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Home , Grand Unified Theory, Supersymmetry breaking, Weak isospin, X and Y bosons

IEKPKA

hepph

March

Grand Unied Theories

and Sup ersymmetry in

Particle and Cosmology

W de Bo er

Inst fur Experimentel le Kernphysik Universitat Karlsruhe

Postfach D Karlsruhe Germany

ABSTRACT

A review is given on the consistency checks of Grand Unied Theories GUT which unify the

electroweak and strong nuclear into a single theory Such theories predict a new kind of

which could provide answers to several op en questions in cosmology The p ossible role of

such a primeval force will b e discussed in the framework of the Theory

Although such a force cannot be observed directly there are several predictions of GUTs

whichcanbeveried at low The Minimal Sup ersymmetric Standard Mo del MSSM

distinguishes itself from other GUTs by a successful prediction of many unrelated phenomena

with a minim um numb er of parameters

Among them a Unication of the couplings constants b Unication of the masses c Existence

of dark d decay e Electroweak breaking at a scale far below the

unication scale

A t of the free parameters in the MSSM to these low constraints predicts the masses of

the as yet unobserved sup erpartners of the SM constrains the unknown top mass to

a range b etween and GeV and requires the second order QCD coupling constanttobe

between and

Published in Progress in and

Email WimdeBo ercernch

Based on lectures at the Herbstschule Maria Laach Maria Laach and the Heisenberg

Landau Summerscho ol Dubna

The possibility that the was generated from noth

ing is very interesting and should be further studied Amost

perplexing question relating to the singularity is this what

preceded the genesis of the universe This question appears

to be absolutely methaphysical but our experience with meta

physics tel ls us that metaphysical questions are sometimes

given answers by physics

A Linde i

Contents

Intro duction

The Standard Mo del

Intro duction

The Standard Mo del

Choice of the Group Structure

Requirementof lo cal gauge invariance

The

Intro duction

Gauge Masses and the Top Mass

Summary on the Higgs mechanism

Running Coupling Constants

Grand Unied Theories

Motivation

Grand Unication

SU predictions

Asymmetry

Quantization

Prediction of sin

W

Sp ontaneous Symmetry Breaking in SU

Relations b etween Quark and Masses

Sup ersymmetry

Motivation

SUSY

The SUSY Mass Sp ectrum

Squarks and Sleptons

and

Higgs Sector

Electroweak Symmetry Breaking

The Big Bang Theory

Intro duction

Predictions from General Relativity

Interpretation in terms of Newtonian Mechanics ii

Time Evolution of the Universe

Temp erature Evolution of the Universe

Flatness Problem

Horizon Problem

Magnetic Monop ole Problem

The smo othness Problem

Ination

Origin of Matter

Summary

Comparison of GUTs with Exp erimental Data

Unication of the Couplings

M Constraint from Electroweak Symmetry Breaking

Z

Evolution of the Masses

Proton Lifetime Constraints

Top Mass Constraints

bquark Mass Constraint

Dark Matter Constraint

Exp erimental lower Limits on SUSY Masses

Decay b s

Fit Strategy

Results

Summary

App endix A

A Intro duction

A Gauge Couplings

A Yukawa Couplings

A RGE for Yukawa Couplings in Region I

A RGE for Yukawa Couplings in Region I I

A RGE for Yukawa Couplings in Region I I I

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

Intro duction

The questions concerning the origin of our universe have long b een thoughtofas

metaphysical and hence outside the realm of physics

However tremendous advances in exp erimental techniques to study b oth the

very large scale structures of the universe with space telescop es as well as the

tiniest building blo cks of matter the and with large accelera

tors allowus to put things together so that the creation of our universe now

has become an area of active research in physics

The two corner stones in this eld are

Cosmology ie the study of the large scale structure and the evolution

of the universe Today the central questions are being explored in the

framework of the Big Bang Theory BBT which provides a

satisfactory explanation for the three basic observations ab out our universe

the Hubble expansion the K microwavebackground radiation and the

density of elements hydrogen helium and the rest for the heavy

elements

ie the study of the building blo cks of Elementary

matter and the interactions between them As far as we know the build

ing blo cks of matter are pointlike particles the quarks and leptons which

can b e group ed according to certain symmetry principles their interactions

have been co died in the socalled SM In this mo del

all forces are describ ed by gauge eld theories which form a marvelous

synthesis of Symmetry Principles and Quantum Theories The latter

combine the classical eld theories with the principles of Quantum Mechan

ics and Einsteins Theory of Relativity

The basic observations b oth in the Microcosm as wellasinthe Macrocosm are

well describ ed by b oth mo dels 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 MatterAntimatter Asymmetry in our universe TOE 19 31 10 Temp. (K) 10 GUT SU(3)xSU(2)xU(1) 28 16 10 10 Energy (GeV) 25 13 10 10 22 10 10 10 19 7 10 ACCELERATORS 10 16 4 10 Electroweak SU(3)xU(1) 10 13 1 10 10 10 -2 10 Nucleo-synthesis 10 7 10 -5 10 +3K -8 4 Galaxies 10 10 -11 1 10 10 10 -43 10 -33 10 -23 10 -13 10 -3 10 7 10 17 (3 ') (1 yr) ( 10 10 yr)

Time (s)

Figure The evolution of the universe and the energy scale of some typical

events ab ove the Planck scale of GeV b ecomes so strong that

one cannot neglect gravity implying the need for a to

describ e all forces Below that energy the well known strong and electroweak

forces are assumed to b e equally strong implying the p ossibilityofaGrandUni

ed Theory GUT with only a single at the unication scale

After sp ontaneous symmetry breaking the gauge b osons of this unied force be

come heavy and freeze out The remaining forces corresp ond to the well known

SU SU U symmetry at lower energies with their coupling con

C L Y

stants changing from the unied value at the GUT scale to the low energy values

this evolution is attributed to calculable radiative corrections Future accelera

tors are exp ected to reach ab out TeV corresp onding to a temp erature of

K which was reached ab out s after the Bang At ab out GeV the

gauge b osons of the electroweak theory freeze out after getting mass through

sp ontaneous symmetry breaking and only the strong and electromagnetic force

play a role Ab out three minutes later the temp erature has dropp ed b elow the

binding energy and the strong force binds the quarks into nuclei nucle nuclear

osynthesis Most of the particles annihilate with their into a large

number of after the energies b ecome to o low to create new par

ticles again After ab out hundred thousand years the temp erature is b elow the

electromagnetic binding energies of atoms so the few remaining and

which did not annihilate form the neutral atoms Then the universe

b ecomes transparent for electromagnetic radiation and the many photons stream

are now observed as away into the universe The photons released at that time

the K microwavebackground radiation Then the neutral atoms start to cluster

slowly intostars and galaxies under the inuence of gravity

Why is our universe so smo oth and isotropic on a large scale

Why is the ratio of photons to in the universe so extremely large

on the order of

What is the origin of dark matter which seems to provide the ma jorityof

mass in our universe

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

Grand Unied Theories GUT in which the known electromagnetic weak

and strong nuclear forces are combined into a single theory hold the promise of

answering at least partially the questions raised ab ove For example they explain

the dierent strengths of the known forces by radiative corrections At high en

ergies all forces are equally strong The Sp ontaneous Symmetry Breaking SSB

of a single unied force into the electroweak and strong forces o ccurs in suchthe

ories through scalar elds which lo ck their phases over macroscopic distances

below the transition temp erature A classical analogy is the buildup of the mag

netization in a ferromagnet below the Curietemp erature ab ove the transition

temp erature the phases of the magnetic dip oles are randomly distributed and

the magnetization is zero but b elow the transition temp erature the phases are

lo cked and the groundstate develops a nonzero magnetization Translated in the

jargon of particle physicists the groundstate is called the vacuum and the scalar

elds develop a nonzero vacuum exp ectation value Such a phase transition

might have released an enormous amount of energy which would cause a rapid

expansion ination of the universe thus explaining simultaneously the origin

of matter its isotropic distribution and the atness of our universe

Given the imp ortance of the questions at stake GUTs have b een 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 unication of the three coupling constants of the electroweak and

strong forces at high energies

be much more turned out to be a disaster for GUTs The proton was found to

stable than predicted and from the precisely measured coupling constants at the

new p ositron LEP at the Europ ean Lab oratory for Elementary

Particle Physics CERN in Geneva one had to conclude that the couplings did

not unify if extrap olated to high energies

However it was shown later that byintro ducing a hitherto unobserved sym

metry called SUSY into the Standard Mo del both

problems disapp eared unication was obtained and the prediction of the proton

life time could b e pushed ab ove the present exp erimental lower limit

The price to b e paid for the intro duction of SUSY is a doubling of the number

of elementary particles since it presupp oses a symmetry between and

b osons ie each particle with even o dd has a partner with odd even

spin These sup ersymmetric partners have not been observed in so the

only way to save Sup ersymmetry is to assume that the predicted particles are

to o heavy to be pro duced by present accelerators However there are strong

theoretical grounds to believe that they can not be extremely heavy and in the

minimal SUSY mo del the lightest socalled Higgs particle will b e relatively light

which implies that it might even be detectable by upgrading the present LEP

accelerator But SUSY particles if they exist should be observable in the

next generation of accelerators since mass estimates from the unication of the

precisely measured coupling constan ts are in the TeV region and the lightest

Higgs particle is exp ected to be of the order of M as will be discussed in the

Z

last chapter

It is the purp ose of the present pap er to discuss the exp erimental tests of

GUTs The following exp erimental constraints have b een considered

Unication of the gauge coupling constants

Unication 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 the minimal SUSY mo del which

can describ e all these indep endent results simultaneously The constraints on the

couplings the unknown topquark mass and the masses of the predicted SUSY

particles will b e discussed in detail

The pap er has been organized as follows In chapters to the Standard

Mo del Grand Unied Theories GUT and Sup ersymmetry are intro duced In

chapter the problems in cosmology will b e discussed and why cosmology cries

for Sup ersymmetry Finallyinchapter the consistency checks of GUTs through

comparison with data are p erformed and in chapter the results are summarized

Chapter

The Standard Mo del

Intro duction

The eld of elementary particles has develop ed very rapidly during the last two

decades after the success of QED as a gauge eld theory of the electromagnetic

force could be extended to the weak and strong forces The success largely

started with the November Revolution in when the charmed quark was

discovered simultaneously at SLAC and Bro okhaven for which B Richter and

SSC Ting were awarded the Nob el prize in This discovery left little doubt

that the p ointlike constituents inside the proton and other are real

existing quarks and not some mathematical ob jects to classify the hadrons as

they were originally prop osed by Gellman and indep endently by Zweig

The existence of the charmed quark paved the way for a symmetry between

quarks and leptons since with one now had four quarks u d c and s

and four leptons e and which tted nicely into the SU U

e

unied theory of the electrow eak interactions prop osed by Glashow Salam and

Weinb erg GSW for the leptonic sector and extended to include quarks as

well as leptons by Glashow Iliop oulis and Maiani GIM as early as

Actually from the absence of avour changing neutral currents they predicted

the with a mass around GeV and indeed the charmed quark

was found four years later with a mass of ab out GeV This discovery b ecame

known as the November Revolution mentioned ab ove

weak interactions had already The unication of the electromagnetic and

been forwarded by Schwinger and Glashow in the sixties Weinb erg and Salam

solved the problem of the heavy gauge b oson masses required in order to explain

the short range of the weak interactions by intro ducing sp ontaneous symmetry

breaking via the Higgsmechanism This intro duced gauge b oson masses without

explicitly breaking the gauge symmetry

The GlashowWeinb ergSalam theory led to three imp ortant predictions

neutral currents ie weak interactions without changing the

In contrast to the charged currents the neutral currents could occur with

Zweig called the constituents aces and b elieved they really existed inside the hadrons

This belief was not shared by the referee of so his pap er was rejected and

circulated only as a CERN preprint alb eit wellknown

leptons from dierent generations in the initial state eg e e

through the exchange of a new neutral gauge b oson

the prediction of the heavy gauge b oson masses around GeV

a scalar neutral particle the Higgs b oson

The rst prediction was conrmed in by the observation of scattering

without in the nal state in the Gargamelle bubble chamber at CERN

Furthermore the predicted parity violation for the neutral currents was observed

in p olarized electrondeuteron scattering and in optical eects in atoms These

successful exp erimental verications led to the award of the Nob el prize in

to Glashow Salam and Weinberg In the second prediction was conrmed

by the discovery of the W and Z b osons at CERN in pp collisions for which C

Rubbia and S van der Meer were awarded the Nob el prize in

The last prediction has not b een conrmed the Higgs b oson is still at large

despite intensive searches It might just be to o heavy to be pro duced with the

present accelerators No predictions for its mass exist within the Standard Mo del

In the sup ersymmetric extension of the SM the mass is predicted to be on the

order 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 a

comparison with available data will b e made

In b etween the of the strong interactions as prop osed byFritzsch

and GellMann had established itself rmly after the discovery of its gauge

eld the in jet pro duction in e e annihilation at the DESY lab ora

tory in Hamburg The colour charge of these which causes the gluon

self has been established rmly at CERNs Large Electron

storage ring called LEP This gluon selfinteraction leads to asymptotic freedom

olitzer thus ex as shown by Gross and Wilcek and indep endently by P

plaining why the quarks can b e observed as almost free p ointlike particles inside

hadrons and why they are not observed as free particles ie they are conned

inside these hadrons This simultaneously explained the success of the Quark

Parton Mo del which assumes quasifree partons inside the hadrons In this case

the cross sections if expressed in dimensionless scaling variables are indep endent

of energy The observation of scaling in deep inelastic lepton scattering

led to the award of the Nob el Prize to Freedman Kendall and Taylor in

Even the observation of logarithmic scaling violations b oth in DIS and e e

annihilation as predicted by QCD were observed and could be used for precise

determinations of the strong coupling constant of QCD

The discovery of the b eauty quark at Fermilab in BataviaUSA in and

the lepton at SLAC b oth in led to the discovery of the third generation of

quarks and leptons of which the exp ected is still missing Recent LEP

data indicate that its mass is around GeV thus explaining why it has not

yet been discovered at the present accelerators The third generation had been

intro duced into the Standard Mo del long before by Kobayashi and Maskawa in

order to b e able to explain the observed CP violation in the system within

the Standard Mo del

From the total decay width of the Z b osons as measured at LEP one con

cludes that it couples to three dierent with a mass b elow M

Z

GeV This strongly suggests that the number of generations of elementary par

ticles is not innite but indeed three since the neutrinos are massless in the

Standard Mo del The three generations have been summarized in table to

gether with the gauge elds which are resp onsible for the low energy interactions

The gluons are b elieved to be massless since there is no reason to assume

that the SU symmetry is broken so one do es not need Higgs elds asso ciated

with the lo w energy strong interactions The apparent short range behaviour of

the strong interactions is not due to the mass of the gauge b osons but to the

gluon selfinteraction leading to connement as will b e discussed in more detail

afterwards

This chapter has b een organized as follows after a short description of the SM

we discuss it shortcomings and unanswered questions They form the motivation

for extending the SM towards a Grand Unied Theory in which the electroweak

and strong forces are unied into a new force with only a single coupling constant

The Grand Unied Theories will b e discussed in the next chapter Although such

unication can only happ en at extremely high energies far ab ove the range of

present accelerators it still has strong implications on low energy physics which

can be tested at present accelerators

The Standard Mo del

Constructing a gauge theory requires the following steps to b e taken

Choice of a symmetry group on the basis of the symmetry of the observed

interactions

Requirement of lo cal gauge invariance under transformations of the sym

metry group

Choice of the Higgs sector to intro duce sp ontaneous symmetry breaking

which allows the generation of masses without breaking explicitly gauge

invariance Massive gauge b osons are needed to obtain the shortrange b e

haviour of the weak interactions Adding adho c mass terms whicharenot

gaugeinvariant leads to nonrenormalizable eld theories In this case the

innities of the theory cannot be absorb ed in the parameters and elds of

the theory With the Higgs mechanism the theory is indeed renormalizable

shown by G t Ho oft as was

of the couplings and masses in the theory in order to relate

the bare charges of the theory to known data The

analysis leads to the concept of running ie energy dep endent coupling

constants which allows the absorption of innities in the theory into the

coupling constants

Interactions

strong electroweak gravitational unied

Theory QCD GSW SUGRA

Symmetry SU SU U SU

Gauge g g photon G Y

b osons gluons W Z b osons GUT b osons

charge colour weak mass

weak hyp ercharge

Table The fundamental forces The question marks indicate areas of intensive

research

Choice of the Group Structure

Groups of particles observed in nature showvery similar prop erties thus suggest

ing the existence of symmetries For example the quarks come in three colours

while the weak interactions suggest the grouping of fermions into doublets This

leads naturally to the SU and SU group structure for the strong and

weak interactions resp ectively The electromagnetic interactions dont change

the quantum numb ers of the interacting particles so the simple U group is

sucient

Consequently the Standard Mo del of the strong and electroweak interactions

is based on the symmetry of the following unitary groups

SU SU U

C L Y

The need for three colours arose in connection with the existence of hadrons

consisting of three quarks with identical quantum numbers According to the

Pauli principle fermions are not allowed to b e in the same state so lab eling them

with dierent colours solved the problem More direct exp erimental evidence

for colour came from the decay width of the and the total hadronic cross

section in e e annihilation Both are prop ortional to the number of quark

number of colours to b e three sp ecies and b oth require the

Although colour was intro duced rst as an adho c for the

reasons given ab ove it b ecame 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 comp onent of the weak

isospin T The charged weak interactions only op erate on lefthanded parti

cles ie particles with the spin aligned opp osite to their momentum negative

helicity so only lefthanded particles are given and right

handed particles are put into singlets see table Righthanded neutrinos

do not exist in nature so within each generation one has matter elds

leftrighthanded leptons and x x leftrighthanded quarks factor for

colour



Unitary transformations rotate vectors but leave their length constant SUN symmetry

groups are Sp ecial Unitary groups with determinant

The electromagnetic interactions originate b oth from the exchange of the

neutral gauge b oson of the SU group as well as the one from the U group

Consequently the charge of the U group cannot b e identical with the electric

charge but it is the socalled weak hyp ercharge Y which is related to the

W

electric charge via the GellMannNishijima relation

Q T Y

W

The quantum number Y is B L for left handed doublets and Q for righthanded

W

singlets where the B for quarks and for leptons while the

L for leptons and for quarks Since T and Q are conserved

Y is also a conserved quantum numb er The electroweak quantum numbers

W

for the sp ectrum are summarized in table

Generations Quantum Numb ers

helicity Q T Y

W

e

e

L L L

L

u c t

d s b

L L L

e

R R R

u c t

R R R

R

d s b

R R R

Table The electroweak quantum numb ers electric charge Q third comp o

nentofweak isospin T and weak hyp ercharge Y of the particle sp ectrum The

W

neutrinos and are the weak isospin partners of the electrone muon

e

and leptons resp ectively The upu downd stranges charmc

bottomb and topt quarks come in three colours which have not been indi

cated The primes for the left handed quarks d s and b indicate the in teraction

eigenstates of the electroweak theorywhich are mixtures of the mass eigenstates

ie the real particles The mixing matrix is the Cabibb oKobayshiMaskawama

trix The weak hyp ercharge Y equals B L for the lefthanded doublets and

W

Q for the righthanded singlets

Figure Global rotations leave the baryon colourless a Lo cal rotations

change the colour lo callythus changing the colour of the baryon b unless the

colour is restored by the exchange of a gluon c

Figure Demonstration of the nonab elian character of the SU rotations

inside a colourless baryon on the lefthand side one rst exchanges a redgreen

gluon which exchanges the colours of the quarks and then a greenblue gluon

on the righthand side the order is reversed The nal result is not the same so

these op erations do not commute

Requirement of lo cal gauge invariance

The Lagrangian L ofafree can b e written as

m L i

where the rst term represents the kinetic energy of the matter eld with

mass m and the second term is the energy corresp onding to the mass m The

EulerLagrange equations for this L yield the Dirac equation for a free fermion

The unitary groups SU N intro duced ab ove represent rotations in N dimensional

space The bases for the space are provided by the eigenstates of the matter elds

which are the colour triplets in case of SU weak isospin doublets in case of

SU and singlets for U

Arbitrary rotations of the states can be represented by

N

X

U expi F exp i F

k k

k

where are the rotation parameters and F the rotation matrices F are the

k k k

eight x GellMann matrices for SU denoted by hereafter and the well

known Pauli matrices for SU denoted by

The Lagrangian is invariant under the SU N rotation if L L where

y

U The mass term is clearly invariant m mU U m

y

since U U for unitary matrices The kinetic term is only invariant under

global transformations ie transformations where is everywhere the same in

k

spacetime In this case U is indep endent of x and can be treated as a constant

y y

multiplying which leads to U U U U

However one could also require local instead of global gauge invariance im

plying that the interactions should b e invariant under rotations of the symmetry

group for each particle separately The motivation is simply that the interactions

should be the same for particles b elonging to the same multiplet of a symmetry

group For example the interaction between a green and a blue quark should

be the same as the interaction between a green and a red quark therefore it

should be allowed to p erform a lo cal colour transformation of a single quark

The consequence of requiring lo cal gauge invariance is dramatic it requires the

intro duction of intermediate gauge b osons whose quantum numb ers completely

determine the p ossible interactions b etween the matter elds as was rst shown

by Yang and Mills in for the isopin symmetry of the strong interactions

Intuitively this is quite clear Consider a consisting of a colour triplet

of quarks in a colourless groundstate A global rotation of all quark elds will

leave the groundstate invariant as shown schematically in g However if a

quark eld is rotated lo cally 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 themselv es and

the lo cal colour variation of the quark eld is restored by the gluons

 

