Asymptotically Free Natural SUSY Twin Higgs

Twin Higgs (TH) models explain the absence of new colored particles responsible for natural electroweak symmetry breaking (EWSB). All known ultraviolet completions of TH models require some non-perturbative dynamics below the Planck scale. We propose a supersymmetric model in which the TH mechanism is introduced by a new asymptotically free gauge interaction. The model features natural EWSB for squarks and gluino heavier than 2 TeV even if supersymmetry breaking is mediated around the Planck scale, and has interesting flavor phenomenology including the top quark decay into the Higgs and the up quark which may be discovered at the LHC.

Introduction.-Models of natural electroweak symmetry breaking (EWSB), such as supersymmetric (SUSY) models [1][2][3][4] and composite Higgs models [5,6], generically predict new light colored particles, called top partners, so that the quantum correction to the Higgs mass is suppressed. Null results of the LHC searches, however, show that new colored particles are heavy, which calls for fine-tuning of the parameters of the theories; this is known as the little hierarchy problem. In light of this fact the idea that the light top partners are not charged under the Standard Model (SM) SU (3) c gauge group has become increasingly attractive. Twin Higgs (TH) models [7] are one of the most studied realizations of the idea.
A crucial ingredient of TH models is an approximate global SU (4) symmetry under which the SM Higgs and its mirror (or twin) partner transform as a fundamental representation. The Higgs boson is a pseudo-Nambu-Goldstone boson associated with the spontaneous breakdown of the SU (4) symmetry. The SU (4) symmetry of the Higgs mass term emerges from a Z 2 symmetry exchanging the SM fields with their mirror counterparts. The light top partners are then charged under the mirror gauge group rather than the SM one. Standard lore says that ultraviolet (UV) completion of TH models involves some non-perturbative dynamics. This is because the quality of the SU (4) symmetry requires a large SU (4) invariant quartic term which points to UV completions based on Composite Higgs models [8][9][10][11]. SUSY UV completions of the TH model also exist [12][13][14][15][16][17]. Acceptable tuning of the electroweak (EW) scale at the level of 5 − 10% can be, however, obtained only with a low Landau pole scale, which requires UV completion by some strong dynamics. SUSY models that are able to keep the tuning at the level of 5 − 10% without resorting to the TH mechanism also require a low cut-off scale [18].
In this Letter we propose a SUSY Twin Higgs model with an asymptotically free SU (4) invariant quartic cou-pling. The model remains perturbative up to around the Planck scale, and does not require any further UV completion below the energy scale of gravity. As a result the yukawa couplings of the SM particles are given by renormalizable interactions.
Setup.-It was proposed in [16] that an SU (4) invariant quartic coupling may be obtained from a D term potential of a new U (1) X gauge symmetry under which the SM and mirror Higgses are charged. The model suffers from a low Landau pole scale of the U (1) X gauge interaction. A model with a non-Abelian SU (2) X gauge symmetry was proposed in [17], so that the Landau pole scale is far above the TeV scale. Still the gauge interaction is asymptotically non-free. In order for the gauge interaction to be perturbative up to a high energy scale of 10 [16][17][18] GeV, the SU (4) invariant quartic coupling at the TeV scale must be small, and the TH mechanism does not work perfectly well; fine-tuning of order one percent is required to obtain a correct EWSB scale.
In this Letter, we present an extension of the model such that the new gauge interaction is asymptotically free. In the model presented in [17], the new gauge symmetry SU (2) X is assumed to be Z 2 neutral, and mirror particles are charged under SU (2) X . We instead assume that SU (2) X has a mirror partner SU (2) X , under which mirror particles are charged. As a result the number of SU (2) X charged fields is reduced, so that the SU (2) X gauge interaction is asymptotically free. A similar group structure in a non-Twin SUSY model was introduced in [19] to achieve asymptotically free gauge theory.
