongly constrained by experiments [46,47]. However, this problem can be solved either by ungauging the mirror hypercharge or by breaking it, giving a mass to the mirror photon. These effects would generate extra Z 2 breaking contributions in the Higgs potential which however do not change the structure of our model drastically. In particular the ungauging of the twin hypercharge leads to a Z 2 breaking quartic via the D-terms, which however is too small to dominate the mirror symmetry breaking in a realistic scenario.
We show the parameter space of the bidoublet model in Fig. 9. The bidoublet coupling λ BD ranges between 0.4 . . . 0.6 depending on the region of the parameter space. This corresponds to the negative shift in κ of order 0.05 . . . 0.1. Among other parameters we scan here on tan β, which however stays around tan β 2 in almost the entire parameter space. We calculate the fine tuning as in the previous section, varying the EW breaking scale v with respect to all the parameters of the hard model. 13 The behavior of the fine tuning isolines in Fig. 9 can be intuitively understood by means of the following approximate formula The idea behind this approximation is that the fine-tuning of f with respect to the SUSY scale is dominated by the running of m Hu as in soft Twin SUSY. The fine-tuning of the EW scale with respect to f is instead accounted by the log-derivative with respect to the hard Z 2 -breaking quartic and the Z 2 -preserving quartic λ BD which is tuned to satisfy the 13 In this case the hard sector parameters include the parameters of the Higgs sector PH =   Higgs mass constraint at fixed f /v and M s . This second contribution to the fine-tuning is shown on Fig. 9 with dashed red contours and it increases at larger M s as expected. The approximate formula in Eq. (3.26) is the SUSY analogue of the FT measure we discussed from the low energy perspective in the right panel of Fig. 4. For M s = 2 TeV the theory is tuned at ∼ 3 − 10%, depending on f /v which is roughly a factor of ∼ 3 − 10 better than the soft Twin SUSY model. Going to larger M s , the fine tuning needed to satisfy m h = 125 GeV becomes larger and introducing hard Z 2 -breaking is not beneficial anymore. That is why we cut the parameter space of Fig. 9 at M s ∼ 3.5 TeV.
Of course the approximate formula in Eq. (3.26) relies on a number of assumptions which are not always satisfied in the full parameter space of the model. The major caveat is for sure the factorization of the fine tuning measure which is broken in the presence of extra quartics by the one-loop corrections to the soft masses proportional to both λ A and λ BD (see for example the RGE of m Hu in Appendix A). These corrections are however always sub-leading compared to the top-stop contribution to the RGE of m Hu as long as both the couplings are smaller or equal to y t and both the singlet and the bi-doublet soft masses are not too large. For this reason we set m S A = m B = 1 TeV in our numerical results.
The white region at low f /v region can be understood by further analyzing the expressions in Eqs. (3.21) and (3.22). First of all we see from Eq. (3.22) that m A decreases with f /v and hence we expect the lower bound m A > 400 GeV to give a lower bound on f /v.
It is then clear that small values of δt β are favored because they can easily raise the mass of m A as one can see again from Eq. (3.22). In this regime we can approximate δt β ≈ 0 and solve (3.22) for σ S A . Plugging back this solution into (3.21) we get from which we see that the lower bound on f /v in order to get λ A > 0 becomes stronger at larger M s (which correspond to a larger δλ u ), explaining the diagonal shape of the boundary of the parameter space on Fig. 9.
This lower bound on f /v is of course not a feature of hard Twin SUSY by itself but of our specific model. It is interesting however that it provides a lower bound on the Twin Higgs mass, which was instead bounded from above in the soft Twin SUSY model of Sec. 3.1. Conversely the mass scale m A of the SUSY Higgses, which was a free parameter in the soft Twin SUSY model, is now fixed by Eq. (3.22) and possibly becomes a promising signature at the LHC as we will discuss in Sec. 4.
