Supersymmetric D-term Twin Higgs

We propose a new type of supersymmetric Twin Higgs model where the SU(4) invariant quartic term is provided by a D-term potential of a new U(1) gauge symmetry. In the model the 125 GeV Higgs mass can be obtained for stop masses below 1 TeV, and a tuning required to obtain the correct electroweak scale can be as low as 20%. A stop mass of about 2 TeV is also possible with tuning of order O10%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O}(10)\% $$\end{document}.


Introduction
The main two pieces of information obtained with the Large Hadron Collider (LHC) so far is the discovery of the Standard Model (SM)-like Higgs boson with a mass of about 125 GeV, and no signs of New Physics close to the electroweak (EW) scale which put strong lower bounds on masses of new particles. The bounds are especially stringent for new colored states, for which they vary between several hundreds of GeV up to about 2 TeV. These bounds threaten many extensions of the SM that aim to solve the hierarchy problem, since naturalness requires that the top quark contribution to the quadratic divergence of the Higgs mass squared is approximately cancelled by the corresponding contribution from top quark partners. If the top quark partners are heavier than the top quark fine-tuning is reintroduced. This is known as the little hierarchy problem.
An interesting solution to the little hierarchy problem is provided by Twin Higgs models [1][2][3][4][5][6] which recently gained renewed interests . In this class of models the SM-like Higgs is a pseudo-Nambu-Goldstone boson of a global SU(4) symmetry, and the Z 2 symmetry relating the SM with a mirror (or twin) SM eliminates the quantum correction to the Higgs mass squared from the explicit breaking of the SU(4) symmetry. A key feature of this scenario is that the top quark partners are not charged under the SM color gauge group and easily evade accelerator bounds.
A successful SUSY Twin Higgs model should possess at least two features. First: a large SU(4) invariant Higgs quartic term λ to suppress the quadratic corrections to the Higgs mass parameter. More precisely, the tuning of a given model is relaxed by a factor 2λ/λ SM , as compared to the corresponding model without the mirror symmetry, where λ SM ≈ 0.13 is the SM Higgs quartic coupling. Second: the Higgs mass of 125 GeV is obtained for stop masses that do not lead to excessive tuning, say no worse than O(10) %. In the limit of arbitrary large λ the second requirement would be automatically satisfied (see eq. (3.6)). However, in realistic models there is some upper bound on λ which does not allow tuning to go away completely. Therefore, when discussing tuning of a given model both features should be taken into account.
Another important point is that in phenomenologically viable Twin Higgs models (SUSY or not) the Z 2 symmetry must be broken. This is because the 125 GeV Higgs couplings measured at the LHC are close to the SM prediction [31] and set a lower bound on the vacuum expectation value (vev) of the mirror Higgs. This results in an irreducible tuning of O(10-50) %, 1 depending on the amount of the Higgs invisible decays to mirror particles and other details of a given model. 2 On the other hand, Z 2 breaking is beneficial as far as the Higgs mass is concerned because in the limit of maximal Z 2 breaking the tree-level Higgs mass is enhanced by a factor √ 2 with respect to the prediction of the Minimal Supersymmetric Standard Model (MSSM). This makes SUSY Twin Higgs models also attractive for relatively light stops -satisfying the current experimental constraints but within the ultimate reach of the LHC. One of the goals of the present paper is to quantify the gain in the Higgs mass and study implications for the stop masses paying particular attention to effects of SU(4) and Z 2 breaking. In particular, we determine parameter space in which tuning does not exceed the irreducible tuning from the Higgs coupling measurements discussed above and calculate upper bounds on stop masses under this assumption.
We find that existing SUSY Twin Higgs models cannot saturate the irreducible tuning. In models proposed so far the SU(4) invariant quartic term is generated by an F -term of a singlet chiral field [4,5,7,8]. The SU(4) invariant quartic term is then maximized for tan β = 1 and decreases as sin 2 (2β) ≈ 4/ tan 2 β. On the other hand, the SU(4) breaking quartic coupling from the EW D-term, which contributes to the Higgs mass, is an increasing function of tan β, and hence a smaller tan β requires a larger stop mass. 3 As a result the 125 GeV Higgs mass is incompatible with a large SU(4) invariant quartic term 1 This irreducible tuning may be evaded by introducing hard Z2 breaking but explicit models of this type require total tuning of O(10) % anyway [8].

