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Dynamical solution to the $mu$ problem at TeV scale


KAIST-TH 99/04 hep-ph/9906363

Dynamical solution to the ? problem at TeV scale
Kiwoon Choi? and Hyung Do Kim?
Department of Physics, Korea Advanced Institute of Science and Technology Taejon 305-701, Korea

arXiv:hep-ph/9906363v1 15 Jun 1999

Abstract
We introduce a new con?ning force (?-color) at TeV scale to dynamically generate a supersymmetry preserving mass scale which would replace the ? parameter in the minimal supersymmetric standard model (MSSM). We discuss the Higgs phenomenology and also the pattern of soft supersymmetry breaking parameters allowing the correct electroweak symmetry breaking within the ?-color model, which have quite distinctive features from the MSSM and also from other generalizations of the MSSM.

Typeset using REVTEX

? kchoi@higgs.kaist.ac.kr ? hdkim@muon.kaist.ac.kr

1

I. INTRODUCTION

The minimal supersymmetric standard model (MSSM) contains two di?erent types of mass scales: (i) soft supersymmetry (SUSY) breaking parameters msoft including the soft scalar and gaugino masses and (ii) the ? parameter in the superpotential W ? ?H1 H2 where H1 and H2 are the MSSM Higgs doublets with opposite hypercharge. In the MSSM point of view, ? is entirely di?erent from msoft since it has nothing to do with SUSY breaking [2]. In order to have correct electroweak symmetry breaking without severe ?ne tuning, both msoft and ? are required to be of order the electroweak scale. Although it is technically natural that both msoft and ? are much smaller than the cuto? scale of the model which may be as large as the Planck scale MP l , one still needs to understand the dynamical origin of these mass scales for deeper understanding of their smallness [1,2]. It is commonly assumed that msoft arises as a consequence of spontaneous SUSY breaking at high energy scales. The explicit relation between msoft and the scale of spontaneous SUSY breaking depends on how the SUSY breaking is transmitted to the MSSM sector: (i) msoft ? F/MP l in the case of gravity mediation with SUSY breaking auxiliary component F [1] and (ii) msoft ? ( 4α )F/MX in the case of gauge mediation [3] by a messenger particle π √ √ F is signi?cantly larger than 1 TeV, F ? 108 TeV for with mass MX . In both cases √ > 20 TeV for gauge-mediated case [4], so it is quite unlikely gravity-mediated case and F ? that SUSY breaking dynamics can be directly probed by future experiments. About the dynamical origin of ?, there have been many interesting suggestions in the literatures [5–10]. Perhaps the most attractive possibility would be that SUSY breaking dynamics provides a dynamical seed for both ? and msoft in a manner to yield ? ? msoft . In most cases, these schemes are again based on high energy dynamics which is hard to be probed by future experiments. In this paper we wish to propose an alternative scheme replacing ? by a new con?ning force (?-color) at TeV scale which would lead to interesting phenomenologies in future experiments. The ? term is essential in the MSSM for several phenomenological reasons. Its absence implies the absence of the associated B -term (B?H1 H2 ) in the scalar potential, leading to H1 = 0 even when nonzero H2 is radiatively induced by the large top quark Yukawa coupling and also to the phenomenologically unacceptable Weinberg-Wilczek axion [2]. The ? term is necessary also to render su?ciently large masses to the Higgsinos. In the ?-color model, Yukawa couplings of H1,2 with the ?-colored matter ?elds generate e?ective ? terms involving the composite Higgs doublets. The unwanted axion is avoided due to the U (1)P Q breaking by the strong ?-color anomaly, and also the correct electroweak symmetry breaking can be achieved by the combined e?ects of the ?-color dynamics and soft SUSY breaking terms. As we will see, the ?-color model is distinguished from the MSSM (and also from many other generalizations of the MSSM) mainly by its Higgs sector. It is distinguished also by the pattern of soft parameters which would allow the correct electroweak symmetry breaking to take place. Some soft parameter values which would lead to a successful electroweak symmetry breaking within the MSSM can not work within the ?-color model, while others which would not work within the MSSM do work in the ?-color model. For instance, in the ?-color case it is not necessary to have a negative mass squared of H1 or H2 for the electroweak symmetry breaking to take place. As another example of the di?erence, a large 2

