Supersymmetry in particle physics: the renormalization group viewpoint

Physics Reports 344 (2001) 309}353

Supersymmetry in particle physics: the renormalization group viewpoint夽 D.I. Kazakov Bogoliubov Laboratory of Theoretical Physics (BLTP), Joint Institute of Nuclear Physics (JINR), 141 980 Dubna and ITEP, Moscow Region, Russia Received June 2000; editor: I. Procaccia

Contents 1. Introduction 1.1. The Standard Model 1.2. RG #ow in the Standard Model 2. Supersymmetry 2.1. Motivations of SUSY 2.2. Global SUSY: algebra and representations 2.3. SUSY Lagrangians 3. Minimal Supersymmetric Standard Model 3.1. The "eld content 3.2. Lagrangian of the MSSM 3.3. Soft SUSY breaking 3.4. Masses 3.5. RG #ow in the MSSM

311 311 314 317 317 317 320 321 322 323 324 325 328

4. Renormalization of softly broken SUSY theories 5. RG #ow for the soft terms 5.1. Radiative electroweak symmetry breaking 6. Infrared quasi-"xed points 6.1. Low tan
regime 6.2. Large tan
regime 7. Higgs boson mass prediction in the SM and MSSM 7.1. The Higgs boson mass in the SM 7.2. The Higgs boson mass in the MSSM 8. Conclusion Acknowledgements References

330 332 333 334 334 338 344 344 345 351 351 351

Abstract An attempt is made to present modern hopes to "nd manifestation of supersymmetry, a new symmetry that relates bosons and fermions, in particle physics from the point of view of renormalization group #ow.

夽 Review talk given at the Conference `Renormalization Group at the Turn of the Millenniuma, Taxco, Mexico, January 1999. E-mail address: [email protected] (D.I. Kazakov).

0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 2 9 - 0

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The Standard Model of particle interactions is brie#y reviewed, the main notions of supersymmetry are introduced. In more detail the RG #ow in the Minimal Supersymmetric Standard Model is considered. Predictions of particle spectrum are obtained based on the RG "xed points.  2001 Elsevier Science B.V. All rights reserved. PACS: 11.10.Gh; 11.10.Hi; 11.30.Pb Keywords: Renormalization group; Supersymmetry; Fixed point; Minimal supersymmetric Standard Model

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1. Introduction 1.1. The Standard Model The Standard Model (SM) of fundamental interactions describes strong, weak and electromagnetic interactions of elementary particles [1]. It is based on a gauge principle, according to which all the forces of Nature are mediated by an exchange of the gauge "elds of a corresponding local symmetry group. The symmetry group of the SM is S; (3)S; (2); (1) ,     

(1.1)

whereas the "eld content is the following: Gauge sector: Spin"1. The gauge bosons are spin 1 vector particles belonging to the adjoint representation of group (1.1). Their quantum numbers with respect to S;(3)S;(2);(1) are: G? : (8, 1, 0) S; (3) g , I A Q   intermediate weak bosons =G : (1, 3, 0) S; (2) g , I *   abelian boson B : (1, 1, 0) ; (1) g , I 7   where the coupling constants are usually denoted by g , g and g, respectively. Q Fermion sector: Spin"1/2. The matter "elds are fermions belonging to the fundamental representation of the gauge group. These are believed to be quarks and leptons at least of three generations. The SM is left}right asymmetric. Left- and right-handed fermions have di!erent quantum numbers. Quarks: gluons







;G uG cG tG , , ,2(3, 2, 1/3) QG " ? " ?* DG * dG * sG * bG *   ? ;G "u , c , t ,2(3H, 1, 4/3) ?0 G0 G0 G0   DG "d , s , b ,2(3H, 1,!2/3) ?0 G0 G0 G0   Leptons:






   ¸ " C , I , O ,2(1, 2,!1) ?* e *  *  *   E "e ,  ,  ,2(1, 1,!2) ?0 0 0 0   i"1, 2, 3 } colour, "1, 2, 3,2 } generation. Higgs sector: Spin"0. In the minimal version of the SM there is one doublet of Higgs scalar "elds




H"

H

H\

(1, 2,!1) ,  

(1.2)

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which is introduced in order to give masses to quarks, leptons and intermediate weak bosons via spontaneous breaking of electroweak symmetry. In Quantum Field Theory framework the SM is described by the following Lagrangian: L"L #L #L ,   7  &  L "!G? G? !=G =G !B B    IJ IJ  IJ IJ  IJ IJ #i¸M ID ¸ #iQM ID Q #iEM ID E ? I ? ? I ? ? I ? # i;M ID ; #iDM ID D #(D H)R(D H) , ? I ? ? I ? I I where

(1.3)

(1.4)

G? "R G? !R G? #g f ?@AG@ GA , IJ I J J I Q I J =G "R =G !R =G #g GHI=H =I , IJ I J J I I J B "R B !R B , IJ I J J I g g D ¸ " R !i G=G #i B ¸ , I ? I I 2 2 I ?






D E "(R #igB )E , I ? I I ? g g g D Q " R !i G=G !i B !i Q ?G? Q , I ? I I I ? 2 6 I 2






 



2 g D ; " R !i gB !i Q ?G? ; , I I I ? I ? 3 2 1 g D D " R #i gB !i Q ?G? D , I I ? I ? 3 I 2 "y* ¸M E H#y" QM D H#y3 QM ; HI #h.c. , 7  ?@ ? @ ?@ ? @ ?@ ? @ where HI "i HR . 

L "!<"mHRH! (HRH) . &  2 L

(1.5)

(1.6)

Here y are the Yukawa and is the Higgs coupling constants, both dimensionless, and m is the only dimensional mass parameter. The Lagrangian of the SM contains the following set of free parameters: E E E E E

three gauge couplings g , g, g; Q three Yukawa matrices y* , y" , y3 ; ?@ ?@ ?@ Higgs coupling constant ; Higgs mass parameter m; number of matter "elds (generations).  We use the usual for particle physics units c" "1.

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All the particles obtain their masses due to spontaneous breaking of S; (2) symmetry group  via a non-zero vacuum expectation value (v.e.v.) of the Higgs "eld




H"

v

0

, v"m/( .

(1.7)

As a result the gauge group of the SM is spontaneously broken down to S; (3)S; (2); (1)NS; (3); (1) . A * 7 A #+ The physical weak intermediate bosons are the linear combinations of the gauge ones = Gi= I , Z "!sin  B #cos  = =!" I I I 5 I 5 I (2

(1.8)

with masses 1 gv, m "m /cos  , tan  "g/g , m " 8 5 5 5 5 (2

(1.9)

while the photon "eld  "cos  B #sin  = I 5 I 5 I

(1.10)

remains massless. The matter "elds acquire masses proportional to the corresponding Yukawa couplings: MS "yS v, MB "yB v, MJ "yJ v, m "(2m . ?@ ?@ ?@ ?@ ?@ ?@ &

(1.11)

Explicit mass terms in the Lagrangian are forbidden because they are not S; (2) symmetrical and  would destroy the renormalizability of the Standard Model. The SM has been constructed as a result of numerous e!orts both theoretical and experimental. At present the SM is extraordinarily successful, the achieved accuracy of its predictions corresponds to the experimental data within 5% [2]. All the particles except for the Higgs boson have been discovered experimentally. However the SM has its natural drawbacks and unsolved problems. Among them are: E large number of free parameters, E formal uni"cation of strong and electroweak interactions, E the Higgs boson has not yet been observed and it is not clear whether it is fundamental or composite, E the problem of CP-violation is not well understood including CP-violation in strong interaction, E #avour mixing and the number of generations are arbitrary, E the origin of the mass spectrum is unclear. The answer to these problems lies beyond the SM.

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1.2. RG yow in the Standard Model The question is: how to go beyond the SM? Apparently, we are talking about new particles, new structures, new interactions and new symmetries. The answer is not obvious. We describe below one of the options and assume that at high energies (small distances) fundamental interactions possess wider symmetries, in particular, new kind of symmetry, symmetry between bosons and fermions, called sypersymmetry. This is a wide subject by itself, however, we would like to look at it from the point of view of renormalization group. Let us try to go along the road o!ered by the renormalization group #ow. Let us take the Lagrangian of the SM and see what happens with its parameters (the couplings) when the energy scale increases. As it follows from Eq. (1.1) one has three gauge couplings corresponding to S;(3) ;S;(2) ;;(1) gauge groups, g , g and g, respectively. In what follows, A * 7 Q it will be more useful to consider the set g , g , g "(3/5g, g, g . Besides, there are three Q    Yukawa couplings y , y and y , which are 3;3 matrices in the generation space and one Higgs 3 " * coupling . To simplify the picture we consider only the third generation Yukawa couplings replacing the Yukawa matrices by their (3, 3) elements. Then the RG equations in the leading one-loop order are [3]: da G "b a, t"log(Q/) , G G dt






(1.12)

 

d> 9 17 9 R "!> 8a # a # a ! > , R    dt 4 20 2 R d> 9 1 3 @ "!> 8a # a # a ! > , @  4  4  2 R dt

(1.13)



9 9 d> O "!> a # a !3> , O 4  4  R dt d 9 9 27 9 9 " a # a a # a ! a ! a #6 > !6>#6  , R R dt 8  20   200  2  10 

(1.14)

where to simplify the formulas we have used the notation  g a, G, G G 4
16


(i"1, 2, 3),

(yI ) > ,  I 16


(k"t, b, ) .

