The Minimal Supersymmetric Fat Higgs Model

LBNL-54469

Roni Harnik,1, 2 Graham D. Kribs,3 Daniel T. Larson,1, 2 and Hitoshi Murayama3

1Theoretical Physics Group, Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720 2Department of Physics, University of California, Berkeley, CA 3Institute for Advanced Study, Princeton, NJ 08540

This work was supported in part by the Director, Office of Science, Office of High Energy Physics, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

DISCLAIMER

This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or The Regents of the University of California.

arXiv:hep-ph/0311349v1 26 Nov 2003 ocr.Ee huhteMS sfrfo being of from level far the at is fine-tuning MSSM on relies the already though the it quantifiable of Even excluded, masses a the becoming and concern. increasingly boson are Higgs the lightest superpartners renormal- on the bounds the lower of via experimental mass The parameters masses [2]. mass their group Higgs because ization light the and be into charginos to feed especially expected also Superpartners, are stops, TeV. 1 about oe MS) npriua,peit ih Higgs light a predicts particular, Standard in boson, Supersymmetric (MSSM), Minimal Model The for explain predictions experiments. can clear and with breaking UV symmetry the electroweak to sensitivity quadratic extreme theory. UV impossible complete a is without SM understand the to in of is origin breaking the squared symmetry Thus electroweak mass physics. new Higgs to the sensitive quadratically theory sensitivity: extreme an (UV) the symmetry with ultraviolet fraught Moreover, to is operators relevant input understanding. by breaking an deeper a just with the without is Unfortunately, consistent mass-squared is processes. flavor- negative current that from Fur- neutral constraints way violation changing and CP a observations squared. and in experimental mass mixings accommodated negative masses, are a fermion single with thermore, a of doublet introduction provides Higgs the in (SM) through electroweak mysteries breaking of Model greatest symmetry parameterization Standard successful the extremely The of an one physics. is particle [1] breaking metry aiona ekly A94720. CA Berkeley, California, ∗ nlaeo bec rmDprmn fPyis nvriyo University Physics, of Department from absence of leave On ti elkonta uesmer eoe the removes supersymmetry that known well is It sym- electroweak of dynamics and mechanism The m oiscae.eklyeu [email protected],dtlarson@s [email protected], [email protected], h ∼ < h oei h ieaue h ig etro h iia Fa minimal the Supersymmetri of Minimal sector the Higgs from different The distinctly im literature. is not that the does in therefore lore invol Model the dynamics Standard GeV strong the fine- inherently 200–450 Supersymmetrizing without the be data despite electroweak maintained easily probl be precision hierarchy can interact with little consistent mass gauge supersymmetric is Higgs new the The solving a masses, through scale. dynamically intermediate broken is symmetry 3 e,frtpsur assls than less masses top-squark for GeV, 130 epeetaclual uesmercter facomposit a of theory supersymmetric calculable a present We oiHarnik, Roni .INTRODUCTION I. 2 eateto hsc,Uiest fClfri,Berkeley California, of University Physics, of Department h iia uesmercFtHgsModel Higgs Fat Supersymmetric Minimal The ainlLbrtr,Uiest fClfri,Berkeley, California, of University Laboratory, National 1 ,2 1, hoeia hsc ru,Ens rad arneBerkel Lawrence Orlando Ernest Group, Physics Theoretical rhmD Kribs, D. Graham 3 nttt o dacdSuy rneo,N 08540 NJ Princeton, Study, Advanced for Institute Dtd eray1 2008) 1, February (Dated: 3 ailT Larson, T. Daniel f e ecn:w alti h sprymti little “supersymmetric the this call we problem”. hierarchy percent: few a ml etraino h togdnmc 6 ] We 7]. [6, a dynamics as strong added the is on breaking perturbation supersymmetry small strongly soft reliable of when is under that knowledge even [5] completely improved theories gauge are the supersymmetric dynamics coupled to UV is thanks and com- theory control IR UV This The thus UV. and plete. the free, in asymptotically fact theory the in renormalizable, confining are coupling a and strong of nature composite of mesons their scale an reveal at the fields coupled Higgs At strongly become scale. to intermediate sector Higgs the Higgs. allow the of yielding mass avoided, the be usual on should The bound coupling upper theory. strong an string that and is unification lore grand idea as theories, theoretical such attractive it supersymmetric to because connections of negatively eliminates completeness and viewed UV been e.g. the usually cutoff, ruins UV has low This a simply or [4]. Standard [3], Supersymmetric e.g. Landau (NMSSM), Next-to-Minimal a Model the as such in behavior, coupling pole to strong leads of invariably type some models beyond-the-MSSM into mass coupling gauge the below scale Yukawa a new scale. at the unification up causes blow mass further to Higgs any coupling the with But, GeV on MSSM. 150 push the about upward to to up comparable GeV 130 fine-tuning to raise from to sufficient is mass is Higgs Yukawa This class the (undetermined) fields. this new Higgs a the of with with interaction idea singlet extra simplest beyond an The the physics add to invoke contributions coupling. additional A to quartic provides only that be is fine-tuning. MSSM but could approach the in masses, mass increase fine-tuned squark drastic Higgs less top unnaturally the an the MSSM with increasing the by In raised mass. Higgs epooeardclrvso fteuullr and lore usual the of revision radical a propose We Higgs heavier significantly a incorporating fact, In the raise to is problem this around way simplest The m eepiil eiyta h model the that verify explicitly We em. cae.eklyeu [email protected] ocrates.berkeley.edu, uig ag opiguicto can unification coupling Gauge tuning. e neetoeksmer breaking. symmetry electroweak in ved tnadModel. Standard c ig oe a asspectrum mass a has model Higgs t ,2 1, l ih ig as otayto contrary mass, Higgs light a ply ft ig oo.Electroweak boson. Higgs “fat” e o htbcmssrn tan at strong becomes that ion n ioh Murayama Hitoshi and ln ihtesuperpartner the with along A94720 CA A94720 CA , ey 3, ∗ s draw significant inspiration from recent proposals to fuse mass µ for the Higgs doublets is smaller in magnitude supersymmetry with technicolor [8, 9], and indeed in than mHu , electroweak symmetry is broken. This so- our model electroweak symmetry is broken dynamically. called radiative breaking of the electroweak symmetry is However, we have physical (composite) Higgs fields in the a very nice feature of the MSSM. low energy effective theory with no a priori restriction However, the phenomenological situation is forcing on the scale of strong coupling, reminiscent of the older some degree of fine-tuning on the MSSM in the following non-supersymmetric composite Higgs models [10] (but fashion. First of all, the Higgs quartic coupling is given without the associated fine-tuning problems). only by D-terms that are determined by the electroweak The outline of this paper is as follows. We first discuss gauge couplings the supersymmetric little hierarchy problem in Sec. II, 2 2 and emphasize that a heavier Higgs mass easily solves g + g′ 2 2 2 VD = ( Hu Hd ) . (2) this problem. We then construct our supersymmetric 8 | | − | | composite Higgs theory in Sec. III. Our basic frame- This implies the natural scale for the Higgs boson mass work is a three-flavor SU(2) theory that s-confines, H is m , and indeed there is a well-known tree-level upper resulting in a low energy effective Lagrangian containing Z limit on the lightest Higgs mass that is precisely m . a dynamically generated superpotential of composite Z The only way to increase the Higgs mass is by using the mesons. By introducing a mass for one flavor below the O(h4) radiative correction to the Higgs quartic coupling. compositeness scale, we show that the mesons acquire t The approximate formula valid for a moderate tan β is expectation values that break electroweak symmetry at a scale that is tunable through this mass parameter. In 2 2 3 4 2 mt˜1 mt˜2 0 Sec. IV we demonstrate how the various energy scales can mh mZ + 2 ht v log 2 . (3) ≃ 4π mt be naturally obtained from the supersymmetry breaking, and in Sec. V we show how fermion masses and mixings Here, v = 174 GeV. Because the Higgs boson has > can be incorporated. We calculate the scalar spectrum not been found up to 115 GeV, this implies mt˜ in Sec. VI. In Sec. VII we briefly comment on the new 500 GeV. On the other hand, the minimization of the∼ phenomenology of this model, emphasizing the unusual scalar potential leads to scalar spectrum. Sec. VIII explains how gauge coupling 2 2 2 unification can be preserved. Finally, we conclude with 1 mHu tan β mHd m2 = µ2 − µ2 m2 , (4) a discussion in Sec. IX. 2 Z − − tan2 β 1 ≃− − Hu − again for moderate tan β. Therefore we need to fine-tune 2 II. SUPERSYMMETRIC LITTLE HIERARCHY the bare mHu and/or µ against the radiative correction PROBLEM in Eq. (1) at the level of