portion of the (tan β, Mm ) space in gauge-mediated SUSY breaking models appears to be incompatible with the ?-color model where Mm is the messenger scale of SUSY breaking, though it can be compatible with the conventional ?-term in the MSSM [10]. A potentially unattractive feature of the ?-color model is that it requires that the ?-color gaugino mass at the messenger scale is signi?cantly smaller than the MSSM soft parameters (by the factor of 1/16π 2). In gauge-mediated SUSY breaking models, such a small ?-color gaugino mass can be achieved if the messenger particles are SU (2)? -singlets. In gravity-mediated case, e.g. string e?ective supergravity in which SUSY breaking is mediated by string moduli, the ?-color gaugino mass is small if the ?-color gauge kinetic function does not depend on the messenger moduli at string tree level. Thus the small ?-color gaugino mass may not be a serious drawback of the model. At any rate, we note that string e?ective supergravity models provide large varieties in the pattern of soft parameters [11,12], which are diverse enough to include those giving the correct Higgs phenomenology within the ?-color model.
II. THE MODEL

The minimal ?-color model includes, in addition to the MSSM gauge and matter multiplets, the ?-color gauge group SU (2)? which con?nes at Λ? ? 1 TeV and also the ?-colored matter super?elds which transform under SU (2)? × SU (2)L × U (1)Y as Yαa = (2, 2)0, X1a = (2, 1)1/2 , X2a = (2, 1)?1/2 , (1) where a = 1, 2 and α = 1, 2 denote the SU (2)? and SU (2)L doublet indices, respectively, and the subscripts of the brackets denote the U (1)Y charge. Obviously these additional matters are free from (both perturbative and global) gauge and gravitational anomalies. The MSSM matter parity can be easily generalized to the ?-color model such that the two MSSM Higgs doublets are even while all other matter multiplets are odd under the generalized matter parity. Then the most general scale-free tree-level superpotential with the generalized matter parity is given by Wtree = λ1 H1 Y X1 + λ2 H2 Y X2 +λd H1 QDc + λu H2 QU c + λl H1 LE c , (2)

where H1,2 , Q, U c , D c , L and E c denote the MSSM ?elds in self-explanatory notation, and all the gauge and generation indices are omitted here. For the ?-colored matter contents of Eq. (1), the holomorphic ?-color scale is given by [13] Λ? = MGU T exp(? 2π 2 θ? + i ), 2 (M g? 4 GU T ) (3)

where g? and θ? are the ?-color gauge coupling and vacuum angle, respectively. Once the extra matter multiplets of (1) carrying SU (2)L × U (1)Y charges are introduced, we lose the uni?cation of gauge couplings at single energy scale. However this may not be a serious drawback of the model since there are many string theory models, e.g. heterotic string theory with a large threshold e?ects [14] and/or Type I strings with di?erent type of Dbranes [15], implying that the gauge couplings at the string or uni?cation scale can take 3

di?erent values. At any rate, we note that α? (MGU T ) ? 1/19 and MGU T ? 1016 GeV lead to Λ? ? 1 TeV, so having Λ? at TeV scale is a plausible possibility. A crucial feature of the ?-color model is that there is no mass parameter in Wtree 1 . Thus at scales above Λ? , all the mass parameters are in the soft SUSY breaking terms which are presumed to be induced by SUSY breaking dynamics at scales far above Λ? . For the scale-free tree level superpotential Wtree = λijk Φi Φj Φk , soft SUSY breaking terms can be written as 1 2 a a ? Lsoft = m2 i |Φi | + (Aijk λijk Φi Φj Φk + Ma λ λ + h.c.) 2 1 2 2 2 2 2 ? ? = m2 Y |Y | + mX1 |X1 | + mX2 |X2 | + ( M? λ λ 2 +A1 λ1 H1 Y X1 + A2 λ2 H2 Y X2 + ... + h.c.),