In conclusion, besides being sub-optimal for the reasons explained above, we believe that the SUSY model described in this section provides a simple existence proof of the hard

Twin SUSY Higgs phenomenology
In this section we discuss the phenomenological signatures of the SUSY Twin Higgs. In particular we focus on the Higgs sector of the theory, which leads to distinctive and model independent signatures at the LHC. The Twin Higgs mechanism implies the existence of an extra scalar in the spectrum, the "twin" Higgs, which is the radial mode of the spontaneously broken SU (4)-symmetry. The phenomenology of the twin Higgs at LHC essentially resembles the one of a scalar singlet which mixes with the SM Higgs and has been already studied e.g. in Refs. [37][38][39][40][41]. Here we recast present and future bounds from LHC and Higgs coupling measurements on the twin Higgs focusing on the particular parameter space of Twin SUSY models (see Appendix B for detailed formulas about the spectrum, the mixing angles and the decay widths). In addition we show that extra MSSM-like Higgses, which are generically present in SUSY UV completions, provide complementary probes on the parameter space of Twin SUSY both at the LHC and in indirect measurements.
We are not going to explore possible extra signatures coming from the twin matter sector. These depend very much on its structure that we leave unspecified in this paper (see CP-even Higgses States:

CP-odd Higgses Charged Higgses States:
A Refs. [26][27][28][29] for some studies about LHC signatures in Twin Higgs scenarios). Moreover, we assume pair production of colored SUSY states to be out of reach of LHC with 300 fb −1 , which roughly corresponds to M s 2 TeV. In the previous section we have shown that this regime is automatically achieved in the simplest soft Twin SUSY model once m h = 125 GeV is imposed, because the Higgs mass constraint bounds M s from below. For hard Twin SUSY models instead, the situation is reversed and the region with M s 2 TeV is accessible at a price of some model building gymnastics. For M s 2 TeV the LHC bounds on extra Higgses should be in principle compared with the canonical SUSY searches for pair produced superparticles. The same comparison should be performed for prospects at HL-LHC with 3000 fb −1 . We leave such a detailed comparison for future works and focus on the physics of the extra Higgses from now on.

The Higgs sector of Twin SUSY
The Higgs sector of the SUSY Twin Higgs exhibits a very rich structure since it contains at least a double copy of the MSSM. In Sec. 3 we have shown that scalar states beyond the visible and hidden Higgs doublets are needed in concrete realizations of Twin SUSY in order to get the required structure of the quartics in the low energy Twin Higgs potential of Eq. (2.1). Since these states are generically required to have a large soft mass of order ∼ M s , we assume that they are sufficiently heavy to be decoupled from the LHC phenomenology.
We summarize the minimal Higgs sector of Twin SUSY in Table 1. The PGB formula of the SM-like Higgs mass in Twin SUSY has already been discussed in Sec. 2, and the expressions beyond PGB can be found in Appendix B. The behavior of the heavy Higgs spectrum is instead described with very good accuracy by the tree-level formulas in Table  1. Here we have assumed that λ 2 λ 2 S f 2 , neglected subleading corrections proportional to λ A and set δt β = 0.
From these expressions we first recognize the twin Higgs h T with a mass squared given by m 2 h T = 4λf 2 , where the SU(4)-invariant quartic λ is given by the SUSY matching condition in Eq. (3.4). We further notice that among the extra Higgs states there is a fully degenerate SU (2)-doublet (H, A, H ± ) with mass set by the CP-odd Higgs mass m 2 A ≡ m 2 A T − λ 2 S f 2 , which becomes light at large f . Because this mass should be nontachyonic, m 2 A T = 2b/s 2β should always be the largest mass scale of the Higgs sector. The other SUSY SU (2)-doublet (H T , A T , H ± T ) has instead larger splittings: the heaviest state is the CP-odd Higgs A T with mass m 2 A T followed by the CP-even H T and the charged Higgses H ± T . The charged Higgses H ± and H ± T are mass-degenerate in this approximation and do not mix because of gauge invariance. As a consequence the charged Higgses H ± have exactly the same couplings as in the MSSM, while H ± T are completely dark and hence uninteresting for collider purposes.
The mixing of the CP-odd and the CP-even SUSY Higgses is controlled by v/f exactly like the one of the SM Higgs h and the twin Higgs h T : A T and H T are purely dark in the limit v/f → 0 while A and H become MSSM-like. 14 The full expressions for both mixing angles and mass eigenvalues can be found in Appendix B.