∆m2 m 2 M In this section we define the supersymmetric little Hu t˜ UV | 2 | 4.8 log . (5) hierarchy problem and propose a simple but unconven- mZ /2 ∼ 500 GeV µIR tional way out. The problem is that the conventional Even for a low UV scale of M = 100 TeV, this already supersymmetric theories are increasingly fine-tuned since UV requires a fine-tuning of 3%. the Higgs boson and/or charginos have not yet been In addition, the null results from searches for charginos discovered. We point out that a composite Higgs will at LEP-II gives a lower bound M > 100 GeV. Assuming solve this problem easily once a suitable UV completion 2 a GUT relation among the gaugino∼ masses, this implies is found. M > 350 GeV. Because M feeds into m through The MSSM provides a simple way to understand elec- 3 3 t˜ renormalization∼ group evolution, this then feeds into troweak symmetry breaking (see, e.g., [11] for a review). m2 , aggravating the situation. Moreover, the MSSM In the supersymmetric limit, electroweak symmetry is Hu potential is rather delicate due to the possible instability not broken. Therefore, electroweak breaking is solely along the D-flat direction H = H . due to the soft supersymmetry breaking effects. It u d The situation would clearly be better if the tree- arises from the renormalization of the up-type Higgs soft level Higgs mass could be raised above the LEP bound. mass-squared that is driven negative by the top Yukawa Modifying Eq. (4), however, necessarily involves ad- coupling. At the one-loop approximation, one finds ditional contributions to the Higgs potential that are 2 not related to the SM gauge couplings. Furthermore, 2 ht 2 MUV ∆mHu 12 2 mt˜ log , (1) reducing the need for (s)top contributions to electroweak ∼− 16π µIR symmetry breaking and the Higgs mass, Eqs. (1) and (3) where MUV (µIR) is the UV (IR) cutoff and ht is the respectively, may help reduce the fine tuning required. top Yukawa coupling. Even with the universal boundary We will see that the Fat Higgs model we propose in this condition mt˜ = mHu , it is easy to see that a large loga- paper achieves both of these aims. rithm between the weak scale and, say, the GUT-scale The simplest extension of the MSSM that raises the 2 makes mHu negative. Assuming the supersymmetric tree-level Higgs mass is the NMSSM. In the NMSSM the

2 µ term is replaced by the superpotential Superfields SU(2)L SU(2)H SU(2)R SU(2)g U(1)R (T 1,T 2) ≡ T 2 2 1 1 0 (T 3,T 4) 1 2 2 1 0 k W = λNH H N 3 (6) (T 5,T 6) 1 2 1 2 1 u d − 3 P 2 1 1 2 1 Q 1 1 2 2 1 where N is neutral under the SM and λ is undetermined. S 1 1 1 1 2 The Higgs quartic coupling therefore depends on λ as S′ 1 1 1 1 2 well as the gauge couplings, potentially allowing for a much higher Higgs mass. Increasing the Higgs mass requires a large λ. This coupling renormalizes upward TABLE I: Field content under SU(2)L × SU(2)H gauge and with increasing energy, eventually encountering a Landau SU(2)R × SU(2)g × U(1)R global symmetries. The U(1)Y pole. At this scale the perturbative description breaks subgroup of SU(2)R is gauged. down and the theory is no longer UV complete. To avoid this problem, it is customary to impose the requirement that all coupling constants, and in particular λ, remain III. THE FAT HIGGS perturbative up to the gauge coupling unification scale. This places an upper bound on λ leading to an upper First we describe the dynamics that leads to elec- bound on the lightest Higgs mass of about 150 GeV troweak symmetry breaking with composite “fat” Higgs [3, 12, 13, 14, 15]. Adding more matter fields can fields. The model is an N = 1 supersymmetric SU(2) relax the bound somewhat, but not much [16, 17, 18]. gauge theory with six doublets, T 1,...,T 6. They carry Even for extensions of the NMSSM with Higgs fields in the quantum numbers given in Table I. other representations this bound is relaxed to at most The tree-level superpotential consists of several terms, mh < 200 GeV [19]. This is the basis for the lore that ∼ the lightest Higgs mass cannot be much higher than in W = W1 + W2 + W3, (7) the MSSM.∗ This is the supersymmetric little hierarchy problem we where intend to solve. We are seeking a theory where the Higgs 1 2 3 4 W1 = yST T + yS′T T (8) mass is allowed to be much larger than mZ at tree-level. 5 6 It is clear that the heart of the problem in the MSSM W2 = mT T (9) − is that the quartic coupling is determined by the gauge 5 5 1 2 T 3 4 T couplings plus radiative corrections. This is the ultimate W3 = y(T ,T )P 6 + y(T ,T )Q 6 (10). source of the tension between the stop masses and the T ! T ! lightest Higgs mass. If the tree-level Higgs mass can be higher there is no need to rely on the radiative corrections The singlet fields S and S′ in W1 are necessary to ensure from the top-stop sector and therefore stops can be light. that electroweak symmetry is indeed broken. W2 is In fact, if all superpartners are around 200–450 GeV, the simply a mass term for the fifth and sixth doublets. natural scale for Higgs soft mass is also around the same The mass parameter m controls the separation between scale, and there is no fine-tuning. the electroweak breaking scale and the compositeness scale, as we will show. Finally, W3 contains the fields In this paper we employ exact results in supersym- P and Q which are two-by-two matrices that transform metric gauge theories to UV complete a strongly coupled as doublets under SU(2)L or SU(2)R and also a global Higgs sector. In our model the Higgs fields are composite SU(2)g. They are present simply to marry off certain bound states of fundamental fields charged under a new, “spectator” composite fields with a mass of order the strongly coupled gauge theory. The supersymmetric compositeness scale. W3 is optional, since these specta- strong dynamics drive electroweak symmetry breaking. tor composite fields also acquire electroweak symmetry The effective theory of the Higgs composites is a vari- breaking masses. But the addition of the P and Q ant of the NMSSM with an arbitrarily strong quartic fields with the above superpotential has the benefit of coupling. Furthermore, unlike the MSSM, the modified minimizing the field content of the low energy effective potential has no flat directions that may cause an theory, which we call the Minimal Supersymmetric Fat instability. Higgs Model. Note also that the overall superpotential is natural if we assign non-anomalous U(1)R charges as shown in Table I. The global symmetries still allow for linear ∗ In [20] a heavier Higgs mass was claimed possible if a stop bound terms in S and S′ in the superpotential that could be state condenses due to a strong trilinear coupling. In [21] a new moderately strong gauge interaction was used to enhance forbidden by an additional Z3 symmetry. This also the Higgs quartic coupling, assuming a large supersymmetry prevents tadpole diagrams for the singlets that could breaking of about 7 TeV in a part of the theory. destabilize the weak scale [22]. Mass terms for the first