For the SM the coe$cients b are G b 0 4/3 1/10  b " b " !22/3 #N 4/3 #N 1/6 . $ &  G  b !11 4/3 0 







(1.15)

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Here N is the number of generations of matter multiplets and N is the number of Higgs $ &  doublets. We use N "3 and N "1 for the minimal SM. $ &  The initial conditions for the couplings can be taken at some scale which is experimentally favourable. Thus, for the gauge couplings one has precise measurement at the Z-boson mass scale obtained at LEP accelerator [4]:  (M )"0.017,  (M )"0.034,  (M )"0.118$0.005 . (1.16)  8  8  8 As for the Yukawa couplings, they are related to the running quark masses by Eq. (1.11), where v is the vacuum expectation value of the Higgs "eld. It can be calculated, for instance, from the Z-boson mass according to Eq. (1.9) and is equal to v"174.1 GeV. Thus, knowing the quark masses one can "nd the values of the Yukawa couplings. One should, however, distinguish between the running and the pole quark masses which are determined experimentally. Having all this in mind and solving Eqs. (1.12) and (1.13) one has the following qualitative picture (see Fig. 1). The behaviour of the Higgs quartic coupling strongly depends on initial conditions which are unknown in this case. We return to this subject later. The qualitative picture presented in Fig. 1 contains an obvious uni"cation pattern. The three gauge couplings are seen to unify at energy of the order of 10}10 GeV and so do the Yukawa couplings > and > . What does it mean? The usual answer is given in the framework of the Grand @ O Uni"cation hypothesis [5]: three-gauge interactions are the three branches of a single-gauge interaction described by a simple gauge group with a single coupling. All quarks and leptons belong to some representation of this group. This explains the equality of gauge and (some) Yukawa couplings at the uni"cation scale. The GUT hypothesis has many far reaching consequences, however, one can see that the uni"cation scale is very high. This is not only di$cult to check experimentally, but creates a big problem, called the hierarchy problem. The point is that in a theory with two so very di!erent scales: M &10 GeV and 5 M &10 GeV, it is very di$cult both to achieve this hierarchy of 10 in a natural way and to %32 preserve it against the radiative corrections. Indeed, due to modern point of view, the mass scales in the SM and in GUT are de"ned by vacuum expectation values of the scalar "elds, called the Higgs "elds. Non-zero v.e.v.'s of these "elds lead to spontaneous breaking of the corresponding gauge symmetry and provide masses to all the particles. So, we have at least two scalar particles with the masses of the order of 10 and 10 GeV. However, these masses obtain the radiative corrections proportional to the masses of the interacting particles. Due to inevitable interaction between the light and heavy "elds the radiative corrections to the light Higgs mass are proportional to the heavy one m&gM , where g is some coupling. Assuming m&10 GeV, M&10 GeV, g&0.1, one gets the radiative correction which is 10 times bigger than the mass itself. This correction obviously spoils the hierarchy unless it is cancelled. A cancellation with a precision &10\ needs a very accurate "ne tuning of the coupling constants. Solution to the "ne-tuning problem has been found in the framework of a revolutionary hypothesis: the existence of a new type of symmetry, the symmetry between bosons and fermions, called supersymmetry.

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Fig. 1. Evolution of the gauge (left) and Yukawa couplings (right) in the Standard Model. The width of the curves corresponds to the experimental error.

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2. Supersymmetry 2.1. Motivations of SUSY Supersymmetry or fermion}boson symmetry has not yet been observed in Nature. This is a purely theoretical invention [6]. Its validity in particle physics follows from common belief in uni"cation. The general idea is a uni"cation of all forces of Nature. It de"nes the strategy: increasing uni"cation towards smaller distances up to l &10\ cm including quantum gravity. However, .J the graviton has spin 2, while the other gauge bosons (photon, gluons, = and Z weak bosons) have spin 1. Uni"cation of spins 2 and 1 gauge forces within unique algebra is forbidden due to the no-go theorems for any symmetry but SUSY. If Q is a generator of SUSY algebra, then Qboson"fermion and Qfermion"boson . Hence starting with the graviton spin 2 state and acting by SUSY generators we get the following chain of states: spin 2Pspin 3/2Pspin 1Pspin 1/2Pspin 0 . Thus, a partial uni"cation of matter (fermions) with forces (bosons) naturally arises out of an attempt to unify gravity with the other interactions. The uniqueness of SUSY is due to a strict mathematical statement that algebra of SUSY is the only graded (i.e. containing anticommutators as well as commutators) Lie algebra possible within relativistic "eld theory [7]. The other motivation of SUSY is the solution of the hierarchy problem mentioned above. At the moment supersymmetry is the only known way to achieve the cancellation of quadratic terms in radiative corrections (also known as the cancellation of the quadratic divergences). Moreover, SUSY automatically cancels quadratic corrections in all orders of perturbation theory [8]. 2.2. Global SUSY: algebra and representations As can be easily seen, supersymmetry transformations di!er from ordinary global transformations as far as they convert bosons into fermions and vice versa. Indeed if we symbolically write SUSY transformation as B" ) f , where B and f are boson and fermion "elds, respectively, and is an in"nitesimal transformation parameter, then from the usual (anti)commutation relations for (fermions) bosons  f, f "0, [B, B]"0 , we immediately "nd  , "0 .

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This means that all the generators of SUSY must be fermionic, i.e. they must change the spin by a half-odd amount and change the statistics. Combined with the usual PoincareH and internal symmetry algebra the Super-PoincareH Lie algebra contains additional generators [6]: [QG , P ]"[QM G , P ]"0 , ? I ? I [QG , M ]"( )@QG , [QM G , M ]"!QM GQ ( )@ Q , ? IJ  IJ ? @ ? IJ  @ IJ ? [QG , B ]"(b )G QH , [QM G , B ]"!QM H (b )G , ? P PH ? ? P ? PH QG , QM HQ "2GH(I) Q P , ? @ ?@ I QG , QH "2 ZGH, Z "aP b , ZGH"Z> , ? @ ?@ GH GH P GH QM G , QM HQ "!2  Q ZGH, [Z , anything]"0 , ? @ ?@ GH , ,
,
Q "1, 2, i, j"1, 2,2, N .

(2.1)

Here P and M are four-momentum and angular momentum operators respectively, B are I IJ P internal symmetry generators, QG and QM G are spinorial SUSY generators and Z are the so-called GH central charges. , ,
,
Q are spinorial indices. In the simplest case one has one spinor generator Q (and the conjugated one QM  ) that corresponds to an ordinary or N"1 sypersymmetry. When ? ? N'1 one has an extended sypersymmetry. An elegant formulation of supersymmetry transformations and invariants can be achieved in the framework of superspace [9]. Superspace di!ers from the ordinary Euclidean (Minkowski) space by addition of two new coordinates,  and M  , which are grassmannian, i.e. anticommuting, ? ? variables  ,  "0, M  , M Q "0, "0, ? @ ? @ ? Thus, we go from space to superspace

M  "0, ,
, ,
Q "1, 2 . ?

Space N Superspace . x ,  , M  I ? ? A SUSY group element can be constructed in superspace in the same way as an ordinary translation in the usual space x

I

G(x, , M )"e \VI.I >F/>FM /M  . It leads to the supertranslation in superspace, x Px #i  !i  M , I I I I P# , M PM #  ,

(2.2)

where and  are grassmannian transformation parameters. Taking them to be local or space}time dependent one gets local translation. And the theory that is invariant under local translations is

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general relativity. Thus, local supersymmetry is just the theory of gravity or supergravity [18]. This way following the gauge principle one gets a uni"ed theory of all four interactions known as SUGRA theory. To de"ne the "elds on a superspace consider representations of the Super-PoincareH group (2.1) [10]. The simplest one is a scalar super"eld F(x, , M ) which is SUSY invariant. Its Taylor expansion in  and M has only several terms due to the nilpotent character of grassmannian parameters. However, this super"eld is a reducible representation of SUSY. To get an irreducible one, we de"ne a chiral super"eld which obeys the equation R DM F"0, where DM "! !iIR . I RM

(2.3)

Its Taylor expansion looks like (y"x#i )
(y, )"A(y)#(2(y)#F(y) "A(x)#iIM R A(x)#M M 䊐A(x) I  i R (x)IM #F(x) . #(2(x)! I (2

(2.4)

The coe$cients are ordinary functions of x being the usual "elds. They are called the components of a super"eld. In Eq. (2.4) one has 2 bosonic (complex scalar "eld A) and 2 fermionic (Weyl spinor "eld ) degrees of freedom. The component "elds A and  are called the superpartners. The "eld F is an auxiliary "eld, it has the `wronga dimension and has no physical meaning. It is needed to close algebra (2.1). One can get rid of the auxiliary "elds with the help of equations of motion. Thus, a super"eld contains an equal number of bosonic and fermionic degrees of freedom. Under SUSY transformation they convert one into another. The product of chiral super"elds
,
, etc., is also a chiral super"eld, while the product of chiral and antichiral ones
>
is a general super"eld. To construct the gauge invariant interactions, we will need a real vector super"eld <"<>. It is not chiral but rather a general super"eld. Its expansion over  and M looks like <(x, , M )"C(x)#i(x)!iM (x) i i # [M(x)#iN(x)]! M M [M(x)!iN(x)] 2 2





i ! IM v (x)#iM M (x)# IR (x) I I 2









i 1 1 ! iM M  # IR (x) # M M D(x)# 䊐C(x) . I 2 2 2

(2.5)

The physical degrees of freedom corresponding to a real vector super"eld are the vector gauge "eld v and its superpartner the Majorana four-component spinor "eld made of two Weyl spinors and I

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M . All other components are unphysical and can be eliminated. Thus, we again have an equal number of bosonic and fermionic degrees of freedom. One can choose a gauge (Wess}Zumino gauge) where C""M"N"0, leaving us with the physical degrees of freedom except for the auxiliary "eld D. In this gauge <"!IM v (x)#iM M (x)!iM M  (x)#M M D(x) , I  <"!M M v (x)vI(x) , I  <"0 , etc. One can de"ne also a "eld strength tensor (as analog of F in gauge theories) IJ = "!DM e\4D e4, = M  "!De\4DM e4 , ?  ? ?  ? which is needed to construct gauge invariant Lagrangians.