In most of the allowed parameter space the twin Higgs h T is lighter than the visible MSSM-like SU (2)-doublet and hence it is the next-to-lightest CP-even scalar after the SMlike Higgs. However, when λ 2 S f 2 m 2 A (corresponding to m 2 , the hierarchy can be inverted and a full MSSM-like SU (2)-doublet forms the first level of extra scalars above the SM-like Higgs. The transition between these two regimes is shown in Fig. 10, where we plot the composition of the next-to-lightest CP-even eigenstate H 2 and its mass for the benchmark values of λ S and tan β. We define the composition of mass ordered CP-even states H I in terms of the gauge eigenstates of the A and B sector as: In the temperature plot in Fig. 10 we show the combination of the mixing matrices s dark ≡ V 2 23 + V 2 24 that indicate how much H 2 is composed by B-sector Higgses. Both mixing angles and the mass eigenvalues are computed numerically taking into account all orders in the λ 2 S f 2 /m 2 A expansion (setting λ A = λ BD = δt β ≈ 0 for simplicity). As expected, H 2 is mainly B-like (i.e. h T ) in the upper left corner of the plot (m A λ S f ), while it is mainly A-like (i.e. H) in the bottom right corner of the plot (m A λ S f ). The four plots in Fig. 10 show how the details of the transition between the two regimes depend on the value of λ S and tan β: comparing the plots row-wise we see that a larger tan β suppresses the radial mode mass hence enlarging the parameter space in which H 2 is mostly B-like (red region). Column-wise, we see that a larger λ S enlarges the parameter space in which H 2 is mostly A-like (blue region), because the twin Higgs mass m h T gets enhanced while the MSSM doublet mass m A gets suppressed.
We are now ready to understand the parametric dependence of the decays of the CPeven Higgses in the (f /v , m A ) plane, which we can numerically compute using the decay rates of the CP-even eigenstates that are collected in Appendix B. In Fig. 11 we show the relevant branching ratios for the next-to-lightest CP-even state H 2 .
Clearly we have two different regions depending if the lightest CP-even state is the twin Higgs h T (red region) or the MSSM-like heavy Higgs H (blue region). In the first case the visible decays of the twin Higgs are into pairs of SM gauge bosons/Higgses and into top pairs. An invisible decay width can also be present if the decays into dark gauge bosons are kinematically open. Since we assume equal gauge couplings in the visible and dark sector, the dark gauge boson masses are fixed and scale approximately linearly with 14 Any mass term that mixes Higgs states in A and B sector must be proportional both to the breaking of SU (2)A and SU (2)B and therefore vanishes in the limit v → 0. f , e.g. m W B ≈ g 2 √ 2 f . Comparing the upper and the lower panel of Fig. 11 we see that a non-zero invisible decay width for the twin Higgs ultimately depends on the value of the SU (4)-invariant coupling λ S , which sets the scale of the twin Higgs mass. In the upper row we take λ S = 0.9, such that the invisible decay channels are kinematically closed while in the second row we take λ S = 1.4. Of course the latter value is favored by fine-tuning arguments as discussed in Sec. 3, however since λ S 1 makes the phenomenology of the twin Higgs radically different we decided to include λ S = 0.9 in our discussion.
The most interesting decay channel to hunt for the twin Higgs at the LHC is certainly the one into Z pairs. In Fig. 11 we see that this branching ratio goes roughly from 25% to 10% in the red area, depending on the region of the parameter space. A 25% branching ratio into Z pairs is the value expected from the Goldstone equivalence principle if one assumes Figure 11. Branching ratio of the second lightest CP-even eigenstate. The four plots correspond to different values of (λ S , tan β). The purple (black dashed) contours denotes the BR into SM tt pairs (the invisible BR into W D and Z D ). The temperature map indicates the BR into SM ZZ.
∼ 100% branching ratio of the twin Higgs into SM gauge bosons/Higgs like was done in [37]. From Fig. 11 we see that this expectation is only (marginally) saturated in the region where m A > 1 TeV and f /v > 5. In the bulk of the SUSY Twin Higgs parameter space the branching ratio into Z pairs (like the ones into W W and hh) is depleted because of an irreducible branching ratio into top pairs (purple contours). The tt channel is suppressed only for very large f /v (i.e. very small mixing) or for very small f /v because of the reduced phase space.