3 four doublets, if present, could be eliminated by shifting this running by neglecting corrections from gauge cou- the fields S and S′. Our model is not sensitive to the plings and the top coupling. The solution to the one-loop precise Yukawa couplings in the superpotential, but for renormalization group equation is‡ simplicity we take a common y that is assumed to have 2 the bare value y0 (1). 2 2π ∼ O λ (t)= 2 2 , (18) SU(2)H becomes strong at a scale ΛH . The theory 2π λ− (0) + t has six doublets, so below ΛH it is described by meson composites M = T iT j, (i, j = 1,..., 6) with a dynam- where t log(ΛH /µ) increases towards the infrared. ij ≡ ically generated superpotential PfM/Λ3. Together with Using the NDA estimate λ(0) 4π, we find, for example, ∼ the tree-level terms, λ(4.5) 2, practically independent of the initial value. ≃ If m is well below ΛH , condensation occurs at the scale 1/2 PfM 4πv0 (mΛH ) ΛH where the theory is weakly Weff = mM + ySM + yS′M Λ3 − 56 12 34 coupled∼ and therefore≪ calculable. k,α l,α + yP Mk,α+4 + yQ Ml+2,α+4 (11) This is one of our important results. In the infrared the theory contains Higgs states with a weakly coupled, where k =1, 2 is an SU(2)L index contracted with P ; l = renormalizable superpotential described by just two pa- 1, 2 is an SU(2)R index contracted with Q; and α =1, 2 is rameters, λ and v0. This is a rather nontrivial result that an SU(2)g index. In terms of the canonically normalized depends on the specific choice of an SU(2)H gauge theory fields this becomes with three flavors. Other choices are not so interesting. For example, precision electroweak constraints tend to 2 W = λ(PfM v M )+ m SM + S′M dyn − 0 56 spect 12 34 severely restrict new gauge interactions beyond the SM k,α l,α at the TeV scale, and thus theories with a dual magnetic +P Mk,α+4 + Q Ml+2,α +4 . (12) description are not good candidates. For an SU(Nc) where Naive Dimensional Analysis (NDA) [23] suggests† theory there is a dynamically generated superpotential when Nf = Nc + 1, and its renormalizability requires mΛ v2 H , (13) Nf 3. This means that an SU(2) gauge theory with 0 ∼ (4π)2 three≤ flavors is the unique choice for this purpose. Λ m y H , (14) spect ∼ 4π IV. SCALES λ(Λ ) 4π. (15) H ∼ The crucial observation is that the scale of electroweak Phenomenologically, the scale of supersymmetry symmetry breaking, v0, is generated dynamically and is breaking soft masses must be near the electroweak scale, controlled by the value of the supersymmetric mass m. λv0 mSUSY, because much larger SUSY-breaking It is useful to change the notation for the meson matrix would∼ lead to fine-tuning, whereas a much smaller SUSY- to make the role of different components clear: breaking scale would have already been observed. Using the parameters of the UV theory, this implies mΛH + 0 2 ∼ H M H M (4πmSUSY) . This coincidence of scales is reminiscent N = M , u = 13 , d = 14 , 56 0 of the µ-problem in the MSSM. Here we show that Hu ! M23 ! Hd− ! M24 ! this can be naturally obtained by a combination of (16) and all the other components of the meson matrix de- the seesaw mechanism [24] and the Giudice–Masiero mechanism [25]. This requires conventional gravity- couple near ΛH . These Higgs fields have the dynamically generated superpotential mediated supersymmetry breaking, which we assume for the discussion in this section. 2 The simplest way to relate ΛH to other scales is W = λN(HdHu v0) . (17) − to employ a superconformal theory where the gauge The Hd and Hu doublets play the role of the MSSM coupling remains constant over many decades in energy. 7 8 Higgs doublets. As advertised, this superpotential forces We introduce two extra doublets T and T to SU(2)H electroweak symmetry breaking without relying on su- for this purpose. We assume T 7 and T 8 transform as a persymmetry breaking effects. vector-like pair under other symmetries so that a super- The strong coupling λ rapidly renormalizes to smaller symmetric mass term can be added to the superpotential values as the energy scale is reduced. We can estimate 7 8 W = m′T T . (19)

† Here, the parameters are deﬁned at the scale ΛH , and hence are ‡ not the bare ones. As we will see in the next section, m ∼ 4πm0, This formula assumes that the spectators decouple at scale ΛH . ′ ′ y ∼ 4πy0, and ΛH ∼ m ∼ 4πm0, due to the superconformal This is justiﬁed in the next section where the superconformal dynamics. dynamics enhances y to y ∼ 4πy0 ∼ 4π.

4 Λ Λ ∼4π H~m' 4 a similar factor. Because the superconformal dynamics is likely to be upset by other strong couplings, the largest 12 composites super- asymp- enhancement factor we consider is 4π. 10 conformal totic The next task is to determine how m of the right size can be generated. First, it is assumed that the heav- EWSB: m freedom 8 Λ 1/2 ier vector-like mass m′ is unrelated to supersymmetry v ~(m H ) 0 4π αΗ breaking and therefore arbitrary. The scale for m′ is 6 presumably set by other flavor symmetries, akin to the coupling 4 right-handed neutrino mass which is protected by lepton λ number. However, the symmetries may conspire to forbid 2 a vector-like mass m for the third flavor, analogous to the left-handed neutrino mass in the neutrino mass matrix. 0.01 0.1 1 10 100 1000 For example, consider a simple U(1) flavor symmetry µ/Λ of charge +1 ( 1) for the third (fourth) flavor. The symmetry is broken− by an order parameter of charge +2. Then m′ is allowed in the superpotential while m is FIG. 1: The renormalization of the couplings in our Fat Higgs not. Nevertheless, mixing between the third and fourth model. The model becomes strong and nearly conformal at flavors is allowed by the symmetries and originates from the scale Λ4, where αH nears 4π. The conformal invariance ′ the supersymmetry breaking due to the Giudice–Masiero is broken by the mass of the extra doublet, m , which makes mechanism. Therefore, the form of the mass matrix for ∼ ′ the theory confine at ΛH m . Below this scale the effective these flavors becomes theory description becomes one of meson composites with a coupling λ that quickly renormalizes down to O(1). When 0 mSUSY 4πv0 ≪ ΛH the mesons condense at weak coupling and the . (22) theory is calculable. mSUSY m′ !

2 The light eigenvalue is given by m = m /m′. After This theory with N = 2 and N = 4 is in the super- SUSY c f the conformal dynamics enhances both m and m′, we conformal window [5]. At some scale Λ the SU(2) 2 4 H naturally obtain mm′ (4πmSUSY) as desired. gauge coupling becomes strong and remains strong all ∼ the way down to m′, the supersymmetric vector-like mass of the extra doublets. At the scale m′ the conformal V. FERMION MASSES symmetry is broken and T 7,8 may be integrated out. Below this scale the theory confines and is effectively the In order to incorporate fermion masses, we follow [9] by three flavor model discussed in the previous section. We adding four additional chiral multiplets that are singlets therefore identify the strong coupling scale ΛH with m′. under SU(2)H but have the same quantum numbers as The renormalization group evolution of the couplings is the Higgs doublets H and H in the MSSM, schematically shown in Fig. 1. u d In addition to determining the scale ΛH , the conformal 1 1 ϕ , ϕ¯ (1, 2, + ), ϕ , ϕ¯ (1, 2, ). (23) dynamics generate large anomalous dimensions which u d 2 d u −2 have the effect of enhancing the couplings of the T fields, and therefore also the couplings of the composite They have the superpotential