(2.6)

(2.7)

2.3. SUSY Lagrangians In the super"eld notation SUSY invariant Lagrangians are the polynomials of super"elds. Having in mind that for component "elds we should have the ordinary terms, the general SUSY invariant Lagrangian has the form [8}10]





L" d dM
>
# d[
#m

#y


]#h.c. , G G G G  GH G H  GHI G H I

(2.8)

where the "rst part is a kinetic term and the second one is a superpotential W. Here instead of taking the proper components we use an integration over the superspace according to the rules of grassmannian integration [11]



d "0, ?



 d " . ? @ ?@

Performing this integration we get in components L"iR M I #AH䊐A #FHF I G G G G G G #[ F #m (A F !  )#y (A A F !  A )#h.c.] GHI G H I G H I G G GH G H  G H or solving the constraints

(2.9)

L"iR M I #AH䊐A !m   !mH M M I G G G G  GH G H  GH G H !y   A !yH M M AH!<(A , A ) , (2.10) GHI G H I GHI G H I G H where <"FHF . Note that because of the renormalizability constraint <4A the superpotential I I should be limited by W4
 as in Eq. (2.8).

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The gauge "eld part of a Lagrangian is





1 1 M ? = M  L" d =?= # dM = ? 4 ? 4 " D!F FIJ!i ID M . (2.11)   IJ I To obtain a gauge-invariant interaction with matter chiral super"elds, consider their gauge transformation
Pe\ E
,
>P
>e E>,

) ,

where  is a gauge parameter (chiral super"eld). It is clear now how to construct both SUSY and gauge-invariant interaction which is equivalent to transition from the usual to covariant derivatives





d dM
>
N d dM
>eE4
. G G G G

Thus, the form of the Lagrangian is practically "xed by symmetry requirements. The only freedom is the "eld content, the value of the gauge coupling g, the Yukawa couplings y and the GHI masses. This allows one to construct a SUSY generalization of the SM.

3. Minimal Supersymmetric Standard Model As has been already mentioned, in SUSY theories the number of bosonic degrees of freedom equals that of fermionic. In the SM one has 28 bosonic and 90 fermionic degrees of freedom. So the SM is in great deal non-supersymmetric. Trying to add some new particles to supersymmetrize the SM, one should take into account the following observations: 1. there are no fermions with quantum numbers of the gauge bosons; 2. Higgs "elds have a non-zero v.e.v.s, hence they cannot be superpartners of quarks and leptons since this would induce a spontaneous violation of baryon and lepton numbers; 3. one needs at least two complex chiral Higgs multiplets to give masses to Up and Down quarks. The latter is due to the form of a superpotential and chirality of matter super"elds. Indeed, the superpotential should be invariant under S;(3);S;(2);;(1) gauge group. If one looks at the Yukawa interaction in the Standard Model, Eq. (1.5), one "nds that it is indeed ;(1) invariant since the sum of hypercharges in each vertex equal zero. In the last term this is achieved by taking the conjugated Higgs doublet HI "i HR instead of H. However, in SUSY H is a chiral super"eld and  hence a superpotential, which is constructed out of chiral "elds, can contain only H but not HI , which is an antichiral super"eld. Another reason for the second Higgs doublet is related to chiral anomalies. It is known that chiral anomalies spoil the gauge invariance and, hence, the renormalizability of the theory. They are cancelled in the SM between quarks and leptons in each generation. However, if one introduces

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a chiral Higgs super"eld, it contains higgsinos, which are chiral fermions, and contain anomalies. To cancel them one has to add the second Higgs doublet with the opposite hypercharge. Therefore, the Higgs sector in SUSY models is inevitably enlarged, it contains an even number of doublets. Conclusion: In SUSY models supersymmetry associates known bosons with new fermions and known fermions with new bosons. 3.1. The xeld content Consider the particle content of the Minimal Supersymmetric Standard Model [12]. According to the previous discussion in the minimal version we double the number of particles (introducing a superpartner to each particle) and add another Higgs doublet (with its superpartner). The particle content of the MSSM is given in Table 1 [13]. In Table 1, a"1, 2,2, 8 and k"1, 2, 3 are S;(3) and S;(2) indices, respectively, and i"1, 2, 3 is the generation index. Hereafter tilde denotes a superpartner of an ordinary particle. Thus, the characteristic feature of any supersymmetric generalization of the SM is the presence of superpartners. If supersymmetry is exact, superpartners of ordinary particles should have the same masses and have to be observed. The absence of them at modern energies is believed to be explained by the fact that their masses are very heavy, that means that supersymmetry should be broken. Hence, if the energy of accelerators is high enough, the superpartners will be created. Table 1 Particle content of the MSSM Super"eld

Bosons

Fermions

S; (3) A

S; (2) *

Gauge G?
Gluon g? Weak =I (=!, Z) Hypercharge B()

Gluino g ? Wino, zino w I (w !, z ) Bino bI ()

8 1 1

0 3 1

0 0 0

1

2

!1

1

1

2

3

2

1/3

3H

1

!4/3

3H

1

2/3

1

2

!1

1

2

1

; (1) 7

Matter ¸ G E G Q G ; G D G Higgs H  H 



¸I "(, e ) G * EI "e G 0 QI "(u , dI ) G * Squarks ;I "u G 0 DI "dI G 0 Sleptons



Higgses



H



H





¸ "(, e) G * E "e G 0 Q "(u, d) G * Quarks ; "uA G 0 D "dA G 0

Leptons



Higgsinos



HI  HI 

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The presence of an extra Higgs doublet in SUSY model is a novel feature of the theory. In the MSSM one has two doublets with the quantum numbers (1, 2,!1) and (1, 2, 1), respectively [14]:











S #iP H>   H v #  H> S    H " " , H " " v #  #iP (2 ,   H\ H  (2   H\ 







(3.1)

where v are the vacuum expectation values of the neutral components. G Hence, one has 8"4#4"5#3 degrees of freedom. As in the case of the SM, 3 degrees of freedom can be gauged away, and one is left with 5 physical states compared to 1 state in the SM. Thus, in the MSSM, as actually in any two Higgs doublet model, one has "ve physical Higgs bosons: two CP-even neutral, one CP-odd neutral and two charged. We consider the mass eigenstates below. 3.2. Lagrangian of the MSSM The Lagrangian of the MSSM consists of two parts; the "rst part is SUSY generalization of the Standard Model, while the second one represents the SUSY breaking as mentioned above: L"L #L , 1317    where L

1317

L

% 

and

"L

% 

1 "  4 13 13 3

7 



(3.3)

#L 7 

#  +  L

(3.2)








d  ¹r =?= # dM ¹r = M ? = M  ? ?



d  dM
ReE 4K  >E 4K  >E 4K 
, G G

" d(W #W )#h.c. . 0 ,0

(3.4) (3.5)

The index R in a superpotential refers to the so-called R-parity which adjusts a `#a charge to all the ordinary particles and a `!a charge to their superpartners [15]. The "rst part of W is R-symmetric = " (y3 QH ;A HG #y" QH DA HG #y* ¸H EA HG #HG HH ) , (3.6) 0 GH ?@ ? @  ?@ ? @  ?@ ? @    where i, j"1, 2, 3 are S;(2) and a, b"1, 2, 3 are the generation indices; colour indices are suppressed. This part of the Lagrangian almost exactly repeats that of the SM except that the "elds are now the super"elds rather than the ordinary "elds of the SM. The only di!erence is the last term which describes the Higgs mixing. It is absent in the SM since we have only one Higgs "eld there. The second part is R-non-symmetric = " ( * ¸G ¸H EA # *Y ¸G QH DA # ¸G HH )# ;A DA DA . ,0 GH ?@B ? @ B ?@B ? @ B ? ?  ?@B ? @ B

(3.7)

324

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These terms are absent in the SM. The reason is very simple: one cannot replace the super"elds in Eq. (3.7) by the ordinary "elds like in Eq. (3.6) because of the Lorentz invariance. These terms have the other property, they violate either lepton (the "rst line in Eq. (3.7)) or baryon number (the second line). Since both e!ects are not observed in Nature, these terms must be suppressed or be excluded. In the minimal version of the MSSM these terms are not included, they are forbidden by R-parity conservation [16]. 3.3. Soft SUSY breaking To introduce supersymmetry breaking as required by the absence of superpartners at modern energies, one has to be careful not to spoil the cancellation of divergencies which allows to solve the hierarchy problem. This is achieved by spontaneous breaking of SUSY in the same way as spontaneous electroweak symmetry breaking. One introduces the "eld whose vacuum expectation value breaks supersymmetry. However, due to a special character of SUSY, this should be a super"eld, whose auxiliary F or D components acquire non-zero v.e.v.'s. This leads to appearance of the so-called soft breaking terms. In the simplest version the soft supersymmetry breaking terms are [17]: !L

  




1 "m   # m  I I ? ?  G 2  ? G



#A[y3 QI ;I A H #y" QI DI A H #y* ¸I EI A H ]#B[H H ]#h.c. , (3.8) ?@ ? @  ?@ ? @  ?@ ? @    where we have suppressed S;(2) indices. Here  are all scalar "elds, I are the gaugino "elds, G ? QI , ;I , DI and ¸I , EI are the squark and slepton "elds, respectively, and H are the S;(2) doublet  Higgs "elds. These terms are obtained via supergravity mechanism and are usually introduced at the GUT scale. We have assumed in Eq. (3.8) the so-called universality of the soft terms, namely, we put all the spin 0 particle masses to be equal to the universal value m , all the spin 1/2 particle (gaugino)  masses to be equal to m and all the cubic and quadratic terms, proportional to A and B, to repeat  the structure of the Yukawa superpotential (3.6). This is an additional requirement motivated by the supergravity mechanism of SUSY breaking as mentioned earlier [18]. Universality is not a necessary requirement and one may consider non-universal soft terms as well. However, it will not change the qualitative picture presented below, so for simplicity, in what follows, we consider the universal boundary conditions. It should be noted that supergravity-induced universality of the soft terms is more likely to be valid at the Planck scale, rather than at the GUT one. This is because a natural scale for gravity is M , while M is the scale for the gauge interactions. However, due to a small di!erence .  %32 between these two scales, it is usually ignored in the "rst approximation resulting in minor uncertainties in the low-energy predictions [19]. The soft terms explicitly break supersymmetry. As will be shown later they lead to the mass spectrum of superpartners di!erent from that of the ordinary particles. Remind that the masses of quarks and leptons remain zero until S;(2) invariance is spontaneously broken.

