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PHYSICAL REVIEW LETTERS 120, 211803 (2018)

Asymptotically Free Natural Supersymmetric Twin Higgs Model

Marcin Badziak Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL–02–093 Warsaw, Poland; Department of Physics, University of California, Berkeley, California 94720, USA; and Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

Keisuke Harigaya Department of Physics, University of California, Berkeley, California 94720, USA and Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Received 8 December 2017; published 23 May 2018)

Twin Higgs (TH) models explain the absence of new colored particles responsible for natural electroweak breaking (EWSB). All known ultraviolet completions of TH models require some nonperturbative 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 breaking is mediated around the Planck scale, and has interesting flavor phenomenology including the top decay into the and the up quark which may be discovered at the LHC.

DOI: 10.1103/PhysRevLett.120.211803

Introduction.—Models of natural electroweak symmetry symmetry requires a large SUð4Þ invariant quartic term which breaking (EWSB), e.g., supersymmetric (SUSY) models points to UV completions based on composite Higgs models [1–4] and composite Higgs models [5,6], generically [8–11]. SUSY UV completions of the TH model also exist predict new light colored particles, called top partners, [12–17]. Acceptable tuning of the electroweak (EW) scale at so that the quantum correction to the Higgs mass is the level of 5%–10% can be, however, obtained only with a suppressed. Null results of the LHC searches, however, low Landau pole scale, which requires UV completion by show that new colored particles are heavy, which calls for some strong dynamics. SUSY models that are able to keep the fine-tuning of the parameters of the theories; this is known tuning at the level of 5%–10% without resorting to the TH as the little hierarchy problem. In light of this fact the idea mechanism also require a low cutoff scale [18]. that the light top partners are not charged under the standard In this Letter we propose a SUSY twin Higgs model with ð3Þ model (SM) SU c gauge group has become increasingly an asymptotically free SUð4Þ invariant quartic coupling. attractive. Twin Higgs (TH) models [7] are one of the most The model remains perturbative up to around the Planck studied realizations of the idea. scale, and does not require any further UV completion A crucial ingredient of TH models is an approximate below the energy scale of gravity. As a result the Yukawa global SUð4Þ symmetry under which the SM Higgs boson couplings of the SM particles are given by renormalizable and its mirror (or twin) partner transform as a fundamental interactions. representation. The Higgs boson is a pseudo-Nambu- Setup.—It was proposed in Ref. [16] that an SUð4Þ associated with the spontaneous break- invariant quartic coupling may be obtained from a D-term down of the SUð4Þ symmetry. The SUð4Þ symmetry of the ð1Þ potential of a new U X gauge symmetry. The model Higgs mass term emerges from a Z2 symmetry exchanging ð1Þ suffers from a low Landau pole scale of the U X gauge the SM fields with their mirror counterparts. The light top ð2Þ interaction. A model with a non-Abelian SU X gauge partners are then charged under the mirror gauge group symmetry was proposed in Ref. [17], so that the Landau rather than the SM one. Standard lore says that ultraviolet pole scale is far above the TeV scale. Still the gauge (UV) completion of TH models involves some nonpertur- interaction is asymptotically nonfree. In order for the gauge ð4Þ bative dynamics. This is because the quality of the SU interaction to be perturbative up to a high energy scale of 1016–18 GeV, the SUð4Þ invariant quartic coupling at the TeV scale must be small, and the TH mechanism does not Published by the American Physical Society under the terms of work perfectly well; fine-tuning of order 1% is required to the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to obtain a correct EWSB scale. the author(s) and the published article’s title, journal citation, In this Letter, we present an extension of the model such and DOI. Funded by SCOAP3. that the new gauge interaction is asymptotically free. In the

