Minimal Non-Abelian Supersymmetric Twin Higgs

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) [1][2][3][4]. 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. [5,6], 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. 1 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.
Early realisations of SUSY UV completion of Twin Higgs scenario [14,15], 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 [35][36][37].
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 [36] or a new U (1) X gauge symmetry whose D-term potential provides a large SU (4) invariant quartic term [37]. 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. [36] 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 Dterm 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 [37] 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 1. 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 bifundamental 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 andS. Except for S andS all matter fields have their mirror partner.
The right-handed top quark is embedded intoQ R and allow for a large enough top yukawa coupling through the superpotential term HQ R Q 3 , where Q 3 is the third generation quark doublet.Ē is necessary in order to cancel the U (1) Y -SU (2) 2 X anomaly. The VEV of φ u is responsible for the masses of the up and charm quarks. Q 1,2,3 ,ū 1,2 ,ē 1,2,3 ,d 1,2,3 and L 1,2,3 are usual MSSM fields. To cancel the gauge anomaly of SU (3) 2 c -U (1) Y and U (1) 3 Y originating from the extra up-type right handed quark inQ R and two extra right-handed leptons in E, 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 We assume that the soft masses of S andS are the same, Otherwise, the magnitude of the VEVs of S andS are different from each other, and give large soft masses to the Higgs doublets through the D-term potential. The VEVs of S and S are given by The constraint on the T (ρ) parameter requires that v S 2.9 TeV in the limit of large tan β and neglecting the effect of mixing between the SM and the mirror Higgses, see Appendix A for a derivation of this constraint and more precise formula. The masses of the SU (2) X gauge bosons are given by After integrating out massive particles with a mass as large as v S , the potential of H and In the SUSY limit, m 2 S = 0, the D-term potential vanishes. In terms of the model parameters M, m S , κ, g X , 2 is given by In the limit where κ g X , 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, 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. 1, which shows that κ g X much below M * . We obtain the same conclusion as long as κ(M * ) > ∼ 1. For κ g X , 2 is We may obtain a sufficiently small 2 , say 2 < ∼ 0.2, for m 2 S > ∼ 0.6g 2 X M 2 . Notice also that for 2 < 1 there is a threshold correction to the soft Higgs mass which is proportional to a new gauge bosons mass squared: 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 2 U 3 .
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 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 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 2 Hu receive a quantum correction from m 2 S , 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 2 S runs to smaller values at higher energies.
Note that the Z 2 symmetry S ↔S is explicitly broken by the above superpotential couplings. Even if we assume the Z 2 symmetry of the soft masses of S andS, we expect a quantum correction to a mass difference of them, This leads a asymmetric VEV of S andS, which give m 2 Hu through the D-term potential, which is always smaller than the direct one-loop quantum correction in Eq. (15).
It is also possible to maintain the Z 2 symmetry. Instead of the coupling in Eqs. (12) and (14), we introduce Here we have assumed that φ d,2 is odd under the Z 2 symmetry. After S andS obtain their VEVs, the mass terms become 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.

Masses of matter particles
We first consider a case where the Z 2 symmetry S ↔S is explicitly broken. A large enough top yukawa coupling is obtained by the superpotential whereQ R = (Q R,1 ,Q R,2 ) T . We give a large mass toQ R,1 by introducing a coupling and identifyQ R,2 with a right-handed top quarkū 3 .
The yukawa couplings of the up and charm quarks originates from the couplings with φ u , 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 , The extra SU (2) X charged particleĒ obtains its mass paired with E 1,2 through the SU (2) X symmetry breaking, Next we consider a case where the Z 2 symmetry is maintained. The top yukawa coupling is obtained by the superpotential One linear combination ofQ R,1 andQ R,2 obtains a Dirac mass term paired with U , We identify the massless combination (Q R,1 +Q R,2 )/ √ 2 ≡ū 3 as a right-handed top quark.
The extra SU (2) X charged particleĒ obtains its mass paired with E 1,2 through the coupling, Here we assume that E 2 is odd under the Z 2 symmetry S ↔S, so that all particles inĒ and E 1,2 obtains their masses.
So far we have assumed that a linear combination ofQ R,1 andQ 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 andū 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.
3 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 [35], where Here To evaluate ∆ f we solve the renormalization group equations (RGEs) of parameters between m stop and Λ.
We assume that the right-handed charm quark is also embedded inQ R . Between m stop and m X we solve MSSM RGEs at the one-loop level appropriately modifying the beta function of m 2 Q 3 . At a scale m X we perform matching by including the threshold correction (11) to m 2 Hu and m 2 U 3 . 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. 2 The yukawa couplings other than the top yukawa are neglected.
As clearly seen from eqs. (28)- (30), for a given value of f there is a lower bound on ∆ v of ∆ v/f . f /v is constrained by the Higgs coupling measurements [38] to be at least 2.3 [39]. 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 = 3v. Nevertheless, the tuning is quite independent of this choice (unless f is so large In fig. 2 we present contours of ∆ v assuming low and high mediation scales of SUSY breaking Λ. 3 Here and hereafter, the stop mass m stop and the gluino mass M 3 refer to the values at the TeV scale. For Λ = 100m stop 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 [40]. In order to assess the impact of this constraint we compute the Higgs mass following closely the procedure described in ref. [37]. 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. 2 that this constraint prefers rather light stop unless tan β 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 tan β = 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 2 Hu . In this region the value of |m 2 Hu | at the mediation scale is somewhat suppressed. For lighter stops (which can be compatible with the Higgs mass constraint for larger tan β) the tuning is dominated by the threshold correction which implies tuning at the level of few percent. It should be noted that finetuning 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 2 Hu 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  In the left panels, the orange contours depict the value of the SU (4) preserving quartic coupling and in the green regions the Landau pole of the SU (2) X gauge coupling constant is below Λ. In the upper (lower) left panels, tan β = 2.5 (3) so that the correct Higgs mass m h = 125 ± 3 GeV (the blue region) is obtained for stop masses close to 2 TeV for the most interesting range of g X . In the right panels, the fine-tuning is shown in the plane m stop -tan β for some fixed values of g X . m X is chosen such that the constraint from EW precision measurements is saturatedsee Appendix for details.  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 lead to smaller tuning. On the other hand, for smaller g X , when the threshold correction is subdominant, it is preferred to have smaller 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. [37]. 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. 2, 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. 2 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. 3. 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. [6] 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 [41] 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. 3 we fix the soft gluino mass to 2 TeV which easily satisfies the LHC constraints even for moderate loop corrections from squarks [42,43]. 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 [44], 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 2 Hu [45]. The soft stop mass and the higgs mass are dominantly generated by the threshold correction around the gluino mass, 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 ∆ v assuming the Dirac gluino is shown in fig. 4. 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 [46]. 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. [47] 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. [48] 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 In the above analysis we have ignored the contribution to the RGE running of m 2 Hu 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 contribu- . The remaining parameters are the same as in fig. 4.
where m 2 i is a soft mass squared of a SU (2) X fundamental. Although this is a two-loop effect, the largeness of m 2 S 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. 6, we show the fine-tuning including this two-loop effect to m 2 Hu , with m 2 S fixed at the value determined by eq. (6), 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 2 S = −m 2Ē 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 2 S at the UV boundary is much smaller  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.