Minimal Non-Abelian Supersymmetric Twin Higgs
Marcin Badziak Keisuke Harigaya
hep-ph




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    diag Spin SOØ O SU U Sp SL trM_ PlInt. J. Mod. Phys. Mod. Phys. Lett. Nucl. Phys. Phys. Lett. Phys. Rev. Phys. Rev. Lett. Prog. Theor. Phys. Z. Phys. df BHinf inf evap eq smM_ Pl GeVm_ stop 0.7cm
     1.35cm
    
     Minimal Non-Abelian Supersymmetric Twin Higgs 1.2cm
    Marcin Badziak^1,2,3 and Keisuke Harigaya^2,3 0.4cm
    ^1 Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL–02–093 Warsaw, Poland
    ^2 Department of Physics, University of California, Berkeley, California 94720, USA
    ^3 Theoretical Physics Group,  Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 1.5cm
    
    
    We propose a minimal supersymmetric Twin Higgs model that can accommodate tuning of the electroweak scale for heavy stops better than 10% with high mediation scales of supersymmetry breaking. A crucial ingredient of this model is a new SU(2)_X gauge symmetry which provides a D-term potential that generates a large SU(4) invariant coupling for the Higgs sector and only small set of particles charged under SU(2)_X, which allows the model to be perturbative around the Planck scale. The new gauge interaction drives the top yukawa coupling small at higher energy scales, which also reduces the tuning.
    
    
    
    
    § INTRODUCTION
    
    
    Supersymmetry (SUSY) provides one of the most promising solutions to the hierarchy problem of the Standard Model (SM) <cit.>. However, the lack of finding of SUSY partners casts serious doubts on whether SUSY can still naturally explain the electroweak (EW) scale. Fine-tuning of the EW scale in minimal SUSY models implied by the LHC searches was recently quantified in refs. <cit.>, which demonstrated that the current limits on stop and gluino masses exclude regions with fine-tuning better than 10%, even if a very low mediation scale of the SUSY breaking of 100 TeV is assumed.[The fine-tuning may be improved if the higgsino mass is not tied to the Higgs mass squared, see e.g. refs. <cit.>. In such a case higgsino could be heavier leading to compressed spectra for which the lower bounds on stops, and gluino are much weaker.]  The fine-tuning quickly 
    gets worse for larger mediation scales due to longer RG running of the soft Higgs mass.
     This is  indication of the little hierarchy problem.
    
    A possible remedy to  the little hierarchy problem is offered by Twin Higgs mechanism <cit.>. In the scenario, the Higgs is a pseudo-Nambu-Goldstone boson of a global SU(4) symmetry emerging from ℤ_2 symmetry exchanging the SM with its mirror (or twin) copy.
    We refer to <cit.> for composite Twin Higgs models, and <cit.> for cosmological aspects of Twin Higgs scenario. 
    
    Early realisations of  SUSY UV completion of Twin Higgs scenario <cit.>, which generate an SU(4) invariant quartic term with an F-term potential of a heavy singlet superfield, are not able to significantly reduce fine-tuning as compared to non-Twin SUSY models <cit.>. 
    It was only very recently that SUSY Twin Higgs models were proposed in which tuning at the level of 10% is possible
    by  introducing either hard ℤ_2 symmetry breaking in the F-term model <cit.> or a new U(1)_X gauge symmetry whose D-term potential provides a large SU(4) invariant quartic term <cit.>. It should be, however, emphasised that the tuning at the level of 10% can be obtained in these models only for a low mediation scale or a low Landau pole scale.
    In the F-term model of ref. <cit.>
    a fine-tuning penalty for a larger mediation scale and hence a longer RG running is severe because the large SU(4) invariant coupling induces growth of the top yukawa coupling at higher energy scales.
    In the D-term model the RG effect of the gauge coupling g_X of the new interaction is to reduce the top yukawa coupling, and the effect of a higher mediation scale is not as severe as the one for the F-term model. However, the RG running of the U(1)_X gauge coupling is fast 
    and hence the Landau pole scale of g_X is as low as 10^5-10^6 GeV for values of g_X that are large enough to guarantee approximate SU(4) symmetry of the Higgs potential.
    While such a low mediation scale or a low Landau pole scale is in principle possible,
    it strongly limits possible schemes of the mediation of the SUSY breaking and UV completions above the Landau pole scale.
    