The SU N groups can b e represented by N N complex matrices A or N real numbers

y 

The unitarity requirementA A imp oses N conditions while requiring the determinant



to be one imp oses one more constraint so in total the matrix is represented by N real

numbers

The colour charge of the gluons is a consequence of the nonab elian character

of SU which implies that rotations in colour space do not commute ie

a b

as demonstrated in g If the gluons would all b e colourless they would

b a

not change the colour of the quarks and their exchange would b e commuting

Mathematically lo cal gauge invariance is intro duced by replacing the deriva

tive with the covariant derivative D which is required to have the following

prop erty

D UD

ie the covariant derivative of the eld has the same transformation prop erties

as the eld in contrast to the normal derivativ e Clearly with this requirement

L is manifestly gauge invariant since in each term of eq the transformation

y

leads to the pro duct U U after substituting D

For innitesimal transformations the covariantderivative can b e written as

ig ig ig

s

D B Y W G

W

where B W and G are the eld quanta mediators of the U SU and

SU groups and g g and g the corresp onding coupling constants

s

The term W can be explicitly written as

i

W W W W W W

i

p

W W W W iW

p

W iW W

W W

The op erators W in the odiagonal elements act as lowering and raising op era

tors for the weak isospin For example they transform an electron into a

and viceversa while the op erator W represents the neutral currentinteractions

between a fermion and antifermion

P

After substituting the GellMann matrices the term G G

k k

k

can be written similarly as

p p

p

p

G G G G

G G iG G iG G

rg rb

B C C B

B C C B

B C C B

p p

B C C B

p

p

G iG G G G iG

G G G G

B C C B

gr gb

B C C B

B C C B

A A

p p

p

p

G G iG G iG

G G G

br bg

This term induces transitions between the colours For example the o

diagonal element G acts like a raising op erator between a green

rg g

and red eld The terms on the diagonal dont change the colour

r

Since the trace of the matrix has to be zero there are only two indep endent



The term originates from Weyl who tried to introduce local gauge invariance for gravity

thus intro ducing the derivative in curved spacetime which varies with the curvature thus

being covariant

gluons which dont change the colour They are linear combinations of the

diagonal matrices and

The W gauge elds cannot represent the mediators of the weak interactions

since 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 b e obtained from the gauge elds after

will be discussed sp ontaneous symmetry breaking via the Higgs mechanism as

in the next section

The Higgs mechanism

Intro duction

The problem of mass for the fermions and weak gauge b osons can be solved by

assuming that masses are generated dynamically through the interaction with a

scalar eld which is assumed to be present everywhere in the vacuum ie the

spacetime in which interactions take place

The vacuum or equivalently the groundstate ie the state with the lowest

potential energy may have a nonzero scalar eld v alue represented by

v expi v is called the vacuum exp ectation value vev The same minimum

is reached for an arbitrary value of the phase so there exists an innity of

dierent but equivalent groundstates This degeneracy of the ground state takes

on a sp ecial signicance in a quantum eld theory b ecause the vacuum is required

to b e unique so the phase cannot b e arbitrarily at each p oint in spacetime Once

a particular value of the phase is chosen it has to remain the same everywhere

ie it cannot change lo cally A scalar eld with a nonzero vev therefore breaks

lo cal gauge invariance More details can be found in the nice intro duction by

Moriyasu

Nature has many examples of broken symmetries Sup erconductivityisawell

known example Below the critical temp erature the electrons bind into Co op er

pairs The density of Co op er pairs corresp onds to the vev Owing to the weak

binding the eective size of a Co op er pair is large ab out cm so every

Co op er pair overlaps with ab out other Co op er pairs and this overlap lo cks

the phases of the wave function over macroscopic distances Sup erconductivity

is a remarkable manifestation of Quantum Mechanics on a truly macroscopic

scale

In the sup erconducting phase the photon gets an eective mass through the

interaction with the Co op er pairs in the vacuum which is apparent in the



An amusing analogy was prop osed by A Salam Anumb er of guests sitting around a round

dinner table all have a serviette on the same side of the plate with complete symmetry As

so on as one guest picks up a serviette say on the lefthand side the symmetry is broken and all

guests have to follow suit and take the serviette on the same side ie the phases are lo cked

together everywhere in the vacuum due to the sp ontaneously broken symmetry



The interaction of the conduction electrons with the lattice pro duces an attractive force

When the electron energies are suciently small ie below the critical temp erature this

attractive force overcomes the Coulomb repulsion and binds the electrons into Co op er pairs

in which the momenta and spins of the electrons are in opp osite directions so the Co op er pair

forms a scalar eld and its quanta havea charge two times the electron charge

Figure Shap e of the Higgs p otential for a and b and

are the real and imaginary parts of the Higgs eld

Meissner eect the magnetic eld has a very short p enetration depth into the

sup erconductor or equivalently the photon is very massive Before the phase

transition the vacuum would have zero Co op er pairs ie a zero vev and the

magnetic eld can p enetrate the sup erconductor without attenuation as exp ected

for massless photons

This example of Quantum Mechanics and sp ontaneous symmetry breaking

in sup erconductivity has been transferred almost literally to elementary particle

physics by Higgs and others For the selfinteraction of the Higgs eld one

considers a p otential analogous to the one prop osed by Ginzburg and Landau for

sup erconductivity

y y

V

where and are constants The p otential has a parab olic shap e if but

takes the shap e of a Mexican hat for as pictured in g In the latter

case the eld free vacuum ie corresp onds to a lo cal maximum thus

forming an unstable equilibrium The groundstate corresp onds to a minimum

with a nonzero value for the eld

s

jj

In sup erconductivity acts like the critical temp erature T ab ove T the

c c

electrons are free particles so their phases can b e rotated arbitrarily at all p oints

in space but b elow T the individual rotational freedom is lost because the

c

electrons form a coherent system in which all phases are lo cked to a certain

value This corresp onds to a single point in the minimum of the Mexican hat

which represents a vacuum with a nonzero vev and a well dened phase thus

dening a unique vacuum The coherent system can still be rotated as a whole

so it is invariant under global but not under lo cal rotations

Masses and the Top Quark Mass

After this general intro duction ab out the Higgs mechanism one has to consider

the number of Higgs elds needed to break the SU U symmetry to

L Y

the U symmetry The latter must have one massless gauge b oson while the

em

W and Z b osons must be massive This can be achieved by cho osing to be a

complex SU doublet with denite hyp ercharge Y

W

x i x

x

x i x

In order to understand the interactions of the Higgs eld with other particles

one considers the following Lagrangian for a scalar eld

y

L D D V

H

The rst term is the usual kinetic energy term for a scalar particle for whichthe

EulerLagrange equations leadtothe KleinGordon equation of motion Instead

of the normal derivative the covariant derivative is used in eq in order to

ensure lo cal gauge invariance under SU U rotations

The vacuum is known to be neutral Therefore the groundstate of has to

be of the form v Furthermore x has to be constant everywhere in order

to have zero kinetic energy ie the derivative term in L disapp ears

H

The quantum uctuations of the eld around the ground state can b e parametrised

as follows if we include an arbitrary SU phase factor

i x

e

v hx

The real elds x are excitations of the eld along the p otential minimum

They corresp ond to the massless Goldstone b osons of a global symmetry in this

case three for the three rotations of the SU group However in a local gauge

theory these massless b osons can be eliminated by a suitable rotation

i x

e x

v hx

Consequently the eld has no physical signicance Only the real eld hx can

be interpreted as a real Higgs particle The original eld with four degrees of

freedom has lost three degrees of freedom these are recovered as the longitudinal

p olarizations of the three heavy gauge b osons

The kinetic part of eq gives rise to mass terms for the vector b osons

which can be written as Y

W

i h i h

y

L g W W W g W W W g B g B

H

or substituting for its vacuum exp ectation 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 v

B g gg

y y y y

gW B g B W gW g B

W gg g

Since mass terms of physical elds have to b e diagonal one obtains the physical

gauge elds of the broken symmetry by diagonalizing the mass term

B

y y

B W U U M U U

W

where U represents a unitary matrix

g g cos sin

W W

p

U

g g sin cos

g g

W W

Consequently the real elds b ecome a mixture of the gauge elds

A B

U

Z W

and the matrix UMU b ecomes a diagonal matrix for a suitable mixing angle

W

In these elds the mass terms have the form

A

A Z M W W

W

M Z

Z

with

M g v

W

g g

v M

Z

Here v is the vacuum exp ectation value of the Higgs p otential which for the

known gauge b oson masses and couplings can b e calculated to be

v GeV

The neutral part of the Lagrangian if expressed in terms of the physical elds

can be written as

neutr

e e e e e e L A g cos g sin

R R L L L L L L L L W W

int

Z g sin e e e e e e g cos

W R R L L L L L L L L W

p



in which case v GeV Sometimes the Higgs eld is normalized by g e

θ W

g

Figure Geometric picture of the relations between the electroweak coupling

constants

The photon eld should only couple to the electron elds and not to the neutrinos

so the terms prop ortional to g cos and g sin should cancel and the coupling

W W

to the electrons has to b e the electric charge e This can b e achieved by requiring

g cos g sin e

W W

Hence

gg g g

p

and e tan sin

W W

g g g

g g

A geometric picture of these relations is shown in g From these rela

tions and the relations between masses and couplings and one nds

the famous relation between the electroweak mixing angle and the gauge b oson

masses

M

W

M cos M or sin

W W Z W

M

Z

The value of M can also be related to the precisely measured muon decay

W

constant G GeV If calculated in the SM one nds

e G

p

sin M

W

W

This relation can be used to calculate the gauge b oson masses from measured

coupling constants G and sin

W

p

M

W

sin

G

W

p

M

Z

sin cos

G

W W

Inserting sin and yields M GeV However these

W Z

relations are only at tree level Radiative corrections dep end on the as yet un

known top mass Fitting the unknown top mass to the measured M mass the

Z

electroweak asymmetries and the cross sections at LEP yields

M stat unk now n H ig g s

top

Also the fermions can interact with the scalar eld alb eit not necessarily with

the gauge coupling constant The Lagrangian for the interaction of the leptons

with the Higgs eld can be written as

h i

e y

L g Le e L

H L R R

Y

Substituting the vacuum exp ectation value for yields

e e e

g v g v g

L

Y Y Y

p p p

e e e v e e e e ee

L L R R L R R L

v e

L

e

The Yukawa coupling constant g is a free parameter which has to be adjusted

Y

p

e

such that m g Thus the coupling g is prop ortional to the mass of v

e Y

Y

the particle and consequently the coupling of the Higgs eld to fermions is pro

p ortional to the mass of the fermion a prediction of utmost imp ortance to the

exp erimental search for the Higgs b oson

Note that the neutrino stays massless with the choice of the Lagrangian since

no mass term for the neutrino app ears in eq

Summary on the Higgs mechanism

In summary the Higgs mechanism assumed the existence of a scalar eld

exp i x After sp ontaneous symmetry breaking the phases are lo cked

over macroscopic distances so the eld averaged over all phases is not zero any

more and develops a vacuum exp ectation value The interaction of the fermions

and gauge b osons with this coherent system of scalar elds gives rise to eec

tive particle masses just like the interaction of the electromagnetic eld with the

Co op er pairs inside a sup erconductor can be describ ed by an eective photon

mass

The vacuum corresp onds to the groundstate with minimal potential energy

and zero kinetic energy At high enough temp eratures the thermal uctuations of

the Higgs particles ab out the groundstate b ecome so strong that the coherence is

lost ie xconstant is not true anymore In other words a phase transition

from the ground state with broken symmetry to the symmetric ground

state takes place In the symmetric phase the groundstate is invariant again under

lo cal SU rotations since the phases can b e adjusted lo cally without changing

the groundstate with In the latter case all masses disapp ear since

they are prop ortional to

Both the fermion and gauge b oson masses are generated through the interac

tion with the Higgs eld Since the interactions are prop ortional to the coupling

constants one nds a relation between masses and coupling constants For the

fermions the Yukawa coupling constant is prop ortional to the fermion mass and

the mass ratio of the W and Z b osons is only dep endent on the electroweak

mixing angle see eq This mass relation is in excellent agreement with ex

p erimental data after including radiative corrections Hence it is the rst indirect

by the interaction evidence that the gauge b osons masses are indeed generated

with a scalar eld since otherwise there is no reason to exp ect the masses of the

charged and neutral gauge b osons to be related in such a sp ecic way via the

couplings

Figure The eectivecharge distribution around an electric charge QED and

colour charge QCD At higher Q one prob es smaller distances thus observing

a larger smaller eective charge ie a larger smaller coupling constant in

QED QCD

Figure Running of the three coupling constants in the Standard Mo del owing

to the dierent space charge distributions compare g

Running Coupling Constants

In a the coupling constants are only eective constants

at a certain energy They are energy or equivalently distance dep endent through

virtual corrections b oth in QED and in QCD

However in QED the coupling constant increases as function of Q while in

QCD the coupling constant decreases A simple picture for this b ehaviour is the

following

The electric eld around a p ointlike electric charge diverges like r In

such a strong eld electronp ositron pairs can be created with a lifetime

determined by Heisenb ergs uncertainty relations These virtual e e pairs

orient themselves in the electric eld thus giving rise to vacuum p olariza

tion just like the atoms in a dielectric are p olarized by an external electric

eld This vacuum p olarization screens the bare charge so at a large

distance one observes only an eectivecharge This causes deviations from

Coulombs law as observed in the wellknown of the energy

levels of the hydrogen If the electric charge is prob ed at higher en

ergies or shorter distances one p enetrates the shielding from the vacuum

alently p olarization deep er and observes more of the bare charge or equiv

one observes a larger coupling constant

In QCD the situation is more complicated the colour charge is surrounded

by a cloud of gluons and virtual q q pairs since the gluons themselves

carry a colour charge one has two contributions a shielding of the bare

charge by the q q pairs and an increase of the colour charge by the gluon

cloud The net eect of the vacuum p olarization is an increase of the total

colour charge provided not to o many q q pairs contribute which is the case

if the number of generations is below see hereafter If one prob es this

charge at smaller distances one p enetrates part of the antishielding thus

observing a smaller colour charge at higher energies So it is the fact that

gluons carry colour themselves which makes the coupling decrease at small

e



R

R



e

a

scr eening antiscr eening



q





R R

 

g

 

   



 

  

     

  

  

R

R

  



  

q

q

b

Figure Lo op corrections in QED a and QCD b In QED only the fermions

contribute in the lo ops which causes a screening of the bare charge In QCD

also the b osons contribute through the gluon selnteraction which enhances the

bare charge This antiscreening dominates over the screening

distances or high energies This prop erty is called asymptotic freedom

and it explains why in deep inelastic leptonnucleus scattering exp eriments

the quarks inside a nucleus app ear quasi free in spite of the fact that they

are tightly b ound inside a nucleus The increase of at large distances

s

explains qualitatively why it is so dicult to separate the quarks inside a

hadron the larger the distance the more energy one needs to separate them

even further If the energy of the colour eld is to o high it is transformed

into mass thus generating new quarks which then recombine with the old

ones to form new hadrons so one always ends up with a system of hadrons

instead of free quarks

The space charges from the virtual pairs surrounding an electric charge and

colour charge are shown schematically in g The dierent vacuum p olar

izations lead to the energy dep endence of the coupling constants sketched in g

The colour eld b ecomes innitely dense at the QCD scale MeV

see hereafter So the connement radius of typical hadrons is O MeV or

ermi cm one F

The vacuum p olarization eects can b e calculated from the lo op diagrams to

the gauge b osons The main dierence between the charge distribution in QED

and QCD originates from the diagrams shown in g In addition one has

to consider diagrams of the typ e shown in g The ultraviolet divergences

Q in these diagrams can be absorb ed in the coupling constants in a

renormalizable theory All other divergences are canceled at the amplitude level

by summing the appropriate amplitudes The rst step in such calculations is

regularization of the divergences ie separating the divergent parts in the the

mathematical expressions The second step is the renormalization of physical + +

Tree Fermion level loop

+

Gauge loop

loop

Figure First order vacuum p olarization diagrams

quantities likecharge and mass to absorb the divergent parts of the amplitudes

ie replace the bare quantities of the theory with measured quantities For

example the lo op corrections to the photon propagator diverge if the momentum

transfer k in the lo op is integrated to innity If one intro duces a cuto for

large values of k one nds for the regularized amplitude of the sum of the Born

term M and the lo op corrections

Q

M e ln ln M for Q m

m m

The divergent part dep ending on the cuto parameter disapp ears if one re

places the bare charge e by the renormalized charge e

R

ln e e

R

m

ie the bare charge o ccurring in the Dirac equation is renormalized to a

measurable quantity e For e one usually takes the Thomson limit for Compton

R R

scattering ie e e for k

T

m

e

with e and m GeV

e

R

After regularization and renormalization to a measured quantity in this case

using the so called on shell scheme ie one uses the mass and charge of a

free electron as measured at low energy one is left with a Q dep endent but

nite part of the vacuum p olarization This can be absorb ed in a Q dep endent

m coupling constant which in case of QED b ecomes for Q

Q

ln Q

m

e

Q

n m

ln and retaining only the If one sums more lo ops this yields terms

m

e

leading logarithms ie nm these terms can be summed to

Q

Q

ln

m

e

since

X

n

x

x

n

Of course the total Q dep endence is obtained by summing over all p ossible

fermion lo ops in the photon propagator

These vacuum p olarization eects are nonnegligible For example at LEP

accelerator energies has increased from its low energy value to or

ab out

The diagrams of g b yield similarly to eq

N Q

s f

Q ln

s s

Note that decreases with increasing Q if N or N

s f f

thus leading to asymptotic freedom at high energy This is in contrast to the

Q dep endence of Q in eq which increases with increasing Q Since

b ecomes innite at small Q one cannot take this scale as a reference scale

s

Instead one could cho ose as renormalization p oint the connement scale ie

ifQ In this case eq b ecomes indep endent of since the

s

in brackets b ecomes negligible so one obtains

Q

s

n

Q

f

ln

The denition of dep ends on the renormalization scheme The most widely

used scheme is the MS scheme which we will use here Other schemes can

be used as well and simple relations b etween the denitions of exist

The higher order corrections are usually calculated with the renormalization

group technique which yields for the dep endence of a coupling constant

The rst two terms in this p erturbative expansion are renormalizationscheme

indep endent Their sp ecic values are given in the app endix The rst order

solution of eq is simple

Q

ln

Q Q Q

where Q is a reference energy One observes a linear relation b etween the change

in the inverse of the coupling constant and the logarithm of the energy The slop e