(4)

where Φi in Lsoft corresponds to the scalar component of the corresponding super?eld, λa are gauginos (λ? and M? are the ?-color gaugino and its mass, respectively), and the ellipsis stands for the terms involving only the MSSM ?elds. In this paper, we will not address the origin of these soft parameters, but take an approach to allow generic forms of soft parameters as long as they are phenomenologically allowed. In this regard, we note that string theories with the SUSY breaking mediated by string moduli show enough varieties in the resulting soft parameters [11,12]. Let us discuss some global symmetries and the associated selection rules which will be useful for the later discussion of the e?ective theory below Λ? . In the limit that Wtree , Lsoft , and the standard model gauge couplings are all turned o?, the model is invariant under the SU (4) global rotation of the four SU (2)? doublets X1a , X2a , Y = (Y1a , Y2a ). The model includes also several global U (1) symmetries whose charge assignments are given by U (1)PQ : (Y, X1 , X2 , H1 , H2 , U c , D c , E c , Λ? ) 1 1 1 1 = (? , ? , ? , 1, 1, ?1, ?1, ?1, ? ), 2 2 2 2 U (1)R : (H1 , H2 , λa , Aijk , Ma ) = (2, 2, 1, ?2, ?2), U (1)? : (Y, X1 , X2 ) = (1, ?1, ?1),

(5)

where the super?elds that do not appear in this charge assignment are understood to have vanishing charge. Note that U (1)PQ is explicitly broken by the strong SU (2)? anomaly as ? 2π 2 indicated by that the holomorphic scale Λ? = MGU T exp(? g2 (M + i θ4 ) carries nonzero GU T ) ? U (1)P Q charge. As a result, its spontaneous breaking at scales below Λ? ? 1 TeV does not lead to any phenomenologically harmful axion. U (1)R is free from the SU (2)? anomaly, however broken by the gaugino masses (Ma ) and A-parameters (Aijk ) carrying ?2 units of U (1)R charge. Finally U (1)? corresponds to the ?-baryon number which is exactly conserved within our framework.

This may be explained by the U (1)R symmetry of Eq. (5) which forbids the bilinear terms such as Y Y , X1 X2 , and H1 H2 in the superpotential.

1

4

III. EFFECTIVE THEORY BELOW Λ?

In the limit that msoft ? Λ? and H1,2 ? Λ? , light degrees of freedom at scales below Λ? correspond to SU (2)? -invariant composite super?elds describing SU (2)? D -?at directions [13]. In our case, the light composite ?elds are given by 0 T Z11 Z12 ? ?T 0 Z21 Z22 ? ? ? ? =? ? ?Z11 ?Z21 0 S ? ?Z12 ?Z22 ?S 0
? ?

ZAB

(6)

obeying the constraint [13]: 1 ? 2, Pf(Z ) = ?ABCD ZAB ZCD = ?αβ Z1α Z2β ? ST = Λ 2 where S? Z1α 1 ab 1 ab ? Y1a Y2b , T ? ? X1a X2b , Λ? Λ? 1 ab 1 ab ? X1a Yαb , Z2α ? ? X2a Yαb . ? Λ? Λ? (7)

(8)

Here a, b and α, β are SU (2)? and SU (2)L doublet indices, respectively. For the composite ?elds normalized to have canonical kinetic terms, the supersymmetric naive dimensional argument (NDA) [16,17] leads to ? ≈ Λ? /4π. Λ (9)

The low energy e?ective action of the composite ?elds ZAB can be expanded in powers of 1/Λ? , more precisely in powers of H1,2 /Λ? and/or of msoft /Λ? , where each term in the expansion is consistent with the symmetries and selection rules discussed in the previous section. The NDA rule [16,17] then provides an order of magnitude estimate of the expansion coe?cients at energy scales around Λ? at which the SU (2)? gauge coupling saturates the < 4π . Let us normalize all super?elds to have the canonical kinetic terms. Then bound g? ? applying the NDA rule together with the symmetries and selection rules of the underlying superpotential, we ?nd the following form of the e?ective superpotential ? 2 ) + a1 Λ( ? λ1 H1 Z1 + λ2 H2 Z2 ) + WMSSM , We? = X (Z1 Z2 ? ST ? Λ (10)