The charged matter content of the model is shown in Table I. The up-type SM and mirror Higgses are embedded into H and H , respectively. The resultant D term potentials of the gauge symmetries are not SU (4) invariant. Once SU (2) X × SU (2) X symmetry is broken down to a diagonal subgroup SU (2) D by a non-zero vacuum expectation value (VEV) of a bi-fundamental Σ, both arXiv:1711.11040v1 [hep-ph] 29 Nov 2017 the SM and mirror Higgses are fundamental representation of SU (2) D , and the D term potential is approximately SU (4) invariant below the symmetry breaking scale. The SU (2) D symmetry is completely broken down by the VEVs of S,S, S ,S .
The right-handed top quark is embedded intoQ R , so that a large enough top yukawa coupling is obtained via the superpotential term W ∼ HQ R Q 3 , where Q i is the i-th generation of left-handed quarks. The right-handed up quark is also embedded intoQ R . The VEV of φ u gives a mass to the charm quark via W ∼ φ uū2 Q 2 . We assume that yukawa couplings HQ R Q 1,2 and φ uū2 Q 1,3 are small so that tree level flavor changing neutral currents (FCNCs) are suppressed. H d gives masses to downtype quarks and charged leptons via We assume that the yukawa couplings involving φ d,1,2 are suppressed; otherwise large FCNCs are induced. H d and φ d,1,2 are the mass partners of H and φ u , for details see [17]. Due to the SU (2) X invariance, after H and φ u obtain their VEVs, one linear combination of the two components inQ R remains massless at the tree level. The one loop quantum correction with a charged wino, charged higgsinos in H and down-type left-handed squarks inside the loop generates the up-quark mass. The mass of the higgsinos in H is given by the SU (2) X symmetry breaking and hence the loop mediate the breaking. The fieldĒ cancels the anomaly of The charged lepton is in general the mixture ofĒ, E,ē and the charged component of L due to possible mixing W ∼ēE.
The number of fundamental representations of SU (2) X is 10. Thus the SU (2) X gauge interaction is asymptotically free, unless g X 3.2 for which two-loop correction changes the sign of the beta function for g X . In Fig. 1, we show the renormalization group (RG) running of the gauge coupling constants and the top yukawa coupling, where we use the NSVZ beta function [20] with the anomalous dimension evaluated at the one-loop level. This explicitly confirms asymptotically-free behavior of the new interaction. Here and hereafter, we approximate the RG running above the SU (2) D symmetry breaking scale by that of the SU (2) X × SU (2) X symmetric theory. This is a good approximation as long as the SU (2) X × SU (2) X breaking scale is within the same order of magnitude as the SU (2) D breaking scale.
The model possesses many new states with non-zero hypercharge which make the appearance of the Landau pole for the hypercharge much lower than in the SM. Nevertheless, this Landau pole appears around 10 18 GeV, as seen from Fig. 1, which is rather close to the Planck scale. The Landau pole scale is pushed up if some of new states are much heavier than the TeV scale. Actually we can give a large Dirac mass term M E1ē1 E 1 + M E2ē2 E 2 . After integrating them out, the electron and muon masses are given by a dimension-5 the Dirac masses may be as large as M E1 ≈ 10 7 GeV, M E2 ≈ 10 9 GeV. The RG running in such a case is also shown in Fig. 1.
Let us evaluate the magnitude of the SU (4) invariant coupling. We assume that Σ obtains its VEV in a SUSY way, e.g. by a superpotential W = Y (Σ 2 − v 2 Σ ), and that Σ is much larger than the TeV scale, say few tens of TeV. Then below the scale v Σ the theory is welldescribed by a SUSY theory with an SU (2) D gauge symmetry. The symmetry breaking of SU (2) D should involve SUSY breaking effect, so that the D term potential of SU (2) D does not decouple after the symmetry breaking. We introduce the superpotential and soft masses, Here we assume that the soft masses of S andS are the same. Otherwise, the asymmetric VEVs of S andS give a large soft mass to the Higgs doublet through the D term potential of SU (2) D . Assuming that all Higgses apart from the SM-like and twin Higgs are heavy, negligible VEV of φ u and integrating out S fields, the SU (4) invariant quartic coupling of the SM Higgs H and the mirror Higgs H is given by where tanβ is the ratio of the up-type Higgs component to the down-type Higgs component in H.