    In the present work, we point out that the Landau pole scale and the mediation scale of the D-term model can be much higher 
    if the SU(4) invariant term is generated by a D-term potential of a new non-abelian gauge symmetry. We construct a consistent model with SU(2)_X
    gauge symmetry with small number of flavors charged under this symmetry.
    The new gauge interaction drives the top yukawa coupling small at higher energy scales, which also helps obtain the EW scale more naturally.
    As a result, the tuning of the EW scale for 2 TeV stops and gluino can be at
    the level
    of 5-10% for mediation scales as high as 10^9-10^13 GeV.
    One can keep perturbativity up to around the Planck scale with tuning better than 5% (for low mediation scales). 
    The model allows for moderate tuning better than few percent with the mediation scale around the Planck scale.
    If the gluino mass is a Dirac one, the tuning may be as good as 10%, which realizes a natural SUSY with a gravity mediation.
    
    
    
    
    § A SUSY D-TERM TWIN HIGGS WITH AN SU(2) GAUGE SYMMETRY
    
    
    In this section we present a SUSY D-term Twin Higgs model <cit.> where the D-term potential of a new SU(2)_X gauge symmetry
    generates the SU(4) invariant quartic coupling.
    We assume a ℤ_2 symmetry exchanging the SM with its mirror copy, and denote mirror objects with supersctripts '.
    
    
    The matter content of the model is shown in Table <ref>.
    In addition to the SU(3)_c× SU(2)_L× U(1)_Y gauge symmetry and its mirror counterpart, we introduce an SU(2)_X gauge symmetry which is neutral under the ℤ_2 symmetry.
    We embed an up-type Higgs H_u into a bi-fundamental of SU(2)_L× SU(2)_X, H, and its mirror partner H_u' into that of SU(2)'_L× SU(2)_X, H'. 
    As we will see later, the D-term potential of SU(2)_X is responsible for the SU(4) invariant quartic coupling of H_u and H_u'.
    The SU(2)_X symmetry is broken by the vacuum expectation value (VEV) of a pair of SU(2)_X fundamental S and S̅.
    Except for S and S̅ all matter fields have their mirror partner.
    
    The right-handed top quark is embedded into Q̅_R and allow for a large enough top yukawa coupling through the superpotential term HQ̅_R Q_3, where Q_3 is the third generation quark doublet. E̅ is necessary in order to cancel the U(1)_Y`-SU(2)_X^2 anomaly.
    The VEV of ϕ_u is responsible for the masses of the up and charm quarks.
    Q_1,2,3, u̅_1,2, e̅_1,2,3, d̅_1,2,3 and L_1,2,3 are usual MSSM fields.
    To cancel the gauge anomaly of 
    SU(3)_c^2 `-U(1)_Y and U(1)_Y^3
    originating from the extra up-type right handed quark in Q̅_R and two extra right-handed leptons in E̅, we introduce U and E_1,2.
    There are three up-type Higgses in H and ϕ_u, so we need to introduce three down-type Higgsses ϕ_d1,2,3.
    Their VEVs are responsible for the masses of down-type quarks and charged leptons.
    
    
    
    
     §.§ SU(2)_X symmetry breaking
    
    We introduce a singlet chiral field Z and the superpotential coupling
    
    
        W = κ Z (S S̅ - M^2).
    
    We assume that the soft masses of S and S̅ are the same,
    
    
        V_ soft = m_S^2 (|S|^2 + |S̅|^2).
    
    Otherwise, the magnitude of the VEVs of S and S̅ are different from each other, and give large soft masses to the Higgs doublets through the D-term potential. The VEVs of S and S̅ are given by
    
    
        S =
        [   0; v_S ], S̅ =
        [ v_S;   0 ],  v_S = √(M^2 - m_S^2 / κ^2).
    
    The constraint on the T (ρ) parameter requires that v_S ≳2.9 TeV in the limit of large tanβ and neglecting the effect of mixing between the SM and the mirror Higgses, see Appendix <ref> for a derivation of this constraint and more precise formula.
    The masses of the SU(2)_X gauge bosons are given by
    
    
        m_X^2 = g_X^2 v_S^2.
    
    
    After integrating out massive particles with a mass as large as v_S, the potential of H and H' is given by
    
    
        1/8g_X^2 ∑_i=1,2,3(  H^†σ^i  H +  H'^†σ^i  H')^2  ( 1 - ϵ^2 ),
        ϵ^2 = m_X^2/2m_S^2 + m_X^2.
    
    In the SUSY limit, m_S^2 =0, the D-term potential vanishes.
    In terms of the model parameters M,m_S, κ, g_X, ϵ^2 is given by
    
    
        ϵ^2 = g_X^2 (m_S^2 - κ^2 M^2)/g_X^2 (m_S^2 - κ^2 M^2) - 2 κ^2 m_S^2
    
    In the limit where κ≪ g_X, ϵ^2=1 and hence the D-term potential decouples.
    In order to obtain a large D-term potential, it is preferable that κ is as large g_X.
    