where WMSSM stands for the Yukawa terms involving only the MSSM super?elds, a1 is a nonperturbative parameter of order unity, and the SU (2)L gauge indices are omitted. Here the Lagrange multiplier super?eld X is introduced to implement the constraint (7). Note that X is not a dynamical ?eld and so does not appear in the K¨ ahler potential. There may be additional terms in We? which are higher order in 1/Λ? , but the NDA rule suggests that the e?ects of such higher order terms are suppressed by more powers of H1,2 /Λ? . As will be argued in the subsequent discussions, msoft and the Higgs VEVs are all comparable to ? ≈ Λ? /4π in our framework, and then the 1/Λ? expansion whose coe?cients obey the Λ 5

NDA rule becomes essentially an expansion in powers of 1/4π . Though not a terribly good approximation, we expect that this expansion is reasonably good and thus the leading order results are not signi?cantly modi?ed by higher order corrections. In the ?-color model, there are four doublet VEVs participating in the electroweak symmetry breaking: H1
2

+ H2

2

+ Z1

2

+ Z2

2

= (178 GeV)2 .

(11)

If any of S and T developes a nonzero VEV, U (1)? will be spontaneously broken, leading to a potentially dangerous Goldstone boson. To avoid this problem, we assume S = T = 0 2 which can be easily achieved by choosing appropriate values of m2 S and mT . Then the 2 2 2 2 ? , and so Z1 + Z2 > 2Λ ? . Furthermore, one would constraint (7) gives Z1 Z2 = Λ ? require H2 not signi?cantly smaller than 100 GeV in order to avoid a too large top quark ? < 110 GeV where the upper limit is Yukawa coupling. Combining these, one ?nds Λ ? ? In most cases, it is phenomenologically desirable to have saturated when Z1 ≈ Z2 ≈ Λ. ? close to its upper limit, and then we have Λ ? ? 1 TeV. Λ ? = 4π Λ (12)

Soft SUSY breaking terms of the composite ?elds ZAB can be similarly expanded in powers of msoft /Λ? (and also of H1,2 /Λ? ) where msoft denote the soft parameters of the ?-colored elementary ?elds renormalized at the NDA scale. At the leading order, we ?nd2 ? Le? soft where m2 S m2 T 2 mZ1 m2 Z2 ?1 A
2 2 = a2 (m2 X1 + mX2 ) + a3 |M? | , 2 = 2a2 m2 Y + a3 |M? | , 2 2 = a2 (m2 X1 + mY ) + a3 |M? | , 2 2 = a2 (m2 X2 + mY ) + a3 |M? | , 2 2 2 2 2 2 2 = m2 S |S | + mT |T | + mZ1 |Z1 | + mZ2 |Z2 | ?1 λ1 Λ ? H1 Z1 + A ?2 λ2 Λ ? H2 Z2 + A ?3 Λ ? 2 X + h.c.), + (A

(13)

= a1 A1 + a4 M? ,

?2 = a1 A2 + a4 M? , A ?3 = a5 M? . A

(14)

Here the nonperturbative parameters ai (i = 2, 3, 4, 5) are again of order unity when the soft parameters of the ?-colored elementary ?elds are renormalized at the NDA scale Λ? at which g? (Λ? ) ? 4π .
The soft SUSY breaking scalar potential includes also the additional A-term: AX (Z1 Z2 ? ST ? 2 ? Λ ), but this can be eliminated by the rede?nition of the F -component of the Lagrange multiplier: FX → FX + AX .
2

6

When it is runned from the messenger scale Mm of SUSY breaking to Λ? , the ?-color gaugino mass is enhanced by the nonperturbative factor ? 16π 2 : M? (Λ? ) ?
2 g? (Λ? )M? (Mm ) ? (4π )2 M? (Mm ). 2 (M ) g? m