Natural electroweak symmetry breaking.-Asymptotic freedom of the new gauge interactions allows the SU (4) invariant coupling of O(1). The tuning of the EW scale arising from heaviness of higgsino, stops and gluino, may be suppressed even by a factor of O(10) by means of the TH mechanism alone. Moreover, large g X strongly suppresses the top yukawa coupling at high energy scales, as seen from Fig. 1, which results in additional suppression of the correction to the Higgs mass parameter from stops and gluino. However, for very large values of g X , close to the perturbativity bound for the SU (2) X interaction, the tuning of the EW scale is dominated by a finite threshold correction from the gauge bosons of the new interaction: For large values of g X , that we are most interested in, the strongest lower mass limit on the new gauge boson mass of m X g X ×4 TeV originates from the mixing between the Z boson and the SU (2) D gauge bosons which breaks custodial symmetry, see [17] for a detailed derivation of this bound using the EW precision observables.
The threshold correction in Eq. (4) is smaller for larger which leads also to smaller SU (4) invariant coupling; some intermediate value of is optimal from the point of view of tuning of the EW scale. Not too small , i.e. not too heavy S fields, is also preferred too avoid a large two-loop correction to m 2 Hu proportional to g 4 X m 2 S . In order to quantify the tuning we use the measure [14] ∆ where the tuning in percent is 100%/∆ v and Here H ≡ v, H ≡ v , and f ≡ √ v 2 + v 2 is the decay constant of the spontaneous SU (4) breaking. ∆ v/f measures the tuning to obtain v < f via explicit soft Z 2 symmetry breaking which is required by the Higgs coupling measurements [21], implying f 2.3v [22]. In our numerical analysis we fix f = 3v. ∆ f measures the tuning to obtain the scale f from the soft SUSY breaking which is analogous to the fine-tuning to obtain the EW scale from the soft SUSY breaking in the MSSM. such that m S = m X at the SU (2) D breaking scale, corresponding to 2 = 1/3, is generated via the RG running with κ = 0.2 at the mediation scale, see [17] for more details of the calculation of ∆ v . The tuning in the plane Λ-g X for m stop = M 3 = 2 TeV at the TeV scale is shown in Fig. 2. The tuning does not depend strongly on tan β so we fix tan β = 3 which leads to the Higgs mass consistent with the Higgs mass measurement within theoretical uncertainties, see [16] for a more detailed discussion of the Higgs mass in SUSY TH models. Wee see that the tuning decreases with increasing g X as a consequence of the TH mechanism as long as g X 2. For larger g X the tuning becomes dominated by the threshold correction in Eq. (4) and the two-loop correction from the soft masses of S fields, so further increasing g X worsens the tuning. For the optimal value of g X ≈ 2 the tuning is only at the level of 5 − 10% even for very large mediation scales. This allows to employ gravitational interactions as a source of SUSY breaking mediation without excessive fine-tuning, in contrast to the MSSM and previously proposed SUSY TH models.
The above discussion of tuning, similarly to all previous papers on SUSY TH models, assumed the soft stop masses at the low scale as an input without paying attention to the question of what kind of SUSY breaking mechanism can realize such a spectrum. Since in this model TH mechanism is at work also for high mediation scales, we calculate the spectrum using simple UV boundary conditions. We assume a universal soft scalar masses m 0 for the SM charged fields at the mediation scale, which explains the smallness of the flavor violation from SUSY particles. m 2 S and m 2 Ξ are determined in the same way as before. We fix all soft trilinear terms A 0 = 0 at the mediation scale. On the other hand, there is no well-motivated choice for gaugino masses since in this model the gauge couplings do not unify. Thus, similarly as before we take gaugino masses at the low scale as input. We fix M 1 = M 2 = 200 GeV and vary M 3 .