0031-9007=18=120(21)=211803(6) 211803-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 120, 211803 (2018) model presented in Ref. [17], the new gauge symmetry level flavor changing neutral currents (FCNCs) are sup- ð2Þ Z SU X is assumed to be 2 neutral, and mirror particles pressed. Hd gives masses to down-type and charged ð2Þ ð2Þ ¯ are charged under SU X. We instead assume that SU X via W ∼ HddQ þ HdLe¯. We assume that the ð2Þ0 has a mirror partner SU X, under which mirror particles Yukawa couplings involving ϕd;1;2 are suppressed; other- ð2Þ are charged. As a result the number of SU X charged wise large FCNCs are induced. For details of the ð2Þ fields is reduced, so that the SU X gauge interaction is extended Higgs sector see the Supplemental Material ð2Þ H ϕ asymptotically free. A similar group structure in a nontwin [20]. Because of the SU X invariance, after and u SUSY model was introduced in Ref. [19] to achieve obtain their VEVs, one linear combination of the two ¯ asymptotically free . components in QR remains massless at the tree level. The The charged matter content of the model is shown in one loop quantum correction with a charged wino, charged Table I. The up-type SM and mirror Higgs bosons are Higgsinos in H, and down-type left-handed squarks inside 0 embedded into H and H , respectively. The resultant the loop generates the up-quark mass. The mass of the D-term potentials of the gauge symmetries are not Higgsinos in H is given by the SUð2Þ symmetry breaking ð4Þ ð2Þ ð2Þ0 X SU invariant. Once SU X × SU X symmetry is and hence the loop mediates the breaking. One cannot broken down to a diagonal subgroup SUð2Þ by a nonzero ¯ D embed a charm quark instead of an up quark into QR vacuum expectation value (VEV) of a bifundamental Σ, because the above loop correction is too small to generate both the SM and mirror Higgs bosons are fundamental the charm quark mass. The field E¯ cancels the anomaly of ð2Þ 2 3 representations of SU D, and the D-term potential is Uð1Þ -SUð2Þ , while E1 2 cancels that of Uð1Þ . The ð4Þ Y X ; Y approximately SU invariant below the symmetry break- charged is, in general, the mixture of E¯ , E, e¯, and the ð2Þ ing scale. The SU D symmetry is completely broken ∼ ¯ ¯ 0 ¯ 0 charged component of L due to possible mixing W eE. down by the VEVs of S, S, S , S . The neutrino mass can be obtained by the seesaw mecha- The right-handed is embedded into Q¯ , so that R nism [21–23] with the superpotential W ∼ ϕuLN. a large enough top Yukawa coupling is obtained via the In Fig. 1, we show the (RG) ∼ H ¯ superpotential W QRQ3, where Qi is the ith generation running of the gauge coupling constants and the top of left-handed quarks. The right-handed up quark is also Yukawa coupling, where we use the NSVZ beta function ¯ embedded into QR. The VEV of ϕu gives a mass to the [24] with the anomalous dimension evaluated at the one- charm quark via W ∼ ϕuu¯ 2Q2. We assume that Yukawa loop level. We see that the new gauge interaction is ¯ couplings HQRQ1;2 and ϕuu¯ 2Q1;3 are small so that tree asymptotically free. Here and hereafter, we approximate ð2Þ the RG running above the SU D symmetry breaking ð2Þ ð2Þ0 scale by that of the SU X × SU X symmetric theory. TABLE I. The charged matter content of the model. In addition ð2Þ to the fields shown in the table, the model contains SUð2Þ - This is a good approximation as long as the SU X × X ð2Þ0 neutral left-handed quarks Q1;2;3, a right-handed charm u¯ 2, right- SU X breaking scale is within the same order of ¯ ð2Þ handed leptons e1;2;3, left-handed leptons L1;2;3, and right-handed magnitude as the SU D breaking scale. neutrino N1;2;3 as well as their mirror partners. 4 ð2Þ ð2Þ0 30 20 10 SU X SU X 3-2-1 - - H 2 ð1; 2; 1=2Þ 3 H0 2 ð1; 2; 1=2Þ Σ 22 g1

S 2 2 g S¯ 2 X S0 2 g3 S¯ 0 2 1 g ¯ 2 ð3¯ 1 −2 3Þ 2 QR ; ; = yMSSM ¯ 0 2 ð3 1 −2 3Þ yt t QR ; ; = 0 E¯ 2 (1; 1; 1) 104 108 1012 1016 E¯ 0 2 (1; 1; 1) /GeV

E1;2 (ð1; 1; −1Þ) 0 FIG. 1. RG running of gX (red), g1 (blue), g2 (yellow), g3 (green) E1 2 ð1; 1; −1Þ ; and the top Yukawa coupling yt (black) for mX ¼ 10 TeV, ϕ ð1 2 1 2Þ u ; ; = mstop ¼ 2 TeV, gXðmXÞ¼2, and tan β ¼ 3. Solid lines correspond 0 ϕu ð1; 2; 1=2Þ to the case where all states beyond the Minimal Supersymmetric ϕ ð1 2 −1 2Þ (MSSM) have masses around mX.Dashedlines Hd, d;1;2 ; ; = 7 9 0 0 ¼ 10 ¼ 10 ϕ ð1 2 −1 2Þ assume ME1 , ME2 GeV, see text for details. Dotted Hd; d;1;2 ; ; = black line corresponds to the running of yt in the MSSM.

