Supersymmetric Standard Models Coupled with Superconformal Field Theories

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Sfermion masses in Nelson-Strassler-type models: Supersymmetric standard models coupled with superconformal field theories

著者 Kobayashi Tatsuo, Terao Haruhiko journal or Physical Review D publication title volume 64 number 7 page range 075003 year 2001-09-01 URL http://hdl.handle.net/2297/3497 PHYSICAL REVIEW D, VOLUME 64, 075003

Sfermion masses in Nelson-Strassler-type models: Supersymmetric standard models coupled with superconformal ﬁeld theories

Tatsuo Kobayashi* Department of Physics, Kyoto University, Kyoto 606-8502, Japan

Haruhiko Terao† Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan ͑Received 12 March 2001; published 4 September 2001͒ We study soft supersymmetric ͑SUSY͒ breaking parameters in the Nelson-Strassler-type models: SUSY standard models coupled with superconformal ﬁeld theories ͑SCFT’s͒. In this type of model, soft SUSY breaking parameters including sfermion masses can be suppressed around the decoupling scale of SCFT’s. We clarify the condition to derive exponential suppression of sfermion masses within the framework of pure SCFT’s. Such behavior is favorable for degeneracy of sfermion masses. However, the realistic sfermion masses are not quite degenerate due to the gauge couplings and the gaugino masses in the standard model sector. We show the sfermion mass spectrum obtained in such models. The aspect of suppression for the soft SUSY breaking parameters is also demonstrated in an explicit model. We also give a mechanism generating the ␮ term of the electroweak scale by a singlet ﬁeld coupled with the SCFT.

DOI: 10.1103/PhysRevD.64.075003 PACS number͑s͒: 12.60.Jv, 11.10.Hi, 12.15.Ff

I. INTRODUCTION heavy sfermion masses,1 and ͑3͒ the alignment of the fermion and the sfermion bases. Recently, supersymmetric standard models ͑SSM’s͒ Understanding the origin of flavor structure, i.e., hierar- coupled with superconformal field theories ͑SCFT’s͒ have chical fermion masses and mixing angles, is one of the most been discussed by Nelson and Strassler ͓8͔. Here the SCFT important issues in particle physics. Actually, several types means the theory realized at a nontrivial IR fixed point. Such of mechanisms to realize the fermion mass matrices have fixed points are known to exist according to the discussions ͓ ͔ 2 been proposed: e.g., the Froggatt-Nielsen mechanism ͓1͔ and given in Ref. 9 . Within this framework, quark and lepton ͑ ͒ some new ideas concerned with extra dimensions ͓2,3͔. fields coupled with the superconformal SC sector have en- hanced anomalous dimensions due to strong gauge and Supersymmetry extension is one of the most attractive Yukawa couplings in the SC sector around IR fixed points. ͑ ͒ ways beyond the standard model SM . In supersymmetric The anomalous dimensions lead to the hierarchically sup- models, realization mechanisms of fermion mass matrices, in pressed Yukawa couplings at low energy in the SSM sector general, affect the sfermion sector. Each realization mecha- even if those are of O(1) at high energy. Thus, this can nism of the flavor structure would lead to a proper pattern of provide one type of mechanism to generate realistic quark the sfermion mass matrices as well as supersymmetry and lepton mass matrices. ͑SUSY͒ breaking trilinear couplings. For example, within the Superconformal IR fixed points have more intriguing as- ͑ ͒ framework of the Froggatt-Nielsen mechanism with gauged pects for renormalization-group RG behavior of SUSY breaking parameters. For example, the IR behavior of softly extra symmetries, sfermion masses have the so-called D-term ͓ ͔ contributions, which are proportional to charges of fermions broken supersymmetric QCD has been studied in Ref. 14 and it has been shown that the gaugino mass and the squark under broken symmetries. Alternatively, if the Yukawa cou- ͑ ͒ masses are exponentially suppressed around the IR fixed plings are subject to infrared IR fixed points in the manner point.3 Furthermore, its dual theory is described in terms of ͓ ͔ of Pendelton and Ross 4 , we have specific relations among dual quarks and singlet ͑meson͒ fields ͓9͔. In the dual side, the soft SUSY breaking terms ͓5͔. Thus, the study of the SUSY breaking trilinear couplings are suppressed. Moreover sfermion sector is interesting to distinguish several types of the soft scalar masses of the singlet fields, as well as the sum realization mechanisms of the flavor structure. Furthermore, of ͑mass͒2 for the dual squark and its conjugate, are found to the sfermion sector has severe constraints due to experiments be suppressed. of flavor changing neutral current ͑FCNC͒ processes as well In Ref. ͓8͔, it has been mentioned that the above behavior as CP physics ͓6͔. FCNC problems can be solved by three of suppressed soft scalar masses around the IR fixed point ways: ͑1͒ degenerate sfermion masses, ͑2͒ decoupling of can be useful to avoid the dangerous FCNC processes. Be- cause it is expected that sfermion masses at least for the first and the second families would be quite suppressed at the *Email address: [email protected] †Email address: [email protected] 1 The recent measurement of the muons anomalous magnetic mo- 2See Ref. ͓10͔ for a review of superconformal theories and their ment ͓7͔ disfavors the decoupling solution at least for the slepton dual descriptions. The IR fixed points have been discussed also in sector, if the deviation from the prediction of the SM is indeed due Refs. ͓11,12,13͔. to superpartners. 3See also Ref. ͓15͔.

0556-2821/2001/64͑7͒/075003͑15͒/$20.0064 075003-1 ©2001 The American Physical Society TATSUO KOBAYASHI AND HARUHIKO TERAO PHYSICAL REVIEW D 64 075003 decoupling scale of the SC sector, and that after decoupling, II. EXACT RESULTS FOR SOFT MASSES IN SCFT the masses receive radiative corrections due to gaugino A. Beta functions masses of the SSM sector, which are flavor blind. Thus, sfermion masses could be degenerate at low energy for any ini- In this section, we are going to discuss the IR behavior of tial condition at high energy. the soft parameters added to generic SCFT’s. In particular, From the above-mentioned phenomenological viewpoint, soft scalar masses will be found to satisfy interesting sum it is quite interesting to study the IR behavior of softly bro- rules. Our argument is based on the explicit form of the beta functions for soft parameters ͓16–23͔. Therefore, we first ken SCFT’s and SSM’s coupled to the SC sector, and to review the exact beta functions of general softly broken su- clarify the conditions leading to exponentially suppressed persymmetric gauge theories in this section. SUSY breaking terms. That would provide useful constraints Let us begin with the gauge coupling and the correspond- for model building. ing gaugino mass. The holomorphic gauge coupling S In this paper, we first study the IR behavior of general ϭ 2 ͑ ͒ softly broken SCFT’s by means of the so-called exact beta 1/2gh satisfies the RG equation RGE : functions. The conditions to realize suppressed SUSY break- dS 1 ing terms will be shown. It is also shown that some fields ␮ ϭ Ϫ ͑ ͒ ␮ ␲2 ͩ 3TG ͚ Ti ͪ , 1 become tachyonic in generic cases. These aspects around the d 16 i IR fixed point are also useful for phenomenology. Next we discuss phenomenological aspects of SSM’s coupled with the where Ti is the Dynkin index and TG denotes the Dynkin ͑ ͒ SC sector. Taking account of the effects due to gaugino index quadratic Casimir of adjoint representation. The masses of SSM’s, the sfermion masses are found to converge physical coupling g is related to the holomorphic coupling to flavor dependent values. We study this flavor dependence through the general formula of the sfermion masses that remained after suppression, and 1 show how much degeneracy between sfermion masses is fi- ␲2 ϩ †͒Ϫ ϭ ϩ ␣ϩ ␣n 8 ͑S S ͚ Ti ln Zi TG ln ͚ an nally achieved at the weak scale. In practice, the range of i ␣ nϾ0 scale where the SSM’s couple with the SC sector must be ϵ ͑␣͒ ͑ ͒ finite to generate the small but nonvanishing Yukawa cou- F , 2 plings. Therefore, the sfermion masses do not totally con- ␣ϭ 2 ␲2 verge at the decoupling scale. We also discuss the amount of where g /8 and Zi denotes the wave-function renor- ␾i convergence by demonstrating the RG flows for explicit malization of the chiral superfield . The coefficients an are models. We also mention that the SCFT may resolve the ␮ the scheme-dependent constants and the Novikov-Shifman- ͑ ͒ ͓ ͔ problem in SSM’s in a natural way. We will show a model in Vainstein-Sakharov NSVZ scheme 24 is given by an ϭ ␣ which soft SUSY breaking mass of a singlet coupled with 0. From this relation the beta function for is given ex- SCFT’s becomes tachyonic and its vacuum expectation value actly as appears of the weak scale automatically. If this singlet ␣ couples to the Higgs fields, then the ␮ term may be gener- d 1 ␤␣ϭ␮ ϭ ͫ 3T Ϫ͚ T ͑1Ϫ␥ ͒ͬ, ͑3͒ G i i ated through the vacuum expectation value of the weak a␮ FЈ͑␣͒ i scale. ␥ This paper is organized as follows. In Sec. II, we study where the anomalous dimension i is defined by the IR behavior of pure SCFT’s with soft SUSY breaking terms, and show the condition for suppressed sfermion d ln Zi ␥ ϭϪ . ͑4͒ masses. We also give speculative considerations on SCFT’s i d ln ␮ with nonrenormalizable couplings and corresponding SUSY breaking terms. In Sec. III, we study SSM’s coupled with the Here we assume the wave-function renormalization to be di- SC sector. In Sec. III A we give a brief review on the setup of agonal just for simplicity. Nelson-Strassler models, and also a constraint for the decou- The gaugino mass can be incorporated with the gauge pling scale of the SC sector is given. In Sec. III B it is found coupling by superfield extension. The holomorphic coupling that gaugino masses in the SSM sector have important mean- is extended as ing for exponential suppression of soft scalar masses, which is also significant from the viewpoint of FCNC constraints. 1 ͑ ͒ ˜Sϭ ͑1ϪM␪2͒. ͑5͒ Within the framework of the minimal SSM MSSM ,we 2g2 show numerically how sfermion masses are strongly degen- h erate at the weak scale. Also, a typical mass spectrum is Here ˜S satisfies the same RGE as S does. On the other hand, shown in Sec. III C. In Sec. IV, we consider explicit models we also extend the wave-function renormalization factors Z showing the desired suppression. After discussing the types i and the physical gauge coupling to real superfields as for such models in Sec. IV A, we demonstrate typical RG flows in an illustrating model in Sec. IV B. The convergence ˜ ϭ˜ ͑ Ϫ 2␪2¯␪2͒ ន of the sfermion masses are also examined there. In Sec. V, Zi Z␾i 1 mi Z␾i , the ␮ problem is discussed. Section VI is devoted to discus- ␣ϭ␣ ϩ ␪2ϩ ¯ ¯␪2ϩ ¯ ϩ⌬ ͒␪2¯␪2 ͑ ͒ sions and the conclusion. ˜ ͓1 M M ͑2MM g ͔, 6