    To estimate the maximal possible value of κ, we solve the renormalization group equation of g_X and κ,
    
    
        d/ dlnμ g_X =   g_X^3/16π^2 1 + 21/16π^2 g_X^2 - 1/8π^2κ^2/1-g_X^2/4π^2,
         d/ dlnμκ =   κ/16π^2 (4 κ^2 - 3 g_X^2),
    
    from a high energy scale M_* towards low energy scales, with a boundary condition at M_* of g_X = κ≃ 2π.
    M_* can be identified with the Landau pole scale. 
    The running of g_X and κ is shown in Fig. <ref>, which shows that κ≃ g_X much below M_*.
    We obtain the same conclusion as long as κ(M_*)  1.
    For κ≃ g_X, ϵ^2 is
    
    
        ϵ^2 ≃g_X^2 M^2 - m_S^2/g_X^2 M^2 + m_S^2
    
    We may obtain a sufficiently small ϵ^2, say ϵ^2  0.2, for m_S^2  0.6 g_X^2 M^2.
    
    Notice also that for ϵ^2<1 there is a threshold correction to the soft Higgs mass which is proportional to
    a new gauge bosons mass squared:
    
        (δ m_H_u^2)_X= 3 g_X^2/64 π^2 m_X^2 ln(ϵ^-2)  ,
    
    which may be a source of tuning of the EW scale. The same threshold correction is present also for the right-handed stop soft mass squared m_U_3^2. 
    
    
    
    
     §.§ SU(4) invariant quartic coupling and μ terms
    
    
    We give masses to H = (H_1,H_2)^T and ϕ_u by pairing them with ϕ_d,1,2,3 through the superpotential terms,
    
    
        W = λ_1 ϕ_d,1 H S + λ_2 ϕ_d,2 HS̅ + m ϕ_u ϕ_d,3
    
    The pairs (H_1,ϕ_d,1), (H_2,ϕ_d,2) and (ϕ_u,ϕ_d,3) obtain masses of λ_1 v_S, λ_2 v_S and m, respectively.
    We assume that λ_2 v_S, m  1 TeV and neglect (H_2,ϕ_d,2) and (ϕ_u,ϕ_d,3) for the dynamics of the electroweak symmetry breaking.
    We identify H_1 and ϕ_d,1 with H_u and H_d in the Higgs sector of the standard SUSY model.
    The μ parameter is given by μ=λ_1 v_S. 
    The SU(4) invariant quartic coupling of (H_u,H_u') is given by
    
    
        V =g_X^2/8 (1-ϵ^2) (|H_u|^2 + |H_u'|^2)^2.
    
    
    As we will see, the VEV of ϕ_u is responsible for the masses of the up and charm quarks, and the neutrinos.
    To give a VEV to ϕ_u, we introduce a coupling
    
    
        W = δ m ϕ_u ϕ_d,1.
    
    Through the F term potential of ϕ_d,1, ϕ_u obtains a tadpole term after H_u obtains its VEV, which induces a non-zero VEV of ϕ_u.
    
    Through the coupling λ_2 (> λ_1), m_H_u^2 receive a quantum correction from m_S^2,
    
    
        Δ m_H_u^2  ≃ - λ_2^2/8π^2m_S^2 L = - (600  GeV)^2 ( λ_2/0.3)^2 m_S^2/(6  TeV)^2L/ ln10^4
    
    where L denotes a log-enhancement through an RGE.
    As long as λ_2 0.4, this contribution is always smaller than that from stops and/or the threshold correction from X, and hence we
    neglect it. Note, however, that even larger values of λ_2 may be possible without introducing tuning if the mediation scale of SUSY breaking
    is relatively low and/or m_S^2 runs to smaller values at higher energies.
     
    
    Note that the Z_2 symmetry S ↔S̅ is explicitly broken by the above superpotential couplings.
    Even if we assume the Z_2 symmetry of the soft masses of S and S̅, we expect a quantum correction to a mass difference of them,
    
    
        Δ m_S^2 ≡ m_S^2 - m_S̅^2 ≃λ_2^2/8π^2m_S^2 L.
    
    This leads a asymmetric VEV of S and S̅, which give m_H_u^2 through the D-term potential,
    
    
        m_H_u^2≃ -ϵ^2/2Δ m_S^2,
    
    which is always smaller than the direct one-loop quantum correction in Eq. (<ref>).
    