Higgs fields. The structure of the superconformal algebra 4 3 determines the anomalous dimensions exactly in terms Wf = Mf (ϕuϕ¯u +ϕ ¯dϕd)+ϕ ¯d(TT )+ϕ ¯u(TT ) ij ij ij of the anomaly-free R-charges. Running from the strong +hu Qiujϕu + hd Qidj ϕd + he Liej ϕd. (24) scale Λ4 down to the scale of conformal breaking ΛH , the wave function of the T ’s is suppressed as where Mf is the mass of ϕ andϕ ¯. The only flavor- ij ij γ∗ violating couplings are the Yukawa couplings hu , hd , ΛH ij Z (20) he . We assume Mf m′ ΛH , possibly due to the ∼ Λ4 ∼ ∼ same flavor symmetries that control the size of m′. where γ = 1/2 is the anomalous dimension. Once Between Λ4 and ΛH m′ the superconformal dynam- ∗ ≃ 1/2 the fields are canonically normalized this leads to an ics enhances the Yukawa couplings by (Λ4/ΛH ) 4π, ∼ enhancement of their couplings. For example, the as described in the previous section. After the ϕ’s are effective mass m′ gets enhanced by a factor of integrated out, the effective dimension-5 superpotential is 1/2 Λ4 . (21) 4π ij 3 ij 4 ΛH Wf = hu Qiuj (TT )+ hd Qidj (TT ) Mf In the low energy theory, any operator that involves one h ij 4 Higgs field, such as the top Yukawa, will be enhanced by + he Liej(TT ) . (25) i 5 Below the compositeness scale ΛH , NDA specifies the For a more general set of parameters it becomes 3 4 replacement (TT ) ΛH Hu/4π, (TT ) ΛH Hd/4π. difficult to solve for the vacuum analytically. We take Using M Λ , the→ superpotential becomes→ advantage of the (small) hierarchy f ∼ H ij W = hij Q u H + h Q d H + hij L e H . (26) λv0 m1,2 gv0,g′v0, (32) f u i j u d i j d e i j d ∼ ≫ One may wonder if the Yukawa couplings are sup- which allows us to drop the MSSM D-terms. If m1/m2 is pressed in the low-energy theory due to the wavefunction very large, the quartic term in the potential is dominated renormalization of the Higgs fields due to the strong by the D-term and cannot be ignored. Our solution for coupling λ. Again using the one-loop renormalization the vacuum state applies for a moderate ratio m1/m2 group equation for simplicity, we find where both Higgs masses m1,2 are much larger than mZ . The fact that the Higgs quartic coupling of the MSSM λ(t) 1/4 is negligible compared to that coming from the strong h(t)= h(0) . (27) dynamics illustrates that our model manifestly solves the λ(0) supersymmetric little hierarchy problem, as anticipated For λ(0) 4π and λ(t) 2, we find that the suppression in Section II. For simplicity we also set A = C = 0 for is only 60%.∼ Therefore∼ the mechanism presented above most of the discussions below. With these approxima- does yield sufficiently large Yukawa couplings. tions, the ground state is m m m H0 = v 1 1 2 1 , (33) u 0 − λ2v2 m VI. HIGGS MASS SPECTRUM r 0 r 2 0 m1m2 m2 H = v0 1 , (34) In this section the mass spectrum of the model is d − λ2v2 m r 0 r 1 calculated. The supersymmetric part of the Higgs N = 0 (35) potential is 2 2 up to corrections of order mZ /m1,2. To leading order in V = λ2 H H v2 2 + λ2 N 2( H 2 + H 2), (28) SUSY | d u − 0| | | | u| | d| A and C we find that N no longer vanishes, together with the D-term contributions that are familiar m m (( A + C)λ2v2 + Am m ) N = 1 2 − 0 1 2 , (36) from the MSSM, λ((λ2v2 m m )(m2 + m2)+ m m m2) 0 − 1 2 1 2 1 2 0 2 2 g g′ 2 2 2 2 2 2 while shifts to Hu and Hd are only O(A , AC, C ). This VD = (Hu†~τHu +Hd†~τHd) + ( Hu Hd ) . (29) 8 8 | | −| | demonstrates that a µ-term is naturally generated, giving Unlike the MSSM, electroweak symmetry breaking is a mass of order mSUSY to the Higgsinos. Our vacuum solution was obtained by assuming that caused by the confining dynamics even in the absence 2 of supersymmetry breaking. Nevertheless, the potential m0,1,2 > 0, and thus arises from dynamical (as opposed also contains soft supersymmetry breaking terms to radiative) breaking of electroweak symmetry. Nev- ertheless, we expect a stable vacuum with electroweak 2 2 2 2 2 2 symmetry breaking even if some or all of (mass)2 are Vsoft = m1 Hd + m2 Hu + m0 N | | | | | | negative because our potential Eq. (28)-(31) is bounded +(AλNH H Cλv2N + h.c.) (30) d u − 0 from below, unlike in the MSSM where there is a possible instability along the D-flat direction. We leave such cases where m ,m ,m , A, C m . 1 2 0 SUSY for a future study. It is instructive to first∼ look at a simple case where It is rather convenient to define m1 = m2 = m0, A = C = 0. In this case, we can define the “Standard Model-like Higgs” H = (H0 + H0)/√2 u d ms = √m1m2 (37) whose potential is simply H0 m tan β h ui = 1 , (38) 1 ≡ H0 m V = λ2 H2 2v2 2 + m2 H 2 h d i 2 4 | − 0| 0| | 1 and then the electroweak breaking scale v 174 GeV is = λ2 H2 2 (λ2v2 m2) H 2 + const. (31) ≃ 4 | | − 0 − 0 | | fixed in terms of the parameters of the model, This is no different from the potential in the minimal λ2v2 m2 v2 =2 0 − s , (39) Standard Model. It is clear that electroweak symmetry λ2 sin2β 2 2 2 is broken so long as λ v0 >m0. The vacuum expectation 2 2 2 with the usual W and Z masses value is v = H = 2(v0 m0/λ ), and the mass of the Higgs bosonh isi λv. There− is an exact custodial SU(2) 2 1 2 2 2 1 2 2 2 p m = g v and m = (g + g′ )v . (40) symmetry in the Higgs sector. W 2 Z 2