075003-2 SFERMION MASSES IN NELSON-STRASSLER-TYPE... PHYSICAL REVIEW D 64 075003

2 ijk where M and mi give the gaugino mass and the soft scalar dh 1 ␤ijkϭ␮ ϭ ͑␥ ϩ␥ ϩ␥ ͒hijkϪ͑␥͑1͒ϩ␥͑1͒ masses, respectively. We have extracted the chiral and the h d␮ 2 i j k i j ˜ ¯ ន antichiral parts of Zi as Z␾i and Z␾i for the wave-function ͑ ͒ ⌬ ϩ␥͑1͒͒ ijk ͑ ͒ renormalization of the antichiral matter fields. Here g is k y . 15 determined by consistency with the extended relation ˜ It has been known that the wave-function superfields Zi are also given by the extension of Z (␣,y ijk,¯y ) ͓19͔: ␲2 ˜ ϩ˜ †͒Ϫ ˜ ϭ ␣͒ ͑ ͒ i ijk 8 ͑S S ͚ Ti ln Zi F͑˜ , 7 i ˜ ϭ ␣ ijk ន ͒ ͑ ͒ Zi Zi͑˜ ,˜y ,y ijk , 16 and is found out to be where the extended Yukawa couplings ˜y ijk are defined by 1 2 2 ijkϭ ijkϩ 1 ͑ 2ϩ 2ϩ 2͒ ijk␪2¯␪2 ͑ ͒ ⌬ ϭ ͚ͫ T m Ϫ͓␣ FЈ͑␣͔͒ЈMM¯ ͬ. ͑8͒ ˜y Y 2 mi m j mk y . 17 g i i ␣FЈ͑␣͒ i ˜ Therefore, the superfields Zi are given explicitly in terms of In the NSVZ scheme, ⌬ is given by ͓20–22͔ g the rigid factor Zi as

␣ ˜Z ϭZ ϩD Z ␪2ϩD¯ Z ¯␪2ϩD Z ␪2¯␪2, ͑18͒ ⌬ ϭϪ ͚ͫ T m2ϪT MM¯ ͬ. ͑9͒ i i 1 i 1 i 2 i g Ϫ ␣ i i G 1 TG i where D1 and D2 are the differential operators deﬁned by ץ ץ The beta function of the gaugino mass can be derived from the extended relation by expanding with ␪2 and found ϭ ␣ Ϫ ijk D1 M h ijk , yץ ␣ץ to be

1 ץ ␣ dM d ln ˜ 1 ϭ ¯ ϩ͑ ¯ ϩ⌬ ͒␣ ϩ ͑ 2ϩ 2ϩ 2͒ mi m j mk ␣ץ ␮ ϭ␮ ͯ ϭ ͫ 3T Ϫ͚ T ͑1Ϫ˜␥ ͒ͬͯ , D2 D1D1 MM g d␮ d␮ ˜␣FЈ͑˜␣͒ G i i 2 ␪2 i ␪2 ץ ץ ϫͩ ijk ϩ ͪ ͑ ͒ 19 . ץijk ¯y ijk ץ y ͒ ͑ 10 y ¯y ijk ␥ where the extended anomalous dimension ˜ i is given by Here it will be helpful for the later discussions to note that ␣ijkϭ͉ ijk͉2 ␲2 ˜ y ˜y /8 satisfies the same form of renormalization ˜ d ln Zi as the rigid one: ˜␥ ϭϪ ϵ␥ ϩ␥͑1͒␪2ϩ¯␥͑1͒¯␪2ϩ␥͑2͒␪2¯␪2. ͑11͒ i d ln ␮ i i i i ␣ijk ϭ␣ijk˜ ˜ ˜ ͑ ͒ ˜ ybare ˜ y ZiZ jZk. 20 Next let us consider the Yukawa couplings and the trilin- Moreover, we also find the beta functions for the scalar ear couplings given by the superpotential masses by the superfield extension as Wϭ 1 ͑y ijkϪhijk␪2͒␾i␾ j␾k. ͑12͒ 6 dm2 d ln ˜Z ␤ ϵ␮ i ϭϪ␮ iͯ m2 ijk i ␮ ␮ The SUSY breaking trilinear coupling h is often written as d d ␪2¯␪2 ijkϭ ijk ijk ijk h y A , where A are called A terms. Because of ϭ␥͑2͒ nonrenormalization of the superpotential, the holomorphic i couplings Y ijkϭy ijkϪhijk␪2 are renormalized by the chiral ϭ ␥ ͑ ͒ D2 i . 21 ˜ superfield Z␾i as B. Renormalization group flows around the infrared stable ijk ϭ ijk˜ ˜ ˜ ͑ ͒ Y bare Y Z␾iZ␾ jZ␾k. 13 fixed points First let us consider the IR fixed point of the rigid beta By noting that the chiral superfields are represented as functions, where the SCFT realizes. The beta functions for the gauge coupling and the Yukawa couplings vanish, i.e., ˜ ϭ 1/2ϩ˜ ͉ ␪2 ͑ ͒ Z␾i Zi Zi ␪2 , 14 ␤ ϭ␤ijkϭ ␣ y 0, when the anomalous dimensions satisfy the following conditions: we can immediately derive the beta functions for the Yukawa couplings and the trilinear couplings as ␥ ϭ Ϫ ͚ Ti i 3TG ͚ Ti , i i dyijk 1 ␤ijkϭ␮ ϭ ͑␥ ϩ␥ ϩ␥ ͒ ijk y ␮ i j k y , ␥ ϩ␥ ϩ␥ ϭ ͑ ͒ d 2 i j k 0 22