Using the formulas from Appendix B, assuming m A λ S f and neglecting the phase space suppressions we get which shows that an irreducible branching ratio into tt for the twin Higgs is somewhat typical for Twin SUSY scenarios, where the branching ratio into ZZ gets suppressed for tan β > 1 (where sin 2β is reduced) and cannot be enhanced arbitrarily by taking a larger λ S because of perturbativity.
In the second row of Fig. 11 we set λ S = 1.4 and the invisible width of the twin Higgs is different from zero (dashed black contours). When the decays into dark gauge bosons is kinematically open, if one would assume 100% decay into (Pseudo)-Goldstones, the Goldstone equivalence theorem would predict a branching ratio of ∼ 0.14 into ZZ and an invisible width of ∼ 3 × 0.14. However, this naive estimate is once again modified by the irreducible decay width into tt (though less relevant than λ S = 0.9 in agreement with formula (4.2)) and by the fact that in most of the Twin SUSY parameter space the twin Higgs mass is not parametrically larger than the dark gauge boson masses. Indeed the invisible width of the radial mode gets sensibly reduced because of phase space suppression, as can be seen from the contours in Fig. 11. The phase space suppression of the invisible channels becomes more important at larger tan β, where the twin Higgs mass is further reduced.
Finally, for fixed value of f /v, decreasing the value of m A implies that the mixing with the MSSM-like CP-even Higgs becomes important and the branching ratio into tt is enhanced with respect to ZZ (and with respect to invisible ones if present). Decreasing m A further the MSSM-like Higgs becomes the next-to-lightest state and as a consequence the decay width into ZZ is suppressed while the one into tt gets rapidly close to 100%.

Probing the SUSY Twin Higgses
From the previous discussion one can infer what are the most promising phenomenological signatures associated to the Higgs sector of the SUSY Twin model. We essentially have two different types of signatures depending on the region of the parameter space: • In the region where m A λ S f the twin Higgs is the next-to-lightest CP-even state (this is the red region in Figs. 10 and 11). The radial mode is copiously produced at the LHC via its mixing with the SM Higgs and the most promising channels to probe it are di-boson final states; in particular di-Higgs (four b-jets) or Z-boson pairs [20,37,42,48]. Indirect bounds on the twin Higgs also arise from modifications of the SM-like Higgs couplings (in particular the ones to gauge bosons).
• In the region where m A λ S f we have a full MSSM-like SU (2) doublet which becomes light. Since having m h = 125 GeV always forces tan β to be quite small, this region of the parameter space is preferably tested by searches for charged MSSM-like Higgses. These can be probed either indirectly through their contribution to b → sγ, or directly hunted at the LHC via tb and bt final states [42,49]. Another interesting channel is provided by the associated production of a CP-even/odd Higgs with tt [42]. However, searches for charged MSSM-like Higgses typically provide a cleaner channel for similar mass scales.
We now provide a preliminary study of the LHC reach on the parameter space of Twin SUSY models, which focus on direct and indirect searches for the twin Higgs and charged MSSM-like Higgses. We present both existing constraints and the future reach of LHC searches and indirect measurements. For LHC prospects we consider projections for the final stage of LHC with 300 fb −1 and for HL-LHC, where 3000 fb −1 are expected [50].
For the ZZ channel we use the results of Ref. [37], where the current and future bounds for a scalar singlet mixing with the SM-Higgs are given as a function of its mass and signal strength into ZZ. We recast these bounds in the Twin SUSY parameter space by computing the production cross section of the radial mode from both gluon fusion and vector boson fusion (VBF) processes and multiplying by the branching ratio into ZZ. 15 For what concerns Higgs coupling deviations, a global fit of Higgs couplings and EWPT in Twin SUSY has already been performed in Ref. [20], where it has been shown that current data cannot probe our region of interest on which we already impose f /v 2.3 and m A 400 GeV. To estimate the HL-LHC prospects on Higgs couplings, one can focus on the measurement of the SM Higgs coupling to Z-bosons. The HL-LHC prospects for the Higgs coupling measurements will ameliorate the current bound on the mixing angle by roughly a factor of 2 (the precise numbers can be found in Refs. [37] and [52]), which leads to a future bound of approximately f /v 4.6. We will comment on such a limit in the following discussion.
