Предсказания масс в рамках МССМ на основе инфракрасных квазификсированных точек тема автореферата и диссертации по физике, 01.04.02 ВАК РФ

Диссертационный Совет К 720.001.01 Лаборатории теоретической физики им. Н. Н. Боголюбова ОИЯИ г. Дубна

Юрчишин Мариан кандидат физико-математических наук 01.04.02 "Теоретическая физика"

Joint Institute for Nuclear Research

Bogoliubov Laboratory of Theoretical Physics

Marian Jurcisin

Mass Predictions in the MSSM Based on the Infrared Quasi-Fixed Points

in Candidacy for the Degree of Doctor of Philosophy

Scientific supervisor:

doctor of physical and mathematical sciences

D.I.Kazakov

КНИГА ИМЕЕТ

Contents

Contents 1

Introduction 3

1 One-Loop Renormalization Group Equations and the Infrared Quasi-Fixed Points in the MSSM 17

1.1 Infrared Quasi-Fixed Points . .............. 17

1.2 Renormalization Group Equations............ 20

1.3 Exact solution of the RGEs in the case of small tan/3 scenario.......................... 23

1.4 Infrared Quasi-Fixed Points Analysis in the Small tan /3 Scenario.......................... 24

1.5 Analysis of the infrared behavior of the RGEs in the case of large tan (3 scenario................ 30

1.6 Iterative Solution of the RGEs

and the IRQFP Behavior................. 37

2 Universality Assumption and the Mass Prediction in the MSSM 45

2.1 The Mass Formulas in the MSSM............ 45

2.1.1 Gaugino-Higgsino Mass Terms.......... 46

2.1.2 Squark and Slepton Masses ........... 47

2.1.3 The Higgs potential and masses of the Higgs

bosons..............................................48

2.2 Masses of Stops, Higgs Bosons and Charginos in the Small tan /3 Case..........................................52

2.3 Mass Prediction in the large tan/3 Scenario..............61

3 Mass Prediction in the MSSM: Non-Universal Case 71

3.1 Small tan /3 regime........................................71

3.2 Large tan/5 Scenario......................................75

4 The Lightest Higgs Boson in the Large tan/3 Scenario Based on the Exact Determination of Top and Bottom Masses 79

Conclusion 83

Apendix A 85

Apendix В 87

Apendix С 92

Bibliography 94

Introduction

The modern theory of the formally unified electromagnetic and weak interactions was established during the 1960's [1, 2, 3]. Without a shadow of a doubt it can be said that it was not only the beginning of the modern consideration of electroweak aspects of the elementary particles but also high energy physics at all. In the end, the main concept of the aforementioned theory, namely the concept of the local gauge group transformations, has led to the formulation of quantum chro-modynamics (QCD), the theory of strong interactions [4, 5, 6]. Combining the two above mentioned theories, the Standard Model of particle physics has obtained its present final form. The Standard Model of particle physics includes description of the electromagnetic, weak and strong interactions based on the local SU(3)c x SU(2)lxU(l)y gauge group. The particle contends of the Standard Model includes leptons, quarks, gauge bosons and a Lorentz scalar SU(2)l doublet. The last one is necessary needed to give masses to the matter fields (leptons and quarks) and the weak gauge fields (W±, Z) through its vacuum expectation value, v, which is obtained by spontaneous symmetry breaking mechanism. The term "spontaneous broken symmetry" is used when the interaction potential of the model is symmetric in respect to the corresponding symmetry but the state with the lowest energy, the ground state or vacuum, is not. To do this, the so-called Higgs boson

doublet is added into the Lagrangian of the Standard Model in a special way, such that after minimization of the corresponding scalar potential the non-trivial equivalent minima appear and by choosing one of them as physical ground state, the breaking of the SJJ{2)l x U(1 )y gauge symmetry to the U(1)em one takes place. Due to the local-ness of the gauge symmetry the so-called Higgs mechanism leads to the appearance of the massive physical Higgs boson h, and the rest of the four degrees of freedom of the primary scalar Higgs doublet are gauged away to give masses to the gauge bosons W±,Z. The matter fields (quarks and leptons) acquire masses through specially introduced Yukawa interaction part of the Lagrangian.