6 The charged Higgs states have mass I II A0 III 800 2 2 2ms H0 mH± = . (41) sin2β N0 H± The singlet state N has both scalar and pseudo-scalar 600 states that are degenerate (within our simplifying as- 0 A 0 sumption of A = C = 0), 0 A H 0 400 0 N 2 2 2 2 2 0 N 0 mN1 = mN2 = λ v + m0 , (42) h H SM H± SM mass (GeV) h0H± while the pseudo-scalar from the Higgs doublets has mass 0 200 λ λ h λ 2 2 2 2 =3 =2 =2 mA0 = λ v + mH± . (43) tanβ=2 tanβ=2 tanβ=1 ms=400GeV ms=200GeV ms=200GeV The neutral scalars from the doublets have a mass matrix m0=400GeV m0=200GeV m0=200GeV 0 2 2 2 2 2 2 2 (λ v + mH± )cos β (λ v mH± ) sin β cos β 2 2 2 2 −2 2 2 , FIG. 2: Sample Higgs spectra in our Fat Higgs model. (λ v m ± ) sin β cos β (λ v + m ± ) sin β 0 − H H ! In Spectrum I the SM-like Higgs is dominantly h (89%), (44) whereas in Spectrum III the SM-like Higgs is purely H0. where the upper (lower) components correspond to the h0 (h0) deﬁned by the expansion H0 = H0 + u d u,d h u,di 0 heavier 0 hu,d/√2. This mass matrix leads to the eigenvalues and the neutral Higgs H is Standard Model- + 0 like, and the custodial SU(2) triplet is (H ,h , H−). 2 2 2 2 λ v + mH± + X For a general tan β, however, the eigenstates and the m 0 = , (45) H 2 vacuum are not aligned, and both neutral states contain 2 2 2 the Standard Model-like Higgs state. In particular, the 2 λ v + mH± X m 0 = − , (46) mixture is maximal if β =3π/8 (tan β =2.414). h 2 where VII. PHENOMENOLOGY 2 2 2 2 2 2 2 2 X = (λ v + m ± ) 4λ v m ± sin 2β . (47) H − H q Since the theory is supersymmetric, we expect super- They are given in terms of the gauge eigenstates by partners just like in the ordinary MSSM. However, there are important distinctions between our Fat Higgs model 0 0 h cos α sin α hu and the MSSM, especially in the Higgs spectrum. This 0 = − 0 , (48) H ! sin α cos α ! hd ! is the main issue we discuss in this section. where α is the mixing angle that in our model is A. Spectrum 2 2 2 (λ v + mH± )cos2β + X tan α = 2 2 2 . (49) (λ v m ± )sin2β − H To study the phenomenology of the Minimal Super- symmetric Fat Higgs model, we pick three points in the The above expressions will receive corrections from the 2 2 2 2 2 parameter space: D-terms at the level of mZ /mSUSY and mZ /λ v . Here we highlight some of the interesting general fea- λ tan β m (GeV) tures of this spectrum. The model contains singlet scalar s I 3 2 400 and pseudo-scalar states N1 and N2 that are not present (50) in the MSSM. In addition, the pseudo-scalar Higgs A0 II 2 2 200 is always heavier than the charged Higgs, which is the III 2 1 200 opposite of the MSSM. When studying the other states, it is useful to consider two limiting cases, λv m ± and m0 is chosen to be the same as ms for simplicity; changing ≪ H λv mH± . In the ﬁrst case, tan α cot β, and hence m0 merely brings the mass of the N1,2 states up and the≫ lighter eigenstate “aligns” with→− the vacuum. In other down independent of the rest of the spectrum (within words, the lighter neutral Higgs h0 is Standard Model- our simplifying assumption A = C = 0). 0 like, while the heavier state H does not have couplings Spectrum I corresponds to the case mH± > λv where + to ZZ or W W −. It forms a custodial SU(2) triplet the lightest neutral Higgs is Standard Model-like, while + 0 0 with the charged Higgs, (H , H , H−). In the second the heavier neutral Higgs H forms a triplet with the case, tan α cot β. If tan β = 1 they are still aligned, charged Higgses H± under the approximate custodial →

7 SU(2) symmetry. The pseudo-scalar Higgs A0 is heavier 0.6 β than both of them. It resembles the spectrum in the m s=400 GeV, tan =2 0 β MSSM when A is heavy, but the relative ordering of the m s=200 GeV, tan =2 β heavy Higgs states is quite different. m s=200 GeV, tan =1 Spectrum II has a smaller supersymmetry breaking 0.4 mh0=235 scale, and both h0 and H0 have significant Standard Model-like Higgs content, approximately 75% and 25%, respectively. Such a large mixing is unusual in the MSSM 0.2 when the masses are this different. Spectrum III is the most unconventional of all. Be- T 0 360 cause of the exact custodial SU(2), the triplet h and H± 210 are degenerate, and they do not contain the Standard 0 Model-like Higgs component. On the other hand, the 263 heavier neutral Higgs H0 is Standard Model-like. The 350 pseudo-scalar Higgs is even heavier. 68% −0.2 525 SM Higgs B. Electroweak constraints 99% −0.4 It is well known that as the SM-like Higgs mass is −0.4 −0.2 0 0.2 0.4 0.6 raised above about 250 GeV the SM without new physics S is increasingly disfavored by electroweak precision data. In our Fat Higgs model, however, there are several FIG. 3: Constraints on S and T parameters from precision contributions to electroweak observables that are around electroweak data at 68% and 99% confidence levels. The plot the same size as the one from a heavier SM-like Higgs. We assumes U = 0. Contributions of the Fat Higgs model to S have calculated the contribution of the Higgs states to the and T are shown along three trajectories where λ is varied electroweak parameters S, T [26]. The analytical results from 2 to 3 in the direction of the arrow. The endpoints are are the same as the MSSM, and we present formulae in labeled with the mass (in GeV) of the lighter component of Appendix A for completeness. the SM-like Higgs. For comparison, the black line shows the We find that the model is consistent with the exper- contributions to S and T for the Standard Model with various imentally allowed region in the S-T plane, described in Higgs masses between 100 GeV and 1 TeV in increments of Appendix B, with no fine-tuning of model parameters. 100 GeV. As an example of this, we present the S and T contributions for three trajectories in parameter space in Fig. 3. 0.5 In two trajectories, the coupling λ(v0) is varied be- m =175 GeV tween 2 and 3 for tan β = 2 and for two different SUSY t breaking scales. The mass of the lightest component 0.4 no LR mixing of the SM Higgs is shown at the endpoints of each trajectory. Spectrum I of Fig. 2 is at the top of the T 0.3 solid trajectory and Spectrum II is at the bottom of the dashed one. Note that these two spectra are in excellent 0.2 agreement with electroweak constraints despite the heavy Higgs masses of 360 and 210 GeV. Furthermore, we find 0.1 that the constraints for S and T are easily satisfied for a significant range of the model parameters, as the two trajectories demonstrate. 100 200 300 400 500 In the third (dotted) trajectory, λ(v ) is varied between m~ (GeV) 0 bL 2 and 3 for tan β = 1. Spectrum III in Fig. 2 is at the top of this line. This trajectory lies mostly within the 99% CL contour despite the unconventional Higgs spectrum. FIG. 4: Contribution of the stop-sbottom sector to the T - However, it is well known that a stop-sbottom splitting parameter. (m2 = m2 + m2) may contribute significantly to T t˜L ˜bL t (see Fig. 4). This contribution may easily be 0.1–0.5 (e.g. [27]) that would bring these points back into the 68% Nevertheless we stress that even with a negligible CL ellipse. The size of this contribution depends on the contribution to T from the stop-sbottom sector, the masses in the stop-sbottom sytem which can generically Higgs mass can be hundreds of GeV, as shown by be different from the SUSY breaking masses in the Higgs trajectories I and II, and yet still stay well within the sector. precision electroweak contours.

8 C. b → sγ constraint 109 108 Another constraint on our model comes from b sγ → transitions mediated by charged Higgs bosons. Consid- 107 ering only the charged Higgs/top quark contribution, the 6 constraint on the charged Higgs mass is mH± > 350 GeV 10 at 99% CL [28]. There are two ways this constraint could (GeV) H 105 be satisfied. The first is to simply raise the charged Higgs Λ mass above the bound. The second is the well-known 104 possibility that the charged Higgs contribution cancels against the chargino-stop contribution, and therefore al- 103 lows a lighter charged Higgs [29]. This of course depends 200 300 400 500 600 on the specific model and parameters for supersymmetry m (GeV) breaking. H

FIG. 5: The compositeness scale is shown as a function of the Higgs mass, ﬁxing tan β = 1. This was determined by ﬁnding the scale ΛH where λ = 4π, using one-loop renormalization D. Search strategies group evolution.

