IEKP�KA������hep�ph������

March ����

Grand Uni�ed Theories

and Supersymmetry in

Particle Physics and Cosmology

W� de Boer�

Inst� f�ur Experimentelle Kernphysik� Universit�at KarlsruhePostfach ����� D���� Karlsruhe � Germany

ABSTRACT

A review is given on the consistency checks of Grand Uni�ed Theories �GUT�� which unify theelectroweak and strong nuclear forces into a single theory� Such theories predict a new kind offorce� which could provide answers to several open questions in cosmology� The possible role ofsuch a �primeval� force will be discussed in the framework of the Big Bang Theory�Although such a force cannot be observed directly� there are several predictions of GUTs�which can be veri�ed at low energies� The Minimal Supersymmetric Standard Model �MSSM�distinguishes itself from other GUTs by a successful prediction of many unrelated phenomenawith a minimum number of parameters�Among them a� Uni�cation of the couplings constants� b� Uni�cation of the masses� c� Existenceof dark matter� d� Proton decay� e� Electroweak symmetry breaking at a scale far below theuni�cation scale�A �t of the free parameters in the MSSM to these low energy constraints predicts the masses ofthe as yet unobserved superpartners of the SM particles� constrains the unknown top mass toa range between � � and ��� GeV� and requires the second order QCD coupling constant to bebetween ����� and ������

�Published in �Progress in Particle and Nuclear Physics �� ����� �����

�Email� Wim�de�Boer�cern�chBased on lectures at the Herbstschule Maria Laach� Maria Laach ���� and the Heisenberg�Landau Summerschool� Dubna �����

�The possibility that the universe was generated from noth�

ing is very interesting and should be further studied� A most

perplexing question relating to the singularity is this� what

preceded the genesis of the universe� This question appearsto be absolutely methaphysical� but our experience with meta�

physics tells us that metaphysical questions are sometimes

given answers by physics��

A� Linde ���

i

Contents

� Introduction �

� The Standard Model� ���� Introduction� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���� The Standard Model � � � � � � � � � � � � � � � � � � � � � � � � � �

����� Choice of the Group Structure� � � � � � � � � � � � � � � � ���� Requirement of local gauge invariance� � � � � � � � � � � � � � � � ����� The Higgs mechanism� � � � � � � � � � � � � � � � � � � � � � � � � ��

����� Introduction� � � � � � � � � � � � � � � � � � � � � � � � � � ������� Gauge Boson Masses and the Top Quark Mass� � � � � � � ������� Summary on the Higgs mechanism� � � � � � � � � � � � � � ��

��� Running Coupling Constants � � � � � � � � � � � � � � � � � � � � ��

� Grand Uni�ed Theories� ����� Motivation � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Grand Uni�cation � � � � � � � � � � � � � � � � � � � � � � � � � � ���� SU�� predictions � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

����� Proton decay � � � � � � � � � � � � � � � � � � � � � � � � � ������� Baryon Asymmetry � � � � � � � � � � � � � � � � � � � � � ������� Charge Quantization � � � � � � � � � � � � � � � � � � � � � ������� Prediction of sin� �W � � � � � � � � � � � � � � � � � � � � ��

��� Spontaneous Symmetry Breaking in SU�� � � � � � � � � � � � � ����� Relations between Quark and Lepton Masses � � � � � � � � � � � ��

� Supersymmetry ����� Motivation � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� SUSY interactions � � � � � � � � � � � � � � � � � � � � � � � � � � ����� The SUSY Mass Spectrum � � � � � � � � � � � � � � � � � � � � � ����� Squarks and Sleptons � � � � � � � � � � � � � � � � � � � � � � � � ����� Charginos and Neutralinos � � � � � � � � � � � � � � � � � � � � � ���� Higgs Sector � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Electroweak Symmetry Breaking � � � � � � � � � � � � � � � � � � �

� The Big Bang Theory ���� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Predictions from General Relativity � � � � � � � � � � � � � � � � ����� Interpretation in terms of Newtonian Mechanics � � � � � � � � � � ��

ii

��� Time Evolution of the Universe � � � � � � � � � � � � � � � � � � � ����� Temperature Evolution of the Universe � � � � � � � � � � � � � � � ���� Flatness Problem � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Horizon Problem � � � � � � � � � � � � � � � � � � � � � � � � � � � ���� Magnetic Monopole Problem � � � � � � � � � � � � � � � � � � � � ���� The smoothness Problem � � � � � � � � � � � � � � � � � � � � � � � ����� In�ation � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Origin of Matter � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Dark Matter � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Summary � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Comparison of GUT�s with Experimental Data �� Uni�cation of the Couplings � � � � � � � � � � � � � � � � � � � � � ���� MZ Constraint from Electroweak Symmetry Breaking � � � � � � ���� Evolution of the Masses � � � � � � � � � � � � � � � � � � � � � � � ���� Proton Lifetime Constraints � � � � � � � � � � � � � � � � � � � � � ���� Top Mass Constraints � � � � � � � � � � � � � � � � � � � � � � � � ��� b�quark Mass Constraint � � � � � � � � � � � � � � � � � � � � � � � ���� Dark Matter Constraint � � � � � � � � � � � � � � � � � � � � � � � ���� Experimental lower Limits on SUSY Masses � � � � � � � � � � � � ���� Decay b� s� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����� Fit Strategy � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���� Results � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Summary�

Appendix A �A�� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��A�� Gauge Couplings � � � � � � � � � � � � � � � � � � � � � � � � � � � �A�� Yukawa Couplings � � � � � � � � � � � � � � � � � � � � � � � � � � ��

A���� RGE for Yukawa Couplings in Region I � � � � � � � � � � ��A���� RGE for Yukawa Couplings in Region II � � � � � � � � � � ��A���� RGE for Yukawa Couplings in Region III � � � � � � � � � ��A���� RGE for Yukawa Couplings in Region IV � � � � � � � � � ���

A�� Squark and Slepton Masses � � � � � � � � � � � � � � � � � � � � � ���A���� Solutions for the squark and slepton masses� � � � � � � � � ���

A�� Higgs Sector � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���A���� Higgs Scalar Potential � � � � � � � � � � � � � � � � � � � � ���A���� Solutions for the Mass Parameters in the Higgs Potential ���

A� Charginos and Neutralinos � � � � � � � � � � � � � � � � � � � � � ���A�� RGE for the Trilinear Couplings in the Soft Breaking Terms � � � ��

References� � �

iii

Chapter �

Introduction

The questions concerning the origin of our universe have long been thought of asmetaphysical and hence outside the realm of physics�

However tremendous advances in experimental techniques to study both thevery large scale structures of the universe with space telescopes as well as thetiniest building blocks of matter � the quarks and leptons � with large accelera�tors allow us �to put things together� so that the creation of our universe nowhas become an area of active research in physics�

The two corner stones in this �eld are�

� Cosmology i�e� the study of the large scale structure and the evolutionof the universe� Today the central questions are being explored in theframework of the Big Bang Theory �BBT �� � � � �� which provides asatisfactory explanation for the three basic observations about our universe�the Hubble expansion the ��� K microwave background radiation and thedensity of elements ���� hydrogen ��� helium and the rest for the heavyelements �

� Elementary Particle Physics i�e� the study of the building blocks ofmatter and the interactions between them� As far as we know the build�ing blocks of matter are pointlike particles the quarks and leptons whichcan be grouped according to certain symmetry principles� their interactionshave been codi�ed in the so�called Standard Model �SM ��� In this modelall forces are described by gauge eld theories��� which form a marveloussynthesis of Symmetry Principles and Quantum Field Theories� The lattercombine the classical �eld theories with the principles of Quantum Mechan�ics and Einstein�s Theory of Relativity�

The basic observations both in theMicrocosm as well as in theMacrocosm arewell described by both models� Nevertheless many questions remain unanswered�Among them�

� What is the origin of mass�

� What is the origin of matter�

� What is the origin of the Matter�Antimatter Asymmetry in our universe�

�

1031

1028

1025

1022

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1016

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10 -43 10 -33 10 -23 10 -13 10 -3 10 7 10 17

Time (s)

Tem

p. (

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TOE

GUT SU(3)xSU(2)xU(1)

ACCELERATORSElectroweak SU(3)xU(1)

Nucleo-synthesis

Atoms+3KGalaxies

10

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rgy

(GeV

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(3 ') (1 yr) ( yr)1010

Figure ���� The evolution of the universe and the energy scale of some typicalevents� above the Planck scale of ���� GeV gravity becomes so strong thatone cannot neglect gravity implying the need for a �Theory Of Everything� todescribe all forces� Below that energy the well known strong and electroweakforces are assumed to be equally strong implying the possibility of a Grand Uni��ed Theory �GUT with only a single coupling constant at the uni�cation scale�After spontaneous symmetry breaking the gauge bosons of this uni�ed force be�come heavy and �freeze out�� The remaining forces correspond to the well knownSU�� C � SU�� L � U�� Y symmetry at lower energies with their coupling con�stants changing from the uni�ed value at the GUT scale to the low energy values�this evolution is attributed to calculable radiative corrections� Future accelera�tors are expected to reach about �� TeV corresponding to a temperature of ����

K which was reached about ����� s after the �Bang�� At about ��� GeV thegauge bosons of the electroweak theory �freeze out� after getting mass throughspontaneous symmetry breaking and only the strong and electromagnetic forceplay a role� About three minutes later the temperature has dropped below thenuclear binding energy and the strong force binds the quarks into nuclei �nucle�osynthesis � Most of the particles annihilate with their antiparticles into a largenumber of photons after the photon energies become too low to create new par�ticles again� After about hundred thousand years the temperature is below theelectromagnetic binding energies of atoms so the few remaining electrons andprotons which did not annihilate form the neutral atoms� Then the universebecomes transparent for electromagnetic radiation and the many photons streamaway into the universe� The photons released at that time are now observed asthe � K microwave background radiation� Then the neutral atoms start to clusterslowly intostars and galaxies under the in�uence of gravity�

�

� Why is our universe so smooth and isotropic on a large scale�

� Why is the ratio of photons to baryons in the universe so extremely largeon the order of �����

� What is the origin of dark matter which seems to provide the majority ofmass in our universe�

� Why are the strong forces so strong and the electroweak forces so weak�

Grand Uni�ed Theories �GUT �� �� in which the known electromagnetic weakand strong nuclear forces are combined into a single theory hold the promise ofanswering at least partially the questions raised above� For example they explainthe di�erent strengths of the known forces by radiative corrections� At high en�ergies all forces are equally strong� The Spontaneous Symmetry Breaking �SSB of a single uni�ed force into the electroweak and strong forces occurs in such the�ories through scalar �elds which �lock� their phases over macroscopic distancesbelow the transition temperature� A classical analogy is the build�up of the mag�netization in a ferromagnet below the Curie�temperature� above the transitiontemperature the phases of the magnetic dipoles are randomly distributed andthe magnetization is zero but below the transition temperature the phases arelocked and the groundstate develops a nonzero magnetization� Translated in thejargon of particle physicists� the groundstate is called the vacuum and the scalar�elds develop a nonzero �vacuum expectation value�� Such a phase transitionmight have released an enormous amount of energy which would cause a rapidexpansion ��in�ation� of the universe thus explaining simultaneously the originof matter its isotropic distribution and the �atness of our universe�

Given the importance of the questions at stake GUT�s have been under in�tense investigation during the last years�

The two directly testable predictions of the simplest GUT namely

� the �nite lifetime of the proton

� and the uni�cation of the three coupling constants of the electroweak andstrong forces at high energies

turned out to be a disaster for GUT�s� The proton was found to be much morestable than predicted and from the precisely measured coupling constants at thenew electron�positron collider LEP at the European Laboratory for ElementaryParticle Physics �CERN� in Geneva one had to conclude that the couplings didnot unify if extrapolated to high energies��� �� ����

However it was shown later that by introducing a hitherto unobserved sym�metry called Supersymmetry �SUSY ��� ��� into the Standard Model bothproblems disappeared� uni�cation was obtained and the prediction of the protonlife time could be pushed above the present experimental lower limit�

The price to be paid for the introduction of SUSY is a doubling of the numberof elementary particles since it presupposes a symmetry between fermions andbosons i�e� each particle with even �odd spin has a partner with odd �even spin� These supersymmetric partners have not been observed in nature so the

�

only way to save Supersymmetry is to assume that the predicted particles aretoo heavy to be produced by present accelerators� However there are strongtheoretical grounds to believe that they can not be extremely heavy and in theminimal SUSY model the lightest so�called Higgs particle will be relatively lightwhich implies that it might even be detectable by upgrading the present LEPaccelerator� But SUSY particles if they exist should be observable in thenext generation of accelerators since mass estimates from the uni�cation of theprecisely measured coupling constants are in the TeV region���� and the lightestHiggs particle is expected to be of the order of MZ as will be discussed in thelast chapter�

It is the purpose of the present paper to discuss the experimental tests ofGUT�s� The following experimental constraints have been considered�

� Uni�cation of the gauge coupling constants�

� Uni�cation of the Yukawa couplings�

� Limits on proton decay�

� Electroweak breaking scale�

� Radiative b� s� decays�

� Relic abundance of dark matter�

It is surprising that one can �nd solutions within theminimal SUSY model whichcan describe all these independent results simultaneously� The constraints on thecouplings the unknown top�quark mass and the masses of the predicted SUSYparticles will be discussed in detail�

The paper has been organized as follows� In chapters � to � the StandardModel Grand Uni�ed Theories �GUT and Supersymmetry are introduced� Inchapter � the problems in cosmology will be discussed and why cosmology �cries�for Supersymmetry� Finally in chapter the consistency checks of GUT�s throughcomparison with data are performed and in chapter � the results are summarized�

�

Chapter �

The Standard Model�

��� Introduction�

The �eld of elementary particles has developed very rapidly during the last twodecades after the success of QED as a gauge �eld theory of the electromagneticforce could be extended to the weak� and strong forces� The success largelystarted with the November Revolution in ���� when the charmed quark wasdiscovered simultaneously at SLAC and Brookhaven for which B� Richter andS�S�C Ting were awarded the Nobel prize in ���� This discovery left little doubtthat the pointlike constituents inside the proton and other hadrons are realexisting quarks and not some mathematical objects to classify the hadrons asthey were originally proposed by Gellman and independently by Zweig��

The existence of the charmed quark paved the way for a symmetry betweenquarks and leptons since with charm one now had four quarks �u� d� c and s and four leptons �e� �� �e and �� which �tted nicely into the SU�� � U�� uni�ed theory of the electroweak interactions proposed by Glashow Salam andWeinberg �GSW �� for the leptonic sector and extended to include quarks aswell as leptons by Glashow Iliopoulis and Maiani �GIM ��� as early as �����Actually from the absence of �avour changing neutral currents they predictedthe charm quark with a mass around ��� GeV and indeed the charmed quarkwas found four years later with a mass of about ��� GeV� This discovery becameknown as the November Revolution mentioned above�

The uni�cation of the electromagnetic and weak interactions had alreadybeen forwarded by Schwinger and Glashow in the sixties� Weinberg and Salamsolved the problem of the heavy gauge boson masses required in order to explainthe short range of the weak interactions by introducing spontaneous symmetrybreaking via the Higgs�mechanism� This introduced gauge boson masses withoutexplicitly breaking the gauge symmetry�

The Glashow�Weinberg�Salam theory led to three important predictions�

� neutral currents i�e� weak interactions without changing the electric charge�In contrast to the charged currents the neutral currents could occur with

�Zweig called the constituents �aces and believed they really existed inside the hadrons�This belief was not shared by the referee of Physical Review� so his paper was rejected andcirculated only as a CERN preprint� albeit well�known �����

�

leptons from di�erent generations in the initial state e�g� ��e � ��ethrough the exchange of a new neutral gauge boson�

� the prediction of the heavy gauge boson masses around �� GeV�

� a scalar neutral particle the Higgs boson�

The �rst prediction was con�rmed in ���� by the observation of �� scatteringwithout muon in the �nal state in the Gargamelle bubble chamber at CERN�Furthermore the predicted parity violation for the neutral currents was observedin polarized electron�deuteron scattering and in optical e�ects in atoms� Thesesuccessful experimental veri�cations��� led to the award of the Nobel prize in ����to Glashow Salam and Weinberg� In ���� the second prediction was con�rmedby the discovery of the W and Z bosons at CERN in p p collisions for which C�Rubbia and S� van der Meer were awarded the Nobel prize in �����

The last prediction has not been con�rmed� the Higgs boson is still at largedespite intensive searches� It might just be too heavy to be produced with thepresent accelerators� No predictions for its mass exist within the Standard Model�In the supersymmetric extension of the SM the mass is predicted to be on theorder of ��� GeV which might be in reach after an upgrading of LEP to ���GeV� These predictions will be discussed in detail in the last chapter where acomparison with available data will be made�

In between the gauge theory of the strong interactions as proposed by Fritzschand Gell�Mann ���� had established itself �rmly after the discovery of its gauge�eld the gluon in ��jet production in e�e� annihilation at the DESY labora�tory in Hamburg� The colour charge of these gluons which causes the gluonself�interaction has been established �rmly at CERN�s Large Electron Positronstorage ring called LEP� This gluon self�interaction leads to asymptotic freedomas shown by Gross and Wilcek ���� and independently by Politzer ���� thus ex�plaining why the quarks can be observed as almost free pointlike particles insidehadrons and why they are not observed as free particles i�e� they are con�nedinside these hadrons� This simultaneously explained the success of the QuarkParton Model which assumes quasi�free partons inside the hadrons� In this casethe cross sections if expressed in dimensionless scaling variables are independentof energy� The observation of scaling in deep inelastic lepton�nucleon scatteringled to the award of the Nobel Prize to Freedman Kendall and Taylor in �����Even the observation of logarithmic scaling violations both in DIS and e�e�

annihilation as predicted by QCD were observed and could be used for precisedeterminations of the strong coupling constant of QCD��� ����

The discovery of the beauty quark at Fermilab in Batavia�USA in ��� andthe � �lepton at SLAC both in ��� led to the discovery of the third generation ofquarks and leptons of which the expected top quark is still missing� Recent LEPdata indicate that its mass is around � GeV���� thus explaining why it has notyet been discovered at the present accelerators� The third generation had beenintroduced into the Standard Model long before by Kobayashi and Maskawa inorder to be able to explain the observed CP violation in the kaon system withinthe Standard Model�

From the total decay width of the Z� bosons as measured at LEP one con�cludes that it couples to three di�erent neutrinos with a mass below MZ�� � ��GeV� This strongly suggests that the number of generations of elementary par�ticles is not in�nite but indeed three since the neutrinos are massless in theStandard Model� The three generations have been summarized in table ��� to�gether with the gauge �elds which are responsible for the low energy interactions�

The gluons are believed to be massless since there is no reason to assumethat the SU�� symmetry is broken so one does not need Higgs �elds associatedwith the low energy strong interactions� The apparent short range behaviour ofthe strong interactions is not due to the mass of the gauge bosons but to thegluon self�interaction leading to con�nement as will be discussed in more detailafterwards�

This chapter has been organized as follows� after a short description of the SMwe discuss it shortcomings and unanswered questions� They form the motivationfor extending the SM towards a Grand Uni�ed Theory in which the electroweak�and strong forces are uni�ed into a new force with only a single coupling constant�The Grand Uni�ed Theories will be discussed in the next chapter� Although suchuni�cation can only happen at extremely high energies � far above the range ofpresent accelerators � it still has strong implications on low energy physics whichcan be tested at present accelerators�

��� The Standard Model

Constructing a gauge theory requires the following steps to be taken�

� Choice of a symmetry group on the basis of the symmetry of the observedinteractions�

� Requirement of local gauge invariance under transformations of the sym�metry group�

� Choice of the Higgs sector to introduce spontaneous symmetry breakingwhich allows the generation of masses without breaking explicitly gaugeinvariance� Massive gauge bosons are needed to obtain the short�range be�haviour of the weak interactions� Adding ad�hoc mass terms which are notgauge�invariant leads to non�renormalizable �eld theories� In this case thein�nities of the theory cannot be absorbed in the parameters and �elds ofthe theory� With the Higgs mechanism the theory is indeed renormalizableas was shown by G� �t Hooft�����

� Renormalization of the couplings and masses in the theory in order to relatethe bare charges of the theory to known data� The Renormalization Groupanalysis leads to the concept of �running� i�e� energy dependent couplingconstants which allows the absorption of in�nities in the theory into thecoupling constants�

�

Interactionsstrong electro�weak gravitational uni�ed �

Theory QCD GSW quantum gravity � SUGRA �Symmetry SU�� SU�� � U�� � SU�� �Gauge g� � � � g� photon G XY �bosons gluons W� Z� bosons graviton GUT bosons�charge colour weak isospin mass �

weak hypercharge

Table ���� The fundamental forces� The question marks indicate areas of intensiveresearch�

����� Choice of the Group Structure�

Groups of particles observed in nature show very similar properties thus suggest�ing the existence of symmetries� For example the quarks come in three colourswhile the weak interactions suggest the grouping of fermions into doublets� Thisleads naturally to the SU�� and SU�� group structure for the strong andweak interactions respectively� The electromagnetic interactions don�t changethe quantum numbers of the interacting particles so the simple U�� group issu!cient�

Consequently the Standard Model of the strong and electroweak interactionsis based on the symmetry of the following unitary� groups�

SU�� C � SU�� L � U�� Y ����

The need for three colours arose in connection with the existence of hadronsconsisting of three quarks with identical quantum numbers� According to thePauli principle fermions are not allowed to be in the same state so labeling themwith di�erent colours solved the problem���� More direct experimental evidencefor colour came from the decay width of the � and the total hadronic crosssection in e�e� annihilation���� Both are proportional to the number of quarkspecies and both require the number of colours to be three�

Although colour was introduced �rst as an ad�hoc quantum number for thereasons given above it became later evident that its role was much more funda�mental namely that it acted as the source of the �eld for the strong interactions�the �colour� �eld just like the electric charge is the source of the electric �eld�

The �charge� of the weak interactions is the third component of the �weak�isospin T�� The charged weak interactions only operate on left�handed parti�cles i�e� particles with the spin aligned opposite to their momentum �negativehelicity so only left�handed particles are given weak isospin � ��� and right�handed particles are put into singlets �see table ��� � Right�handed neutrinosdo not exist in nature so within each generation one has �� matter �elds� ��� left�right �handed leptons and �x� ��x� left�right �handed quarks �factor � forcolour �

�Unitary transformations rotate vectors� but leave their length constant� SU�N symmetrygroups are Special Unitary groups with determinant ���

�

The electromagnetic interactions originate both from the exchange of theneutral gauge boson of the SU�� group as well as the one from the U�� group�Consequently the �charge� of the U�� group cannot be identical with the electriccharge but it is the so�called weak hypercharge �YW � which is related to theelectric charge via the Gell�Mann�Nishijima relation�

Q " T� #�

�YW ����

The quantum number YW is �B�L for left handed doublets and �Q for righthandedsinglets where the baryon number B"��� for quarks and � for leptons while thelepton number L "� for leptons and � for quarks� Since T� and Q are conservedYW is also a conserved quantum number� The electro�weak quantum numbersfor the elementary particle spectrum are summarized in table ����

Generations Quantum Numbershelicity �� �� �� Q T� YW

��ee

�L

����

�L

����

�L

���

�������

����

L �ud�

�L

�cs�

�L

�tb�

�L

�������

�������

������

eR �R �R �� � ��

RuRdR

cRsR

tRbR

�������

��

�������

Table ���� The electro�weak quantum numbers �electric charge Q third compo�nent of weak isospin T� and weak hypercharge YW of the particle spectrum� Theneutrinos �e �� and �� are the weak isospin partners of the electron�e muon�� and tau�� leptons respectively� The up�u down�d strange�s charm�c bottom�b and top�t quarks come in three colours which have not been indi�cated� The primes for the left handed quarks d� s� and b� indicate the interactioneigenstates of the electro�weak theory which are mixtures of the mass eigenstatesi�e� the real particles� The mixing matrix is the Cabibbo�Kobayshi�Maskawa ma�trix� The weak hypercharge YW equals B � L for the left�handed doublets and�Q for the right�handed singlets�

�

Figure ���� Global rotations leave the baryon colourless �a � Local rotationschange the colour locally thus changing the colour of the baryon �b unless thecolour is restored by the exchange of a gluon �c �

Figure ���� Demonstration of the non�abelian character of the SU�� rotationsinside a colourless baryon� on the left�hand side one �rst exchanges a red�greengluon which exchanges the colours of the quarks and then a green�blue gluon�on the right�hand side the order is reversed� The �nal result is not the same sothese operations do not commute�

��

��� Requirement of local gauge invariance�

The Lagrangian L of a free fermion can be written as�

L " i$����$�m$$� ����

where the �rst term represents the kinetic energy of the matter �eld $ withmass m and the second term is the energy corresponding to the mass m� TheEuler�Lagrange equations for this L yield the Dirac equation for a free fermion�

The unitary groups SU�N introduced above represent rotations inN dimensional�

space� The bases for the space are provided by the eigenstates of the matter �eldswhich are the colour triplets in case of SU�� weak isospin doublets in case ofSU�� and singlets for U�� �

Arbitrary rotations of the states can be represented by

U " exp��i� � �F " exp��iN���Xk�

k � Fk ����

where k are the rotation parameters and Fk the rotation matrices� Fk are theeight �x� Gell�Mann matrices for SU�� denoted by � hereafter and the wellknown Pauli matrices for SU�� denoted by � �

The Lagrangian is invariant under the SU�N rotation if L�$� "L�$ where$� " U$� The mass term is clearly invariant� m$

�$� " m$U yU$ " m$$

since U yU " � for unitary matrices� The kinetic term is only invariant underglobal transformations i�e� transformations where k is everywhere the same inspace�time� In this case U is independent of x and can be treated as a constantmultiplying $ which leads to� $U y����U$ " $U yU����$ " $����$�

However one could also require local instead of global gauge invariance im�plying that the interactions should be invariant under rotations of the symmetrygroup for each particle separately� The motivation is simply that the interactionsshould be the same for particles belonging to the same multiplet of a symmetrygroup� For example the interaction between a green and a blue quark shouldbe the same as the interaction between a green and a red quark� therefore itshould be allowed to perform a local colour transformation of a single quark�The consequence of requiring local gauge invariance is dramatic� it requires theintroduction of intermediate gauge bosons whose quantum numbers completelydetermine the possible interactions between the matter �elds as was �rst shownby Yang and Mills in ���� for the isopin symmetry of the strong interactions�

