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Joint Institute for Nuclear Research
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
Bogoliubov Laboratory of Theoretical Physics
Marian Jurcisin
cj
КНИГА И.МССТ
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 d
scenario................................................... 23
1.4 Infrared Quasi-Fixed Points Analysis in the Small tan d
Scenario................................................... 24
1.5 Analysis of the infrared behavior of the RGEs in the
case of large tan /3 scenario.............................. 30
1.0 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
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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 0 Case................................................ 52
2.3 Mass Prediction in the large tan0 Scenario................ 61
3 Mass Prediction in the MSSM: Non-Universal Case 71
3.1 Small tan 0 regime........................................ 71
3.2 Large tan # Scenario...................................... 75
4 The Lightest Higgs Boson in the Large tan# Scenario Based on the Exact Determination of Top and Bottom
Masses 79
Conclusion 83
Apendix A 85
Apendix B 87
Apendix C 92
Bibliography 94
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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 elect roweak 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)e x SU(2)l x U{\)y gauge group. The particle contends of the Standard Model includes leptons, quarks, gauge bosons and a Lorentz scalar SU(2)i 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
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doublet is added into the Lagratigian of the Standard Model in a special way, such that after minimization of the corresponding scalar potential the non-trivia! equivalent minima appear and by choosing one of them as physical ground state, the breaking of the SU(2)i x U{\)y gauge symmetry to the U(1)em one takes place. Due to the localness 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 Wh,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 (LEF, 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
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scale (w 1016 GeV) or at the Planck scale (« 1019 GcV). If we suppose that there is no new scale (new physics) between the elcctroweak 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 chrornodynamics. 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 phenomenological 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 supersymmetry.
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 supcrsyinmetry. The main problem of this model was the fact that it was not renormalisable. First super-symmctric renormalisable model was presented by Wess and Zumino (12]. Their model is now known as Wess-Zumino model and considers
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