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3.4. Masses With given values of m , m , , > , > , > , A, and B at the GUT scale, one can solve the   R @ O corresponding RG equations thus linking the values at the GUT and electroweak scales. Substituting these parameters into the mass matrices one can predict the mass spectrum of superpartners [12,20]. 3.4.1. Gaugino}higgsino mass terms The mass matrix for the gauginos, the superpartners of the gauge bosons, and for higgsinos, the superpartners of the Higgs bosons, is non-diagonal, thus leading to their mixing. The mass terms look like "!M M !M!(M MA#h.c.) ,   ? ?  %  }&   where , a"1, 2,2, 8, are the Majorana gluino "elds and ? BI 

(3.9)

L





"

= I 




, "

= I >

(3.10) HI > HI   HI   are, respectively, the Majorana neutralino and Dirac chargino "elds. The neutralino mass matrix is M  0

0



!M cos
sin M sin
sin 8 5 8 5 M cos
cos !M sin
cos 8 5 8 5 . (3.11) 0 !

M  !M cos
sin M cos
cos 8 5 8 5 M sin
sin !M sin
cos ! 0 8 5 8 5 The physical neutralino masses M  G are obtained as eigenvalues of this matrix after diagonaliQ zation. For charginos one has M"






(2M sin
 5 .  (2M cos
5 This matrix has two chargino eigenstates ! with mass eigenvalues  M "[ M ##2M  5   G((M !)#4M cos 2
#4M (M ##2M  sin 2
)] .  5 5   MA"

M

(3.12)

(3.13)

3.4.2. Squark and slepton masses The non-negligible Yukawa couplings cause a mixing between the electroweak eigenstates and the mass eigenstates of the third generation particles. The mixing matrices for the m , m  and R @ m  are O m  m (A ! cot
) R* R R , (3.14) m (A ! cot
) m  R R R0






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m (A ! tan
) @ @ , m  @0




m (A ! tan
) O O m  O0

m  @* m (A ! tan
) @ @

m  O* m (A ! tan
) O O



(3.15)



(3.16)

with m  "m  #m#(4M !M ) cos 2
, R* / R  5 8 m  "m  #m!(M !M ) cos 2
, R0 3 R  5 8 m  "m  #m!(2M #M ) cos 2
, 5 8 @* / @  m  "m  #m#(M !M ) cos 2
, @0 " @  5 8 m  "m  #m!(2M !M ) cos 2
, O* * O  5 8 m  "m  #m#(M !M ) cos 2
O0 # O 5 8 and the mass eigenstates are the eigenvalues of these mass matrices. 3.4.3. The Higgs potential As has been already mentioned, the Higgs potential in MSSM is totally de"ned by superpotential (and the soft terms). Due to the structure of W the Higgs self-interaction is given by the D-terms, while the F-terms contribute only to the mass matrix. The tree level potential is < (H , H )"m H #m H !m (H H #h.c.)           g g#g (H !H )# H>H  , #   2   8

(3.17)

where m "m  #, m "m  #. At the GUT scale m "m "m # , m "!B .  &  &       Notice, that the Higgs self-interaction coupling in Eq. (3.17) is "xed and is de"ned by the gauge interactions as opposite to the SM. Potential (3.17), in accordance with supersymmetry, is positively de"nite and stable. It has no non-trivial minimum di!erent from zero. Indeed, let us write the minimization condition for potential (3.17) 1 < g#g "m v !m v # (v !v )v "0 ,        2 H 4  1 < g#g "m v !m v # (v !v )v "0 ,        2 H 4  where we have introduced the notation v

H ,v "v cos
, H ,v "v sin
, v"v #v , tan
,  .       v 

(3.18) (3.19)

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Solution of Eqs. (3.18) and (3.19) can be expressed in terms of v and sin
: 4(m !m tan
)   v" , (g#g)(tan
!1)

2m  sin 2
" . (3.20) m #m   One can easily see from Eq. (3.20) that if m "m "m # , v happens to be negative, i.e. the     minimum does not exist. In fact, real positive solutions to Eqs. (3.18) and (3.19) exist only if the following conditions are satis"ed [13]: m #m '2m , m m (m , (3.21)       which is not the case at the GUT scale. This means that spontaneous breaking of the S;(2) gauge invariance, which is needed in the SM to give masses for all the particles, does not take place in the MSSM. This strong statement is valid, however, only at the GUT scale. Indeed, going down with energy the parameters of potential (3.17) are renormalized. They become the `runninga parameters with the energy scale dependence given by the RG equations. The running of the parameters leads to a remarkable phenomenon known as a radiative spontaneous symmetry breaking which we discuss below. Provided conditions (3.21) are satis"ed the mass matrices at the tree level are CP-odd components P and P :   tan
1 R< " m , (3.22) M"  RP RP G G 1 cot
G H & T CP-even neutral components S and S :   tan
!1 cot
!1 R< M" " m # M cos
sin
. (3.23)  8 RS RS G G !1 cot
!1 tan
G H & T Charged components H\ and H>:



























tan
1 R< " (m #M cos
sin
) . M "  5 RH>RH\ G G 1 cot
G H & T Diagonalizing the mass matrices one gets the mass eigenstates [13]: G"!cos
P #sin
P , Goldstone boson PZ ,    A"sin
P #cos
P , Neutral CP"!1 Higgs ,   G>"!cos
(H\)H#sin
H>, Goldstone boson P=> ,   H>"sin
(H\)H#cos
H>, Charged Higgs ,   h"!sin S #cos S , SM Higgs boson CP"1 ,   H"cos S #sin S , Extra heavy Higgs boson ,   where the mixing angle  is given by




tan 2"!tan 2


m #M  8 m !M  8



.

(3.24)

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The physical Higgs bosons acquire the following masses [12]: CP-odd neutral Higgs A: m "m #m ,    Charge Higgses H!:

m ! "m #M , &  5

(3.25)

CP-even neutral Higgses H, h: m "[m #M $((m #M )!4m M cos 2
] , &F   8  8  8

(3.26)

where as usual g g#g M " v, M " v . 5 8 2 2 This leads to the once celebrated SUSY mass relations: m ! 5M , & 5 m 4m 4M , F  & m 4M cos 2
4M , F 8 8 m#m "m #M . F &  8 Thus, the lightest neutral Higgs boson happens to be lighter than Z boson, that clearly distinguishes it from the SM one. Though we do not know the mass of the Higgs boson in the SM, there are several indirect constraints leading to the lower boundary of m1+5135 GeV [31]. After F including the radiative corrections the mass of the lightest Higgs boson in the MSSM, m , increases. F We consider it in more detail below. 3.5. RG yow in the MSSM If one compares the RG #ow in the SM and the MSSM, one "nds additional contributions from superpartners to the RG equations. Consider the gauge couplings. In the SM the RG #ow is given by Eqs. (1.12). We have mentioned already in Section 1 that it o!ers the uni"cation pattern supporting the GUT hypothesis. However, if one looks at the curves more attentively, one "nds that the situation is not that good. Indeed, let us consider the solution to the RG equations in more detail. The result is demonstrated in the left part of Fig. 2, which shows the evolution of an inverse of the couplings as function of a logarithm of energy [21]. In this presentation the evolution becomes a straight line in "rst order. The second-order corrections are small and do not cause any visible deviation from a straight line. Fig. 2 clearly demonstrates that within the SM the coupling constants uni"cation at a single point is impossible. It is excluded by more than 8 standard deviations [21}23]. This result means that the uni"cation can only be obtained if new physics enters between the electroweak and the Planck scales.

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Fig. 2. Evolution of the inverse of the three coupling constants in the Standard Model (left) and in the supersymmetric extension of the SM (MSSM) (right). Only in the latter case uni"cation is obtained. The SUSY particles are assumed to contribute only above the e!ective SUSY scale M of about 1 TeV, which causes the change in slope in the evolution 1317 of couplings. The thickness of the lines represents the error in the coupling constants [21].

In the MSSM the slopes of the RG evolution curves are modi"ed. The coe$cients b in Eq. (1.12) G now are





 b



0

2

3/10

b " b " !6 #N (3.27) 2 #N 1/2 , G $ &   b !9 2 0  where use N "3 and N "2, which corresponds to the MSSM. $ &  It turns out that within the SUSY model perfect uni"cation can be obtained if the SUSY masses are of the order of 1 TeV. This is shown on the right part of Fig. 2; the SUSY particles are assumed to contribute e!ectively to the running of the coupling constants only for energies above the typical SUSY mass scale, which causes the change in the slope of the lines near 1 TeV. From the "t requiring uni"cation one "nds for the breakpoint M and the uni"cation point M [21]: 1317 %32 M "10 ! !  GeV , 1317 M "10 ! !  GeV , %32 \ "(26.3$1.9$1.0) . (3.28) %32 The "rst error originates from the uncertainty in the coupling constant, while the second one is due to the uncertainty in the mass splittings between the SUSY particles. For SUSY models, the dimensional reduction DR scheme is used [24]. This uni"cation of the gauge couplings was considered as the "rst `evidencea for supersymmetry, especially since M was found in the range preferred by the "ne-tuning arguments. 1317

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It should be noted, that the uni"cation of the three curves at a single point is not that trivial as it may seem from the existence of three free parameters (M ,M and  ). The reason is 1317 %32 %32 simple: when introducing new particles one in#uences all three curves simultaneously, thus giving rise to strong correlations between the slopes of the three lines. For example, adding new generations and/or new Higgs doublets never yield uni"cation.