Using the above assumptions we show in Fig. 3 the contours of masses for the lightest stop and the lightest firstgeneration squark other than the right-handed up squark in the plane m 0 -M 3 . The lightest stop is mostly righthanded and roughly degenerate with the right-handed up squark. An important constraint on the parameter space is provided by the condition of correct EWSB since the top yukawa coupling is much smaller during the RG evolution than in MSSM so the negative corrections from stops and gluino to m 2 Hu [23,24] are smaller. This suppression is only partly compensated by the negative correction from the S fields. In consequence, for too large m 0 , m 2 Hu is positive at the low scale. This can be easily circumvented by assuming m Hu smaller than m 0 at the mediation scale. Even without this assumption there are parts of parameter space that give EWSB as well as vi- able sparticle spectrum. In this example gluino is slightly heavier than squarks and the tuning at the level of 5% can be achieved with this simple UV boundary condition leading to squarks and gluino masses that comfortably satisfy the LHC constraints. It may be possible to reduce tuning even more if there exist some correlations between the soft SUSY breaking parameters in the UV leading to a focus point [25] in which the overall correction to m 2 Hu is small. Flavor and collider phenomenology-We should emphasize that asymptotic freedom for g X is obtained thanks to a small number of SM fermions charged under SU (2) X . This implies non-trivial flavor structure of the model which may impact flavor observables. As explained before, we have assumed a flavor structure in yukawa couplings to suppress most of the tree-level FC-NCs. Tree-level FCNCs are, however, unavoidable in the top sector as we embed the right-handed up quark inQ R .
The heavy Higgs in H which we call H 2 couples to quarks via L = y t H 2ūR Q 3 . We expect non-negligible t → hu decays through mixing between the SM-like Higgs h and the neutral component in H 2 which we call H 0 2 . The resultant h-t-u coupling λ htu is as large as m 2 Z /m 2 H2 . The current upper limit on BR(t → hu) is 2 × 10 −3 corresponding to λ htu of about 0.1 [26,27], which implies a lower bound on m H2 of few hundred GeV. The future sensitivity of the High-Luminosity LHC to BR(t → hu) is around 10 −4 [28] so this process will serve as an im-portant probe of the model.
Flavor violation in the top sector has also impact on the rare decays of mesons. We find that the strongest constraint comes from a possible deviation in BR(b → sγ) due to one-loop corrections involving the charged component of H 2 , that we refer to as H ± 2 , and the up quark, which is not suppressed by the GIM mechanism [29]. Translating the bound obtained in [30] for type-II Two-Higgs-Doublet model using the loop function in [31], we obtain the lower bound m H2 200 GeV.
The heavy Higgs H 2 is produced in proton colliders via the process u + g → H + 2 b, H 0 2 t involving the strong interaction and the top yukawa coupling, with a dominant decay mode H + 2 → ub, H 0 2 → ut,ūt. None of existing searches give relevant constraints on the masses of these new Higgses.
The right-handed up squark is almost degenerate in mass with the right-handed stop and decays mainly to the top/bottom quark and a higgsino. The signal resembles the one of the right-handed stop but with a much larger cross-section.
Discussion.-We have presented the first SUSY model which accommodates tuning of the EW scale 5 − 10% for stops and gluino heavier than 2 TeV, satisfying current LHC constraints, even if SUSY breaking mediation occurs close to the Planck scale. This is achieved in the novel UV completion of TH mechanism which is at work thanks to a new asymptotically free SU (2) X gauge interaction. The model predicts that the right-handed up squark is degenerate with the right-handed stop and has a very different decay pattern than in the MSSM. This scenario may be also tested via flavor-violating top decays which are generically correlated with deviations from the SM prediction for the b → sγ decay.