dep ends on the sign of which is p ositive for QED but negative for QCD thus

leading to asymptotic freedom in the latter case The second order corrections

are so small that they do not change this conclusion Higher order terms dep end

on the renormalization prescription In higher orders there are also corrections

from Higgs particles and gauge b osons in the lo ops Therefore the running of

a given coupling constant dep ends slightly on the value of the other coupling

constants and the Yukawa couplings These higher order corrections cause the

RGE equations to be coupled so one has to solve a large number of coupled

dierential equations All these equations are summarized in the app endix

Chapter

Grand Unied Theories

Motivation

The Standard Mo del describ es all observed interactions b etween elementary par

ticles with astonishing precision Nevertheless it cannot be considered to be

the ultimate theory b ecause the many unanswered questions remain a problem

Among them

The Gauge Problem

Why are there three indep endent symmetry groups

The Parameter Problem

How can one reduce the number of free parameters At least from

the couplings the mixing parameters the Yukawa couplings and the Higgs

p otential

The Fermion Problem

Why are there three generations of quarks and leptons What is the origin

of the the symmetry between quarks and leptons Are they comp osite

particles of more fundamental ob jects

The Charge Quantization Problem

Why do protons and electrons have exactly opp osite electric charges

The

Why is the weak scale so small compared with the GUT scale ie why is

M M

W P l anck

The Finetuning Problem

Radiative corrections to the Higgs masses and gauge boson masses have

quadratic divergences For example M OM In other words

H P l anck

the corrections to the Higgs masses are many orders of magnitude larger

than the masses themselves since they are exp ected to be of the order of

the electroweak gauge b oson masses This requires extremely unnatural

netuning in the parameters of the Higgs p otential This netuning

problem is solved in the sup ersymmetric extension of the SM as will be

discussed afterwards

Grand Unication

The problems mentioned ab ove can be partly solved by assuming the symmetry

groups SU SU U are part of a larger group G ie

C L Y

G SU SU U

C L Y

The smallest group G is the SU group so the minimal extension of the

SM towards a GUT is based on the SU group Throughout this pap er wewill

only consider this minimal extension The group G has a single coupling constant

for all interactions and the observed dierences in the couplings at low energy are

caused by radiative corrections As discussed b efore the strong coupling constant

decreases with increasing energy while the electromagnetic one increases with

energy so that at some high energy they will b ecome equal Since the changes

with energy are only logarithmic eq the unication scale is high namely

of the order of GeV dep ending on the assumed particle content in

the lo op diagrams

In the SU group the particles and antiparticles of the rst generation

can be t into the plet and plet

C C C

u u u d

d

g g

g b r

B B C C

C C

C

B B C C

u u u d

d

r r

b g

r

B B C C

C C

C

B B C C

p

u u u d

d

b b

r g b

B B C C

B B C C

e

u u u e

A A

g r b

d d d e

g r b e

L

The sup erscript C indicates the charge conjugated particle ie the

and all particles are chosen to be lefthanded since a lefthanded antiparticle

transforms like a righthanded particle Thus the sup erscript C implies a right

handed singlet with weak isospin equal zero

With this multiplet structure the sum of the quantum numbers Q T and Y

is zero within one multiplet as required since the corresp onding op erators are

represented by traceless matrices

Note that there is no space for the antineutrino in these multiplets so within

the minimal SU the neutrino must be massless since for a massive particle

the righthanded helicity state is also present Of course it is p ossible to put a

righthanded neutrino into a singlet representation

SU rotations can b e represented by matrices Lo cal gauge invariance

requires the intro duction of N gauge elds the mediators which

cause the interactions between the matter elds The gauge elds transform

under the adjoint representation of the SU group which can be written in

G cannot b e the direct pro duct of the SU SU and U groups since this would not

represent a new unied force with a single coupling constant but still require three indep endent

coupling constants



The bar indicates the complementary representation of the fundamental representation ν u e u e u e X Y Y

u d π d d π d u π

d d u u u u

Figure GUT proton decays through the exchange of X and Y gauge b osons

matrix form as compare eqns and

B

C C

p

X Y G G G

C B

B

C C

C B

p

G G G X Y

C B

C B

B

C C

p

C B

G G G X Y

C B



C B

W B

p p

C B

X X X W

A



W B

p p

W Y Y Y

The Gs represent the gluon elds of equation while the W s and B s are the

gauge elds of the SU symmetry groups The X and Y s are new gauge b osons

which represent interactions in which quarks are transformed into leptons and

viceversa as should be apparent if one op erates with this matrix on the plet

Consequently the X Y b osons which couple to the electron neutrino and

dquark must have electric charge

SU  predictions

Proton decay

The X and Y gauge b osons can intro duce transitions b etween quarks and leptons

thus violating lepton and baryon number This can lead to the following proton

and decays see g

p e n e

p e n e

p e n

p e n

p n K

p

K p

The decays with in the nal state are allowed through avour mixing

ie the interaction eigenstates are not necessarily the mass eigenstates



The dierence b etween lepton and baryon numb er BL is conserved in these transitions

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

M

X

p

m

p

The proton mass m to the fth p ower originates from the phase space in case the

p

nal states are much lighter than the proton which is the case for the dominant

decay mo de p e After this prediction of an unstable proton in grand

unied theories a great deal of activity develop ed and the lower limit on the

proton life time increased to

yrs

p

for the dominant decay mo de p e From equation this implies for

see chapter

M GeV

X

From the extrap olation of the couplings in the SU mo del to high energies

one exp ects the unication scale to b e reached well b elow GeV so the proton

lifetime measurements exclude the minimal SU mo del as a viable GUT As

will b e discussed later the sup ersymmetric extension of the SU mo del has the

unication p ointwell ab ove GeV

The heavy gauge b osons resp onsible for the unied force cannot be pro duced

with conventional accelerators but energies ab ove were easily accessible

during the birth of our universe This could have led to an excess of matter over

antimatter right at the b eginning since the X and Y b osons can decayinto pure

matter eg X uu which is allowed b ecause the charge of the X b oson is

As pointed out by Sakharov such an excess is p ossible if both C and

CP are violated if the baryon number B is violated and if the pro cess go es

through a phase of nonequilibrium All three conditions are p ossible within the

SU mo del The nonequilibrium phase happ ens if the hot universe co ols down

and arrives at a temp erature to o low to generate X and Y b osons anymore so

only the decays are p ossible Since the CP violation is exp ected to b e small the

excess of matter over antimatter will b e small so most of the matter annihilated

with antimatter into enormous number of photons This would explain why the

number of photons over baryons is so large

N

N

b

However later it was realized that the electroweak phase transition may wash

out any BL excess generated by GUTs One then has to explain the observed

baryon asymmetry by the electroweak which is actively studied + + Scalar Fermion loop loop

+ Gauge boson loop

loop

Figure Radiative corrections to particle masses

Charge Quantization

From the fact that quarks and leptons are assigned to the same multiplet the

charges must be related since the trace of any generator has to be zero For

example the charge op erator Q on the fundamental representation yields

TrQ Trq q q e

d d d

or in other words in SU the electric charge of the dquark has to be of

the charge of an electron Similarlyonends the charge of the uquark is of

the p ositron charge so the total charge of the proton uud has to be exactly

opp osite to the charge of an electron

Prediction of sin

W

If the SU and U groups have equal coupling constants the electroweak

mixing angle can b e calculated easily since it is given by the ratio g g g

which would be for equal coupling constants However the see eq

argument is slightly more subtle since for unitary transformations the rotation

matrices have to b e normalized such that

Tr F F

k l kl

This normalization is not critical in case one has indep endent coupling constants

for the subgroups since a wrong normalization for a rotation matrix can al

ways be corrected by a redenition of the corresp onding coupling constants as

is apparent from equation This freedom is lost if one has a single coupling

constant so one has to b e careful ab out the relative normalization It turns out

that the GellMann and Pauli rotation matrices of the SU and SU groups

have the correct normalization but the normalization of the weak hyp ercharge

op erator needs to b e changed Dening Y CT and substituting this into

W

the GellMannNishijima relation yields

Q T CT

Requiring the same normalization for T and T implies from equation

Tr Q C Tr T

or inserting numb ers from the plet of SU yields

Tr Q

C

Tr T

Replacing in the covariant derivative eq Y with CT implies

W

g CT g T or

g Cg

where C from eq With this normalization the electroweak mixing

angle after unication b ecomes

g g C

sin

W

g g g g C C

The manifest disagreement with the exp erimental value of at low energies

brought the SU mo del originally into discredit until it was noticed that the

running of the couplings between the unication scale and low energies could

reduce the value of sin considerably As we will show in the last chapter

W

with the very precise measurement of sin at LEP unication of the three

W

coupling constants within the SU mo del is excluded and just as in the case

of the proton life time sup ersymmetry comes to the rescue and unication is

p erfectly p ossible within the sup ersymmetric extension of SU

Note that the prediction of sin is not sp ecic to the SU mo del

W

but is true for any group with SU SU U as subgroups implying

C L Y

that Q T and Y are generators with traces equal zero and thus leading to the

W

predictions given ab ove

Sp ontaneous Symmetry Breaking in SU 

The SU symmetry is certainly broken since the new force corresp onding to

ould lead to very rapid proton decay if the exchange of the X and Y b osons w

these new gauge b osons were massless As mentioned ab ove from the limit on

the proton life time these SU gauge b oson have to be very heavy ie masses

ab ove GeV The generation of masses can be obtained again in a gauge

invariant way via the Higgs mechanism The Higgs eld is chosen in the adjoint

representation and the minimum can b e chosen in the following way

C B

C B

C B

C B

v

C B

C B

A

The XY gauge b osons of the SU group require a mass

M M g v

X Y

after eating of the scalar elds in the adjoint representation thus providing

the longitudinal degrees of freedom The eld is invariant under the rotations

of the SU SU U group so this symmetry is not broken and the

C L Y

corresp onding gauge b osons including the W and Z b osons remain massless

after the rst stage of SU symmetry breaking

The usual breakdown of the electroweak symmetry to SU U is

C em

achieved by a plet of Higgs elds for which the minimum of the eective

potential can be chosen at

B C

B C

C B

C B

v

C B

C B

A

The fourth and fth comp onentof corresp ond to the SU doublet

of the SM see eq Since the total charge in a representation has to b e zero

again the rst triplet of complex elds in which transforms as and

must have charge jj Since they couple to all fermions with mass

they can induce proton decay

e u

u d H

d

e

Such decays can be suppressed only by suciently high masses of the coloured

Higgs triplet These can obtain high masses through interaction terms between

and

Note that from eq has to be of the order of M while has

X

to b e of the order of M since

W

M g v

W

and

M

W

M

Z

cos

W

p

or more precisely v G GeV The minimum of the Higgs p otential

F

involves b oth and Despite this mixing the ratio v v has to

b e preserved hierarchy problem Radiative corrections sp oil usually such a ne

tuning so SU is in trouble As will b e discussed later also here sup ersymmetry

oers solutions for b oth this netuning and the hierarchy problem

Relations between Quark and Lepton Masses

The Higgs plet can b e used to generate fermion masses Since the plet of

the matter elds contains b oth leptons and downtyp e quarks their masses are

related while the uptyp e quark masses are free parameters At the GUT scale

one exp ects

m m

d e

m m

s

m m

b

Unfortunately the masses of the light quarks have large uncertainties from the

binding energies in the hadrons but the bquark mass can b e correctly predicted

from the mass after including radiative corrections see g for typical

graphs

Since the corrections from graphs involving the strong coupling constant

s

are dominant one exp ects in rst order

m m

b s b

O O

m M

s X

More precise formulae are given in the app endix and will be used in the last

chapter in a quantitative analysis since the bquark mass gives a rather strong

constraint on the evolution of the couplings and through the radiative corrections

involving the Yukawa couplings on the top quark mass

Chapter

Sup ersymmetry

Motivation

Sup ersymmetry presupp oses a symmetry between fermions and b osons

which can be realized in nature only if one assumes each particle with spin j

has a sup ersymmetric partner with spin j This leads to a doubling of the

particle sp ectrum see table which are assigned to two sup ermultiplets the

vector multiplet for the gauge and the chiral multiplet for the matter

elds Unfortunately the sup ersymmetric particles or sparticles have not been

observed so far so either sup ersymmetry is an elegant idea which has nothing

to do with reality or sup ersymmetry is not an exact symmetry in whichcasethe

sparticles can be heavier than the particles Many p eople opt for the latter way

out since there are many go o d reasons to b elieve in sup ersymmetry

SUSY solves the netuning problem

As mentioned b efore the radiative corrections in the SU mo del have

quadratic divergences from the diagrams in g whichleadtoM

H

O M where M is a cuto scale typically the unication scale if no

X

X

other scales intro duce new physics b eforehand

However in SUSY the lo op corrections contain both fermions F and

b osons B in the lo ops which according to the Feynman rules contribute

VECTOR MULTIPLET CHIRAL MULTIPLET

J J J J

C C C C

g g Q U D Q U D

L L

L L L L

C C

W W W W L E L E

L L

L L

B B H H H H

Table Assignment of gauge elds to the vector sup ereld and the matter

elds to the chiral sup ereld

with an opp osite sign ie

jO M M O jM M

SU SY F B H

where M isatypical SUSY mass scale In other words the netuning

SU SY

problem disapp ears if the SUSY partners are not to o heavy compared with

the known fermions An estimate of the required SUSY breaking scale can

be obtained by considering that the masses of the weak gauge b osons and

Higgs masses are b oth obtained by multiplying the vacuum exp ectation

value of the Higgs eld see previous chapter with a coupling constant so

one exp ects M M Requiring that the radiative corrections are not

W H

much larger than the masses themselves ie M M or replacing

W W

M by M M O yields after substitution into eq

W H H

M GeV

SU SY

SUSY oers a solution for the hierarchy problem

The p ossible explanation for the small ratio M M is simple

W X

in SUSY mo dels large radiative corrections from the topquark Yukawa

coupling to the Higgs sector drive one of the Higgs masses squared nega

tive thus changing the shap e of the eective p otential from the parab olic

shap e to the Mexican hat see g and triggering electroweak symme

try breaking Since radiative corrections are logarithmic in energy this

automatically leads to a large hierarchy between the scales In the SM one

could invoke a similar mechanism for the triggering of electroweak symme

try breaking but in that case the quadratic divergences in the radiative

corrections would upset the argument

In the MSSM the electroweak scale is governed by the starting values of

the parameters at the GUT scale and the topquark mass This strongly

constrains the SUSY mass sp ectrum as will b e discussed in the last chapter

SUSY yields unication of the coupling constants

After the precise measurements of the SU SU U coupling

C L Y

constants the p ossibility of coupling constant unication within the SM

could b e excluded since after extrap olation to high energies the three cou

pling constants would not meet in a single point This is demonstrated in

the upp er part of g which shows the evolution of the inverse of the

couplings as function of the logarithm of the energy In this presentation

the evolution b ecomes a straight line in rst order as is apparent from the

solution of the RGE eqn The second order corrections whichhave

been included in g by using eqs A from the app endix are so

small that they cause no visible deviation from a straight line

A single unication pointis excluded by more than standard deviations

The curve meets the crossing p oint of the other two coupling constants

only for a starting value at M while the measured value is

s Z

This is an exciting result since it means unication can

only be obtained if new physics enters between the electroweak and the

Planck scale Unification of the Couplings of the Electromagnetic, Weak and Strong Forces