4. Renormalization of softly broken SUSY theories To "nd the RG #ow for the soft terms one has to know how they are renormalized. Remarkably that the renormalizations in softly broken SUSY theories follow a simple pattern which is completely de"ned by an unbroken theory [25}28]. The main idea is that a softly broken supersymmetric gauge theory can be considered as a rigid SUSY theory imbedded into external space}time-independent super"eld, so that all couplings and masses become external super"elds. The crucial statement is that the singular part of e!ective action depends on external super"eld, but not on its derivatives, so that one can calculate it when the external "eld is a constant, i.e. in a rigid theory [29]. This approach to a softly broken sypersymmetric theory allows one to use remarkable mathematical properties of N"1 SUSY theories such as non-renormalization theorems, cancellation of quadratic divergences, etc. The renormalization procedure in a softly broken SUSY gauge theory can be performed in the following way [27]: One takes renormalization constants of a rigid theory, calculated in some massless scheme, substitutes instead of the rigid couplings (gauge and Yukawa) their modi"ed expressions, which depend on a grassmannian variable, and expand over this variable. This gives renormalization constants for the soft terms. Di!erentiating them with respect to a scale one can "nd corresponding renormalization group equations. In fact, as it has been shown in [30] this procedure works at all stages. One can make the abovementioned substitution on the level of the renormalization constants, RG equations, solutions to these equations, approximate solutions, "xed points, "niteness conditions, etc. Expanding then over a grassmannian variable one obtains corresponding expressions for the soft terms. We demonstrate now how this procedure works in the MSSM. Using notation introduced above the modi"ed couplings in the MSSM are (", "M ) (4.1) a "a (1#M #M M #(M M M # G )) , G G G G G G ? >I "> (1!A !AM #(A AM # )) , (4.2) I I I I I I I where M are the gaugino masses, A are the trilinear scalar couplings,  are the certain G I I combinations of soft squark and slepton masses entering the Yukawa vertex and  G are the SUSY ? ghost soft terms  "m  #m  #m ,  "m  #m  #m , R / 3 & @ / " &  "m  #m  #m ,  G "M#m  G G EF O * # & ? and m  is the soft scalar ghost mass, which is eliminated by solving the RG equation. In one-loop EF order m  "0. EF

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To get now the RG equations for the soft terms one just has to take the corresponding RG equations for the rigid couplings and perform the grassmannian expansion. The one-loop RG equations for the MSSM couplings are [34]: da G "b a , G G dt








16 13 d> 3 "!> a #3a # a !6> !> , * 3   15  3 " dt



d> 16 7 " "!> a #3a # a !> !6> !> , " 3   15  3 " * dt



9 d> * "!> 3a # a !3> !4> . *  " * 5  dt

(4.3)

Performing the grassmannian expansion one "nds: dM G "b a M , G G G dt 16 13 dA 3 " a M #3a M # a M #6> A #> A ,     3 3 " " 3 15   dt 16 7 dA " " a M #3a M # a M #6> A #> A #> A ,   15   " " 3 3 * * 3   dt 9 dA * "3a M # a M #3> A #4> A ,   5   " " * * dt dB 3 "3a M # a M #3> A #3> A #> A .   3 3 " " * * dt 5  




dm  / "! dt



16 1 a M #3a M # a M !> (m  #m  #m  #A )   15   3 / 3 & 3 3  



!> (m  #m  #m  #A ) , " " / " &







 

 

dm  3 "! dt

16 16 a M # a M !2> (m  #m  #m  #A ) , 3 / 3 & 3 3   15  

dm  " "! dt

16 4 a M # a M !2> (m  #m  #m  #A ) , " / " & " 3   15  





1 dm  * "! 3 a M # a M !> (m  #m  #m #A ) ,   * * # & * 5   dt dm  # "! dt





12 a M !2> (m  #m  #m  #A ) , * * # & * 5  

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d 1 "! 3 a # a !(3> #3> #> ) ,  3 " * dt 5 



1 dm  & "! 3 a M # a M !3> (m  #m  #m #A )   " / " & " dt 5  



!> (m  #m  #m  #A ) , * * # & *








1 dm  & "! 3 a M # a M !3> (m  #m  #m #A ) ,   5   3 / 3 & 3 dt

(4.4)

where we have already substituted the solution m  "0 in the one-loop order. (Note that to get the EF RG equation for the individual squark and slepton masses one needs to know the anomalous dimensions for the corresponding "elds.)

5. RG 6ow for the soft terms Having all the RG equations, one can "nd now the RG #ow for the soft terms. To see what happens at lower scales one has to run the RG equations for the mass parameters from GUT to the EW scale. Let us take some initial values of the soft masses at the GUT scale in the interval between 10 and 10 GeV consistent with SUSY scale suggested by uni"cation of the gauge couplings (3.28). This leads to the following RG #ow of the soft terms shown in Fig. 3 [20] (note that we perform the running of soft parameters in the opposite direction, from GUT to EW scale).

Fig. 3. An example of evolution of sparticle masses and soft supersymmetry breaking parameters m "m  # and  & m "m  #.  &

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One should mention the following general features common to any choice of initial conditions: (i) The gaugino masses follow the running of the gauge couplings and split at low energies. The gluino mass is running faster than the others and is usually the heaviest due to the strong interaction. (ii) The squark and slepton masses also split at low energies, the stops (and sbottoms) being the lightest due to relatively big Yukawa couplings of the third generation. (iii) The Higgs masses (or at least one of them) are running down very quickly and may even become negative. To calculate the masses one has also to take into account the mixing between various states (see Eqs. (3.11), (3.12), (3.14)}(3.16)). 5.1. Radiative electroweak symmetry breaking The running of the Higgs masses leads to the phenomenon known as a radiative electroweak symmetry breaking. By this we mean the following: at the GUT energy scale both the Higgs mass parameters m and m are positive and the Higgs potential has no non-trivial minima. However,   when running down to the EW scale due to the radiative corrections they may change sign so that the potential develops a non-trivial minimum. At this minimum the electroweak symmetry happens to be spontaneously broken. Thus, contrary to the SM where one has to choose the negative sign of the Higgs mass squared `by handa, in the MSSM the e!ect of spontaneous symmetry breaking is triggered by the radiative corrections. Indeed, one can see in Fig. 3 that m (or both m and m ) decreases when going down from the    GUT scale to the M scale and can even become negative. This is the e!ect of the large top (and 8 bottom) Yukawa couplings in the RG equations. As a result, at some value of Q conditions (3.21) are satis"ed, so that the non-trivial minimum appears. This triggers spontaneous breaking of the S;(2) gauge invariance. The vacuum expectations of the Higgs "elds acquire non-zero values and provide masses to the quarks, leptons, S;(2) gauge bosons, and additional masses to their superpartners. This way one obtains also the explanation of why the two scales are so much di!erent. Due to the logarithmic running of the parameters one needs a long `running timea to get m to be negative  when starting from a positive value of the order of M scale &10}10 GeV. 1317 6. Infrared quasi-5xed points Examining the RG equations for the Yukawa couplings one "nds that they possess the infrared "xed points. This is a very typical behaviour for RG equations. In this section we give a short description of the infrared quasi-"xed points (IRQFP) [33] in the MSSM. They play an important role in predictions of the mass spectrum. As in the previous section we consider the RG #ow in the direction from GUT to EW scale, the running parameter being t"log M /Q. This corresponds to the opposite sign in RG equations %32 (4.3) and (4.4).

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Fig. 4. The infrared quasi-"xed points for  "> /a (a) and  "A /M (b). The x-axis shows the values of  . R R   R  

6.1. Low tan
regime Consider "rst the low tan
regime. In this case, the only important Yukawa coupling is the top-quark one, all the others can be put equal to zero and the RG equations can be solved analytically [34] a > E (t)   R a (t)" , > (t)" , (6.1) G R 1#a b t 1#6> F (t)  G  R where 13 16 E (t)" (1#b a t)ARG @G , c " , 3, , R G  RG 3 15 G R F (t)" E (t) dt . R R  In the IR regime solution (6.1) possesses a quasi-"xed point. Indeed taking the limit > "  > (0)PR one can drop 1 in the denominator of Eq. (6.1) and obtain the IRQFP R E (t) , (6.2) >(t)N>$." R R 6F (t) R which is independent of the initial condition [33,35]. Though perturbation theory is not valid for > '1, it does not prevent us from using the "xed R point (6.2) since it attracts any solution with > 'a or, numerically, for > '0.1/4
. Thus, for    a wide range of initial values > is driven to the IR quasi-"xed point given by Eq. (6.2) which R








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335

numerically corresponds to y (M )+1.125. It is useful to introduce the ratio  ,> /a since the R 8 R R  strong coupling is the leading one in the IR regime. At the "xed point  (M )+0.84 and is R 8 approached in the IR regime when Q decreases. The behaviour of (t) is shown in Fig. 4 [36]. To get the solutions for the soft terms it is enough to perform the substitution aPa and >P>I and expand over  and . Expanding the gauge coupling in (6.1) up to  one has (hereafter we assume M "m ) G  m  . (6.3) M (t)" G 1#b a t G  Performing the same expansion for the Yukawa coupling one "nds [30]






t dE 6> A R!   !m (tE !F ) . (6.4) A (t)"  E dt R R R 1#6> F 1#6> F R  R  R To get the solution for the  term one has to make expansion over  and . This leads to








d t dE (A #m 6> (tE !F )) 6> dE  !A R !   R R #m   #  t R ,  (t)"   dt E dt R (1#6> F ) 1#6> F dt 1#6> F R  R  R  R (6.5) With analytic solutions (6.4), (6.5) one can analyse asymptotics and, in particular, "nd the infrared quasi-"xed points which correspond to > PR  t dE tE !F R! R R , A$."!m (6.6) R  E dt F R R tE !F  d t dE t dE R R # R ! R . $."m (6.7) R  F dt E dt F dt R R R The FP solutions (6.6), (6.7) can be directly obtained from a "xed point for the rigid Yukawa coupling (6.2) by grassmannian expansion. This explains, in particular, why "xed point solutions for the soft couplings exist if they exist for the rigid ones and with the same stability properties [37]. The behaviour of  "A /M as a function of a for a "xed ratio > /a "5 is shown in Fig. 4b.  R     One can observe the strong attraction to the IR stable quasi-"xed point  +!0.62 [36].  One can also write down solutions for the individual masses. This can be obtained using the grassmannian expansion of solutions for the corresponding super"eld propagators. For the "rst two generations one has























m 16 1 f #3f # f , m  * "m #    15  / 3  2

 

m 16 16 m  0 "m #  f # f , 3  3  15  2 m 16 4 f # f , m  0 "m #   " 3  15  2



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m 3 m  "m #  3f # f , &   2 5  m 3 m * "m #  3f # f ,   15  * 2 m 12 f m 0 "m #   # 5  2