Intuitively this is quite clear� Consider a hadron consisting of a colour tripletof quarks in a colourless groundstate� A global rotation of all quark �elds willleave the groundstate invariant as shown schematically in �g� ���� However if aquark �eld is rotated locally the groundstate is not colourless anymore unless a�message� is mediated to the other quarks to change their colours as well� The�mediators� in SU�� are the gluons which carry a colour charge themselves andthe local colour variation of the quark �eld is restored by the gluons�

�The SU�N groups can be represented by N�N complex matrices A or N� real numbers�The unitarity requirement �Ay � A�� imposes N� conditions� while requiring the determinantto be one imposes one more constraint� so in total the matrix is represented by N� � � realnumbers�

��

The colour charge of the gluons is a consequence of the non�abelian characterof SU�� which implies that rotations in colour space do not commute i�e� �a�b "�b�a as demonstrated in �g� ���� If the gluons would all be colourless they wouldnot change the colour of the quarks and their exchange would be commuting�

Mathematically local gauge invariance is introduced by replacing the deriva�tive �� with the covariant derivative D� which is required to have the followingproperty�

D�$� " UD$� ����

i�e� the covariant derivative of the �eld has the same transformation propertiesas the �eld in contrast to the normal derivative� Clearly with this requirementL is manifestly gauge invariant since in each term of eq� ��� the transformationleads to the product U yU " � after substituting �� � D�

For in�nitesimal transformations the covariant derivative can be written as����

D� " �� #ig�

�B�YW #

ig

��W� � �� #

igs��G� � ��� ���

where B�� �W� and �G� are the �eld quanta ��mediators� of the U�� SU�� andSU�� groups and g�� g and gs the corresponding coupling constants�

The term �W� � �� can be explicitly written as�

W ���� #W �

��� #W ���� " W �

�

�� �� �

�#W �

�

�� �ii �

�#W �

�

�� �� ��

�"

�W �

� W �� � iW �

�

W �� # iW �

� �W ��

��

W ��

p�W�p

�W� �W ��

�����

The operatorsW� in the o��diagonal elements act as lowering� and raising opera�tors for the weak isospin� For example they transform an electron into a neutrinoand vice�versa while the operator W �

� represents the neutral current interactionsbetween a fermion and antifermion�

After substituting the Gell�Mann matrices � the term G� � � "P�

k�Gk�kcan be written similarly as�

�BBBBBBB�

G�� #

�p�G� G� � iG� G � iG�

G�� # iG� �G� # �p

�G� G� � iG�

G� # iG� G� # iG� � �p

�G�

�CCCCCCCA

�BBBBBBB�

G�� #

�p�G�

p�Grg

p�Grb

p�Ggr �G� # �p

�G�

p�Ggb

p�Gbr

p�Gbg � �p

�G�

�CCCCCCCA

���� This term induces transitions between the colours� For example the o��

diagonal element Grg acts like a raising operator between a green %g " ��� �� � and red %r " ��� �� � �eld� The terms on the diagonal don�t change the colour�Since the trace of the matrix has to be zero there are only two independent

�The term originates from Weyl� who tried to introduce local gauge invariance for gravity�thus introducing the derivative in curved space�time� which varies with the curvature� thusbeing covariant�

��

gluons which don�t change the colour� They are linear combinations of thediagonal matrices �� and ���

The W� gauge �elds cannot represent the mediators of the weak interactionssince the latter have to be massive� Mass terms for W� such as M�W�W

�are not gauge invariant as can be checked from the transformation laws for the�elds� The real �elds �� Z�� and W� can be obtained from the gauge �elds afterspontaneous symmetry breaking via the Higgs mechanism as will be discussedin the next section�

��� The Higgs mechanism�

����� Introduction�

The problem of mass for the fermions and weak gauge bosons can be solved byassuming that masses are generated dynamically through the interaction with ascalar �eld which is assumed to be present everywhere in the vacuum i�e� thespace�time in which interactions take place�

The vacuum or equivalently the groundstate i�e� the state with the lowestpotential energy may have a non�zero �scalar �eld value represented by % "v exp�i� � v is called the vacuum expectation value �vev � The same minimumis reached for an arbitrary value of the phase � so there exists an in�nity ofdi�erent but equivalent groundstates� This degeneracy of the ground state takeson a special signi�cance in a quantum �eld theory because the vacuum is requiredto be unique so the phase cannot be arbitrarily at each point in space�time� Oncea particular value of the phase is chosen it has to remain the same everywherei�e� it cannot change locally� A scalar �eld with a nonzero vev therefore breakslocal gauge invariance�� More details can be found in the nice introduction byMoriyasu����

Nature has many examples of broken symmetries� Superconductivity is a wellknown example� Below the critical temperature the electrons bind into Cooperpairs�� The density of Cooper pairs corresponds to the vev� Owing to the weakbinding the e�ective size of a Cooper pair is large about ��� cm so everyCooper pair overlaps with about ��� other Cooper pairs and this overlap �locks�the phases of the wave function over macroscopic distances� �Superconductivityis a remarkable manifestation of Quantum Mechanics on a truly macroscopicscale� �����

In the superconducting phase the photon gets an e�ective mass through theinteraction with the Cooper pairs in the �vacuum� which is apparent in the

�An amusing analogy was proposed by A� Salam� A number of guests sitting around a rounddinner table all have a serviette on the same side of the plate with complete symmetry� Assoon as one guest picks up a serviette� say on the lefthand side� the symmetry is broken and allguests have to follow suit and take the serviette on the same side�� i�e� the phases are lockedtogether everywhere in the �vacuum due to the �spontaneously broken symmetry �

�The interaction of the conduction electrons with the lattice produces an attractive force�When the electron energies are su�ciently small� i�e� below the critical temperature� thisattractive force overcomes the Coulomb repulsion and binds the electrons into Cooper pairs�in which the momenta and spins of the electrons are in opposite directions� so the Cooper pairforms a scalar �eld and its quanta have a charge two times the electron charge�

��

Figure ���� Shape of the Higgs potential for �� � � �a and �� � � �b � �� and�� are the real and imaginary parts of the Higgs �eld�

Meissner e�ect� the magnetic �eld has a very short penetration depth into thesuperconductor or equivalently the photon is very massive� Before the phasetransition the vacuum would have zero Cooper pairs i�e� a zero vev and themagnetic �eld can penetrate the superconductor without attenuation as expectedfor massless photons�

This example of Quantum Mechanics and spontaneous symmetry breakingin superconductivity has been transferred almost literally to elementary particlephysics by Higgs and others����� For the self�interaction of the Higgs �eld oneconsiders a potential analogous to the one proposed by Ginzburg and Landau forsuperconductivity�

V �% " �� %y% # ��%y% � ����

where �� and � are constants� The potential has a parabolic shape if �� � � buttakes the shape of a Mexican hat for �� � � as pictured in �g� ���� In the lattercase the �eld free vacuum i�e� % " � corresponds to a local maximum thusforming an unstable equilibrium� The groundstate corresponds to a minimumwith a nonzero value for the �eld�

j%j "s�����

�����

In superconductivity �� acts like the critical temperature Tc� above Tc theelectrons are free particles so their phases can be rotated arbitrarily at all pointsin space but below Tc the individual rotational freedom is lost because theelectrons form a coherent system in which all phases are locked to a certainvalue� This corresponds to a single point in the minimum of the Mexican hatwhich represents a vacuum with a nonzero vev and a well de�ned phase thus

��

de�ning a unique vacuum� The coherent system can still be rotated as a wholeso it is invariant under global but not under local rotations�

����� Gauge Boson Masses and the Top Quark Mass�

After this general introduction about the Higgs mechanism one has to considerthe number of Higgs �elds needed to break the SU�� L � U�� Y symmetry tothe U�� em symmetry� The latter must have one massless gauge boson while theW and Z bosons must be massive� This can be achieved by choosing % to be acomplex SU�� doublet with de�nite hypercharge �YW " � �

%�x "

���� �x #i��� �x ����x #i����x

������

In order to understand the interactions of the Higgs �eld with other particlesone considers the following Lagrangian for a scalar �eld�

LH " �D�% y�D�% � V �% �����

The �rst term is the usual kinetic energy term for a scalar particle for which theEuler�Lagrange equations lead to the Klein�Gordon equation of motion� Insteadof the normal derivative the covariant derivative is used in eq� ���� in order toensure local gauge invariance under SU�� � U�� rotations�

The vacuum is known to be neutral� Therefore the groundstate of % has tobe of the form ��v � Furthermore %�x has to be constant everywhere in orderto have zero kinetic energy i�e� the derivative term in LH disappears�

The quantum �uctuations of the �eld around the ground state can be parametrisedas follows if we include an arbitrary SU�� phase factor�

% " ei�� x����

��

v # h�x

� �����

The �real �elds ��x are excitations of the �eld along the potential minimum�They correspond to the massless Goldstone bosons of a global symmetry in thiscase three for the three rotations of the SU�� group� However in a local gaugetheory these massless bosons can be eliminated by a suitable rotation�

%� " e�i�� x����%�x "

��

v # h�x

� �����

Consequently the �eld � has no physical signi�cance� Only the real �eld h�x canbe interpreted as a real �Higgs particle� The original �eld � with four degrees offreedom has lost three degrees of freedom� these are recovered as the longitudinalpolarizations of the three heavy gauge bosons�

The kinetic part of eq� ���� gives rise to mass terms for the vector bosonswhich can be written as �YW " � �

LH "�

�

h�g�W �

��� #W ���� #W �

��� # g�B�

�%iy h�

g�W ���� #W ���� #W ���� # g�B��%i

�����

��

or substituting for % its vacuum expectation value v one obtains from the o��diagonal terms �by writing the � matrices explicitly see eq� ���

�gv

�

� ��W �

� � # �W �

� ��

����

and from the diagonal terms�

�

�

�v�

�

���gW �y

� # g�By�

� ��gW �� # g�B�

�"

�

�

�v�

�

��By�W

�y�

�� #g�� �gg��gg� g�

��B�

W ��

�

����� Since mass terms of physical �elds have to be diagonal one obtains the �physical�gauge �elds of the broken symmetry by diagonalizing the mass term�

�By�W

�y�

�U��U M U��U

�B�

W ��

������

where U represents a unitary matrix

U "�p

g�� # g�

�g g�

�g� g

��

cos �W sin �W� sin �W cos �W

������

Consequently the real �elds become a mixture of the gauge �elds��A�

Z�

�" U

�B�

W ��

������

and the matrix UMU�� becomes a diagonal matrix for a suitable mixing angle�W �

In these �elds the mass terms have the form

M�WW

�� W

�� #�

��A�� Z�

�� �� M�

Z

��A�

Z�

������

with

M�W "

�

�g�v� �����

M�Z "

g� � # g�

�v� �����

Here v is the vacuum expectation value of the Higgs potential which for theknown gauge boson masses and couplings can be calculated to be��

v � ��� GeV �����

The neutral part of the Lagrangian if expressed in terms of the physical �eldscan be written as�

Lneutrint " �A��g� cos �W �eR�

�eR #�

��L�

��L #�

�eL�

�eL � �

�g sin �W ��L�

��L � eL��eL �

#Z��g� sin �W �eR�

�eR #�

��L�

��L #�

�eL�

�eL #�

�g cos �W ��L�

��L � eL��eL �

�����

�Sometimes the Higgs �eld is normalized by ��p� in which case v � �� GeV�

�

θW

g

g

e

Figure ���� Geometric picture of the relations between the electroweak couplingconstants�

The photon �eld should only couple to the electron �elds and not to the neutrinosso the terms proportional to g� cos �W and g sin �W should cancel and the couplingto the electrons has to be the electric charge e� This can be achieved by requiring�

g� cos �W " g sin �W " e ����

Hence

tan �W "g�

g� sin� �W "

g��

g� # g��and e "

gg�pg� # g��

�����

A geometric picture of these relations is shown in �g� ���� From these rela�tions and the relations between masses and couplings ����� and ���� one �ndsthe famous relation between the electroweak mixing angle and the gauge bosonmasses�

MW " cos �W �MZ or sin� �W " �� M�W

M�Z

�����

The value of MW can also be related to the precisely measured muon decayconstant G� " ������ � ���� GeV��� If calculated in the SM one �nds�

G�p�"

e�

� sin� �WM�W

�����

This relation can be used to calculate the gauge boson masses from measuredcoupling constants G� and sin �W �

M�W "

p�G�

� �

sin� �W�����

M�Z "

p�G�

� �

sin� �W cos� �W�����

�����

Inserting sin� �W " ��� and �� " ����� yields MZ"�� GeV� However theserelations are only at tree level� Radiative corrections depend on the as yet un�known top mass� Fitting the unknown top mass to the measured MZ mass theelectroweak asymmetries and the cross sections at LEP yields�����

Mtop " ������� �stat ������ �unknown Higgs �����

��

Also the fermions can interact with the scalar �eld albeit not necessarily withthe gauge coupling constant� The Lagrangian for the interaction of the leptonswith the Higgs �eld can be written as�

LH�L " �geYhL�eR # eR�

yLi �����

Substituting the vacuum expectation value for % yields

�geYp�

��L� eL

��v

�eR # eR ��� v

��LeL

��"�geY vp

��eLeR # eReL� "

�geY vp�ee

����� The Yukawa coupling constant geY is a free parameter which has to be adjustedsuch that me " geY v

p�� Thus the coupling gY is proportional to the mass of

the particle and consequently the coupling of the Higgs �eld to fermions is pro�portional to the mass of the fermion a prediction of utmost importance to theexperimental search for the Higgs boson�

Note that the neutrino stays massless with the choice of the Lagrangian sinceno mass term for the neutrino appears in eq� �����

����� Summary on the Higgs mechanism�

In summary the Higgs mechanism assumed the existence of a scalar �eld % "%� exp �i��x � After spontaneous symmetry breaking the phases are �locked�over macroscopic distances so the �eld averaged over all phases is not zero any�more and % develops a vacuum expectation value� The interaction of the fermionsand gauge bosons with this coherent system of scalar �elds % gives rise to e�ec�tive particle masses just like the interaction of the electromagnetic �eld with theCooper pairs inside a superconductor can be described by an e�ective photonmass�

The vacuum corresponds to the groundstate with minimal potential energyand zero kinetic energy� At high enough temperatures the thermal �uctuations ofthe Higgs particles about the groundstate become so strong that the coherence islost i�e� %�x " constant is not true anymore� In other words a phase transitionfrom the ground state with broken symmetry �% " � to the symmetric ground�state takes place� In the symmetric phase the groundstate is invariant again underlocal SU�� rotations since the phases can be adjusted locally without changingthe groundstate with � % �" �� In the latter case all masses disappear sincethey are proportional to � % �" ��

Both the fermion and gauge boson masses are generated through the interac�tion with the Higgs �eld� Since the interactions are proportional to the couplingconstants one �nds a relation between masses and coupling constants� For thefermions the Yukawa coupling constant is proportional to the fermion mass andthe mass ratio of the W and Z bosons is only dependent on the electroweakmixing angle �see eq� ���� � This mass relation is in excellent agreement with ex�perimental data after including radiative corrections� Hence it is the �rst indirectevidence that the gauge bosons masses are indeed generated by the interactionwith a scalar �eld since otherwise there is no reason to expect the masses of thecharged and neutral gauge bosons to be related in such a speci�c way via thecouplings�

��

Figure ���� The e�ective charge distribution around an electric charge �QED andcolour charge �QCD � At higher Q� one probes smaller distances thus observinga larger �smaller e�ective charge i�e� a larger �smaller coupling constant inQED �QCD �

��

Figure ��� Running of the three coupling constants in the Standard Model owingto the di�erent space charge distributions �compare �g� ����

��� Running Coupling Constants

In a Quantum Field Theory the coupling constants are only e�ective constantsat a certain energy� They are energy or equivalently distance dependent throughvirtual corrections both in QED and in QCD�

However in QED the coupling constant increases as function of Q� while inQCD the coupling constant decreases� A simple picture for this behaviour is thefollowing�

� The electric �eld around a pointlike electric charge diverges like ��r� Insuch a strong �eld electron�positron pairs can be created with a lifetimedetermined by Heisenberg�s uncertainty relations� These virtual e�e� pairsorient themselves in the electric �eld thus giving rise to vacuum polariza�tion just like the atoms in a dielectric are polarized by an external electric�eld� This vacuum polarization screens the �bare� charge so at a largedistance one observes only an e�ective charge� This causes deviations fromCoulomb�s law as observed in the well�known Lamb shift of the energylevels of the hydrogen atom� If the electric charge is probed at higher en�ergies �or shorter distances one penetrates the shielding from the vacuumpolarization deeper and observes more of the bare charge or equivalentlyone observes a larger coupling constant�

� In QCD the situation is more complicated� the colour charge is surroundedby a cloud of gluons and virtual qq pairs� since the gluons themselvescarry a colour charge one has two contributions� a shielding of the barecharge by the qq pairs and an increase of the colour charge by the gluoncloud� The net e�ect of the vacuum polarization is an increase of the totalcolour charge provided not too many qq pairs contribute which is the caseif the number of generations is below � see hereafter� If one probes thischarge at smaller distances one penetrates part of the �antishielding� thusobserving a smaller colour charge at higher energies� So it is the fact thatgluons carry colour themselves which makes the coupling decrease at small

��

� ��� �����R

���� ��

���e�

e�

�a�

�b�

�� ��� �� �

���

���

R

����� �� �����g���R

����

q�����q

q������ �� ������

���

���

R

screening antiscreening

����� �� ��������R

����

������ ���� ��� � � � ���� ��� �

����� �� ���������

���

R

Figure ���� Loop corrections in QED �a and QCD �b � In QED only the fermionscontribute in the loops which causes a screening of the bare charge� In QCDalso the bosons contribute through the gluon sel�nteraction which enhances thebare charge� This antiscreening dominates over the screening�

distances �or high energies � This property is called asymptotic freedomand it explains why in deep inelastic lepton�nucleus scattering experimentsthe quarks inside a nucleus appear quasi free in spite of the fact that theyare tightly bound inside a nucleus� The increase of s at large distancesexplains qualitatively why it is so di!cult to separate the quarks inside ahadron� the larger the distance the more energy one needs to separate themeven further� If the energy of the colour �eld is too high it is transformedinto mass thus generating new quarks which then recombine with the oldones to form new hadrons so one always ends up with a system of hadronsinstead of free quarks�

The space charges from the virtual pairs surrounding an electric charge andcolour charge are shown schematically in �g� ���� The di�erent vacuum polar�izations lead to the energy dependence of the coupling constants sketched in �g���� The colour �eld becomes in�nitely dense at the QCD scale & � ��� MeV�see hereafter � So the con�nement radius of typical hadrons is O���� MeV orone Fermi ������ cm �

The vacuum polarization e�ects can be calculated from the loop diagrams tothe gauge bosons� The main di�erence between the charge distribution in QEDand QCD originates from the diagrams shown in �g� ���� In addition one hasto consider diagrams of the type shown in �g� ���� The ultraviolet divergences�Q� � � in these diagrams can be absorbed in the coupling constants in arenormalizable theory� All other divergences are canceled at the amplitude levelby summing the appropriate amplitudes� The �rst step in such calculations isthe regularization of the divergences i�e� separating the divergent parts in themathematical expressions� The second step is the renormalization of physical

��

G a u g eb o s o nl o o p

+

H i g g s l o o p

++

F e r m i o n l o o p

T r e el e v e l

Figure ���� First order vacuum polarization diagrams�

quantities like charge and mass to absorb the divergent parts of the amplitudesi�e� replace the �bare� quantities of the theory with measured quantities� Forexample the loop corrections to the photon propagator diverge if the momentumtransfer k in the loop is integrated to in�nity� If one introduces a cuto� �� forlarge values of k one �nds for the regularized amplitude of the sum of the Bornterm M� and the loop corrections����

M� " e����

�ln

���m�

��� #

�lnQ�

m�

�M� for Q� �� m� ����

The divergent part depending on the cuto� parameter �� disappears if one re�places the �bare� charge e by the renormalized charge eR�

e�R e����

�ln

���m�

������

i�e� the �bare� charge occurring in the Dirac equation is renormalized to ameasurable quantity eR� For eR one usually takes the Thomson limit for Comptonscattering i�e� �e� �e for k � ��

�T "�

�

�

m�e

�����

with " e�R�� " ������� and me " ������ GeV�After regularization and renormalization to a measured quantity �in this case

using the so called �on shell� scheme i�e� one uses the mass and charge of afree electron as measured at low energy one is left with a Q� dependent but�nite part of the vacuum polarization� This can be absorbed in a Q� dependentcoupling constant which in case of QED becomes for Q� �� m��

�Q� "

�� #

�lnQ�

m�e

������

��

If one sums more loops this yields terms � ��� n�ln Q�

m�e m and retaining only the

leading logarithms �i�e� n"m these terms can be summed to�

�Q� "

��� ���

ln Q�

m�e

�����

since �Xn�

xn "�

�� x �����

Of course the total Q� dependence is obtained by summing over all possiblefermion loops in the photon propagator�

These vacuum polarization e�ects are non�negligible� For example at LEPaccelerator energies has increased from its low energy value ����� to ����� orabout ��

The diagrams of �g� ���b yield similarly to eq� �����

s�Q� " s��

�

� #

s���

�

���� �Nf

�

lnQ�

��

��������

Note that s decreases with increasing Q� if �� � �Nf�� � � or Nf � �thus leading to asymptotic freedom at high energy� This is in contrast to theQ� dependence of �Q� in eq� ���� which increases with increasing Q�� Since s becomes in�nite at small Q� one cannot take this scale as a reference scale�Instead one could choose as renormalization point the �con�nement scale� & i�e� s �� if Q� � &� In this case eq� ���� becomes independent of � since the �in brackets becomes negligible so one obtains�

s�Q� "

�

���� �nf� lnQ�

��

�����

The de�nition of & depends on the renormalization scheme� The most widelyused scheme is the MS scheme���� which we will use here� Other schemes canbe used as well and simple relations between the de�nitions of & exist�����

The higher order corrections are usually calculated with the renormalizationgroup technique which yields for the � dependence of a coupling constant �

��

��" ��

� # �� � # ��

# �����

The �rst two terms in this perturbative expansion are renormalization�schemeindependent� Their speci�c values are given in the appendix� The �rst ordersolution of eq� ���� is simple�

�

�Q� "

�

�Q�� � �� ln�

Q�

Q��

�����

where Q�� is a reference energy� One observes a linear relation between the change

in the inverse of the coupling constant and the logarithm of the energy� The slopedepends on the sign of �� which is positive for QED but negative for QCD thus

��

leading to asymptotic freedom in the latter case� The second order correctionsare so small that they do not change this conclusion� Higher order terms dependon the renormalization prescription� In higher orders there are also correctionsfrom Higgs particles and gauge bosons in the loops� Therefore the running ofa given coupling constant depends slightly on the value of the other couplingconstants and the Yukawa couplings� These higher order corrections cause theRGE equations to be coupled so one has to solve a large number of coupleddi�erential equations� All these equations are summarized in the appendix�

��

Chapter �

Grand Uni�ed Theories�

��� Motivation

The Standard Model describes all observed interactions between elementary par�ticles with astonishing precision� Nevertheless it cannot be considered to bethe ultimate theory because the many unanswered questions remain a problem�Among them�

� The Gauge ProblemWhy are there three independent symmetry groups�

� The Parameter ProblemHow can one reduce the number of free parameters� �At least �� fromthe couplings the mixing parameters the Yukawa couplings and the Higgspotential�

� The Fermion ProblemWhy are there three generations of quarks and leptons� What is the originof the the symmetry between quarks and leptons� Are they compositeparticles of more fundamental objects�

� The Charge Quantization ProblemWhy do protons and electrons have exactly opposite electric charges�

� The Hierarchy ProblemWhy is the weak scale so small compared with the GUT scale i�e� why isMW � ����� MP lanck�

� The Fine�tuning ProblemRadiative corrections to the Higgs masses and gauge boson masses havequadratic divergences� For example 'M�

H � O�M�P lanck � In other words

the corrections to the Higgs masses are many orders of magnitude largerthan the masses themselves since they are expected to be of the order ofthe electroweak gauge boson masses� This requires extremely unnatural�ne�tuning in the parameters of the Higgs potential� This ��ne�tuning�problem is solved in the supersymmetric extension of the SM as will bediscussed afterwards�

��

��� Grand Uni�cation

The problems mentioned above can be partly solved by assuming the symmetrygroups SU�� C � SU�� L � U�� Y are part of a larger group G i�e�

G � SU�� C � SU�� L � U�� Y ����

The smallest group G is the SU�� group����� so the minimal extension of theSM towards a GUT is based on the SU�� group� Throughout this paper we willonly consider this minimal extension� The group G has a single coupling constantfor all interactions and the observed di�erences in the couplings at low energy arecaused by radiative corrections� As discussed before the strong coupling constantdecreases with increasing energy while the electromagnetic one increases withenergy so that at some high energy they will become equal� Since the changeswith energy are only logarithmic �eq� ���� the uni�cation scale is high namelyof the order of ���� � ���� GeV depending on the assumed particle content inthe loop diagrams�

In the SU�� group���� the �� particles and antiparticles of the �rst generationcan be �t into the ��plet� and ���plet�

� "

�BBBBBB�

dCgdCrdCbe�

��e

�CCCCCCA

�� "�p�

�BBBBBB�

� #uCb �uCr �ug �dg�uCb � #uCg �ur �dr#uCr �uCg � �ub �db#ug #ur #ub � �e�#dg #dr #db #e� �

�CCCCCCAL

����

The superscript C indicates the charge conjugated particle i�e� the antiparticleand all particles are chosen to be left�handed since a left�handed antiparticletransforms like a right�handed particle� Thus the superscript C implies a right�handed singlet with weak isospin equal zero�

With this multiplet structure the sum of the quantum numbers Q T� and Yis zero within one multiplet as required since the corresponding operators arerepresented by traceless matrices�

Note that there is no space for the antineutrino in these multiplets so withinthe minimal SU�� the neutrino must be massless since for a massive particlethe right�handed helicity state is also present� Of course it is possible to put aright�handed neutrino into a singlet representation�

SU�� rotations can be represented by ��� matrices� Local gauge invariancerequires the introduction of N� � � " �� gauge �elds �the �mediators� whichcause the interactions between the matter �elds� The gauge �elds transformunder the adjoint representation of the SU�� group which can be written in

�G cannot be the direct product of the SU��� SU� and U�� groups� since this would notrepresent a new uni�ed force with a single coupling constant� but still require three independentcoupling constants�

�The bar indicates the complementary representation of the fundamental representation�

�

e

π

X

u

u

d

dd

νe

π

Y

u

ud d

e

π

Y

u

ud u

uu

Figure ���� GUT proton decays through the exchange of X and Y gauge bosons�

matrix form as �compare eqns� ��� and ��� �

�� "