Furthermore if the soft SUSY breaking at Λ? is dominated by M? , the renormalization group evolution makes the other soft parameters of the ?-colored ?elds at Λ? , i.e. m2 X1,2 , 2 mY and A1,2 , to be comparable to M? (Λ? ) also. Thus if M? were comparable to the soft parameters of the MSSM ?elds at Mm , there will arise a 16π 2 -hierarchy between the MSSM soft parameters and the soft parameters of the ?-colored ?elds at the NDA scale Λ? , and thus the same hierarchy between the MSSM soft parameters and the soft parameters of the composite ?elds ZAB = {S, T, Z1 , Z2 }. In order to provide a consistent framework, the soft parameters of both ZAB and the MSSM ?elds at the electroweak scale are required to be ? comparable to Λ . This means that at Mm the ?-color gaugino mass must be smaller than 4π the MSSM soft parameters by the factor of 161 : π2 < M? (Mm ) ? 1 msoft (Mm ). 16π 2 (15)

In gauge-mediated SUSY breaking models [3], such a small ?-color gaugino mass can be achieved if the messenger particles are SU (2)? -singlets. In gravity-mediated case, e.g. string e?ective supergravity models in which SUSY breaking is mediated by string moduli, M? (Mm ) is small if the ?-color gauge kinetic function does not depend on the messenger moduli at string tree level [11,12].
IV. HIGGS PHENOMENOLOGY

The key di?erence between the ?-color model and the MSSM is in the Higgs sector. To see this, let us consider the neutral Higgs sector of the model in more detail. For notational simplicity, in this section let Z1,2 and H1,2 denote the neutral components of the corresponding composite and elementary Higgs doublets. Due to the exact ?-baryon 2 symmetry (U (1)? ), one can always adjust the parameters of the model, e.g. m2 S and mT , to have S = T = 0. We then have ?ve neutral complex scalar ?eld ?uctuations (δ Φ = Φ ? Φ ) with masses of order the electroweak scale: the two composite singlet Higgs ?uctuations δS and δT , the two elementary doublet Higgs ?uctuations δH1 and δH2 , and ?nally one linear combination of the composite Higgs doublet ?uctuations δZ1 and δZ2 obeying the constraint Z1 δZ1 + Z2 δZ2 = 0. In particular, we have three physical scalar and two pseudo-scalar particles arising from the neutral components of the doublet Higgs ?uctuations. To study the electroweak symmetry breaking and the Higgs mass spectrum, let us consider the scalar potential of the Higgs doublets while setting S and T to their vanishing VEVs. We ?rst have the F -term potential arising from the superpotential: ? H1 |2 + |XZ1 + λ2 Λ ? H2 | 2 VF = |XZ2 + λ1 Λ ? 2 Z 1 | 2 + | λ2 Λ ? 2 Z2 |2 ? (FX (Z1 Z2 ? Λ ? 2 ) + h.c.). +| λ 1 Λ 7 (16)

and also the contribution from soft SUSY breaking: ?1 λ1 Λ ? H1 Z1 + A ?2 λ2 Λ ? H2 Z2 + A ?3 Λ ? 2 X + h.c.) Vsoft = (A 2 2 2 2 2 2 2 +m2 H1 |H1 | + mH2 |H2 | + mZ1 |Z1 | + mZ2 |Z2 | . Then the quation of motion for the auxiliary ?eld X yields X =? leading to ? Z 1 | 2 + | λ2 Λ ? Z 2 | 2 + | λ1 Λ ? H 1 | 2 + | λ2 Λ ? H2 | 2 VF + Vsoft = |λ1 Λ ? ? ? ?? ? ?2 2 | (λ 1 H 1 Z 2 + λ2 H 2 Z 1 )Λ + A 3Λ | ? 2 ) + h.c.) ? ? (FX (Z1 Z2 ? Λ |Z1 |2 + |Z2 |2 ?1 λ1 Λ ? H1 Z1 + A ?2 λ2 Λ ? H2 Z2 + +h.c.) +(A
2 2 2 2 2 2 2 +m2 H1 |H1 | + mH2 |H2 | + mZ1 |Z1 | + mZ2 |Z2 | . ? ? ? ?? Λ ? ?2 (λ 1 H 1 Z 2 + λ2 H 2 Z 1 )Λ + A 3 , |Z1 |2 + |Z2 |2