The LHC prospects for direct probe of the MSSM charged Higgses are taken from Ref. [42] where the process pp → H +t b → tbtb (and charged conjugate) is employed. Notice that we can use these bounds without any further modification since the visible charged Higgses do not mix with the dark sector ones.
Concerning the indirect searches through b → sγ, we parametrize the BSM contributions as modifications to the Wilson coefficients ∆C 7  and we derive the bounds allowing for 2σ deviations from the experimental value, summing in quadrature the theoretical and experimental uncertainties. The contribution of the charged MSSM-like Higgs to this observable implies a lower bound on its mass for a fixed value of tan β. In particular, the existing limits set a lower bound of around m H + 500 GeV for tan β 2, and hence does not constraint our parameter space. In order to estimate the future prospects for this observable, we optimistically assume that both the theoretical and the experimental uncertainties get reduced by a factor of 2. 16 The bound on the charged MSSM-like Higgs mass is then significantly improved and goes up to m H + ≥ 775 GeV for tan β 2. 15 To estimate the 8 TeV cross section we use the Higgs production cross section for the gluon fusion and the VBF processes for equivalent mass [51] and weight them with the appropriate mixing angles. For the √ s = 13 TeV and √ s = 14 TeV cross section we just re-weight the √ s = 8 TeV cross sections for gluon fusion and VBF with the corresponding parton luminosities ratios taken from Ref. [51]. 16 M. Misiak private communication. In Fig. 12 we project the current and future bounds from the searches described above in the (f /v , m A ) plane. Analogously to Figs. 10 and 11 we show the results for four different choices of λ S = {1.4 , 0.9} and tan β = {1.5 , 2.5}. From these plots we can see that direct searches for the twin Higgs in the ZZ channel give a lower bound on f /v as long as m A λ S f . In contrast direct and indirect searches on the MSSM charged Higgses set a lower bound on m A independently on the value of f . We first discuss the features of the searches targeting the MSSM-like scalars. Comparing the plots row-wise we see that direct LHC searches on the charged Higgs depend significantly on tan β, which sets the couplings totb. The projected bound from b → sγ is also dependent on tan β and gets slightly stronger for very low values of tan β. Note that here we used the complete expression for the charged Higgs mass which includes also O(v 2 ) corrections. Conversely, comparing plots column-wise one can see that varying λ S does not change the physics of the MSSM charged Higgs as expected.
Now we focus on the region probed by the di-boson resonance searches. The shape of the exclusion lines from the ZZ final states presents a rich structure, which depends on multiple effects controlled by λ S and tan β. The overall reach is affected by variations of the coupling to tops (the main production mechanism is gluon fusion), variations in the structure of the mass spectrum (cf. Fig. 10) and in the branching ratio to ZZ (c.f. Fig. 11). However, independently on the values of λ S and tan β, the exclusion regions exhibit a peculiar horn-like shape. The horn is located where the next-to-lightest eigenstate changes from mostly twin-like to mostly MSSM-like In such region, because of the transition, there is a local enhancement in the A-sector components of the H 2 state and the sensitivity is increased. Then, in the region where the H 2 state is very much MSSM-like, the sensitivity drops because of the reduced branching ratio into gauge bosons. In the low m A region of the plots (close to m A ∼ 500 GeV) the reach of HL-LHC is again enhanced because the very small branching ratio of H 2 into gauge bosons is compensated by a sizeable enhancement of the production cross section for such low masses.
In the first row of Fig. 12 we fix λ S = 0.9 such that the invisible decay channels are closed. Comparing the plots row-wise we see that the bounds from LHC are very similar. The reason is that for lower tan β the enhancement of the top coupling, and hence of the gluon fusion channel, is partially compensated by a larger twin Higgs mass (cf. the contours of Fig. 10). The transition between the twin Higgs regime and the MSSM-like regime is not the same in the two plots and reflects the discussion of the previous subsection.
In the second row of Fig. 12 we fix λ S = 1.4 and the invisible decay channel for the twin Higgs opens up, depleting the signal strength into ZZ with respect to the λ S = 0.9 case. However, in the bottom right plot the larger value of tan β reduces m h T and closes the invisible decay into dark gauge bosons exactly where the small bump in the exclusion curves is, roughly where m h T ∼ 2m W D (compare with contours in Figure 10).