075003-3 TATSUO KOBAYASHI AND HARUHIKO TERAO PHYSICAL REVIEW D 64 075003 for each Yukawa coupling. We may wonder that these con- the IR region. Note that this does not always mean that these ditions are insufficient to determine the fixed points, since couplings are irrelevant in Wilson’s sense, since they are the Yukawa couplings are complex in general. However the dimensionful. phase of the Yukawa coupling is not renormalized by the real We also regard the ␪2¯␪2 components of the extended cou- wave-function renormalization. Also the anomalous dimen- plings as the infinitesimal variations. Since M and hijk vanish sions are actually independent of the phases, since they are at the IR regime, the variations given by found to satisfy ͓18͔ 1 ␦␣ϭ 2␪2¯␪2 , ͚ Timi ͒ ␣͑ ␥ץ ␥ץ ijk ϭ ͑ ͒ FЈ i * 23 . ץijk ¯y ijk ץ y y ¯y ijk ␦␣ijkϭ␣ijk͑m2ϩm2ϩm2͒␪2¯␪2 ͑26͒ As a result the phases of the Yukawa couplings are com- y y* i j k pletely undetermined in all order of perturbation. This is satisfy Eq. ͑24͒. This shows that ⌺ T m2 as well as m2 ␪ i i i i similar to the behavior of a parameter in generic gauge ϩm2ϩm2 corresponding to the Yukawa couplings y ijk, de- ͑ ͒ j k theories. On the other hand, however, the ir relevance of crease exponentially towards the IR regime. By using the IR the couplings is concerned only with evolution of their ab- behavior of the soft parameters clarified so far, it is seen that solute values. Therefore, we should rather consider the real the beta functions for soft scalar masses also decrease expo- ␣ijkϭ͉ ijk͉2 ␲2 couplings y y /8 . nentially. Consequently we find that the soft scalar masses Now we assume the existence of the IR attractive non- approach the constant values satisfying trivial fixed points (␣ ,␣ijk).4 Then generic low-energy ef- * y* fective theories turn out to be SCFT’s subject to these fixed 2ϭ ͚ Timi 0, points. Around the IR attractive fixed points, both the gauge i coupling and the Yukawa couplings should be irrelevant. If we take infinitesimal variations from the fixed point: ␣ 2ϩ 2ϩ 2ϭ ͑ ͒ mi m j mk 0 27 ϭ␣ ϩ␦␣, ␣ijkϭ␣ijkϩ␦␣ijk , then the variations are sub- * y y* y ject to the linear differential equations for each Yukawa coupling. Each IR value mi is heavily dependent on the initial soft parameters. However the relations -among them must be universal. In the case where the anoma ␣␤ץ ␣␤ץ ␣␦d ␮ ϭͩ ͪ ␦␣ϩͩ ͪ ␦␣ijk , ␥ ͑ ͒ , ijk y lous dimensions i are completely determined by Eq. 22␣ץ ␣ץ d␮ * y * the above Eq. ͑27͒ lead to the vanishing IR soft scalar masses for the corresponding masses. This happens when- lmn lmn ␣␤ץ ␣␤ץ lmn␣␦ d y y y ever the anomalous dimensions of the fields can be uniquely ␮ ϭͩ ͪ ␦␣ϩͩ ͪ ␦␣ijk , ͑24͒ ijk y determined from an R symmetry, since the dimension of the␣ץ ␣ץ d␮ * y * field must be given by the R charge in SCFT ͓8͔. We mention the dual SQCD as a special case. The theory where the asterisk represents evaluation at the fixed point. contains the magnetic quark pairs (q,¯q) and a gauge singlet The irrelevance of these couplings means that the eigenval- M and the Yukawa coupling of them is unique, Wϭyqq¯ M. ues of these equations are all positive. Therefore, the soft masses of them should behave as5 Next let us consider the IR behavior of the gaugino mass ͓ ͔ and the trilinear couplings 14,15 . As we have already seen, ␮→0 the beta functions for these couplings can be obtained by the ͑m2,m2,m2 ͒ → m2͑1,Ϫ1,0͒. ͑28͒ ␣ ␣ijk q ¯q M Grasmannian expansion. The extended couplings ˜ and ˜ y satisfy the same form of the RG equations: 2ϭ 2 If we assume mq m¯q as the initial condition, then all scalar masses are exponentially suppressed. d˜␣ ␮ ϭ␤ ␣ ␣ ͒ ␣͑˜ ,˜ y , d␮ C. Higher-dimensional interactions The higher-dimensional operators can be turned into ones d˜␣ijk ␮ y ϭ␤ijk ␣ ␣ ͒ ͑ ͒ relevant to the large anomalous dimensions at the fixed point. ␣ ͑˜ ,˜ y . 25 d␮ y Therefore, we should include such operators as well to find the IR stable fixed points in general. However, we cannot As a result, ␣ M and Ϫ¯y hijk/8␲2, as well as their com- * ijk* apply the RG framework for the renormalizable theories dis- plex conjugates, are found to satisfy the same linear differ- cussed so far. If the Wilson RG, respecting the gauge sym- ␦␣ ␦␣ijk ͑ ͒ ential equations for and y given by Eq. 24 around metries and supersymmetry simultaneously were found, it the fixed point. Therefore, both M and hijk acquire negative would give a suitable framework instead. Here we naively anomalous dimensions and decrease exponentially towards

5Similar discussions of suppressing the sfermion masses in the 4We do not consider the possibility of ﬁxed lines. dual side have also been done in Ref. ͓14͔.

075003-4 SFERMION MASSES IN NELSON-STRASSLER-TYPE... PHYSICAL REVIEW D 64 075003

assume such a framework and somewhat speculatively dis- as the necessary condition. If the fixed point is IR attractive, cuss the IR behavior of the soft parameters around such a then all the eigenvalues of the linearized beta functions for fixed point. the infinitesimal variation from the fixed point values must Suppose that the superpotential of SCFT also contains be positive. higher-dimensional operators such as Now we shall consider incorporating the SUSY breaking parameters by applying the spurion method. We introduce i i2 i 1 y 1 ¯ n the chiral superfield Y i1i2¯inϭy i1i2¯inϪhi1i2¯in␪2 and the ϭ ␾i1␾i2 ␾in ͑ ͒ W ͚ ␮nϪ3 . 29 ␬ ˜ ␣ ͑ ͒ n! ¯ real superfields ˜ n adding to Zi and ˜ defined by Eq. 6 . ˜ Here suppose the wave function superfield Zi is simply given The nonrenormalization for the superpotential and the gauge by the extension as coupling may well be supposed to remain intact.6 Then we write the Wilsonian effective Lagrangian as ˜Z ϭZ ͑˜␣,˜y ij¯k,yន ,˜␬ ͒, ͑34͒ i i ij¯k n ij k 1 where the extended couplings ˜y ¯ are defined by 4 i i† 2 ␣ Lϭ ͵ d ␪ K͑␾ ,␾ ,V͒ϩ ͵ d ␪ tr W W␣ 2 1 2 2 2 16g ij¯kϭ ij¯kϩ ͑ ϩ ϩ ϩ ͒ ij¯k␪2␪2 ͑ ͒ h ˜y Y 2 mi m j ¯ mk y ¯ . 35 ϩ ͵ d2␪ W͑␾i͒ϩH.c., ͑30͒ The reasoning of this extension is the same for the Yukawa coupling. Then the beta functions for ͉˜y ij¯k͉2 as well as ˜␣ can be given by extending the couplings in the rigid beta where the superpotential W is given by Eq. ͑29͒. The Ka¨hler functions for ͉˜y ij¯k͉2. Since the fixed point is IR attractive, potential K given generally as the ␪2¯␪2 term in the extended couplings given by Eq. ͑35͒ decreases exponentially again. Namely, we could obtain the ␾i ␾i† ͒ϭ ␾i† ϪV␾iϩ ␬ O ͑ ͒ extended sum rule at IR as K͑ , ,V Zi e ͚ n n 31 2ϩ 2ϩ ϩ 2→ ͑ ͒ mi m j ¯ mk 0. 36 O contains generic operators n allowed by symmetries. It should be noted that the wave-function renormalization fac- III. SUPERSYMETRIC STANDARD MODELS COUPLED ␬ tors Zi also depend on the effective couplings n as well as WITH SUPERCONFORMAL FIELD THEORIES other couplings in the Wilson RG. The gauge beta function is given in the same way as Eq. A. Yukawa hierarchy ͑ ͒ ␥ 3 except for the fact that the anomalous dimension i is Here we give a brief review on the mechanism to realize defined from the generalized wave-function renormalization hierarchically suppressed Yukawa couplings following Ref. i1i2 in Zi . The beta functions for the couplings y ¯ are also ͓8͔. We assume two sectors: One is the SSM sector, which ϭ ϫ ϫ given by has the gauge group GSM SU(3) SU(2) U(1)Y or an extended group, and three families of quarks and leptons as i1i2¯in i i i dy 1 well as Higgs fields Hu,d . The ith family of them are de- ␤ 1 2¯ nϭ␮ ϭ͑nϪ3͒y i1i2¯inϩ ͑␥ ϩ␥ y d␮ 2 i1 i2 noted by qi , representatively, and they have ordinary ij i j Yukawa couplings y u,dqLqRHu,d . The other sector is the SC ϩ ϩ␥ ͒y i1i2¯in. ͑32͒ ¯ in sector, which has the gauge group GSC and matter fields, which are denoted by ⌽r representatively. The SC-sector The beta functions for ␬ are unknown though. All these beta matter fields also have their couplings n r r r r r r i functions are required to vanish at the fixed points. If the ␭Ј 1 2¯ n⌽ 1⌽ 2 ⌽ n, and the first two families of q are ¯ r fixed point action contains the higher-dimensional interaction assumed to have Yukawa couplings with ⌽ , i.e., rsi r s i i1i2 in␾i1␾i2 ␾in ␭ ⌽ ⌽ q . In the small tan ␤ scenario, the bottom quark y ¯ ¯ , then we impose and tau lepton, as well as the down sector Higgs field Hd , must be coupled to ⌽r. Altogether we have the following ␥ ϭ Ϫ ͚ Ti i* 3TG ͚ Ti , superpotential: i i ϭ ij i j ϩ␭Јr1r2 rn⌽r1⌽r2 ⌽rnϩ␭rsi⌽r⌽s i W y u,dqLqRHu,d ¯ ¯ q . ␥*ϩ␥*ϩ ϩ␥*ϭϪ2͑nϪ3͒ ͑33͒ ͑37͒ i1 i2 ¯ in i The SSM matter fields q and Hu,d are assumed to be singlets ⌽r under GSC . Hence, some of the SC matter fields must 6Perturbative nonrenormalization theorem applied to the non- have nontrivial representations under GSM to allow Yukawa ␭rsi⌽r⌽s renormalizable theories has been presented in Ref. ͓25͔.Onthe couplings qi . The gauge couplings of the SSM sec- ϭ other hand nonrenormalization is not maintained nonperturbatively tor and the SC sector are denoted by ga (a 1,2,3) and gЈ, in general, e.g., the Affleck-Dine-Seiberg superpotential. We as- respectively, and the gauge group GSC is assumed to be sume that such corrections are absent in the following argument. strongly coupled. On top of that, as mentioned in Sec. II, it is