If we do not consider the problem which is related to the masses of neutrinos, the Standard Model is in very good agreement with precision electroweak tests (LEP, SLC, Tevatron, HERA). The only completely unknown part of the Standard Model is the Higgs sector [7] which is still rather mysterious and experimental confirmation of its existence is impatiently expected.

Although beyond doubt the Standard Model is at high level precision successful from the experimental point of view, nevertheless it is not free of theoretical drawbacks. First of all the very existence of the Higgs bosons, a spin zero particles, is problematic in the theories like the Standard Model. Despite of their exceptionally nice property, namely, their possibility to have nonvanishing vacuum expectation values without breaking Lorentz invariance, they have another property which cannot be consider as nice. The problem is related to the fact that their masses are subject to quadratic divergences in the framework of perturbation theory. More precisely, no matter how the Standard Model is successful experimentally, it has to be an effective theory of a more fundamental one at least at the grand unification

scale 1016 GeV) or at the Planck scale 1019 GeV). If we suppose that there is no new scale (new physics) between the electroweak scale 102 GeV) and the grand unification one than, due to the quadratic divergences, the natural mass of the Higgs boson will be the typical value of the fundamental scale (it is related to the fact that the value of the grand unification scale is directly connected to the masses of corresponding Higgs bosons of the Grand Unification Model under consideration). However, from the phenomenological point of view, the mass of the Standard Model Higgs boson is needed to be closed to the electroweek scale. Of course, to avoid this problem one can make the so-called fine tuning in the each order of perturbation theory but despite of this possibility theories where such adjustments of incredible accuracy have to be made are usually called " unnatural". The above problem of the existence of such two different scales in the theory is called the hierarchy problem [8].

Among further theoretical problems of the Standard Model belong the following: the model has a lot of parameters; only formal unification of the electromagnetic and weak interactions is presented (there are still three coupling constants); the origin of the mass spectrum of the particles is unknown; the nature of the Higgs boson is not understood; number of generations is not fixed in the model and last but not least possibility of the unification of gravity with the other three interactions in the framework of basic principles of the Standard Model is missing. The solution of the aforementioned problems doubtless lies beyond the Standard Model.

Possible way how to explain unanswered questions of the Standard Model related to the Higgs boson is to consider so-called technicolor or composite models where the Higgs sector of the Standard Model is replaced by a strongly interacting gauge system [8, 9]. The

electroweak symmetry breaking is obtained by the appearing of the fermion-antifermion bound states in the same way as in the ordinary quantum chromodynamics. However, such type of models has its own unsolvable problems and it seems that they are not on the right way to a more fundamental theory and we shall not discuss them deeper here.

Another possibility how to render the Standard Model natural are so-called supersymmetric extensions of the Standard Model and phe-nomenological analysis of the minimal of such extensions, called Minimal Supersymmetric Extension of the Standard Model (the MSSM), will be the main subject of the present thesis. But first we have to give some general ideas about this new symmetry of the world of elementary particles.

In the Standard Model the scale of the breakdown of the weak interaction is entirely given by the vacuum expectation value of scalar particles, therefore, one needs a symmetry that could imply vanishing of masses of scalar particles and so to protect them from quadratic divergences. Such symmetry exists and it is only one known in the framework of ordinary quantum field theories. This symmetry is commonly known as super symmetry.

Supersymmetry, as a new symmetry in physics, was introduced in the very beginning of 1970's as a pure theoretical construction to extend the Poincare group of the space-time transformations by introduction to the algebra of the generators of the Poincare group of new grassmannian generators [10]. In Ref .[11] was considered what we now call a non-linear realization of supersymmetry. The main problem of this model was the fact that it was not renormalisable. First super-symmetric renormalisable model was presented by Wess and Zumino [12]. Their model is now known as Wess-Zumino model and considers

a spin | particle in interaction with two spin 0 particles in the same multiplet. The limitations of the famous Coleman-Mandula [13] no-go theorem had been avoid by the introduction of a fermionic symmetry operator Q which carries spin Thus, their space-time transformation properties are those of the Weyl spinors. Such operators obey anticommutator relations with each other what was not considered in the Coleman-Mandula no-go theorem. Supersymmetric extension of gauge theories was immediately done too [14].

Because of fermionic character of the supersymmetry generators, they have to transform bosons into fermions and vice versa:

Q | boson) =| fermion) Q | fermion) =| boson).

Therefore, the supersymmetry is also known as symmetry between bosons and fermions.