Standard Model

Minimal Supersymmetric

Model

Figure Evolution of the inverse of the three coupling constants in the Stan

dard Mo del SM top and in the sup ersymmetric extension of the SM MSSM

b ottom Only in the latter case unication is obtained The SUSY particles are

assumed to contribute only ab ove the eective SUSY scale M of ab out one

SU SY

TeV which causes the change in slop e in the evolution of the couplings The

CL for this scale is indicated by the vertical lines dashed The evolution of the

couplings was calculated in second order see section A of the app endix with

the constants and calculated for the MSSM ab ove M in the b ottom

i ij SU SY

part and for the SM elsewhere The thickness of the lines represents the error

in the coupling constan ts

Figure distribution for M andM

SU SY GU T

It turns out that within the SUSY mo del p erfect unication can b e obtained

if the SUSY masses are of the order of one TeV This is shown in the b ottom

part of g the SUSY particles are assumed to contribute eectively

to the running of the coupling constants only for energies ab ovethetypical

SUSY mass scale which causes the change in the slop e of the lines near one

TeV From a t requiring unication one nds for the breakp oint M

SU SY

and the unication p oint M

GU T

M GeV

SU SY

M GeV

GU T

GU T

where g The rst error originates from the uncertainty in

GU T

the coupling constant while the second error is due to the uncertainty in

the mass splittings between the SUSY particles The distributions of

M and M for the t in the b ottom part of g are shown in g

SU SY GU T

These gures are an up date of the published gures using the newest

values of the coupling constants as shown in the gure

sp ectrum with a single Note that the parametrisation of the SUSY mass

mass scale is not adequate and leads to uncertainties However the errors

in the coupling constants mainly in are large and the uncertainties from

s

mass splittings between the sparticles are more than a factor two smaller

see eq In the last chapter the unication including a more detailed

treatmentofthe mass splittings will b e studied

One can ask What is the signicance of this observation For many p eople

it was the rst evidence for sup ersymmetry esp ecially since M was

SU SY

found in the range where the netuning problem do es not reapp ear 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 unication

of the coupling constants as exemplied in gs and reached the

TopTen of the citation list thus leading to discussions in practically all

p opular journals

NonSUSY enthusiasts were considering unication obvious with a total

of three free parameters M and M and three equations

GU T GU T SU SY

one can naively always nd a solution The latter statement is certainly

free not true searching for other typ es of new physics with the masses as

particles yields only rarely unication esp ecially if one requires in addition

that the unication scale is ab ove GeV in order to be consistent with

the proton lifetime limits and below the Planck scale in order to be in the

regime where gravity can be neglected From the mo dels tried only

a handful yielded unication The reason is simple intro ducing new

particles usually alters all three couplings simultaneously thus giving rise

to strong correlations between the slop es of the three lines For example

adding a fourth family of particles with an arbitrary mass will never yield

unication since it changes the slop es of all three coupling by the same

amount so if with three families unication cannot b e obtained it will not

work with four families either even if one has an additional free parameter

Nevertheless unication do es not prove sup ersymmetry it only gives an

interesting hint The real pro of would b e the observation of the sparticles

Unication with gravity

The spacetime symmetry group is the Poincare group Requiring lo cal

gauge invariance under the transformations of this group leads to the Ein

stein theory of gravitation Lo calizing b oth the internal and the spacetime

symmetry groups yields the YangMills gauge elds and the gravitational

elds This paves the way for the unication of gravity with the strong

and electroweak interactions The only nontrivial unication of an inter

nal symmetry and the spacetime symmetry group is the sup ersymmetry

group so sup ersymmetric theories automatically include gravity Un

fortunately sup ergravity mo dels are inherently nonrenormalizible whic h

prevents up to now clear predictions Nevertheless the sp ontaneous sym

metry breaking of sup ergravity is imp ortant for the low energy sp ectrum

of sup ersymmetry The most common scenario is the hidden sector

scenario in which one p ostulates two sectors of elds the visible sector

containing all the particles of the GUTs describ ed b efore and the hidden

sector which contains elds which lead to symmetry breaking of sup ersym

metry at some large scale One assumes that none of the elds in the

SU SY

hidden sector contains quantum numb ers of the visible sector so the two

sectors only communicate via gravitational interactions Consequently the

eective scale of sup ersymmetry breaking in the visible sector is suppressed

by a po wer of the Planck scale ie

n

SU SY

M

SU SY

n

M

P l anck ~c ~c d s d s ~ ~ ~ H W W u ν u ν

~μ ~μ

Figure Examples of proton decay in the minimal sup ersymmetric mo del via

wino and exchange

where n is mo deldep endent eg n in the Polonyi mo del Thus the

SUSY breaking scale can be large ab ove GeV while still pro ducing

a small breaking scale in the visible sector In this case the netuning

problem can b e avoided in a natural way and it is gratifying to see that the

rst exp erimental hints for M are indeed in the mass range consistent

SU SY

with eq

The hidden sector scenario leads to an eective lowenergy theory with

explicit soft breaking terms where soft implies that no new quadratic di

The softbreaking terms in stringinspired su vergences are generated

p ergravity mo dels have been studied recently in refs A nal theory

which simultaneously solves the problem and ex

plains the origin of sup ersymmetry breaking needs certainly a b etter un

derstanding of sup erstring theory

The unication scale in SUSY is large

As discussed in chapter the limits on the proton lifetime require the

unication scale to be ab ove GeV which is the case for the MSSM

In addition one has to consider proton decay via graphs of the typ e shown

in g These yield a strong constraint on the mixing in the Higgs

sector as will b e discussed in detail in the last chapter

Prediction of dark matter

The lightest sup ersymmetric particle LSP cannot decayinto normal mat

ter b ecause of Rparity conservation see the next section for a denition

of Rparity In addition Rparity forbids a coupling b etween the LSP and

normal matter

Consequently the LSP is an ideal candidate for dark matter which is

b elieved to account for a large fraction of all mass in the universe see next

chapter The mass of the dark matter particles is exp ected to be below

one TeV

SUSY interactions

The quantum numb ers and the gauge couplings of the particles and sparticles

have to b e the same since they b elong to the same multiplet structure

The interaction of the sparticles with normal matter is governed byanewmul

tiplicative quantum number called Rparity which is needed in order to prevent

baryon and lepton number violation Remember that quarks leptons and Hig

gses are all contained in the same chiral sup ermultiplet which allows couplings

between quarks and leptons Such transitions which could lead to rapid proton

decay are not observed in nature Therefore the SM particles are assigned a p os

itive Rparity and the sup ersymmetric partners are Ro dd Requiring Rparity

conservation implies that

sparticles can be pro duced only in pairs

the lightest sup ersymmetric particle is stable since its decay into normal

matter would change Rparity

the interactions of particles and sparticles can be dierent For example

the photon couples to electronp ositron pairs but the do es not

couple to selectronsp ositron pairs since in the latter case the Rparity

would change from to

The SUSY Mass Sp ectrum

Obviously SUSY cannot be an exact symmetry of nature or else the sup ersym

metric partners would have the same mass as the normal particles As mentioned

ab ove the sup ersymmetric partners should b e not to o heavy since otherwise the

hierarchy problem reapp ears

Furthermore if one requires that the breaking terms do not in tro duce quadratic

divergences only the socalled soft breaking terms are allowed

Using the sup ergravity inspired breaking terms which assume a common mass

m for the and another common mass m for the scalars leadstothe

following breaking term in the Lagrangian in the notation of ref

X X

L m j j m

Breaking i

i

h i

u c d c e c

Am h Q U H h Q D H h L E H Bm H H

a a a

ab b ab b ab b

Here

ude

h are the Yukawa couplings a b run over the generations

ab

Q are the SU doublet quark elds

a

c

U are the SU singlet chargeconjugated upquark elds

a

c

D are the SU singlet chargeconjugated downquark elds

b

L are the SU doublet lepton elds

a

c

E are the SU singlet chargeconjugated lepton elds

a

H are the SU doublet Higgs elds

are all scalar elds

i

are the gaugino elds

The last two terms in L originate from the cubic and quadratic terms

B r eak ing

A B and as free parameters In total we now have in the sup erp otential with

three couplings and ve mass parameters

i

m m t At B t

with the following b oundary conditions at M t

GU T

scalars m m m m m m

Q U D L E

gauginos M m i

i

couplings i

i GU T

Here M M and M are the gauginos masses of the U SU and SU

groups In N sup ergravity one exp ects at the Planck scale B A With

these parameters and the initial conditions at the GUT scale the masses of all

SUSY 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 corrections

from virtual gluons like the ones in g for quarks which makes them heavier

than the sleptons at low energies These radiative corrections can be calculated

from the corresp onding RGE which have been assembled in the app endix The

solutions are

t m m cos M m

Z E

L

m t m m cos M

Z

L

m t m m cos M

E Z

R

m t m m cos M

U Z

L

m t m m cos M

D Z

L

m t m m cos M

U Z

R

m t m m cos M

D Z

R

where is the mixing angle between the two Higgs doublets which will be de

ned more precisely in section The co ecients dep end on the couplings as

shown explicitly in the app endix They were calculated for the parameters from

the typical t shown in table M GeV and

GU T GU T

sin For the third generation the Yukawa coupling is not necessar

W

ily negligible If one includes only the correction from the top Yukawa coupling

Y one nds

t

m t m

b D

R R

m t m m m

b D

L L

m m m m t m

U t t

R R

m m m t m m

t U t

L L

For large values of the mixing angle tan in the Higgs sector the bquark Yukawa coupling

can b ecome large to o However since the limits on the proton lifetime limit tan to rather

small values see last chapter this option is not further considered here

The numerical factors have b een calculated for A and the explicit dep en

t

dence on the couplings can be found in the app endix Note that only the left

handed bquark gets corrections from the topquark Yukawa coupling through a

lo op with a charged Higgsino and a topquark The subscripts L or R do not

indicate the helicity since the squarks and sleptons have no spin The lab els

just indicate in analogy to the nonSUSY particles if they are SU doublets

or singlets The mass eigenstates are mixtures of the L and R

states Since the mixing is prop ortional to the Yukawa coupling we will only

consider the mixing for the top quarks After mixing the mass eigenstates are

using the same numerical input as for the light quarks

q

m t m m m m m A m tan

t

t t t

t t t

L R



L R

i h

cos M m m m

Z t

r

h i

m m cos M m A m tan

t

t

Z

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

t

m

A M A

t Z t

m

M

Z

Note that for large values of A or combined with a small tan the

t

splitting b ecomes large and one of the stop masses can b ecome very small since

the stop mass lower limit is ab out GeV this yields a constraint on the

p ossible values of m m tan and

t

Charginos and Neutralinos

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

t

i

m M t

i

i

Numerically at the weak scale t lnM M one nds see g

GU T Z

M g m

M M m

Z

M M m

Z

Since the obtain corrections from the strong coupling constant they

grow heavier than the gauginos of the SU SU U group

C L Y

The calculation of the mass eigenstates is more complicated since b oth Hig

gsinos and gauginos are spin particles so the mass eigenstates are in general

mixtures of the weak interaction eigenstates The mixing of the and 600

500 χ0 mass [GeV] i χ+/- i 400

300

q~

200 ~ eL

M3

100 M2

M1 0 2 4 6 8 10 12 14 16

log10 Q

Figure Typical running of the squark q slepton e and gaugino

L

M M M masses solid lines The dashed lines indicate the running of

the four neutralinos and two charginos

gauginos whose mass eigenstates are called charginos and neutralinos for the

charged and neutral elds resp ectively can be parametrised by the following

Lagrangian

c

L M M M hc

Gaug inoH ig g sino a a

where a are the Ma jorana elds and

a

B

C B

C B

W W

C B

C B

H H

A

H

are the Ma jorana and Dirac elds resp ectively Here all the

terms in the Lagrangian were assembled into matrix notation similarly to the

mass matrix for the mixing b etween B and W in the SM eq The mass

matrices can b e written as

M M cos sin M sin sin

Z W Z W

B C

M M cos cos M sin cos

B C

Z W Z W

M B C

A

M cos sin M cos cos

Z W Z W

M sin sin M sin cos

Z W Z W

p

M M sin

W

c

p

M

M cos

W

The last matrix leads to two chargino eigenstates with mass eigenvalues

q

M M sin M cos M M M M M

W W

W

The dep endence on the parameters at the GUT scale can be estimated by sub

stituting for M and their values at the weak scale M M m and

Z

M In the case favoured by the t discussed in chapter one

Z

nds M M inwhichcasethecharginos eigenstates are approximately

Z

M and

The four neutralino mass eigenstates are denoted by i with

i

 

masses M M The sign of the mass eigenvalue corresp onds to the

 

CP quantum number of the Ma jorana neutralino state

In the limiting case M M M one can neglect the odiagonal ele

Z

ts and the mass eigenstates b ecome men

p p

B W H H H H

i

with eigenvalues jM j jM j jj and jj resp ectively In other words the bino

and neutral wino do not mix with each other nor with the Higgsino eigenstates

in this limiting case As we will see in a quantitative analysis the data indeed

prefer M M M so the LSP is binolike which has consequences for dark

Z

matter searches

Higgs Sector

The Higgs sector of the SUSY mo del has to b e extended with resp ect to the one

of the SM for two reasons

the Higgsinos have spin which implies they contribute to the gauge

unless one has pairs of Higgsinos with opp osite hyp ercharge so

in addition to the Higgs doublet with Y one needs a second one with

W

Y

W

H H

H H

H H

The intro duction of the second Higgs doublet solves simultaneously the

problem that a single doublet can give mass to only either the up or down

typ e quarks as is apparent from the fact that only the neutral comp onents

have a non zero vev since else the vacuum would not be neutral So one

can write

v

H H

v

In the SM the conjugate eld can give mass to the other typ e However

sup ersymmetry is a spinsymmetry in which the matter and Higgs elds

are contained in the same chiral sup ermultiplet This forbids couplings

between matter elds and conjugate Higgs elds With the two Higgs elds

intro duced ab ove H generates mass to the downtyp e matter elds while

H generates mass for the uptyp e matter elds

The sup ersymmetric mo del with two Higgs doublets is called the Minimal Su

p ersymmetric Standard Mo del MSSM The mass sp ectrum can b e analyzed by

considering again the expansion around the vacuum exp ectation value given by

eq

p

H cos h sin iA sin iG sin v

H

H sin G cos

H cos G sin

H

p

H sin h cos iA cos iG sin v

Here H h and A represent the uctuations around the vacuum corresp onding to

the real Higgs elds while the Gs represent the Goldstone elds which disapp ear

in exchange for the longitudinal p olarization comp onents of the heavy gauge

b osons The imaginary and real sectors do not mix since they have dierent

CPeigenvalues and are the mixing angles in these dierent sectors The

mass eigenvalues of the imaginary comp onents are CPo dd so one is left with

neutral CPeven Higgs b osons H and h CPo dd neutral Higgs b osons A

and CPeven charged Higgs b osons

The complete tree level p otential for the neutral Higgs sector assuming colour

and charge conservation reads

g g

H V H hc H H j m jH j m jH j m j jH jH

Note that in comparison with the p otential in the SM the rst terms do not have

arbitrary co ecients anymore but these are restricted to be the gauge coupling

constants in sup ersymmetry again b ecause the Higgses b elong to the same chiral

multiplet as the matter elds The last three terms in the p otential arise from

the soft breaking terms with the following b oundary conditions at the GUT scale

Bm m m m m

where is the value of at the GUT scale Since generates mass for the

Higgsinos one exp ects to b e small compared with the GUT scale Alow value

can b e obtained dynamically if one adds a singlet scalar eld to the MSSM This

will not be considered further Instead is considered to be a free parameter to

be determined from data see chapter

From the p otential one can derive easily the ve Higgs masses in terms of

these parameters by diagonalization of the mass matrices

V

H

M

ij

j j

where is a generic notation for the real or imaginary part of the Higgs eld

i

Since the Higgs particles are quantum eld oscillations around the minimum

eq has to be evaluated at the minimum One nds zero masses for the

Goldstone b osons These wouldb e Goldstone b osons G and G are eaten by

the SU gauge b osons For the masses of the ve remaining Higgs particles one

nds

CPo dd neutral Higgs A

m m m

A

Charged Higgses H

m m M



A H W

CPeven neutral Higgses H h

q

m m M m M cos m M

Hh A Z A Z A Z

By convention m m The mixing angles and are related by

H h

m M

A Z

tan tan

m M

A Z



In principle one should consider the running of the gauge couplings b etween the electroweak

scale and the mass scale of the Higgs b osons However since the Higgs b osons are exp ected to

be belowtheTeV mass scale this running is small and can b e neglected if one considers all

other sources of uncertainty in the MSSM x C ~ C ~ t L t L t L H H 2 2 + h2 h h t t t

Scalar Fermion

loop loop

Figure Corrections to the Higgs selfenergy from Yukawa typ e interactions

v and v have b een chosen real and p ositive which implies

Furthermore the electroweak breaking conditions require tan so

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

relations



m M

W

H

m m M

h A H

m M cos M

h Z Z

m m m M

h H A Z

After including radiative corrections the lightest neutral Higgs m b ecomes con

h

siderably heavier and these relations are not valid anymore The mass formulae

including the radiative corrections are given in the app endix

Electroweak Symmetry Breaking

The coupling plays an imp ortant role in the shap e of the p otential and con

sequently in the pattern of electroweak symmetry breaking which o ccurs if the

minimum of the potential is not obtained for H H In the

adho c by requiring the co ecient of the SM this condition could be intro duced

quadratic term to be negative In sup ersymmetry this term is restricted by the

gauge couplings A nontrivial minimum can only be obtained by the soft

V

breaking terms if the mass matrix for the Higgs sector given by M

ij

H H

i j

has a negative eigenvalue This is obtained if the determinant is negative ie

jm tj m t m t

In order that the new minimum is below the trivial minimum with H

H one has to require in addition V v v V V which

H H H

is fullled if

m tm t jm tj

If one compares eqns and and notices from eq that m m

one realizes that these conditions cannot b e fullled simultaneously at least not

at the GUT scale

However at lower energies there are substantial radiative corrections which

can cause dierences between m and m since the rst one involves mass cor

rections prop ortional to the top Yukawa coupling Y while for the latter these

t

corrections are prop ortional to the b ottom Yukawa coupling Typical diagrams

are shown in g From the RGE for the mass parameters in the Higgs

potential one nds at the weak scale

t

m t m m

m t m m

A m m A m

t t

m t m m A m

t

The co ecients were evaluated for the parameters of the t to the exp erimental

data central column of table in chapter The explicit dep endence of the

co ecients on the coupling constants is given in the app endix The co ecients of

the last three terms in m dep end on the top Yukawa coupling This dep endence

disapp ears 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 the

diagram with the top quarks dominates in this case m decreases much faster

than m with decreasing energy and the p otential takes the form of a mexican

hat as so on as conditions and are satised Since At is exp ected

to be small the dominant negative contribution is prop ortional to m see eq

so the electroweak breaking scale is a sensitive function of b oth the initial

conditions the top Yukawa coupling and the gaugino masses

The minimum of the p otential can be found by requiring

V g g

m v m v v v v

j jH

g g V

m v m v v v v

j jH

Here we substituted

H v v cos H v v sin

where

v

v v v v GeV tan

v

From the minimization conditions given ab ove one can derive easily

n o

v m m tan

g g tan

m m m sin

g g m m tan

M v

Z

tan

g

M v M cos

W

W Z

The derivation of these formulae including the onelo op radiative corrections

is given in the app endix

Chapter

The Big Bang Theory

Intro duction

In the s Hubble discovered that most galaxies showed a redshift in the visible

sp ectra implying that they were moving awayfromeach other This observation

is one of the basic building blo cks of the Big Bang Theory which assumes the

universe is expanding thus solving the problem that a static universe cannot be

stable according the Einsteins equations of general relativity An expanding

universe will co ol down so at the b eginning the universe might have b een hot

The remnants of the radiation of such a hot universe can still b e observed to day

as microwave background radiation corresp onding to a temp erature of a few de

grees This radiation was rst predicted by Gamow but accidentally observed

in by Penzia and Wilson from Bell Lab oratories as noise in microwave

antennas used for communication with early satellites Such an antenna is only

sensitive to a single frequency Recently the whole sp ectrum was measured by

the COBE satellite and it was found to be indeed describable by a black body

radiation as shown in g from ref The deviation from p erfect isotropy

if one ignores the dip ole anisotropy from the Doppler shift caused by the move

ment of the earth through the microwave background is a few times This

has strong implications for theories concerning the clustering of galaxies since

this radiation was released so on after the bang and hardly interacted after

wards so the inhomogeneities in this radiation are prop ortional to the density

uctuations in the early universe These density uctuations are the seeds for

the nal formation of galaxies As will b e discussed later such small anisotropies

have strong implications for the mo dels trying to understand the formation of

dark matter in the universe Direct evidence that galaxies and the nature of the

the temp erature from the microwave background is indeed the temp erature of

the universe came from the measurement of the temp erature of gas clouds deep

in space As it happ ens the rotational energy levels of cyanogen CN are such

that the K background radiation can excite these From the detection

of the relative p opulation of the groundstate and the higher levels the excitation

temp erature was determined to be T K which is in excel

CN

They were awarded the Nob el prize for this discovery in



Cosmic Background Explorer

Figure Sp ectrum of the microwave background radiation as measured by

the COBE satellite The curve is the black body radiation corresp onding to a

temp erature of K

lent agreement with the direct measurement of the microwave background of

Kby COBE

Other evidence that the universe was indeed very hot at the b eginning came

from the measurement of the natural abundance of the light elements the uni

verse consists for out of hydrogen helium and for the remaining

elements Both the rarity of heavy elements and the large abundance of helium

are hard to explain unless one assumes ahot universe at the b eginning

The reason for the low abundance of the heavier elements in a hot universe

is simple they are cracked by the intense radiation around The abundance of

the light elements is plotted in g as function of the ratio of primordial

baryons and photons from ref Agreement with exp erimental observations

can only b e obtained for in the range

The very heavy elements can be pro duced only at much lower temp eratures

but high pressure so it is usually assumed that the heavy elements on earth and

in our b o dies were co oked by the high pressure inside the cores of collapsing stars

which explo ded as sup ernovae and put large quantities of these elements into the