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and sufficiently light stops that do not lead to large fine-tuning. We also find that the higgsino mass is required to be small to suppress the singlet-Higgs mixing, which would otherwise reduce the Higgs mass. Motivated by these findings we propose a new type of supersymmetric Twin Higgs model where the SU(4) invariant quartic term is provided by a D-term potential of a new U(1) X gauge symmetry. In this setup the SU(4) invariant quartic term grows with tan β, which does not conflict with the Higgs mass constraint. We discuss the Landau pole constraints and show that the SU(4) invariant quartic term can be large enough to minimize the tuning in the regime where the model is under perturbative control. We present scenarios in which the tuning of the EW scale is solely determined by the irreducible one while the LHC constraints on sparticle masses are satisfied. In the least tuned region stops are within the reach of the LHC. Even if no sparticles are found at the end of the high-luminosity run of the LHC the tuning of the model may be still better than 10 %.
The rest of the paper is organized as follows. In section 2 we briefly review the F -term Twin Higgs model, introduce the D-term model and discuss constraints from perturbativity. In section 3 we discuss the impact of the Higgs mass on SUSY Twin Higgs models in a quite general effective field theory framework assuming that the only source of the treelevel SU(4) breaking quartic term is the EW D-term potential. In section 4 we discuss the fine-tuning of SUSY Twin Higgs models in detail. We show that the non-decoupling effect of the singlet have a substantial impact on the Higgs mass, which worsens fine-tuning in the F -term model, while analogous effects are almost absent in the D-term model. We quantify the naturalness of the D-term model in several scenarios. We briefly discuss differences in the heavy Higgs spectrum and phenomenology between F -term and D-term models. We reserve section 5 for our concluding remarks.

SUSY Twin Higgs models
In this section we briefly review a SUSY Twin Higgs model in which an SU(4) invariant quartic term is generated via an F -term potential and introduce a new class of SUSY Twin Higgs models in which an SU(4) invariant quartic term is generated via a D-term potential.

F -term Twin Higgs
A SUSY realisation of the Twin Higgs mechanism was first proposed in refs. [4,5] which used an F -term of a singlet chiral superfield S to generate the SU(4) invariant quartic term. The F -term Twin Higgs model was analysed in light of the Higgs boson discovery in ref. [7], and more recently in ref. [8]. The SU(4) invariant part of the F -term model is given by the following superpotential and soft SUSY breaking terms:

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Note that the SU(4) symmetry is automatically realised by the Z 2 symmetry. At tree level, the SU(4) symmetry is explicitly broken by the EW D-term potential: The above terms are Z 2 invariant. In phenomenologically viable models the Z 2 symmetry must be broken. This is obtained by introducing soft scalar masses: The Twin Higgs mechanism may relax fine-tuning only if the SU(4) invariant quartic term λ is larger than the SM Higgs quartic coupling. In this model this coupling is given, after integrating out a heavy singlet and heavy Higgs bosons, by So large λ prefers large λ S and small tan β. However, there is an upper bound on λ S and a lower bound on tan β. The former constraint comes from the requirement of perturbativity. Avoiding a Landau pole below 10 (100) times the singlet mass scale requires λ S below about 1.9 (1.4). A lower bound on tan β originates from the Higgs mass constraint which we discuss in more detail in the following sections.

D-term Twin Higgs
As an alternative to the F -term Twin Higgs model we propose a model in which a large SU(4) invariant quartic term originates from a non-decouping D-term of a new U(1) X gauge symmetry. Such a non-decoupling D-term may be present if the mass of a scalar field responsible for the breaking of the U(1) X gauge symmetry is dominated by a SUSY breaking soft mass, see appendix for details. Such models were considered in the context of non-twinned SUSY in refs. [32][33][34][35][36][37][38][39][40][41]. The non-decoupling D-term potential can be written as where is a model-dependent parameter in the range between 0 and 1. We refer to the appendix for explicit model that naturally allows for 1 which maximizes the magnitude of the D-term potential. This term gives the following SU(4) invariant coupling: A crucial difference with the F -term model is that λ is now maximized in the limit of large tan β which makes it easier to satisfy the Higgs mass constraint. This merit of a D-term generated SU(4) invariant quartic term was recently noted also in ref. [8]. The magnitude of λ is still bounded from above to avoid too low a Landau pole scale so it is not guaranteed that fine-tuning is considerably relaxed. The beta function of the U(1) X gauge coupling constant depends on the charge assignment of particles in the visible and mirror sectors. Let us first assume that the U(1) X