    
    It is also possible to maintain the Z_2 symmetry.
    Instead of the coupling in Eqs. (<ref>) and (<ref>), we introduce
    
    
        W = H ( λ_1 ϕ_d,1 + λ_3 ϕ_d,3 )( S + S̅) + λ_2  Hϕ_d,2 ( S - S̅)  + ϕ_u ( m_1ϕ_d,1 + m_3 ϕ_d,3)
    
    Here we have assumed that ϕ_d,2 is odd under the Z_2 symmetry.
    After S and S̅ obtain their VEVs, the mass terms become
    
    
        W = v_S (λ_1 ϕ_d,1 + λ_3 ϕ_d,3 ) (H_1 - H_2) + λ_2 v_S ϕ_d,2 (H_1 + H_2) + ϕ_u ( m_1ϕ_d,1 + m_3 ϕ_d,3).
    
    We assume that λ_2 v_S, m_i  1 TeV.
    Then (H_1 + H_2) /√(2) and ϕ_u obtain a large mass paired with  ϕ_d,2 and a linear combination of ϕ_d,1 and ϕ_d,3, respectively, and 
    are irrelevant for the dynamics of the electroweak symmetry breaking.
    H_u ≡ (H_1 - H_2)/√(2) obtains a mass of O(λ_2,3 v_S) paired with another linear combination of ϕ_d,1 and ϕ_d,3 which we call H_d.
    
    
    
    
    
     §.§ Masses of matter particles
    
    
    We first consider a case where the Z_2 symmetry S↔S̅ is explicitly broken.
    A large enough top yukawa coupling is obtained by the superpotential
    
    
        W = y_t  HQ̅_R Q_3 →  y_t  (H_2Q̅_R,1 - H_1 Q̅_R,2) Q_3,
    
    where Q̅_R = ( Q̅_R,1 , Q̅_R,2)^T.
    We give a large mass to Q̅_R,1 by introducing a coupling
    
    
        W = y Q̅_R U S,
    
    and identify Q̅_R,2 with a right-handed top quark u̅_3.
    
    The yukawa couplings of the up and charm quarks originates from the couplings with ϕ_u,
    
    
        W = y_u,ijϕ_u Q_i u̅_̅j̅.
    
    The left-handed neutrino masses are obtained in a similar manner once right-handed neutrinos are introduced.
    The yukawa couplings of the down-type quarks and the charged leptons is given by couplings with ϕ_d,i,
    
    
        W = y_d,ijkϕ_d,i Q_j d̅_k + y_e,ijkϕ_d,i L_j e̅_k.
    
    
    The extra SU(2)_X charged particle E̅ obtains its mass paired with E_1,2 through the SU(2)_X symmetry breaking,
    
    
        W  = E̅ (y_E,1 E_1 + y_E,2 E_2 ) S +  E̅ (y̅_E,1 E_1 + y̅_E,2 E_2 ) S̅.
    
    
    
    Next we consider a case where the Z_2 symmetry is maintained. The top yukawa coupling is obtained by the superpotential
    
    
        W = y_t  HQ̅_R Q_3 →  y_t H_u 1/√(2)(Q̅_R,1 +  Q̅_R,2) Q_3.
    
    One linear combination of Q̅_R,1 and Q̅_R,2 obtains a Dirac mass term paired with U,
    
    
        W = y Q̅_R U (S + S̅) → y v_S 1/√(2) ( Q̅_R,1 - Q̅_R,2 )U.
    
    We identify the massless combination (Q̅_R,1 +Q̅_R,2 )/√(2)≡u̅_3 as a right-handed top quark.
    The extra SU(2)_X charged particle E̅ obtains its mass paired with E_1,2 through the coupling,
    
    
        W  = E̅ (y_E,1 E_1 + y_E,2 E_2 ) S +  E̅ (y_E,1 E_1 - y_E,2 E_2 ) S̅.
    
    Here we assume that E_2 is odd under the Z_2 symmetry S ↔S̅, so that all particles in E̅ and E_1,2 obtains their masses.
    
    So far we have assumed that a linear combination of Q̅_R,1 and Q̅_R,2 obtains a large mass paired with U.
    It is also possible to identify the linear combination with the right-handed charm quark. In such a model U and u̅_2 are not necessary.
    The mass of the right-handed scharm is predicted to be as large as that of the right-handed stop.
    This choice is beneficial for a high mediation scale, as it makes the SU(3)_c and U(1)_Y coupling constants relatively smaller, reducing the fine-tuning from the gluino and the bino.
    
    
    
    
    
    § FINE-TUNING OF THE ELECTROWEAK SCALE
    
    
    
    
    Let us now discuss fine-tuning of the EW scale in the model. We quantify the degree of fine-tuning by introducing the measure <cit.>,
    
    
        Δ_v ≡Δ_f ×Δ_v/f,
    
    where
    
        Δ_v/f = 1/2( f^2/v^2 -2), 
           Δ_f =   max_i ( |∂ ln f^2/∂ ln x_i(Λ)|, 1 ) .
    