0 0 The neutral Higgs scalars h and H each may have a E. Cosmology significant component of the Standard Model-like Higgs boson and can thus be discovered by the standard search The NMSSM is known to have a cosmological problem methods, in particular the “gold-plated” signal h SM due to the spontaneous breaking of its Z symmetry that ZZ 4ℓ at the LHC. However, the decay modes H0 → 3 0 0→ + → produces domain walls (see [30] for a recent discussion h h , H H− may also be open, and their partial decay 2 on this issue). Interestingly, our Fat Higgs model does widths are all comparable and proportional to λ m 0 . H not contain such a symmetry and is free from the domain In the strict custodial SU(2) limit, when tan β = 1, the wall problem. The Z3 symmetry used to forbid the linear 1,2 triplet Higgses are produced only in pairs. In particular, terms in S and S′ acts on T with charge +1 and 3,4 there is no production process of the neutral state by T with charge 1, and hence all Higgs fields N, Hu,d, + + − itself, such as e e− Zh or ud¯ W h, when h does quarks, and leptons are neutral under this Z . On the → → 3 not contain the Standard Model-like Higgs component. other hand, R-parity can be imposed consistently and Nevertheless, they do have the top Yukawa coupling and thus the lightest supersymmetric particle is a candidate thus can be produced from the gluon fusion at the LHC. for cold dark matter. Their decays depend very sensitively on the superpartner Finally, note that the charge assignments given in spectrum and the Higgs spectrum. Table I for T 5,6 and the P ’s and Q’s lead to fractionally charged spectators with electric charge 1/2. The In order to positively establish that our model correctly ± describes the Higgs sector, numerous other measurements lightest stable one does not decay, and this could lead to will be needed: the complete mass spectrum, branching problems in early universe cosmology. There are several fractions, and Higgs self-couplings. For this purpose, an options: The first is to simply assume that these particles + are not produced after reheating by restricting the reheat e e− Linear Collider would be a great asset. temperature to be much lower than their mass. A second Once the Higgs mass is measured, we know its quartic possibility is to change the charge assignments of T 5,6 coupling λ. Because of the renormalization group evolu- so that they carry 1/2 hypercharge, and therefore all tion, a lower compositeness scale corresponds to a heavier the low energy composites± have integral charge. We will Higgs mass. This is shown in Fig. 5. The limit of the low see below, however, that leaving the charge assignment compositeness scale is of course of special interest, where as given in Table I allows the simplest interpretation of we may have direct access to the composite dynamics of gauge coupling unification in the model. the Higgs. Because this limit has its own special issues, we will defer the discussion of this case, the “Fattest Higgs”, to a separate paper.§ VIII. UNIFICATION

In this section we complete the discussion of our Fat Higgs model by showing that gauge coupling unification § The limit m → m′ is identical to Technicolorful Supersymme- try [9] except for the presence of the P and Q fields. Our Fat can be easily preserved despite the composite nature of Higgs model is hence an “analytic continuation” of Technicolorful the Higgs fields and the strong coupling. The effective Supersymmetry. theory below the compositeness scale (and below Mf and

9 mspect) has the same matter content as the NMSSM, thus This is the same result obtained for a gauge mediation the gauge couplings run exactly like the MSSM gauge model with three sets of 5 + 5 messengers. couplings until the compositeness scale is reached. That Like gauge mediation, these extra fields have the the couplings can unify above the compositeness scale is appearance of “completing” the would-be incomplete nontrivial, and to show this we will step through each SU(5) matter representations. For example, a gauge contribution to the beta functions in the high energy mediation model with only messenger quark doublets theory (well above the compositeness scale). Qm + Qm is sufficient to communicate supersymmetry The one-loop beta functions for the SM gauge cou- breaking, but as these fields do not form a complete plings are¶ SU(5) representation, additional messenger fields filling up the 10m and 10m must be added to preserve gauge d MSSM 3 ga = ba + ∆ba ga , (51) coupling unification. Unlike gauge mediation, however, dt the extra color triplets D(D) cannot be in the same MSSM where ba are the MSSM contributions and the GUT representation as ϕu,d, ϕu,d, otherwise dimension-6 ∆ba characterize differences between our model and the triplet-induced proton decay will be too fast. In any case, MSSM. Above the compositeness scale our model has no we have shown that adding three pairs of color triplets to fundamental Higgs fields. However, the SU(2)H doublets our model does not affect the dynamics and yet provides T 1,...,T 4 give exactly the same contribution to the an existence proof that gauge coupling unification can beta functions as the two Higgs doublets of the MSSM. work just as well as in the MSSM. Thus the selection of SU(2)H as the strong gauge group It is also important to emphasize that we do not has the interesting side-effect that the fundamental and expect large threshold corrections from passing through composite states give precisely the same contribution the strong coupling/superconformal sector. Due to holo- to the SM gauge beta functions. Since T 5,...,T 8 are morphy, the low-energy gauge couplings are determined neutral under the SM, they do not contribute to the only by the bare mass of the heavy particles that are running of the SM gauge couplings. integrated out [31]. This can also be seen by noting Our model has two new sectors that contribute to ∆ba: that in supersymmetric theories both the exact NSVZ (1) the P and Q fields that marry off spectator composite beta function [32, 33] and the decoupling mass depend fields, and (2) the extra doublets ϕu,d, ϕu,d needed to on the wave-function renormalization factor, which drops generate fermion masses and mixings. out from the final result. Therefore, gauge coupling The first contribution consists of the P and Q fields, unification is unaffected even in the presence of strong which yields SU(2)H dynamics in which the standard model gauge groups SU(3) SU(2) U(1) are perturbatively 3 c × L × Y ∆b1 = , ∆b2 =1 , ∆b3 = 0 (52) coupled. The dominant effect is therefore the threshold 5 correction resulting from potential differences between which corresponds to two SU(2)L doublets and four fields the mass of the color triplets, the mass mspect of the with hypercharge 1/2. spectators, and the mass Mf of the extra doublets ϕ. The second contribution,± from the extra doublets Suppose that the same flavor symmetry that ensures the T 7T 8 doublets acquire the mass m could also be used ϕu,d, ϕu,d needed for fermion masses, is ′ to determine the color triplet masses. In this case the 6 threshold corrections are no larger than log m′/Mf or ∆b1 = , ∆b2 =2 , ∆b3 = 0 (53) 5 log m′/mspect, of the same order of magnitude as the MSSM or GUT threshold corrections, which is much which corresponds to four SU(2)L doublets with hypercharge 1/2. smaller than the leading log Munif/MZ in the MSSM. The total± of (52) and (53) is We have shown that gauge coupling unification can be preserved with a small number of additional matter 9 fields, but it is obvious that we cannot embed the matter ∆b = , ∆b =3 , ∆b =0 . (54) 1 5 2 3 content into a single four-dimensional GUT group. Uni- CouplingX unificationX requires usX to add additional fication of the gauge couplings therefore could be due to string unification or orbifold GUT unification in five [34] matter at the ΛH m′ Mf scale. For instance, we can add three vector-like∼ ∼ pairs of chiral multiplets (or four [35]) dimensions, where the matter content does not need to fall into a GUT representation [36]. Di(Di), (i = 1, 2, 3), with the quantum numbers of the right-handed down quarks, i.e. triplets under SU(3) with U(1)Y quantum numbers 1/3. Then ± IX. DISCUSSION

∆b1 = ∆b2 = ∆b3 =3 . (55) X X X We have constructed a supersymmetric composite Higgs theory that solves the supersymmetric little hierarchy problem. Electroweak symmetry is broken dynami- ¶ We use the SU(5) GUT normalization b1 = (3/5)bY . cally through a new gauge interaction that gets strong