Supersymmetry generators obey anticommutation relations of the following general form

{Я1Щ} = 26

where P^ are generators of space-time translations, a and /3 are spinor indices, i and j take their values from 1 to TV and N denotes the number of supersymmetry generators in the model. In what follows we shall consider only the simplest supersymmetry with N — 1. In this terminology the Standard Model is theoretical model of N — 0 type.

In Ref. [15] was shown that the supersymmetry algebras, are the only graded Lie algebras of symmetries of the S—matrix that are consistent with relativistic quantum field theory (possible extentions to include so-called central charges are understood). Besides the standard concepts of quantum field theory allow for supersymmetry without any further assumptions.

Now when we return to the problem of the Higgs mass one can immediately see that it is possible to solve it in an elegant way within the framework of supersymmetric quantum field models. In the supersym-metric theories the cancellation of quadratic divergences in all orders of perturbation theory takes place immediately. The reason for such "miraculous" cancellation is rather simply understood. A supersym-metry multiplet necessarily has to contain an equal number of bosonic and fermionic degrees of freedom and if supersymmetry in the model is rigid then their masses have to be equal each other. If, for example, a chiral symmetry forbids fermion masses then the corresponding boson partners are also massless in the rigid supersymmetric model. Now if one applies this idea directly in the Standard Model, one can immediately have the Higgs boson mass to be in the needed region of masses.

The main motivation for application of supersymmetry in particle physics comes from this fact that supersymmetry might render the Standard Model natural. The situation is rather similar to the one with a gauge symmetries, which prevent spin 1 particles to have nonzero masses [16]. In this case, nonzero masses arise as a consequence of a spontaneous breakdown of the corresponding gauge symmetry. The same idea can be used in the case with supersymmetric models. We know that our world is not supersymmetric at our energy levels which can be seen on the particle contents (or spectrum of masses). For example, charge scalar particles with mass and quantum numbers of electron are not found. Therefore, supersymmetry has to be broken in nature and a theoretical model has to contain a mechanism for the breaking of supersymmetry that splits the masses of the different members of the supermultiplets in the way which can be accepted by phenomenology and experiments. On the other hand,

we are interested in a supersymmetry breaking mechanism which in addition immediately induces the scale of the breakdown of the weak interactions. However, because we still want to have theory without quadratic divergences, this breaking should not be arbitrary. Thus, we are interested in so-called soft supersymmetry breaking mechanisms [17]. In this case, the cancellations between corresponding Feynman diagrams are not exact and originally degenerate spectrum of masses is splitting but these cancellations still take place in such a way that one has finite results. We shall return to the problem of supersymmetry breaking later and first we introduce the model which will play the central role in the present thesis.

As was already mentioned, the simplest supersymmetric extension of the Standard Model is so-called Minimal Supersymmetric Extension of the Standard Model (the MSSM) [18, 19, 20, 21]. It is the model of N = 1 type, therefore, to each particle one has to introduce one corresponding superparticle. The MSSM is minimal extension of the Standard Model in the supersymmetric direction as for particle contents. To describe it explicitly we have to introduce two types of supermultiplets (or superfields). Namely, chiral and vector supermul-tiplets. The chiral supermultiplet contents complex scalar field (p and Weyl spinor ifr

(<P,4>)-

It is also called matter supermultiplet because it is related to the matter fields of the Standard Model. The second (vector) supermultiplet (also called gauge supermultiplet due to its connection with gauge fields of the Standard Model) is given by gauge boson vector field V^ and its superpartner gaugino Л which is fermion:

(A

Now one has to introduce to each matter field in the SM correspond-

ing superpartner and built up supermultiplet. The same has to be done with gauge fields. In addition, one also needs to introduce at least two Higgs supermultiplets to give masses to both Up and Down quarks. It is the price we have to pay to make the Standard Model supersymmetric.