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charges of the MSSM particles and the mirror particles are a linear combination of U(1) Y and U(1) B−L charges, so that the gauge anomaly is cancelled solely by introducing the right-handed neutrinos, (2.8) Then the beta function of the U(1) X gauge coupling constant is given by d dlnµ The scale of the Landau pole is maximized when x = −1/2, which we assume in the following. In this case, b X = −14. For fraternal Twin Higgs models [26], where the mirror of the first and the second generations are not introduced, b X = −10.
Denoting the mass of the U(1) X gauge boson as m X , the scale of the Landau pole M c is given by We expect that the Twin Higgs theory has a UV completion at the scale M c . 4 We require that M c is larger than the mediation scale of the SUSY breaking which we assume throughout the article to be Λ = 100m stop , where m stop is the soft mass of stops. In order to avoid the experimental constraints on m X , to be discussed later, the mass of X is typically expected to be a factor of between 5 to 10 larger than the stop masses. This requires M c 10m X which sets an upper bound on g X (m X ) of about 1.6 (1.9) for the mirror (fraternal) Twin Higgs model.
The constraint is relaxed if the U(1) X charge is flavor dependent. For example, it is possible that the first and the second generation fermions are U(1) X neutral, and their yukawa couplings are generated via mixing between these fermions and heavy U(1) X charged fermions. Then the renormalization group (RG) running of the U(1) X gauge coupling constant is significant only above the masses of those heavy fermions, and below those mass scales b X = −6, which allows values of g X (m X ) up to about 2.4 if one requires M c 10m X . In this type of models, the experimental lower bound on m X which is discussed later is also significantly relaxed. Throughout this paper we refer to this class of models as flavor non-universal SUSY D-term Twin Higgs models. Such a construction is also motivated by the observed hierarchy of fermions masses and explains why the SM fermions of the third generation are much heavier than those of the first two generations. Nevertheless, to also explain the observed hierarchy among the first two generations of the SM fermions ala Froggatt-Nielsen [42], additional horizontal symmetry would be required, see e.g. refs. [43][44][45][46][47][48] for the ideas of SUSY model building in this direction and its relation to possible solutions of the SUSY flavor problem.
Before going to a disscussion of full SUSY Twin Higgs models it is instructive to discuss general effective theory with heavy MSSM-like Higgs doublets and other states decoupled. In such a case the Higgs potential depends only on the SM-like Higgs and its mirror partner: (3.1) The first two terms are both Z 2 and SU(4) symmetric, ∆λ preserves Z 2 but breaks SU(4), while ∆m 2 breaks both Z 2 and SU(4) symmetry. One could also consider a hard Z 2 breaking quartic term which in our setup is subdominant, see ref. [8] for discussion of effects of hard Z 2 breaking. The vevs of the Higgs fields and the masses of them are given by The above formulae are independent of whether the UV completion is supersymmetric or not. In SUSY models the SU(4) symmetry is generically broken at tree level by the EW D-term potential of eq. (2.3) which in the above framework corresponds to Note that ∆λ SUSY grows as a function of tan β from zero (for tan β = 1) up to 0.07 in the large tan β limit. Thus for lower tan β the observed Higgs mass gives a stronger lower bound on masses of stops which dominate the radiative corrections to the Higgs mass. Let us first discuss the Higgs mass at the tree level. In the limit of an exact Z 2 symmetry and a large SU(4) preserving quartic coupling, λ ∆λ, the tree-level Higgs mass is the same as in MSSM. However, in phenomenologically viable models the Z 2 symmetry must be broken. Moreover, corrections to the Higgs mass of order O(∆λ/λ) are often non-negligible in realistic SUSY Twin Higgs models. After taking these effects into account the tree-level Higgs mass in SUSY Twin Higgs models is approximately given by where the first term is the effect of Z 2 breaking while the second term corresponds to the correction of order O(∆λ/λ), which is negative, and f 2 ≡ v 2 + v 2 . We see that in the limit v f and λ ∆λ the tree-level Higgs mass is enhanced by a factor of √ 2 with respect to the MSSM Higgs mass which in large tan β limit turns out to be very close to the observed Higgs mass of 125 GeV. This is another virtue of SUSY Twin Higgs models. While large hierarchy between v and f , which introduces the fine-tuning of is not preferred from the view point of naturalness, the ratio f /v above about two or three, which is required by the Higgs coupling measurements, leads to a significant boost of the tree-level Higgs mass. Terms O(∆λ/λ) also reduce the Higgs mass and in the limit of λ ∆λ, in which the SU(4) symmetry is not even approximately realized, the Higgs mass is the same as that in the MSSM.