,

where






1 1 f " 1! . G b (1#b a t) G G  The third generation masses get the contribution from the top Yukawa coupling [38]: m 0 "m  0 , @ " m * "m  * #/6 , / @ m 0 "m  0 #/3 , R 3 m * "m  * #/6 , / R m  "m  #/2 , & & where  is related to  (6.5) by R d t dE R . " ! !m R   dt E dt R There is no obvious infrared attractive "xed point for m  . However, one can take the linear & combination m "m  #2m  which together with m  shows the IR "xed point behaviour in & & > & the limit > PR.  m$. 1 m m$. & +!0.12  #3.4 . > +!0.73, (6.8) 2 m M M    In Eq. (6.8) one has only weak dependence on the ratio m /m . One can "nd the IR quasi-"xed   point m  /M +!0.40 which corresponds to m /m "0. As for the combination m , the    > & dependence on initial conditions disappears completely, as it follows from (6.8). The situation is illustrated in Fig. 5 [36]. Consider now the squark masses. In the limit > PR these solutions are driven to the IRQFPs R 1 m m $. m $.  #5.8 . / +0.12 3 +0.48, (6.9) 2 m M M    As it follows from Eq. (6.9), the solution for m  /M becomes independent of the initial conditions 3  m /m and A /m , when the top-quark Yukawa coupling is initially large enough. As a result,     the solutions of RGEs are driven to the "xed point (6.9) for a wide range of m /m (Fig. 6b). As for   m , the dependence on initial conditions does not completely disappear; however, it is rather weak / like in the case of m  and approaches the value m  /M +0.69 (Fig. 6a) [36]. /  &
















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337

Fig. 5. The infrared quasi-"xed points for m  /M (a) and (m  #2m  )/M (b). &  & & 

Fig. 6. The infrared quasi-"xed points for m  /M (a) and m  /M (b). /  3 

The bilinear SUSY breaking parameter B does not exhibit a "xed-point behaviour in the limit >





B B 1 A  !  !0.8 . +0.35 M m 2m   

(6.10)

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Table 2 Comparison of one- and two-loop IRQFPs

One-loop level Two-loop level

A /M R 

m  /M 3 

m /M > 

m  /M & 

m  /M / 

!0.62 !0.59

0.48 0.48

!0.73 !0.72

!0.40 !0.46

0.69 0.75

It is clear that neither B /m nor A /m may be neglected. As a consequence, no "xed point     behaviour for the ratio B/M is observed.  We have considered the one-loop RGEs. It is interesting to see, however, how our results are modi"ed when two-loop RGEs are used. For comparison we present the two-loop IRQFP values [39] together with our one-loop results in Table 2. As one can see from this table, the di!erence between the one- and two-loop results is negligible for A /M , m  /M and m /M . As for m  /M and m  /M , the two-loop corrections to the "xed  /  R  3  >  & points are about two times as small as deviations from them. As it was mentioned above, such corrections have a negligible impact on our main results. 6.2. Large tan
regime We now give a short description of the infrared behaviour of the RGEs in the MSSM for the large tan
regime. While with a single Yukawa coupling the analytical solution to the one-loop RG equations has been known for long, for increasing number of Yukawa couplings it has been obtained quite recently [40] in the form that allows iterative representation. One can write down the one-loop RG equations (4.3) as da G "b a , G G dt

(6.11)






d> I "!>  c a ! a > , I IG G IJ J dt G J where

(6.12)

b "33/5, 1,!3 , G c "13/15, 3, 16/3 , c "7/15, 3, 16/3 , c "9/5, 3, 0 , RG @G OG a "6, 1, 0 , a "1, 6, 1 , a "0, 3, 4 . RJ @J OJ Then the general solution to Eqs. (6.11) and (6.12) can be written in the form [40] a  a" , G 1#b a t G  >u I I , > " I 1#a >R u II I  I

(6.13) (6.14)

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where the functions u obey the integral system of equations I E R u" , R (1#6>R u ) @  @

E @ u " , @ (1#6>R u )(1#4>R u ) R  R O  O

E O u" O (1#6>R u ) @  @

(6.15)

and the functions E are given by I  E "  (1#b a t)AIG @G . I G  G

(6.16)

Let us stress that Eqs. (6.13) and (6.14) give the exact solution to Eqs. (6.11) and (6.12), while the u 's in Eqs. (6.15), although solved formally in terms of the E 's and >'s as continued integrated I I I fractions, should in practice be solved iteratively. Let us now perform the substitution (4.1), (4.2) in (6.13)}(6.15) and expand over  and . Then the linear term in  will give us the solution for M and A and the  terms the ones for  . The G I I resulting exact solutions look similar to those for the rigid couplings (6.13)}(6.15) [41] M G , M" G 1#b at G G

(6.17)

A/>#a u e II I I , A "!e # I I I I 1/>#a u I II I

(6.18)

(A)/>!/>#a u  I I I II I I ,  " #A#2e A ! I I I I I I 1/>#a u I II I

(6.19)

where the new functions e and  have been introduced which obey the iteration equations I I 1 dEI Au !u e R# @ @ @ @ , e" R E d 1/>#6u R @ @ Au !u e Au !u e 1 dEI R R# O O O O , @# R R e " @ E d 1/>#6u 1/>#4u R R O O @ 1 dEI Au !u e O #3 @ @ @ @ , e" O E d 1/>#6u O @ @




  

Au !u e  1 dEI 1 dEI Au !u e R #2 R @ @ @ @ #7 @ @ @ @ " R E d d d 1/> #6u 1/>#6u E @ @ @ @ R R








 


! (#(A)) u !2A u e # u  @ @ @ @ @ @ @ @

1 #6 u , @ > @

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Au !u e Au !u e R R R R# O O O O 1/>#6u 1/>#4u R R O O  Au !u e  O O #5 O O 1/>#4u O O Au !u e O O O O 1/>#4u O O 1 ! (#(A)) u !2A u e # u  #6 u R R R R R R R R R > R 1 ! (#(A)) u !2A u e # u  #4 u , O O O O O O O O O > O Au !u e  1 dEI Au !u e 1 dEI O #6 O @ @ @ @ # 27 @ @ @ @ " O E d d 1/>#6u E d 1/>#6u @ @ @ @ O O 1 #6 u . (6.20) ! 3 (#(A)) u !2A u e # u  @ @ @ @ @ @ @ @ @ > @ Here the variations of EI should be taken at ""0. When solving Eqs. (6.15) and (6.20) in the I nth iteration one has to substitute in the r.h.s. the (n!1)th iterative solution for all the corresponding functions. The solutions for the individual soft masses are linearly expressed through 's [38] 1 dEI 1 dEI @ #2 @  " @ E d d E d @ @ Au !u e R R #7 R R 1/>#6u R R Au !u e R R #2 R R 1/>#6u R R

m   "m #m   /



  
   
 
   
 

128f #87f !11f 17( !)#20( !)!5( !)   # R R @ @ O R , 122 122

m   "m #m 3  

42( !)!8( !)#2( !) 144f !108f #144/5f R R @ @ O R ,   # 122 122

m   "m #m   "

112f !84f #112/5f !8( !)#48( !)!12( !)   # R R @ @ O R , 122 122

m  "m #m &  

!9( !)#54( !)#17( !) !240f !3f !57/5f R R @ @ O R ,   # 122 122

m  "m #m   &

!272f #21f !89/5f 63( !)!12( !)#3( !)   # R R @ @ O R , 122 122

m  "m #m   *

80f #123f !103/5f 3( !)!18( !)#35( !)   # R R @ @ O R , 122 122

m  "m #m #  

160f !120f #32f 6( !)!36( !)#70( !)   # R R @ @ O R . 122 122

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Solutions (6.13)}(6.15), (6.17)}(6.20) have a nice property since they contain explicit dependence on initial conditions and one can trace this dependence in the "nal results. This is of special importance for the non-universal case since one can see which of the parameters is essential and which is washed out during the evolution. In particular, the solution for the Yukawa couplings exhibit the "xed point behaviour when the initial values are large enough. More precisely, in the regime >, >, >PR with "xed "nite ratios >/>"r , >/>"r , it is legitimate to drop 1 in R @ O R @  @ O  the denominators of Eqs. (6.14) and (6.15) in which case the exact Yukawa solutions go to the IRQFP de"ned by [41] u$. >$." I I a u$. II I

(6.21)

E E E @ O R , u$." , u$." u$." @ O R (u$.)(u$.) (u$.) (u$.) R O @ @

(6.22)

with

extending the IRQFP (6.2) to three Yukawa couplings. What is worth stressing here is that both the dependence on the initial condition for each Yukawa as well as the e!ect of Yukawa nonuni"cation, (r , r ), have completely dropped out of the runnings.   This in turn leads to the IRQFPs for the soft terms. Disappearance of > in the FP solution I naturally leads to the disappearance of A and  in the soft term "xed points. I I Below we present the result of numerical analysis. We begin with Yukawa couplings and assume the equality of the Yukawa couplings of the third generation at the GUT scale: > (M )" R %32 > (M )"> (M ). @ %32 O %32 In Figs. 4a}c the numerical solutions of the RGEs are shown for a wide range of initial values of  (M )" (M )" (M ) from the interval 0.2, 5, where  "> /a . One can clearly R %32 @ %32 O %32 G G  see the IRQFP-type behaviour when the parameter  at the GUT scale is big enough [42]. G We have found the following values of the Yukawa couplings y at the M scale: G 8 y 3 0.787, 1.048, y 3 0.758, 0.98, y 3 0.375, 0.619 . R @ O Comparing y and y one can see that the ratio belongs to a very narrow interval R @ y /y 3 1.039, 1.069. R @ Now, we proceed with the discussion of RGEs for trilinear scalar couplings, A , i"(t, b, ). The G results are shown in Figs. 4d and e for the following quantities  G "A /M , i"(t, b) for di!erent G   initial values at the GUT scale and for  (M )"5. One can see the strong attraction to the "xed G %32 points [42]. The question of stability of these IRQFPs becomes important for further consideration. Analysing their stability under the change of the initial conditions for  (M ) one "nds remarkable G %32 stability, which allows to use them as "xed parameters at the M scale. In Fig. 7f a particular 8 example of stability of IRQFP for A is shown. As a result one has the following IRQFP values for R the parameters  G :   R +!0.619,  @ +!0.658,  O +0.090 .   