075003-5 TATSUO KOBAYASHI AND HARUHIKO TERAO PHYSICAL REVIEW D 64 075003 expected that the SC sector has a nontrivial IR ﬁxed point. Here we assume that the gauge couplings ga of the SSM sector are weak compared with gЈ. Then we neglect ga and ij Yukawa couplings y u,d of the SSM sector for calculations of rsi r r r the ﬁxed point for gЈ, ␭ , and ␭Ј 1 2¯ n, that is, ␤␣Јϭ␤␭ ϭ␤␭Јϭ0, where 1 ␤␣ ϭ ͫ 3T Ϫ͚ T ͑1Ϫ␥ ͒ͬ, Ј G r r F͑␣Ј͒ r

␤rsiϭ␭rsi ␥ ϩ␥ ϩ␥ ͒ ͑ ͒ ␭ ͑ r s i , 38

r r r ␤ 1 2¯ nϭ␭Јr1r2¯rn͑␥ ϩ␥ ϩ ϩ␥ ͒. ␭Ј r1 r2 ¯ rn FIG. 1. Blowup of g3 . ␥ Through this procedure, the anomalous dimensions i of the SSM matter fields qi are obtained by fixed-point values of The decoupling energy scale M c should not be as low as gЈ, ␭rsi and ␭Јr1r2¯rn, and are in general, large. In particu- the weak scale. One constraint for M comes from the fact lar, the anomalous dimension ␥ is fixed to be a definite c i that ⌽r are charged under G , and the inclusion of such value in the case discussed in Sec. II, and also in that case SM the corresponding sfermion mass is exponentially sup- extra matter fields change beta-function coefficients of GSM pressed. Thus, we have the following beta function of y ij : to be asymptotically nonfree. In that case, the gauge cou- u,d plings would be strong at a high-energy scale and compa- ␤ij ϭ 1 ij ͑␥ ϩ␥ ϩ␥ ͒ ͑ ͒ rable with gЈ of the SC sector. Then, the above fixed-point 2 y , 39 yu,d u,d Li Rj Hu,d calculations, neglecting ga , are not reliable, and the above and the Yukawa coupling y ij at the decoupling energy scale mechanism to produce hierarchical suppression Yukawa couplings, would be spoiled. For example, here we assume that M c of the SC sector is obtained: the gauge couplings of GSM should not blow up below the ͑␥ ϩ␥ ϩ␥ ͒ Li Rj H /2 ͑ ͒ ϭ ϫ 16 M c u,d grand unified theory GUT scale M X 2 10 GeV. Then y ij ͑M ͒ϭy ij ͑M ͒ͩ ͪ , ͑40͒ u,d c u,d 0 M we take the case that the beta-function coefficient of SU(3) 0 ϭϪ ͑ ͒ is obtained by b3 3 just like the MSSM below M c and ij ⌽ where y u,d(M 0) is an initial condition at M 0 . The factor up to M Z , and above M c , extra matter fields a contribute ij ϭϪ ϩ M c /M 0 gives the suppression factor. Thus, even if y (M 0) to it as b3 3 x. Figure 1 shows the curve of (M c ,x) ϭ ͑ ͒ O(1) for most of i,j , we can have hierarchical Yukawa corresponding to the gauge coupling g3 , which blows up at ␥ 7 matrices by powers of large anomalous dimensions i . M X . We have used the one-loop beta function. The region ij Note that y (M C) itself is not a fixed point and its value is above the curve corresponds to the region, where g3 blows not fixed, but its suppression factor is fixed. Here large up below M X . anomalous dimensions play a role similar to U(1) charges of Also the gauge coupling unification is spoiled if we add the Froggatt-Nielsen mechanism with an extra U(1) symme- generic extra matter fields. However, the coupling unification try. Resultant Yukawa matrices have the same form as the still holds at M X in the case where we add extra matter such Froggatt-Nielsen mechanism. To obtain realistic Yukawa ma- that the beta-function coefficients shift universally from the ␥ MSSM MSSM trices, we need nondegenerate anomalous dimensions i values of MSSM, b →b ϩx at M . We assume this Þ␥ a a c j . situation in this whole section. A value of the unified cou- The decoupling energy scale M c is obtained by mass pling ␣ changes from the value for the MSSM, and in gen- r X terms of ⌽ . In general, families can have different decou- eral, it becomes strong. pling energy scales with each other, because they couple with In the previous section, we have seen that soft scalar r different fields ⌽ . However, here we restrict ourselves to masses are exponentially suppressed around the IR fixed the universal decoupling scale M c for simplicity. The discus- point in the case that the corresponding anomalous dimensions in the following sections can be extended to the case ␥ ͑ ͒ sion i is determined definitely by Eq. 38 . That is favorable with nonuniversal decoupling scales. Such mass terms for for FCNC constraints because after such suppressions at M c , decoupling can be generated by another dynamical mecha- we have radiative corrections due to the gaugino masses of nism. the SSM sector, which are flavor blind. Actually, such a possibility has been mentioned in Ref. ͓8͔. However, in the previous section, we have considered the pure SC sector. It is 7This form is similar to Yukawa couplings with power-law behav- important to study the effects of finite gauge couplings and ior due to Kaluza-Klein modes in extra dimensions, where extra gaugino masses of the SSM sector for realistic models. That dimensions actually play a role similar to anomalous dimensions is the purpose of Sec. III B. Actually, we shall show that the ͓3͔, and in this case FCNC problems could be solved by the IR gauge couplings and the gaugino masses of the SSM sector alignment mechanism ͓26͔. play an important role.

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⌫ B. Degeneracy of sfermion masses The constant i can be obtained from fixed-point values of In Sec. II, we have shown that within the framework of the gauge and Yukawa couplings of the SC sector by fixing a ͑ ͒ ␥ ⌫ р pure SCFT’s soft scalar masses as well as gaugino masses model from Eq. 21 , and it is O( i), i.e., i O(1). It is ⌫ and A parameters, decrease exponentially at M in the case important that i is flavor dependent because anomalous di- c ␥ where the corresponding anomalous dimensions are deter- mensions i are flavor-dependent to realize hierarchical mined definitely. That is favorable from the viewpoint of Yukawa couplings. Thus, the difference between sfermion FCNC problems because that would provide degenerate sfer- masses, e.g., the first and the second families, is obtained by mion masses at the weak scale by flavor-blind radiative corrections due to gaugino masses of the SM sector. However, 2 2 2 1 1 ͑ ͒Ϫ ͑ ͒ϭ ˜ ␣ ͑ ͒ ͑ ͒ͩ Ϫ ͪ m2 M c m1 M c C f a a M c M a M c ⌫ ⌫ , in a realistic case, we have to examine two points for SSM’s 2 1 coupled with SCFT’s: One is that we have to take into ac- ͑42͒ count effects due to gauge couplings and gaugino masses of the SM sector. The other point is that a running region is where Cia is denoted by C˜f a , because the quadratic Casimir ⌫ ⌫ finite. The former point is considered in this section, while in is common. Here 2 would be smaller than 1 to obtain ⌫ Sec. IV B the latter shall be discussed by use of an illustrat- realistic Yukawa matrices. Naturally, we would have 1/ 2 Ϫ ⌫ ϭ ⌫ ing model. 1/ 1 O(1/ 2). Below M c we have only flavor-blind ra- For concreteness, we consider the case where M c is less diative corrections. Hence, the mass difference is estimated ͑ ͒ than M X , and below M C we have the same matter content as as Eq. 42 at any scale below M c . Actually this difference is Ͼ ␣ the MSSM. It is possible to assume M C M X , that is, the suppressed by the one-loop factor a compared with the ini- Nelson-Strassler mechanism works above M X . It is easy to tial value, which is favorable for FCNC constraints. How- extend the following calculations to such cases, although re- ever, whether that is indeed suppressed enough for FCNC sults are GUT model dependent. constraints, depends on radiative corrections between M c ⌫ Here we denote gaugino masses of the SM sector as M a . and the weak scale, and an explicit value for 2 . ϭ We assume the universal gaugino mass M a M 1/2 at the Before estimating nondegeneracy explicitly for the GUT scale M X . Recall that the gauge coupling is unified at MSSM, we comment on the fact that we have neglected the M X in the case where the beta-function coefficients SC gaugino mass MЈ and A parameters A␭ , which corre- ϭ MSSMϩ MSSMϩ MSSMϩ (b1 ,b2 ,b3) (b1 x,b2 x,b3 x) and we are spond to trilinear couplings among the SC sector and the SM ␣ considering such a case. It holds that M a / a is a RG invari- sector. In pure SCFT’s without effects of the SM gaugino ant. Suppose that the theory is regarded as SCFT at the scale masses M a , all of them decrease exponentially as discussed Ͻ␮Ͻ ␣ of M c M X . In the RG equations of soft scalar masses in Sec. II. However, for nonvanishing aM a , they converge ϭ ␣ ϭ ␣ we ignore the gaugino mass and A parameters A␭ of the SC on MЈ O( aM a) and A␭ O( aM a). The RG equations 2 Ј2 sector because they decrease rapidly. We shall come back to of soft scalar masses squared mi include the terms of M 2 ␣ 2 ͑ ͒ this point later. Then the RG equations for soft scalar masses and A␭. These are small compared with aM a in Eq. 41 by ␣ are written down as the loop factor a . That justifies our above calculations. Here we study the degeneracy of sfermion masses explic- dm2 ␮ i ϭM 2Ϫ ␣ 2 itly for the MSSM. Sfermion masses in the MSSM are ob- ␮ ijm j Cia aM a d tained at M c , Ϫ ϭM ͑m2ϪM 1C ␣ M 2͒, ͑41͒ ij j jk ka a a 1 16 1 2 ͑ ͒ϭ ͫ ␣ 2ϩ ␣ 2ϩ ␣ 2ͬ͑ ͒ mQi M C ⌫ 3M 3 3 2M 2 1M 1 M C , Qi 3 15 where Cia is a quadratic Casimir. In the pure SCFT limit ͑ ͒ ␣ → 43 a 0, the second term vanishes and soft scalar masses con- tinue to decrease exponentially. However, the exponentially 1 16 16 suppressing behavior is stopped by the second term. The m2 ͑M ͒ϭ ͫ ␣ M 2ϩ ␣ M 2ͬ͑M ͒, ͑44͒ ␣ 2 ui C ⌫ 3 3 3 15 1 1 C evolution of aM a is small compared with the exponential ui ␣ 2 running of the soft scalar masses. Thus, the term aM a could be treated as a constant during the exponential running of the 2 1 16 2 4 2 2 ͑ ͒ϭ ͫ ␣ ϩ ␣ ͬ͑ ͒ ͑ ͒ soft scalar masses. However, the finite-size effect of ␣ M is mdi M C ⌫ 3M 3 1M 1 M C , 45 a a ui 3 15 2 important. The soft mass squared mi converges on 1 3 m2 ͑M ͒ϭ ͫ3␣ M 2ϩ ␣ M 2ͬ͑M ͒, ͑46͒ Cia Li C ⌫ 2 2 1 1 C 2→ ␣ ͑ ͒ 2͑ ͒ Li 5 mi ⌫ a M c M a M c , i 1 12 2 ͑ ͒ϭ ͫ ␣ 2ͬ͑ ͒ ͑ ͒ where we denote mei M C ⌫ 1M 1 M C . 47 ei 5 C ia ϭMϪ1 To be explicit, here we write radiative corrections due to ⌫ ij C ja . i gaugino masses between M c and M Z ,