heavens They clustered into galaxies under the inuence of gravity

The ratio of helium and hydrogen is determined by the number of

available for fusion into deuterium and subsequently into helium at the freezeout

temp erature of ab out MeV or KAt these high temp eratures no complex

nuclei can exist only free protons and neutrons They can be converted into

en p Note each other via charged weak interactions like ep n and

e e

that this is the same interaction which is resp onsible for the decay of a free

neutron into a proton electron and antineutrino The weak interactions maintain

Figure Big Bang nucleosynthesis predictions for the primordial abundance of

the light elements as function of the primordial ratio of baryons and photons

From ref

thermal equilibrium b etween the protons and neutrons as long as the densityand

temp erature are high enough thus leading to a Boltzmann distribution

n

Qk T

e

p

where Q m m c MeV is the energy dierence b etween the states

n p

Thermal equilibrium is not guaranteed anymore if the weak interaction rates

are slower than the expansion rate of the universe given by the Hubble constant

ie freezeout o ccurs when Ht This happ ens by the time the temp erature

is ab out K or MeV Then the ratio np is ab out Since the photon

energies at these temp eratures are to o lowtocracktheheavier nuclei nuclei can

H n H form through reactions like n p H H p H and

which in turn react to form He The latter is a very stable nucleus which hardly

can be cracked so the chain essentially stops till all neutrons are bound inside

He Heavier nuclei are hardly pro duced at this stage since there are no stable

elements with or nuclei so as so on as He catches another nuclei it will decay

Consequently the np ratio as determined from the Boltzmann distribution

yields a He mass fraction Y

He

np

Y

he

np

Figure The Z lineshap e for dierent number of neutrino typ es The data

black points exclude more than three typ es with mass b elow M GeV

Z

Exp erimentally the mass fraction Y of He is The mass fractions

P

of deuterium He and Li are many orders of magnitude smaller The

concentration of the latter elements is a strong function of the primordial baryon

density see g since at high enough density all the deuterium will fuse into

He thus eliminating the comp onents for He and Li

As said ab ove freezeout o ccurs if Ht Thus the expansion rate H t

around T MeV determines the He abundance The expansion rate in turn is

determined by the fraction of relativistic particles like neutrinos light

etc Roughly for each additional sp ecies the primordial He abundance increases

by as shown in g for a neutron halflife time of minutes from ref

The neutron lifetime is not negligible on the scale of the rst three minutes

so it has to be taken into account If one wants to reconcile the abundance of

all light elements there can only be three neutrino sp ecies see g with

practically no ro om for other weakly interacting relativistic particles like light

photinos Present collider data conrm that there are indeed only three light

neutrinos

N

The strongest constraint comes from the Z resonance data at LEP An

example of the quality of the data is shown in g Note that collider data

limit the number of neutrino generations while nucleosynthesis is sensitive to

all kinds of light particles where light means ab out one MeV or less Happily



The presently accepted value is minutes

enough the collider data require the lightest neutralino to b e ab ove GeV

so there is no conict b etween cosmology and sup ersymmetry

Note that this is a b eautiful example of the interactions between cosmology

and 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 conservative

upp er limit on the He abundance to be and at the LEP accelerator one

observes that the number of light neutrinos is indeed three Alternatively one

can combine the accelerator data and the abundance of the light elements to

p ostdict the primordial helium abundance to be and use it to obtain an

upp er limit on the baryonic density

b c

where is the critical density needed for a at universe The critical density

c

will be calculated in section Thus LEP data in combination with baryoge

nesis strengthens the argument that we need nonbaryonic dark matter in a at

universe for which Other arguments for dark matter will b e discussed in

c

section

In spite of the marvelous successes of this mo del of the universe many ques

tions and problems remain as mentioned in the Intro duction However GUTs

can provide amazingly simple solutions at least in principle since many details

are still op en

These problems will be discussed more quantitatively in the next sections

starting with Einsteins equations in a homogeneous and isotropic universe con

ditions whichhave b een well veried in the presentuniverse and whichmakethe

solutions to Einsteins equations particularly simple Esp eciallyitis easy to see

that a phase transition can lead to ination the key in all present cosmological

theories

Predictions from General Relativity

At large distances the univ erse is homogeneous ie one nds the same mass

density everywhere in the universe typically

kg m

univ

which corresp onds to hydrogen atoms p er cubic meter In comparison an

extremely go o d vacuum of Nm at K contains ab out molecules

p er cubic meter Of course the volume to b e averaged over should b e chosen to b e

much larger than the size of clusters of galaxies Furthermore the same density

and temp erature is observed in all directions ie the universe is very isotropic If

no p oint and no direction is preferred in the universe the p ossible geometry of the

ie universe b ecomes very simple the curvature hastobethe same everywhere

instead of a curvature tensor one needs only asinglenumber usually written as

K tkR t where R t is the socalled scale factor This factor can b e used

to dene dimensionless timeindep endent comoving co ordinates in an expanding

universe the prop er or real distance D tbetween two galaxies scales as

D t R td

where d is the distance at a given time t The factor k intro duced ab ove denes

the sign of the curvature k implies no curvature ie a at universe while

k corresp onds to a space with a p ositive curvature spherical and

k corresp onds to a space with a negativecurvature hyp erb olic

The movement of a galaxy in a homogeneous universe can b e compared to the

molecules in a gas the stars are just the atoms of a and the molecules

are homogeneously distributed Dierentiating equation results in

v R td

or substituting d from eq results in the famous relation b etween the velo cit y

and the distance of two galaxies

R t

v D t H tD t

R t

where H tisthe famous Hubble constant

This relation between the velo city and the distance of the galaxies was rst

observed exp erimentally by Hubble in the s He observed that all neigh

b ouring galaxies showed a redshift in the sp ectral lines of the light emitted by

sp ecic elements and the redshift was roughly prop ortional to the distance So

this was the rst evidence that w e are living in an expanding universe which

mighthave been created by a Big Bang

The Hubble relation is a direct consequence of the homogeneity and

isotropy of the universe since the scale factor cannot be a constant in that case

This follows directly from Einsteins eld equations of general relativity which

can be written as

G

R t utptR t

c

kc G R t

ut

R t c R

where G Nm kg is the gravitational constant R and R are the

derivatives of R with resp ect to time p is the pressure and u is the energy density

A static universe in which the derivatives and pressure are zero implies ut

so a static universe cannot exist unless one intro duces additional p otential

energy in the universe eg Einsteins cosmological constant At present there is

no exp erimental evidence for such a term

Interpretation in terms of Newtonian Me

chanics

Since the energy density and corresp ondingly the curvature is small in our

present universe relativistic eects 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 be

related the average density

M R

For an expanding universe the total mechanical energy of the mass shell can be

written as the sum of the kinetic and p otential energy

GM m

mR E

tot

R

R

mR G

R

The expression in brackets is just the left hand side of eq so the sum of

kinetic and p otential energy determines the sign of the curvature k If k

then E implying that the universe will recollapse under the inuence of

tot

gravity Big Crunch just like a ro cket which is launched with a sp eed below

the escap e velo city will return to the earth k on the other hand implies

that the universe will expand and co ol forever Big Chill In case of k the

total energy equals zero at Euclidean space in which case the gravitational

energy or equivalently the mass density is sucient to halt the expansion

The rst eld equation eq follows from the second equation eq

by dierentiation and taking into account that in an expanding universe energy

is converted into gravitational p otential energy when the volume increases by

an innitesimal amount V then the remaining energy in the gas decreases by

an amount pV where p denotes the pressure Therefore

phy s

R t

ptV t tV t E t p

phy s phy s

R t

Here weusedV V R t where V is the volume in comoving timeindep endent

phy s

co ordinates analogous to eq On the other hand follows from E t

utV t

phy s

R t

E tutV t utV t

phy s phy s

R t

Combining eqs and results in

R t

utpt u t

R t

Substituting this equation for u t after dierentiation 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 and

pressure are prop ortional ie p c where for cold nonrelativistic

Figure Evolution of the radius of the universe for a closed C at F or

op en O universe with and without ination From ref

matter p and for a relativistic hot gas as follows from elementary

Thermo dynamics From eq given ab ove it follows immediately that

R

and substituting this into eq results in



R t

if we neglect the curvature term ie either R is large or k small As we will see

b oth are true after the inationary phase of the universe

hot r el ativ istic tt R t D R

Ht

inf l ation t t t R e const

hot r el ativ istic t t t R t D R

col d non relativ tt R t D R

Table The time dep endence of the scale factor and energy density during

various stages in the evolution of the universe Typically t s t

s and t yrs The constants D are integration constants

i

The time dep endence has b een summarized in table for various stages of

the universe The inationary period will be discussed in the next section One

observes that the scale factor vanishes at some time t and the energy density

b ecomes innite at that time This singularity explains the p opular name Big

erse The solutions for R t are shown Bang theory for the evolution of the univ

graphically in g for three cases a at universe k an op en universe

k and a closed universe k from ref

An op en or at universe will expand forever since the kinetic energy is larger

than the gravitational attraction A closed universe will recollapse The lifetime

of a closed universe with p and a total mass M of cold nonrelativistic

matter is

M MG

s t

c

M

P

so the present lifetime of the universe of at least yrs gives a strong upp er

limit on the density of the universe

The lifetime can easily b e calculated if weassumeaatuniverse from table

it follows that R t t for most of the time Substituting this and its time

derivative into the denition of the Hubble constant eq results in

H t

t

With the presently accepted value of the measured Hubble constant

km

h s h yrs H h

s Mpc

where h indicates the exp erimental uncertainty h one nds for the

age of the universe

t H h yrs

univ er se

The critical density which is the density corresp onding to a at universe can

be calculated from eqn by requiring E or equivalen tly k and

tot

substituting for RR the Hubble constant see eq

H

h kg m

c

G

where the numerical value of H from eq was used

The size of the observable universe the horizon distance D can b e calculated

h

in the following way the prop er distance between two points is R td eq

where d is the distance in comoving co ordinates Light propagates on the light

cone This can be studied most easily by considering the time in comoving

co ordinates with

dt R td

In comoving co ordinates the distance d light can propagate is cd so d

h h

R R

c d c dtR torthe prop er distance D R td equals

h h

Z

t

dt

D cR t

h

R t

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

yields

D ct cH th m

h

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

Temp erature Evolution of the Universe

In the previous section the scale factor and the energy density were calculated

as function of time The energy density has two comp onents the energy density

from the photon radiation in the microwave background and the energy

rad

density of the nonrelativistic matter At present is negligible but

matter rad

at the b eginning of the universe it was the dominating energy Assuming this

radiation to b e in thermal equilibrium with matter implies a black b o dy radiation

with a frequency distribution given by Plancks law and an energy density

aT

rad

where a J m K Since R see table one nds from

rad

eq

T R t

from which follows immediately RR TT and R T Substituting

these expressions and eq into the second eld equation eq leads to

T aG

T

T c

since the term with kc is only prop ortional to T so it can b e neglected at high

temp eratures Integrating eq yields

s s

s s c

p

T MeV K

aG t t

t

p

so the temp erature drops as t

From this equation one observes that ab out one microsecond after the Big

Bang the temp erature has dropp ed from a value ab ove the Planck temp erature

corresp onding to an energy of GeV to a temp erature of ab out one GeV

so after ab out one microsecond the temp erature is already to o low to generate

protons and after ab out one second the lightest matter particles the electrons

are frozen out After ab out three minutes the temp erature is so low that the

light elements b ecome stable and after ab out years atoms can form

t all matter b ecomes neutral and the photons can escap e These At this momen

are the photons which are still around in the form of the microwavebackground

radiation

After the discovery of the microwave background radiation the Big Bang

theory gained widespread acceptance Nevertheless the simplest mo del as for

mulated here has several serious problems which can only be solved by the

socalled inationary mo dels These mo dels have the bizarre prop erty that the

expansion of the universe go es faster than the sp eed of light This is not a contra

diction of sp ecial relativity since these regions are causally disconnected so no

information will b e transmitted Sp ecial relativity do es not restrict the velo cities

of causally disconnected ob jects Howev er such inationary scenarios require

the intro duction of a scalar eld eg the Higgs eld discussed in the previous

chapter For certain conditions of the p otential of this eld the gravitational

Figure The atness of the universe after ination is easily understo o d if one

thinks ab out the ination of a ballo on

force b ecomes repulsive as can be derived directly from the Einstein equations

given ab ove This will b e discussed in more detail after a short summary of the

main problems and questions of the simple Big Bang theory

Flatness Problem

At present we do not know if the universe is op en or closed but exp erimentally

the ratio of the actual density to the critical density is b ound as follows

c

The luminous matter contributes only ab out to but from the dynamics of

the galaxies one estimates that the galaxies contribute between and so

the lower limit on stems from these observations The upp er limit is obtained

from the lower limit on the lifetime of the universe From the dating of the oldest

stars and the elements one knows that the universe is at least years old

which gives an upp er limit on the Hubble constant eq and consequently

on the density This do es lo ok like a p erfectly acceptable number and the

universe might even be p erfectly at since is not excluded However

it can be shown easily that grows with time as t and for the present

lifetime t s this number b ecomes big unless very sp ecial initial conditions

limit the prop ortionality constant to b e exceedingly small This constant can b e

calculated easily From eq and the denition of eq one nds

c

kc

t

R tH 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 P

ABab tr

x

Figure The spacetime diagram of the microwave background radiation

which was released at the time t Photons observed in point P from opp osite

r

directions traveled some yrs with hardly anyinteractions from the p oints A

and B resp ectively The distance light could have traveled b etween the Big Bang

and t is only ab whichismuch smaller than the distance AB Consequently the

r

points A and B could never have been in causal contact with each other Nev

ertheless the radiation from A and B have the same temp erature although the

horizon ab is much smaller horizon problem The problem can b e solved if one

assumes the region of causal contact was much larger than ab through ination

of spacetime via a phase transition

to zero for t very large implies that k must have b een very close to zero right

from the b eginning Remember that k is prop ortional to the sum of p otential

and kinetic energy in the nonrelativistic approximation One can show that

in order for to lie in the range close to now implies that in the early universe

j j M T or for T M

P

P

j j

In other words if the density of the initial universe was above the critical density

say by the universe would have collapsed long ago On the other hand

c

would the density have been below the critical density by a similar amount the

present density in the universe would have b een negligible small and life could

not 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 time

t years when the microwave background was released Nevertheless the

temp erature of the microwave background radiation is the same in all directions

How did these photons thermalize after b eing emitted some years ago One

should realize that the density in the universe is exceedingly low so photons

from opp osite directions have traveled some lightyears without interac

tions Since the distance scales as t these regions were ab out light years

apart at the time they were released ie the distance AB in g which

is two orders of magnitude larger than the horizon of the universe at that time

distance ab so no signal could have b een transmitted Nevertheless the tem

p erature dierence TT between these regions is less than as shown bythe

recent COBE data As with the atness problem one can imp ose an accidental

temp erature isotropy in the universe as an initial condition but with all the hefty

uctuations during the Big Bang this is a very unsatisfactory explanation As

we will see later ination solves b oth problems in a very elegantway

Magnetic Monop ole Problem

Magnetic monop oles are predicted by GUTs as top ological defects in the Higgs

eld after sp ontaneous symmetry breaking the vacuum obtains a nonzero vac

uum exp ectation value in a given region Dierent regions may have dierent

orientations of the phases of the Higgs eld and the b orderlines of these regions

have the prop erties exp ected for magnetic monop oles Unfortunately the

magnetic monop ole density is very small if not zero Their absence has to be

explained in any theory based on GUTs with SSB The rst attackwas made by

Alan Guth who invented ination for this problem Although the original mo del

did not solve the monop ole problem it provided a p erfectly reasonable solution

for the horizon and atness problem An alternative version of ination the so

called new ination which was invented by AD Linde and indep endently by

Albrecht and Steinhardt provided also a solution of the monop ole problem

as will b e discussed in section

The smo othness Problem

Our universe has density inhomogeneities in the form of galaxies On a large scale

the sp ectrum of inhomogeneities is approximately scaleinvariant which can be

understo o d in the inationary scenario as follows the ination smo othens out

any inhomogeneities which might have b een present in the initial conditions

Then in the course of the phase transition inhomogeneities are generated by the

quantum uctuations of the Higgs eld on a v ery small scale of length namely

the scale where quantum eects are imp ortant These density uctuations are

then enlarged to an astronomical scale by ination and they stay scale invariant

as is obvious if one thinks ab out a little circle on a ballo on which stays a circle

after ination but just on a larger scale

Ination

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 p ositive However the situation

can change drastically if the universe undergo es a rstorder phase transition In

Grand Unied Theories such phase transitions are exp ected eg the highly sym

metric phase mighthave b een an SU symmetric state while the less symmetric

state corresp onds to the SU SU U symmetry of the Standard

C L Y

Mo del The description of the sp ontaneous symmetry breaking by the Higgs

mechanism leads to a sp ecic picture of this phasetransition the Higgs eld is

a scalar eld which lls the vacuum with a p otential energy V The value of

the potential is temp erature dep endent at high temp eratures the minimum o c

curs for but for temp eratures belo w the critical temp erature the ground

state ie the state with the lowest energy is reached for a value of the eld

This is completely analogous to other phase transitions eg in sup ercon

ductivity the scalar eld corresp onds to the density of spin Co op er pairs or in

ferromagnetism it would be the magnetization

If during the expansion of the universe the energy density falls b elowthe en

ergy density of this scalar eld something dramatic can happ en the deceleration

can b ecome an acceleration leading to a rapid expansion of the universe usually

called ination

This can be understo o d as follows if the vacuum is lled with this p otential

energy of the scalar eld with an energy density the work W done during

vac

the expansion is pV However the gain in energy is V since the p otential

vac

energy of the vacuum do es not change a void stays a void as long as no phase

transition 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 stay

constant ie pV V or

vac

p

vac

This is the famous equation of state in case of a p otential dominated vacuum In

this case eq reduces to

GRt R

This equation has the solution

t

R t e

where

s

G

As we have seen in the previous chapter the symmetry breaking of a GUT

happ ens at an energy of GeV The energy density at this energy is extremely

high

E

GU T

Jm u c

hc

where the p owers are derived from dimensional analysis Inserting this result into

eq yields

s

Thus the universe inates extremely rapidly its diameter doubles every ln

s The behaviour of the Hubble constant during the inationary era is then

determined by eq which yields

H t

Clearly the inationary scenario provides in an elegantway solutions for many

of the shortcomings of the BBT

The rapid expansion removes all curvature in spacetime thus providing a

solution for the atness problem

After expansion the universe reheats b ecause of the quantum uctuations

around the new minimum of the vacuum thus thermalizing a region much

larger than the visible region at that time This explains why all visible

regions 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 monop oles since

after suciently large ination the monop oles are diluted to a negligible

level

However the whole idea of ination only works if one assumes the ination to go

smo othly from a single homogenous region to a large homogenous region many

times the size of our universe socalled new ination This requires rather sp e

cial conditions for the shap e of the p otential as was p ointed out by Linde and

indep endently by Albrecht and Steinhardt after the original intro duction of

ination by Guth The problem is that the corresp onding scalar elds pro

viding the potential energy of the vacuum have to be weakly interacting since

otherwise the phase transition will involve only microscopic distances according

to the uncertainty relation The Higgs elds providing sp ontaneous symmetry

breaking are interacting to o strongly so one has to intro duce additional weakly

interacting scalar elds It is nontrivial to combine the requirement of a neg

ligible small cosmological constant which represents the p otential energy of the