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Fig. 7. The infrared quasi "xed points for  "> /a , i"t, b,  (a, b, c),  G "A /M , i"t, b (d, e) and  R with G G   G    R (a )"1 for di!erent initial values of  (a ) (f).   R 

The last step in the investigation of the RGEs is the calculation of the soft mass parameters. As one can see from Figs. 8a and b there exist IRQFPs [42] m  /M +!0.306, m  /M +!0.339 . &  &  The numbers correspond to the initial condition m /m "0. Later the initial values for the ratio   m /m belonging to the following interval m /m 3 0, 2 are considered.     In Figs. 8c}e the infrared behaviour of the soft SUSY breaking squark masses is shown. One can immediately see that all masses have IRQFPs which are used in the next section to "nd the mass spectrum. For further analysis only the squark masses are important. As for sleptons they also have an attractive infrared behaviour but it does not in#uence the mass spectrum of the Higgs bosons in which we are interested in below and we do not show them explicitly.

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Fig. 8. The infrared quasi-"xed points for mass parameters.

Numerical values of the ratios are the following [42]: m /M +0.58, m /M +0.52, m /M +0.53 , /  3  "  obtained for m /m "0. One again "nds a very weak dependence on initial values of the Yukawa   couplings. The behaviour of the bilinear SUSY breaking parameter B is the same as in low tan
case. The ratio B/M does not exhibit the infrared quasi-"xed point behaviour.  Thus, one can see that solutions of RGEs for all MSSM SUSY breaking parameters (the only exception is the parameter B) are driven to the infrared attractive "xed points if the Yukawa couplings at the GUT scale are large enough. Our analysis is constrained by the one-loop RG equations. The di!erence between one- and two-loop IRQFPs is similar to the low tan
case (see Table 1) and is less than 10%. At the same time the deviations from the IRQFPs obtained by one-loop RGEs are also of the same order which de"nes the accuracy of our predictions. The only place where it really matters is the prediction of the lightest Higgs boson mass where all the proper corrections are taken into account.

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7. Higgs boson mass prediction in the SM and MSSM 7.1. The Higgs boson mass in the SM The last unobserved particle from the Standard Model is the Higgs boson. Its discovery would allow one to complete the SM paradigm and con"rm the mechanism of spontaneous symmetry breaking. On the contrary, the absence of the Higgs boson would awake doubts about the whole picture and would require new concepts. Experimental limits on the Higgs boson mass come from a direct search at LEP II and Tevatron and from indirect "ts of electroweak precision data, "rst of all from the radiative corrections to the W and top quark masses. A combined "t of modern experimental data gives [43] m "78> GeV , (7.1) F \ which at the 95% con"dence level leads to the upper bound of 260 GeV (see Fig. 9). At the same time, recent direct searches at LEP II for the c.m. energy of 189 GeV give the lower limit of almost 95 GeV [43]. From theoretical point of view low Higgs mass could be a hint for physics beyond the SM, in particular for the supersymmetric extension of the SM. Within the Standard Model the value of the Higgs mass m is not predicted. However, one can F get the bounds on the Higgs mass [31,32]. They follow from the behaviour of the quartic coupling which obeys the RG equation (1.14). Since the quartic coupling grows with rising energy inde"nitely, an upper bound on m follows F from the requirement that the theory be valid up to the scale M or up to a given cut-o! scale .   below M [31]. The scale  could be identi"ed with the scale at which a Landau pole .  develops. The upper bound on m depends mildly on the top-quark mass through the impact of the F top-quark Yukawa coupling on the running of the quartic coupling . On the other hand, the requirement of vacuum stability in the SM (positivity of ) imposes a lower bound on the Higgs boson mass, which crucially depends on the top-quark mass as well as on the cut-o!  [31,32]. Again, the dependence of this lower bound on m is due to the e!ect of the R top-quark Yukawa coupling on the quartic coupling in Eq. (1.14), which drives to negative values at large scales, thus destabilizing the standard electroweak vacuum. From the point of view of LEP and Tevatron physics, the upper bound on the SM Higgs boson mass does not pose any relevant restriction. The lower bound on m , instead, is particularly F important in view of search for the Higgs boson at LEP II and Tevatron. For m &174 GeV and R  (M )"0.118 the results at "10 GeV or at "1 TeV can be given by the approximate Q 8 formulae [32]:

 

m '72#0.9[m !174]!1.0 F R

 

 (M )!0.118 Q 8 , "10 GeV , 0.006

m '135#2.1[m !174]!4.5 F R

 (M )!0.118 Q 8 , "1 TeV , 0.006

where the masses are in units of GeV.  The last run of LEP II at 200 GeV c.m. energy has increased this bound up to 103 GeV [44].

(7.2) (7.3)

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345

Fig. 9. The  distribution as function of the Higgs mass from the SM "t to the electroweak precision observables and the top mass. The shaded area is excluded by the direct searches. Fig. 10. Strong interaction and stability bounds on the SM Higgs boson mass.  denotes the energy scale up to which the SM is valid.

Fig. 10 [45] shows the perturbativity and stability bounds on the Higgs boson mass of the SM for di!erent values of the cut-o!  at which new physics is expected. We see from Fig. 10 and Eqs. (7.2), (7.3) that indeed for m &174 GeV the discovery of a Higgs particle at LEP II would R imply that the Standard Model breaks down at a scale  well below M or M , smaller for %32 .  lighter Higgs. Actually, if the SM is valid up to &M or M , for m &174 GeV only %32 .  R a small range of values is allowed: 134(m (&200 GeV. For m "174 GeV and m (100 GeV F R F (i.e. in the LEP II range) new physics should appear below the scale &a few to 100 TeV. The dependence on the top-quark mass however is noticeable. A lower value, m K170 GeV, would R relax the previous requirement to &10 TeV, while a heavier value m K180 GeV would R demand new physics at an energy scale as low as 10 TeV. 7.2. The Higgs boson mass in the MSSM It has been already mentioned that in the MSSM the mass of the lightest Higgs boson is predicted to be less than the Z-boson mass. This is, however, the tree level result and the masses acquire the radiative corrections. With account of the radiative corrections the e!ective Higgs bosons potential is < "< #< , &  

(7.4)

where < is given by Eq. (3.17) and in one-loop order 






m 3 1 (!1)(I (2J #1)c m log I ! . < " I I I   Q 2 64
 I

(7.5)

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Here the sum is taken over all the particles in the loop, J is the spin and m is the "eldI I dependent mass of a particle at the scale Q. These radiative corrections vanish when supersymmetry is not broken and are positive in softly broken case. The leading contribution comes from (s)top loops














m  3 m  3 m 3 3 m  log R ! #m  log R ! !2m log R ! < " R R   32
 R Q Q 2 2 Q 2



.

(7.6)

Contributions from the other particles are much smaller [46}48]. These corrections lead to the following modi"cation of the tree level relation for the lightest Higgs mass m  m  3gm R log R R . (7.7) m+M cos 2
# F 8 m 16
M R 5 One "nds that the one-loop correction is positive and increases the mass value. Two-loop corrections have the opposite e!ect but are smaller and result in slightly lower value of the Higgs mass. To "nd out numerical values of these corrections one has to determine the masses of all superpartners. This means that one has to know the initial conditions for the soft parameters. Fortunately, due to the IRQFP solutions the dependence on initial conditions may disappear at low energies. This allows one to reduce the number of unknown parameters and make predictions. Due to extreme importance of the Higgs mass predictions for experimental searches this problem has been the subject of intense investigation. There is a considerable amount of papers devoted to this topic (see e.g. [47}49,52,57,60]). And though the initial assumptions and the strategy are di!erent, the general conclusions are very similar. In what follows, we accept the strategy advocated in Refs. [36,42]: As input parameters one takes the known values of the top-quark, bottom-quark and -lepton masses, the experimental values of the gauge couplings, and the mass of the Z-boson [1]. To reduce the arbitrariness of the soft terms we use the "xed-point values for the Yukawa couplings and SUSY breaking parameters. The value of tan
is determined from the relations between the running quark masses and the Higgs v.e.v.'s in the MSSM m "y v sin
, (7.8) R R m "y v cos
, (7.9) @ @ m "y v cos
. (7.10) O O The Higgs mixing parameter  is de"ned from the minimization conditions for the Higgs potential and requirement of radiative EWSB. As an output we determine the mass spectrum of superpartners and of the Higgs bosons as functions of the only free parameter, namely m , which is directly related to the gluino mass M .   Varying this parameter within the experimentally allowed range, one gets all the masses as functions of this parameter (see Table 3). For low tan
the value of sin
is determined from Eq. (7.8), while for high tan
it is more convenient to use the relation tan
"(m /m )y /y , since the ratio y /y is almost a constant in the R @ @ R R @ range of possible values of y and y . R @

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Table 3 The values of input parameters and the output of "xed point analysis Input