Due to the large value of the top Yukawa couplings the quantum correction by the top and the stop significantly affect the Higgs mass. We take into account the quantum correction by computing the Coleman-Weinberg (CW) potential of the Higgs fields. We include contributions from top and stop from both visible and mirror sectors. In reliable prediction of the Higgs mass proper choice of a renormalization scale for the top Yukawa coupling y t is crucial since the correction to the Higgs mass is proportional to y 4 t . It is well known that in MSSM the Higgs mass calculated at one loop level grossly overestimates the full result if y t at the top mass scale is used, see e.g. refs. [50][51][52]. In ref. [52] it was shown that the dominant two-loop effects in the computation of the Higgs mass can be accommodated by using in the one loop result the RG running top mass at a scale µ t ≡ √ m t m stop . Since the RG running at one loop in the visible and mirror sector is independent from each other, we expect that using in the CW potential y t matched to the top mass at a scale µ t will also accommodate the leading two-loop corrections. Therefore, in our calculations we adopt the RG-improved procedure of ref. [52] using their formulae with m t (m t ) = 165 GeV.
Since we do not include corrections other than that from top/stop loops, some nonnegligible theoretical uncertainties may still be present even after the RG improvement. We estimate this uncertainty by comparing our result in the limit f = v/ √ 2 and λ ∆λ, in which the MSSM Higgs mass should be recovered, with SOFTSUSY [53] computation of the MSSM Higgs mass for a degenerate sparticle spectrum (but with heavy MSSM-like Higgs decoupled) and find that our procedure still overestimates the Higgs mass by about 5 (3) GeV for the stop masses of 1 TeV (400 GeV). These numbers are in good agreement with findings of ref. [52]. In Twin Higgs model this overestimation may be even larger expecially for m stop f , because in such a case also the mirror stop contributes substantially to the Higgs mass. On top of that, there are additional contributions to the Higgs mass arising from mass splitings in sparticle spectrum, which are unavoidable given strong LHC bounds on the gluino mass, that typically result in further reduction of the Higgs mass in MSSM. On the other hand, the Higgs mass may be enhanced by few GeV by stop mixing effects (not included in our computation) with only a minor increase in tuning caused by the stop sector. 5 Having all of the above in mind we substract 5 GeV from the Higgs mass obtained using the above procedure and assume theoretical uncertainty of 3 GeV.
In the left panel of figure 1 the region preferred by the measured Higgs mass is presented in the plane m stop -tan β. It is clear from this plot that much lighter stops are sufficient to satisfy the Higgs mass constraint than in the MSSM even for a SU(4) preserving quartic coupling of similar size as the one from the SU(4) breaking EW D-term. In particular, a lower bound on tan β is much weaker but it should be emphasized that values of tan β 3 cannot accommodate the measured Higgs mass for sub-TeV stops even for large λ. The preferred range of stop masses does not depend strongly on f /v as long as it is above about 2.5, i.e. in a region preferred by the Higgs coupling measurements, as seen from the right panel of figure 1.
The Higgs mass larger than the MSSM one also results in a rather strong upper bound on stop masses for large tan β. In fact in the limit of large tan β and large λ for f = 3v the stops must be lighter than about 400 GeV, as seen in figure 1. While 400 GeV stops may be still consistent with the LHC constraints if the LSP mass is heavier than about 300 GeV [56,57], the 400 GeV left-handed sbottom (which has a similar mass to the lefthanded stop) is already excluded by the LHC [58]. The upper bound on the stop/sbottom masses may be relaxed to about 600 GeV for f = 2.3v -the smallest value of f consistent with the data [8], which may evade the current constraints if the LSP mass is above about 500 GeV. The upper bound may be further relaxed if non-decoupling effects of the remaining scalars are important but one should keep in mind that generically the LHC has the ability to set an upper bound on tan β.
In SUSY UV completions one generally expects that λ depends on tan β. This is the case in models where the SU(4) invariant quartic term is generated from F -term as well as in the case of D-term generated λ that we propose in the present paper. In figure 2 we plot the Higgs mass in the plane tan β-λ for several values of the stop masses. As expected from the previous discussion, the lighter stops are the larger tan β and λ are preferred by the Higgs mass constraint. In figure 2 we also present maximal values of λ as a function of tan β in the F -term and D-term Twin Higgs models under assumption that the Landau pole scale is at least ten times larger than the singlet mass (the X gauge boson mass) in the F -term (D-term) models. We see that for m stop up to 1 TeV the maximal value of λ is definitely larger in the D-term Twin Higgs models than in the F -term one, especially in the flavor non-universal version of the former, so the improvement in tuning as compared to non-twinned SUSY models is better in the D-term model. Larger λ in the F -term model than the D-term one may be obtained for m stop = 2 TeV but is not large enough to prevent tuning from the stop sector which in the leading-log approximation given by where Λ is the messenger scale that we take to be 100m stop . Figure 2 emphasizes that in the F -term model it is hard to saturate the irreducible tuning even in the decoupling limit. Moreover, this situation gets much worse after taking into account non-decoupling effects, as we discuss in the next section. On the other hand, the D-term model can saturate this tuning even if f /v is as small as 2.3 which corresponds to ∆ v/f ≈ 1.6 so essentially no tuning exists at all. In the D-term case non-decoupling effects are much less important than in the F -term one and the total tuning can be O(10) % as we show in the next section.