    Here f ≡√(v^2 + v^'2) is the decay constant of the spontaneous SU(4) breaking. Δ_v/f measures the fine-tuning to obtain v < f via explicit soft ℤ_2 symmetry breaking. Δ_f measures the fine-tuning to obtain the scale f from the soft SUSY breaking which is analogous to the fine-tuning to obtain the electroweak scale from the soft SUSY breaking in the MSSM.
    x_i(Λ) are the parameters of the theory evaluated at the mediation scale of the SUSY breaking Λ.
    We include the important seven parameters, m_H_u^2, m_Q_3^2, m_u̅_3^2, M_1^2, M_2^2, M_3^2 and μ^2.
    To evaluate Δ_f we solve the renormalization group equations (RGEs) of parameters between  and Λ.
    We assume that the right-handed charm quark is also embedded in Q̅_R.
    Between  and m_X we solve MSSM RGEs at the one-loop level
    appropriately modifying the beta function of m_Q_3^2. 
    At a scale m_X we perform matching by including the threshold correction (<ref>) to m_H_u^2 and m_U_3^2. 
    Above m_X we solve the RGEs (that include the effects of non-MSSM states) at least at the one-loop level. The RGEs of the gauge couplings are solved
    at the two-loop level,
    but set, for simplicity, κ=0.[Non-zero κ slightly slows down the running of g_X but the impact on Δ_v and the scale of  the Landau pole is negligible. ]
    The yukawa couplings other than the top yukawa are neglected.
    
    As clearly seen from eqs. (<ref>)-(<ref>),  for a given value of f there is a lower bound on Δ_v of Δ_v/f. f/v is constrained by the Higgs coupling measurements <cit.> to be at least 2.3<cit.>. The latter value has been obtained neglecting invisible decays of the Higgs to mirror particles, which are generically non-negligible, so in our numerical analysis we use less extremal value of  f=3 v. Nevertheless, the tuning is quite independent of this choice (unless f is so large that Δ_v/f determines Δ_v). 
    
    
    
    
    
    In fig. <ref> we present contours of Δ_v assuming low and high mediation scales of SUSY breaking Λ.
    [In the figure we shade the parameter region where the Landau pole scale of the gauge coupling g_X is above Λ. It is also possible that the SUSY breaking is mediated above the Landau pole scale, but we cannot calculate the fine-tuning measure unless we specify the description of the model above the Landau pole scale.]
    Here and hereafter, the stop mass m_ stop and the gluino mass M_3 refer to the values at the TeV scale.
    For Λ=100 tuning at the level of 10% can be obtained for the stop masses as large as 3 TeV, as seen from the upper left panel. An
    important constraint on the parameter space is provided by the Higgs mass measurement <cit.>. In order to assess the impact of this
    constraint we compute the Higgs mass following closely the procedure described in ref. <cit.>. The blue bands show the parameter region with m_h = 125 ± 3 GeV, where the error is a theoretical one. It can be seen from the upper right
    panel of fig. <ref> that this constraint prefers rather light stop unless tanβ is small enough.  Since we are most interested in stop
    masses that easily avoid current or even potential future LHC constraint we set for the low scale mediation case tanβ=2.5 which  implies the
    stop masses in the range between about 1.5 and 3 TeV. This range narrows to between 1.7 and 2 TeV if one demands tuning better than 10%.
    Interestingly, tuning is minimised for intermediate values of the stop masses which is a consequence of some cancellation between the threshold 
    correction from X and corrections from stops and gluino to m_H_u^2. In this region the value of |m_H_u^2| at the mediation scale is somewhat
    suppressed. 
    For lighter stops (which can be compatible with the Higgs mass constraint for larger tanβ) the tuning is dominated by the threshold correction 
    which implies tuning at the level of few percent. 
    It should be noted that fine-tuning of the EW scale is minimized at some intermediate value of g_X of about 1.5-2 even though perturbativity
    constraint allows for g_X as large as about 2.5. This is because for appropriately large g_X the tuning is dominated by the threshold correction
    to m_H_u^2 from the new gauge bosons. Since the latter must be rather heavy for large g_X due to EW precision constraints, the threshold
    correction dominates for g_X≳2 and the tuning gets worse with increasing g_X in spite of larger SU(4) invariant coupling. In fact, for very
    large value of g_X there is essentially no tuning of the EW scale from stops and gluino but the overall tuning is at the level of few percent. In
    the region of large g_X, where the threshold correction dominates the fine-tuning, larger values of ϵ lead to smaller tuning. On the other
    hand, for smaller g_X, when the threshold correction is subdominant, it is preferred to have smaller ϵ to suppress corrections 
    from stops and gluino by larger SU(4) invariant coupling. 
    
