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Way to the interaction unification

1) Introduction

2) Interactions and their character

3) Symmetry and conservation laws

4) Symmetry violation

5) Quantum electrodynamics

6) Quantum chromodynamics

7) Unification of electromagnetic and weak interactions

8) Standard model

9) Grand unification

10) Supersymmetric theories

Candidate for observation of Higgs boson production at experiment DELPHI on LEP accelerator at CERN (year 2000).

Build up accelerator LHC

Interaction – term describing possibility of energy and momentum exchange or possibility of creation and anihilation of particles

The known interactions:

1) Gravitation 2) Electromagnetic3) Strong4) Weak

Description by field – scalar or vector variable, which is function of space-time coordinates, it describes behavior and properties of particles and forces acting between them.

Quantum character of interaction– energy and momentum transfer through v discrete quanta

Exchange character of interactions– caused by particle exchange

Real particle– particle for which it is valid

Virtual particle– temporarily existed particle, it is not valid relation (they exist thanks Heisenberg uncertainty principle):

Search of unified theory description of forces (interactions)

Started by Maxwel theory of electromagnetic field → unification of electric and magnetic phenomena description

Great importance of symmetries: gauge symmetry – measurable effects of force field existence do not change for certain changes of scalar or vector potentials describing field

Microscopic description of electromagnetic interaction → quantum description = quantum electrodynamics (QED)

Unified description of electromagnetic and weak interactions – electroweak interaction

Strong interaction – quantum chromodynamics (QCD)

String theory = searched final theory?

Interaction intensity α

unified

SU(5)

weak SU(2)

electromagnetic U(1)

Energy E [GeV]

Interactions and their character

Four known interactions:

*) Effective value given by large masses of W+, W- a Z0 bosons

Interaction intensity given by interaction coupling constant – its magnitude changes with increasing of transfer momentum (energy). Variously for different interactions → equalizing of coupling constant for high transferred momenta (energies)

Exchange character of interactions:

Mediate particle – intermedial bosons

Range of interaction depends on mediate particle mass

Magnitude of coupling constant on their properties (also mass)

Example of graphical representation of exchange interaction nature during inelastic electron scattering on proton with charm creation using Feynman diagram

Leveling of coupling constants for high transferred momentum (high energies)

Symmetry – constancy of some properties during change of others → constancy (invariance) against some change (transformation)

1) Space-time symmetries

2) Intrinsic symmetries

1) Accurate symmetries

2) Approximate (broken) symmetries

1) Continuous symmetries

2) Discrete symmetries

Relation between symmetries and conservation laws (Nether-Theorem)

- Accurate symmetries:

1) Symmetry of natural laws against translation in space – momentum conservation law.

2) Symmetry of natural laws against translation in time – energy conservation law.

3) Symmetry of natural laws against rotation (orientation change) in space – angular momentum conservation law

4) Symmetryof natural laws against charge sign change (symmetry in charge space)

– charge conservation law

B) Approximate symmetries:

1) Symmetry of natural laws against mirror inversion – parity conservation law (P-symmetry)

x → -x , y → -y, z → -z

2) Symmetry of natural laws against exchange of particles by antiparticles and vice versa – conservation law of C-symmetry

Q → -Q, B → -B, L → -L, S → -S, …

3) Symmetry of natural laws against time inversion – conservation law of T-symmetryt → -t.

1) Symmetry of natural laws against simultaneous mirror inversion and exchange of particles by antiparticles – conservation law of CP symmetry

2) Symmetry of natural laws against simultaneous mirror inversion, exchange of particles by antiparticles and efflux direction change – conservation law of CPT symmetry

What are results of symmetry violation:

Violation of P symmetry → world in the mirror can be distinguished from world

Violation of C symmetry → antiworld is distinguished from world

Violation of T symmetry → direction of efflux is not equivalent

Violation of CP symmetry → antiworld in mirror is distinguished from world

CPT theorem– CPT symmetrical is each theory, which is invariant against Lorentz transformation. Its consequences:

1) Integral spin → Bose-Einstein statistics, half-integral spin → Fermi-Dirac statistics

2) Identity of masses and lifetimes of particles and antiparticles

3) All intrinsic quantum numbers are opposite for antiparticles then for particles

Intrinsic symmetry in charge spaces – conservation laws of isospin, baryon and lepton numbers, strangeness, charm, …

They are mostly only approximate and they conserve only for some interactions

Gauge symmetry – conservation of properties during changes of some quantity at space points.

1) Global transformation – the same change at all points

2) Local transformation – different change at different points

Requirement of achievement of gauge symmetry in elementary particle physics → necessity of introduction of compensating fields – they describe action of the forces.

Gauge theory introduces interaction between particles and it determines their properties

Symmetry violation

Some symmetries are not fully accurate → symmetry violation → violation of appropriate conservation law

Violation of isospin symmetry (in electromagnetic and weak):

Example of evidence: difference between neutron and proton

Violation of P symmetry (parity):

Macroworld - asymmetries exist (heart on the left side – on the right side in the mirror ...) – result of random processes

Microworld - (common physical laws) – strict conservation?