Using all above described facts one can write down the particle contents of the MSSM. First we need the following multiplets of matter:

L leptons LL = sleptons Ъ = QL (1,2,-1)

Ё Ir = ед Ir = ёд (1Дг2)

Q quarks Яь = G)l squarks QL = (l)£ (3,2,1/3)

и Ur = ur UR = UR (3,1,4/3)

b Dr = dR Dr = dR (3,1,-2/3)

#i higgsinos Ф\1Щ)Ь Higgses (1.2,-1)

Я2 (Ht) (1)2,1)

where supermuptiplets are given by capital letters with hat and super-partners of the Standard Model particles are defined by letters with tilde. In the last column are presented transformation properties under the gauge group SUC(3) x SUL{2) x UY{ 1). Needed gauge multiplets are the following:

G gluons Gl gluinos GJ (8,1,0)

V SU(2)-gauge bosons AJ, gauginos AJ, (1,3,0) V' [/(l)-gauge boson gauginos (1,1,0)

where the notation is as in the case of the matter supermultiplets. The Lagrangian of the MSSM is built up of three parts and can be

written as follows (see e.g.[21] for notation):

(i)

where L

'gauge

is defined as

L

gauge

SU(3),SU(2),U(1) *

£ 1 [тг WaWa + TrW*Wa] (2)

4

Matter

and Lyukawa part which is responsible for the generation of masses of the particles has the following form:

Lyukawa = h^QaUpH2 + h*pQaDpHi + h!apLalpHi + д#1#2. (3)

The supersymmetry breaking part, ЬвгеаЫпд> will be given later.

It has to be mentioned that it is possible to include into the Lyukawa part of the Lagrangian of the MSSM some other interaction terms which are consistent with the symmetry of the model (these terms are absent in the Standard Model). Namely

However, it is easy to see that such terms shall cause violation of the lepton number L (first two interactions) and the baryon number В (the last term). To solve the above problem one needs to introduce a new symmetry to forbid the problematic interactions. It is enough to involve so-called R—parity [22, 23]. The R—parity quantum number is defined as

where S denotes spin of the particle. Each Standard Model particle has R = 0 and for superpartners one has R = ±1. Although there is still some experimental room for models without i?-parity, in what follows, we suppose that R—parity is presented in the MSSM.

LLE, LQD, UDD.

(4)

R = (_!)3(i?-£)+25

(5)

As was already mentioned the supersymmetry has to be broken in the nature and we are interesting in soft supersymmetry breaking to protect its nice properties. There are several possible mechanisms for the breaking of supersymmetry. In general they can be split into two groups: explicit and spontaneous supersymmetry breaking mechanisms. Explicit breaking mechanisms are not interesting for us because they contain a lot of arbitrariness. As for spontaneous super-symmetry breaking mechanisms we shall work with so-called gravity mediated supersymmetry breaking or a supergravity induced mechanism of supersymmetry breaking. This mechanism is based on effective non-renormalizable interactions arising as low energy limit of supergravity theories. In this case supersymmetry is local (supergravity), so spontaneous breaking of supersymmetry is connected with the appearance of the Goldstone particle which is a Goldstone fermion. With the help of so-called super Higgs effect this particle is absorbed into the additional component of the spin | particle, gravitino, which becomes massive.

How does it work? In this case so-called hidden sector scenario is commonly assumed [24, 25, 26, 27, 28]. According to this scenario, there have to exist two separate sectors. First, the usual matter belongs to the "visible" sector, on the other hand, the second one, "hidden" sector, contains fields which lead to the necessary breaking of supersymmetry. The two sectors can interact with each other only via gravity. Supersymmetry is broken in the hidden sector by concrete mechanism at some scale Asusy which leads to the supersymetry breaking in the visible sector at the scale

д n

MsuSY

1V1 Planck

where the concrete value of n depends on the supersymmetry breaking mechanism in the hidden sector. We shall not discuss it more deeply

now but we can stress that in all mechanisms to have phenomeno-logically accepted Msusy of the order of 1 TeV one needs Asusy of the order of 1010 GeV. At such energy scales the gravity cannot be neglected and one has to consider local supersymmetry, also called supergravity and so we have to break supergravity. In these models the partner of the spin 2 graviton is a spin | particle: the gravitino. If supergravity is broken at a scale Л susy, the gravitino will obtain mass due to the super Higgs mechanism [29, 30] and the value of its mass will be closed to the Msusy■ Moreover, the breakdown of super-gravity can induce the breakdown of the weak interaction of the order of gravitino mass.