F -term vs D-term Twin Higgs beyond decoupling limit
In this section we give a more detailed analysis of F -term and D-term Twin Higgs models, going beyond the decoupling limit. We quantify the degree of fine-tuning by introducing the measure, Here x i (Λ) are the parameters of the theory evaluated at the mediation scale of the SUSY breaking.
To evaluate ∆ f we solve the renormalization group equations (RGEs) of parameters between m stop and Λ at the one-loop level.

F -term Twin Higgs model
It was already noted in ref. [7] that fine-tuning is not minimized for the maximal value of λ S that avoids the Landau pole because in such a case the tuning from large soft singlet mass dominates that from stops. Instead, the fine-tuning is minimized for some intermediate value of λ S in the range between 1 and 1.5 which results in ∆ v ∼ 50 ÷ 100, i.e. 1 ÷ 2% fine-tuning [7]. This result for fine-tuning in the F -term Twin Higgs model was obtained for µ = 500 GeV, m S = 1 TeV and m stop = 2 TeV and was confirmed recently in ref. [8]. However, we find that the fine-tuning in the F -term Twin Higgs model is even more severe due to the Higgs mixing with the singlet which gives a negative contribution to the Higgs mass. The Higgs-singlet mixing is proportional to λ S vµ. For large λ S (which is crucial in the Twin Higgs mechanism) and moderate values of µ (which naturally is close to f ) the mixing is sizable and cannot be neglected in the Higgs mass calculation for the singlet mass of 1 TeV. This is demonstrated in figure 3 from which it is clear that the correct Higgs mass requires, for µ = 500 GeV, the singlet mass of at least 2 TeV, while for m S = 1 TeV the Higgs direction turns out to be tachyonic. However, for values of m S above 2 TeV the fine-tuning from the heavy singlet dominates the one from stops. In consequence, the finetuning is worse than 1%. The problem is less severe for smaller values of µ which, however, is constrained from below, µ 100 GeV by null results of the LEP chargino searches [49]. It can be seen from the right panel of figure 3 that for µ = 100 GeV there is small impact of the 1 TeV singlet on the Higgs mass and the fine-tuning can still be at the level of 1 ÷ 2%. Note that for µ = 100 GeV, m S can be very small without making the Higgs direction tachyonic because for small m S the physical singlet mass is dominated by that given by the mirror Higgs vev, λ S f . 6 The Higgs-singlet mixing may be reduced if a non-vanishing singlet A-term of the form A λ λH u H d S is introduced. In such a case this mixing can be suppressed for any value of µ if A λ ≈ µ tan β. However, if µ is not small this implies large 6 The presence of the singlet also modifies the Higgs couplings but in the regions of parameter space consistent with the observed Higgs mass the singlet component of the SM-like Higgs is at most few percent so the constraints from the Higgs coupling measurements are easily satisfied. The Hu and H d components of the singlet-like state are also at most few percent so there are no meaningful constraints from direct LHC searches for additional scalars. A λ which again generates large fine-tuning. We conclude that in the F -term Twin Higgs model fine-tuning at least at the level of few percent is required, and a very light higgsino is a signature of the smallest fine-tuning, similarly as in the MSSM, in spite of the Twin Higgs mechanism.