(17)

(18)

(19)

There is also the D -term potential 1 VD = (g 2 + g ′2 )(|H1 |2 ? |H2 |2 ? |Z1 |2 + |Z2 |2 )2 . 8 Putting these together, V = VF + VD + Vsoft , (21) (20)

we see that the Higgs potential takes a form very di?erent from that of the MSSM or of other generalizations of the MSSM. Since the Higgs potential takes so di?erent form, the soft parameter ranges for successful electroweak symmetry breaking can be di?erent also. Some soft parameter ranges which would not lead to the correct electroweak symmetry breaking within the MSSM, e.g. positive 2 m2 H1 and mH2 at the electroweak scale, can successfully generate the symmetry breaking in the ?-color framework, while some others which would work in the MSSM do not work within the ?-color framework. To see this more explicitly, let us consider the case that all Higgs doublet VEVs can be chosen to be real. Then the vacuum stability condition includes ?2V ? Re(H2 )2 which corresponds to 2 | λ2 Λ | 2 ?
2 g 2 + g ′2 2 | λ2 Λ | 2 Z 1 2 2 2 2 2 (3H2 ? H1 + Z1 ? Z2 ) ≥ 0, + 2 m + H 2 2 2 Z1 + Z2 2

≥ 0,

(22)

(23)

where all Higgs ?elds mean their VEVs which are assumed to be real. Combining this with Eqs. (7) and (11) which imply (mZ = the Z -boson mass) ? |2 < Z 2 + Z 2 < 4m2 , 2| Λ ? 1 2 ? Z 8 (24)

one easily ?nds (with g 2 + g ′2 ≈ 0.5)
2 2 < 3m2 + |Λ ? | 2 (| λ 2 | 2 ? 1 ? | λ 2 | Z 1 ) ? 1 Z 2 ? m2 H2 ? Z 2m2 2 2 Z ?| | λ2 Λ 2 < 3m2 + (2|λ2 |2 ? 1)|Λ ? |2 . ? )| Λ | ? ( Z ? m Z

(25)

< 1.5, the above limit gives For |λ2 | ?

2 > m2 H2 ? ?(174 GeV) ,

(26)

which is in con?ict with the large portion of the (tan β, Mm ) space in gauge mediated SUSY breaking models [10] where Mm is the messenger scale of SUSY breaking. This shows that the ? color model can be incompatible with certain soft parameter ranges which would be ?ne with the conventional ? term in the MSSM. Since the Higgs potential of the ?-color model is too complicate to get analytic vacuum solutions for generic parameter values, here we consider two cases one of which allows an analytic solution, while the other requires numerical analysis. The ?rst case is when the parameters renormalized at the electroweak scale are all real and obey λ1 ≈ λ2 , ?1 ≈ A ?2 , m2 ≈ m2 , A Z1 Z2 2 ?A ?1 > 0. ≈ mH2 ≈ λ1 Λ (27)

m2 H1

In this case, it is straightforward to ?nd that the Higgs potential has a (local) minimum at ?, H1 ≈ H2 ≈ Z1 ≈ Z2 ≈ Λ (28)

where all parameters are assumed to be real. The neutral components of the four Higgs ? 2 contain three physical scalar and two doublets, H1,2 and Z1,2 , constrained as Z1 Z2 = Λ pesudoscalar particles. After a tedious but still straightforward computation, we ?nd the scalar mass eigenvalues are given by
2 2 2 2 (scalar mass)2 ≈ (m2 H1 , m1 + m2 , m1 ? m2 ),

(29)

where
2?2 2 2 ′2 ? 2 m2 1 = 2λ1 Λ + mZ1 + (g + g )Λ ? 4 + 2m2 λ2 Λ ? 2 + 2λ2 (g 2 + g ′2)Λ ? 2 + m4 m4 = 2λ4 Λ 2

(30)

?

1 2?2 2 4λ1 Λ mH1

Z1 1

1

Z1

?