    
    
    It is interesting to compare the fine-tuning of the present model to that in the model where an SU(4) invariant coupling originates from a
    non-decoupling D-term of U(1)_X gauge symmetry proposed in ref. <cit.>. For the stop mass below about 1 TeV, the U(1)_X is less
    tuned with tuning even better than 20%. This is because the threshold correction from the X gauge bosons  in the U(1)_X case is three times
    smaller than in the case of SU(2)_X. As the stop mass increases the tuning in the U(1)_X model gets worse and already for 2 TeV stops the tuning
    in the SU(2)_X model becomes better than in the  U(1)_X model due to larger SU(4) invariant coupling which suppresses the correction from stops.
    
    
    
    The biggest advantage of the SU(2)_X model is that RGE running of g_X is relatively slow so the Landau pole scale, for given g_X, is much higher
    than in the  U(1)_X model. 
    For example in the case of Λ=10^16 GeV presented in the lower panels of fig. <ref>, values of g_X up to about 1.2 are possible
    without the Landau pole below Λ.  In the previously proposed SUSY Twin Higgs models it is was not possible to keep perturbativity up to such
    high scale. We see from fig. <ref> that for Λ=10^16 GeV the fine-tuning  better than few % can be obtained for the stop masses as
    large as 2 TeV. This is obviously worse than in the low-scale mediation case discussed before but for high-scale mediation there are more possible
    mechanisms of the mediation of the SUSY breaking. The fine-tuning is also much better than in the MSSM  with high-scale mediation. This is due to
    suppression of the corrections from stops and gluino (which dominates tuning for high mediation scales) by 
    the SU(4) invariant coupling but also because a large value of g_X efficiently drives the top yukawa coupling to smaller values at higher scales. 
    Dependence of fine-tuning on the mediation scale for 2 TeV stops is presented in fig. <ref>. 
    We see that moderate tuning of few percent can be obtained for high mediation scales. For high mediation scales the tuning is dominated by the
    correction from the gluino so the tuning crucially depends on the gluino mass limits. It was recently emphasised in ref. <cit.>  that
    one should convert running soft masses to pole masses when assessing the impact of experimental constraints on naturalness of SUSY models. It was
    shown that the loop corrections <cit.> from 2 TeV squarks increase the gluino pole mass by 10% as compared to the soft mass.  For
    heavier 1st/2nd generation of squarks, as experimentally preferred, the correction may be much larger e.g. 20% for 10 TeV squarks.
    In the left panel of fig. <ref> we fix the soft gluino mass to 2 TeV which easily satisfies the LHC constraints even for moderate loop
    corrections from squarks <cit.>.
    In such a case, 5% tuning is possible with the mediation scale, being below the Landau pole scale, as high as  𝒪(10^12) GeV. For
    M_3=2.5 TeV, presented in the right panel, for which the gluino is definitely outside of the LHC reach <cit.>, mediation scale of order
    𝒪(10^10) GeV can still allow for better than 5% tuning. Notice also a sharp increase in tuning when the mediation scale approaches the
    Planck scale. This originates from the fact that U(1)_Y gauge coupling constant runs rather fast due to many new states carrying hypercharge and
    eventually enters non-perturbative regime around the Planck scale. In consequence, bino strongly dominates fine-tuning when the mediation scale is
    close to the Landau pole for U(1)_Y.  
    
    The fine-tuning for high mediation scales is even better if the gluino obtains its mass paired with an adjoint chiral superfield by a supersoft
    operator, 
    due to the absence of the log-enhanced correction to m_H_u^2 <cit.>.
    The soft stop mass and the higgs mass are dominantly generated by the threshold correction around the gluino mass,
    
    
        m_ stop^2 ≃   1/16M_3^2, 
        
        m_ H_u^2 ≃   3y_t(M_3)^2 /4π^2 m_ stop lnM_3/m_ stop .
    
    In non-Twin models the fine-tuning may be at a few % level even if the stop mass is as large as 2 TeV, which is further improved by the Twin-Higgs
    mechanism.
    The contour of Δ_v assuming the Dirac gluino is shown in fig. <ref>.
    For the stop mass of 2 TeV, 𝒪(10)% tuning is possible even if the mediation scale is as high as 10^16 GeV.
    Note that in Dirac gluino models the large log enhancement of the quantum correction to the Higgs mass squared is already absent.
    Thus the improvement of the fine-tuning by the Twin higgs mechanism simply originates from a large SU(4) invariant coupling. For g_X =1 - 1.5, the improvement is by a factor of 2-4.
    