�BBBBBBBB�

G�� � �Bp��

G�� G�� XC� Y C

�

G�� G�� � �Bp��

G�� XC� Y C

�

G�� G�� G�� � �Bp��

XC� Y C

�

X� X� X�W �p�# �Bp

��W�

Y� Y� Y� W� �W �p�# �Bp

��

�CCCCCCCCA

����

The G�s represent the gluon �elds of equation ��� while the W �s and B�s are thegauge �elds of the SU�� symmetry groups� TheX and Y �s are new gauge bosonswhich represent interactions in which quarks are transformed into leptons andvice�versa as should be apparent if one operates with this matrix on the ��plet�Consequently the X �Y bosons which couple to the electron �neutrino andd�quark must have electric charge ��� ���� �

��� SU��� predictions

����� Proton decay

The X and Y gauge bosons can introduce transitions between quarks and leptonsthus violating lepton and baryon number�� This can lead to the following protonand neutron decays �see �g� ��� �

p� e�� n� e��

p� e��� n� e���

p� e��� n� ���

p� e�� n� ��

p� �� n� ��K�

p� ���

p� ��K�

����

The decays with kaons in the �nal state are allowed through �avour mixingi�e� the interaction eigenstates are not necessarily the mass eigenstates�

�The di�erence between lepton and baryon number B�L is conserved in these transitions�

��

For the lifetime of the nucleon one writes in analogy to muon decay�

�p � MX

��m�p

����

The proton mass mp to the �fth power originates from the phase space in case the�nal states are much lighter than the proton which is the case for the dominantdecay mode� p � e��� After this prediction of an unstable proton in granduni�ed theories a great deal of activity developed and the lower limit on theproton life time increased to����

�p � � � ���� yrs ���

for the dominant decay mode p � e��� From equation ��� this implies �for � " ���� see chapter

MX ���� GeV ����

From the extrapolation of the couplings in the SU�� model to high energiesone expects the uni�cation scale to be reached well below ���� GeV so the protonlifetime measurements exclude the minimal SU�� model as a viable GUT� Aswill be discussed later the supersymmetric extension of the SU�� model has theuni�cation point well above ���� GeV�

����� Baryon Asymmetry

The heavy gauge bosons responsible for the uni�ed force cannot be producedwith conventional accelerators but energies above ���� were easily accessibleduring the birth of our universe� This could have led to an excess of matter overantimatter right at the beginning since the X and Y bosons can decay into purematter e�g� X � uu which is allowed because the charge of the X boson is���� As pointed out by Sakharov���� such an excess is possible if both C andCP are violated if the baryon number B is violated and if the process goesthrough a phase of non�equilibrium� All three conditions are possible within theSU�� model� The non�equilibrium phase happens if the hot universe cools downand arrives at a temperature too low to generate X and Y bosons anymore soonly the decays are possible� Since the CP violation is expected to be small theexcess of matter over antimatter will be small so most of the matter annihilatedwith antimatter into enormous number of photons� This would explain why thenumber of photons over baryons is so large�

N�

Nb

� ���� ����

However later it was realized that the electroweak phase transition may washout any �B#L excess generated by GUT�s� One then has to explain the observedbaryon asymmetry by the electroweak baryogenesis which is actively studied�����

��

+

+

+F e r m i o n l o o p

G a u g eb o s o nl o o p

G a u g i n o l o o p

S c a l a r l o o p

Figure ���� Radiative corrections to particle masses�

����� Charge Quantization

From the fact that quarks and leptons are assigned to the same multiplet thecharges must be related since the trace of any generator has to be zero� Forexample the charge operator Q on the fundamental representation yields�

TrQ " Tr�qd� qd� qd� e� � " � ����

or in other words in SU�� the electric charge of the d�quark has to be ��� ofthe charge of an electron� Similarly one �nds the charge of the u�quark is ��� ofthe positron charge so the total charge of the proton �"uud has to be exactlyopposite to the charge of an electron�

����� Prediction of sin�W

If the SU�� and U�� groups have equal coupling constants the electroweakmixing angle can be calculated easily since it is given by the ratio g� ���g�# g� � �see eq� ���� which would be ��� for equal coupling constants� However theargument is slightly more subtle since for unitary transformations the rotationmatrices have to be normalized such that

Tr FkFl " �kl �����

This normalization is not critical in case one has independent coupling constantsfor the subgroups since a �wrong� normalization for a rotation matrix can al�ways be corrected by a rede�nition of the corresponding coupling constants asis apparent from equation ��� This freedom is lost if one has a single couplingconstant so one has to be careful about the relative normalization� It turns outthat the Gell�Mann and Pauli rotation matrices of the SU�� and SU�� groupshave the correct normalization but the normalization of the weak hyperchargeoperator needs to be changed� De�ning ���YW " CT� and substituting this intothe Gell�Mann�Nishijima relation ��� yields�

Q " T� # CT� �����

��

Requiring the same normalization for T� and T� implies from equation �����

Tr Q� " �� # C� Tr T �� �����

or inserting numbers from the ��plet of SU�� yields�

� # C� "Tr Q�

Tr T ��

"� � ��� # �

� � ��� "�

� �����

Replacing in the covariant derivative �eq� �� ���YW with CT� impliesg�CT� g�T� or�

g� " Cg�� �����

where C� " ��� from eq� ����� With this normalization the electroweak mixingangle after uni�cation becomes�

sin� �W "g��

�g� # g� � "

g���C�

�g�� # g���C� "

�

� # C�"

�

� �����

The manifest disagreement with the experimental value of ���� at low energiesbrought the SU�� model originally into discredit until it was noticed that therunning of the couplings between the uni�cation scale and low energies couldreduce the value of sin� �W considerably� As we will show in the last chapterwith the very precise measurement of sin� �W at LEP uni�cation of the threecoupling constants within the SU�� model is excluded and just as in the caseof the proton life time supersymmetry comes to the rescue and uni�cation isperfectly possible within the supersymmetric extension of SU�� �

Note that the prediction of sin� �W " ��� is not speci�c to the SU�� modelbut is true for any group with SU�� C � SU�� L � U�� Y as subgroups implyingthat Q T� and YW are generators with traces equal zero and thus leading to thepredictions given above�

��� Spontaneous Symmetry Breaking in SU���

The SU�� symmetry is certainly broken since the new force corresponding tothe exchange of the X and Y bosons would lead to very rapid proton decay ifthese new gauge bosons were massless� As mentioned above from the limit onthe proton life time these SU�� gauge boson have to be very heavy i�e� massesabove ���� GeV� The generation of masses can be obtained again in a gaugeinvariant way via the Higgs mechanism� The Higgs �eld is chosen in the adjointrepresentation �� and the minimum � %� � can be chosen in the following way�

� %� �" v�

�BBBBBB�

��

���

�

���

�CCCCCCA

����

The �� XY gauge bosons of the SU�� group require a mass�

M�X " M�

Y "��

�g��v

�� �����

��

after (eating� �� of the �� scalar �elds in the adjoint representation thus providingthe longitudinal degrees of freedom� The �eld %� is invariant under the rotationsof the SU�� C � SU�� L � U�� Y group so this symmetry is not broken and thecorresponding gauge bosons including the W and Z bosons remain massless�after the �rst stage of SU�� symmetry breaking�

The usual breakdown of the electroweak symmetry to SU�� C � U�� em isachieved by a ��plet %� of Higgs �elds for which the minimum of the e�ectivepotential can be chosen at�

� %� �" v�

�BBBBBB�

�����

�CCCCCCA

�����

The fourth and �fth component of %� correspond to the SU�� doublet �%��%� of the SM �see eq� ���� � Since the total charge in a representation has to be zeroagain the �rst triplet of complex �elds in %� which transforms as ��� ��� and���� � �� must have charge j���j� Since they couple to all fermions with massthey can induce proton decay�

u# d� H�� � e� # u�e # d

�����

Such decays can be suppressed only by su!ciently high masses of the colouredHiggs triplet� These can obtain high masses through interaction terms between%� and %��

Note that from eq� ���� � %� � has to be of the order of MX while %� hasto be of the order of MW since

M�W "

�

��g�v�

� �����

and

MZ "MW

cos �W�����

or more precisely v� " ��pGF" ��� GeV� The minimum of the Higgs potential

involves both %� and %�� Despite this mixing the ratio v��v� � ����� has tobe preserved �hierarchy problem � Radiative corrections spoil usually such a �ne�tuning so SU�� is in trouble� As will be discussed later also here supersymmetryo�ers solutions for both this �ne�tuning and the hierarchy problem�

��� Relations between Quark and LeptonMasses

The Higgs ��plet %� can be used to generate fermion masses� Since the ��plet ofthe matter �elds contains both leptons and down�type quarks their masses arerelated while the up�type quark masses are free parameters� At the GUT scale

��

one expects�

md " me �����

ms " m� �����

mb " m� �����

Unfortunately the masses of the light quarks have large uncertainties from thebinding energies in the hadrons but the b�quark mass can be correctly predictedfrom the � �mass after including radiative corrections �see �g� ��� for typicalgraphs �

Since the corrections from graphs involving the strong coupling constant sare dominant one expects in �rst order����

mb

m�" O

� s�mb

s�MX

�" O�� �����

More precise formulae are given in the appendix and will be used in the lastchapter in a quantitative analysis since the b�quark mass gives a rather strongconstraint on the evolution of the couplings and through the radiative correctionsinvolving the Yukawa couplings on the top quark mass�

��

Chapter �

Supersymmetry

��� Motivation

Supersymmetry��� ��� presupposes a symmetry between fermions and bosonswhich can be realized in nature only if one assumes each particle with spin jhas a supersymmetric partner with spin j����� This leads to a doubling of theparticle spectrum �see table ��� which are assigned to two supermultiplets� thevector multiplet for the gauge bosons and the chiral multiplet for the matter�elds� Unfortunately the supersymmetric particles or �sparticles� have not beenobserved so far so either supersymmetry is an elegant idea which has nothingto do with reality or supersymmetry is not an exact symmetry in which case thesparticles can be heavier than the particles� Many people opt for the latter wayout since there are many good reasons to believe in supersymmetry�

� SUSY solves the �ne�tuning problemAs mentioned before the radiative corrections in the SU�� model havequadratic divergences from the diagrams in �g� ��� which lead to 'M�

H �O�M�

X where MX is a cuto� scale typically the uni�cation scale if noother scales introduce new physics beforehand�

However in SUSY the loop corrections contain both fermions �F andbosons �B in the loops which according to the Feynman rules contribute

VECTOR MULTIPLET CHIRAL MULTIPLET

J " � J " ��� J " ��� J " �

g )g QL� UCL � D

CL

)QL� )UCL � )D

CL

W��W � )W�� )W � LL� ECL

)LL� )ECL

B )B )H�� )H� H�� H�

Table ���� Assignment of gauge �elds to the vector super�eld and the matter�elds to the chiral super�eld�

��

with an opposite sign i�e�

'M�H � O� jM�

B �M�F j � O����� M�

SUSY ����

where MSUSY is a typical SUSY mass scale� In other words the �ne�tuningproblem disappears if the SUSY partners are not too heavy compared withthe known fermions� An estimate of the required SUSY breaking scale canbe obtained by considering that the masses of the weak gauge bosons andHiggs masses are both obtained by multiplying the vacuum expectationvalue of the Higgs �eld �see previous chapter with a coupling constant soone expects MW � MH � Requiring that the radiative corrections are notmuch larger than the masses themselves i�e� 'MW � MW or replacingMW by MH 'MH � O���� yields after substitution into eq� ����

MSUSY � ��� GeV ����

� SUSY o�ers a solution for the hierarchy problemThe possible explanation for the small ratio M�

W�M�X � ����� is simple

in SUSY models� large radiative corrections from the top�quark Yukawacoupling to the Higgs sector drive one of the Higgs masses squared nega�tive thus changing the shape of the e�ective potential from the parabolicshape to the Mexican hat �see �g� ��� and triggering electroweak symme�try breaking����� Since radiative corrections are logarithmic in energy thisautomatically leads to a large hierarchy between the scales� In the SM onecould invoke a similar mechanism for the triggering of electroweak symme�try breaking but in that case the quadratic divergences in the radiativecorrections would upset the argument�

In the MSSM the electroweak scale is governed by the starting values ofthe parameters at the GUT scale and the top�quark mass� This stronglyconstrains the SUSY mass spectrum as will be discussed in the last chapter�

� SUSY yields uni�cation of the coupling constantsAfter the precise measurements of the SU�� C � SU�� L � U�� Y couplingconstants the possibility of coupling constant uni�cation within the SMcould be excluded since after extrapolation to high energies the three cou�pling constants would not meet in a single point� This is demonstrated inthe upper part of �g� ��� which shows the evolution of the inverse of thecouplings as function of the logarithm of the energy� In this presentationthe evolution becomes a straight line in �rst order as is apparent from thesolution of the RGE �eqn� ���� � The second order corrections which havebeen included in �g� ��� by using eqs� A��� from the appendix are sosmall that they cause no visible deviation from a straight line�

A single uni�cation point is excluded by more than � standard deviations�The curve �� � meets the crossing point of the other two coupling constantsonly for a starting value at s�MZ " ���� while the measured value is��� � �������� This is an exciting result since it means uni�cation canonly be obtained if new physics enters between the electroweak and thePlanck scale�

��

Unification of the Couplings of theElectromagnetic, Weak and Strong Forces

Standard Model

MinimalSupersymmetric

Model

Figure ���� Evolution of the inverse of the three coupling constants in the Stan�dard Model �SM �top and in the supersymmetric extension of the SM �MSSM �bottom � Only in the latter case uni�cation is obtained� The SUSY particles areassumed to contribute only above the e�ective SUSY scale MSUSY of about oneTeV which causes the change in slope in the evolution of the couplings� The ��C�L� for this scale is indicated by the vertical lines �dashed � The evolution of thecouplings was calculated in second order �see section A�� of the appendix withthe constants �i and �ij calculated for the MSSM above MSUSY in the bottompart and for the SM elsewhere � The thickness of the lines represents the errorin the coupling constants� ��

Figure ���� �� distribution for MSUSY andMGUT �

It turns out that within the SUSY model perfect uni�cation can be obtainedif the SUSY masses are of the order of one TeV� This is shown in the bottompart of �g� ���� the SUSY particles are assumed to contribute e�ectivelyto the running of the coupling constants only for energies above the typicalSUSY mass scale which causes the change in the slope of the lines near oneTeV� From a �t requiring uni�cation one �nds for the breakpoint MSUSY

and the uni�cation point MGUT ��� ���

MSUSY " ��� � �� � � GeV ����

MGUT " ����� � �� � �� GeV ����

GUT�� " �� � �� � ��� ����

where GUT g����� The �rst error originates from the uncertainty inthe coupling constant while the second error is due to the uncertainty inthe mass splittings between the SUSY particles� The �� distributions ofMSUSY and MGUT for the �t in the bottom part of �g� ��� are shown in �g����� These �gures are an update of the published �gures using the newestvalues of the coupling constants as shown in the �gure����

Note that the parametrisation of the SUSY mass spectrum with a singlemass scale is not adequate and leads to uncertainties� However the errorsin the coupling constants �mainly in s are large and the uncertainties frommass splittings between the sparticles are more than a factor two smaller�see eq� ��� � In the last chapter the uni�cation including a more detailedtreatment of the mass splittings will be studied�

One can ask� �What is the signi cance of this observation� For many peopleit was the �rst �evidence� for supersymmetry especially since MSUSY wasfound in the range where the �ne�tuning problem does not reappear �see eq���� � Consequently the results triggered a revival of the interest in SUSY

�

as was apparent from the fact that ref� ���� with the �� �t of the uni�cationof the coupling constants as exempli�ed in �gs� ��� and ��� reached theTop�Ten of the citation list thus leading to discussions in practically allpopular journals�����

Non�SUSY enthusiasts were considering uni�cation obvious� with a totalof three free parameters �MGUT � GUT and MSUSY and three equationsone can naively always �nd a solution� The latter statement is certainlynot true� searching for other types of new physics with the masses as freeparticles yields only rarely uni�cation especially if one requires in additionthat the uni�cation scale is above ���� GeV in order to be consistent withthe proton lifetime limits and below the Planck scale in order to be in theregime where gravity can be neglected� From the ��� models tried onlya handful yielded uni�cation����� The reason is simple� introducing newparticles usually alters all three couplings simultaneously thus giving riseto strong correlations between the slopes of the three lines� For exampleadding a fourth family of particles with an arbitrary mass will never yielduni�cation since it changes the slopes of all three coupling by the sameamount so if with three families uni�cation cannot be obtained it will notwork with four families either even if one has an additional free parameter�Nevertheless uni�cation does not prove supersymmetry it only gives aninteresting hint� The real proof would be the observation of the sparticles�

� Uni�cation with gravityThe space�time symmetry group is the Poincar*e group� Requiring localgauge invariance under the transformations of this group leads to the Ein�stein theory of gravitation� Localizing both the internal and the space�timesymmetry groups yields the Yang�Mills gauge �elds and the gravitational�elds� This paves the way for the uni�cation of gravity with the strongand electroweak interactions� The only non�trivial uni�cation of an inter�nal symmetry and the space�time symmetry group is the supersymmetrygroup so supersymmetric theories automatically include gravity����� Un�fortunately supergravity models are inherently non�renormalizible whichprevents up to now clear predictions� Nevertheless the spontaneous sym�metry breaking of supergravity is important for the low energy spectrumof supersymmetry����� The most common scenario is the hidden sector

scenario���� in which one postulates two sectors of �elds� the visible sectorcontaining all the particles of the GUT�s described before and the hiddensector which contains �elds which lead to symmetry breaking of supersym�metry at some large scale &SUSY � One assumes that none of the �elds in thehidden sector contains quantum numbers of the visible sector so the twosectors only communicate via gravitational interactions� Consequently thee�ective scale of supersymmetry breaking in the visible sector is suppressedby a power of the Planck scale i�e�

MSUSY � &nSUSY

Mn��P lanck

� ���

��

H~

u

d

W~

s

νμ~

c~

u

d

W~

s

νμ~

c~

Figure ���� Examples of proton decay in the minimal supersymmetric model viawino and Higgsino exchange�

where n is model�dependent �e�g� n " � in the Polonyi model � Thus theSUSY breaking scale can be large above ���� GeV while still producinga small breaking scale in the visible sector� In this case the �ne�tuningproblem can be avoided in a natural way and it is gratifying to see that the�rst experimental hints for MSUSY are indeed in the mass range consistentwith eq� ����

The hidden sector scenario leads to an e�ective low�energy theory withexplicit soft breaking terms where soft implies that no new quadratic di�vergences are generated����� The soft�breaking terms in string�inspired su�pergravity models have been studied recently in refs� ����� A �nal theorywhich simultaneously solves the cosmological constant problem���� and ex�plains the origin of supersymmetry breaking needs certainly a better un�derstanding of superstring theory�

� The uni�cation scale in SUSY is largeAs discussed in chapter � the limits on the proton lifetime require theuni�cation scale to be above ���� GeV which is the case for the MSSM�In addition one has to consider proton decay via graphs of the type shownin �g� ���� These yield a strong constraint on the mixing in the Higgssector���� as will be discussed in detail in the last chapter�

� Prediction of dark matterThe lightest supersymmetric particle �LSP cannot decay into normal mat�ter because of R�parity conservation �see the next section for a de�nitionof R�parity � In addition R�parity forbids a coupling between the LSP andnormal matter�

Consequently the LSP is an ideal candidate for dark matter���� which isbelieved to account for a large fraction of all mass in the universe �see nextchapter � The mass of the dark matter particles is expected to be belowone TeV����

��� SUSY interactions

The quantum numbers and the gauge couplings of the particles and sparticleshave to be the same since they belong to the same multiplet structure�

��

The interaction of the sparticles with normal matter is governed by a newmul

tiplicative quantum number called R�parity which is needed in order to preventbaryon� and lepton number violation� Remember that quarks leptons and Hig�gses are all contained in the same chiral supermultiplet which allows couplingsbetween quarks and leptons� Such transitions which could lead to rapid protondecay are not observed in nature� Therefore the SM particles are assigned a pos�itive R�parity and the supersymmetric partners are R�odd� Requiring R�parityconservation implies that�

� sparticles can be produced only in pairs

� the lightest supersymmetric particle is stable since its decay into normalmatter would change R�parity�

� the interactions of particles and sparticles can be di�erent� For examplethe photon couples to electron�positron pairs but the photino does notcouple to selectron�spositron pairs since in the latter case the R�paritywould change from �� to #��

��� The SUSY Mass Spectrum

Obviously SUSY cannot be an exact symmetry of nature� or else the supersym�metric partners would have the same mass as the normal particles� As mentionedabove the supersymmetric partners should be not too heavy since otherwise thehierarchy problem reappears�

Furthermore if one requires that the breaking terms do not introduce quadraticdivergences only the so�called soft breaking terms are allowed�����

Using the supergravity inspired breaking terms which assume a common massm�� for the gauginos and another common mass m� for the scalars leads to thefollowing breaking term in the Lagrangian �in the notation of ref� ���� �

LBreaking " �m��

Xi

j�ij� �m��

X�

���� ����

� Am�

hhuabQaU

cbH� # hdabQaD

cbH� # heabLaE

cbH�

i�Bm� ��H�H�� ����

Herehu�d�eab are the Yukawa couplings a� b " �� �� � run over the generationsQa are the SU�� doublet quark �eldsU ca are the SU�� singlet charge�conjugated up�quark �elds

Dcb are the SU�� singlet charge�conjugated down�quark �elds

La are the SU�� doublet lepton �eldsEca are the SU�� singlet charge�conjugated lepton �elds

H��� are the SU�� doublet Higgs �elds�i are all scalar �elds�� are the gaugino �elds

The last two terms in LBreaking originate from the cubic and quadratic termsin the superpotential with A B and � as free parameters� In total we now have

��

three couplings i and �ve mass parameters

m�� m��� ��t � A�t � B�t

with the following boundary conditions at MGUT �t " � �

scalars � )m�Q " )m�

U " )m�D " )m�

L " )m�E " m�

�� ����

gauginos � Mi " m��� i " �� �� �� �����

couplings � ) i�� " ) GUT � i " �� �� � �����

Here M� M� and M� are the gauginos masses of the U�� SU�� and SU�� groups� In N " � supergravity one expects at the Planck scale B " A� �� Withthese parameters and the initial conditions at the GUT scale the masses of allSUSY particles can be calculated via the renormalization group equations�

��� Squarks and Sleptons

The squark and slepton masses all have the same value at the GUT scale� How�ever in contrast to the leptons the squarks get additional radiative correctionsfrom virtual gluons �like the ones in �g� ��� for quarks which makes them heavierthan the sleptons at low energies� These radiative corrections can be calculatedfrom the corresponding RGE which have been assembled in the appendix� Thesolutions are�

)m�EL

�t " " m�� # ���m�

�� � ��� cos��� M�Z �����

)m��L�t " " m�

� # ���m��� # �� cos��� M�

Z �����

)m�ER

�t " " m�� # ���m�

�� � ��� cos��� M�Z �����

)m�UL

�t " " m�� # m�

�� # ��� cos��� M�Z �����

)m�DL

�t " " m�� # m�

�� � ��� cos��� M�Z ����

)m�UR

�t " " m�� # �m�

�� # ��� cos��� M�Z �����

)m�DR

�t " " m�� # �m�

�� � ��� cos��� M�Z� �����

where � is the mixing angle between the two Higgs doublets which will be de��ned more precisely in section ��� The coe!cients depend on the couplings asshown explicitly in the appendix� They were calculated for the parameters fromthe typical �t shown in table �� � GUT " ����� MGUT " �� � ���� GeV andsin� �W " ����� � For the third generation the Yukawa coupling is not necessar�ily negligible� If one includes only the correction from the top Yukawa couplingYt

� one �nds�

)m�bR�t " " )m�

DR�����

)m�bL�t " " )m�

DL� ���m�

� � ���m��� �����

)m�tR�t " " )m�

UR#m�

t � ��m�� � ���m�

�� �����

)m�tL�t " " )m�

UL#m�

t � ���m�� � ���m�

�� �����

�For large values of the mixing angle tan� in the Higgs sector� the b�quark Yukawa couplingcan become large too� However� since the limits on the proton lifetime limit tan� to rathersmall values �see last chapter� this option is not further considered here�

��

The numerical factors have been calculated for At�� " � and the explicit depen�dence on the couplings can be found in the appendix� Note that only the left�handed b�quark gets corrections from the top�quark Yukawa coupling through aloop with a charged Higgsino and a top�quark� The subscripts L or R do notindicate the helicity since the squarks and sleptons have no spin� The labelsjust indicate in analogy to the non�SUSY particles if they are SU�� doubletsor singlets� The mass eigenstates are mixtures of the L and R weak interactionstates� Since the mixing is proportional to the Yukawa coupling we will onlyconsider the mixing for the top quarks� After mixing the mass eigenstates are��using the same numerical input as for the light quarks �

)m�t���

�t " "�

�

�)m�tL

# )m�tR�q� )m�

tL � )m�tR

� # �m�t �Atm� # �� tan� �

� �

�

h�m�

� # ��m��� # �m�

t � ��� cos��� M�Z

i

��

�

rh�m�

�� # ��m�� � �� cos��� M�

Z

i�# �m�

t �Atm� # �� tan� �

�����

where the values of At and � at the weak scale can be calculated as�

At�MZ " �At�� # ��m��

m������

��MZ " ����� �����

Note that for large values of At�� or � combined with a small tan� thesplitting becomes large and one of the stop masses can become very small� sincethe stop mass lower limit is about � GeV���� this yields a constraint on thepossible values of mt� m��� tan � and ��

��� Charginos and Neutralinos

The solutions of the RGE group equations for the gaugino masses are simple�

Mi�t ") i�t

) i�� m�� ����

Numerically at the weak scale �t " � ln�MGUT�MZ " one �nds �see �g� ��� �

M��)g � ��m��� �����

M��MZ � ��m��� �����

M��MZ � ��m�� �����

Since the gluinos obtain corrections from the strong coupling constant � theygrow heavier than the gauginos of the SU�� C � SU�� L � U�� Y group�

The calculation of the mass eigenstates is more complicated since both Hig�gsinos and gauginos are spin ��� particles so the mass eigenstates are in generalmixtures of the weak interaction eigenstates� The mixing of the Higgsinos and

��

χi0

χi+/-

log10 Q

mas

s [G

eV]

M3

M2

M1

eL~

q~

0

100

200

300

400

500

600

2 4 6 8 10 12 14 16

Figure ���� Typical running of the squark �)q slepton �)eL and gaugino�M�� M�� M� masses �solid lines � The dashed lines indicate the running ofthe four neutralinos and two charginos�

��

gauginos whose mass eigenstates are called charginos and neutralinos for thecharged and neutral �elds respectively can be parametrised by the followingLagrangian�

LGaugino�Higgsino " ��

�M�

�a�a � �

� �M ���� � �M c�� # hc

where �a� a " �� �� � �� are the Majorana gluino �elds and

� "

�BBBB�

)B)W �

)H��

)H��

�CCCCA � � "

�)W�

)H�

��

are the Majorana neutralino and Dirac chargino �elds respectively� Here all theterms in the Lagrangian were assembled into matrix notation �similarly to themass matrix for the mixing between B and W � in the SM eq� ���� � The massmatrices can be written as�����

M �� "

�BBB�

M� � �MZ cos � sin �W MZ sin � sin �W� M� MZ cos � cos �W �MZ sin� cos �W

�MZ cos � sin �W MZ cos � cos �W � ��MZ sin � sin �W �MZ sin� cos �W �� �

�CCCA

�����

M c� "

�M�

p�MW sin�p

�MW cos � �

������

The last matrix leads to two chargino eigenstates )����� with mass eigenvalues

M���� "

�

�

�M�

� # �� # �M�W �

q�M�

� � �� � # �MW cos� �� # �M�

W �M�� # �� # �M�� sin ��

����� The dependence on the parameters at the GUT scale can be estimated by sub�stituting for M� and � their values at the weak scale� M��MZ � ��m�� and��MZ � ����� � In the case favoured by the �t discussed in chapter one�nds � �� M� � MZ in which case the charginos eigenstates are approximatelyM� and ��

The four neutralino mass eigenstates are denoted by )��i �i " �� �� �� � withmasses M� ��

� � � � � M� ��� The sign of the mass eigenvalue corresponds to the

CP quantum number of the Majorana neutralino state�In the limiting case M��M�� � �� MZ one can neglect the o��diagonal ele�

ments and the mass eigenstates become�

)��i " � )B� )W���p�� )H� � )H� �

�p�� )H� # )H� � �����

with eigenvalues jM�j� jM�j� j�j� and j�j respectively� In other words the binoand neutral wino do not mix with each other nor with the Higgsino eigenstatesin this limiting case� As we will see in a quantitative analysis the data indeedprefer M��M�� � � MZ so the LSP is bino�like which has consequences for darkmatter searches�

��

��� Higgs Sector

The Higgs sector of the SUSY model has to be extended with respect to the oneof the SM for two reasons�

� the Higgsinos have spin ��� which implies they contribute to the gaugeanomaly unless one has pairs of Higgsinos with opposite hypercharge soin addition to the Higgs doublet with YW"� one needs a second one withYW"���

H���� ���� "�

H��

H��

�� H���� �� � "

�H�

�

H��

������

� The introduction of the second Higgs doublet solves simultaneously theproblem that a single doublet can give mass to only either the up� or down�type quarks as is apparent from the fact that only the neutral componentshave a non zero vev since else the vacuum would not be neutral� So onecan write�

� H� �"

�v��

�� � H� �"

��v�

� �����

In the SM the conjugate �eld can give mass to the other type� Howeversupersymmetry is a spin�symmetry in which the matter � and Higgs �eldsare contained in the same chiral supermultiplet� This forbids couplingsbetween matter �elds and conjugate Higgs �elds� With the two Higgs �eldsintroduced above H� generates mass to the down�type matter �elds whileH� generates mass for the up�type matter �elds�

The supersymmetric model with two Higgs doublets is called the Minimal Su�persymmetric Standard Model �MSSM � The mass spectrum can be analyzed byconsidering again the expansion around the vacuum expectation value given byeq� �����

H� "

�v� #

�p��H� cos � h� sin # iA� sin� � iG� sin�

H� sin� �G� cos �

�����

H� "

�H� cos � #G� sin�

v� #�p��H� sin # h� cos # iA� cos � # iG� sin�

������

Here H� h and A represent the �uctuations around the vacuum corresponding tothe real Higgs �elds while the G�s represent the Goldstone �elds which disappearin exchange for the longitudinal polarization components of the heavy gaugebosons� The imaginary and real sectors do not mix since they have di�erentCP�eigenvalues� and � are the mixing angles in these di�erent sectors� Themass eigenvalues of the imaginary components are CP�odd so one is left with� neutral CP�even Higgs bosons H� and h� � CP�odd neutral Higgs bosons A�and � CP�even charged Higgs bosons�

��

The complete tree level potential for the neutral Higgs sector assuming colourand charge conservation reads�

V �H�� � H

�� "

g� # g��

��jH�

� j� � jH�� j� � #m�

�jH�� j� #m�

�jH�� j� �m�

��H��H

�� # hc

����� Note that in comparison with the potential in the SM the �rst terms do not havearbitrary coe!cients anymore but these are restricted to be the gauge couplingconstants in supersymmetry again because the Higgses belong to the same chiralmultiplet as the matter �elds�� The last three terms in the potential arise fromthe soft breaking terms with the following boundary conditions at the GUT scale�

m���� " m�

��� " ��� � #m��� m

���� " �B��� m�� �����

where ��� is the value of � at the GUT scale� Since � generates mass for theHiggsinos one expects � to be small compared with the GUT scale� A low � valuecan be obtained dynamically if one adds a singlet scalar �eld to the MSSM� Thiswill not be considered further� Instead � is considered to be a free parameter tobe determined from data �see chapter �

From the potential one can derive easily the �ve Higgs masses in terms ofthese parameters by diagonalization of the mass matrices�

M�ij "

�

�

��VH��j��j

�����

where �i is a generic notation for the real or imaginary part of the Higgs �eld�Since the Higgs particles are quantum �eld oscillations around the minimumeq� ���� has to be evaluated at the minimum� One �nds zero masses for theGoldstone bosons� These would�be Goldstone bosons G� and G� are �eaten� bythe SU�� gauge bosons� For the masses of the �ve remaining Higgs particles one�nds�����CP�odd neutral Higgs A�

m�A " m�

� #m�� �����

Charged Higgses H��m�

H� " m�A #M�

W �����

CP�even neutral Higgses H� h�

m�H�h "

�

�

�m�

A #M�Z �

q�m�

A #M�Z

� � �m�AM

�Z cos� ��

�����

By convention mH � mh� The mixing angles and � are related by

tan � " �m�A #M�

Z

m�A �M�

Z

tan �� �����

�In principle one should consider the running of the gauge couplings between the electroweakscale and the mass scale of the Higgs bosons� However� since the Higgs bosons are expected tobe below the TeV mass scale� this running is small and can be neglected� if one considers allother sources of uncertainty in the MSSM�

��

~L

t

+

F e r m i o n l o o p

x

S c a l a r l o o p

H 2

h2t

ht

ht

t L

C

H 2

tL

C~

Figure ���� Corrections to the Higgs self�energy from Yukawa type interactions�

v� and v� have been chosen real and positive which implies � � � � ���Furthermore the electroweak breaking conditions require tan� � � so

�� � � � �� �����

From the mass formulae at tree level one obtains the once celebrated SUSY massrelations�

mH� MW ����

mh � mA �MH �����

mh �MZ cos �� �MZ �����

m�h #m�

H " m�A #M�

Z �����

After including radiative corrections the lightest neutral Higgs mh becomes con�siderably heavier and these relations are not valid anymore� The mass formulaeincluding the radiative corrections are given in the appendix�

�� Electroweak Symmetry Breaking

The coupling � plays an important role in the shape of the potential and con�sequently in the pattern of electroweak symmetry breaking which occurs if theminimum of the potential is not obtained for � H� �"� H� �"�� In theSM this condition could be introduced ad�hoc by requiring the coe!cient of thequadratic term to be negative� In supersymmetry this term is restricted by thegauge couplings����� A non�trivial minimum can only be obtained by the softbreaking terms if the mass matrix for the Higgs sector given by M�

ij "��V

�Hi�Hj

has a negative eigenvalue� This is obtained if the determinant is negative i�e�

jm���t j� � m�

��t m���t �����

In order that the new minimum is below the trivial minimum with� H� �"�H� �"� one has to require in addition VH�v�� v� � VH��� � � VH���� whichis ful�lled if

m���t #m�

��t �jm���t j �����

�

If one compares eqns� ���� and ���� and notices from eq� ���� that m� " m�one realizes that these conditions cannot be ful�lled simultaneously at least notat the GUT scale�

However at lower energies there are substantial radiative corrections whichcan cause di�erences between m� and m� since the �rst one involves mass cor�rections proportional to the top Yukawa coupling Yt�� while for the latter thesecorrections are proportional to the bottom Yukawa coupling� Typical diagramsare shown in �g� ���� From the RGE for the mass parameters in the Higgspotential one �nds at the weak scale�

���t " " ������� �����

m���t " " m�

� # ������� # ���m��� �����

m���t " " ����m�

� # ������� � ���m���

����At�� m�m�� � ���At�� �m�

� �����

m���t " " ��m�

��� # ������ m�� # ���At�� m���� �����

The coe!cients were evaluated for the parameters of the �t to the experimentaldata �central column of table �� in chapter � The explicit dependence of thecoe!cients on the coupling constants is given in the appendix� The coe!cients ofthe last three terms in m� depend on the top Yukawa coupling� This dependencedisappears if the masses of the stop and top quarks in the diagrams of �g� ���are equal� However if the stop mass is heavier the negative contribution of thediagram with the top quarks dominates� in this case m� decreases much fasterthan m� with decreasing energy and the potential takes the form of a mexicanhat as soon as conditions ���� and ���� are satis�ed� Since A�t is expectedto be small the dominant negative contribution is proportional to m�� �see eq����� so the electroweak breaking scale is a sensitive function of both the initialconditions the top Yukawa coupling and the gaugino masses�

The minimum of the potential can be found by requiring�

�V

�jH�� j

" �m��v� � �m�

�v� #g� # g

��

��v�� � v�� v� " �

�V

�jH�� j

" �m��v� � �m�

�v� �g� # g

��

��v�� � v�� v� " �

Here we substituted

� H� � v� " v cos �� � H� � v� " v sin��

wherev� " v�� # v�� �v � ��� GeV � tan� v�

v�

From the minimization conditions given above one can derive easily�

v� "�

�g� # g�� �tan� � � �

nm�

� �m�� tan

� �o

����

�m�� " �m�

� #m�� sin �� �����

��

M�Z

g� # g��

�v� " �

m�� �m�

� tan� �

tan� � � ������

M�W g�

�v� " M�

Z cos� �W �����

The derivation of these formulae including the one�loop radiative correctionsis given in the appendix�

��

Chapter �

The Big Bang Theory

��� Introduction

In the �����s Hubble discovered that most galaxies showed a redshift in the visiblespectra implying that they were moving away from each other� This observationis one of the basic building blocks of the Big Bang Theory��� which assumes theuniverse is expanding thus solving the problem that a static universe cannot bestable according the Einstein�s equations of general relativity� An expandinguniverse will cool down so at the beginning the universe might have been hot�The remnants of the radiation of such a hot universe can still be observed todayas microwave background radiation corresponding to a temperature of a few de�grees� This radiation was �rst predicted by Gamow but accidentally observedin ��� by Penzia and Wilson from Bell Laboratories� as noise in microwaveantennas used for communication with early satellites� Such an antenna is onlysensitive to a single frequency� Recently the whole spectrum was measured bythe COBE� satellite and it was found to be indeed describable by a black bodyradiation as shown in �g� ��� �from ref� ��� � The deviation from perfect isotropyif one ignores the dipole anisotropy from the Doppler shift caused by the move�ment of the earth through the microwave background is a few times ����� Thishas strong implications for theories concerning the clustering of galaxies sincethis radiation was released soon after the �bang� and hardly interacted after�wards so the inhomogeneities in this radiation are proportional to the density�uctuations in the early universe� These density �uctuations are the seeds forthe �nal formation of galaxies� As will be discussed later such small anisotropieshave strong implications for the models trying to understand the formation ofgalaxies and the nature of the dark matter in the universe� Direct evidence thatthe temperature from the microwave background is indeed the temperature ofthe universe came from the measurement of the temperature of gas clouds deepin space� As it happens the rotational energy levels of cyanogen �CN are suchthat the �K background radiation can excite these molecules� From the detectionof the relative population of the groundstate and the higher levels the excitationtemperature was determined to be TCN " �������������� K���� which is in excel�

�They were awarded the Nobel prize for this discovery in ������Cosmic Background Explorer�

��

Figure ���� Spectrum of the microwave background radiation as measured bythe COBE satellite� The curve is the black body radiation corresponding to atemperature of ��� K�

lent agreement with the direct measurement of the microwave background of���� ���� K by COBE�����

Other evidence that the universe was indeed very hot at the beginning camefrom the measurement of the natural abundance of the light elements� the uni�verse consists for ��� out of hydrogen ��� helium and �� for the remainingelements� Both the rarity of heavy elements and the large abundance of heliumare hard to explain unless one assumes a hot universe at the beginning�

The reason for the low abundance of the heavier elements in a hot universeis simple� they are cracked by the intense radiation around� The abundance ofthe light elements is plotted in �g� ��� as function of the ratio � of primordialbaryons and photons �from ref� ���� � Agreement with experimental observationscan only be obtained for � in the range �� � � ������

The very heavy elements can be produced only at much lower temperaturesbut high pressure so it is usually assumed that the heavy elements on earth andin our bodies were cooked by the high pressure inside the cores of collapsing starswhich exploded as supernovae and put large quantities of these elements into theheavens� They clustered into galaxies under the in�uence of gravity�

The ratio of helium and hydrogen is determined by the number of neutronsavailable for fusion into deuterium and subsequently into helium at the freeze�outtemperature of about � MeV or ���� K�At these high temperatures no complexnuclei can exist only free protons and neutrons� They can be converted intoeach other via charged weak interactions like ep � �en and en � �ep� Notethat this is the same interaction which is responsible for the decay of a freeneutron into a proton electron and antineutrino� The weak interactions maintain

��

Figure ���� Big Bang nucleosynthesis predictions for the primordial abundance ofthe light elements as function of the primordial ratio � of baryons and photons�From ref� �����

thermal equilibrium between the protons and neutrons as long as the density andtemperature are high enough thus leading to a Boltzmann distribution�

n

p" e�QkT � ����

where Q " �mn �mp c� " ��� MeV is the energy di�erence between the states�

Thermal equilibrium is not guaranteed anymore if the weak interaction rates +are slower than the expansion rate of the universe given by the Hubble constanti�e� freeze�out occurs when + � H�t � This happens by the time the temperatureis about ��� K or ��� MeV� Then the ratio n�p is about ���� Since the photonenergies at these temperatures are too low to crack the heavier nuclei nuclei canform through reactions like n#p�� H#� �H#p�� H#� and �H#n�� H#�which in turn react to form He� The latter is a very stable nucleus which hardlycan be cracked so the chain essentially stops till all neutrons are bound insideHe� Heavier nuclei are hardly produced at this stage since there are no stableelements with � or � nuclei so as soon as He catches another nuclei it will decay�Consequently the n�p ratio ��� as determined from the Boltzmann distributionyields a He mass fraction YHe

Yhe "�n�p

n�p# �� �

� ����

��

Figure ���� The Z� lineshape for di�erent number of neutrino types� The data�black points exclude more than three types with mass below MZ�� � �� GeV�

Experimentally the mass fraction YP of He is ��� �� ����� The mass fractionsof deuterium �He and �Li are many orders of magnitude smaller�� ��� Theconcentration of the latter elements is a strong function of the primordial baryondensity �see �g� ��� since at high enough density all the deuterium will fuse intoHe thus eliminating the �components� for �He and �Li�

As said above freeze�out occurs if + � H�t � Thus the expansion rate H�t around T � � MeV determines the He abundance� The expansion rate in turn isdetermined by the fraction of relativistic particles like neutrinos light photinosetc� Roughly for each additional species the primordial He abundance increasesby �� as shown in �g� ��� for a neutron half�life time of ��� minutes �from ref����� �� The neutron lifetime is not negligible on the scale of the �rst three minutesso it has to be taken into account� If one wants to reconcile the abundance ofall light elements there can only be three neutrino species �see �g� ��� withpractically no room for other weakly interacting relativistic particles like lightphotinos� Present collider data con�rm that there are indeed only three lightneutrinos�����

N� " ���� ��� ����

The strongest constraint comes from the Z� resonance data at LEP����� Anexample of the quality of the data is shown in �g� ���� Note that collider datalimit the number of neutrino generations while nucleosynthesis is sensitive toall kinds of light particles where light means about one MeV or less� Happily

�The presently accepted value is ����� ���� minutes�����

��

enough the collider data require the lightest neutralino to be above ���� GeV����so there is no con�ict between cosmology and supersymmetry�

Note that this is a beautiful example of the interactions between cosmologyand elementary particle physics� from the Big Bang Theory the number of rela�tivistic neutrinos is restricted to three �or four if one takes the more conservativeupper limit on the He abundance to be ���� and at the LEP accelerator oneobserves that the number of light neutrinos is indeed three� Alternatively onecan combine the accelerator data and the abundance of the light elements to�postdict� the primordial helium abundance to be ��� and use it to obtain anupper limit on the baryonic density�����

�b � ���c� ����

where �c is the critical density needed for a �at universe� The critical densitywill be calculated in section ���� Thus LEP data in combination with baryoge�nesis strengthens the argument that we need non�baryonic dark matter in a �atuniverse for which � " �c� Other arguments for dark matter will be discussed insection �����

In spite of the marvelous successes of this model of the universe many ques�tions and problems remain as mentioned in the Introduction� However GUT�scan provide amazingly simple solutions at least in principle since many detailsare still open�

These problems will be discussed more quantitatively in the next sectionsstarting with Einsteins equations in a homogeneous and isotropic universe con�ditions which have been well veri�ed in the present universe and which make thesolutions to Einstein�s equations particularly simple� Especially it is easy to seethat a phase transition can lead to in�ation the key in all present cosmologicaltheories�

��� Predictions from General Relativity

At large distances the universe is homogeneous i�e� one �nds the same massdensity everywhere in the universe typically

�univ " ��� � ����� kg�m�� ����

which corresponds to ��� ��� hydrogen atoms per cubic meter� �In comparison anextremely good vacuum of ����N�m� at ��� K contains about � ���� moleculesper cubic meter� Of course the volume to be averaged over should be chosen to bemuch larger than the size of clusters of galaxies� Furthermore the same densityand temperature is observed in all directions i�e� the universe is very isotropic� Ifno point and no direction is preferred in the universe the possible geometry of theuniverse becomes very simple� the curvature has to be the same everywhere i�e�instead of a curvature tensor one needs only a single number usually written asK�t " k�R��t where R�t is the so�called scale factor� This factor can be usedto de�ne dimensionless time�independent �comoving coordinates in an expandinguniverse� the proper �or real distance D�t between two galaxies scales as

D�t " R�t d� ���

��

where d is the distance at a given time t�� The factor k introduced above de�nesthe sign of the curvature� k " � implies no curvature i�e� a �at universe whilek " #���� corresponds to a space with a positive curvature �spherical andk " �� corresponds to a space with a negative curvature �hyperbolic �

The movement of a galaxy in a homogeneous universe can be compared to themolecules in a gas� the stars are just the atoms of a molecule and the moleculesare homogeneously distributed� Di�erentiating equation �� results in

v " ,R�t d� ����

or substituting d from eq� �� results in the famous relation between the velocityand the distance of two galaxies�

v ",R�t

R�t D�t H�t D�t � ����

where H�t is the famous Hubble constant�This relation between the velocity and the distance of the galaxies was �rst

observed experimentally by Hubble in the �����s� He observed that all neigh�bouring galaxies showed a redshift in the spectral lines of the light emitted byspeci�c elements and the redshift was roughly proportional to the distance� Sothis was the �rst evidence that we are living in an expanding universe whichmight have been created by a �Big Bang��

The Hubble relation ��� is a direct consequence of the homogeneity andisotropy of the universe since the scale factor cannot be a constant in that case�This follows directly from Einstein�s �eld equations of general relativity whichcan be written as�

-R�t " �� G

�c��u�t # �p�t R�t � ����

,R�t �

R��t � � G

�c�u�t " �kc

�

R�� �����

where G " � � �����Nm��kg� is the gravitational constant ,R and -R are thederivatives of R with respect to time p is the pressure and u is the energy density�

A static universe in which the derivatives and pressure are zero implies u�t "� so a static universe cannot exist unless one introduces additional potentialenergy in the universe e�g� Einstein�s cosmological constant� At present there isno experimental evidence for such a term�����

��� Interpretation in terms of Newtonian Me

chanics

Since the energy density and correspondingly the curvature is small in ourpresent universe relativistic e�ects can be neglected and the �eld equations ���and ���� have a simple interpretation in terms of Newtonian mechanics� Consider

��

a spherical shell with radius R and mass m� The mass inside this sphere can berelated the average density ��

M "�

�R�� �����

For an expanding universe the total mechanical energy of the mass shell can bewritten as the sum of the kinetic and potential energy�

Etot "�

�m ,R� � GMm

R

"�

�mR�

,R�

R�� �

�G�

� �����

The expression in brackets is just the left hand side of eq� ���� so the sum ofkinetic � and potential energy determines the sign of the curvature k� If k " �then Etot � � implying that the universe will recollapse under the in�uence ofgravity ��Big Crunch� just like a rocket which is launched with a speed belowthe escape velocity will return to the earth� k " �� on the other hand impliesthat the universe will expand and cool forever ��Big Chill� � In case of k " � thetotal energy equals zero ��at Euclidean space in which case the gravitationalenergy or equivalently the mass density is su!cient to halt the expansion�

The �rst �eld equation �eq� ��� follows from the second equation �eq� ���� by di�erentiation and taking into account that in an expanding universe energyis converted into gravitational potential energy� when the volume increases byan in�nitesimal amount 'V then the remaining energy in the gas decreases byan amount p'Vphys where p denotes the pressure� Therefore

,E�t " �p�t ,Vphys�t " ��,R�t

R�t p�t Vphys�t �����

Here we used Vphys " V�R��t where V� is the volume in comoving time�independent

coordinates analogous to eq� ��� On the other hand follows from E�t "u�t Vphys�t

,E�t " ,u�t Vphys�t # �,R�t

R�t u�t Vphys�t �����

Combining eqs� ���� and ���� results in�

,u�t " ��,R�t

R�t �u�t # p�t �����

Substituting this equation for ,u�t after di�erentiation of eq� ���� yields eq� ����

��� Time Evolution of the Universe

To �nd out how the universe will evolve in time one needs to know the equa�tion of state which relates the energy density to pressure� Usually energy andpressure are proportional i�e� p " �c� where " � for cold non�relativistic

��

Figure ���� Evolution of the radius of the universe for a closed �C �at �F oropen �O universe with and without in�ation� From ref� ����

matter �p " � and " ��� for a relativistic hot gas as follows from elementaryThermodynamics� From eq� ���� given above it follows immediately that

� � R�� ���� ����

and substituting this into eq� ���� results in�

R � t�

������ � �����

if we neglect the curvature term i�e� either R is large or k small� As we will seeboth are true after the in�ationary phase of the universe�

hot relativistic t � t� R � t�� � " D��R

inflation t� � t � t� R � eHt � " const

hot relativistic t� � t � t� R � t�� � " D��R

cold non� relativ t � t� R � t�� � " D��R�

Table ���� The time dependence of the scale factor and energy density duringvarious stages in the evolution of the universe� Typically t� � ���� s t� � �����

s and t� � ��� yrs� The constants Di are integration constants�

The time dependence has been summarized in table ��� for various stages ofthe universe� The in�ationary period will be discussed in the next section� Oneobserves that the scale factor vanishes at some time t " � and the energy densitybecomes in�nite at that time� This singularity explains the popular name �BigBang� theory for the evolution of the universe� The solutions for R�t are showngraphically in �g� ��� for three cases� a �at universe �k " � an open universe�k " �� and a closed universe �k " � �from ref���� �

�

An open or �at universe will expand forever since the kinetic energy is largerthan the gravitational attraction� A closed universe will recollapse� The lifetimeof a closed universe with p � ���� and a total mass M of cold non�relativisticmatter is����

tc "�MG

�� M

MP����s� �����

so the present lifetime of the universe of at least ���� yrs gives a strong upperlimit on the density of the universe�

The lifetime can easily be calculated if we assume a �at universe� from table��� it follows that R�t � t�� for most of the time� Substituting this and its timederivative into the de�nition of the Hubble constant �eq� ��� results in�

H�t "�

�t �����

With the presently accepted value of the measured Hubble constant�

H " ��� h� �km

s Mpc � h� �� ���� ��s�� � h� ����� yrs��� �����

where h indicates the experimental uncertainty ��� � h� � � one �nds for theage of the universe�

tuniverse " ���H " ����h� � ����yrs �����

The critical density which is the density corresponding to a �at universe canbe calculated from eqn� ���� by requiring Etot " � �or equivalently k " � andsubstituting for ,R�R the Hubble constant �see eq� ��� �

�c "�H�

�G" � � ����� h�� kg�m�� �����

where the numerical value of H from eq� ���� was used�The size of the observable universe the horizon distance Dh can be calculated

in the following way� the proper distance between two points is R�t d �eq� �� where d is the distance in comoving coordinates� Light propagates on the light�cone� This can be studied most easily by considering the time � in comovingcoordinates with

dt " R�t d� �����

In comoving coordinates the distance dh light can propagate is cd� so dh cRd� " c

Rdt�R�t or the proper distance Dh " R�t dh equals�

Dh " cR�t Z t�

�

dt�

R�t� �����

For most of the time R�t " at�� �see table ��� � Substituting this into eq����� yields�

Dh " �ct " �c�H�t " ��h��� � ���� m� �����

where for the lifetime t of the universe eqs� ���� and ���� were used�

��

��� Temperature Evolution of the Universe

In the previous section the scale factor and the energy density were calculatedas function of time� The energy density has two components� the energy densityfrom the photon radiation in the microwave background �rad and the energydensity of the non�relativistic matter �matter At present �rad is negligible butat the beginning of the universe it was the dominating energy� Assuming thisradiation to be in thermal equilibrium with matter implies a black body radiationwith a frequency distribution given by Planck�s law and an energy density

�rad " aT � ����

where a " ��� ������ J m��K� Since �rad � ��R�see table ��� one �nds fromeq� ����

T � ��R�t � �����

from which follows immediately� ,R�R " � ,T�T and R�� � T �� Substitutingthese expressions and eq� ��� into the second �eld equation �eq� ���� leads to�

�,T

T

��"

�aG

�c�T � �����

since the term with kc� is only proportional to T � so it can be neglected at hightemperatures� Integrating eq� ���� yields�

T "

��c�

��aG

��� �p

t" �� � ���� K �

s� s

t" �� MeV

s� s

t� �����

so the temperature drops as ��pt

From this equation one observes that about one microsecond after the BigBang the temperature has dropped from a value above the Planck temperaturecorresponding to an energy of ���� GeV to a temperature of about one GeVso after about one microsecond the temperature is already too low to generateprotons and after about one second the lightest matter particles the electronsare �frozen� out� After about three minutes the temperature is so low that thelight elements become stable and after about ��� years atoms can form�

At this moment all matter becomes neutral and the photons can escape� Theseare the photons which are still around in the form of the microwave backgroundradiation�

After the discovery of the microwave background radiation the Big Bangtheory gained widespread acceptance� Nevertheless the simplest model as for�mulated here has several serious problems which can only be solved by theso�called in�ationary models� These models have the bizarre property that theexpansion of the universe goes faster than the speed of light� This is not a contra�diction of special relativity since these regions are causally disconnected so noinformation will be transmitted� Special relativity does not restrict the velocitiesof causally disconnected objects� However such in�ationary scenarios requirethe introduction of a scalar �eld e�g� the Higgs �eld discussed in the previouschapter� For certain conditions of the potential of this �eld the gravitational

��

Figure ���� The �atness of the universe after in�ation is easily understood if onethinks about the in�ation of a balloon�

force becomes repulsive as can be derived directly from the Einstein equationsgiven above� This will be discussed in more detail after a short summary of themain problems and questions of the simple Big Bang theory�

��� Flatness Problem

At present we do not know if the universe is open or closed but experimentallythe ratio of the actual density to the critical density is bound as follows����

�� � . " ���c � � �����

The luminous matter contributes only about �� to . but from the dynamics ofthe galaxies one estimates that the galaxies contribute between ��� and ��� sothe lower limit on . stems from these observations� The upper limit is obtainedfrom the lower limit on the lifetime of the universe� From the dating of the oldeststars and the elements one knows that the universe is at least ���� years oldwhich gives an upper limit on the Hubble constant �eq� ���� and consequentlyon the density���� This does look like a perfectly acceptable number and theuniverse might even be perfectly �at since . " � is not excluded� Howeverit can be shown easily that . � � grows with time as t�� and for the presentlifetime t � ����s this number becomes big unless very special initial conditionslimit the proportionality constant to be exceedingly small� This constant can becalculated easily� From eq� ���� and the de�nition of �c �eq� ���� one �nds�

.�t � � "kc�

R��t H��t �����

Since R � t�� for the longest time of the universe �see table ��� and H�t � ��t�eq� ���� one observes that . � � � kt��� For this number to come out close

��

t

x

P

trA Ba b

Figure ��� The space�time diagram of the microwave background radiationwhich was released at the time tr� Photons observed in point P from oppositedirections traveled some ���� yrs with hardly any interactions from the points Aand B respectively� The distance light could have traveled between the Big Bangand tr is only ab which is much smaller than the distance AB� Consequently thepoints A and B could never have been in causal contact with each other� Nev�ertheless the radiation from A and B have the same temperature although thehorizon ab is much smaller �horizon problem � The problem can be solved if oneassumes the region of causal contact was much larger than ab through in�ationof space�time via a phase transition�

to zero for t very large implies that k must have been very close to zero rightfrom the beginning� Remember that k is proportional to the sum of potentialand kinetic energy in the non�relativistic approximation� One can show��� thatin order for . to lie in the range close to � now implies that in the early universej.� �j � �����M�

P�T� or for T �MP

j.� �j.