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8 ␣2͑M ͒ 2 ͑ ͒Ϫ 2 ͑ ͒ϭ ͫ 3 Z Ϫ ͬ 2͑ ͒ mQi M Z mQi M C ␣2͑ ͒ 1 M 3 M c 9 3 M C 3 ␣2͑M ͒ ϩ ͫ Ϫ 2 Z ͬ 2͑ ͒ ͑ ͒ 1 ␣2͑ ͒ M 2 M c 48 2 2 M C

1 ␣2͑M ͒ ϩ ͫ Ϫ 1 Z ͬ 2͑ ͒ ͑ ͒ 1 ␣2͑ ͒ M 1 M c , 49 198 1 M C

8 ␣2͑M ͒ 2 ͑ ͒Ϫ 2 ͑ ͒ϭ ͫ 3 Z Ϫ ͬ 2͑ ͒ mui M Z mui M C ␣2͑ ͒ 1 M 3 M c 9 3 M C 2 8 ␣ ͑M ͒ ⌬˜ ⌬˜ ϩ ͫ Ϫ 1 Z ͬ 2͑ ͒ FIG. 2. Q and d against M c . 1 ␣2͑ ͒ M 1 M c , 99 1 M C ͑16/3͒␣3͑M ͒ϩ͑16/15͒␣3͑M ͒ ͑50͒ ⌬ ϭ 3 c 1 c ͑ ͒ ˜u Ϫ͑ ͒⌬␣2ϩ͑ ͒⌬␣2 , 57 8/9 3 8/99 1 8 ␣2͑M ͒ 2 ͑ ͒Ϫ 2 ͑ ͒ϭ ͫ 3 Z Ϫ ͬ 2͑ ͒ ͑ ͒␣3͑ ͒ϩ͑ ͒␣3͑ ͒ mdi M Z mdi M C 2 1 M 3 M c 16/3 3 M c 4/15 1 M c 9 ␣ ͑M ͒ ⌬˜ϭ ͑ ͒ 3 C d Ϫ͑ ͒⌬␣2ϩ͑ ͒⌬␣2 , 58 8/9 3 2/99 1 ␣2͑M ͒ 2 1 Z 2 ϩ ͫ Ϫ ͬ ͑ ͒ 3 3 1 2 M 1 M c , ␣ ͑ ͒ϩ͑ ͒␣ ͑ ͒ 99 ␣ ͑M ͒ 3 2 M c 3/5 1 M c 1 C ⌬˜ϭ ͑ ͒ L ͑ ͒⌬␣2ϩ͑ ͒⌬␣2 , 59 ͑51͒ 3/2 2 1/22 1 ͑ ͒␣3͑ ͒ ␣ ͑ ͒ ␣2͑ ͒ 12/5 1 M c 66 1 M C 3 2 M Z ⌬ ϭ ϭ 2 ͑ ͒Ϫ 2 ͑ ͒ϭ ͫ Ϫ ͬ 2͑ ͒ ˜e 2 2 , m M Z m M C 1 2 M M c ͑ ͒⌬␣ 5 1Ϫ͓␣ ͑M ͒/␣ ͑M ͔͒ Li Li 2 ␣ ͑M ͒ 2 2/11 1 1 Z 1 C 2 C ͑60͒ ␣2͑ ͒ 1 1 M Z ϩ ͫ Ϫ ͬ 2͑ ͒ where ⌬␣2ϭ␣2(M )Ϫ␣2(M ). Recall that we have as- 1 ␣2͑ ͒ M 1 M c , i i C i Z 22 1 M C ϭ sumed gaugino mass uniﬁcation M a M 1/2 at the GUT scale 2 ͑52͒ ⌬˜ ⌬ M x . It should be noted that f , and therefore m˜f , may be predicted independently of the SM gaugino masses. ␣2͑ ͒ 2 1 M Z Figure 2 shows ⌬˜ and ⌬˜ against M . We omitted to m2 ͑M ͒Ϫm2 ͑M ͒ϭ ͫ 1Ϫ ͬM 2͑M ͒. Q d c ei Z ei C ␣2͑ ͒ 1 c show ⌬ , because it is almost same as ⌬˜ . As a result, this 11 1 M C ˜u d ͑53͒ mechanism can realize favorable degeneracy between squark ⌫ Ͼ masses for large M c . For i 0.1, we could avoid the FCNC We have assumed to have exactly the MSSM matter content problem. On the other hand, the FCNC problem would be ⌫ below M c . These radiative corrections are quite large com- serious for smaller values of i . ⌬ ⌬ pared with the initial values at M c . Thus, the nondegeneracy, Similarly, Fig. 3 shows ˜L and ˜c against M c . We have a 2 2 2 2 2 ⌫ ⌬m˜ ϭ(m˜ Ϫm˜ )/m˜ , where m˜ is an average value of good degeneracy between left-handed sleptons. For i f f 2 f 1 f av f av Ͼ them, is obtained as 0.1, we could avoid the FCNC problem. However, for the

2 C˜ ␣ ͑M ͒M ͑M ͒ 2 f a a c a c 1 1 ⌬m ϭ ͩ Ϫ ͪ . ͑54͒ ˜f 2 ⌫ ⌫ ͑ ͒ 2 1 m˜f M Z

To estimate such nondegeneracy, we deﬁne

␣ ͑ ͒ 2͑ ͒ C˜f a a M c M a M c ⌬˜ϭ . ͑55͒ f 2͑ ͒ m˜f M Z

To be explicit, we use

͑16/3͒␣3͑M ͒ϩ3␣3͑M ͒ϩ͑1/15͒␣3͑M ͒ ⌬ ϭ 3 c 2 c 1 c ͑ ͒ Q˜ 2 2 2 , 56 Ϫ͑8/9͒⌬␣ ϩ͑3/2͒⌬␣ ϩ͑1/198͒⌬␣ ⌬ ⌬ 3 2 1 FIG. 3. ˜L and ˜e against M c .

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FIG. 5. Ratios of sfermion masses to M at the weak scale. The ⌬ ϭ ϭϪ 2 ϭϪ 2 3 FIG. 4. ˜e against M c with S 0, S M 1, and S 10M 1. three solid lines correspond to mQ˜ /M 3 , m˜L /M 3 , and m˜e /M 3 , respectively. The two dotted ͑upper and lower͒ lines correspond to right-handed slepton the degeneracy is not strong compared ͑ ͒ ratios of gaugino masses to M 3 M 2 /M 3 and M 1 /M 3 . with squarks and left-handed sleptons. The reason is that the radiative correction due to the bino is not large compared We have assumed the universal gaugino mass M a(M X) ϭ with the others. In this case, we would be faced with the M 1/2 and can relax the condition. However, all of the ⌫ ϳ FCNC problem for i O(0.1). above results on degeneracy of sfermion masses are similar, We have ignored contributions to the RG equations due to because only one of the gaugino masses contributes almost the U(1)YD term. However, such a contribution would be dominantly to each fermion mass degeneracy, i.e., M 3 , M 2 , sizable, in particular, for the right-handed slepton masses. and M 1 contribute to the degeneracy of squark masses, left- Therefore, we also discuss contributions due to the U(1)YD handed slepton masses and right-handed slepton masses, re- term. Including such effects, the right-handed slepton mass spectively. 2 squared m˜ei at M c is obtained: We have assumed that the SC region is below M X . Alter- natively, We can take the possibility that the SC region is 1 5 between M X and the Planck scale, and the Nelson-Strassler m2 ͑M ͒ϭ ␣ ͓4M 2ϪS͔͑M ͒, ͑61͒ ei C ⌫ 3 1 1 C mechanism would work in some GUT model. Such a case ei can be studied similarly and we may have a signiﬁcant where Sϭtr Ym2, i.e., change for the slepton masses. This GUT scenario shall be i discussed elsewhere ͓27͔.