10 at an intermediate scale. The composite Higgs fields APPENDIX A: HIGGS SECTOR have a dynamically generated superpotential that has CONTRIBUTION TO S AND T a form similar to the NMSSM, and hence solves the µ-problem, but with no restriction on the coupling λ. Here we provide expressions for the perturbative con- This allows the tree-level Higgs mass to be much higher, tribution of the Higgs sector to S and T . The Higgs 200-450 GeV, solving the supersymmetric little hierarchy 0 0 0 sector consists of mass eigenstates H±, H , h , and A problem. The usual lore about upper bounds on the which fit into an SU(2)L doublet, lightest Higgs boson mass in supersymmetric theories is therefore obviously violated. With hindsight we see that H+ requiring perturbativity of the Higgs sector was simply , (A1) 1 H0 + iA0 too restrictive. To the best of our knowledge, the Fat √2 ! Higgs model provides the first explicit example where the Higgs sector is composite and yet the dynamics are where H˜ 0 = cos(β α)h0 e sin(β α)H0, and a Standard fully calculable and UV complete. − − − Model-like neutral scalar h˜0, which is orthogonal to H0. There are several interesting future avenues of research. The Higgsinos get a vector-like mass of order mSUSY, We used the Giudice-Masiero mechanism to determine so their contribution to S and T is negligible. h˜0 wille certain mass scales, and therefore supergravity-mediation contribute much like a heavy Standard Model Higgs. was implicit. For generic choices of the supergravity- However, since it is not a pure mass eigenstate, the exact mediated contributions we have the regular supersym- contribution is fairly complicated. We approximate this metric FCNC problem. One solution is a flavor sym- contribution by taking the weighted sum of the two-loop 5 metry, e.g. U(3) in [37]. Another possibility is to contributions extracted from the electroweak observables implement one of several flavor-blind supersymmetry discussed in Appendix B. breaking mechanisms such as gauge mediation [38], Below we summarize the one-loop contribution of the anomaly mediation [39] (supplemented by U(1) D-terms scalar Higgs doublet. Note that the various components to make it viable with UV insensitivity [40]) or its 4D of the doublet all have different masses, and the neutral realization [41], or gaugino mediation in five [42] or scalar H0 is itself a linear combination of two mass four [43] dimensions. It remains to be seen whether eigenstates. Taking this mixing into account, we find: these mediation mechanisms achieve an acceptable mass e spectrum and electroweak symmetry breaking. These 2 ∆S = sin (β α)F (m ± ,m 0 ,m 0 ) methods of supersymmetry breaking would also require H H A 2 − +cos (β α)F (m ± ,m 0 ,m 0 ). (A2) a different mechanism to naturally determine the scales. − H h A It would also be interesting to explore unification further in this model, such as whether SU(2)H can be unified where the function F is defined by with the other SM gauge groups. Finally, we have shown that the Higgs mass spectrum F (m1,m2,m3) is quite unusual. It is important to study specifically 1 1 (1 x)m2 + xm2 = dx x(1 x)log 2 3 . (A3) how our Fat Higgs model can be distinguished from more − 2 2π 0 − m1 conventional supersymmetric models at future collider Z experiments. Clearly more work is needed. We can- Similarly for T , we find not overemphasize the importance of next generation experiments being able to analyze their data with as few 1 2 ± 0 0 theoretical assumptions as possible. ∆T = 2 2 sin (β α)G(mH ,mH ,mA ) 16πmW sW − 2 +cos (β α)G(m ± ,m 0 ,m 0 ) . (A4) − H h A where the function G is defined as Acknowledgments

G(m1,m2,m3) 2 2 We thank Jens Erler, Markus Luty, Michael Peskin = m2I(m3,m2,m1,m2)+ m3I(m2,m3,m1,m3) and Aaron Pierce for discussions. RH and DTL thank 2 2 m I(m2,m1,m1,m1) m I(m3,m1,m1,m1) the Institute for Advanced Study for hospitality. GDK is − 1 − 1 a Frank and Peggy Taplin Member and thanks them for (A5) their generous support of the School of Natural Sciences. This work was supported by the Institute for Advanced in terms of the integral I, Study, funds for Natural Sciences, as well as in part 1 2 2 by the DOE under contracts DE-FG02-90ER40542 and (1 x)m1 + xm2 I(m1,m2,m3,m4)=2 dx x log − 2 2 . DE-AC03-76SF00098 and in part by NSF grant PHY- 0 (1 x)m3 + xm4 0098840. Z − (A6)

11 APPENDIX B: S-T CONTOURS +0.0036S 0.0025T +0.00052lh, − 1 Γl = 84.011+0.12∆α− +0.009∆mt The experimental constraints on the S-T plane can 2 0.19S +0.78T 0.054lh 0.021lh, (B1) be easily computed approximately using the following − − − method. We follow the path of Marciano [44] and where l = log(m /100 GeV). Perelstein–Peskin–Pierce [45] to focus on only three H H 2 2 lept For the experimental values, we use [45] observables, mW , Γl, and s sin θeff from the asymmetries as they are the most∗ ≈ accurately measured m = (80.425 0.034) GeV, (B2) and sensitive observables to the oblique corrections. W ± Expressions for these observables, including their s2 = 0.23150 0.00016, (B3) ∗ ± approximate mt, α, and mH dependence, have been Γl = (83.984 0.086) MeV. (B4) computed by Degrassi and Gambino [46]. We add to ± those expressions the dependence on S and T as found Then χ2 is defined as in Appendix B of Peskin–Takeuchi [26]. The LEP (5) 2 Electroweak Working Group recommends ∆α (mZ )= 2 2 2 had 2 (mW 80.425) (s 0.23150) 0.02761(36), including the BES data as discussed in χ = − 2 + − 2 1 0.034 0.00016 section 16.3 of [47], which implies α− (mZ ) = 128.945 ±1 2 1 2 2 0.049. Expanding to linear order in ∆mt and ∆α− (Γl 83.984) (∆α− ) (∆mt) 1 + − + + , (B5) about mt = 174.3 GeV and α− = 128.945 leads to the 0.0862 0.0492 5.12 expressions which is first minimized with respect to ∆α 1 and ∆m m = 80.380+0.13∆α 1 +0.0061∆m 0.29S − t W − t for each (S,T ). This expression for χ2 yields contours −2 +0.44T +0.34U 0.058lh 0.008lh, that agree very well with those by the Particle Data 2 − 1 − s = 0.23140 0.0026∆α− 0.000032∆mt Group [48]. ∗ − −