In the end, the effective low-energy theory at Msusy is a supersym-metric gauge theory with explicit soft supersymmetry breaking terms and a massive gravitino of the order of Msusy■ The Lbreaking in the MSSM has the following general form:

2 1 ~ ~

~~ L breaking = +(nml/2E^a

i * a

+ ^[h^bqaucbh2 + h°bqadcbh i + hLJaecbhx\ + B[y,hih2] + h.c.). (6)

Here ifi are all scalar fields, Aa are the gaugino fields, h\ and are the SU(2) doublet Higgs fields. In the (6) we have assumed so-called universality of the soft terms, namely, we put all spin 0 particles masses to be equal to the universal value mo, all the spin \ particles masses to be equal to m 1/2 and all trilinear couplings are equal to A.

The reason why these terms are called soft is related to the fact that corresponding operators have dimension less than four and thus do not influence the running of the couplings of the superpotential and do not spoil the cancellation of quadratic divergences in the model.

It is supposed that the soft terms are generated at the Planck scale,

Mpianck5 and first, they have to be driven to the GUT scale by corresponding renormalization group equations. However, usually it is assumed that running between Planck scale and GUT scale can be neglected in the first approximation and can be taken into account through the idea of non-universality of the soft terms. Detail investigation of its phenomenological consequences will be discussed in the present thesis.

Supersymmetry has fascinated many physicists since its discovery but despite of its attractiveness we have no phenomenological evidence that it might be relevant for a description of nature. On the other hand, a few non-trivial results was achieved within the supersymmetry extensions of the Standard Model. First of all, it is the hierachy problem as it was discussed above. Another non-trivial conclusion is that the unification of coupling constants of the Standard Model (not possible in the framework of the Standard Model itself) can be reached within supersymmetry [31, 32, 33, 34]. Thus, grand unification is the natural process in the supersymmetric theories.

This thesis are devoted to the investigation of the mass spectrum of the MSSM based on the concept of the infrared quasi-fixed points. The main attention is paid to the Higgs sector, especially to the lightest Higgs boson, which is the most interesting from experimental point of view.

The structure of the thesis is the following:

In Chapter 1 we study behaviour of the Yukawa couplings and the soft supersymmetry breaking parameters when driven from the high-energy scale (GUT scale) to the low-energy scale (electroweak scale). Corresponding analytical solution of the one-loop renormalization group equations for the low tan (3 are analyzed from the point of view of possible infrared quasi-fixed point behaviour. In the case of

large tan /3 numerical and iterative procedures are used in the analysis of the renormalization group equations. The restriction of the parameter freedom due to infrared quasi-fixed points is investigated in details.

In Chapter 2 we use the infrared quasi-fixed points obtained in the Chapter 1 when universality of soft supersymmetry breaking parameters is supposed for determination of the masses of sparticles and the Higgs bosons in both cases: with small tan /3 scenario and large tan /3 one, respectively. Main interest is paid to the lightest CP-even neutral Higgs boson, which is the most interesting from experimental point of view. It is shown that under such assumptions the small tan/3 case is completely excluded experimentally by LEP II data. On the other hand, the large tan/3 is still opened and will be the main object of interest for Tevatron and LHC experiments.

In Chapter 3 we suppose the non-universality of the Yukawa couplings and the soft supersymmetry breaking parameters. We use the analysis which was made in the Chapter 1 for calculation of the masses of the lightest Higgs boson in both the cases: small and large tan/3 case. It is shown that, if one suppose more or less natural non-universality conditions, the small tan /3 scenario is completely excluded by modern experimental data too. The situation is different in the large tan/3 case where the large mass window for the lightest Higgs boson still exists.

In Chapter 4 we show that the infrared quasi-fixed points of the Yukawa couplings are not preferred in the large tan/3 scenario. It is shown that actual mass of the lightest Higgs boson is about 10 GeV smaller than one which was calculated by assumption of infrared quasi-fixed point attraction. Anyway, the experimental data allow large tan /3 scenario in the MSSM.

In Conclusion we list the main results obtained in this thesis.

In Appendices we present some useful formulas.

Results of the thesis were presented on the seminars of Bogoliubov Laboratory of Theoretical Physics, JINR, on the seminars of Institute of Experimental Physics, Kosice, Slovakia, on the conference "Small Triangle Meeting on Theoretical Physics" (Kosice, 1998), on the "European School of High Energy Physics" (Casta Papiernicka, 1999), on the conference "Hadron Structure 2000" (Stara Lesna, 2000), on the "Conference of Young Scientists of BLTP" (Dubna, 2000).