vacuum with a vacuum lled with Higgs elds to generate masses This re

quires large cancellations of p ositive and negativecontributions which o ccur eg

in unbroken sup ersymmetric theories However these theories if they describ e

the real world have to be broken Details on these problems are discussed by

Olive and in the recent text books by Borner and Kolb and Turner In

spite of these problems the arguments in favour of ination are so strong that it

has become the only acceptable paradigm of present cosmology

Origin of Matter

As discussed ab ove matter in our universe consists largely of hydrogen

and helium typically The absence of antimatter can be

explained if one has phase transitions and among others CPviolation as dis

cussed in chapter At present the nuclei dominate the energy density in the

Figure Mo dels including hot dark matter HDM can describ e the large

scale structure of the universe as prob ed by the galaxy surveys QDOT and

COBE temp erature anisotropy b etter than mo dels 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 energy

density of the many photons around decreases T while the energy densityof

the nuclei decreases T see eq

Large amounts of matter can be created from the energy release during the

inationary phase of the universe This can b e easily estimated as follo ws After

ination the universe has a macroscopic size typically the size of a fo otball or

larger The large energy density inthisvolume see eq yields a total energy

many times the mass in our present universe

Thus within the inationary scenario the universe could originate as a quan

tum uctuation starting from absolute nothing ie a state devoid of space

time and matter with a total energy equal to zero Most matter was created after

the inationary phase from the decay of the eld quanta of the elds resp onsable

for the ination Of course a quantum description of spacetime can b e discussed

only in the context of quantum gravity so these ideas must b e considered sp ecu

lativeuntil a theory of quantum gravity is formulated and proven by exp eriment

Nevertheless it is fascinating to contemplate that physical laws may determine

olution of our universe but they may remove also the need for not only the ev

assumptions ab out the initial conditions

Dark Matter

The visible matter is clustered in large galaxies which are themselves clustered in

clusters and sup erclusters with immense voids in b etween From the movements

of the galaxies one is forced to conclude that there must be much more matter

than the observed visible matter if we want to stick to Newtonian mechanics

The most impressive evidence for the dark ie not visible matter comes from the

socalled at rotation curves the orbital velo cities of luminous matter around

the central of spiral galaxies remain constant out to the far edges of the galaxies

in apparent contradiction to velo city distributions exp ected from Kepplers law

M r

v r G

r

where r is the radial distance to the centre of the galaxy M r the mass of the

galaxy inside a sphere with radius r and G the gravitational constant From

p

this law one exp ects the velo cities to decrease with r if the mass is concen

trated in the centre which is certainly the case for the visible matter Velo cities

indep endent of r imply M r r to be constant or M r r Such a behaviour

is exp ected for weakly interacting matter like neutrinos or photinos

since strongly interacting matter would be attracted to the centre by gravity

interact lo ose energy and concentrate in the centre just like the visible matter

do es Also the dynamical prop erties of galaxies in large clusters require large

amounts of dark matter To make the sp eeds work out consistently one has to

assume that the total density of dark matter is an order of magnitude more than

the visible matter Recent reviews for the exp erimental evidence of dark matter

can be found in ref

From the concentration of light elements as shown in g one has to

conclude that the total baryonic density is only ab out of the critical density

see eq The critical density is the density for a at universe which naturally

o ccurs after ination Consequently in the inationary scenario the dark matter

makes up ab out of the total mass in the universe and it has to be non

baryonic

Possible candidates for dark matter are the MACHOs which have been

observed recently through their microlensing eect on the light of stars behind

them But since they cluster in heavy compact ob jects they are likely to be

remnants of collapsed stars or light dwarfs which have to o little mass to start

nuclear burning In these cases they would b e baryonic and make up of the

critical density required by baryogenesis Note that the visible baryonic matter

in stars represents at most of the critical densit y

So one needs additional dark matter if one b elieves in ination and takes the

value of from baryogenesis Candidates for this additional dark matter

b

are neutrinos with a mass in the eV range

m eV

Such small masses are exp erimentally not excluded but they would be rela

tivistic or hot Unfortunately al l dark matter cannot b e relativistic since this



Massive Compact Halo Ob jects

is inconsistent with the extremely small anisotropy in the microwavebackground

as observed by the COBE satellite

This anisotropy in the temp erature is prop ortional to the anisotropy in the

mass density at the time of release of this radiation shortly after the Big Bang

Through gravity the galaxies were formed around these uctuations seeds in

the mass density So the present structure of the universe has to follow from

the sp ectrum of uctuations in the early universe which can be prob ed by the

microwave background anisotropy The b est t is obtained for a mixture of

cold and hot dark matter as shown in g from ref

The lightest sup ersymmetric particles are ideal candidates for cold dark mat

ter provided they are not to o numerous and to o heavy Otherwise they would

provide a densityabove the critical density in whichcasetheuniverse

would b e closed and the lifetime would b e very short see g The LSP can

annihilate suciently rapidly into fermionantifermion pairs if the masses of the

SUSY particles are not to o heavy as will b e discussed in chapter

Summary

The Big Bang theory is remarkably successful in explaining the basic observations

of the universe ie the Hubble expansian the microwave background radiation

and the abundance of the elements From the measured Hubble constant one can

derivesuch basic quantities as the the size and the age of the universe Neverthe

less many questions remain unanswered They can be answered by p ostulating

phase transitions during the evolution of the universe from the Planck temp era

ture of K to the K observed to day Among the questions

The MatterAntimatter Asymmetry in our Universe

As rst sp elled out by Sakharov any theory trying to explain the pre

p onderance of matter in our universe must necessarily implement

Baryon and Lepton number violation

C and CP violation

Thermal nonequilibrium conditions as exp ected after phase transi

tions

The Dominance of Photons over Baryons

If the excess of matter originates from small CPviolation eects most

matter and antimatter will have enough time to annihilate into photons

thus providing an explanation why the number of photons as observed in

the K microwave background radiation is ab out to times as high

as the number of baryons in our universe

Ination

An inationary phase ie a rapid expansion generated by a p otential

energy term which according to Einsteins equations of General Relativity

provides a repulsive instead of attractive gravitational force is the only

viable explanation to solve the following problems

Horizon Problem

The fact that the observed temp erature of the K microwave back

ground radiation is to a very high degree the same in all directions can

only b e explained if we assumes that all regions were in causal contact

with each other at the b eginning However the size of our universe is

larger than the horizon ie the distance light could have traveled

since the b eginning Therefore one can only explain the temp erature

isotropy if one assumes that all regions were in causal contact at the

b eginning and that space expanded faster than the sp eed of light This

is indeed the case in the inationary scenario

Flatness Problem

Exp erimentally the observed density in our universe is close to the

socalled critical density which is the densit y where the total energy

of the universe is zero ie the kinetic energy of the expanding uni

verse is just comp ensated by the gravitational p otential energy This

corresp onds to a at universe ie zero curvature The inationary

scenario naturally explains why the universe is so at the rapid ex

pansion by more than orders of magnitude drives all curvature to

zero

Magnetic Monop ole Problem

Magnetic monop oles are predicted by GUTs Their absence in our

universe is explained by the inationary mo dels if one assumes the

ination to go smo othly from a single homogeneous region to a large

homogeneous region many times the size of our universe socalled new

ination

The Smo othness Problem

Exp erimentally the cosmic background radiation shows the features in

accord with the HarrisonZeldovich scaleinvariant sp ectrum n

which is the sp ectrum exp ected after ination The scale invariance

can b e understo o d as follows the ination smo othens out out anyin

homogeneities whichmighthave b een 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 small

scale of length namely the scale where quantum eects are imp or

tant These density uctuations are then enlarged to an astronomical

scale by ination and they stay scale invariant as is obvious if one

thinks ab out eg a little circle on a ballo on which stays a circle after

ination but just on a larger scale

In Grand Unied Theories the conditions needed for the inationary Big Bang

Theory are naturally met at least two phase transitions which generate mass

and thus provide nonequilibrium conditions are exp ected one at the unication

scale of GeV ie at temp erature of ab out K and one at the electroweak

K Furthermore a p otential energy term in scale ie a temp erature of ab out

the vacuum is exp ected from the scalar elds in these theories which are needed



Just like blowing up a ballo on removes all wrinkles and curvature from the surface

to generate particle masses in a gaugeinvariant way In the minimal mo del at

least scalar elds are required Unfortunately none have b een discovered so

far so little is known ab out the scalar sector

Nevertheless the arguments in favour of ination are so strong that it has

b ecome the only acceptable paradigm of present cosmology Exp erimental obser

vation of scalar elds would provide a great b o ost in the acceptance of the role

of scalar elds b oth in cosmology and particle physics

Chapter

Comparison of GUTs with

Exp erimental Data

In this chapter the various low energy GUT predictions are compared with data

The most restrictive constraints are the coupling constant unication combined

with the lower limits on the proton lifetime They exclude the SM

as well as many other mo dels with either a more complicated Higgs

sector or mo dels in which one searches for the minimum numb er of new particles

required to full the constraints mentioned ab ove From the many mo dels tried

only a few yielded unication at the required energies but these mo dels have

particles intro duced adho c without the app ealing prop erties of Sup ersymmetry

Therefore we will concentrate here on the sup ersymmetric mo dels and ask if the

predictions of the simplest ie minimal mo dels are consistent with all the

constraints describ ed in the previous chapters The relevant RG equations for

the running of the couplings and the masses are given in the app endix Assuming

soft symmetry breaking at the GUT scale all SUSY masses can be expressed in

terms of parameters and the masses at low energies are then determined bythe

well known Renormalization Group RG equations So many parameters cannot

b e derived from the unication condition alone However further constraints can

be considered

M predicted from electroweak symmetry breaking

Z

bquark mass predicted from the unication 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

Exp erimental lower limits on SUSY masses

Constraints from b s decays

Of course in many of the references given ab ove several constraints are stud

ied simultaneously since considering one constraint at a time yields only one

relation between parameters Trying to nd complete solutions with only a few

constraints requires then additional assumptions like naturalness noscale mo d

els xed ratios for gaugino and scalar masses or a xed ratio for the Higgs

mixing parameter and the scalar mass or combinations of these assumptions

Several ways to study the constraints simultaneously have b een pursued One

can either sample the whole parameter space in a systematic or random wayand

check the regions which are allowed by the exp erimental constraints

Alternatively one can try a statistical analysis in which all the constraints

are implemented in a denition and try to nd the most probable region of

the parameter space by minimizing the function

In the rst case one has to ask which weight should one give to the various

regions of parameter space and how large is the parameter space Some sample

the space only logarithmicallythus emphasizing the low energy regions oth

ers provide a linear sampling In the second case one is faced with

the diculty that the function to be minimized is not monotonous b ecause of

the exp erimental limits on the particle masses proton lifetime relic density and

so on At the transitions where these constraints b ecome eective the derivative

of the function is not dened Fortunately go o d 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 for

the allowed regions of the parameter space and can calculate condence levels

The results of such an analysis will be presented after a short description of

the exp erimental input values Other analysis have obtained similar mass sp ectra

for the predicted particles in the MSSM or extended versions

of the MSSM

Unication of the Couplings

In the SM based on the group SU SU U the couplings are dened

as

g cos

W

g sin

W

g

s

where g g and g are the U SU and SU coupling constants the rst

s

two coupling constants are related to the ne structure constantby see g

p

cos g sin g e

W W

The factor of in the denition of has been included for the prop er nor

malization at the unication p oint see eq The couplings when dened

as eective values including lo op corrections in the gauge b oson propagators

b ecome energy dep endent running A running coupling requires the sp eci

cation of a renormalization prescription for which one usually uses the mo died

minimal subtraction MS scheme

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

M

Z

sin

MS

The value of is given in ref and the value of sin has been been

MS

taken 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 the

uncertainty from the top quark We have not used the smaller error of for

agiven value of m since the t was only done within the SM not the MSSM so

t

we prefer to use the more conservative error including the uncertainty from m

t

The value corresp onds to the value at M as determined from quantities

Z

calculated in the Next to Leading Log Approximation These quantities are

less sensitive to the renormalization scale which is an indicator of the unknown

higher order corrections they are the dominant uncertainties in quantities relying

on second order QCD calculations This value is in excellent agreement with

s

a preliminary value of from a t to the Z cross sections and

asymmetries measured at LEP for which the third order QCD corrections

have b een calculated to o the renormalization scale uncertainty is corresp ondingly

small

The top quark mass was simultaneously tted and found to b e

GeV M

top

where the rst error is statistical and the second error corresp onds to a variation

of the Higgs mass b etween and GeV The central value corresp onds to a

Higgs mass of GeV

For SUSY mo dels the dimensional reduction DR scheme is a more appro

priate renormalization scheme This scheme also has the advantage that all

thresholds can b e treated by simple step approximations Thus unication o ccurs

in the DR scheme if all three meet exactly at one point This crossing

i

pointthengives the mass of the heavy gauge b osons The MS and DR couplings

dier by a small oset

C

i

DR MS

i i

where the C are the quadratic Casimir co ecients of the group C N for

i i

SUN and for U so stays the same Throughout the following we use

the DR scheme for the MSSM

M Constraint from Electroweak Symme

Z

try Breaking

As discussed in chapter the electroweak breaking in the MSSM is triggered by

the large negative corrections to the mass of one of the Higgs doublets After

including the onelo op corrections to the Higgs p otential the following

expression for M can be found see app endix

Z

m m tan

Z

M

Z

tan

f m f m g m

t t t

f m f m m A m cot

t t t t Z

M m m

t t

W

where m and m are the mass parameters in the Higgs p otential tan is the

mixing angle b etween the Higgs doublets and the function f has b een dened in

the app endix The corrections are zero if the top and stop quark masses are

Z

identical ie if sup ersymmetry would be exact They grow with the dierence

m m so these corrections become unnaturally large for large values of the

t

t

stop masses as will be discussed later In addition to relation one nds

from the minimzation of the p otential a relation between tan and m see

app endix so requiring electroweak breaking eectively 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 universal

masses of the gauginos and scalar particles at the GUT scale resp ectively and

constrains the masses of the Higgsinos At lower energies the masses of the

SUSY particles start to dier from these universal masses due to the radiative

corrections Eg the coloured particles get contributions prop ortional to

s

while the noncoloured ones get contributions dep ending on from gluon lo ops

the electroweak coupling constants only The evolution of the masses is given by

the renormalization group equations whichhave b een summarized in the

app endix Approximate numerical mass formulae for the squarks and sleptons

mass mixing between the top quarks gauginos and the Higgs mass parameters

are given in chapter The exact formulae can be found in the app endix

Proton Lifetime Constraints

GUTs predict proton decay and the present lower limits on the proton lifetime

yield quite strong constraints on the GUT scale and the SUSY parameters As

p

mentioned at the b eginning the direct decay p e via schannel exchange

requires the GUT scale to b e ab ove GeV This is not fullled in the Standard

Mo del SM but always fullled in the Minimal Sup ersymmetric Standard Mo del

MSSM Therefore we do not consider this constraint However the decays via

box diagrams with winos and Higgsinos predict much shorter lifetimes esp ecially

K From the present exp erimental lower limit of in the preferred mo de p

yr for this decay mo de Arnowitt and Nath deduce an upp er limit on

the parameter B which is prop ortional to

p

B M M GeV

H GU T



Here M is the Higgsino mass which is exp ected to be ab ove M else it

H GU T



would induce to o rapid proton decay If M would b ecome much larger than

H



M onewould enter the nonp erturbative regime Arnowitt and Nath giv e

GU T

the following acceptable range

M M

H GU T



To obtain a conservative upp er limit on B we allow M to b ecome an order of

H



magnitude heavier than M so we require

GU T

B GeV

The uncertainties from the unknown heavy Higgs mass are large compared

with the contributions from the rst and third generation which contribute

through the mixing in the CKM matrix Therefore we only consider the sec

ond order generation contribution which can be written as

B sin m m GeV

g

q

One observes that the upp er limit on B favours small gluino masses m large

g

squark masses m and small values of tan To full this constraint requires

q

tan

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

in case m is not to o large since m m and m m m see

g

q

eq The constraint can always be fullled for very large values of m

However the netuning constraint implies m GeV or m

g

GeV In this case eq requires m to be ab ove a few hundred GeV if m

b ecomes of the order of GeV or below as will b e discussed below

Top Mass Constraints

The top mass can be expressed as

Y t v sin m

t t

M

X

where the running of the Yukawa coupling as function of t log is given

Q

by

Y E t

t

Y t

t

Y F t

t

One observes that Y t b ecomes indep endent of Y for large values of Y

t t t

implying an upp er limit on the top mass Requiring electroweak symmetry

breaking implies a minimal value of the top Yukawa coupling typically Y

t

O In this case the term Y F t in the denominator of is much

t

larger than one since F t at the weak scale where t In this case

Y tE tF t so from eq it follows

t

E t

m v sin GeV sin

t

F t

where E and F are functions of the couplings onlysee app endix The physical

p ole mass is ab out larger than the running mass

s

pol e

M m GeV sin

t

t

The electroweak breaking conditions require eq hence

the equation ab ove implies for the MSSM approximately

pol e

M GeV

t

which is consistent with the exp erimental value of GeV as determined at

LEP see eq Although the latter value was determined from a t using the

SM one do es not exp ect shifts outside the errors if the t would be made for

the MSSM

As will b e shown in chapter for such large top masses the bquark mass b e

comes a sensitive function of m and of the starting values of the gauge couplings

t

at M

GU T

bquark Mass Constraint

arbitrary in the As discussed in chapter the masses of the uptyp e quarks are

SU mo del but the masses of the downtyp e quarks are related to the lepton

masses within a generation if one assumes unication of the Yukawa couplings at

the GUT scale This do es not work for the light quarks but the ratio of bquark

and lepton masses can be correctly predicted by the radiative corrections to

the masses

To calculate the exp erimentally observed mass ratio the second order renor

malization group equations for the running masses havetobeused These equa

tions are integrated b etween the value of the physical mass and M

GU T

For the running mass of the bquark we used

m GeV

b

This mass dep ends on the choice of scale and the v alue of m Consequently

s b

we have assigned a rather conservative error of GeV instead of the prop osed

value of GeV Notethatthe running mass in the MS scheme is related

pol e

to the physical p ole mass M by

b

s s

pol e pol e

M m M

b

b b

pol e

so m corresp onds to M GeVWe ignore the running of m below

b

b

m and use for the p ole mass M GeV

b

Dark Matter Constraint

As discussed in chapter there is abundant evidence for the existence of non

relativistic neutral nonbaryonic dark matter in our universe The lightest su

p ersymmetric particle LSP is supp osedly stable and would b e an ideal candidate

for dark matter

implies an The present lifetime of the universe is at least years which

upp er limit on the expansion rate see eq and corresp ondingly on the total

relic abundance compare eq Assuming h one nds that for each

relic particle sp ecies

h

This bound can only be ob eyed if most of the LSPs annihilated into fermion

antifermion pairs whichinturnwould annihilate into photons again As will b e

shown b elow the LSP is most likely a gauginolike neutralino In this case

f dep ends most sensitively on the mass of the the annihilation rate f

lightest tchannel exchanged h m m Consequently the

f

upp er limit on the relic density implies an upp er limit on the sfermion mass

However as discussed in chapter the neutralinos are mixtures of gauginos and

higgsinos The higgsino comp onent also allows schannel exchange of the Z

and Higgs b osons The size of the Higgsino comp onent dep ends on the relative

sizes of the elements in the mixing matrix esp ecially on the mixing angle

tan and the size of the parameter in comparison to M m and M

m Consequently the relic density is a complicated function of the SUSY

parameters esp ecially if one takes into account the resonances and thresholds

in the annihilation cross sections but in general one nds a large region

in parameter space where the universe is not overclosed In the preferred

gauginolike neutralino region the relic density constraint translates into an upp er