[1] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967), A. Salam, ibid. D 58, 119905 (1998)] [arXiv:hep-ph/9703317]. Originally printed in *Svartholm: Elementary Particle [16] T. Moroi and Y. Okada, Mod. Phys. Lett. A 7, 187 Theory, Proceedings Of The Nobel Symposium Held 1968 (1992). At Lerum, Sweden*, Stockholm 1968, 367-377. [17] G. L. Kane, C. F. Kolda and J. D. Wells, Phys. Rev. [2] R. Barbieri and G. F. Giudice, Nucl. Phys. B 306, 63 Lett. 70, 2686 (1993) [arXiv:hep-ph/9210242]. (1988). [18] T. Moroi and Y. Okada, Phys. Lett. B 295, 73 (1992). [3] H. E. Haber and M. Sher, Phys. Rev. D 35, 2206 (1987). [19] J. R. Espinosa and M. Quirós, Phys. Rev. Lett. 81, 516 [4] J. A. Casas, J. R. Espinosa and I. Hidalgo, arXiv:hep- (1998) [arXiv:hep-ph/9804235]. ph/0310137. [20] G. F. Giudice and A. Kusenko, Phys. Lett. B 439, 55 [5] K. A. Intriligator and N. Seiberg, Nucl. Phys. Proc. (1998) [arXiv:hep-ph/9805379]. Suppl. 45BC, 1 (1996) [arXiv:hep-th/9509066]. [21] P. Batra, A. Delgado, D. E. Kaplan and T. M. P. Tait, [6] N. Arkani-Hamed and R. Rattazzi, Phys. Lett. B 454, arXiv:hep-ph/0309149. 290 (1999) [arXiv:hep-th/9804068]. [22] J. Bagger and E. Poppitz, Phys. Rev. Lett. 71, 2380 [7] M. A. Luty and R. Rattazzi, JHEP 9911, 001 (1999) (1993) [arXiv:hep-ph/9307317]. [arXiv:hep-th/9908085]. [23] H. Georgi, A. Manohar and G. W. Moore, Phys. Lett. [8] M. A. Luty, J. Terning and A. K. Grant, Phys. Rev. D B 149, 234 (1984); H. Georgi and L. Randall, Nucl. 63, 075001 (2001) [arXiv:hep-ph/0006224]. Phys. B 276, 241 (1986); M. A. Luty, Phys. Rev. D [9] H. Murayama, arXiv:hep-ph/0307293. 57, 1531 (1998) [arXiv:hep-ph/9706235]; A. G. Cohen, [10] D. B. Kaplan, H. Georgi and S. Dimopoulos, Phys. Lett. D. B. Kaplan and A. E. Nelson, Phys. Lett. B 412, 301 B 136, 187 (1984); H. Georgi and D. B. Kaplan, Phys. (1997) [arXiv:hep-ph/9706275]. Lett. B 145, 216 (1984); M. J. Dugan, H. Georgi and [24] T. Yanagida, in Proceedings of the Workshop on Unified D. B. Kaplan, Nucl. Phys. B 254, 299 (1985). Theory and Baryon Number of the Universe, eds. O. [11] S. P. Martin, “A supersymmetry primer,” in Perspectives Sawada and A. Sugamoto (KEK, 1979) p.95; M. Gell- on Supersymmetry, edited by G.L. Kane, World Scien- Mann, P. Ramond and R. Slansky, in Supergravity, eds. tific, arXiv:hep-ph/9709356. P. van Nieuwenhuizen and D. Freedman (North Holland, [12] M. Drees, Phys. Rev. D 35, 2910 (1987). Amsterdam, 1979). [13] J. R. Espinosa and M. Quirós, Phys. Lett. B 279, 92 [25] G. F. Giudice and A. Masiero, Phys. Lett. B 206, 480 (1992). (1988). [14] J. R. Espinosa and M. Quirós, Phys. Lett. B 302, 51 [26] M. E. Peskin and T. Takeuchi, Phys. Rev. D bf 46, 381 (1993) [arXiv:hep-ph/9212305]. (1992). [15] M. Cvetiˇc, D. A. Demir, J. R. Espinosa, L. L. Everett and [27] M. Drees and K. Hagiwara, Phys. Rev. D 42, 1709 (1990). P. Langacker, Phys. Rev. D 56, 2861 (1997) [Erratum- [28] P. Gambino and M. Misiak, Nucl. Phys. B 611, 338

12 (2001) [arXiv:hep-ph/0104034]. 027 (1998) [arXiv:hep-ph/9810442]. [29] S. Bertolini, F. Borzumati and A. Masiero, Nucl. Phys. B [40] N. Arkani-Hamed, D. E. Kaplan, H. Murayama and 294, 321 (1987); S. Bertolini, F. Borzumati, A. Masiero Y. Nomura, JHEP 0102, 041 (2001) [arXiv:hep- and G. Ridolfi, Nucl. Phys. B 353, 591 (1991). ph/0012103]; I. Jack and D. R. Jones, Phys. Lett. B 482, [30] R. B. Nevzorov, K. A. Ter-Martirosyan and 167 (2000) [arXiv:hep-ph/0003081]. M. A. Trusov, Published in Dubna 2001, Supersymmetry [41] M. A. Luty and R. Sundrum, Phys. Rev. D 65, 066004 and unification of fundamental interactions, 265-268 (2002) [arXiv:hep-th/0105137]; Phys. Rev. D 67, 045007 [arXiv:hep-ph/0110010]. (2003) [arXiv:hep-th/0111231]; R. Harnik, H. Murayama [31] N. Arkani-Hamed and H. Murayama, Phys. Rev. D 57, and A. Pierce, JHEP 0208, 034 (2002) [arXiv:hep- 6638 (1998) [arXiv:hep-th/9705189]. ph/0204122]. [32] V. A. Novikov, M. A. Shifman, A. I. Vainshtein and [42] D. E. Kaplan, G. D. Kribs and M. Schmaltz, Phys. Rev. V. I. Zakharov, Nucl. Phys. B 229, 381 (1983); Nucl. D 62, 035010 (2000) [arXiv:hep-ph/9911293]; Z. Chacko, Phys. B 260, 157 (1985) [Yad. Fiz. 42, 1499 (1985)]; M. A. Luty, A. E. Nelson and E. Pontón, JHEP 0001, Phys. Lett. B 166, 329 (1986) [Sov. J. Nucl. Phys. 43, 003 (2000) [arXiv:hep-ph/9911323]. 294.1986 YAFIA,43,459 (1986 YAFIA,43,459-464.1986)]. [43] C. Csáki, J. Erlich, C. Grojean and G. D. Kribs, [33] N. Arkani-Hamed and H. Murayama, JHEP 0006, 030 Phys. Rev. D 65, 015003 (2002) [arXiv:hep-ph/0106044]; (2000) [arXiv:hep-th/9707133]. H. C. Cheng, D. E. Kaplan, M. Schmaltz and W. Skiba, [34] Y. Kawamura, Prog. Theor. Phys. 105, 999 (2001) Phys. Lett. B 515, 395 (2001) [arXiv:hep-ph/0106098]. [arXiv:hep-ph/0012125]; L. J. Hall and Y. Nomura, Phys. [44] W. J. Marciano, arXiv:hep-ph/0003181. Published in the Rev. D 65, 125012 (2002) [arXiv:hep-ph/0111068]. proceedings of 5th International Conference on Physics [35] C. Csáki, G. D. Kribs and J. Terning, Phys. Rev. D Potential and Development of Muon Colliders (MUMU 65, 015004 (2002) [arXiv:hep-ph/0107266]; H. C. Cheng, 99), San Francisco, California, 15-17 Dec 1999. K. T. Matchev and J. Wang, Phys. Lett. B 521, 308 [45] M. Perelstein, M. E. Peskin and A. Pierce, arXiv:hep- (2001) [arXiv:hep-ph/0107268]. ph/0310039. [36] A. Hebecker and J. March-Russell, Nucl. Phys. B 625, [46] G. Degrassi and P. Gambino, Nucl. Phys. B 567, 3 (2000) 128 (2002) [arXiv:hep-ph/0107039]; L. J. Hall, H. Mu- [arXiv:hep-ph/9905472]. rayama and Y. Nomura, Nucl. Phys. B 645, 85 (2002) [47] LEP Electroweak Working Group, LEPEWWG [arXiv:hep-th/0107245]. 2003-01, “A combination of preliminary electroweak [37] L. J. Hall and L. Randall, Phys. Rev. Lett. 65, 2939 measurements and constraints on the standard model,” (1990). http://lepewwg.web.cern.ch/LEPEWWG/stanmod/ [38] M. Dine, A. E. Nelson, Y. Nir and Y. Shirman, Phys. winter2003/w03_ew.ps.gz Rev. D 53, 2658 (1996) [arXiv:hep-ph/9507378]. [48] K. Hagiwara et al. [Particle Data Group Collaboration], [39] L. Randall and R. Sundrum, Nucl. Phys. B 557, Phys. Rev. D 66, 010001 (2002). 79 (1999) [arXiv:hep-th/9810155]; G. F. Giudice, M. A. Luty, H. Murayama and R. Rattazzi, JHEP 9812,