In summary, we see that searches for the twin Higgs into ZZ final states at the LHC are extremely interesting in Twin SUSY constructions, because perturbativity of the SU (4)invariant coupling λ S and tan β > 1 give an upper bound on the twin Higgs mass for fixed value of f /v and fixed m A . This has to be contrasted with strongly coupled UV completions where usually the strongest constraint comes from Higgs coupling measurements. Note that the prospects for Higgs coupling measurements at HL-LHC reach roughly the value of f /v 4.6 as we mentioned before. For the largest λ S we considered, which corresponds to the largest m h T compatible with perturbativity, the Twin SUSY phenomenology starts to be similar to strongly coupled UV completions with the Higgs coupling measurements having a reach not very far from the direct searches (see left bottom plot of Fig. 12). However, for lower values of λ S Higgs coupling measurements are never competitive with the direct resonance search as can be seen from the other plots on Fig. 12. Finally, the region with large f /v features a radial mode too heavy to be probed in ZZ searches at the LHC, even at High Luminosity. For small m A , the spectrum here resembles the usual MSSM and MSSM-like Higgses searches will explore this region for m A 1 TeV.
Looking back at the parameter space of the soft and the hard Twin SUSY models of Sec. 3, we can roughly compare the sensitivity of the direct searches at the LHC in these two particular cases. In the simplest soft Twin SUSY model of Fig. 5 and Fig. 7 we found a strong upper bound on f /v for M s ≈ 2 TeV, which, together with the quite large value of tan β ≈ 3.5, gives a stringent upper bound on the mass of the twin Higgs 400 − 700 GeV, depending on the particular choice of δt β and m A T . A considerable amount of parameter space of soft Twin SUSY is then likely to be probed by ZZ searches already with 100 fb −1 of data. The mass of the MSSM-like Higgses depends instead on m A T , which is unconstrained in this model and can always be taken to be heavy.
The hard Twin SUSY model we presented in Fig. 9 has instead a lower bound on the Twin Higgs mass which increases at large M s . This lower bound is still pretty mild for M s ≈ 2 TeV and Twin Higgs searches are certainly promising to explore these models. Quite interestingly the masses of the MSSM-like Higgses cannot be arbitrarily decoupled in this model as far as natural values of δt β are considered and the charged Higgses can easily be below the TeV leaving some hope for indirect signatures in b → sγ or the direct searches at the LHC.

Conclusions
In this work we have performed a systematic study of perturbative UV completions of the Twin Higgs mechanism based on Supersymmetry. In this context we showed that breaking the Z 2 mirror symmetry with large quartics can be beneficial in terms of fine tuning, leading to theories with colored states decoupled from the LHC and tuned to the level of ∼ 10%.
In order to explore the role of Z 2 -breaking quartics, we performed a detailed comparative study between hard and soft breaking of the Z 2 -symmetry from the effective field theory point of view in Sec. 2. This study provides a complete picture of the parameter space of the Twin Higgs and opens up new model building avenues. The main result is that hard Z 2 -breaking models can lead to a gain in fine tuning with respect to soft Z 2 -breaking at the price increasing the SM-like Higgs mass, which is usually predicted to be too high. Once the Higgs mass constraint is satisfied, the fine tuning of hard Z 2 -breaking models can still be around a factor of ∼ 5 better compared to soft Z 2 -breaking.
We have studied explicit SUSY UV completions of both soft and hard Z 2 -breaking models. We performed our analysis both numerically, solving directly the EWSB conditions and the Higgs mass constraint in the UV theory, and analytically, using a simple tree level matching to the original Twin Higgs model after the SUSY states are decoupled. Both models are not saturating the parametric gain in fine-tuning that we naïvely expect from Twin SUSY theories with respect to standard SUSY scenarios, but provide simple existence proofs of both mechanisms of mirror symmetry breaking. By comparing these two simple SUSY models we recover the gain in fine tuning of the hard Z 2 -breaking model as obtained in the effective field theory. As a summary plot we show in Fig. 13 the allowed FT range in the soft and the hard model as a function of the scale of the colored particles, marginalized over the rest of the parameters. The Twin SUSY constructions discussed here are only valid up to a scale of roughly ∼ 100 TeV. The final goal would be to build full UV completions up to the GUT scale where both the SUSY-breaking and the Z 2 -breaking are generated dynamically and then mediated to the MSSM and its mirror copy. Specifying the mechanism of mediation will give a definite prediction on the SUSY spectrum of both the visible and the mirror sector. Further studies in this direction are definitely worthwhile.