bound of ab out GeV on m except for large m where some SUSY

masses b ecome much larger than TeV and are therefore disfavoured by the

netuning criterion see eq

Exp erimental lower Limits on SUSY Masses

SUSY particles have not b een found so far and from the searches at LEP one

knows that the lower limit on the charged leptons and charginos is ab out half

the Z mass GeV and the Higgs mass has to b e ab ove GeV The

lower limit on the lightest neutralino is GeV while the sneutrinos have

ve GeV These limits require minimal values for the SUSY mass to b e ab o

parameters

There exist also limits on squark and gluino masses from the hadron

but these limits dep end on the assumed decay mo des Furthermore if one takes

the limits given ab ove into account the constraints from the limits of all other

particles are usually fullled so they do not provide additional reductions of the

parameter space in case of the minimal SUSY mo del

Decay b s

Recently CLEO has published an upp er bound for this transition b s

Furthermore a central value of and a lower limit of

can b e extracted from the observed pro cess B K and assuming

that the branching ratio for this pro cess is using lattice calculations

In the SM the transition b s can happ en through onelo op diagrams with a

quark charge and a charged gauge b oson SUSY allows for additional lo ops

involving a charged Higgs and the charginos and neutralinos

In SUSY large cancellations occur since the W t and H t lo ops have an

opp osite sign as compared to the t lo op and all lo ops are of the same order

of magnitude

Kane et al nd acceptable rates for the b s transition in the MSSM for

a large range of parameter space even if they include constraints from electroweak

symmetry breaking and unication of gauge and Yukawa couplings They do

not nd as strong lower limits on the charged Higgs b oson masses as others

Similar conclusions were reached by Borzumati who used the more complete

calculations including the avour changing neutral curren ts It turns out

that with the present errors the combination of all constraints discussed ab ove

are more restrictive than the limits on b s so we have not included it in the

analyses discussed b elow

Fit Strategy

As mentioned b efore given the ve parameters in the MSSM plus and

GU T

M all other SUSY masses the bquark mass and M can be calculated by

GU T Z

p erforming the complete evolution of the couplings including all thresholds

The proton lifetime prefers small values of tan eq while all SUSY

masses are exp ected to be below TeV from the netuning 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 A

t

at M was kept mostly at zero but the large radiative corrections to it were

GU T

taken into account so at lower energies it is unequal zero Varying A b etween

t

m and m did not change the results signicantly so the following results

are quoted for A

t

The remaining four parameters M and Y were tted

GU T GU T t

with the MINUIT programby minimizing the following function

X

M M

Z Z

i

MSSM

i

i

i

M

Z

Z

m

b

b

B

for B

B

D mmm

for D

D

M M

exp

for M M

exp

M

The rst term is the contribution of the dierence between the three calculated

and measured gauge coupling constants at M and the following two terms are

Z

the contributions from the M mass and m mass constraints The last three

Z b

terms imp ose constraints from the proton lifetime limits from electroweak sym

metry breaking ie D V v v V see eq and from

H H

exp erimental lower limits on the SUSY masses The top mass or equiv alently

the top Yukawa coupling enters sensitively into the calculation of m and M

b Z

Instead of the top Yukawa coupling one could have taken the top mass as a

parameter However if the couplings are evolved from M downwards it is

GU T

more convenient to run also the Yukawa coupling downward since the RGE of

the gauge and Yukawa couplings form a set of coupled dierential equations in

second order see app endix Once the Yukawa coupling is known at M the

GU T

top mass can b e calculated at anyscale The top mass can b e taken as an input

parameter to o using the value from the LEP data Unfortunatelythevalue from

the LEP data eq is not yet very precise compared with the range exp ected

in the MSSM eq so it do es not provide a sensitive constraint Instead we

prefer to t the Yukawa coupling in order to obtain the most probable top mass

in 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 ab ove

The following errors w ere attributed are the exp erimental errors in the

i

coupling constants as given ab ove GeV GeV while and

b B D

were set to GeV The values of the latter errors are not critical since the

M

corresp onding terms in the numerator are zero in case of a go o d t and even for

the CL limits these constraints could b e fullled and the was determined

by the other terms for which one knows the errors

DR scheme all three couplings must cross at For unication in the

i

a single unication point M Thus in these mo dels one can t the cou

GU T

plings at M by extrap olating from a single starting point at M back to

Z GU T

M for each of the s and taking into account all light thresholds The tting

Z i

program will then adjust the starting values of the four high energy parame

ters M and Y until the ve lo w energy values three coupling

GU T GU T t

constants M and m are hit The t is rep eated for all values of m and

Z b

m between and GeV and tan between and Alternatively ts

were p erformed in which m was left free to o

The light thresholds are taken into account by changing the co ecients of

the RGE at the value Q m where the threshold masses m are obtained

i i

analytical solutions of the corresp onding RGE see section These from the

solutions dep end on the integration range which was chosen between m and

i

M However since one do es not know m at the b eginning an iterative

GU T i

pro cedure has to b e used one rst uses M as a lower integration limit calculates

Z

m and uses this as lower limit in the next iteration Of course since the coupling

i

constants are running the latter have to b e iterated to o so the values of m

i i

have to be used for calculating the mass at the scale m Usually three

i

solution iterations are enough to nd a stable

Following Ellis Kelley and Nanop oulos the p ossible eects from heavy

thresholds are set to zero since proton lifetime forbids Higgs triplet masses to

be below M see eq These heavy thresholds have been considered by

GU T

other authors for dierent assumptions

SUSY particles inuence the evolution only through their app earance in the

lo ops so they enter only in higher order Therefore it is sucient to consider

the lo op corrections to the masses in rst order in which case simple analytical

solutions can be found even if the onelo op correction to the Higgs p otential

from the top Yukawa coupling is taken into account see app endix There is one

exception the corrections to the b ottom and tau mass are compared directly

with data which implies that the second order solutions have to b e taken for the

RGE predicting the ratio of the bottom and tau mass Since this ratio involves

the top Yukawa coupling Y the RGE for Y has to b e considered in second order

t t

to o These second order corrections are imp ortant for the b ottom mass since the

strong coupling constant b ecomes large at the small scale of the b ottom mass

ie m

s b

In total one has to solve a system of coupled dierential equations

second order onesfor the gauge couplings Y and Y Y and rst order

t b

masses and parameters in the Higgs sector see app endix The ones for the

second order ones are solved numerically taking into account the thresholds of

the light particles using the iteration pro cedure discussed ab ove Note that from

the starting values of all parameters at M one can calculate all light thresholds

GU T

from the simple rst order equations b efore one starts the numerical integration

of the ve second order equations Consequently the program is fast in nding

the optimum solution even if b efore each iteration the light thresholds have to

be recalculated

Results

The upp er part of g shows the evolution of the coupling constants in

the MSSM for two cases one for the minimum value of the function given

in eq solid lines and one corresp onding to the CL upp er limit

of the thresholds of the light SUSY particles dashed lines The p osition of

the light thresholds is shown in the b ottom part as jumps in the rst order

co ecients which are increased according to the entries in table A as so on

as a new threshold is passed Also the second order co ecients are changed

corresp ondingly see table A but their eect on the evolution is not visible

in the top gure in contrast to the rst order eects which change the slop e of

the lines considerably in the top gure One observes that the changes in the

coupling constants occur in a rather narrow energy regime so qualitatively this

picture is very similar to g in which case all sparticles were assumed to

be degenerate at an eective SUSY mass scale M Since the running

SU SY

the couplings dep ends only logarithmically on the sparticle masses the of

CL upp er 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 of

m and tan as well as the tted parameters M m Y and

GU T GU T t

The program DDEQMR from the CERN library was used for the solution of these coupled

second order dierential equations

the corresp onding top mass after running down Y from M to m are shown

t GU T t

at the top and given these parameters the corresp onding masses of the SUSY

particles can b e 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 go o d

solution can be found This is indeed the case The upp er and lower limits in

table will b e discussed below

Only the value of the top Yukawa coupling is given since for the ratio of

b ottom 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 quark

sector only the small contribution from the top Yukawa coupling is taken into

account since for the range of tan considered all other Yukawa contributions

are negligible

As mentioned b efore varying A between m and m do es not in

t

uence the results very much so its value at the unication scale was kept at

to the large radiative corrections but its nonzero value at lower energies due

was taken into account The ts are shown for p ositivevalues of the Higgs mix

ing paramter but similar values are obtained for negative values of with an

equally go o d value for the t

The parameters m m and are correlated as shown in g where

the value of is shown for all combinations of m and m between and

GeV One observes that increases with increasing m and m The strong

correlation b etween m and originates mainly from the electroweak symmetry

breaking condition but also from the fact that the thresholds in the running of

the 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 the

solid line of simply b ecause the thresholds are to o dierent the thresholds

in are mainly determined by m while the thresholds for the upp er two

lines include the winos and higgsinos to o so one obtains automatically a p ositiv e

correlation b etween and m

The value is acceptable in the whole region except for the regions where

either m or m or b oth b ecome very small as shown in g The increase

in this corner is completely due to the constraint from the proton lifetime see

section This plot was made for tan For larger values the region

excluded by proton decay quickly increases for tan practically the whole

region 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

p erimental lower limits on the SUSY masses and the requirement of radiative

symmetry breaking In the minimum the value is zero but one notices a long

valley where the is only slowly increasing Consequently the upp er limits on

the sparticle masses which grow with increasing values of and m become

y several TeV as shown in table The CL upp er limits were obtained b

requiring an increase in of so the rather The upp er limits are a sensitive

function of the central value of decreasing the central value of by two

s s

standard deviations ie can increase the thresholds of sparticles several

TeV Acceptable ts are only obtained for input values between and

s

if the error is kept at Outside this range all requirements cannot b e

met simultaneously any more so the MSSM predicts in this range

s

As discussed previously sparticle masses in the TeV range sp oil the cancella

tion of the quadratic divergences This can be seen explicitly in the corrections

to M is exactly zero if the masses of stop and top quarks are identical

Z Z

but 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 M are

Z

small If one requires that only solutions are allowed for which the corrections

to M are not large compared with M itself one has to limit the mass of the

Z Z

heaviest stop quark to ab out one TeV The corresp onding CL upp er limits

of the individual sparticles masses are given in the right hand column of table

The correction to M is times M in this case The limits are obtained

Z Z

by scanning m and m till the value increases by while optimizing the

values of tan Y and M The lower limits on the SUSY pa

GU T t GU T

rameters are shown in the left column of table The lowest values of m

GeV are required to have simultaneously a sneutrino mass GeV and m

ab ove GeV and a wino mass ab ove GeV If the proton lifetime is included

the minimum value of either m or m have to increase see g Since the

squarks and gauginos are much more sensitivetom one obtains the lower lim

its by increasing m The minimum value for m is ab out GeV in this case

But in b oth cases the increase for the lower limits is due to the bmass which

is predicted to b e GeV from the parameters determining the lower limits

The bquark mass is a strong function of b oth tan and m as shown in g

t

this dep endence originates from the W t lo op to the b ottom quark The

horizontal band corresp onds to the mass of the bquark after QCD corrections

m GeV see eq Since also M is a strong function of

b Z

the same parameters the requirementofgaugeandYukawa coupling unication

together with electroweak symmetry breaking strongly constrains the SUSY par

ticle sp ectrum A typical t with a equal zero is given in the central column

of table but is should b e noted that the values in the other columns provide

acceptable ts to o at the CL

The mass of the lightest Higgs particle called h in table is a rather strong

All function of m asshown in g for various choices of tan m and m

t

other parameters were optimized for these inputs and after the t the values of

the Higgs and top mass were calculated and plotted One observes that the mass

of the lightest Higgs particle varies between and GeV and the top mass

between and GeV Furthermore it is evident that tan almost uniquely

determines the value of m since even if m and m are varied b etween and

t

GeV one nds the practically the same m for a given tan and the value

t

of m varies b etween and GeV if tan is varied b etween and This

t

range is in excellen t agreement with the estimates given in eq if one takes

pol e

into account that M m see eq

t

t

Note the strong correlation b etween tan and m in g for a given value

t

of tan m is constrained to a vary narrow range almost indep endent of m

t

and m Furthermore one observes a rather strong p ositive correlation between

m ig g s and all other parameters tan m and m originating from the

h

lo op corrections to the p otential

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

lifetime is ob eyed and extreme netuning is to b e avoided ie m TeV

t

m GeV

m GeV

GeV

tan

m GeV from g

t

s

The corresp onding constraints on the SUSY masses are see table for

details

GeV

Z W GeV

g GeV

q GeV

t GeV

t GeV

e GeV

L

e GeV

R

GeV

L

H GeV

H GeV

H GeV

H GeV

H GeV

A GeV

h GeV from g

The lower limits will all increase as so on as the LEP limits on sneutrinos

winos and the lightest Higgs increase

The lightest Higgs particle is certainly within reach of exp eriments at present

or future accelerators Its observation in the predicted mass range of

to GeV would be a strong case in supp ort of this minimal version of a

sup ersymmetric grand unied theory i α

1/ 60

50

40

30

20

10 90% C.L. upper limit 0 0 5 10 15 10log Q i

β 10 β 8 THRESHOLDS β 6 1 4 2 0 β -2 2 -4 β -6 3 -8 -10 0 5 10 15

10log Q

Figure Evolution of the inverse of the three couplings in the MSSM The line

ab ove M follows the prediction from the sup ersymmetric SU mo del The

GUT

SUSY thresholds have been indicated in the lower part of the curve they are

treated as step functions in the rst order co ecients in the renormalization

group equations which corresp ond to a change in slop e in the evolution of the

couplings in the top gure The dashed lines corresp ond to the CL upp er

limit for the SUSY thresholds 2500

2000 [GeV]

μ 1500

1000

500

m 1000 1/2 900 1000 [GeV] 800 900 700 800 700 600 600 [GeV] 500 500 m 0 400 400 300 300 200 200

100 100

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

2 1 χ 0.8

0.6

0.4

0.2

m 1000 1/2 900 1000 [GeV] 800 900 700 800 700 [GeV] 600 600 m 0 500 500 400 400 300 300 200 200

100 100

Figure The of the t as function of m and m for tan The sharp

increase in in the corner is caused by the lower limit on the proton lifetime GeV m for and m een w correlation b et The

2

m χ 1/2

Figure 0.5 1.5 2.5 [GeV] 1 2 3 3500 3000 2500 2000 1500 1000 500 1000 1500 2000 2500 3000 3500 μ [GeV] 4000 1400 1200 in %

Z 1000 800 600

corr. M 400 200

m 1000 1/2 900 1000 [GeV] 800 900 700 800 700 600 600 [GeV] 500 500 m 0 400 400 300 300 200 200

100 100

Figure The onelo op correction factor to M as function of m and m

Z

] 5.2 GeV [

b 5 m

4.8

4.6

4.4

4.2

4

3.8 tan β = 10. tan β = 1.2 3.6 100 110 120 130 140 150 160 170 180 190 200 [ ]

mt GeV

Figure The correlation b etween m and m for m GeV and twovalues

b t

of tan The hatched area indicates the exp erimental value for m

b 160 m = 1000 [GeV] 5.0 150 0 [GeV] m = 100 [GeV] higgs 140 0 m

130 2.0 1000 GeV 120

110 β 100 tan = 1.2

90

80 m1/2 = 100 GeV

70

60

50 130 140 150 160 170 180 190 200

mtop [GeV]

Figure The mass of the lightest Higgs particle as function of the top quark

mass for values of tan between and and values of m and m between

and GeV The parameters of M and Y are optimized

GU T GU T t

for each choice of these parameters the corresp onding values of the top and

lightest Higgs mass are shown as symbols For small values of m the Higgs

increases with m as shown for a string of points each representing a mass

step of GeV in m for a given value of m which is increasing in steps

of GeV starting with the low values for the lowest strings At high values

of m the value of m b ecomes irrelevant and the string shrinks to a point

other parameters Note the strong p ositive correlation between m and all

hig g s

the highest value of the Higgs mass corresp onds to the maximum values of the

input parameters ie tan m m GeV this value do es not

corresp ond to the minimum More likely values corresp ond to m

higgs

GeV for m m and tan

Symbol Lower limits Typical t CLUpp er limits

Constraints GEY GEYP GEYPF GEY P GEYPF

Fitted SUSY parameters

m

m

tan

Y

t

m

t

GU T



M

GU T

SUSY masses in GeV

Z

W

g

e

L

e

R

L

q

L

q

R

b

L

b

R

t

t

H

H

H

h

H

A

H

Table Values of SUSY masses and parameters for various constraints

Ggauge coupling unication Eelectroweak symmetry breaking YYukawa

coupling unication PProton lifetime constraint Fnetuning constraint

Constraints in brackets indicate that they are fullled but not required The

value 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 the

evolution of the universe from the Planck temp erature of K to the K

observed to day 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 monop ole problem and the smo othness problem

see chapter

In Grand Unied Theories GUT phase transitions are exp ected one at the

unication scale of GeV ie at a temp erature of ab out K and one at

the electroweak scale ie at a temp erature of ab out K Furthermore scalar

elds which are a prerequisite for ination are included in GUTs In the min

imal mo del at least scalar elds are required Unfortunately none have been

discovered so far so little is known ab out the scalar sector although the veri

cation of the relation between the couplings and the masses of the electroweak

gauge b osons indeed are indirect evidence that their mass is generated by the

interaction with a scalar eld Exp erimental observation of these scalar elds

would provide a great boost for cosmology and particle physics First estimates

of the required mass sp ectra of the scalar elds can be obtained by comparing

the exp erimental consequences of Grand Unied Theories GUT with lowenergy

phenomenology

One of the interesting discoveries of LEP was the fact that within the Stan

dard Mo del SM unication of the gauge couplings could be excluded see g

In contrast the minimal sup ersymmetric extension of the SM MSSM

provided p erfect unication This observation b o osted the interest in Sup ersym

metry enormously esp ecially since the MSSM was not designed to provide

to have very unication but it was invented many years ago and turned out

interesting prop erties

Sup ersymmetry automatically provides gravitational interactions thus paving

the road for a Theory Of Everything

The symmetry between b osons and fermions alleviates the divergences in

the radiative corrections in which case these corrections can be made re

sp onsible for the electroweak symmetry breaking at a muchlower scale than

the GUT scale

The lightest sup ersymmetric partner LSP is a natural candidate for non

relativistic dark matter in our universe

Other nonsup ersymmetric mo dels can yield unication to obut they do not ex

hibit the elegant symmetry prop erties of sup ersymmetry they oer 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 Sup ersymmetric Standard Mo del MSSM mo del has manypre

dictions which can b e compared with exp eriment even in the energy range where

the predicted SUSY particles are out of reach Among these predictions

M

Z

m

b

Proton decay

Dark matter

It is surprising that in addition to the unication of the coupling c onstants the

minimal sup ersymmetric mo del can full all exp erimental constraints from these

predictions As far as we know sup ersymmetric mo dels are the only ones which

are consistent with all these observations simultaneously Within the MSSM the

evolution of the universe can be traced back to ab out seconds after the

bang as sketched in g If we b elieve in the inationary scenario even

the actual creation of the universe is describable by physical laws In this view

the universe would originate as a quantum uctuation starting from absolute

nothing ie a state devoid of space time and matter with a total energy equal

to zero Indeed estimates of the total p ositive nongravitational energy and

negative p otential energy are ab out equal in our universe ie according to this

view the universe is the ultimate free lunch All this mass was generated from

the p otential energy of the vacuum which also caused the inationary phase

Of course a quantum description of spacetime can be discussed only in the

context of quantum gravity so these ideas must be considered sp eculative until

a renormalizable theory of quantum gravity is formulated and proven by ex

p eriment Nevertheless it is fascinating to contemplate that physical laws may

determine not only the evolution of our universe but they may remove also the

need for assumptions ab out the initial conditions

rom the exp erimental constraints at low energies the mass sp ectra for the F

SUSY particles can be predicted see table in the previous chapter The

lightest Higgs particle is certainly within reach of exp eriments at present or future

accelerators Its observation in the predicted mass range of to GeV would

b e a strong case in supp ort of this minimal version of the sup ersymmetric grand

unied theory Discovering also the heavier SUSY particles implies that the

known strong electromagnetic and weak forces were all unied into a single

primeval force during the birth of our universe Future exp eriments will tell SU(3)