    
    In some UV completions of the Dirac gluino, the fine-tuning may be worse and at the O(1)% level <cit.>.
    For example in gauge mediated models, a tachyonic soft mass term of the adjoint chiral superfield larger than the Dirac gluino mass is often generated. See ref. <cit.> for a pedagogical discussion. 
    To prevent the instability of the adjoint field one needs to cancel the tachyonic mass by additional large soft mass or a supersymmetric mass of the adjoint, which leads to fine-tuning. See ref. <cit.> for a gauge mediated model free from this problem.
    In gravity mediated model the tachyonic mass is not necessarily larger than the Dirac gluino mass.
    Our D-term model, together with the Dirac gluino, realizes the natural SUSY even for the gravity mediation.
    
    
    
    
    The wino and the bino masses are also bounded from above by naturalness. The constraint is stronger than that in the MSSM as we add extra SU(2)_L
    and/or U(1)_Y
    charged fields which makes the corresponding gauge couplings and gaugino masses growing faster with the renormalization scale. Fine-tuning from bino
    and wino may be very large especially for high mediation scales. In the left panel of fig. <ref> we fix Λ=10^17 GeV and present
    contours of fine-tuning in the plane M_1-M_2. We see that bino as light as 700 GeV induces tuning at the level of 1 % for this mediation scale.
    The tuning from wino is slightly smaller but still 1 TeV wino results in about 1 % tuning. From the comparison of
    figs. <ref> and <ref> we see that in order not to increase tuning by more than a factor of two, bino (wino) must be lighter than
    about 400 (600) GeV. Thus, one generally expects all neutralinos to be light and the LSP to be a mixture of bino, wino and
    higgsino. Assuming majorana gluino, the impact of wino and bino on the tuning is less pronounce but in order not to increase tuning by more than a
    factor of two their masses are still expected to be below about 1 TeV, cf. figs. <ref> and <ref>. For smaller mediation scales
    the tuning from bino and wino is milder. The tuning from bino is subdominant unless Λ≳10^16 GeV. In the right panel of
    fig. <ref> we present tuning from wino as a function of the mediation scale. We see that even for small mediation scale wino mass should
    generally be below 1 TeV in order not to dominate tuning.  
    The bounds on the masses is avoided if the wino and the bino also obtain Dirac masses. 
    Interestingly, with an additional SU(2)_L adjoint paired with wino, the SU(2)_L gauge coupling constant also blows up around the Planck
    scale. 
    
    
    
    
    
    In the above analysis we have ignored the contribution to the RGE running of m_H_u^2 proportional to the SU(2)_X gauge coupling constant.
    As long as the SU(2)_X gaugino mass is suppressed, one-loop contributions are negligible. At the two loop level, there is a contribution,
    
    
        d/ dlnμ m_H_u^2 ⊃3 g_X^4/256π^4∑_im_i^2,
    
    where m_i^2 is a soft mass squared of a SU(2)_X fundamental.
    Although this is a two-loop effect, the largeness of m_S^2 required to obtain a large non-decoupling SU(4) invariant quartic and the largeness of 
    g_X around the Landau pole scale can make this 
    contribution non-negligible.
    In the left panel of fig. <ref>, we show the fine-tuning including this two-loop effect to m_H_u^2, with m_S^2 fixed at the value
    determined by eq. (<ref>), while ignoring contribution from other SU(2)_X
    charged fields.
    The fine-tuning gets worse than the case ignoring the two-loop effect, especially when the mediation scale is close to the Landau pole scale, while it
    remains the same if the mediation scale 
    is much smaller than the Landau pole scale.
    We note, however, that the two-loop effect strongly depends on the boundary condition of soft masses at the mediation scale and might be much smaller.
    For example, if m_S^2 = - m_E̅^2 at the boundary, the two-loop effect is suppressed.
    This special boundary condition should be explained by a UV completion of our model.
    It is also possible that m_S^2 at the UV boundary is much smaller than around the m_X scale, and is generated through the RG
    running.
    Actually couplings with the fields Z and/or E_1,2 can generate a non-zero and positive m_S^2, if the soft masses of them are negative. For
    example, in the right panel of fig. <ref> we show tuning under the assumption that m_S^2 vanishes at the mediation scale and gets
    renormalized to appropriately large value determined by eq. (<ref>) at the EW scale via the interaction (<ref>) (by suitable
    choice of the soft mass for Z) with κ =0.3 at the UV boundary. In this case the impact of m_S^2 on tuning is rather small unless the mediation scale is very close to the Landau pole scale.
    