D-term Twin Higgs model
In the D-term Twin Higgs model there are no effects that significantly affect the prediction for the Higgs mass in the decoupling limit analysed in section 3 so the Higgs mass is determined by the value of λ, tan β and f /v. Let us know discuss fine-tuning in this model and show that it is significantly better than the one in the F -term Twin Higgs model. Apart from the usual tuning from stops, the tuning may also arise from a threshold correction to the soft Higgs mass which is proportional to a new gauge boson mass squared: It is important to note that, in contrast to the F -term model, this correction does not depend on the cut-off scale. However, it does depend on a model-dependent parameter which characterizes the size of the mass splitting in the vector supermultiplet. The same parameter enters the effective SU(4)-preserving quartic coupling: Therefore, small values of are preferred to maximize the SU(4)-preserving quartic term but this enhances the threshold correction to the soft Higgs mass of eq. (4.2). There is a

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lower bound on the size of this correction which comes from searches for additional U(1) gauge bosons. For large values of g X the most stringent constraint comes from searches for off-shell production of the X boson in dimuon final states at LEP which gives a lower bound of m X 4350 GeV ×g X [36]. 7 Since the limit is stronger for larger g X the fine-tuning is not necessarily smaller for larger g X .
In order to minimize fine-tuning we demand that the fine-tuning due to the threshold correction of eq. (4.2) does not exceed the fine-tuning due to SUSY particles (dominated by stops, higgsino and gluino) For a given value of g X and m stop , as well as gluino and higgsino masses, the fine-tuning is minimized for the smallest value of m X allowed by experiments and chosen such that the inequality in eq. (4.4) is saturated. The optimal value of 2 is O(0.1) among most of the parameter space. 8 Such values of do not require unusual hierarchies in underlying model, see appendix for details where we present an explicit model generating a non-decoupling D-term potential and show that other possible sources of tuning are less important than those included in our numerical analysis. We show the resulting contours of fine-tuning in the plane m stop -g X in the left panels of figure 4. We find that the effect of the U(1) X gauge coupling constant on the RG running of the yukawa coupling is important, as it reduces the top yukawa coupling at higher energy scales. As a result fine-tuning tends to be better for larger g X despite of larger m X . This is another advantage with respect to the F -term model, where the singlet effects tend to increase the top Yukawa coupling at higher energy scales. The magnitude of tuning is different between the mirror and fraternal Twin Higgs models because the RG running of the U(1) X gauge coupling constant is faster in the latter case so for a given Landau pole scale g X must be smaller. The top left panel of figure 4 shows that in the mirror model with tanβ = 10, ∆ v can be as small as about 8 for the stop masses up to about 1.2 TeV which corresponds to ∆ f 2 i.e. essentially no finetuning in f . Moderate tuning of the EW scale of 10 % can be obtained for stops as heavy as about 1.4 TeV. In the fraternal model the same level of fine-tuning as in the mirror model may be obtained for stops heavier by few hundred GeV, as seen from the bottom left panel of figure 4. For both mirror and fraternal models the tuning is dominated by the higgsino for the stop masses below about 1 TeV. This is because we set µ = 500 GeV to evade the LHC constraints on sub-TeV sbottoms [58]. Therefore, the current constraints on the stop/sbottom masses [56,57] do not introduce fine-tuning in f in the D-term model. It may be even possible to have only moderate tuning of O(5÷10) % even if there is no sign of stops/sbottoms at the end of the high-luminosity run of the LHC. A useful reference point to compare is the non-twinned version of the MSSM with non-decoupling U(1) D-term in 7 The LHC constraints on mX are becoming competitive with the LEP one, especially for smaller values of gX . However, we found that for gX 1 the recent LHC constraints [59] are still weaker than the LEP one. 8 For large stop masses 2 is much smaller than 0.1 but the tuning would not be significantly different if = 0.1 is taken instead because for such value of the SU(4) invariant coupling is already close to maximal, see eq. (4.3).  In the left panels, where tan β = 10, the orange contours depict the value of the SU(4) preserving quartic coupling and in the green regions the Landau pole of the U(1) X gauge coupling constant is below Λ. In the right panels, at each point of the plane m stop -tan β, g X is fixed to the maximal value that allows the messenger scale to be below the Landau pole. In the blue region the Higgs mass is in agreement with the measured value and several blue contours of the Higgs mass are also shown. In the top (bottom) panels mirror (fraternal) Twin Higgs model is assumed.