Output

m (pole)"173.8$5.2 GeV R m (pole)"4.94$0.15 GeV @ m (pole)"1.7771$0.0005 GeV O  (M )"0.118$0.005  8  (M )"0.034  8  (M )"0.017  8 M "91.187$0.007 GeV 8

> ,> ,> R @ O A ,A ,A R @ O tan
,  m  , m  , m  / 3 " m  , m  * # m  , m  & & m , m , m , m! F  & &

N

For the evaluation of tan
one "rst needs to determine the running top and bottom-quark masses. One can "nd them using the well-known relations to the pole masses (see e.g. [50}52]), including both QCD and SUSY corrections. For the top-quark one has m  R . (7.11) m (m )" R R 1#(m /m ) #(m /m ) R R /!" R R 1317 Then, the following procedure is used to evaluate the running top mass. First, only the QCD correction is taken into account and m (m ) is found in the "rst approximation. This allows one to R R determine both the stop masses and the stop mixing angle. Next, having at hand the stop and gluino masses, one takes into account the stop/gluino corrections. For the bottom quark the situation is more complicated because the mass of the bottom quark m is essentially smaller than the scale M and so one has to take into account the running of this @ 8 mass from the scale m to the scale M . The procedure is the following [51,53,54]: one starts with @ 8 the bottom-quark pole mass, m "4.94$0.15 [55] and "nds the SM bottom-quark mass at the @ scale m using the two-loop QCD corrections @ m  @ . (7.12) m (m )1+" @ @ 1#(m /m ) @ @ /!" Then, evolving this mass to the scale M and using a numerical solution of the two-loop SM RGEs 8 [51,54] with  (M )"0.12 one obtains m (M ) "2.91 GeV. Using this value one can calculate  8 @ 8 1+ the sbottom masses and then return back to take into account the SUSY corrections from massive SUSY particles m (M )1+ @ 8 m (M )" . (7.13) @ 8 1#(m /m ) @ @ 1317 When calculating the stop and sbottom masses one needs to know the Higgs mixing parameter . For determination of this parameter one uses the relation between the Z-boson mass and the low-energy values of m  and m  which comes from the minimization of the Higgs potential: & & M m  # !(m  # ) tan
 &  8 #" & , (7.14) tan
!1 2

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where  and  are the radiative corrections [47]. Large contributions to these functions come   from stops and sbottoms. This equation allows one to obtain the absolute value of , the sign of  remains a free parameter. Whence the quark running masses and the  parameter are found, one can determine the corresponding values of tan
with the help of Eqs. (7.8) and (7.9). This gives in low and high tan
cases, respectively [36,42,56]: tan
"1.47$0.15$0.05 for '0 , tan
"1.56$0.15$0.05 for (0 , tan
"69.3$0.6$0.3

for '0 ,

tan
"38.1$0.9$0.4

for (0 .

The deviations from the central value are connected with the experimental uncertainties of the top-quark mass,  (M ) and uncertainty due to the "xed point values of y (M ) and y (M ).  8 R 8 @ 8 Having all the relevant parameters at hand it is possible to estimate the masses of the Higgs bosons. With the "xed point behaviour the only dependence left is on m or the gluino mass M .   It is restricted only experimentally: M '144 GeV [43] for arbitrary values of the squarks masses.  Let us start with low tan
case. The masses of CP-odd, charged and CP-even heavy Higgses increase almost linearly with M . The main restriction comes from the experimental limit on the  lightest Higgs boson mass. It excludes (0 case and for '0 requires the heavy gluino mass M 5750 GeV. Subsequently, one obtains [36]  m '844 GeV, m ! '846 GeV, m '848 GeV, for '0 ,  & & i.e. these particles are too heavy to be detected in the nearest experiments. For high tan
already the requirement of positivity of m excludes the region with small M . In   the most promising region M '1 TeV (m '300 GeV) for the both cases '0 and (0 the   masses of CP-odd, charged and CP-even heavy Higgses are also too heavy to be detected in the near future [42] m '1100 GeV for '0, m '570 GeV for (0 ,   m ! '1105 GeV for '0, m ! '575 GeV for (0 , & & m '1100 GeV for '0, m '570 GeV for (0 . & & The situation is di!erent for the lightest Higgs boson h, which is much lighter. Radiative corrections in this case are crucial and increase the value of the Higgs mass substantially [48,47]. They have been calculated up to the second loop order [58,60]. As can be seen from Eq. (7.7) the one-loop correction is positive increasing the tree level value almost up to 100% and the second one is negative thus decreasing it a little bit. We use in our analysis the leading two-loop contributions evaluated in Ref. [48]. As has been already mentioned these corrections depend on the masses of squarks and the other superpartners for which we substitute the values obtained above from the IRQFPs.

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349

Fig. 11. (A) The dependence of the mass of the lightest Higgs boson h on M "(m  m  ) (shaded area) for '0, low 1317 R R tan
. The dashed line corresponds to the minimum value of m "90 GeV allowed by experiment. (B), (C) The mass of the F lightest Higgs boson h as function of M for di!erent signs of , large tan
. The curves (a,b) correspond to the upper 1317 limit of the Yukawa couplings and to m /m "0 (a) or to m /m "2 (b). The curves (c, d) correspond to the lower     limit of the Yukawa couplings and to m /m "0 (c) or to m /m "2 (d). Possible values of the mass of the lightest     Higgs boson are inside the areas marked by these lines.

The results depend on the sign of parameter . It is not "xed since the requirement of EWSB de"nes only the value of . However, for low tan
negative values of  lead to a very small Higgs mass which is already excluded by modern experimental data, so further on we consider only the positive values of . Fot high tan
both signs of  are allowed. Consider "rst the low tan
regime. At the upper part of Fig. 11 it is shown the value of m for F '0 as a function of the geometrical mean of stop masses } this parameter is often identi"ed with

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a supersymmetry breaking scale M . One can see that the value of m quickly saturates close to 1317 F &100 GeV. For M of the order of 1 TeV the value of the lightest Higgs mass is [36] 1317 m "(94.3#1.6#0.6$5$0.4) GeV for M "1 TeV , (7.15) F 1317 where the "rst uncertainty comes from the deviations from the IRQFPs for the mass parameters, the second one is related to that of the top-quark Yukawa coupling, the third re#ects the uncertainty of the top-quark mass of 5 GeV, and the last one comes from that of the strong coupling. One can see that the main source of uncertainty is the experimental error in the top-quark mass. As for the uncertainties connected with the "xed points, they give much smaller errors of the order of 1 GeV. The obtained result (7.15) is very close to the upper boundary, m "97 GeV, obtained in F Refs. [52,57]. Note, however, that the uncertainties mentioned above as well as the upper bound [52,57] are valid for the universal boundary conditions. Loosing these requirement leads to increase of the upper bound values of the Higgs mass in case of low tan
up to &102 GeV [59]. For the high tan
case the lightest Higgs is slightly heavier, but the di!erence is crucial for LEP II. The mass of the lightest Higgs boson as a function of M is shown in the lower part of Fig. 11. 1317 One has the following values of m at a typical scale M "1 TeV (M +1.3 TeV) [42]: F 1317  m "128.2!0.4!7.1$5 GeV for '0 , F m "120.6!0.1!3.8$5 GeV for (0 . F The "rst uncertainty is connected with the deviations from the IRQFPs for mass parameters, the second one with the Yukawa coupling IRQFPs, and the third one is due to the experimental uncertainty in the top-quark mass. One can immediately see that the deviations from the IRQFPs for mass parameters are negligible and only in#uence the steep fall of the function on the left, which is related to the restriction on the CP-odd Higgs boson mass m . In contrast with the low tan
 case, where the dependence on the deviations from Yukawa "xed points was about 1 GeV, in the present case it is much stronger. The experimental uncertainty in the strong coupling constant  is Q not included because it is negligible compared to those of the top-quark mass and the Yukawa couplings and is not essential here contrary to the low tan
case. One can see that for large tan
the masses of the lightest Higgs boson are typically around 120 GeV that is too heavy for observation at LEP II. Note, however, that the uncertainties increase if one considers the non-universal boundary conditions which decreases the lower boundary for the Higgs mass. At the same time recent experimental data has practically excluded the low tan
scenario and in the next year the situation will be completely clari"ed. Thus, one can see that in the IRQFP approach all the Higgs bosons except for the lightest one are found to be too heavy to be accessible in the nearest experiments. This conclusion essentially coincides with the results of more sophisticated analyses. The lightest neutral Higgs boson, on the contrary is always light. The situation may improve a bit if one considers more complicated models with enlarged Higgs structure [61]. However, it does not change the generic feature of SUSY theories, the presence of light Higgs boson.  Lep II is now increasing its energy and may possibly reach the lower bound of high tan
scenario predictions.

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In the near future with the operation of the Tevatron and LHC proton}antiproton accelerators with the c.m. energy of 2 and 15 TeV, respectively, the energy range from 100 to 1000 GeV will be scanned and supersymmetry will be either discovered or the minimal version will be abandoned in particle physics. The Higgs boson might be the "rst target in search of SUSY.

8. Conclusion We have attempted to show how following the pattern of RG #ow one can explore physics lying beyond the Standard Model of fundamental interactions. The methods are essentially based on renormalization group technique in the framework of quantum "eld theory. The running of parameters is the key ingredient in attempts to go beyond the range of energies accessible to modern accelerators, to "nd manifestation of new physics. In the absence of solid experimental facts the RG #ow happens to be the only source of de"nite predictions. The peculiarity of the moment is that these predictions are the subject of experimental tests today, and in the near future, so one may hope to check the ideas described above at the turn of the Millennium.

Acknowledgements The author is grateful to S. Codoban, A. Gladyshev, M. Jurc\ is\ in, V. Velizhanin, K. TerMartirossian, R. Nevzorov, W. de Boer and G. Moultaka for useful discussions and to S. Codoban for help in preparing the manuscript. Special thanks to the organizers of the conference D. O'Connor and C. Stephens. Financial support from RFBR grants C 99-02-16650 and C 9615-96030 is kindly acknowledged.

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