Chapter 1

One-Loop Renormalization Group Equations and the Infrared Quasi-Fixed Points in the MSSM

In this Chapter, we give description of the concept of infrared quasi-fixed points (IRQFPs) in the framework of the MSSM. First, we introduce the corresponding system of the one-loop renormalization group equations (RGEs) and we discuss their analytical (small tan/3 case) and numerical (large tan /3 case) solutions. The concept of IRQFP is introduced and the analysis of IRQFP behaviour of RGEs is presented in details. Their influence on the restriction of the parameter space of the MSSM is analyzed. Results of this investigation will be used in calculation of the mass spectrum in the subsequent chapters.

1.1 Infrared Quasi-Fixed Points

Although the MSSM is the simplest supersymmetric extension of the Standard Model, it contains a large number of free parameters. The parameter freedom of the MSSM comes mostly from soft supersymmetry breaking part of Lagrangian, which is needed to obtain a phe-nomenologically acceptable mass spectrum of particles. At the same

time, a large number of free parameters decrease the predictive power of a theory. A common way to reduce this freedom is to make some assumptions at a high-energy scale (GUT or Planck scale). Then, treating the MSSM parameters as running variables and using the RGEs, one can derive their values at a low-energy scale.

The usual assumption is the so-called universality of soft-breaking terms at high energy. As was discussed in Introduction, within a su-pergravity induced supersymmetry breaking mechanism universality seems to be very natural and leads at low energies to a softly broken supersymmetric theory which depends on five free parameters (for completeness we repeat their definition here): a common scalar mass mo, a common gaugino mass m\/2, a common trilinear scalar coupling A, a supersymmetric Higgs-mixing mass parameter /z, and a bilinear Higgs coupling B. These parameters are defined at the high-energy scale and are treated as initial conditions for the RGEs. Using minimization conditions of the Higgs potential the last two parameters can be eliminated in favor of the electroweak symmetry breaking scale, v2 = v\ + v\ = (174.1 GeV)2, and the ratio tan/3 = v2/vi, where vi and i)2 are the vevs of the Higgs fields. However, the sign of // is unknown and is a free parameter of the theory (details see in the subsequent chapters).

This five-dimensional parameter space can be further restricted. Namely, some low-energy MSSM parameters are insensitive to their initial values. This allows one to find their low-energy values without detailed knowledge of the physics at high energy. To do this one has to examine the infrared behavior of RGEs for these parameters and to find possible infrared fixed points.

Notice, however, that the true infrared fixed points, discussed, e.g., in an earlier paper by Pendleton and Ross [35] are reached only in the

asymptotic regime. For the "running time" given by logMqUT/M^ they are reached only by a very narrow range of solutions. This problem has been resolved by consideration of more complicated fixed solutions like invariant lines, surfaces, etc. [36, 37, 38]. Such solutions turned to be strongly attractive and for the above-mentioned "running time", the wide range of solutions of RGEs ended their evolution on these fixed manifolds.

Here we are interested in another possibility connected with the so-called infrared quasi-fixed points (IRQFPs) first found out by Hill [39, 40] and then widely studied by several authors [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. These fixed points differ from Pendleton-Ross ones at the intermediate scale and usually give upper (or lower) boundary for the relevant solutions.

It has to be mentioned that these fixed points are not always relevant since the parameters of a theory do not necessarily have their maximum or minimum allowable values. For example, in the Standard Model the Hill fixed point for the top-quark Yukawa coupling corresponds to the pole top mass m^ole = 230 GeV [38], which is excluded by modern experimental data [59, 60].

As was shown in Refs. [45, 46, 47, 49], the situation is quite different in the MSSM. Here the top-quark running mass is given by the simple equation

mt = htv sin /3. (1.1)

In this case, the Hill fixed point corresponds to physical values of the top mass when tan (5 is chosen appropriately.

It is remarkable that imposing the constraint of bottom-tau unification and radiative electroweak symmetry breaking leads to the value of the top Yukawa coupling close to its quasi-fixed point value [49, 61]. This serves as an additional argument in favor of the Hill-type quasi-

fixed points.

In the subsequent sections, we present detailed analysis of the possible IRQFP behaviour in the MSSM but first we present the full set of one-loop RGEs of the MSSM.

1.2 Renormalization Group Equations

We begin our investigation of the one-loop RGEs of the MSSM with equations for the Yukawa couplings. It is a self-consistent system of differential equations (together with well-known RGEs for gauge couplings).