    
    
    Even though naturalness does not require sparticles to be within discovery reach of the LHC (perhaps except for wino if the mediation scale is high
    enough) it does require that the twin Higgs boson is relatively
    light.
    The mass of the twin Higgs boson is well approximated by 2√(λ)f with λ being the SU(4) invariant coupling. Both λ and f
    are constrained from above by naturalness. For example, demanding better than 10% tuning f must be below about 4.5. This upper bound on f is
    quite generic for Twin Higgs models unless hard ℤ_2 breaking is non-negligible. The upper bound on λ is specific for this model and
    is set by the requirement of not too large threshold correction from X gauge bosons. We find that better  than 10% tuning requires
    λ≲0.5 which, together with the upper bound on f, leads to the upper bound on the twin Higgs boson mass of about 1 TeV. The twin Higgs
    tends to be lighter for a larger Landau pole scale.  For recent studies of the phenomenology  of the twin Higgs boson we refer the 
    reader to refs. <cit.>. It is also noteworthy that in this model MSSM-like Higgs bosons and their mirror counterparts are not required to be light by naturalness because H_d is not charged under SU(2)_X. 
    
    
    
    
    
    
    
    § SUMMARY
    
    
    We proposed a new SUSY Twin Higgs model in which an SU(4) invariant quartic term originates from a D-term potential of a new SU(2)_X gauge symmetry. The choice of the non-abelian gauge symmetry, together with a minimal number of flavors charged under SU(2)_X, makes the running of the new gauge coupling constant rather slow allowing for a large SU(4) invariant quartic term without generating a low-scale Landau pole. The Twin Higgs mechanism, together with the negative contribution from the new gauge coupling to the RG running of the top yukawa coupling, allows for tuning of the EW scale better than 10% for high mediation scales up to 𝒪(10^9) GeV even for sparticle spectra that may be outside of the ultimate LHC reach.
    If the gluino obtains a Dirac mass term, tuning of 10% is possible even if  the mediation scale is around the Planck scale.
    The model may be tested at the LHC by searching for a twin Higgs boson whose mass is bounded from above by naturalness and is anti-correlated with the
    Landau Pole scale.
    In parts of parameter space with tuning better than 10% the twin Higgs boson is expected to be lighter than about 1 TeV. All electroweakinos are
    expected to be rather light, with masses in the sub-TeV region, especially if the mediation scale of SUSY breaking is high. 
    
    
    
    
    § ACKNOWLEDGMENTS
    
    This work has been partially supported by National Science Centre, Poland, under research grant DEC-2014/15/B/ST2/02157, by
    the Office of High Energy Physics of the U.S. Department of Energy
    under Contract DE-AC02-05CH11231, and by the National Science Foundation
    under grant PHY-1316783. MB acknowledges support from the
    Polish 
    Ministry of Science and Higher Education through its programme Mobility Plus (decision no. 1266/MOB/IV/2015/0). 
    The work of KH was in part performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.
    
    
    
    § ELECTROWEAK PRECISION MEASUREMENTS
    
    
    We use the so-called S,T,U parametrization <cit.> to constrain the parameter space of our model.
    We follow the method presented in <cit.>, where the observables shown in Table <ref> are used to constrain S,T,U.
    We take U=0 and show the constraint on (S,T) in Fig. <ref>.
    
    
    
    
    The Higgs multiplet H is charged under SU(2)_X. After the electroweak symmetry breaking scale, Z boson mixes with the SU(2)_X gauge bosons.
    The mixing breaks the custodial symmetry and we expect a severe constraint from the electroweak precision measurement.
    After integrating out the SU(2)_X gauge bosons, we obtain the effective dimension 6 operator,
    
    
        L_ eff = g_X^2/8 m_X^2( H_u^† D_μ H_u - (D_μ H_u)^† H_u )^2.
    
    This generates a non-zero T parameter,
    
    
        T  =  1/2αc_W^2/g_2^2g_X^2 m_Z^2/ m_X^2sin^2β .
    
    The dependence on tanβ originates from subdominant H_d component of the Higgs.  
    S and U parameters are negligibly small. Comparing this result with fig. <ref>, we obtain the constraint,
    
    
        m_X / g_X > 4.1  TeVsin^2β.
    
    
    
    The mixing between the SM-like Higgs and the mirror Higgs also contributes to  S and T parameters,
    
    
        S =  1/12π s_γ^2  lnm_h'^2/m_h^2,
        
        T = - 3/16π c_w^2 s_γ^2  lnm_h'^2/m_h^2.
    
    where γ is the mixing angle between the SM Higgs and the mirror Higgs.
    Note that the sign of the T parameter is negative, and hence this contribution relaxes the constraint on the SU(2)_X symmetry breaking scale.
    In Sec. <ref>, we use the constraint including the Higgs mixing to determine the magnitude of the threshold correction around the SU(2)_X symmetry breaking scale.
    
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