Our analysis has unveiled several important aspects of the parameter space of Twin SUSY models, highlighting in particular the special role of the Higgs mass in constraining the parameter space of both soft and hard Twin SUSY. To improve the analysis it would be desirable to have a more precise computation of the SM Higgs mass including at least 2-loop QCD corrections. The implementation of Twin SUSY models in a package like SARAH [55] can bring them to a similar precision we have for standard SUSY scenarios.
We hope that our study elucidated the basic building blocks of Twin SUSY constructions and their connection to the original Twin Higgs proposal. With these tools at hand it will be easier to construct "optimal" models of Twin SUSY, in attempt of maximizing their parametric gain in fine tuning with respect to standard SUSY scenarios..
We also sketched the phenomenology of the Twin SUSY Higgses.
In particular we showed that LHC searches for a resonance going into di-bosons can probe a large portion of the parameter space of Twin Higgs models, independently on how the mirror symmetry is broken. This is a consequence of the requirement of perturbativity on the SU (4)-invariant quartic coupling which gives an upper bound on the mass of the twin Higgs at fixed SU (4)-breaking VEV f , and it is a distinctive feature of perturbative UV completions of the Twin Higgs mechanism.
It would be interesting to further characterize the signatures associated to the invisible decays of the twin Higgs. These can potentially distinguish it from a generic singlet that mixes with the SM Higgs. Indeed, the invisible branching ratio of the twin Higgs is likely to be enhanced when extra light SUSY states are present in the mirror sector making it discoverable at the LHC in final states with large missing transverse energy (MET).
Interesting signatures can also arise from MSSM-like Higgses which become light at large f . Twin SUSY models then provide an additional motivation to extend the search program for extra MSSM-like Higgses in the small tan β regime both at LHC and at future colliders [42,56,57]. It would be interesting to investigate in detail alternative final states to probe CP-odd and CP-even MSSM-like scalars, such as the ones suggested in Ref. [42] including for instance ttA/H production. The possibility of having extra invisible decay channels into the mirror sector can also give rise to final states with MET, enriching further the landscape of accessible topologies.
The phenomenology of the extended Higgs sector is just the tip of an iceberg of possible phenomenological questions one can ask in the context of Twin SUSY. By specifying the detailed structure of both the SUSY and the mirror spectrum one can explore further the phenomenological implications of Twin SUSY. For instance it would be interesting to investigate allowed cosmological scenarios as it has been done for the original Twin Higgs proposal [31][32][33][34][35][36], identifying possible dark matter candidates (both symmetric and asymmetric), viable mechanisms for baryogenesis etc. Of course extra LHC signatures other than the ones associated to extra Higgses can arise once the full spectrum is specified (see for example Refs. [26][27][28][29]) and might require interesting new detecting techniques [58,59]. All these instances are likely to have an impact on the fine tuning of Twin SUSY models establishing a unique interplay between cosmology, collider searches and naturalness which is a distinctive features of NN scenarios.
with the auxiliary functions Moreover one has

B Appendix: Higgs sector spectrum
In this appendix we give further details about the (SUSY) Twin Higgs model. First we fully solve the Twin Higgs model as a linear sigma model deriving exact formulas for the VEVs, the masses and the mixing in order to study the validity regime of the PGB approximation. Then we provide the analytical formulae describing the phenomenology of the Twin SUSY Higgs sector when some hierarchy is present between the twin Higgs and the SUSY Higgses (i.e. for λ S f m A ): mass eigenvalues, eigenvectors and decay widths.