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The model possesses many new states with nonzero coupling at high energy scales, as seen from Fig. 1, which ð1Þ hypercharge, which make the Landau pole scale of U Y results in additional suppression of the correction to the much lower than in the SM. Nevertheless, this Landau pole Higgs mass parameter from stops and gluino. However, 18 appears around 10 GeV, as seen from Fig. 1, which is for very large values of gX, the tuning of the EW scale is rather close to the Planck scale. The Landau pole scale is dominated by a finite threshold correction from the gauge pushed up if some of the new states are much heavier than bosons of the new interaction: the TeV scale. Actually, we can give a large Dirac mass 2 M e¯1E1 þ M e¯2E2. After integrating them out, the 2 g 2 −2 E1 E2 ðδm Þ ¼ 3 X m lnðϵ Þ: ð4Þ electron and muon masses are given by a dimension-5 Hu X 64π2 X term W ∼ ðSϕ EL¯ þ S¯ϕ EL¯ Þ=M .ForOð1Þ coupling of d i d i E1 For large values of g , which we are most interested in, the W ∼ SEE¯ þ ϕ EL¯ X d , the Dirac masses may be as large as strongest lower mass limit on the new gauge boson mass of ≈ 107 ≈ 109 ME1 , ME2 GeV. The RG running in such a mX ≳ gX × 4 TeV originates from the mixing between the case is also shown in Fig. 1. Z boson and the SUð2Þ gauge bosons which breaks ð4Þ D Let us evaluate the magnitude of the SU invariant custodial symmetry; see Ref. [17] for a detailed derivation Σ coupling. We assume that obtains its VEV vΣ in a SUSY of this bound using the EW precision observables. The ∼ ðΣ2 − 2 Þ way, e.g., by a superpotential W Y vΣ , where Y is a threshold correction in Eq. (4) is smaller for larger ϵ which chiral multiplet, and that vΣ is much larger than the TeV leads also to smaller SUð4Þ invariant coupling; some scale, say few tens of TeV. Then below the scale vΣ the intermediate value of ϵ minimizes the tuning of the EW ð2Þ theory is well described by a SUSY theory with an SU D scale. Not too small ϵ, i.e., not too heavy S fields, is also ð2Þ 2 gauge symmetry. The symmetry breaking of SU D preferred to avoid a large two-loop correction to m Hu should involve the SUSY breaking effect, so that the D- g4 m2 ð2Þ proportional to X S. term potential of SU D does not decouple after the In order to quantify the tuning we use the measure [14] symmetry breaking. We introduce chiral multiplets Ξ, Ξ0, and the superpotential Δv ≡ Δf × Δv=f; ð5Þ W ¼ κΞðSS¯ − M2ÞþκΞ0ðS0S¯ 0 − M2Þ; ð1Þ where the tuning in percent is 100%=Δv and κ 2 where , M are constants and soft masses 1 f Δ ¼ − 2 ; v=f 2 v2 V ¼ m2ðjSj2 þjS¯j2 þjS0j2 þjS¯ 0j2Þ: ð2Þ soft S ∂ 2 Δ ¼ ln f 1 ð Þ ¯ f maxi ; : 6 Here we assume that the soft masses of S and S are the ∂ ln xiðΛÞ same. Otherwise, the asymmetric VEVs of S and S¯ give a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 2 02 large soft mass to the Higgs doublet through the D-term Here hHi ≡ v, hH i ≡ v , and f ≡ v þ v is the decay ð2Þ ð4Þ Δ potential of SU D. Splitting between the soft masses by constant of the spontaneous SU breaking. v=f mea- quantum corrections does not introduce a large soft Higgs sures the tuning to obtain v

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2 2 M3=2 TeV, mstop=2 TeV, tan =3, f=3v, =1 3 gX=2, tan =3, f=3v, =1/3 3.0 3000 2500 100 40 2800 50 2500 30 2.5 30 2600 30