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which tuning is worse by a factor 2λ/λ SM . We see from figure 4 that in the D-term mirror (fraternal) Twin Higgs model the tuning may be smaller by a factor of about 3 (4).
It should be noted, however, that for tan β = 10 the Higgs mass constraint prefers rather light stops in the range between about 500 and 700 GeV because heavier stops overshoot the experimental value of the Higgs mass. Such light stops (and sbottoms) are not in conflict with the current LHC results for the higgsino mass of 500 GeV that we use in our calculation of fine-tuning but this indicates that they may be discovered relatively soon. One can imagine extensions of the present model in which there are negative contributions to the Higgs mass that may lift the stop masses required to obtain the 125 GeV Higgs without altering tuning of the model. For example, one may consider an extension of the D-term Twin Higgs model by a singlet that couples in the superpotential only to the visible Higgs bosons in the NMSSM-like way i.e. λ S SH u H d . Such coupling generates a mixing between a singlet and the Higgs. The mixing effects between a relatively light singlet, say below 1 TeV, and the Higgs can provide a necessary reduction of the Higgs mass without reduction of the SU(4) preserving quartic term. 9 We leave a detailed analysis of such a model for future work but we expect that the presence of such singlet does not introduce additional tuning. For example, the reduction of the Higgs mass by 15 GeV for the singlet mass of 500 GeV and µ = 500 GeV, as used in our numerical example, requires λ S ≈ 0.3 which results in a subdominant correction to the soft Higgs mass from the singlet.
Alternatively, one can obtain the correct Higgs mass for heavier stops without extending the D-term model by reducing tan β, as seen from the right panels of figure 4. Smaller tan β reduces the tree-level Higgs mass but it also reduces the SU(4) preserving quartic couplings, so for given stop masses the tuning gets slightly worse for smaller tan β. In order to increase the stop masses up to 1 (2) TeV consistently with the 125 GeV Higgs mass, tan β must be reduced to about 4 (3) which increases tuning by only about 20 (50) % as compared to the tan β = 10 case. In consequence, even after taking the Higgs mass constraint into account tuning better than 10 % can be obtained for the stop masses up to about 1.2 (1.3) TeV in the mirror (fraternal) case.
Let us also discuss the flavor non-universal model in which the first two generations of SM and mirror fermions are neutral under U(1) X . In such a case the X gauge boson production at colliders is strongly suppressed and there is no relevant constraint on m X . In addition, below the scale of U(1) X charged fermions masses (which we reasonably assume to be at least two orders of magnitude above m stop ), the RG running of g X is slower. As a result larger values of g X may be obtained and the tuning may be further relaxed. We see from the left panel of figure 5 that in this case for tan β = 10 the tuning better than 25 % can be obtained even for the stop masses around 1 TeV, while for 2 TeV stops the tuning may be better than 10 %. Since λ can be as large as about 0.5, the tuning may be relaxed by a factor of about 8 with respect to a non-twinned model with a nondecoupling D-term. Similarly as in the previous cases, the Higgs mass is overshot unless some negative contribution to the Higgs mass is introduced. Nevertheless, even if no such

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the discussion on H below. The mass formula for the mirror Higgs h is the same in both models but for a given f in the D-term model one expects it to be somewhat heavier due to a larger achievable value of the SU(4) preserving quartic coupling. In the least-tuned region with f 3v and λ ∼ 0.3 ÷ 0.5 the mirror Higgs mass in the D-term model is in the range of 500 to 700 GeV.
Mirror CP-even heavy Higgs H . The main difference between the D-term and Fterm models is that in the former one there is large mass degeneracy between H and H , see u,d . This should be contrasted with the F -term model, where A mixes with A through the quartic coupling generated by the F term of the singlet. For this reason we do not discuss the phenomenology of A , but we note that there may be a mixing if one introduce additional interactions, e.g. a superpotential coupling with a singlet field like in the F -term model.
Mirror charged Higgs H ± . H ± does not mix with any SM particles as long as the U(1) EM symmetry is unbroken.