B.1 The Twin Higgs as a linear sigma model
In Sec. 2 we have explored the parameter space of the Twin Higgs in complete generality, integrating out the radial mode and obtaining a non-linear sigma model description of the SM Higgs as a PGB of the spontaneously broken SU (4)-symmetry. Here we repeat the same exercise working directly at the level of the linear sigma model with both the radial mode/twin Higgs and the SM-like Higgs in the spectrum. We obtain fully general formulas for the mass of the twin Higgs and the SM-like Higgs mass and for their mixing. We show how the PGB formulas are obtained expanding these expressions at leading order in k, σ λ. The Twin potential can be defined in terms of five parameters {λ, ρ, κ, m 2 ,μ 2 } The minimization conditions set the VEVs of H A and H B (v A and v B ) as a function of these parameters. Defining v 2 A = v 2 and v 2 B = f 2 − v 2 in analogy with the PGB approximation, one obtains with the shorthand notation σ ≡ −2λμ 2 /m 2 v 2 = − m 2 4 −σ + κ (2 − σ/λ) λρ + κ (2λ + ρ + κ) , f 2 = − m 2 4 2ρ + κ (4 − σ/λ) λρ + κ (2λ + ρ + κ) .
where we have defined g 2 ew = g 2 + g 2 . Apart from the soft Z 2 -breaking terms ∆m 2 u,d , hard breaking terms λ 2 A and bi-doublet terms λ 2 BD , we have included the terms δλ u and δρ u that are generated at one-loop from the stop/top sector, with We then minimize the potential and trade the parameters ∆m 2 u , ∆m 2 d , m 2 u , m 2 d for v 2 , f 2 and t β ≡ tan β A ≈ tan β B . We do not take into account corrections proportional to δ tan β = tan β A − tan β B , which typically lead only to small corrections. We can then compute the mass eigenvalues in the CP-even sector, in the CP-odd sector and for the charged Higgses. These latter results are particularly simple, since the CP-odd and charged Higgs mass matrices are 2 × 2 matrices that can be diagonalized exactly. In order to better understand the phenomenology in Sec. 4 we plot in Fig. 15 Fig. 15 we focus on the B-components into the dark sector and denote with solid blue/red lines the u/d-components the next-to-lightest eigenvalues. The dashed lines refer instead to the third lightest eigenvalue with the same color coding (blue for u-and red for d-components). In the right panel of Fig. 15 we plot with the same color coding the A-components into the visible sector (blue for u-and red for d-components).
From these plots it is clear that the role of the eigenvalues is inverted in the transition region around λ S f ≈ m A . For λ S f m A the next-to-lightest eigenstate is the twin-Higgs, while for λ S f m A it is MSSM-like. In the transition region with λ S f ≈ m A all states are sizeably mixed one each other.
The visible sector content of the extra Higgs states determines their production cross section at the LHC and their visible branching ratio.
For small λ S f m A the twin Higgs is the next-to-lightest Higgs and it gets sizeable couplings to the visible sector through its component along h A u . As a consequence it can be produced at the LHC in gluon fusion and it gives rise to interesting signals in di-boson channels. In the transition region, when λ S f ≈ m A , there is an enhancement of the h A u components as it can be seen from the solid blue line in the right panel of Fig. 15  then expect a slight improvement in the sensitivity on the twin Higgs searches at LHC into di-bosons. For λ S f m A the next-to-lightest state becomes MSSM-like and the sensitivity of the di-boson searches drops since MSSM-like have very small branching ratio into gauge bosons and at the same time the twin Higgs has a small mixing angle with the visible sector as it can be seen from the dashed lines in the right panel Fig. 15. This argument explains the horn-like shape of the di-boson exclusion regions in Fig. 12.

B.3 Approximate expressions for eigenvalues and eigenvectors
We have seen that in the regime λ 2 S f 2 m 2 A the collider phenomenology of the model is characterised by a light twin Higgs with interesting signatures. In this regime we can expand the expressions for masses and mixings for large m A (or equivalently large m A T ) and obtain approximate analytical results. From the scalar potential (B.10) we get for the spectrum, keeping only the leading order terms in g 2 ew , λ 2 A , λ 2 BD , δλ u , δρ u and f 2 /m 2 A T : To calculate a given coupling all permutations have to be summed over, e.g. the Lagrangian coupling L = A h T hh h T hh is given by A h T hh =