20 2400 2000 15 25 2.0 8 X /GeV g

3 2200 20 2000 10 M 30 2000 1.5 50 1500 No EWSB 1800 1500 100 1.0 1600

4 6 8 10 12 14 16 18 0 500 1000 1500 2000 log[ /GeV] m0 /GeV

FIG. 3. Masses of the lightest (right-handed) stop and the right- FIG. 2. Fine-tuning Δv of the model in the plane Λ-gX for handed up squark (blue) and the lightest squark of the first mstop ¼ 2 TeV, tan β ¼ 3, f ¼ 3v, μ ¼ 500 GeV, M1 ¼ M2 ¼ 2 200 GeV and the soft gluino mass M3 ¼ 2 TeV. We fix ϵ ¼ 1=3 generation other than the right-handed up squark (orange) as a function of gluino soft mass parameter M3 defined at the stop which corresponds to mS ¼ mX. mass scale and universal soft scalar mass m0 at the UV scale of 16 2 Λ ¼ 10 GeV for gX ¼ 2 and ϵ ¼ 1=3. Red contours depict Δv. analysis we fix tan β ¼ 3 which reproduces the observed Higgs mass within theoretical uncertainties for the stop masses of about 2 TeV; see Ref. [16] for more details. The unify. Thus, similarly as before we take gaugino masses at Λ − ¼ ¼ 2 M1 ¼ M2 ¼ 200 M3 tuning in the plane gX for mstop M3 TeV at the the low scale as input, with GeV and TeV scale is shown in Fig. 2. We see that the tuning as a variable, and include their RG running. Using the above assumptions we show in Fig. 3 the decreases with increasing gX due to the TH mechanism as contours of masses for the lightest stop and the lightest long as gX ≲ 2. For larger gX the tuning becomes domi- nated by the threshold correction in Eq. (4) and the two- first-generation squark other than the right-handed up loop correction from the soft masses of S fields, so further squark in the plane m0-M3. The lightest stop is mostly right handed and roughly degenerate with the right-handed increasing gX worsens the tuning. For the optimal value up squark. An important constraint on the parameter space of gX ≈ 2 the tuning is only at the level of 5%–10% even for very large mediation scales. This allows us to employ is provided by the condition of correct EWSB since the top gravitational interactions as a source of SUSY breaking Yukawa coupling is much smaller during the RG evolution mediation without excessive fine-tuning, in contrast to the than in MSSM, so the negative corrections from stops and gluino to m2 [28,29] are smaller. This suppression is only MSSM and previously proposed SUSY TH models. Hu The above discussion of tuning, similarly to all previous partly compensated by the negative correction from the S 2 fields. In consequence, for too large m0, m is positive at papers on SUSY TH models, assumed the soft stop masses Hu at the low scale as an input without paying attention to the the low scale. This can be easily circumvented by assuming question of what kind of SUSY breaking mechanism can mHu smaller than m0 at the mediation scale. Even without realize the spectrum. Since in this model the TH mecha- this assumption there are parts of parameter space that give nism is at work also for high mediation scales, we calculate EWSB as well as the viable sparticle spectrum. In this the spectrum using simple UV boundary conditions. We example, gluino is slightly heavier than squarks and the assume a universal soft scalar masses m0 for the SM tuning at the level of 5% can be achieved with this simple charged fields at the mediation scale, which explains the UV boundary condition leading to squarks and gluino 2 smallness of the flavor violation from SUSY particles. mS masses that comfortably satisfy the LHC constraints. In the 2 and mΞ are determined in the same way as before. We fix all Supplemental Material [20] we present a typical mass soft trilinear terms A0 ¼ 0 at the mediation scale. On the spectrum. other hand, there is no well-motivated choice for gaugino Flavor and collider phenomenology.—Asymptotic free- masses since in this model the gauge couplings do not dom for gX is obtained thanks to a small number of SM

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ð2Þ fermions charged under SU X. This implies a nontrivial No. DEC-2014/15/B/ST2/02157 and No. 2017/26/D/ST2/ flavor structure of the model which may impact flavor 00225, by the Office of High Energy Physics of the U.S. observables. As explained before, we have assumed a Department of Energy under Contract No. DE-AC02- flavor structure in Yukawa couplings to suppress most of 05CH11231, and by the National Science Foundation the tree-level FCNCs. Tree-level FCNCs are, however, under Grant No. PHY-1316783. M. B. acknowledges sup- unavoidable in the top sector as we embed the right-handed port from the Polish Ministry of Science and Higher ¯ up quark in QR. Education through its programme Mobility Plus (decision The heavy Higgs in H which we call H2 couples to quarks No. 1266/MOB/IV/2015/0). via L ¼ ytH2u¯ RQ3. We expect non-negligible t → hu decays through mixing between the SM-like Higgs h and 0 the neutral component in H2 which we call H2. 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