2 2 2 ϭ 2 Ϫ 2 ϩ͚ ͑ Ϫ 2 ϩ Ϫ ϩ 2 ͒ C. Mass spectrum S mHu mHd mQ˜ i 2m˜ui m˜di m˜Li m˜ci . i ͑62͒ Here we show representative mass spectra in the case where we have the exactly same matter content below M C as At M , the fields that do not couple to the SC sector, e.g., the MSSM and the gaugino masses are unified at M X , c ϭ top squark and Higgs fields, have nonsuppressed soft scalar M a(M X) M 1/2 . As we saw in Sec. III C, sfermion masses masses, and these masses contribute to the initial value of can be quite suppressed at M c for the fields that couple with S(M c), which is, in general, not suppressed and would be the SC sector and whose anomalous dimensions are deter- 2 mined definitely. Namely, we have no-scale type of initial O(M a). In addition, the radiative corrections including the S effect are obtained: conditions for such fields. Thus magnitudes of sfermion masses of this type are calculated only by radiative correc- 2 ͑ ͒ ͑ ͒ 2 ␣ ͑M ͒ tions between M c and the weak scale 48 – 53 . Figure 5 2 ͑ ͒Ϫ 2 ͑ ͒ϭ ͫ Ϫ 1 Z ͬ 2͑ ͒ mei M Z mei M C 1 ␣2͑ ͒ M 1 M c shows ratios of sfermion masses to M 3 at the weak scale. 11 1 M C The three solid lines correspond to mQ˜ /M 3 , m˜L /M 3 , and ϭ 1 ␣ ͑M ͒ m˜e /M 3 , respectively. We have taken S 0. Also the two ϩ ͫ 1 Z Ϫ ͬ ͑ ͒ ␣ ͒ 1 S M c . dotted lines show ratios of gauginos to M 3 . The upper and 11 1͑M C the lower correspond to M 2 /M 3 and M 1 /M 3 , respectively. ͑63͒ Note that the right-handed slepton is lighter than the B-ino. In this case the lightest sypersymmetric particle ͑LSP͒ would ⌬ ϭ Figure 4 shows ˜e including these effects for S(M c) 0, be slepton and the ordinary no-scale type initial condition Ϫ 2 Ϫ 2 ͓ ͔ M 1(M c) and 10M 1(M c). We have a slight suppression has the same problem 28,29 , although we have to take ⌬ of ˜e , but that is not drastic enough to change its order. mass eigenvalues and it depends on the overall magnitude of ⌫ ϭ Thus, for i O(0.1), we would still have the serious FCNC soft masses. However, the U(1)Y D term has a sizable effect problem. In this case, we may be required to take the degen- as discussed in Sec. III B. Figure 6 shows m˜e /M 3 for ␥ ϭ␥ ⌫ ϭ⌫ ϭϪ 2 erate case with e1 e2 and e1 e2 . That would con- S(M c) 0.5M 3(M Z). In this case the LSP is the strain the form of the lepton Yukawa matrix. neutralino.

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and P must be equal, since QP forms a GSC singlet. There- fore, the dimensions of GSM representations of Q and P are also necessarily the same. Taking into account the fact that q carries GSM charges, the possible types of models seem to be rather limited. We shall enumerate a few simple examples below.

1. Chiral SU(5) model

The SC-gauge group GSC is SU(Nc) and the SM-gauge group GSM is SU(5). We introduce the following chiral ﬁelds assigned the representations under (GSC ,GSM):

͑ ¯ ͒ ͑¯ ¯ ͒ ͑ ͒ ͑ ͒ ϭ Ϫ 2 Q: Nc ,5 , P: Nc ,5 , q: 1, 10 . 64 FIG. 6. m˜e /M 3 for S 0 and 0.5M 3. ϭ␭ The masses of the sfermion, which do not couple with the The superpotential is given by W qQP, and the IR fixed ϭ SC sector, e.g., top squark masses ͑and sbottom and stau point is found to exist for Nc 2,3. masses for the large tan ␤ scenario͒ depend on their initial In this class of models, the SC-gauge nonsinglet fields Q and conditions. It is natural to assume their masses are of P belong to the same dimensional representations of the SM- gauge group and, therefore, their anomalous dimensions are O͓M a(M X)͔ or O͓M a(M C)͔. Note that the ratio ͑ equal.8 All scalar masses, m2 , m2 , and m2 converge to 0, M 3(M Z)/M 1/2 is less than 3 which is expected in the ordi- Q P q ͒ irrespective of initial values. nary MSSM if M c is lower than M X , because the unified gauge coupling becomes large by adding extra matter fields. We have a large mass gap between the stau and the other 2. L-R symmetric SU(3) model sleptons if the stau couple with the SC sector. On the other ϭ ϭ Suppose GSC SU(Nc) and GSM SU(3) and introduce hand, whether the top squark is lighter than the other squarks depends on the initial condition. Anyway, we can predict ͑ ͒ ¯ ͑¯ ¯ ͒ ͑¯ ͒ ¯ ͑ ¯ ͒ definitely the mass spectrum for the sfermions coupling with Q: Nc,3 , Q: Nc ,3 , P: Nc,3 , P: Nc ,3 , the SC sector for fixed M c . Also, we could relax the condi- ϭ ͒ ¯ ͒ ͑ ͒ tion with the universal gaugino mass M a(M X) M 1/2 . qL :͑1,3 , qR :͑1,3 . 65

IV. ANALYSES OF SQUARK MASSES IN EXPLICIT ϭ␭ ¯ Also the superpotential is defined as W (qLQP MODELS ϩ ¯ ϭ qRQP). The IR fixed point is found to exist for Nc 3. A. Models with suppressed soft parameters The anomalous dimensions of Q and Q¯ , and also P and ¯P, ␥ ϩ␥ The models based on the SCFT with exponentially sup- are the same by the left-right symmetry. Therefore, Q P pressed scalar masses are favorable phenomenologically in is fixed by the fixed-point equation given by Eq. ͑22͒.In 2 ϭ 2 2 avoiding the flavor problems. In this section, we consider the such cases, however, we need to assume mQ mQ¯ , m P perturbatively renormalizable theories enjoying this property. 2 ϭm¯ for exponential suppression of the scalar masses m Indeed we could consider also many varieties by using the P qL and m . Note that m2 or m2 is not reduced to 0, though the SCFT’s with higher-dimensional operators as discussed in qR Q P Sec. II. However, in that case, we should start with the assum of them decreases exponentially. sumption that there exists such an IR fixed point, because of For these types of models, we cannot introduce two the lack of RG frameworks applicable to nonrenormalizable quarks with distinct anomalous dimensions in a single SC- 9 theories. Therefore, we shall restrict ourselves to the renor- gauge sector. In other words, we need to assume a different malizable theories. Then the types of models with suppressed SC-gauge theory for every quark or lepton to be given large scalar masses are found to be rather limited as follows. anomalous dimension. There may be some exceptional cases Suppose a quark ͑lepton͒ q couples to the SCFT through where the Yukawa interactions composed only of the SC- Yukawa interaction qQP. Then we search for the models in gauge nonsinglet fields are also allowed. In this paper we are ␥ ϩ␥ not going to explore such possibilities. Hereafter we discuss which Q P is uniquely determined by the fixed-point conditions in terms of the anomalous dimensions. In this the IR behavior of the soft scalar masses by considering the case, the squark ͑slepton͒ mass decreases exponentially, as models similar to the above examples. shown in Sec. II. Now the interactions are limited to the Yukawa type in renormalizable theories. Here, let us also assume that there are no Yukawa terms composed only of 8The SU(3)3 model in Ref. ͓8͔ belongs to this class. ␥ ϩ␥ 9 nonsinglet fields under GSC . Then Q P must be deter- The hierarchy of Yukawa couplings can be generated by assum- mined by the condition for vanishing gauge beta function. ing different decoupling scales M c instead of the anomalous dimen- This means that the gauge beta function should depend only sions. In such cases we may make several quarks couple to a com- ␥ ϩ␥ on Q P . On the other hand, the quadratic Casimirs of Q mon SC sector.