� ����� �����

In other words if the density of the initial universe was above the critical densitysay by ������c the universe would have collapsed long ago� On the other handwould the density have been below the critical density by a similar amount thepresent density in the universe would have been negligible small and life couldnot exist�

�� Horizon Problem

Since the horizon increases linearly with time but the expansion only with t��most of the presently visible universe was causally disconnected at the timet " ��� years when the microwave background was released� Nevertheless thetemperature of the microwave background radiation is the same in all directions�How did these photons thermalize after being emitted some ���� years ago� One

�

should realize that the density in the universe is exceedingly low so photonsfrom opposite directions have traveled some � � ���� lightyears without interac�tions� Since the distance scales as t�� these regions were about ��� light yearsapart at the time they were released i�e� the distance AB in �g� �� whichis two orders of magnitude larger than the horizon of the universe at that time�distance ab so no signal could have been transmitted� Nevertheless the tem�perature di�erence 'T�T between these regions is less than ���� as shown by therecent COBE data� As with the �atness problem one can impose an accidentaltemperature isotropy in the universe as an initial condition but with all the hefty�uctuations during the Big Bang this is a very unsatisfactory explanation� Aswe will see later in�ation solves both problems in a very elegant way�

��� Magnetic Monopole Problem

Magnetic monopoles are predicted by GUT�s as topological defects in the Higgs�eld� after spontaneous symmetry breaking the vacuum obtains a non�zero vac�uum expectation value in a given region� Di�erent regions may have di�erentorientations of the phases of the Higgs �eld and the borderlines of these regionshave the properties expected for magnetic monopoles����� Unfortunately themagnetic monopole density is very small if not zero� Their absence has to beexplained in any theory based on GUT�s with SSB� The �rst attack was made byAlan Guth who invented in�ation for this problem� Although the original modeldid not solve the monopole problem it provided a perfectly reasonable solutionfor the horizon and �atness problem� An alternative version of in�ation the so�called new in�ation which was invented by A�D� Linde��� and independently byAlbrecht and Steinhardt��� provided also a solution of the monopole problemas will be discussed in section �����

��� The smoothness Problem

Our universe has density inhomogeneities in the form of galaxies� On a large scalethe spectrum of inhomogeneities is approximately scale�invariant which can beunderstood in the in�ationary scenario as follows� the in�ation smoothens outany inhomogeneities which might have been present in the initial conditions�Then in the course of the phase transition inhomogeneities are generated by thequantum �uctuations of the Higgs �eld on a very small scale of length namelythe scale where quantum e�ects are important� These density �uctuations arethen enlarged to an astronomical scale by in�ation and they stay scale invariantas is obvious if one thinks about a little circle on a balloon which stays a circleafter in�ation but just on a larger scale�

��� In�ation

The deceleration in the universe is given by eq� ���� In case the energy den�sity only consists of kinetic and gravitational energy the sign of -R is negative

�

since both the pressure and energy density are positive� However the situationcan change drastically if the universe undergoes a �rst�order phase transition� InGrand Uni�ed Theories such phase transitions are expected� e�g� the highly sym�metric phase might have been an SU�� symmetric state while the less symmetricstate corresponds to the SU�� C � SU�� L � U�� Y symmetry of the StandardModel� The description of the spontaneous symmetry breaking by the Higgsmechanism leads to a speci�c picture of this phase�transition� the Higgs �eld � isa scalar �eld which �lls the vacuum with a potential energy V �� � The value ofthe potential is temperature dependent� at high temperatures the minimum oc�curs for � " � but for temperatures below the critical temperature the groundstate i�e� the state with the lowest energy is reached for a value of the �eld� " �� This is completely analogous to other phase transitions e�g� in supercon�ductivity the scalar �eld corresponds to the density of spin � Cooper pairs or inferromagnetism it would be the magnetization�

If during the expansion of the universe the energy density falls below the en�ergy density of this scalar �eld something dramatic can happen� the decelerationcan become an acceleration leading to a rapid expansion of the universe usuallycalled �in�ation��

This can be understood as follows� if the vacuum is �lled with this potentialenergy of the scalar �eld with an energy density �vac the work W done duringthe expansion is p'V � However the gain in energy is �vac'V since the potentialenergy of the vacuum does not change �a �void� stays a �void� as long as no phasetransition takes place so an increase in volume implies an increase in energy�Since no external energy is supplied the total energy of the system must stayconstant i�e� p'V # �vac'V " �� or

p " ��vac �����

This is the famous equation of state in case of a potential dominated vacuum� Inthis case eq� ���� reduces to�

-R "�

�GR�t � �����

This equation has the solution�

R�t � et� � �����

where

� "

s�

�G� ����

As we have seen in the previous chapter the symmetry breaking of a GUThappens at an energy of ���� GeV� The energy density at this energy is extremelyhigh�

u " �c� "EGUT

� hc �" ����� Jm��� �����

where the powers are derived from dimensional analysis� Inserting this result intoeq� ��� yields�

� " �����s �����

�

Thus the universe in�ates extremely rapidly� its diameter doubles every � ln �s� The behaviour of the Hubble constant during the in�ationary era is thendetermined by eq� ��� which yields�

H�t "�

� �����

Clearly the in�ationary scenario provides in an elegant way solutions for manyof the shortcomings of the BBT�

� The rapid expansion removes all curvature in space�time thus providing asolution for the �atness problem�

� After expansion the universe reheats because of the quantum �uctuationsaround the new minimum of the vacuum thus thermalizing a region muchlarger than the visible region at that time� This explains why all visibleregions in the present universe were in causal contact during the time the��� K background microwave radiation was released�

� The rapid expansion explains the absence of magnetic monopoles sinceafter su!ciently large in�ation the monopoles are diluted to a negligiblelevel�

However the whole idea of in�ation only works if one assumes the in�ation to gosmoothly from a single homogenous region to a large homogenous region manytimes the size of our universe �so�called new in�ation � This requires rather spe�cial conditions for the shape of the potential as was pointed out by Linde��� andindependently by Albrecht and Steinhardt��� after the original introduction ofin�ation by Guth���� The problem is that the corresponding scalar �elds pro�viding the potential energy of the vacuum have to be weakly interacting sinceotherwise the phase transition will involve only microscopic distances accordingto the uncertainty relation� The Higgs �elds providing spontaneous symmetrybreaking are interacting too strongly so one has to introduce additional weaklyinteracting scalar �elds� It is non�trivial to combine the requirement of a neg�ligible small cosmological constant which represents the potential energy of thevacuum with a vacuum �lled with Higgs �elds to generate masses� This re�quires large cancellations of positive and negative contributions which occur e�g�in unbroken supersymmetric theories� However these theories if they describethe real world have to be broken� Details on these problems are discussed byOlive���� and in the recent text books by B-orner��� and Kolb and Turner���� Inspite of these problems the arguments in favour of in�ation are so strong that ithas become the only acceptable paradigm of present cosmology�

���� Origin of Matter

As discussed above matter in our universe consists largely of hydrogen ���� and helium ���� �typically ���� nucleons � The absence of antimatter can beexplained if one has phase transitions and among others CP�violation as dis�cussed in chapter �� At present the nuclei dominate the energy density in the

�

Figure ���� Models including ��� hot dark matter �HDM can describe the largescale structure of the universe as probed by the galaxy surveys �QDOT andCOBE temperature anisotropy better than models with only dark matter �DM �From Schaefer and Sha� �����

universe in contrast to the �rst ��� years when the energy density of the radi�ation dominated� The reason for this change is simply the fact that the energydensity of the many photons around decreases � T while the energy density ofthe nuclei decreases � T � �see eq� ��� �

Large amounts of matter can be created from the energy release during thein�ationary phase of the universe� This can be easily estimated as follows� Afterin�ation the universe has a macroscopic size typically the size of a football orlarger� The large energy density in this volume �see eq� ���� yields a total energymany times the mass in our present universe�

Thus within the in�ationary scenario the universe could originate as a quan�tum �uctuation starting from absolute �nothing� i�e� a state devoid of spacetime and matter with a total energy equal to zero� Most matter was created afterthe in�ationary phase from the decay of the �eld quanta of the �elds responsablefor the in�ation� Of course a quantum description of space�time can be discussedonly in the context of quantum gravity so these ideas must be considered specu�lative until a theory of quantum gravity is formulated and proven by experiment�Nevertheless it is fascinating to contemplate that physical laws may determinenot only the evolution of our universe but they may remove also the need forassumptions about the initial conditions�

�

���� Dark Matter

The visible matter is clustered in large galaxies which are themselves clustered inclusters and superclusters with immense voids in between� From the movementsof the galaxies one is forced to conclude that there must be much more matterthan the observed visible matter if we want to stick to Newtonian mechanics�The most impressive evidence for the dark i�e� not visible matter comes from theso�called �at rotation curves ����� the orbital velocities of luminous matter aroundthe central of spiral galaxies remain constant out to the far edges of the galaxiesin apparent contradiction to velocity distributions expected from Keppler�s law�

v��r " GM�r

r� �����

where r is the radial distance to the centre of the galaxy M�r the mass of thegalaxy inside a sphere with radius r and G the gravitational constant� Fromthis law one expects the velocities to decrease with ��

pr if the mass is concen�

trated in the centre which is certainly the case for the visible matter� Velocitiesindependent of r imply M�r �r to be constant or M�r � r� Such a behaviouris expected for weakly interacting matter like neutrinos gravitinos or photinossince strongly interacting matter would be attracted to the centre by gravityinteract loose energy and concentrate in the centre just like the visible matterdoes� Also the dynamical properties of galaxies in large clusters require largeamounts of dark matter� To make the speeds work out consistently one has toassume that the total density of �dark� matter is an order of magnitude more thanthe visible matter� Recent reviews for the experimental evidence of dark mattercan be found in ref� �����

From the concentration of light elements as shown in �g� ��� one has toconclude that the total baryonic density is only about ��� of the critical density�see eq� ��� � The critical density is the density for a �at universe which naturallyoccurs after in�ation� Consequently in the in�ationary scenario the dark mattermakes up about ��� of the total mass in the universe and it has to be non�baryonic�

Possible candidates for dark matter are the MACHO�s which have beenobserved recently through their microlensing e�ect on the light of stars behindthem���� But since they cluster in heavy compact objects they are likely to beremnants of collapsed stars or light dwarfs which have too little mass to startnuclear burning� In these cases they would be baryonic and make up ��� of thecritical density required by baryogenesis� Note that the visible baryonic matterin stars represents at most �� of the critical density�

So one needs additional dark matter if one believes in in�ation and takes thevalue of .b " �� from baryogenesis� Candidates for this additional dark matterare neutrinos with a mass in the eV range����

� � m� � �� eV �����

Such small masses are experimentally not excluded���� but they would be rela�tivistic or �hot�� Unfortunately all dark matter cannot be relativistic since this

�Massive Compact Halo Objects�

�

is inconsistent with the extremely small anisotropy in the microwave backgroundas observed by the COBE satellite ����

This anisotropy in the temperature is proportional to the anisotropy in themass density at the time of release of this radiation shortly after the Big Bang�Through gravity the galaxies were formed around these �uctuations ��seeds� inthe mass density� So the present structure of the universe has to follow fromthe spectrum of �uctuations in the early universe which can be probed by themicrowave background anisotropy� The best �t is obtained for a mixture of ���cold and ��� hot dark matter��� as shown in �g� ��� �from ref� ���� �

The lightest supersymmetric particles are ideal candidates for cold dark mat�ter provided they are not too numerous and too heavy� Otherwise they wouldprovide a density above the critical density�� � �� in which case the universewould be closed and the lifetime would be very short �see �g� ��� � The LSP canannihilate su!ciently rapidly into fermion�antifermion pairs if the masses of theSUSY particles are not too heavy as will be discussed in chapter �

���� Summary

The Big Bang theory is remarkably successful in explaining the basic observationsof the universe i�e� the Hubble expansian the microwave background radiationand the abundance of the elements� From the measured Hubble constant one canderive such basic quantities as the the size and the age of the universe� Neverthe�less many questions remain unanswered� They can be answered by postulatingphase transitions during the evolution of the universe from the Planck tempera�ture of ���� K to the ��� K observed today� Among the questions�

� The Matter�Antimatter Asymmetry in our UniverseAs �rst spelled out by Sakharov���� any theory trying to explain the pre�ponderance of matter in our universe must necessarily implement�

� Baryon� and Lepton number violation�

� C� and CP� violation�

� Thermal non�equilibrium conditions as expected after phase transi�tions�

� The Dominance of Photons over BaryonsIf the excess of matter originates from small CP�violation e�ects mostmatter and antimatter will have enough time to annihilate into photonsthus providing an explanation why the number of photons as observed inthe �K microwave background radiation is about ��� to ���� times as highas the number of baryons in our universe�

� In�ationAn in�ationary phase i�e� a rapid expansion generated by a potentialenergy term which according to Einstein�s equations of General Relativityprovides a repulsive instead of attractive gravitational force is the onlyviable explanation to solve the following problems�

� Horizon ProblemThe fact that the observed temperature of the ��� K microwave back�ground radiation is to a very high degree the same in all directions canonly be explained if we assumes that all regions were in causal contactwith each other at the beginning� However the size of our universe islarger than the �horizon� i�e� the distance light could have traveledsince the beginning� Therefore one can only explain the temperatureisotropy if one assumes that all regions were in causal contact at thebeginning and that space expanded faster than the speed of light� Thisis indeed the case in the in�ationary scenario�

� Flatness ProblemExperimentally the observed density in our universe is close to theso�called critical density which is the density where the total energyof the universe is zero i�e� the kinetic energy of the expanding uni�verse is just compensated by the gravitational potential energy� Thiscorresponds to a ��at� universe i�e� zero curvature� The in�ationaryscenario naturally explains why the universe is so �at� the rapid ex�pansion by more than �� orders of magnitude drives all curvature tozero��

� Magnetic Monopole ProblemMagnetic monopoles are predicted by GUT�s� Their absence in ouruniverse is explained by the in�ationary models if one assumes thein�ation to go smoothly from a single homogeneous region to a largehomogeneous region many times the size of our universe �so�called newin�ation �

� The Smoothness ProblemExperimentally the cosmic background radiation shows the features inaccord with the Harrison�Zel�dovich scale�invariant spectrum �n"� ����which is the spectrum expected after in�ation� The scale invariancecan be understood as follows� the in�ation smoothens out out any in�homogeneities which might have been present in the initial conditions�Then in the course of the phase transition inhomogeneities are gener�ated by the quantum �uctuations of the Higgs �eld on a very smallscale of length namely the scale where quantum e�ects are impor�tant� These density �uctuations are then enlarged to an astronomicalscale by in�ation and they stay scale invariant as is obvious if onethinks about e�g� a little circle on a balloon which stays a circle afterin�ation but just on a larger scale�

In Grand Uni�ed Theories the conditions needed for the in�ationary Big BangTheory are naturally met� at least two phase transitions which generate massand thus provide non�equilibrium conditions are expected� one at the uni�cationscale of ���� GeV i�e� at temperature of about ���� K and one at the electroweakscale i�e� a temperature of about ���� K� Furthermore a potential energy term inthe vacuum is expected from the scalar �elds in these theories which are needed

�Just like blowing up a balloon removes all wrinkles and curvature from the surface�

�

to generate particle masses in a gauge�invariant way� In the minimal model atleast �� scalar �elds are required� Unfortunately none have been discovered sofar so little is known about the scalar sector�

Nevertheless the arguments in favour of in�ation are so strong that it hasbecome the only acceptable paradigm of present cosmology� Experimental obser�vation of scalar �elds would provide a great boost in the acceptance of the roleof scalar �elds both in cosmology and particle physics�

�

Chapter

Comparison of GUTs with

Experimental Data

In this chapter the various low energy GUT predictions are compared with data�The most restrictive constraints are the coupling constant uni�cation combinedwith the lower limits on the proton lifetime� They exclude the SM� �� ��as well as many other models��� �� �� with either a more complicated Higgssector or models in which one searches for the minimum number of new particlesrequired to ful�l the constraints mentioned above� From the many models triedonly a few yielded uni�cation at the required energies but these models haveparticles introduced ad�hoc without the appealing properties of Supersymmetry�Therefore we will concentrate here on the supersymmetric models and ask if thepredictions of the simplest i�e� minimal models��� are consistent with all theconstraints described in the previous chapters� The relevant RG equations forthe running of the couplings and the masses are given in the appendix� Assumingsoft symmetry breaking at the GUT scale all SUSY masses can be expressed interms of � parameters and the masses at low energies are then determined by thewell known Renormalization Group �RG equations� So many parameters cannotbe derived from the uni�cation condition alone However further constraints canbe considered�

� MZ predicted from electroweak symmetry breaking��� �� �� � ����

� b�quark mass predicted from the uni�cation of Yukawa couplings��� �� ����

� Constraints from the lower limit on the proton lifetime ��� � ����

� Constraints on the relic density in the universe �� ���

� Constraints on the top mass ��� �� �� ����

� Experimental lower limits on SUSY masses ��� ����

� Constraints from b� s� decays��� �� �� ���

Of course in many of the references given above several constraints are stud�ied simultaneously since considering one constraint at a time yields only onerelation between parameters� Trying to �nd complete solutions with only a few

�

constraints requires then additional assumptions like naturalness no�scale mod�els �xed ratios for gaugino� and scalar masses or a �xed ratio for the Higgsmixing parameter and the scalar mass or combinations of these assumptions�

Several ways to study the constraints simultaneously have been pursued� Onecan either sample the whole parameter space in a systematic or random way andcheck the regions which are allowed by the experimental constraints�

Alternatively one can try a statistical analysis in which all the constraintsare implemented in a �� de�nition and try to �nd the most probable region ofthe parameter space by minimizing the �� function�

In the �rst case one has to ask� which weight should one give to the variousregions of parameter space and how large is the parameter space� Some samplethe space only logarithmically thus emphasizing the low energy regions��� oth�ers provide a linear sampling��� �� �� ��� In the second case one is faced withthe di!culty that the function to be minimized is not monotonous because ofthe experimental limits on the particle masses proton lifetime relic density andso on� At the transitions where these constraints become e�ective the derivativeof the �� function is not de�ned� Fortunately good minimizing programs in mul�tidimensional parameter space which do not rely on the derivatives exist�����The advantage of such a statistical analysis is that one obtains probabilities forthe allowed regions of the parameter space and can calculate con�dence levels�The results of such an analysis���� will be presented after a short description ofthe experimental input values� Other analysis have obtained similar mass spectrafor the predicted particles in the MSSM��� �� � �� ��� or extended versionsof the MSSM�����

��� Uni�cation of the Couplings

In the SM based on the group SU�� � SU�� � U�� the couplings are de�nedas�

� " ���� g����� " � ��� cos� �W � " g���� " � sin� �W � " g�s���

���

where g� � g and gs are the U�� SU�� and SU�� coupling constants� the �rsttwo coupling constants are related to the �ne structure constant by �see �g� ��� �

e "p� " g sin �W " g� cos �W ���

The factor of ��� in the de�nition of � has been included for the proper nor�malization at the uni�cation point �see eq� ���� � The couplings when de�nedas e�ective values including loop corrections in the gauge boson propagatorsbecome energy dependent ��running� � A running coupling requires the speci��cation of a renormalization prescription for which one usually uses the modi�edminimal subtraction �MS scheme�����

In this scheme the world averaged values of the couplings at the Z� energy are

���MZ " ����� �� ���

sin� �MS " ������ ����� ���

� " ����� ��� ���

��

The value of �� is given in ref� ���� and the value of sin� �MS has been beentaken from a detailed analysis of all available data by Langacker and Polonsky����which agrees with the latest analysis of the LEP data����� The error includes theuncertainty from the top quark� We have not used the smaller error of ����� fora given value of mt since the �t was only done within the SM not the MSSM sowe prefer to use the more conservative error including the uncertainty from mt�

The � value corresponds to the value at MZ as determined from quantitiescalculated in the �Next to Leading Log Approximation������ These quantities areless sensitive to the renormalization scale which is an indicator of the unknownhigher order corrections� they are the dominant uncertainties in quantities relyingon second order QCD calculations� This s value is in excellent agreement witha preliminary value of ���� � ��� from a �t to the Z� cross sections andasymmetries measured at LEP���� for which the third order QCD correctionshave been calculated too� the renormalization scale uncertainty is correspondinglysmall�

The top quark mass was simultaneously �tted and found to be�����

Mtop " ���� ������ ��� GeV� ��

where the �rst error is statistical and the second error corresponds to a variationof the Higgs mass between � and ���� GeV� The central value corresponds to aHiggs mass of ��� GeV�

For SUSY models the dimensional reduction DR scheme is a more appro�priate renormalization scheme����� This scheme also has the advantage that allthresholds can be treated by simple step approximations� Thus uni�cation occursin the DR scheme if all three ��i �� meet exactly at one point� This crossingpoint then gives the mass of the heavy gauge bosons� The MS and DR couplingsdi�er by a small o�set

�

DRi"

�

MSi

� Ci

�����

where the Ci are the quadratic Casimir coe!cients of the group �Ci " N forSU�N and � for U�� so � stays the same � Throughout the following we usethe DR scheme for the MSSM�

��� MZ Constraint from Electroweak Symme

try Breaking

As discussed in chapter � the electroweak breaking in the MSSM is triggered bythe large negative corrections to the mass of one of the Higgs doublets����� Afterincluding the one�loop corrections to the Higgs potential�� ��� the followingexpression for MZ can be found �see appendix �

M�Z " �

m�� �m�

� tan� � �'�

Z

tan� � � �� ���

'�Z "

�g�

���m�

t

M�W

f� )m�

t� # f� )m�t� # �m�

t # �A�tm

�� � �� cot� �

f� )m�t� � f� )m�

t�

)m�t� � )m�

t�

����

��

where m� and m� are the mass parameters in the Higgs potential tan� is themixing angle between the Higgs doublets and the function f has been de�ned inthe appendix� The corrections 'Z are zero if the top� and stop quark masses areidentical i�e� if supersymmetry would be exact� They grow with the di�erence)m�t �mt

� so these corrections become unnaturally large for large values of thestop masses as will be discussed later� In addition to relation �� one �ndsfrom the minimzation of the potential a relation between tan� and m� �seeappendix so requiring electroweak breaking e�ectively reduces the original �free mass parameters to only ��

��� Evolution of the Masses

In the soft breaking term of the Lagrangian m� and m�� are the universalmasses of the gauginos and scalar particles at the GUT scale respectively and� constrains the masses of the Higgsinos� At lower energies the masses of theSUSY particles start to di�er from these universal masses due to the radiativecorrections� E�g� the coloured particles get contributions proportional to s

�

from gluon loops while the non�coloured ones get contributions depending onthe electroweak coupling constants only� The evolution of the masses is given bythe renormalization group equations��� ��� which have been summarized in theappendix� Approximate numerical mass formulae for the squarks and sleptonsmass mixing between the top quarks gauginos and the Higgs mass parametersare given in chapter ��The exact formulae can be found in the appendix�

��� Proton Lifetime Constraints

GUT�s predict proton decay and the present lower limits on the proton lifetime�p yield quite strong constraints on the GUT scale and the SUSY parameters� Asmentioned at the beginning the direct decay p � e�� via s�channel exchangerequires the GUT scale to be above ���� GeV� This is not ful�lled in the StandardModel �SM but always ful�lled in the Minimal Supersymmetric Standard Model�MSSM � Therefore we do not consider this constraint� However the decays viabox diagrams with winos and Higgsinos predict much shorter lifetimes especiallyin the preferred mode p � �K�� From the present experimental lower limit of���� yr���� for this decay mode Arnowitt and Nath���� deduce an upper limit onthe parameter B which is proportional to ���p�

B � ���� ���MH���MGUT GeV�� ����

Here MH� is the Higgsino mass which is expected to be above MGUT � else itwould induce too rapid proton decay� If MH� would become much larger thanMGUT one would enter the non�perturbative regime� Arnowitt and Nath���� givethe following acceptable range�

� � MH��MGUT � �� ����

��

To obtain a conservative upper limit on B we allow MH� to become an order ofmagnitude heavier than MGUT so we require

B � ���� ��� GeV�� ����

The uncertainties from the unknown heavy Higgs mass are large comparedwith the contributions from the �rst and third generation which contributethrough the mixing in the CKM matrix� Therefore we only consider the sec�ond order generation contribution which can be written as���� �

B " ��� ��� � sin��� �m�g�m��q ��

� GeV�� ����

One observes that the upper limit on B favours small gluino masses m�g largesquark masses m�q and small values of tan�� To ful�l this constraint requires

tan� � �� ����

for the whole parameter space and requires a minimal value of the parameter m�

in case m�� is not too large since m�g � ��m�� and m��q � m�

� # �m��� �see

eq� ���� � The constraint can always be ful�lled for very large values of m���However the �netuning constraint ��� implies m�g � ���� GeV or m�� � ���GeV� In this case eq� ��� requires m� to be above a few hundred GeV if m��

becomes of the order of ��� GeV or below as will be discussed below�

��� Top Mass Constraints

The top mass can be expressed as�

mt� " �� � Yt�t v

� sin��� � ����

where the running of the Yukawa coupling as function of t " log�M�

X

Q� is given

by�����

Yt�t "Yt�� E�t

� # Yt�� F �t ���

One observes that Yt�t becomes independent of Yt�� for large values of Yt�� implying an upper limit on the top mass��� ���� Requiring electroweak symmetrybreaking implies a minimal value of the top Yukawa coupling typically Yt�� O����� � In this case the term Yt�� F �t in the denominator of �� is muchlarger than one since F �t � ��� at the weak scale where t � � In this caseYt�t " E�t �F �t so from eq� ��� it follows�

m�t "

�� � E�t

F �t v� sin��� � ���� GeV � sin��� � ����

where E and F are functions of the couplings only�see appendix � The physical�pole mass is about � larger than the running mass��� �����

Mpolet " mt

�� #

�

�

s

� ���� GeV sin�� ����

��

The electroweak breaking conditions require �� � � � �� �eq� ���� � hencethe equation above implies for the MSSM approximately�

��� � Mpolet � ��� GeV� ����

which is consistent with the experimental value of � GeV as determined atLEP �see eq� � � Although the latter value was determined from a �t using theSM one does not expect shifts outside the errors if the �t would be made forthe MSSM�

As will be shown in chapter for such large top masses the b�quark mass be�comes a sensitive function of mt and of the starting values of the gauge couplingsat MGUT �

��� bquark Mass Constraint

As discussed in chapter � the masses of the up�type quarks are arbitrary in theSU�� model but the masses of the down�type quarks are related to the leptonmasses within a generation if one assumes uni�cation of the Yukawa couplings atthe GUT scale� This does not work for the light quarks but the ratio of b�quarkand � �lepton masses can be correctly predicted by the radiative corrections tothe masses��� ����

To calculate the experimentally observed mass ratio the second order renor�malization group equations for the running masses have to be used� These equa�tions are integrated between the value of the physical mass and MGUT �

For the running mass of the b�quark we used�����

mb " ���� �� GeV ����

This mass depends on the choice of scale and the value of s�mb � Consequentlywe have assigned a rather conservative error of ��� GeV instead of the proposedvalue of ��� GeV����� Note that the running mass �in the MS scheme is relatedto the physical �pole mass Mpole

b by������

mb " Mpoleb

��� �

�

s� ����

s �� ���� Mpole

b � ����

so mb " ��� corresponds to Mpoleb � � GeV� We ignore the running of m� below

mb and use for the pole mass� M� " ������ ����� GeV������

�� Dark Matter Constraint

As discussed in chapter � there is abundant evidence for the existence of non�relativistic neutral non�baryonic dark matter in our universe� The lightest su�persymmetric particle �LSP is supposedly stable and would be an ideal candidatefor dark matter�

The present lifetime of the universe is at least ���� years which implies anupper limit on the expansion rate �see eq� ���� and correspondingly on the total

��

relic abundance �compare eq� ���� � Assuming h� � �� one �nds that for eachrelic particle species �����

. h�� � � ����

This bound can only be obeyed if most of the LSP�s annihilated into fermion�antifermion pairs which in turn would annihilate into photons again� As will beshown below the LSP is most likely a gaugino�like neutralino ��� In this casethe annihilation rate ���� � ff depends most sensitively on the mass of thelightest �t�channel exchanged sfermion� . h

�� � m

f�m� ������ Consequently the

upper limit on the relic density implies an upper limit on the sfermion mass�However as discussed in chapter � the neutralinos are mixtures of gauginos andhiggsinos� The higgsino component also allows s�channel exchange of the Z�

and Higgs bosons� The size of the Higgsino component depends on the relativesizes of the elements in the mixing matrix ���� especially on the mixing angletan � and the size of the parameter � in comparison to M� � ��m�� and M� ���m��� Consequently the relic density is a complicated function of the SUSYparameters especially if one takes into account the resonances and thresholdsin the annihilation cross sections����� but in general one �nds a large regionin parameter space where the universe is not overclosed���� In the preferredgaugino�like neutralino region the relic density constraint translates into an upperbound of about ���� GeV on m� ��� except for large m�� where some SUSYmasses become much larger than � TeV and are therefore disfavoured by the�ne�tuning criterion �see eq� ��� �

��� Experimental lower Limits on SUSYMasses

SUSY particles have not been found so far and from the searches at LEP oneknows that the lower limit on the charged leptons and charginos is about halfthe Z� mass ��� GeV ���� and the Higgs mass has to be above � GeV����� Thelower limit on the lightest neutralino is ���� GeV���� while the sneutrinos haveto be above �� GeV����� These limits require minimal values for the SUSY massparameters�

There exist also limits on squark and gluino masses from the hadron colliders����but these limits depend on the assumed decay modes� Furthermore if one takesthe limits given above into account the constraints from the limits of all otherparticles are usually ful�lled so they do not provide additional reductions of theparameter space in case of the minimal SUSY model�

��� Decay b� s�

Recently CLEO has published an upper bound for this transition b � s� ��� � ��������� Furthermore a central value of �� � ��� and a lower limit of������ can be extracted from the observed process B � K������� and assumingthat the branching ratio for this process is ��� �using lattice calculations ������

In the SM the transition b� s� can happen through one�loop diagrams with aquark �charge ��� and a charged gauge boson� SUSY allows for additional loops

��

involving a charged Higgs and the charginos and neutralinos��� �� �� ����In SUSY large cancellations occur since the W � t and H� � t loops have anopposite sign as compared to the ��� )t loop and all loops are of the same orderof magnitude�

Kane et al���� �nd acceptable rates for the b� s� transition in the MSSM fora large range of parameter space even if they include constraints from electroweaksymmetry breaking and uni�cation of gauge and Yukawa couplings� They donot �nd as strong lower limits on the charged Higgs boson masses as others�����Similar conclusions were reached by Borzumati����� who used the more completecalculations including the �avour changing neutral currents����� It turns outthat with the present errors the combination of all constraints discussed aboveare more restrictive than the limits on b� s� so we have not included it in theanalyses discussed below�

��� Fit Strategy

As mentioned before given the �ve parameters in the MSSM plus GUT andMGUT all other SUSY masses the b�quark mass and MZ can be calculated byperforming the complete evolution of the couplings including all thresholds�

The proton lifetime prefers small values of tan � �eq� ��� while all SUSYmasses are expected to be below � TeV from the �ne�tuning argument �see eq���� �

Therefore the following strategy was adopted� m� and m�� were varied be�tween � and ���� GeV and tan � between � and ��� The trilinear coupling At�� at MGUT was kept mostly at zero but the large radiative corrections to it weretaken into account so at lower energies it is unequal zero� Varying At�� between#�m� and ��m� did not change the results signi�cantly so the following resultsare quoted for At�� �

The remaining four parameters � GUT � MGUT � �� and Yt�� � were �ttedwith the MINUIT program���� by minimizing the following �� function�

�� "�Xi�

� ��i �MZ � ��MSSMi�MZ

�

��i

#�MZ � ���� �

��Z

#�mb � ��� �

��b

#�B � ��� �

��B�for B � ���

#�D�m�m�m� �

��D�for D � �

#� )M � )Mexp

�

���M�for )M � )Mexp

�

The �rst term is the contribution of the di�erence between the three calculatedand measured gauge coupling constants at MZ and the following two terms arethe contributions from the MZ�mass and mb�mass constraints� The last threeterms impose constraints from the proton lifetime limits from electroweak sym�metry breaking i�e� D " VH�v�� v� � VH��� � � � �see eq� ���� and fromexperimental lower limits on the SUSY masses� The top mass or equivalentlythe top Yukawa coupling enters sensitively into the calculation of mb and MZ �Instead of the top Yukawa coupling one could have taken the top mass as aparameter� However if the couplings are evolved from MGUT downwards it ismore convenient to run also the Yukawa coupling downward since the RGE ofthe gauge and Yukawa couplings form a set of coupled di�erential equations insecond order �see appendix � Once the Yukawa coupling is known at MGUT thetop mass can be calculated at any scale� The top mass can be taken as an inputparameter too using the value from the LEP data� Unfortunately the value fromthe LEP data �eq� ���� is not yet very precise compared with the range expectedin the MSSM �eq� ��� so it does not provide a sensitive constraint� Instead weprefer to �t the Yukawa coupling in order to obtain the most probable top massin the MSSM� As it turns out the resulting parameter space from the minimiza�tion of this �� includes the space allowed by the dark matter and the b � s�constraints which have been discussed above�

The following errors were attributed� �i are the experimental errors in thecoupling constants as given above �b"��� GeV �B"���� GeV while �D and� �M were set to �� GeV� The values of the latter errors are not critical since thecorresponding terms in the numerator are zero in case of a good �t and even forthe ��� C�L� limits these constraints could be ful�lled and the �� was determinedby the other terms for which one knows the errors�

For uni�cation in the DR scheme all three couplings ��i �� must cross ata single uni�cation point MGUT ������ Thus in these models one can �t the cou�plings at MZ by extrapolating from a single starting point at MGUT back toMZ for each of the i�s and taking into account all light thresholds� The �ttingprogram���� will then adjust the starting values of the four high energy parame�ters �MGUT � GUT � � and Yt�� until the �ve low energy values �three couplingconstants MZ and mb are �hit�� The �t is repeated for all values of m� andm�� between � and ���� GeV and tan � between � and ��� Alternatively �tswere performed in which m�� was left free too�

The light thresholds are taken into account by changing the coe!cients ofthe RGE at the value Q " mi where the threshold masses mi are obtainedfrom the analytical solutions of the corresponding RGE �see section ��� � Thesesolutions depend on the integration range which was chosen between mi andMGUT � However since one does not know mi at the beginning an iterativeprocedure has to be used� one �rst usesMZ as a lower integration limit calculatesmi and uses this as lower limit in the next iteration� Of course since the couplingconstants are running the latter have to be iterated too so the values of i�mi have to be used for calculating the mass at the scale mi��� ����� Usually threeiterations are enough to �nd a stable solution�

Following Ellis Kelley and Nanopoulos�� the possible e�ects from heavythresholds are set to zero since proton lifetime forbids Higgs triplet masses to

��

be below MGUT �see eq� ��� � These heavy thresholds have been considered byother authors for di�erent assumptions���� �� �����

SUSY particles in�uence the evolution only through their appearance in theloops so they enter only in higher order� Therefore it is su!cient to considerthe loop corrections to the masses in �rst order in which case simple analyticalsolutions can be found even if the one�loop correction to the Higgs potentialfrom the top Yukawa coupling is taken into account �see appendix � There is oneexception� the corrections to the bottom and tau mass are compared directlywith data which implies that the second order solutions have to be taken for theRGE predicting the ratio of the bottom and tau mass� Since this ratio involvesthe top Yukawa coupling Yt the RGE for Yt has to be considered in second ordertoo� These second order corrections are important for the bottom mass since thestrong coupling constant becomes large at the small scale of the bottom massi�e� s�mb � ���

In total one has to solve a system of �� coupled di�erential equations� �second order ones�for the � gauge couplings Yt and Yb�Y� and �� �rst orderones �for the masses and parameters in the Higgs sector see appendix � Thesecond order ones are solved numerically� taking into account the thresholds ofthe light particles using the iteration procedure discussed above� Note that fromthe starting values of all parameters atMGUT one can calculate all light thresholdsfrom the simple �rst order equations before one starts the numerical integrationof the �ve second order equations� Consequently the program is fast in �ndingthe optimum solution even if before each iteration the light thresholds have tobe recalculated�

���� Results

The upper part of �g� �� shows the evolution of the coupling constants inthe MSSM for two cases� one for the minimum value of the �� function givenin eq� ��� �solid lines and one corresponding to the ��� C�L� upper limitof the thresholds of the light SUSY particles �dashed lines � The position ofthe light thresholds is shown in the bottom part as jumps in the �rst order �coe!cients which are increased according to the entries in table A�� as soonas a new threshold is passed� Also the second order coe!cients are changedcorrespondingly �see table A�� but their e�ect on the evolution is not visiblein the top �gure in contrast to the �rst order e�ects which change the slope ofthe lines considerably in the top �gure� One observes that the changes in thecoupling constants occur in a rather narrow energy regime so qualitatively thispicture is very similar to �g� ��� in which case all sparticles were assumed tobe degenerate at an e�ective SUSY mass scale MSUSY ����� Since the runningof the couplings depends only logarithmically on the sparticle masses the ���C�L� upper limits are as large as several TeV as shown by the dashed lines in�g� �� and more quantitatively in table ��� In this table the initial choices ofm� and tan� as well as the �tted parameters GUT � MGUT � m��� �� Yt�� �and

�The program DDEQMR from the CERN library was used for the solution of these coupledsecond order di�erential equations�

��

the corresponding top mass after running down Yt from MGUT to mt are shownat the top and given these parameters the corresponding masses of the SUSYparticles can be calculated� Their values are given in the lower part of the table�Note we �tted here �ve parameters with �ve constraints so the ��"� if a goodsolution can be found� This is indeed the case� The upper and lower limits intable �� will be discussed below�

Only the value of the top Yukawa coupling is given since for the ratio ofbottom and tau mass only the ratio of the Yukawa couplings enters not their ab�solute values� For the running of the gauge couplings and the mixing in the quarksector only the small contribution from the top Yukawa coupling is taken intoaccount since for the range of tan� considered all other Yukawa contributionsare negligible�

As mentioned before varying At�� between #�m� and ��m� does not in��uence the results very much so its value at the uni�cation scale was kept at� but its non�zero value at lower energies due to the large radiative correctionswas taken into account� The �ts are shown for positive values of the Higgs mix�ing paramter � but similar values are obtained for negative values of � with anequally good �� value for the �t�

The parameters m�� m�� and � are correlated as shown in �g� �� wherethe value of � is shown for all combinations of m� and m�� between ��� and ����GeV� One observes that � increases with increasing m� and m��� The strongcorrelation between m�� and � originates mainly from the electroweak symmetrybreaking condition but also from the fact that the thresholds in the running ofthe gauge couplings all have to occur at a similar scale� For example from �g��� it is obvious that the dashed lines for �� � and �� � will not meet with thesolid line of �� � simply because the thresholds are too di�erent� the thresholdsin �� � are mainly determined by m�� while the thresholds for the upper twolines include the winos and higgsinos too so one obtains automatically a positivecorrelation between � and m���

The �� value is acceptable in the whole region except for the regions whereeither m� or m�� or both become very small as shown in �g� ��� The increasein this corner is completely due to the constraint from the proton lifetime �seesection �� � This plot was made for tan� " �� For larger values the regionexcluded by proton decay quickly increases� for tan� " �� practically the wholeregion is excluded�

One notices from �g� �� already a strong correlation between � and m���This is explicitly shown in �g� ��� The steep walls originate from the ex�perimental lower limits on the SUSY masses and the requirement of radiativesymmetry breaking� In the minimum the �� value is zero but one notices a longvalley where the �� is only slowly increasing� Consequently the upper limits onthe sparticle masses which grow with increasing values of � and m�� becomeseveral TeV as shown in table ��� The ��� C�L� upper limits were obtained byrequiring an increase in �� of ���� so the rather The upper limits are a sensitivefunction of the central value of s� decreasing the central value of s by twostandard deviations �i�e� ����� can increase the thresholds of sparticles severalTeV� Acceptable �ts are only obtained for input s values between ����� and����� if the error is kept at ����� Outside this range all requirements cannot be

��

met simultaneously any more so the MSSM predicts s in this range�As discussed previously sparticle masses in the TeV range spoil the cancella�

tion of the quadratic divergences� This can be seen explicitly in the correctionsto MZ � 'Z is exactly zero if the masses of stop� and top quarks are identicalbut the corrections grow quickly if the degeneracy is removed as shown in �g���� For the SUSY masses at the minimum value of �� the corrections to MZ aresmall� If one requires that only solutions are allowed for which the correctionsto MZ are not large compared with MZ itself one has to limit the mass of theheaviest stop quark to about one TeV� The corresponding ��� C�L� upper limitsof the individual sparticles masses are given in the right hand column of table��� The correction to MZ is times MZ in this case� The limits are obtainedby scanning m� and m�� till the �

� value increases by ��� while optimizing thevalues of tan�� �� � GUT � Yt�� and MGUT � The lower limits on the SUSY pa�rameters are shown in the left column of table ��� The lowest values of m� " ��GeV and m��"�� GeV are required to have simultaneously a sneutrino massabove �� GeV and a wino mass above �� GeV� If the proton lifetime is includedthe minimum value of either m� or m�� have to increase �see �g� �� � Since thesquarks and gauginos are much more sensitive to m�� one obtains the lower lim�its by increasing m�� The minimum value for m� is about ��� GeV in this case�But in both cases the �� increase for the lower limits is due to the b�mass whichis predicted to be �� GeV from the parameters determining the lower limits�

The b�quark mass is a strong function of both tan� and mt as shown in �g��� this dependence originates from the W � t loop to the bottom quark� Thehorizontal band corresponds to the mass of the b�quark after QCD corrections�mb " ��� � �� GeV �see eq� ��� � Since also MZ is a strong function ofthe same parameters the requirement of gauge and Yukawa coupling uni�cationtogether with electroweak symmetry breaking strongly constrains the SUSY par�ticle spectrum� A typical �t with a �� equal zero is given in the central columnof table �� but is should be noted that the values in the other columns provideacceptable �ts too at the ��� C�L��

The mass of the lightest Higgs particle called h in table �� is a rather strongfunction of mt as shown in �g� �� for various choices of tan� m� and m��� Allother parameters were optimized for these inputs and after the �t the values ofthe Higgs and top mass were calculated and plotted� One observes that the massof the lightest Higgs particle varies between � and ��� GeV and the top massbetween ��� and ��� GeV� Furthermore it is evident that tan� almost uniquelydetermines the value of mt since even if m�� and m� are varied between ��� and���� GeV one �nds the practically the same mt for a given tan� and the valueof mt varies between ��� and ��� GeV if tan � is varied between ��� and �� Thisrange is in excellent agreement with the estimates given in eq� ��� if one takesinto account that Mpole

t � ��mt �see eq� ��� �Note the strong correlation between tan� and mt in �g� ��� for a given value

of tan � mt is constrained to a vary narrow range almost independent of m��

and m�� Furthermore one observes a rather strong positive correlation betweenmhiggs and all other parameters �tan�� m�� and m�� originating from theloop corrections to the potential�

In summary the following parameter ranges are allowed �if the limit on proton

��

lifetime is obeyed and extreme �netuning is to be avoided i�e� )mt� � � TeV �

��� � m� � ���� GeV

�� � m�� � ��� GeV

��� � � � ���� GeV

� � tan � � ��

��� � mt � ��� GeV �from �g� ���

���� � s � ����

The corresponding constraints on the SUSY masses are �see table �� fordetails �

�� � ����)� � ��� GeV

�� � ���� )Z � ��� � )W � �� GeV

��� � )g � ���� GeV

��� � )q � ���� GeV

��� � )t� � ��� GeV

��� � )t� � ���� GeV

�� � )eL � ��� GeV

��� � )eR � ��� GeV

��� � )�L � �� GeV

��� � ���� )H� � ��� GeV

��� � ��� )H� � ��� GeV

��� � ��� � )H� � ��� GeV

��� � H� � ���� GeV

��� � H � ���� GeV

��� � A � ���� GeV

� � h � ��� GeV �from �g� ���

The lower limits will all increase as soon as the LEP limits on sneutrinoswinos and the lightest Higgs increase�

The lightest Higgs particle is certainly within reach of experiments at presentor future accelerators���� ����� Its observation in the predicted mass range of� to ��� GeV would be a strong case in support of this minimal version of asupersymmetric grand uni�ed theory�

��

10log Q

1/α i

90% C.L. upper limit

β1

β2

β3

β THRESHOLDS

10log Q

βi

0

10

20

30

40

50

60

0 5 10 15

-10-8-6-4-202468

10

0 5 10 15

Figure ��� Evolution of the inverse of the three couplings in the MSSM� The lineabove MGUT follows the prediction from the supersymmetric SU�� model� TheSUSY thresholds have been indicated in the lower part of the curve� they aretreated as step functions in the �rst order � coe!cients in the renormalizationgroup equations which correspond to a change in slope in the evolution of thecouplings in the top �gure� The dashed lines correspond to the ��� C�L� upperlimit for the SUSY thresholds�

��

m 0 [G

eV]m

1/2 [GeV]

μ [G

eV]

100200

300400

500600

700800

9001000

100200

300400

500600

700800

9001000

500

1000

1500

2000

2500

Figure ��� The �tted MSSM parameter � as function of m� and m�� for tan � "��

��

m 0 [G

eV]m

1/2 [GeV]

χ2

100200

300400

500600

700800

9001000

100200

300400

500600

700800

9001000

0.2

0.4

0.6

0.8

1

Figure ��� The �� of the �t as function of m� and m��for tan� " �� The sharpincrease in �� in the corner is caused by the lower limit on the proton lifetime�

��

χ2

m1/2 [GeV]

μ [GeV

]

1000

1500

2000

2500

3000

3500

4000

500

1000

1500

2000

2500

3000

3500

0.5

1

1.5

2

2.5

3

Figure ��� The correlation between m�� and � for m�"��� GeV�

��

m 0 [G

eV]m

1/2 [GeV]

corr

. MZ in

%

100200

300400

500600

700800

9001000

100200

300400

500600

700800

9001000

200

400

600

800

1000

1200

1400

Figure ��� The one�loop correction factor to MZ as function of m� and m���

�

mt [GeV]

mb [G

eV]

tan β = 10.

tan β = 1.2

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

100 110 120 130 140 150 160 170 180 190 200

Figure �� The correlation between mb and mt for m�"��� GeV and two valuesof tan �� The hatched area indicates the experimental value for mb�

��

mtop [GeV]

mhi

ggs [

GeV

]m0 = 100 [GeV]

m0 = 1000 [GeV]

m1/2 = 100 GeV

1000 GeV

tanβ = 1.2

2.0

5.0

50

60

70

80

90

100

110

120

130

140

150

160

130 140 150 160 170 180 190 200

Figure ��� The mass of the lightest Higgs particle as function of the top quarkmass for values of tan � between ��� and � and values of m� and m�� between��� and ���� GeV� The parameters of �� MGUT � GUT and Yt�� are optimizedfor each choice of these parameters� the corresponding values of the top andlightest Higgs mass are shown as symbols� For small values of m�� the Higgsmass increases with m� as shown for a �string� of points each representing astep of ��� GeV in m� for a given value of m�� which is increasing in stepsof ��� GeV starting with the low values for the lowest strings� At high valuesof m�� the value of m� becomes irrelevant and the �string� shrinks to a point�Note the strong positive correlation between mhiggs and all other parameters�the highest value of the Higgs mass corresponds to the maximum values of theinput parameters i�e� tan� " � m� " m�� " ���� GeV� this value does notcorrespond to the minimum ��� More likely values correspond to mhiggs � ��GeV for m�� " ��� m� " ��� and tan� " ��

��

Symbol Lower limits Typical �t ��� C�L�Upper limits

Constraints GEY GEY�P GEY��PF� GEY� �P� GEY��P��F

Fitted SUSY parameters

m� � �� ��� �� ��

m�� �� �� ��� ���� ��

� ��� ��� �� � ����

tan� ��� ��� �� ��� ���

Yt��� ���� � ������ ������ ������ �����

mt �� ��� � ��� ���

���GUT � �� � �� ���� ���� ����

MGUT ��� ���� ��� ���� ��� ���� ��� ���� ��� ����

SUSY masses in �GeV�

������� �� �� �� ��� ���

�����Z� �� �� � �� � ���

��� � �W � � � �� � ���

�g ��� ��� �� ���� ����

�eL �� �� ��� ���� ���

�eR �� �� ��� ��� �

�L � �� ��� ���� ���

�qL ��� � � ���� ����

�qR ��� � � � ���� ����

�bL ��� ��� � ���� ���

�bR ��� � � � ���� ����

�t� ��� � � �� ���� ���

�t� ��� � ��� ���� ����

�����H�� ��� ��� ��� ���� ���

����H�� ��� ��� �� ���� ���

��� � �H�� ��� ��� ��� ���� ���

h ��� �� � � � ���

H � � ��� �� ���� ����

A � � ��� � ���� ����

H� ��� ��� � ���� ���

Table ��� Values of SUSY masses and parameters for various constraints�G"gauge coupling uni�cation� E"electroweak symmetry breaking� Y"Yukawacoupling uni�cation� P"Proton lifetime constraint� F"�netuning constraint�Constraints in brackets indicate that they are ful�lled but not required� Thevalue of the lightest Higgs h can be lower than indicated �see text �

��

Chapter �

Summary�

Many of the questions posed by cosmology suggest phase transitions during theevolution of the universe from the Planck temperature of ���� K to the ��� Kobserved today� Among them the baryon asymmetry in our universe and in��ation which is the only viable solution to explain the horizon problem the�atness problem the magnetic monopole problem and the smoothness problem�see chapter � �

In Grand Uni�ed Theories �GUT phase transitions are expected� one at theuni�cation scale of ���� GeV i�e� at a temperature of about ���� K and one atthe electroweak scale i�e� at a temperature of about ��� K� Furthermore scalar�elds which are a prerequisite for in�ation are included in GUT�s� In the min�imal model at least �� scalar �elds are required� Unfortunately none have beendiscovered so far so little is known about the scalar sector although the veri��cation of the relation between the couplings and the masses of the electroweakgauge bosons indeed are indirect evidence that their mass is generated by theinteraction with a scalar �eld� Experimental observation of these scalar �eldswould provide a great boost for cosmology and particle physics� First estimatesof the required mass spectra of the scalar �elds can be obtained by comparingthe experimental consequences of Grand Uni�ed Theories �GUT with low energyphenomenology�

One of the interesting �discoveries� of LEP was the fact that within the Stan�dard Model �SM uni�cation of the gauge couplings could be excluded �see �g���� � In contrast the minimal supersymmetric extension of the SM �MSSM provided perfect uni�cation� This observation boosted the interest in Supersym�metry enormously especially since the MSSM was not �designed� to provideuni�cation but it was invented many years ago and turned out to have veryinteresting properties�

� Supersymmetry automatically provides gravitational interactions thus pavingthe road for a �Theory Of Everything��

� The symmetry between bosons and fermions alleviates the divergences inthe radiative corrections in which case these corrections can be made re�sponsible for the electroweak symmetry breaking at a much lower scale thanthe GUT scale�

��

� The lightest supersymmetric partner �LSP is a natural candidate for non�relativistic dark matter in our universe�

Other non�supersymmetric models can yield uni�cation toobut they do not ex�hibit the elegant symmetry properties of supersymmetry they o�er no explana�tion for dark matter and no explanation for the electroweak symmetry breaking�Furthermore the quadratic divergences in the radiative corrections do not cancel�

The Minimal Supersymmetric Standard Model �MSSM model has many pre�dictions which can be compared with experiment even in the energy range wherethe predicted SUSY particles are out of reach� Among these predictions�

� MZ �

� mb�

� Proton decay�

� Dark matter�

It is surprising that in addition to the uni�cation of the coupling c onstants theminimal supersymmetric model can ful�l all experimental constraints from thesepredictions� As far as we know supersymmetric models are the only ones whichare consistent with all these observations simultaneously� Within the MSSM theevolution of the universe can be traced back to about ����� seconds after the(bang� as sketched in �g� ���� If we believe in the in�ationary scenario eventhe actual creation of the universe is describable by physical laws� In this viewthe universe would originate as a quantum �uctuation starting from absolute�nothing� i�e� a state devoid of space time and matter with a total energy equalto zero� Indeed estimates of the total positive non�gravitational energy andnegative potential energy are about equal in our universe i�e� according to thisview the universe is the ultimate �free lunch�� All this mass was generated fromthe potential energy of the vacuum which also caused the in�ationary phase�

Of course a quantum description of space�time can be discussed only in thecontext of quantum gravity so these ideas must be considered speculative untila renormalizable theory of quantum gravity is formulated and proven by ex�periment� Nevertheless it is fascinating to contemplate that physical laws maydetermine not only the evolution of our universe but they may remove also theneed for assumptions about the initial conditions�

From the experimental constraints at low energies the mass spectra for theSUSY particles can be predicted �see table �� in the previous chapter � Thelightest Higgs particle is certainly within reach of experiments at present or futureaccelerators� Its observation in the predicted mass range of � to ��� GeV wouldbe a strong case in support of this minimal version of the supersymmetric granduni�ed theory� Discovering also the heavier SUSY particles implies that theknown strong electromagnetic and weak forces were all uni�ed into a single�primeval� force during the birth of our universe� Future experiments will tell�

��

SU(3)

SU(2)

U(1)

QED

Figure ���� Possible evolution of the radius of the universe and the couplingconstants� Before t " ����� s spontaneous symmetry breaking occurs whichbreaks the symmetry of the GUT into the well known symmetries at low energies�In the mean time the universe in�ates to a size far above the distance light couldhave traveled as indicated by the dashed line� From ������

��

Acknowledgments�

I want to thank sincerely Ugo Amaldi Hermann F-urstenau Ralf Ehret andDmitri Kazakov for their close collaboration in this exciting �eld� Without theirenthusiasm work and sharing of ideas many of our common results presentedin this review would not have been available� Furthermore I thank John El�lis Gian Giudice Howie Haber Gordy Kane Stavros Katsanevas Sergey Ko�valenko Hans K-uhn Jorge Lopez Dimitri Nanopoulos Pran Nath Dick RobertsLeszek Roszkowski Mikhail Shaposhnikov William Trischuk and Fabio Zwirnerfor helpful discussions and�or commenting parts of the manuscript�

Last but not least I want to thank Prof� Faessler for inviting me to a seminarin T-ubingen and his encouragement to write down the results presented there inthis review�

��

Appendix A

A�� Introduction

In this appendix all the Renormalization Group Equations �RGE for the evo�lution of the masses and the couplings are given� SUSY particles in�uence theevolution only through their appearance in the loops so they enter only in higherorder� Therefore it is su!cient to consider the loop corrections to the masses onlyin �rst order in which case a simple analytical solution can be found even if theone�loop correction to the Higgs potential from the top Yukawa coupling is takeninto account� There is one exception� the corrections to the bottom and tau massare compared directly with data which implies that the second order solutionshave to be taken for the RGE predicting the ratio of the bottom and tau mass�Since this ratio involves the top Yukawa coupling Yt the RGE for Yt has to beconsidered in second order too� These second order corrections are important forthe bottom mass since the strong coupling constant becomes large at the smallscale of the bottom mass i�e� s�mb � ���

So in total one has to solve a system of �� coupled di�erential equations ��second order �� �rst order �

� � second order equations for the running of the gauge coupling constants i� i " �� ��

� � second order equations for the running of the top Yukawa coupling Yt andthe ratio of bottom and tau Yukawa coupling Rb� �

� � �rst order equation for the masses of the left�handed doublet of an u�typeand d�type squark pair Q�

� � �rst order equation for the masses of the right�handed up�type squarksU �

� � �rst order equation for the masses of the right�handed down�type squarksD�

� � �rst order equation for the masses of the left�handed doublet of sleptonsL�

� � �rst order equation for the masses of the right�handed singlet of a chargedlepton E�

� � �rst order equations for the � mass parameters of the Higgs potential�m�� m�� m�� and � �

� � �rst order equations for the �Majorana masses of the gauginos �M��M�

and M� �

� � �rst order equation for the trilinear coupling between left� and righthanded squarks and the Higgs �eld At where the subscript indicates thatone only considers this coupling for the third generation�

��

Note that the absolute values of the bottom and tau Yukawa couplings need notto be known if one neglects their small contribution to the running of the gaugecouplings� If one wants to include these one has to integrate the RGE for Yb andY� separately �they are given below too instead of the RGE for their ratio only�Integrating the ratio has the advantage that the boundary condition at MGUT isknown to be one if one assumes Yukawa coupling uni�cation�

The particle masses are related directly to the Yukawa couplings�

Yt�mt "h�t

�� �� mt " ht�mt v sin � �A��

Yb�mb "h�b

�� �� mb " hb�mb v cos � �A��

Y� �m� "h��

�� �� m� " h� �m� v cos � �A��

It follows thatYtYb

"m�

t

m�b

�

tan� � �A��

For tan� � �� as required by proton decay limits one observes that Yb is at leastan order of magnitude smaller than Yt for the values of mt considered� Henceits contribution is indeed negligible in the running of the gauge couplings sobelow we will only consider the contribution of Yb in the ratio of Yb�Y� which isindependent of the absolute value of Yb�� �

Below we collect all the RGE�s in a coherent notation and consider the coef��cients for the various threshold regions i�e� virtual particles with mass mi areconsidered to contribute to the running of the gauge coupling constants e�ec�tively only for Q values above mi� Thus the thresholds are treated as simple stepfunctions in the coe!cients of the RGE�

Furthermore all �rst order solutions are given in an analytical form includingthe corrections from the top Yukawa coupling������ Note that a more demandinganalysis requires a numerical solution of the �rst �ve second order equations inwhich the coe!cients are changed according to the thresholds found as analyticalsolutions of the �rst order equations for the evolution of the masses� The resultsgiven in chapter all use the numerical solution of these second order equation��

Using the supergravity inspired breaking terms which assume a common massm�� for the gauginos and another common mass m� for the scalars leads to thefollowing breaking term in the Lagrangian�

LBreaking " �m��

Xi

j�ij� �m��

X�

���� �A��

� Am�

hhuabQaU

cbH� # hdabQaD

cbH� # heabLaE

cbH�

i�Bm� ��H�H�� �A�

Here

�The program DDEQMR from the CERN library was used for the solution of these coupledsecond order di�erential equations�

��

hu�d�eab are the Yukawa couplings a� b " �� �� � run over the generations

Qa are the SU�� doublet quark �elds

U ca are the SU�� singlet charge�conjugated up�quark �elds

Dcb are the SU�� singlet charge�conjugated down�quark �elds

La are the SU�� doublet lepton �elds

Eca are the SU�� singlet charge�conjugated lepton �elds

H��� are the SU�� doublet Higgs �elds

�i are all scalar �elds

�� are the gaugino �eldsThe last two terms in LBreaking originate from the cubic and quadratic terms

in the superpotential with A B and � as free parameters� In total we now havethree couplings i and �ve mass parameters�

m�� m��� ��t � A�t � B�t

with the following boundary conditions at MGUT �t " � �

scalars � )m�Q " )m�

U " )m�D " )m�

L " )m�E " m�

�� �A��

gauginos � Mi " m��� i " �� �� �� �A��

couplings � ) i�� " ) GUT � i " �� �� � �A��

Here M� M� and M� are the gauginos masses of the U�� SU�� and SU�� groups�In N " � supergravity one expects at the Planck scale B " A� ��

With these parameters and the initial conditions at the GUT scale the massesof all SUSY particles can be calculated via the renormalization group equations�

A�� Gauge Couplings

The following de�nitions are used�

) i " i�

�A���

t " ln�M�

GUT

Q� �A���

�i " bi ) GUT �A���

fi�t "�

�i

��� �

�� # �it �

��A���

hi�t "t

�� # �it � �A���

where i �i"�� denote the three gauge coupling constants of U�� SU�� andSU�� respectively GUT is the common gauge coupling at the GUT scale MGUT and biare the coe!cients of the RGE as de�ned below�

�

The second order RGE�s for the gauge couplings including the e�ect of theYukawa couplings are��� �� ����

d) idt

" �bi ) �i � ) �i

��X

j

bij ) j � aiYt

�A � �A���

where a� "���� a� " � a� " � for SUSY and a� "

����� a� "

��� a� " � for the SM�

The �rst order coe!cients for the SM are������

bi "

�BBB�b�

b�

b�

�CCCA "

�BBB�

�

��������

�CCCA#NFam

�BBB�

���

���

���

�CCCA#NHiggs

�BBB�

����

��

�

�CCCA � �A��

while for the supersymmetric extension of the SM �to be called MSSM in thefollowing ������

bi "

�BBB�b�

b�

b�

�CCCA "

�BBB�

�

���

�CCCA#NFam

�BBB�

�

�

�

�CCCA#NHiggs

�BBB�

����

���

�

�CCCA � �A���

Here NFam is the number of families of matter supermultiplets and NHiggs isthe number of Higgs doublets� We use NFam " � and NHiggs " � or � whichcorresponds to the minimal SM or minimal SUSY model respectively�

The second order coe!cients are�

bij "

�BBBB�

� � �

� �����

�

� � ����

�CCCCA#NFam

�BBBB�

����

��

��

��

��

�

����

��

���

�CCCCA#NHiggs

�BBBB�

���

���

�

���

���

�

� � �

�CCCCA

�A��� For the SUSY model they become�

bij "

�BBBB�

� � �

� ��� �

� � ���

�CCCCA#NFam

�BBBB�

����

��

����

��

�� �

����

� ���

�CCCCA#NHiggs

�BBBB�

���

���

�

���

��

�

� � �

�CCCCA

�A��� The contributions for the individual thresholds to bi and bij are listed in tables

A�� �from ref� �� and A�� respectively�The running of each i depends on the values of the two other coupling

constants if the second order e�ects are taken into account� However thesee�ects are small because the bij�s are multiplied by j�� � ���� Higher ordersare presumably even smaller�

If the small Yukawa couplings are neglected the RGE�s A��� can be solvedby integration to obtain �i��

� at a scale �� for a given i�� �

�i��� "

�� � ln �

��

��#

�

i�� #����

ln

��� �i��

� # ������� i�� # �����

�����A���

��

with

�� "���

�bi #

bij�

j�� #bik�

k��

��A���

�� "�� � bii�� �

�A���

This exact solution to the second order renormalization group equation canbe used to calculate the coupling constants at an arbitrary energy if they havebeen measured at a given energy i�e� one calculates i��

� from a given i�� �This transcendental equation is most easily solved numerically by iteration� Ifthe Yukawa couplings are included their running has to be considered too andone can solve the coupled equations of gauge couplings and Yukawa couplingsonly numerically�

A�� Yukawa Couplings

In order to calculate the evolution of the Yukawa coupling for the b quark inthe region between mb and MGUT one has to consider four di�erent thresholdregions�

� Region I between the typical sparticle masses MSUSY and the GUT scale�

� Region II between MSUSY and the top mass mt�

� Region III between mt and MZ �

� Region IV between MZ and mb�

A���� RGE for Yukawa Couplings in Region I

The second order RGE for the three Yukawa couplings of the third generation inthe regions between MSUSY and MGUT are������

dYtdt

" Yt

��

�) � # �) � #

��

��) � � Yt

����b� #

���

� ) �� � ��b� #

�

� ) �� � �

��

��b� #

��

��� ) �� � �) �) � � ��

��) �) � � ) �) �

��) �Yt � ) �Yt �

�) �Yt # ��Y �

t

�A���

dYbdt

" Yb

��

�) � # �) � #

�

��) � � Yt

����b� #

���

� ) �� � ��b� #

�

� ) �� � �

�

��b� #

��

��� ) �� � �) �) � � �

�) �) � � ) � ) �

��

�) �Yt # �Y �

t

�A���

dY�dt

" Y�

�#�) � #

�

�) �

���b� # �

� ) �� � �

�

�b� #

��

�� ) �� �

�

�) �) �

�A���

��

If one assumes Yukawa coupling uni�cation for particles belonging to the samemultiplet i�e� Yb " Y� at the GUT scale one can calculate easily the RGE for

the ratio Rb� �t " mb�m� "qYb�t �Y� �t �

dRb�

dt" Rb�

��

�) � � �

�) � � �

�Yt

����b� #

�

� ) �� # �

�

�b� #

��

�� ) �� � �) �) � � �

�) �) � #

�

�) �) �

��

�) �Yt #

�

�Y �t

�A��

A���� RGE for Yukawa Couplings in Region II

For the region between MSUSY and mt one �nds������

dYtdt

" Yt

��) � #

�

�) � #

��

��) � � �

�Yt

#���) �� #��

�) �� �

����

��) �� � �) �) � � ��

��) �) � #

�

��) �) �

��) �Yt � ���

�) �Yt � ���

��) �Yt # ��Y �

t

�A���

dYbdt

" Yb

��) � #

�

�) � #

�

�) � � �

�Yt

#���) �� #��

�) �� #

���

��) �� � �) �) � � ��

��) �) � #

��

��) �) �

��) �Yt � ��

�) �Yt � ��

��) �Yt #

�

�Y �t

�A���

dY�dt

" Y�

��

�) � #

�

�) � � �Yt

#��

�) �� �

����

���) �� �

��

��) �) �

���) �Yt � ��

�) �Yt � ��

�) �Yt #

��

�Y �t

�A���

dRb�

dt" Rb�

��) � � ) � #

�

�Yt

#��) �� #��

��) �� �

�

�) �) � � ��

��) � ) � #

��

��) �) �

#�) �Yt � �

��) �Yt #

��

��) �Yt � ��

�Y �t

�A���

A���� RGE for Yukawa Couplings in Region III

For the region between mt and MZ one �nds�

dYbdt

" Yb

��) � #

�

�) � #

�

�) �

#���) �� #��

�) �� #

���

��) �� � �) �) � � ��

��) �) � #

��

��) �) �

�A���

dY�dt

" Y�

��

�) � #

�

�) �

��

#��

�) �� �

����

���) �� �

��

��) �) �

�A���

dRb�

dt" Rb� ��) � � ) �

#��) �� #��

��) �� �

�

�) � ) � � ��

��) �) � #

��

��) �) �

�A���

A���� RGE for Yukawa Couplings in Region IV

dRb�

dt" Rb�

��) � � ) � # ��) �� #

��

��) �� �

��

��) �) �

�A���

A�� Squark and Slepton Masses

Using the notation introduced at the beginning the RGE equations for thesquarks and sleptons can be written as�����

d )m�L

dt"

��) �M

�� #

�

�) �M

��

�A���

d )m�E

dt" �

��

�) �M

�� �A��

d )m�Q

dt" �

�

�) �M

�� # �) �M

�� #

�

��) �M

�� � �i�Yt� )m

�Q # )m�

U #m�� # A�

tm�� � ��

�A���

d )m�U

dt"

��

�) �M

�� #

�

��) �M

��

� �i��Yt� )m

�Q # )m�

U #m�� # A�

tm�� � �� �A���

d )m�D

dt"

��

�) �M

�� #

�

��) �M

��

�A���

The �i� factor ensures that this term is only included for the third generation�

A���� Solutions for the squark and slepton masses�

The solutions for the RGE given above are �����

)m�EL

" m�� #m�

�� ) GUT

��

�f��t #

�

��f��t

� cos��� M�

Z��

�� sin� �W �A���

)m��L

" m�� #m�

�� ) GUT

��

�f��t #

�

��f��t

# cos���

�

�M�

Z �A���

)m�ER

" m�� #m�

�� ) GUT

�

�f��t

� cos��� M�

Z sin� �W �A���

)m�UL

" m�� #m�

�� ) GUT

��

�f��t #

�

�f��t #

�

��f��t

� cos��� M�

Z���

�#

�

�sin� �W

�A���

)m�DL

" m�� #m�

�� ) GUT

��

�f��t #

�

�f��t #

�

��f��t

� cos��� M�

Z��

�� �

�sin� �W

�A���

���

)m�UR

" m�� #m�

�� ) GUT

��

�f��t #

�

��f��t

# cos��� M�

Z��

�sin� �W �A���

)m�DR

" m�� #m�

�� ) GUT

��

�f��t #

�

��f��t

� cos��� M�

Z��

�sin� �W �A��

�A���

For the third generation the e�ect of the top Yukawa coupling needs to be takeninto account in which case the solution given above are changed to������

)m�bR

" )m�DR

�A���

)m�bL

" )m�DL

#��

��m�

� � �� �m�� �

�

�) GUT

�f��t #

�

�f��t

m�

��

�A���

)m�tR

" )m�UR

# ���

��m�

� � �� �m�� �

�

�) GUT

�f��t #

�

�f��t

m�

��

#m�

t�A���

)m�tL

" )m�UL

#��

��m�

� � �� �m�� �

�

�) GUT

�f��t #

�

�f��t

m�

��

#m�

t�A���

A non�negligible Yukawa coupling causes a mixing between the weak interactioneigenstates� The mass matrix is�����

�� )m�

tR�ht�At m� jH�

� j# �jH�� j

�ht�At m� jH�� j# �jH�

� j )m�tL

�A �A���

and the mass eigenstates are�

)m�t��� "

�

�

�)m�tL# )m�

tR�q� )m�

tL � )m�tR

� # �m�t �Atm� # � cot� �

�A���

A�� Higgs Sector

A���� Higgs Scalar Potential

The MSSM has two Higgs doublets �Q " T� # YW�� �

H���� ���� "�� H�

�

H��

�A � H���� �� � "

�� H�

�

H��

�A �

The tree level potential for the neutral sector can be written as�

V �H�� � H

�� " m�

�jH�� j� #m�

�jH�� j� �m�

��H��H

�� # hc #

g� # g��

��jH�

� j� � jH�� j� �

�A��� with the following boundary conditions at the GUT scale m�

� " m�� " �� #

m��� m

�� " � B�m� where the value of � is the one at the GUT scale�

���

The renormalization group equations for the mass parameters in the Higgspotential can be written as�����

d��

dt" ��) � #

�

�) � � Yt �

� �A���

dm��

dt" ��) �M

�� #

�

�) �M

�� # ��) � #

�

�) � � Yt �

� �A��

dm��

dt" ��) �M

�� #

�

�) �M

�� # ��) � #

�

�) � �

� � �Yt� )m�Q # )m�

U #m�� # A�

tm��

�A���

dm��

dt"

�

��) � #

�

�) � � Yt m

�� # ��m�YtAt � ���) �M� #

�

�) �M� �A���

A���� Solutions for the Mass Parameters in the Higgs Po�

tential

The solutions for the RGE given above are �����

���t " q�t ����� �A���

m���t " m�

� # ���t #m��� ) GUT �

�

�f��t #

�

��f��t �A��

m���t " q�t ����� #m�

��e�t # At�� m�m��f�t #m���h�t � k�t At��

� �A��

m���t " q�t m�

��� # r�t ��� m�� # s�t At�� m���� �A��

where

q�t "�

�� # Yt�� F �t ��� # ��t

� �b���� # ��t � ��b��

h�t "�

��

�

D�t � �

k�t "�Yt�� F �t

D��t

f�t " �Yt�� H��t

D��t

D�t " � # Yt�� F �t

e�t "�

�

G��t # Yt�� G��t

D�t #

�H��t # Yt�� H�t �

�D��t #H��t

�

s�t "�Yt�� F �t

D�t q�t

r�t "

��Yt�� H��t

D�t �H��t

�q�t

E�t " �� # ��t �� �b���� # ��t

�b��� # ��t �� ��b��

F �t "

tZ�

E�t� dt�

H��t " ) GUT ��

�h��t # �h��t #

��

��h��t

���

H��t " tE�t � F �t

H�t " F �t H��t �H��t

H��t " ) GUT ���

�f��t # f��t � ��

��f��t

H��t "

tZ�

H�� �t

� E�t� dt�

H��t " ) GUT ��h��t #�

�h��t

H��t " ) GUT ���

�f��t # f��t � �

�f��t

G��t " F��t � �

�H�

� �t

G��t " F��t � F�t � �H��t H�t # �F �t H���t � �H��t

F��t " ) GUT ��

�f��t #

�

��f��t

F��t " F �t F��t �tZ

�

E�t� F��t� dt�

F�t "

tZ�

E�t� H��t� dt�

The functions fi and hi have been de�ned before� The Higgs mass spectrum canbe obtained from the potential given above by diagonalizing the mass matrix�

M�ij "

�

�

��VH��j��j

�A��

where �i is a generic notation for the real or imaginary part of the Higgs �eld�Since the Higgs particles are quantum �eld oscillations around the minimumeq� A�� has to be evaluated at the minimum� The mass terms at tree levelhaven been given in the text� However as discovered a few years ago the radia�tive corrections to the Higgs mass spectrum are not small and one has to takethe corrections from a heavy top quark into account� In this case the e�ectivepotential for the neutral sector can be written as����

V �H�� � H

�� " m�

�jH�� j� #m�

�jH�� j� �m�

��H��H

�� # hc #

g� # g��

��jH�

� j� � jH�� j� �

#�

���

)mt��ln

)m�t�

Q�� �

� # )m

t��ln)m�t�

Q�� �

� �m

t �lnm�

t

Q�� �

�

��

where )mti are �eld dependent masses which are obtained from eqns� A��� bysubstituting m�

t " h�t H��

The minimum of the potential can be found by requiring�

�V

�jH�� j

" �m��v� � �m�

�v� #g� # g

��

��v�� � v�� v�

���

#�

��h�t��Atm�v� # �v�

f� )m�t� � f� )m�

t�

)m�t� � )m�

t�

" � �A��

�V

�jH�� j

" �m��v� � �m�

�v� �g� # g

��

��v�� � v�� v�

#�

��

�h�tAtm��Atm�v� # �v�

f� )m�t� � f� )m�

t�

)m�t� � )m�

t�

#��f� )m�t� # f� )m�

t� � �f�m�t �h

�t v�o" �� �A��

where

f�m� " m��lnm�

m�t

� � �A�

From the minimization conditions given above one obtains�

v� "�

�g� # g�� �tan� � � �

�m�

� �m�� tan

� � �A��

� �h�t��

�f� )m�

t� # f� )m�t� � �f�m�

t � tan� � # �A�

tm�� tan

� � � �� f� )m�

t� � f� )m�t�

)m�t� � )m�

t�

��

�m�� " �m�

� #m�� sin �� #

�h�t sin ��

��

nf� )m�

t� # f� )m�t� � �f�m�

t �A��

#�Atm� # � tan� �Atm� # � cot� f� )m�

t� � f� )m�t�

)m�t� � )m�

t�

�

From the above equations one can derive easily�

M�Z " �

m�� �m�

� tan� � �'�

Z

tan� � � �� �A��

'�Z "

�g�

���m�

t

M�W cos� �

f� )m�

t� # f� )m�t� # �m�

t # �A�tm

�� � �� cot� �

f� )m�t� � f� )m�

t�

)m�t� � )m�

t�

�

�A���

Here all mi are evaluated at MZ using eqns A���A��� Only the splittingin the stop sector has been taken into account since this splitting depends onthe large Yukawa coupling for the top quark �see the mixing matrix �eq� A��� �More general formulae are given in ref� ����� The Higgs masses corresponding tothis one loop potential are����

m�A " m�

� #m�� #'�

A� �A���

'�A "

�g�

���m�

t

M�W sin� �

f� )m�

t� # f� )m�t� # �m�

t # �A�tm

�� # ��

f� )m�t� � f� )m�

t�

)m�t� � )m�

t�

��A���

m�H� " m�

A #M�W #'�

H � �A���

'�H " � �g�

���m

t��

sin �M�W

h� )m�t� � h� )m�

t�

)m�t� � )m�

t�

�A���

m�h�H "