2 2m2 Z1 mH1

+ (g + g ) Λ ,

2

′2 2 ? 4

and also the pseudoscalar mass eigenvalues
2 2 ?2 (pseudoscalar mass)2 ≈ (m2 H1 , 2mH1 + 2λ1 Λ ).

(31)

? ≈ 90 GeV is ?xed by Eq.(11), and the Higgs spectrums are The ? color con?ning scale Λ distributed in hundred GeV range if the soft parameters are also in few hundred GeV range. The lightest Higgs mass can be large enough to satisfy the current experimental lower bound, 9

particularly when the one-loop corrections involving the large top Yukawa coupling are taken into account. Di?erent types of VEVs and spectrums are obtained by alleviating the relations among 2 the parameters given in (27). Note that the conditions in (27), especially m2 H1 ≈ mH2 at the electroweak scale, are di?cult to be achieved in the popular minimal supergravity model or gauge mediated models, though possible in string theory models with moduli-mediated SUSY breaking [11,12]. As another example of the successful Higgs phenomenology, we consider the parameter values at the electroweak scale: ?1 ≈ A ?2 ≈ 70 GeV, λ1 ≈ 0.5, λ2 ≈ 0.9, A 2 2 2 2 m2 m2 Z1 ≈ mZ2 ≈ mH1 ≈ (270 GeV) , H2 ≈ ?(30 GeV) . We then ?nd the Higgs VEVs given by H1 ≈ 7 GeV, H2 ≈ 120 GeV, (33) (32)

? ≈ 90 GeV, Z1 ≈ Z2 ≈ Λ and also the masses of the three scalar and two pseudoscalar neutral Higgs particles scalar mass = (90, 270, 390) GeV pseudoscalar mass = (92, 270) GeV.

(34)

If we include the loop corrections involving the top Yukawa coupling, the scalar mass can be increased by 10 ? 30 GeV depending on the top-stop mass ratio.
V. CONCLUSION

In this paper, we introduced a new con?ning force, the ?-color, at TeV scale to replace the ? parameter in the MSSM superpotential. Below the ?-color scale, the model predict composite Higgs doublets and singlets whose mass spectrum has been analyzed for certain parameter range. The ? color model has very distinctive electroweak symmetry breaking mechanism which di?ers entirely from the conventional radiatively generated one. Electroweak symmetry is broken by the ? color dynamics together with soft SUSY breaking terms. The soft parameter ranges for successful electroweak symmetry breaking can be quite di?erent from the MSSM and other generalizations of the MSSM. Some soft parameter ranges which would not lead to the correct electroweak symmetry breaking within the 2 MSSM, e.g. positive m2 H1 and mH2 at the electroweak scale, can successfully generate the symmetry breaking in the ?-color framework, while some others which would work in the MSSM do not work within the ?-color framework. It would be fair to ?nally summarize the potentially unattractive features of the ?color model which have been noticed in sections III and IV. First, we lose the uni?cation of gauge couplings at single scale due to the extra ?-colored matter multiplets carrying SU (2)L × U (1)Y charges. However this may not be so serious in view of that many string theory scenarios imply that generically gauge couplings at the string or uni?cation scale can take di?erent values. Second, in order to implement the electroweak symmetry breaking 10

without ?ne tuning, it is required that the ?-color gaugino mass at the SUSY breaking messenger scale Mm is smaller than the MSSM soft parameters by the factor of 1/16π 2. Such a small ?-color gaugino mass can be easily achieved within gauge-mediated and/or gravity-mediated SUSY breaking models. In particular, string e?ective supergravity models would give a small ?-color gaugino mass if the ?-color gauge kinetic function does not depend on the messenger moduli at string tree level [11,12]. Finally the model does not provide a rationale for ? ? msoft since these two mass scales have di?erent dynamical origin. Even with these features, it appears to be worthwhile to study the phenomenological aspects of the ?-color model in view of its very rich phenomenologies at TeV scale.
ACKNOWLEDGMENTS

We thanks S. Y. Choi and H. B. Kim for helpful discussions.

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