SU(2)

QED

U(1)

Figure Possible evolution of the radius of the universe and the coupling

constants Before t s sp ontaneous symmetry breaking o ccurs which

breaks the symmetry of the GUT into the well known symmetries at low energies

In the mean time the universe inates to a size far ab ove the distance light could

have traveled as indicated by the dashed line From

Acknowledgments

I want to thank sincerely Ugo Amaldi Hermann Furstenau Ralf Ehret and

Dmitri Kazakov for their close collab oration in this exciting eld Without their

enthusiasm work and sharing of ideas many of our common results presented

in this review would not have b een available Furthermore I thank John El

lis Gian Giudice Howie Hab er Gordy Kane Stavros Katsanevas Sergey Ko

valenko Hans Kuhn Jorge Lop ez Dimitri Nanop oulos Pran Nath Dick Rob erts

Leszek Roszkowski Mikhail Shap oshnikov William Trischuk and Fabio Zwirner

for helpful discussions andor commenting parts of the manuscript

Last but not least I want to thank Prof F aessler for inviting me to a seminar

in Tubingen and his encouragement to write down the results presented there in

this review

App endix A

A Intro duction

In this app endix all the Renormalization Group Equations RGE for the evo

lution of the masses and the couplings are given SUSY particles inuence the

evolution only through their app earance in the lo ops so they enter only in higher

order Therefore it is sucient to consider the lo op corrections to the masses only

in rst order in which case a simple analytical solution can b e found even if the

onelo op correction to the Higgs p otential from the top Yukawa coupling is taken

into account There is one exception the corrections to the b ottom and tau mass

are compared directly with data which implies that the second order solutions

have to be taken for the RGE predicting the ratio of the bottom and tau mass

a coupling Y the RGE for Y has to be Since this ratio involves the top Yukaw

t t

considered in second order to o These second order corrections are imp ortantfor

the b ottom mass since the strong coupling constant b ecomes large at the small

scale of the b ottom mass ie m

s b

So in total one has to solve a system of coupled dierential 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 Y and

t

the ratio of b ottom and tau Yuka wa coupling R

b

rst order equation for the masses of the lefthanded doublet of an utyp e

and dtyp e squark pair Q

rst order equation for the masses of the righthanded uptyp e squarks

U

rst order equation for the masses of the righthanded downtyp e squarks

D

rst order equation for the masses of the lefthanded doublet of sleptons

L

rst order equation for the masses of the righthanded singlet of a charged

lepton E

rst order equations for the mass parameters of the Higgs p otential

m m m and

rst order equations for the Ma jorana masses of the gauginos M M

and M

rst order equation for the trilinear coupling between left and right

handed squarks and the Higgs eld A where the subscript indicates that

t

one only considers this coupling for the third generation

Note that the absolute values of the b ottom and tau Yukawa couplings need not

to b e known if one neglects their small contribution to the running of the gauge

couplings If one wants to include these one has to integrate the RGE for Y and

b

Y separately they are given b elow to o instead of the RGE for their ratio only

Integrating the ratio has the advantage that the b oundary condition at M is

GU T

known to b e one if one assumes Yukawa coupling unication

The particle masses are related directly to the Yukawa couplings

h

t

Y m m h m v sin A

t t t t t

h

b

Y m m h m v cos A

b b b b b

h

m h m v cos A Y m

It follows that

Y m

t

t

A

Y m tan

b

b

For tan as required by proton decay limits one observes that Y is at least

b

an order of magnitude smaller than Y for the values of m considered Hence

t t

its contribution is indeed negligible in the running of the gauge couplings so

below we will only consider the contribution of Y in the ratio of Y Y whichis

b b

indep endent of the absolute value of Y

b

Below we collect all the RGEs in a coherent notation and consider the co ef

cients for the various threshold regions ie virtual particles with mass m are

i

considered to contribute to the running of the gauge coupling constants eec

tively only for Q values ab ove m Thus the thresholds are treated as simple step

i

functions in the co ecients of the RGE

Furthermore all rst order solutions are given in an analytical form including

the corrections from the top Yukawa coupling Note that a more demanding

analysis requires a numerical solution of the rst ve second order equations in

which the co ecients are changed according to the thresholds found as analytical

solutions of the rst order equations for the evolution of the masses The results

given in chapter all use the numerical solution of these second order equation

Using the sup ergravity inspired breaking terms which assume a common mass

m for the gauginos and another common mass m for the scalars leadstothe

following breaking term in the Lagrangian

X X

j j m A L m

i Breaking

i

i h

c e c d c u

Bm H HA H L E H h Q D H h Q U h Am

a a a

b ab b ab b ab

Here

The program DDEQMR from the CERN library was used for the solution of these coupled

second order dierential equations

ude

h are the Yukawa couplings a b run over the generations

ab

Q are the SU doublet quark elds

a

c

U are the SU singlet chargeconjugated upquark elds

a

c

D are the SU singlet chargeconjugated downquark elds

b

L are the SU doublet lepton elds

a

c

E are the SU singlet chargeconjugated lepton elds

a

H are the SU doublet Higgs elds

are all scalar elds

i

are the gaugino elds

The last two terms in L originate from the cubic and quadratic terms

B r eak ing

in the sup erp otential with A B and as free parameters In total we now have

three couplings and ve mass parameters

i

m m t At B t

with the following b oundary conditions at M t

GU T

scalars m m m m m m A

Q U D L E

gauginos M m i A

i

couplings i A

i GU T

Here M M and M are the gauginos masses of the U SU and SU

groupsIn N sup ergravity one exp ects at the Planck scale B A

With these parameters and the initial conditions at the GUT scale the masses

of all SUSY particles can b e calculated via the renormalization group equations

A Gauge Couplings

The following denitions are used

i

A

i

M

GU T

A t ln

Q

b A

i i GU T

A f t

i

t

i i

t

h t A

i

t

i

where i denote the three gauge coupling constants of U SU andS U

i

resp ectively is the common gauge coupling at the GUT scale M and b

GU T GU T i

are the co ecients of the RGE as dened below

The second order RGEs for the gauge couplings including the eect of the

Yukawa couplings are

X

d

i

A

b A b a Y

i ij j i t

i i

dt

j

a a for SUSY and a a a for the SM where a

The rst order co ecients for the SM are

b

B B C B C C B C

B B C B C C B C

b B B C A B C N C B C N

b

i Higgs Fam

A A A A

b

the SM to be called MSSM in the while for the sup ersymmetric extension of

following

b

B C B C C B C B

B C B C C B C B

B C A B C N C B C N b B

b

Higgs Fam i

A A A A

b

Here N is the number of families of matter sup ermultiplets and N is

Fam Higgs

the number of Higgs doublets We use N and N or which

Fam Higgs

corresp onds to the minimal SM or minimal SUSY mo del resp ectively

The second order co ecients are

C B B C C B

C B B C C B

C B B C C B

N b N

Higgs ij Fam

C B B C C B

A A A

A

For the SUSY mo del they b ecome

C C B B C B

C C B B C B

C C B B C B

N b N

Higgs ij Fam

C C B B C B

A A A

A

The contributions for the individual thresholds to b and b are listed in tables

i ij

A from ref and A resp ectively

The running of each dep ends on the values of the two other coupling

i

constants if the second order eects are taken into account However these

eects are small b ecause the b s are multiplied by Higher orders

ij j

are presumably even smaller

If the small Yukawa couplings are neglected the RGEs A can be solved

by integration to obtain at ascale for a given

i

i

i

A ln ln

i

i i

with

b b

ij ik

b A

i j k

b

ii

A

This exact solution to the second order renormalization group equation can

be used to calculate the coupling constants at an arbitrary energy if they have

been measured at a given energy ie one calculates from a given

i i

This transcendental equation is most easily solved numerically by iteration If

the Yukawa couplings are included their running has to be considered to o and

gauge couplings and Yukawa couplings one can solve the coupled equations of

only numerically

A Yukawa Couplings

In order to calculate the evolution of the Yukawa coupling for the b quark in

the region between m and M one has to consider four dierent threshold

b GU T

regions

Region I between the typical sparticle masses M and the GUT scale

SU SY

Region I I between M and the top mass m

SU SY t

Region I I I between m and M

t Z

Region IV between M and m

Z b

A RGE for Yukawa Couplings in Region I

The second order RGE for the three Yukawa couplings of the third generation in

the regions b etween M and M are

SU SY GU T

dY

t

Y Y

t t

dt

b b b

Y Y Y Y A

t t t

t

dY

b

Y Y

b t

dt

b b b

Y Y A

t

t

dY

Y

dt

b A b

If one assumes Yukawa coupling unication for particles b elonging to the same

multiplet ie Y Y at the GUT scale one can calculate easily the RGE for

b

q

the ratio R t m m Y tY t

b b b

dR

b

R Y

b t

dt

b b

Y Y A

t

t

A RGE for Yukawa Couplings in Region II

For the region b etween M and m one nds

SU SY t

dY

t

Y Y

t t

dt

Y Y Y Y A

t t t

t

dY

b

Y Y

b t

dt

Y Y Y Y A

t t t

t

dY

Y Y

t

dt

Y Y Y Y A

t t t

t

dR

b

R Y

b t

dt

Y A Y Y Y

t t t

t

A RGE for Yukawa Couplings in Region III

For the region b etween m and M one nds

t Z

dY

b

Y

b

dt

A

dY

Y

dt

A

dR

b

R

b

dt

A

A RGE for Yukawa Couplings in Region IV

dR

b

A R

b

dt

A Squark and Slepton Masses

Using the notation intro duced at the b eginning the RGE equations for the

squarks and sleptons can be written as

dm

L

M M A

dt

dm

E

M A

dt

dm

Q

M M M Y m m m A m

i t

Q U t

dt

A

dm

U

M M Y m m m A m A

i t

Q U t

dt

dm

D

M M A

dt

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

i

A Solutions for the squark and slepton masses

The solutions for the RGE given ab ove are

cos M f t f t sin A m m m

W GU T

Z E

L

m m m f t f t cos M A

GU T

Z

L

m m m f t cos M sin A

GU T W

E Z

R

f t f t f t cos M sin m m m

W GU T

Z U

L

A

m m m cos M f t f t f t sin

GU T W

D Z

L

A

f t f t cos M sin A m m m

W GU T

Z U

R

m m m f t f t cos M sin A

GU T W

D Z

R

A

For the third generation the eect of the top Yukawa coupling needs to b e taken

into account in which case the solution given ab ove are changed to

m m A

b D

R R

f t m f t m m A m m

GU T

b D

L L

f t m m f t m Am m m

GU T

t t U

R R

m m m m f t m mA f t

GU T

t U t

L L

A nonnegligible Yukawa coupling causes a mixing b etween the weak interaction

eigenstates The mass matrix is

m h A m jH j jH j

t t

t

R

A

A

h A m jH j jH j m

t t

t

L

and the mass eigenstates are

q

A m A m cot m m m m m

t

t t t

t t t

R L



L R

A Higgs Sector

A Higgs Scalar Potential

The MSSM has two Higgs doublets Q T Y

W

H H

A A

H H

H H

The tree level p otential for the neutral sector can be written as

g g

jH j jH j V H H m jH j m jH j m H H hc

A

with the following b oundary conditions at the GUT scale m m

m m Bm where the value of is the one at the GUT scale

The renormalization group equations for the mass parameters in the Higgs

potential can b e written as

d

Y A

t

dt

dm

M M Y A

t

dt

dm

M M Y m m m A m

t

Q U t

dt

A

dm

m Y A M Y m M A

t t t

dt

A Solutions for the Mass Parameters in the Higgs Po

tential

The solutions for the RGE given ab ove are

t q t A

m t m tm f t f t A

GU T

m t q t m etA m m f tm ht k tA A

t t

m t q tm r tm stA m A

t

where

b b



q t t t

Y F t

t

ht

D t

Y F t

t

k t

D t

Y H t

t

f t

D t

D t Y F t

t

G tY G t H tY H t

t t

H t et

D t D t

Y F t

t

q t st

D t

Y H t

t

r t H t q t

D t

b b b

 

E t t t t

t

Z

E t dt F t

H t h th t h t

GU T

H t tE t F t

H t F tH t H t

H t f tf t f t

GU T

t

Z

H t H t E t dt

h t H t h t

GU T

H t f tf t f t

GU T

H t G t F t

G t F t F t H tH tF tH t H t

F t f t f t

GU T

t

Z

F t F tF t E t F t dt

t

Z

F t E t H t dt

The functions f and h have b een dened b efore The Higgs mass sp ectrum can

i i

be obtained from the p otential given ab ove by diagonalizing the mass matrix

V

H

M A

ij

j j

where is a generic notation for the real or imaginary part of the Higgs eld

i

Since the Higgs particles are quantum eld oscillations around the minimum

eq A has to be evaluated at the minimum The mass terms at tree level

haven b een given in the text However as discovered afewyears ago the radia

tive corrections to the Higgs mass sp ectrum are not small and one has to take

this case the eective the corrections from a heavy top quark into account In

potential for the neutral sector can be written as

g g

V H H m jH j m jH j m H H hc jH j jH j

m m m

t t t

m ln m ln m ln

t t t

Q Q Q

where m are eld dep endent masses which are obtained from eqns A by

ti

substituting m h H

t t

The minimum of the p otential can be found by requiring

V g g

m v m v v v v

jH j

f m f m

t t

h A m v v A

t

t

m m

t t

g g V

v v v m v m v

jH j

f m f m

t t

A m A m v v h

t t

t

m m

t t

o

A f m f m f m h v

t t t t

where

m

A f m m ln

m

t

From the minimization conditions given ab ove one obtains

v m m tan A

g g tan

h f m f m

t t t

f m f m f m tan A m tan

t t t t

m m

t t

n

h sin

t

m m m sin f m f m f m A

t t t

f m f m

t t

A m tan A m cot

t t

m m

t t

From the ab ove equations one can derive easily

m m tan

Z

M A

Z

tan

m f m f m g

t t t

f m f m m A m cot

t t t t Z

M cos m m

t t

W

A

Here all m are evaluated at M using eqns AA Only the splitting

i Z

in the stop sector has been taken into account since this splitting dep ends on

the large Yukawa coupling for the top quark see the mixing matrix eq A

More general formulae are given in ref The Higgs masses corresp onding to

this one lo op p otential are

m m m A

A A

g m f m f m

t t t

f m f m m A m A

A t t t t

m m

M sin

t t

W

m m M A



H A W H

hm hm g m

t t t

A

H

m m

sin M

t t

W

h

M m m

Z A hH

v

u

u

m M m M cos

u

A Z A Z

u

u

A

cos M sin M sin M cos M

u

Z A Z A

t

sin M M

Z A

m A m cot g

t

t

dm m A

t t

m m

sin M

t t

W

m A m A m cot m g m m

t t

t t t t

ln ln

m m m m

sin M

t

t t t

W

A m A m cot

t t

m A dm

t t

m m

t t

m m A m A m cot A m cot g

t t t

t t

ln dm m

t t

m m m m m

sin M

t t t t t

W

A

where

m m

hm ln

m m m

q q

m m m

dm m ln

m m m

and m is the mass of a light squark

q

A Charginos and Neutralinos

The RGE group equations for the gaugino masses of the SU SU and U

groups are simple

dM

i

b M A

i i i

dt

with as b oundary condition at M M t m The solutions are

GU T i

t

i

M t m A

i

i

Since the gluinos obtain corrections from the strong coupling constant they

grow heavier than the gauginos of the SU group There is an additional

complication to calculate the mass eigenstates since b oth Higgsinos and gauginos

are spin particles so the mass eigenstates are in general mixtures of the weak

interaction eigenstates

The mixing of the Higgsinos and gauginos whose mass eigenstates are called

charginos and neutralinos for the charged and neutral elds can b e parametrized

by the following Lagrangian

c

L M M M hc

Gaug inoH ig g sino a a

where a are the Ma jorana gluino elds and

a

B

B C

B C

B C

W W

B C

A

B C

B C

H H

A

H

arethe Ma jorana neutralino and Dirac chargino elds resp ectively Here all the

terms in the Lagrangian were assembled into matrix notation similarly to the

mass matrix for the mixing b etween B and W in the SM eq The mass

matrices can b e written as

M M cos sin M sin sin

Z W Z W

B C

B C

B C

M M cos cos M sin cos

Z W Z W

B C

M

B C

B C

M cos sin M cos cos

Z W Z W

A

M sin sin M sin cos

Z W Z W

A

p

M sin M

W

c

A

A M

p

M cos

W

The last matrix has two chargino eigenstates with mass eigenvalues

q

M M M M cos M M M sin M

W W W

A

i The four mass eigenstates of the neutralino mass matrix are denoted by

i

 

with masses M M The sign of the mass eigenvalue

 

neutralino state corresp onds to the CP quantum number of the Ma jorana

In the limiting case M M M one can neglect the odiagonal ele

Z

ments and the mass eigenstates b ecome

p p

B W H H H H A

i

with eigenvalues jM j jM j jj and jj resp ectively In other words the bino

and neutral wino do not mix with each other nor with the Higgsino eigenstates

in this limiting case As we will see in a quantitative analysis the data indeed

prefers M M M so the LSP is binolike which has consequences for dark

Z

matter searches

A RGE for the Trilinear Couplings in the Soft

Terms Breaking

The Lagrangian for the soft breaking terms has two free parameters A and B for

the trilinear coupling and the mixing between the two Higgs doublets resp ec

tively

Since the A parameter always o ccurs in conjunction with a Yukawa coupling

we will only consider the trilinear coupling for the third generation called A

t

The evolutions of the A and B parameters are given by the following RGE

M M dA M

t

Y A A

t t

dt m m m

dB M M

Y A A

t t

dt m m

The B parameter can be replaced by tan through the minimization conditions

of the p otential The solution for A t is

t

m

Y H A

t t

H A A t

t

Y F t m Y F t

t t

P ar ticl e b b b

g

l

l

l

r

w

q t

t

l

t

r

h

H

t

S tandar d M odel

M inimal SU SY

Table A Contributions to the rst order co ecients of the RGE for the gauge

coupling constants

P ar ticl es b

ij

B C

B C

B C

g

B C

A

B C

B C

B C

w

B C

A

B C

B C

B C

q l

B C

A

B C

B C

H eav y H ig g ses

B C

B C

A

and H ig g sinos

Table A Contributions to the second order co ecients of the RGE for the

gauge couplings

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