�

�

hm�

A #M�Z #'�� #'��

���

�

vuuuuuut�m�

A #M�Z #'�� #'��

� ��m�AM

�Z cos� �� � ��'��'�� �'�

��

���cos� �M�Z # sin� �M�

A '�� ���sin� �M�Z # cos� �M�

A '��

�� sin ���M�Z #M�

A '��

�������A���

'�� "�g�

��m

t

sin� �M�W

��Atm� # � cot�

)m�t� � )m�

t�

��d� )m�

t�� )m�t� � �A��

'�� "�g�

��m

t

sin� �M�W

ln�

)m�t� )m

�t�

mt

#�Atm��Atm� # � cot�

)m�t� � )m�

t�

ln�)m�t�

)m�t�

#

Atm��Atm� # � cot�

)m�t� � )m�

t�

��d� )m�

t�� )m�t�

�� � �A���

'�� "�g�

��m

t

sin� �M�W

��Atm� # � cot�

)m�t� � )m�

t�

ln�

)m�t�

)m�t�

#Atm��Atm� # � cot�

)m�t� � )m�

t�

d� )m�t�� )m

�t�

��

�A���

where

h�m� "m�

m� � )m�q

lnm�

)m�q

�

d�m��� m

�� " �� m�

� #m��

m�� �m�

�

lnm�

�

m��

�

and )m�q is the mass of a light squark�

A�� Charginos and Neutralinos

The RGE group equations for the gaugino masses of the SU�� SU�� and U�� groups are simple�

dMi

dt" �bi ) iMi �A���

with as boundary condition at MGUT � Mi�t " � " m��� The solutions are�

Mi�t ") i�t

) i�� m�� �A���

Since the gluinos obtain corrections from the strong coupling constant � theygrow heavier than the gauginos of the SU�� group� There is an additionalcomplication to calculate the mass eigenstates since both Higgsinos and gauginosare spin ��� particles so the mass eigenstates are in general mixtures of the weakinteraction eigenstates�

The mixing of the Higgsinos and gauginos whose mass eigenstates are calledcharginos and neutralinos for the charged and neutral �elds can be parametrizedby the following Lagrangian�

LGaugino�Higgsino " ��

�M�

�a�a � �

� �M ���� � �M c�� # hc

���

where �a� a " �� �� � �� are the Majorana gluino �elds and

� "

�BBBBBB�

)B

)W �

)H��

)H��

�CCCCCCA� � "

�� )W�

)H�

�A �

arethe Majorana neutralino and Dirac chargino �elds respectively� Here all theterms in the Lagrangian were assembled into matrix notation �similarly to themass matrix for the mixing between B and W � in the SM eq� ���� � The massmatrices can be written as �����

M �� "

�BBBBBB�

M� � �MZ cos � sin �W MZ sin � sin �W

� M� MZ cos � cos �W �MZ sin� cos �W

�MZ cos � sin �W MZ cos � cos �W � ��MZ sin � sin �W �MZ sin� cos �W �� �

�CCCCCCA

�A���

M c� "

�� M�

p�MW sin�p

�MW cos � �

�A �A���

The last matrix has two chargino eigenstates )����� with mass eigenvalues

M���� "

�

�

�M�

� # �� # �M�W �

q�M�

� � �� � # �MW cos� �� # �M�

W �M�� # �� # �M�� sin ��

�A���

The four mass eigenstates of the neutralino mass matrix are denoted by )��i �i "�� �� �� � with masses M� ��

� � � � � M� ��� The sign of the mass eigenvalue

corresponds to the CP quantum number of the Majorana neutralino state�In the limiting case M��M�� � �� MZ one can neglect the o��diagonal ele�

ments and the mass eigenstates become�

)��i " � )B� )W���p�� )H� � )H� �

�p�� )H� # )H� � �A���

with eigenvalues jM�j� jM�j� j�j� and j�j respectively� In other words the binoand neutral wino do not mix with each other nor with the Higgsino eigenstatesin this limiting case� As we will see in a quantitative analysis the data indeedprefers M��M�� � � MZ so the LSP is bino�like which has consequences for darkmatter searches�

A� RGE for the Trilinear Couplings in the Soft

Breaking Terms

The Lagrangian for the soft breaking terms has two free parameters A and B forthe trilinear coupling and the mixing between the two Higgs doublets respec�tively�

��

Since the A parameter always occurs in conjunction with a Yukawa couplingwe will only consider the trilinear coupling for the third generation called At�The evolutions of the A and B parameters are given by the following RGE�����

dAt

dt"

��

�) �M�

m�# �) �

M�

m�#

��

��) �M�

m�

� YtAt �A���

dB

dt" �

�) �M�

m�#

�

�) �M�

m�

� �YtAt �A��

The B parameter can be replaced by tan� through the minimization conditionsof the potential� The solution for At�t is�

At�t "At��

� # Yt�� F �t #m��

m�

�H� � Yt�� H�

� # Yt�� F �t

��A���

Particle b� b� b�

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Bibliography

��� G� B-orner The early Universe Springer Verlag ����� �

��� E�W� Kolb and M�S� Turner The early Universe Addison�Wesley ����� �

��� A�D� Linde Particle Physics and In�ationary Cosmology Harwood Aca�demic Publishers Chur Switzerland ����� �A�D� Linde In�ation and Quantum Cosmology Academic Press San DiegoUSA ����� �

��� T� Padmanabhan Structure Formation in the Universe Cambridge Uni�versity Press ����� �P�J�E� Peebles Principles of Physical Cosmology Princeton UniversityPress N�J� ����� �A�R� Liddle and D�H� Lyth Phys� Rep� ��� ����� ��M�S� Turner Toward the in�ationary paradigm� Lectures on In�ationary

Cosmology in �Gauge Theory and the early Universe Eds� P� Galeotti andD�N� Schramm Kluwer Academic Publishers Dordrecht The Netherlands���� p� ��Ya� B� Zeldovich and I�D� Novikov Relativistic Astrophysics Vol� II Univ�of Chicago Press Chicago ����� �M� Berry Principles of Cosmology and Gravitation Adam Hilger Bristol����� �S� Weinberg Gravitation and Cosmology John Wiley / Sons USA ����� �

��� More popular accounts can be found in� S� W� Hawking Black Holes andBaby Universes and Other Essays Bantam Books New York ����� �M� Riordan and D�N� Schramm The Shadows of Creation W�H� Freemanand Company New York USA ����� �J� Silk The Big Bang W�H� Freeman and Company New York USA����� �L�Z� Fang and X�S� Li Creation of the Universe Singapore World Scienti�c����� �B� Parker Creation Plenum Press New York ���� �

�� S�L� Glashow Nucl� Phys� �� ��� ���� S� Weinberg Phys� Rev� Lett� � ��� ���� S� Salam in Elementary Particle Theory �� Stockholm� ���� �

��� Details and references can be found in standard text books� E�g��P� Renton Electroweak Interactions Cambridge univ� Press ����� �

���

G� Kane Modern Elementary Particle Physics Addison Wesley Publ�Comp� �����D�H� Perkins Introduction to High Energy Physics Addison Wesley Publ�Comp� �����C� Quigg Gauge Theories of the strong� weak and electromagnetic ForcesBenjamin�Cummings Publ� Comp� Inc� �����K� Moriyasu An elementary Primer for Gauge Field Theory World Scien�ti�c ����� �

��� G�G� Ross Grand Uni�ed Theories �Addison�Wesley Publishing CompanyReading MA ����� �

��� P� Langacker Phys� Rep� �� ����� ����

���� J� Ellis S� Kelley D� V� Nanopoulos Phys� Lett� B� ����� ����

���� U� Amaldi W� de Boer H� F-urstenau Phys� Lett� B� ����� ����

���� P� Langacker M� Luo Phys� Rev� D�� ����� ����

���� Introductions and original references can be found in the following text�books�J� Wess and J� Bagger Introduction to Supersymmetry Princeton Univer�sity Press Princeton NJ �����R�N� Mohapatra Uni cation and Supersymmetry Springer Verlag NewYork �����P�C� West Introduction to Supersymmetry and Supergravity World Scien�ti�c Singapore �����H�J�W� M-uller�Kirsten and A� Wiedemann Supersymmetry Vol� � WorldScienti�c Singapore �����P�P� Srivastava Supersymmetry and Super elds Adam�Hilger PublishingBristol England ����P�G�O� Freund Introduction to Supersymmetry Cambridge UniversityPress Cambridge England ����P� Nath R� Arnowitt and A�H� Chamseddine Applied N�� SupergravityWorld Scienti�c Singapore �����

���� Reviews and original references can be found in�H�E� Haber Lectures given at Theoretical Advanced Study Institute Uni�versity of Colorado June ���� Preprint Univ� of Sante Cruz SCIPP ������see also SCIPP ������Perspectives on Higgs Physics G� Kane �Ed� World Scienti�c Singapore����� �Int� Workshop on Supersymmetry and Uni cation P� Nath �Ed� WorldScienti�c Singapore ����� �Phenomenological Aspects of Supersymmetry W� Hollik R� R-uckl and J�Wess �Eds� Springer Verlag ����� �R� Barbieri Riv� Nuovo Cim� �� ����� ��A�B� Lahanus and D�V� Nanopoulos Phys� Rep� ��� ����� ��

���

H�E� Haber and G�L� Kane Phys� Rep� ��� ����� ���M�F� Sohnius Phys� Rep� ��� ����� ���H�P� Nilles Phys� Rep� �� ����� ��P� Fayet and S� Ferrara Phys� Rep� �� ����� ����

���� G� Zweig CERN�Reports �����TH� � ���� �G� Zweig CERN�Reports ����TH��� ���� �

��� S�L� Glashow J� Iliopoulos and L� Maiani Phys� Rev� D� ����� �����

���� H� Fritzsch and M� Gell�Mann �th Int� Conf� on High Energy PhysicsChicago ����� H� Fritzsch M� Gell�Mann and H� Leutwyler Phys� Lett� ��B ����� ���

���� D�J� Gross and F� Wilczek Phys� Rev� Lett� � ����� �����

���� H�D� Politzer Phys� Rev� Lett� � ����� ����

���� A�D� Martin R�G� Roberts W�J� Stirlin Phys� Lett� B� ����� ����M� Virchaux A� Milsztajn Phys� Lett� B��� ����� ����

���� DELPHI Coll� P� Abreu et al� Phys� Lett� B��� ����� ����W� de Boer and T� Ku0maul Karlsruhe preprint IEKP�KA���� ����� �

���� The LEP Collaborations� ALEPH DELPHI L� and OPAL Phys� Lett���B ����� ����An update is given in CERN�PPE��������

���� G� �t Hooft Nucl� Phys� B �� ����� ���ibid� B� ����� ���� ibid� B������ ���G� �t Hooft M� Veltman Nucl� Phys� B�� ����� ����

���� F� London Super�uids Vol� � Wiley New York �����

���� P�W�B� Higgs Phys� Lett� �� ���� ����T�W�B� Kibble Phys� Rev� �� ���� ����P�W� Anderson Phys� Rev� �� ���� ����F� Englert and R� Brout Phys� Rev� �� ���� ����

��� J�D� Bjorken and S�D� Drell Relativistic Quantum Mechanics Mac GrawHill New York ���� �

���� G� �t Hooft M� Veltman Nucl� Phys� B� ����� ����M� Dine J� Sapirstein Phys� Rev� Lett� �� ����� ��K�G� Chetyrkin A�L� Kataev F�V� Tkachov Phys� Lett� B�� ����� ����W� Celmaster R�J� Gensalves Phys� Rev� Lett� �� ����� ���W�A� Bardeen A� Buras D� Duke T� Muta Phys� Rev� D �� ����� �����

���� D�W� Duke R�G� Roberts Phys� Reports �� ����� ����

���� H� Georgi S�L Glashow Phys� Rev� Lett� �� ����� ����A�J� Buras J� Ellis M�K� Gaillard D�V� Nanopoulos Nucl� Phys� B�������� �

���

���� Review of Particle Properties Phys� Rev� D�� ����� �

���� A�D� Sakharov ZhETF Pis�ma � ���� ���

���� For reviews and original refs� see M�E� Shaposhnikov CERN�TH��������A�G� Cohen D�B� Kaplan and A�E� Nelson San Diego preprint UCSD�������M� Dine and S� Thomas Santa Cruz preprint SCIPP ������F�R� Klinkhamer Nucl� Phys� A�� ����� ��c�

���� M� Chanowitz J� Ellis and M� Gaillard Nucl� Phys� B��� ����� ���A� J� Buras J� Ellis M�K� Gaillard and D�V� Nanopoulos Nucl� Phys�B��� ����� �K� Inoue A� Kakuto H� Komatsu and S� Takeshita Prog� Theor� Phys�� ����� ����D�V� Nanopoulos and D�A� Ross Nucl� Phys� B��� ����� ���� Phys� Lett�B� � ����� ���� Phys� Lett� ��� ����� ���L� E� Ib*a)nez and C� L*opez Phys� Lett� ��B ����� ��� Nucl� Phys� B�������� ����

���� K� Inoue A� Kakuto H� Komatsu and S� Takeshita Prog� Theor� Phys�� ����� ���� ERR� ibid� � ����� ����L�E� Ib*a)nez C� Lop*ez Phys� Lett� ��B ����� ��� Nucl� Phys� B�������� ����L� Alvarez�Gaum*e J� Polchinsky and M� Wise Nucl� Phys� ��� ����� ����J� Ellis J�S� Hagelin D�V� Nanopoulos K� Tamvakis Phys� Lett� ���B����� ����G�Gamberini G� Ridol� and F� Zwirner Nucl� Phys� B��� ����� ����

���� U� Amaldi W� de Boer P�H� Frampton H� F-urstenau J�T� Liu Phys� Lett�B��� ����� ����

��� H� F-urstenau Ph�D� thesis Univ� of Karlsruhe IEKP�KA����� and pri�vate communication�

���� N� Hall New Scientist April ����� ���J�S� Stirling Physics World May ����� ���D� P� Hamilton Science ��� ����� ����G�G� Ross Nature ��� ����� ���S� Dimopoulos S�A� Raby and F� Wilczek Physics Today October ����� ���W� de Boer and J� K-uhn Phys� Bl� �� ����� ����

���� D�Z� Freedman P� van Niewenhuizen S� Ferrara Phys� Rev� D� ���� �����P� van Niewenhuizen Phys� Rep� �C ����� ����S� Deser B� Zumino Phys� Lett� �B ���� ����P� van Niewenhuizen Phys� Rep� �C ����� ����

���

���� R� Barbieri S� Ferrara C�A� Savoy Phys� Lett� ��B ����� ����H��P� Nilles M� Srednicki and D� Wyler Phys� Lett� �� B ����� ��� ibid����B ����� ����E� Cremmer P� Fayet and L� Girardello Phys� Lett� ���B ����� ���L�E� Ib*a)nez Phys� Lett� ���B ����� ��� Nucl�Phys� B��� ����� ����J� Ellis D�V� Nanopoulos K� Tamvakis Phys� Lett� ���B ����� ����L� Alvarez�Gaum*e J� Polchinski M� Wise Nucl� Phys� B��� ����� ����L�E� Ib*a)nez and G�G� Ross Phys� Lett� ���B ����� ����L�E� Ib*a)nez C� Lop*ez C� Mu)nos Nucl� Phys� B�� ����� ����

���� L� Hall J� Lykken and S� Weinberg Phys� Rev� D�� ����� �����S� K� Soni and H�A� Weldon Phys� Lett� ��B ����� ����

���� L� Girardello and M�T� Grisaru Nucl� Phys� B�� ����� ����

���� V�S� Kaplunovsky and J� Louis Phys� Lett� B� ����� ���L� E� Ib*a)nez �Madrid Autonoma U� FTUAM������� L� E� Ib*a)nez and D�L-ust Nucl� Phys� B��� ����� ����A� Brignole L�E� Ib*a)nez C� Mu)noz �Madrid Autonoma U� FTUAM����� Aug �����J� L� Lopez D�V� Nanopoulos A� Zichichi CERN�TH�������REV�CERN�TH�������� CTP�TAMU������� CTP�TAMU�������

���� S� Weinberg Rev� Mod� Phys� � ����� � and references therein�

���� R� Arnowitt and P� Nath Phys� Rev� Lett� ����� ���� Phys� Lett� B�������� ���For a review see P� Langacker Univ� of Penn� Preprint UPR������T�

���� J� Ellis et al� Nucl� Phys� B��� ����� ����

��� S� Dimopoulos Phys� Lett� B�� ����� ����

���� R� Barbieri and G�F� Giudice Nucl� Phys� B� ����� ��

���� DELPHI Coll� R� Ker-anen Search for a light Stop DELPHI Note ������PHYS ����

���� K� A� Olive Phys� Rep� � ����� ����

���� K�C� Roth D� M� Meyer and I� Hawkins ApJ ��� ����� L��

���� G�F� Smoot et al� Astrophys� J� Lett� � ����� L��

���� D�N� Schramm in The Birth and Early Evolution of our Universe Phys�Scripta T� ����� ��� G� Steigman ibid� p� ���

���� J� Gribbon New Scientist March ���� p� ���

���� The LEP Collaborations� Phys� Lett� B�� ����� ����

���� G� �t Hooft Nucl� Phys� B� ����� ���A� M� Polyakov JETP Lett� � ����� ����

���

��� A� Guth and P� Steinhardt in The new Physics P� Davis �Ed� CambridgeUniversity Press ����� p� ���A� Albrecht Cosmology for High Energy Physicists Fermilab�Conf�������A Proc� of the ���� Theoretical Advanced Studies Institute Sante Fe NewMexico�

���� R� K� Schaefer and Q� Sha� A Simple Model of Large Structure FormationBartol preprint BA������ �Oct� ���� �

���� J�R� Primack D� Seckel and B�A� Sadoulet Rev� Nucl� Part� Sci� �� ����� ����

���� A� Tyson Physics Today �June ���� ����K� Pretzl Bern Preprint BUHE������

��� C� Alcock et al� Nature �� ����� ��� E� Aubourg et al� ibid� p� ���

��� M�S� Turner Toward the in�ationary paradigm� Lectures on In�ationary

Cosmology in �Gauge Theory and the early Universe� Eds� P� Galeotti andD�N� Schramm Kluwer Academic Publishers Dordrecht The Netherlands���� p� ����

��� A�N� Taylor and M� Rowan�Robinson Nature ��� ����� �������K� M� Gorski R� Stompor R� Juszkiewicz Kyoto Univ� Preprint YITP�U����� Dec �����R� K� Schaefer Q� Sha� Phys� Rev� �� ����� �����J� Silk A� Stebbins UC Berkeley Preprint CFPA�TH������ ����� �

��� G� Steigman K�A� Olive D�N� Schramm M�S� Turner Phys� Lett� B������ ���J� Ellis K� Enquist D�V� Nanopoulos S� Sarkar Phys� Lett� B�� ���� ����

��� G� L� Kane C� Kolda L� Roszkowski and J�D� Wells Univ� of MichiganPreprint UM�TH�������

��� R�G� Roberts and Roszkowski Phys� Lett� B� ����� ����

�� J� Ellis S� Kelley D�V� Nanopoulos Nucl� Phys� B��� ����� ���

��� P� Langacker M� Luo Phys� Rev� D�� ����� ����

��� H� Murayama T� Yanagida Preprint Tohoku University TU���� ����� �T�G� Rizzo Phys� Rev� D�� ����� �����T� Moroi H� Murayama T� Yanagida Preprint Tohoku University TU��������� �

��� S� Dimopoulos H� Georgi Nucl� Phys� B�� ����� ����N� Sakai Z� Phys� C�� ����� ����A�H� Chamseddine R� Arnowitt P� Nath Phys� Rev� Lett� � ����� ����

���� G�G� Ross and R�G� Roberts Nucl� Phys� B��� ����� ����

���

���� L� E� Ib*a)nez and G� G� Ross CERN�TH������� ����� appeared in Per

spectives on Higgs Physics G� Kane �Ed� p� ��� and references therein�

���� R� Arnowitt and P� Nath Phys� Rev� D� ����� �����

���� S� Kelley J� L� Lopez D�V� Nanopoulos Phys� Lett� B��� ����� ����V� Barger M� Berger and P� Ohmann Phys� Rev� D�� ����� �����P� Langacker and N� Polonsky �Penn U� UPR�����T May �����

���� H� Arason et al� Phys� Rev� D� ����� �����

���� H� Arason et al� Phys� Rev� Lett� � ����� �����

��� M� Drees and M� M� Nojiri Phys� Rev� D�� ����� ���J� L� Lopez D�V� Nanopoulos and H� Pois Phys� Rev� D�� ����� ����P� Nath and R� Arnowitt Phys� Rev� Lett� � ����� ���J� L� Lopez D�V� Nanopoulos and K� Yuan Phys� Rev� D�� ����� ���

���� P� Langacker N� Polonski Univ� of Pennsylvania Preprint UPR�����T����� �

���� B� Pendleton and G�G� Ross Phys� Lett� B� ����� ����C�T� Hill Phys� Rev� D�� ����� ���M� Carena M� Olechowski S� Pokorski C�E�M� Wagner CERN�TH������� ����� �V� Barger M�S� Berger and P� Ohmann Phys� Lett� B��� ����� ����

���� D� Buskulic et al� ALEPH Coll� Phys� Lett� B��� ����� ����

���� R� Barbieri and G� Giudice Phys� Lett� B� ����� ��R� Garisto and J�N� Ng Phys� Lett� B��� ����� ����

���� J� L� Lopez D�V� Nanopoulos G� T� Park Phys� Rev� D�� ����� ����

���� J�L� Hewett Phys� Rev� Lett� � ����� �����V� Barger M�S� Berger and R�J�N� Phillips Phys� Rev� Lett� � ����� ����M�A� Diaz Phys� Lett� B� � ����� ����

���� M� Carena S� Pokorski C�E�M� Wagner Nucl� Phys� B� ����� ���M� Olechowski S� Pokorski Nucl� Phys� B� � ����� ���� and privatecommunication�

���� J�L� Lopez D�V� Nanopoulos and A� Zichichi CERN�TH������ CERN�TH��������

���� F�M� Borzumati in Phenomenological Aspects of Supersymmetry W� HollikR� R-uckl and J� Wess �Eds� Springer Verlag ����� �

��� J� Ellis and F� Zwirner Nucl� Phys� B��� ����� ����

���

���� F� James M� Roos MINUIT Function Minimization and Error AnalysisCERN Program Library Long Writeup D��� Release ���� March �����Our �� has discontinuities due to the experimental bounds on various quani�tities which become �active� only for speci�c regions of the parameterspace� Consequently the derivatives are not everywhere de�ned� The op�tion SIMPLEX which does not rely on derivatives can be used to �nd themonotonous region and the option MIGRAD to optimize inside this region�

���� W� de Boer R� Ehret and D� Kazakov Contr� to the Int� Conf� on HighEnergy Physics Cornell USA ����� Preprint Karlsruhe IEKP�������

���� R� Arnowitt and P� Nath Phys� Lett� B� ����� �� ERRATUM�ibid�B� � ����� ���� Phys� Rev� Lett� � ����� ��� Phys� Rev� Lett� ����� ����

���� S�P� Martin and P� Ramond Univ� of Florida preprint UFIFT�HEP������June ���� �D�J� Castano E�J� Piard and P� Ramond Univ� of Florida preprintUFIFT�HEP������ �August ���� �

���� J� L� Lopez D�V� Nanopoulos A� Zichichi CTP�TAMU������ ����� �CTP�TAMU������ ����� � CERN�TH������� ����� � CERN�TH�������REV ����� � CERN�TH������� ����� �J� L� Lopez et al� Phys� Lett� B� ����� ���

���� G� Degrassi S� Fanchiotti A� Sirlin Nucl� Phys� B��� ����� ���

���� P� Langacker N� Polonsky Phys� Rev� D�� ����� �����

���� For a review see e�g� S� Bethke Univ� of Heidelberg preprint HD�PY ����Lectures at the Scottish Univ� Summerschool in Physics �August ���� �

���� I� Antoniadis C� Kounnas K� Tamvakis Phys� Lett� ��B ����� ����

��� J� Ellis G� Ridol� F� Zwirner Phys� Lett� B�� ����� ����A� Brignole J� Ellis G� Ridol� and F� Zwirner Phys� Lett� B��� ����� ����H�E� Haber R� Hemp�ing Phys� Rev� Lett� ����� ���J� R� Espinosa M� Quiros Phys� Lett� B� ����� ����M� Drees M� M� Nojiri Phys� Rev� D�� ����� �����Z� Kunszt and F� Zwirner Nucl� Phys� B ��� ����� ��P� H� Chankowski S� Pokorski J� Rosiek MPI�PH������ ����� � MPI�PH������� ����� � ERRATUM DFPD����TH���ERR�

���� L�E� Ib*a)nez C� Lopez and C� Mu)noz Nucl� Phys� B�� ����� ����

���� R� Arnowitt and P� Nath Northeastern University Preprint NUB�TH��������

���� J� Gasser and H� Leutwyler Phys� Rep� ��C ����� ���S� Narison Phys� Lett� B�� ����� ����

��

����� N� Gray D�J� Broadhurst W� Grafe and K� Schilcher Z� Phys� C�� ����� ���

����� H� Marsiske SLAC�PUB����� ����� �J�Z� Bai et al� Phys� Rev� Lett� ����� �����

����� L� Roszkowski Univ� of Michigan Preprint UM�TH������

����� K� Griest and D� Seckel Phys� Rev� D�� ����� �����

����� E� Thorndike et al� CLEO Collaboration CLEO Preprints CLN ��������CLEO �����

����� C� Bernard P�Hsieh and A� Soni Washington Univ� preprint HEP�������July ���� �

���� S� Bertolini F� Borzumati A�Masiero and G� Ridol� Nucl� Phys� B�������� ��� and references therein�N� Oshimo Nucl� Phys� B� � ����� ���

����� F� Borzumati DESY preprint DESY �������

����� I� Antoniadis C� Kounnas K� Tamvakis Phys� Lett� ��B ����� ����

����� F� Anselmo L� Cifarelli A� Peterman and A� Zichichi Il Nuovo Cimento� � ����� ���� and references therein�

����� R� Barbieri L�J� Hall Phys� Rev� Lett� � ����� ����

����� F� Anselmo L� Cifarelli and A� Zichichi Il Nuovo Cimento � �A ����� ���� and references therein�

����� P� Janot Higgs Boson Search at a ��� GeV e�e Collider Orsay preprintLAL�������A� Sopczak Higgs Boson Discovery Potential at LEP��� CERN preprintCERN�PPE������K� Fujii SUSY at JLC KEK preprint ������

����� J�F�Gunion H�E� Haber G� Kane S� Dawson The Higgs Hunter�s Guide�Addison Wesley ����� and references therein�

����� U� Amaldi W� de Boer H� F-urstenau to be published�

����� W� de Boer R� Ehret and D� Kazakov to be published�

���� V� Barger M�S� Berger and P� Ohmann Phys� Rev� D�� ����� �����

����� M� B� Einhorn D� R� T� Jones Nucl� Phys� B� ����� ����M� E� Machacek M� T� Vaughn Nucl� Phys� B��� ����� ��� Nucl� Phys�B�� ����� ���� Nucl� Phys� B�� ����� ���

����� J�E� Bj-orkman and D�R�I� Jones Nucl� Phys� B��� ����� ��� and refer�ences therein�

����� M� Fischler and J� Oliensis Phys� Lett� �� B ����� ����

���