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Ã B. Sfermion mass convergence in the SU„3…SC SU„3…C The superpotential is defined by model ϭ␭͑ ¯ ϩ ¯ ͒ϩ ͑ ͒ In Sec. III B we have evaluated the flavor dependence of W q1QP ¯q1PQ y i¯qiqiH. 69 squark masses. In this discussion we have assumed that soft Here we have simplified the Yukawa couplings to the diago- scalar masses in the SC sector converge sufficiently. How- nal ones. In this toy model only the Yukawa coupling of the ever, the range of scale where the theory is regarded as a 2 first generation y is suppressed. Also we assume m2 ϭm SCFT, must be finite, otherwise the Yukawa couplings are 1 Q Q¯ 2 ϭ 2 suppressed out too much. Therefore, soft scalar masses will and m P m¯P. not converge completely either at the decoupling scale M c . Below we analyze the RG flows of the various couplings The degree of the convergence is related to the suppression numerically by substituting the anomalous dimensions in the for the Yukawa couplings. First let us estimate roughly how exact beta functions with those evaluated in one-loop pertur- much the squark masses converge. bation. The anomalous dimensions are given by Suppose that the theory is regarded as a SCFT at the scale Ͻ␮Ͻ⌳ ␥ ϭ␥ ϭϪ 8 ␣Јϩ ␣ Ϫ 8 ␣ ͑ ͒ of M c c . In this region the soft scalar masses are Q P 3 2 ␭ 3 , 70 subject to Eq. ͑41͒ ignoring the gaugino mass and A param- 8 8 ␥ ϭ6␣␭Ϫ ␣ϩ␣ , ␥ ϭϪ ␣ϩ␣ ͑iϭ2,3͒, eter in the SC sector. Then the speed of convergence is given q1 3 y1 qi 3 yi by the smallest eigenvalue ␰ of the matrix M. This eigen- ͑71͒ value is found to be of the same order of the anomalous ␾ ␥ ϭ ͑␣ ϩ␣ ϩ␣ ͒ ͑ ͒ dimensions of i . Let us define the deviation of the squark H 3 y1 y2 y3 , 72 mass from the convergent value by ␦m2ϭm2 i i ␣ ϭ 2 ␲2 ␣ϭ 2 ␲2 ␣ ϭ͉␭͉2 ␲2 ␣ Ϫ ⌫ ␣ 2 where Ј gЈ /8 , g /8 , ␭ /8 , and y (Ci / i) 3M 3. Then the deviation at M c , which is esti- i ϭ͉ ͉2 ␲2 mated roughly as y i /8 . It is straightforward to derive the beta functions for all couplings by using formula shown in Sec. II. Here let Ϫ␰ ͑⌳ ͒ ␦ 2͑ ͒ϭ i ln c /M c ␦ 2͑⌳ ͒ ͑ ͒ mi M c e mi c , 66 us write down only the beta functions for soft parameters in the SC sector: ␣ 2 has to be much less than 3M 3 in order that the formula for the squark masses, given in the previous section, is valid. dMЈ 3␣Ј͑2Ϫ3␣Ј͒ 6␣Ј2 ␮ ϭϪ ͓ ϩ ͔ ЈϪ ␥͑1͒ ␦ 2 ␣ 2 2 1 2␥ M Q , Also if mi is found to be much larger than 3M 3, the d␮ ͑1Ϫ3␣Ј͒ Q 1Ϫ3␣Ј squark masses may not be degenerate enough so as to avoid ͑73͒ the flavor problem. The ratio of the Yukawa couplings is determined by the dA␭ ␮ ϭϪ͑2␥͑1͒ϩ␥͑1͒͒, ͑74͒ anomalous dimension of the quarks. By noticing that the d␮ Q q1 ␰ eigenvalue i is found to be the same order as the anomalous dimension, we evaluate ␦m2 also as 2 2 i dmQ dmP ␮ ϭ␮ ϭ␥͑2͒ , ͑75͒ d␮ d␮ Q y ͑M ͒ mq ␦ 2͑ ͒ϳ ii c 2͑⌳ ͒ϳ i 2͑⌳ ͒ ͑ ͒ mi M c ⌳ ͒ mi c mi c , 67 y ii͑ c mq 2 3 dm1 ␮ ϭ␥͑2͒ , ͑76͒ d␮ q1 where m denotes the quark mass of the ith generation. The qi ␦ 2 (1) (2) deviation mi for the second generation should be especially where ␥ and ␥ are obtained by the superfield extension ϳ Ϫ2 2 suppressed by a factor similar to ms /mb O(10 ). There- discussed in Sec. II. By neglecting terms of O(␣ )orof ␣ fore, there a large uncertainty in the squark mass due to this O( y1) as negligible amounts, they are given by deviation at M c for the second generation may remain. If the ␥͑1͒ϭϪ 8 ␣Ј ЈϪ ␣ Ϫ 8 ␣ ͑ ͒ squark mass is the same order as the SM-gaugino mass at Q 3 M 2 ␭A␭ 3 M, 77 ⌳ c , this uncertainty is supposed to be much larger than the convergent value evaluated in Sec. IV A. Therefore the SM- ␥͑1͒ϭϪ ␣ Ϫ 8 ␣ ͑ ͒ q1 6 ␭A␭ 3 M, 78 gaugino mass is required to be fairly larger than the squark masses at ⌳ . ␥͑2͒ϭϪ 8 ␣Ј͑ ͉ Ј͉2ϩ⌬ ͒ϩ ␣ ͉͑ ͉2ϩ 2 ϩ 2 ϩ 2͒ c Q 3 2 M gЈ 2 ␭ A␭ mQ m P m1 In practice the above argument is rather bold. In the fol- Ϫ 16 ␣͉ ͉2 ͑ ͒ lowing, we shall demonstrate the RG flows for the squark 3 M , 79 masses and their converging behavior explicitly in a concrete ␥͑2͒ϭ ␣ ͉͑ ͉2ϩ 2 ϩ 2 ϩ 2͒Ϫ 16 ␣͉ ͉2 ͑ ͒ model and examine the convergence. Suppose both of the q1 6 ␭ A␭ mQ m P m1 3 M , 80 SC-gauge and SM-gauge groups are SU(3) and introduce ⌬ ϭ ␣Ј ͉ Ј͉2Ϫ 2 Ϫ 2 ͑ ͒ the following chiral fields: where gЈ 3 ( M mQ m P) as defined by Eq. 9 . The fixed points are found at A:(␣Ј ,␣␭ )ϭ(5/16,1/6) ϭ͑ ˜ ͒ ¯ ϭ͑¯ ͒ ϭ͑ ͒ ¯ ϭ͑¯ ¯ ͒ * * Q 3,3 , Q 3,3 , P 3,3 , P 3,3 , and B:(␣Ј ,␣␭ )ϭ(3/16,0). Point A is the IR attractive * * fixed point and the anomalous dimensions there are found to ϭ ͒ ϭ ¯ ͒ ϭ ͒ ϭ ͒ ͑ ͒ qi ͑1,3 , ¯qi ͑1,3 ͑i 1,2,3 , H ͑1,1 . 68 be

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FIG. 7. The running couplings (y 1 ,MЈ,A␭) are shown in ratio to their initial values by a solid, a dashed, and a long-dashed line, FIG. 8. RG flows for (m2,m2 ϩm2 ) shown by solid and dashed respectively. tϭlog (␮/⌳ ). 1 Q P 10 c lines, respectively. The long-dashed line gives ␣M 2. ␥ ϭ␥ ϭϪ 1 , ␥ ϭ1. Q P 2 q1 ⌳ ␦ 2 * * * given at c , the deviation mi is found to remain about ten times larger than the converging value at M . Thus we con- In the region that MЈ and A␭ are suppressed to negligible c amounts, the RG evolution of the sfermion masses are given clude that the strongly degenerate squark mass spectrum by evaluated in Sec. III, is indeed achieved irrespective of the ␣Ј2ϩ ␣ ␣ initial sfermion masses, if the SM-gaugino mass is fairly d m2 ϩm2 16 4 ␭ 4 ␭ m2 ϩm2 ␮ ͩ Q P ͪ ϭͩ * * *ͪ ͩ Q P ͪ larger than them. ␮ 2 3 d m1 6␣␭ 6␣␭ m1 In practice, the theories must become SCFT’s at a certain * * scale in order to generate finite ratio among Yukawa cou- 16 2 plings. Therefore, we have also performed similar observa- Ϫ ␣͉M͉2ͩ ͪ . ͑81͒ ␣Ј ␣ 3 1 tions by assuming the initial vales of and ␭ off the fixed ⌳ Ͼ⌳ point at the higher-energy scale 0 c . However, the re- 2 ϩ 2 2 2 sults obtained on the convergence for the sfermion masses, Note that mQ m P but not each of mQ and m P converges to O(␣͉M͉2) in this model. When ␣͉M͉2 is treated as a con- are not significantly changed. stant, the eigenvalues of this coupled equation are found to be ͑2.64͒, ͑0.59͒. Indeed the smaller one ␰ϭ0.59 is close to ␮ the anomalous dimension 1/2. Therefore, degrees of suppres- V. GENERATION OF TERM BY A SINGLET sion for the Yukawa coupling and the scalar masses are al- So far, we have discussed the cases where the large most the same in this model. It is also expected that the anomalous dimensions for quarks and leptons are determined 2→ ␣ 2 2 ϩ 2 scalar masses converge as m1 0.78 M ,mQ m P definitely. It has been seen that the corresponding sfermion →4.55␣M 2. masses are exponentially suppressed and converge to nonva- Now we present the results obtained by numerical analy- nishing values due to effects of an SM-gaugino mass. It will ses of the RG equations. In Fig. 7, the aspect of suppression be shown that this converging value for (mass)2 can be nega- ϭ ␮ ⌳ for (y 1 ,MЈ,A␭) are shown with respect to t log10( / c). tive for a singlet field coupled with the SC sector. The order Here we set ␣Ј and ␣␭ on the IR fixed point. The initial of the tachyonic (mass)2 is fixed to O(␣M 2), namely, the values for other couplings are chosen as follows: MЈϭA␭ weak scale, irrespectively of the bare scalar mass. On the ϭ ϭ ␣ϭ ␲ ␣ ϭ ␲2 1.0, M 5.0, 1/(48 ), and y1 1/(8 ). The value other hand, the weak scale mass term ͑␮ term͒ in the super- of ␣ refers to the GUT gauge coupling. It is seen that the symmetric SM has no theoretical grounds and poses the so- Yukawa coupling is smoothly suppressed. If we suppose M c called ␮ problem ͓30͔. It has been discussed sometimes that to be the scale that the Yukawa coupling is suppressed by a singlet can explain the ␮ term by developing its vacuum Ϫ2 ϭ ⌳ Ϫ 10 , then tc log10(M c / c) is found to be 2.01. expectation value ͑VEV͒ of the weak scale ͓31,32͔. In this Next we examine the RG flows of the sfermion masses by section we propose another solution for the ␮ problem by varying the initial values and observe the converging behav- considering a singlet field coupled with the SC sector. ior. Figure 8 shows the RG flows obtained by varying the Suppose that a singlet field S is coupled to the SC sector 2 ͓ ͔ 2 10 initial value for m1 between 0.0, 2.0 with setting mQ through the superpotential ϭ 2 ϭ m P 1.0. It is seen that the sfermion masses converge to the values of O(␣͉M͉2), though the coefficients are slightly 2→ ␣ 2 shifted from the above naive estimation: m1 0.3 M .Itis 10Here we assume the bare ␮ term is absent in the superpotential. 2 found also that the range of m1 shrinks to about 5% of initial Indeed the␮ term may be prohibited by imposing a discrete sym- one at M c . Actually we obtain almost the same results for metry. However the discrete symmetries lead to a cosmological any setting for the initial couplings. For generic initial sfer- problem by forming domain walls in general. Alternatively we may mion masses of the same order of the SM-gaugino mass introduce an extra U(1) gauge symmetry to forbid the bare ␮ term.