Conclusions
We proposed a new SUSY UV completion of the Twin Higgs model in which the SU(4) invariant quartic term λ is provided by a D-term potential from a new U(1) X gauge symmetry. In this setup λ is maximized at large tan β, which makes it possible to accommodate the 125 GeV Higgs mass simultaneously with the value of λ as large as about 0.5, and to greatly relax tuning of the EW scale. We found that the current LHC constraints can be satisfied with tuning better than 20 %, while 2 TeV stops, which would be beyond the reach of the LHC, may imply only moderate tuning of about 10 %. This should be compared with the model in which λ originates from an F -term of a new singlet that results in the tuning of 2 % at best.

JHEP06(2017)065
We also discussed implications of the measured Higgs mass on the stop mass scale in a general SUSY Twin Higgs model in which the only source of the tree level SU(4) breaking quartic term is the EW D-term potential. In particular, we found that in the large tan β limit of such models the Higgs mass is larger than the measured value unless the stops are lighter than about 500 GeV. This region is already quite constrained by the LHC experiments and not too light LSP is required to evade the bounds. These findings are especially interesting in the context of the D-term Twin Higgs model proposed in this article, in which the least tuned region has large tan β and is expected to be covered by the LHC stop/sbottom searches in the near future. Nevertheless, light stops are not a firm prediction of this model since the 125 GeV Higgs mass can be obtained for heavier stops with smaller tan β. For example, 1 (2) TeV stops require tan β of about 4 (3) which makes tuning worse only by about 30 (60) % as compared to the large tan β result. Alternatively, the correct Higgs mass for heavy stops and large tan β can be obtained by introducing a negative contribution to the Higgs mass which can originate, for example, from a mixing with a non-decoupled singlet.
If all the sparticles are pushed above the LHC reach, the only way to probe Twin Higgs models is via searches for additional Higgs bosons. We identified several differences in the heavy Higgs spectrum between the D-term and F -term model, which may help to distinguish them if several new scalars are found in the future.

A D-term potential and correction to Higgs soft masses
In this appendix we discuss a model to break the U(1) X gauge symmetry, and the resulting D term potential of the Higgs doublets as well the soft masses of them. We introduce chiral multiplets Z, P andP , whose U(1) charges are 0, +q and −q, respectively, and the following superpotential, Otherwise, the asymmetric VEVs of P andP give large soft masses to the Higgs doublets through the D-term potential. The VEVs of P andP are given by The mass of the U(1) X gauge boson is given by In the SUSY limit, m 2 P κ 2 M 2 , the D term potential of the U(1) X charged particles vanishes after integrating out P andP . In fact, after integrating out the scalar components of P andP , we obtain the D term potential of the Higgs doublets, It can be seen that V D vanishes for m 2 P = 0. From the above we determine the value of 2 introduced in eq. (2.6): We see that ∼ O(0.1) does not require m P much larger than m X . Although the RG running of the Higgs doublets from P andP vanishes due to the identical soft masses for P andP , the threshold correction around the U(1) X symmetry breaking scale necessarily gives the correction to the Higgs doublets. At the one-loop level, we find Here we assume that the SUSY breaking contribution to the gaugino mass and the soft masses of Higgs doublets are negligible. The results in eqs. (A.5) and (A.7) are consistent with the model-independent discussion in ref. [36]. Let us comment on the possible sources of the fine-tuning in addition to ∆ f due to the threshold correction. In the above analysis we assume that the soft masses of P andP are identical. This can be guaranteed by a C symmetry P ↔P in the sector which generates the soft masses. The symmetry is explicitly broken by the SM SU(2) × U(1) gauge interaction and the yukawa interactions. Once the soft masses of P andP are different with each other, there are extra contributions to m 2 Hu , so the tuning ∆ f may become worse. Among the source of the breaking of the symmetry P ↔P , the top yukawa coupling is the largest, and we expect that the following magnitude of the soft mass difference is unavoidable, This results in the asymmetric VEVs of P andP , P 2 − P 2 −∆m 2 −m 2 P + κ 2 M 2 2g 2 X κ 2 q 2 M 2 − 2g 2 X q 2 m 2 P + κ 2 m 2 P = − 2g 2 X q 2 ∆m 2 , (A.9) -19 -