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␥ ϭϪ 8 ␣Јϩ ␣ ϩ␣ Ϫ 8 ␣ ͑ ͒ Wϭ␭SQQ¯ . ͑82͒ P 3 2 ␭ ␭Љ 3 , 87

8 ␥ ϭ6␣␭Ϫ ␣, ͑88͒ Here we also assume that (Q,Q¯ ) carries SM-gauge charges. q1 3 2 By assuming m2 ϭm , the form of the RG equations for the ␥ ϭ ␣ ϩ ␣ ͑ ͒ Q Q¯ S 3 ␭Ј 3 ␭Љ . 89 scalar masses are given generally as The IR fixed-point couplings are found at ␣Ј ϭ3/16, ␣␭ * * 2 ϭ␣␭Ј ϭ1/6. By ignoring the gaugino mass and A parameter dmQ * ␮ ϭ͑ ϩ ͒ 2 ϩ 2Ϫ ␣͉ ͉2 of the SC sector again, the RG equations for the scalar ␮ a 2b mQ bmS CQ M , d masses are given by 2 m2 ϩm2 13/4 2/3 1/3 m2 ϩm2 dmS Q P Q P ␮ ϭ2cm2 ϩcm2, ͑83͒ d 2 2 d␮ Q S ␮ ͩ mq ͪ ϭ 110ͩ mq ͪ 1 ͩ ͪ 1 ␮ d 2 2 mS 101 mS where the gaugino mass and A parameter have been ignored 2 again. The coefficients a, b, c, and CQ are positive and de- 16 termined by group-theoretical factors. Ϫ ␣M 2ͩ 1 ͪ , ͑90͒ ¯ ϭ 3 If Q and Q are SM-gauge singlets, or CQ 0, the scalar 0 2 2 masses mQ and mS are reduced to 0 exponentially. However ␣͉ ͉2 where the fixed-point couplings are used. From this equation, the correction by the SM-sector gaugino M , makes the it is found that the scalar masses converge as scalar masses converge to nonvanishing values. We can treat 2 ϩ 2 the gaugino mass as well as the gauge coupling in the SM- mQ m P 16/27 16 sector as constants, since their evolution is slow enough. 2 2 ͩ mq ͪ → ␣M ͩ 11/27 ͪ . ͑91͒ Then the scalar masses converge to 1 3 2 Ϫ16/27 mS CQ m2 → ␣͉M͉2, Thus it is seen that the singlet becomes tachyonic indeed. Q a As another possibility for a singlet to develop the weak scale VEV, we may consider the SCFT’s whose anomalous 2CQ dimensions are not uniquely determined by the fixed-point m2→Ϫ ␣͉M͉2. ͑84͒ ͑ ͒ S a conditions 22 . In such cases, the sfermion masses converge to certain values of the same order as the initial masses. Here we should note that the singlet S becomes tachyonic, Therefore, the singlet field can be driven to be tachyonic by the Yukawa coupling to the SC sector. However, the converg- irrespective of the initial values of the scalar masses. The ing values depend on the initial conditions and, hence, it is singlet mass remains to be tachyonic and also appears in ͓␣ 2 ͔ not automatic for the singlet to become tachyonic contrary to O M (M c) at the weak scale. Now we suppose that the the above case. bare supersymmetric mass term is forbidden by an extra U(1) gauge symmetry and that S carries this U(1) charge so VI. CONCLUSIONS AND DISCUSSIONS as to couple with Higgs fields through Yukawa interaction We have studied soft SUSY breaking parameters in the SHuHd. Also the potential for S is supplied by the D term of this gauge interaction. Therefore, the ␮ term can be gener- Nelson-Strassler type of models: SSM’s coupled with ated by the VEV ͗S͘ of electroweak scale induced by the SCFT’s. We have clarified the condition to derive the expo- tachyonic mass. nential suppression of sfermion masses within the framework of pure SCFT’s, i.e., we have suppressed sfermion masses The singlet field generating the ␮ term of the weak scale for the fields whose anomalous dimensions are determined can be incorporated with the SC sector inducing Yukawa definitely. suppression. Let us demonstrate this here by introducing a ϫ In a realistic case with nonvanishing gauge couplings of singlet to the SU(3)SC SU(3)C model analyzed in Sec. IV. ␣ 2 the SM sector, however, the terms aM a in RGE’s of sfer- We extend the superpotential of the SCFT as mion masses, play an important role in realizing degenerate sfermion masses. The sfermion masses converge on Wϭ␭͑q Q¯ Pϩ¯q ¯PQ͒ϩy ¯q q Hϩ␭ЈSQ¯ Q ␣ 2 ␥ ϭ␥ 1 1 ij i j O( aM a) and these are flavor dependent unless i j .We have shown explicitly how much degeneracy we have be- ϩ␭Љ ¯ ϩ 2 ͑ ͒ SPP SH . 85 tween sfermion masses in the MSSM. For squarks we can have suppression strong enough to avoid the FCNC problem. At the IR fixed point, the anomalous dimensions are fixed to On the other hand, for sleptons, such suppression is weak. ␥ ϭ␥ ϭϪ ␥ ϭ␥ ϭ ␭Јϭ␭Љ be Q P 1/2, q1 S 1. It is seen that at the For squarks, this mechanism is attractive even if we could ␥ ϭ␥ fixed point from Q P . not obtain sufficiently realistic Yukawa matrices only by the Below we examine the RG equations by applying the Nelson-Strassler mechanism, i.e., it might be useful to intro- anomalous dimensions obtained by one-loop perturbation: duce a SC sector in order only to suppress initial nondegeneracy between squark masses. ␥ ϭϪ 8 ␣Јϩ ␣ ϩ␣ Ϫ 8 ␣ ͑ ͒ Q 3 2 ␭ ␭Ј 3 , 86 We have assumed that the SC region is below M X .Itis

075003-13 TATSUO KOBAYASHI AND HARUHIKO TERAO PHYSICAL REVIEW D 64 075003 also possible that the SC region is above M X and the Nelson- are determined definitely. Otherwise, squark and slepton Strassler mechanism would work within the GUT frame- masses are of the same order as the initial values. Suppose work. Such a case can be studied similarly and we would that soft SUSY breaking terms appear only in the SC sector have a significant change for the slepton masses. Such a including squarks and sleptons coupled with this sector, GUT scenario shall be discussed elsewhere ͓27͔. while the SM sector has no SUSY breaking terms, that is, ϭ 2 Also we have discussed the possibility for generating the M a 0 for the gaugino masses of the SM sector and mi ␮ term. We can have naturally the singlet fields, which have ϭ0 for the stop as well as for the sbottom and stau for the tachyonic masses of O(M Z) and whose VEV’s generate the large tan ␤ scenario. In this case, the gaugino and the stop supersymmetric mass term of the Higgs fields. It might be field of the SM sector gain masses due to higher loop effects possible that a similar mechanism generates mass terms of from the SC sector. Thus, those masses are suppressed by the SC matter fields, so that they would decouple the SC loop factors compared with the squark masses of the first and sector from the SM sector. This decoupling scale of the SC the second families. This is one of the possibilities to realize ϭ sector is O(M Z), M c O(M Z). That has the problem of the the decoupling solution. However, note that although squark blowup of ga as discussed in Sec. III A, if those are charged masses of the first and the second families appear in the same 2 under GSM . order as initial values in general, the sign of ͑mass͒ as well Moreover, an application to the neutrino sector is interest- as the values, are totally dependent on initial sfermion ing. Since the right-handed neutrino is the GSM singlet, we masses in the SC sector. We must choose the initial condi- have less limitation for model building. Such an application tions to avoid tachyonic sfermion masses. will be studied elsewhere. We have studied mainly the degenerate solution for the ACKNOWLEDGMENTS FCNC problem. Finally we comment on the decoupling solution. It has been shown that the sfermion masses exponen- The authors would like to thank J. Kubo and K. Yoshioka tially damp in the case where their anomalous dimensions for useful discussions.

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