Evidence of parity nonconservation:

Important equation between momenta (vector) and angular momenta (pseudovector)

Vector transformation during mirroring:

Pseudovector (axial vector) transformationduring mirroring:

1) Decay of mesons K+ and K-:

Meson spins: I=0, orbital moments of πmeson systems: l = 0→ parity after decay is given by intrinsic parities of πmesons (are pseudoscalars with parity П(π+) = П(π -) = П(π0) = –1)

Two possible decays for K+:

K+ → π+ + π0 → П = П(π+)∙П(π0) = 1

K+ → π + + π+ + π - → П = П(π+)∙П(π+) ∙П(π -) = -1

for K-:

K- → π- + π0 → П = П(π -)∙П(π0) = 1

K- → π + + π - + π - → П = П(π+)∙П(π -) ∙П(π -) = -1

Because П(K+) = П(K-) = -1 → decay to two π does not conserved parity

at the mirror

60Co decay

2) Asymmetry of electron emission direction during beta decay against spin direction - firstly for 60Co – C.S.Wu 1957:

60Co → 60Ni + e- + anti-νe

Polarization by strong magnetic field → enhanced emission of electrons to opposite direction against magnetic field (spin) direction

3) Only left-handed neutrina exist (helicity –1) and right-handed antineutrina (helicity +1) → only P transformation → left-handed neutrino to right-handed neutrino

Orientation of nucleus spin and electron momentum at the normal world and at mirror world

Occurs only for weak interactions → very small effects → world in the mirror differs from normal world only a little

Neutrino

Violation of C symmetry:

Example: only left-handed neutrina and right-handed antineutrina exist → only C transformation → left-handed neutrino transforms to left-handed antineutrino

Neutrino

at the mirror

Simultaneously C and P transformation → left-handed neutrino transforms to right-handed antineutrino → CP symmetry conserves

Orientation of neutrino spin andmomentum at normal world and mirror world

Violation of C symmetry and P symmetry compensate mutually almost fully → violation of CP symmetry is even smaller

Evidence of CP symmetry violation:

K0 and anti-K0 differ only in strangeness – strangeness is not conserve for weak interaction → oscillation between K0 and anti-K0 states.

Decay of the K0 and anti-K0 system will have two:

components: K0L → π0 + π0 + π0 (τ = 5.17∙10-8s, CP = -1)K0S → π0 + π0 (τ = 0 .89∙10-10s, CP = 1)

Weak component of decay K0L → π0 + π0, which violets CP symmetry

Even larger effect occurs for B0 and anti-B0 mesons and some other decays connected with B mesons → first results from Fermilab confirm violation of CP near to standard model predictions

Violation of T symmetry:

In the case of CPT symmetry conservation → violation of T symmetry compensates CP symmetry violation → equivalence of these phenomena

Conservation of CPT symmetry – its violation was not observed up to now

Electromagnetic interaction description:

Macroworld- Maxwell theory of electromagnetic field → classic electrodynamics – description by fields:

Satisfy Maxwell equations:

I. series:

(at vacuum)

II. series:

Gauge invariance:

The E and B fields are not changed during such cases of potential transformation

Microworld - quantum description → necessity of quantum electrodynamics (QED) building

Spectrum of absolutely black body + photoeffect electromagnetic field is quantized – quantum of electromagnetic field = photon

Building started by Dirac equation:

QED describes interaction of Dirac charged fields with quantized electromagnetic field

Mainly description of interaction of charged leptons (mainly electrons and positrons) and photons.

Hadrons influence of strong interaction

Mathematical apparatus of quantum electrodynamics:

Decreasing with distance, weak interaction between fields (constant α = e2/ħc = 1/137 is small) → possibility of separation of interacting fields to electromagnetic and electron-positron & possibility of perturbation theory using – numerical results are expand with order of α

Infinite number of degrees of freedom → perturbation theory leads to divergent series

Elimination of divergences and obtaining of right finite values of physical quantities using redefinition of masses, charges and coupling constants - renormalisation

Perturbative terms of higher order in α2 α3 … – radiation corrections

Searching of perturbation theory in relativistic invariant form

Simplification of mathematic apparatus - Feynman graphs:

Laws for construction and interpretation of Feynman diagrams:

Elementary QED vertex (straight lines with arrow represent electron, wave line photon)

1) Energy, momentum and charge are conserved at vertexes

2) Unbroken straight lines with arrow in the time direction represent fermions, arrows against time direction represent antifermions

3) Dashed, wavy and helical lines represent bosons

4) Lines which have one end on diagram boundary represent free (real) particles incoming to or exiting from reaction

5) Line connecting two vertexes (internal lines) mostly represents virtual particles. Exception is

representation of real and unstable particle, which is compound state of incoming particles to

reaction

6) Time arrow of internal lines is not determined. Diagrams with arrows in opposite direction are

same

7) Each outside particle should have marked out momentum

Searching of combination of vertexes representing given process

The simplest diagrams with the smallest number of vertexes for given processes – basic the lowest diagram – the lowest order of perturbation theory

Calculations of cross sections and ratios between transition probabilities using diagrams:

Each vertex contributes to transition amplitude A by magnitude ~e Scattering of e on e – two vertexes

Cross section: σ ~ A2 ~ α2

Constant α =1/137 higher order of diagram higher power of α smaller influence of diagram perturbation theory can be used

Basic the lowest diagrams of process of scattering of electron on electron (one diagram) and electron on positron (Bhabha scattering – two diagrams)

Electron scattering on nucleus:

1) A ~ Zα

2) Virtual photon transfers momentum q → A ~ 1/q2

3) q depends on scattering angle θ → we determine

dσ/dq2

And then:

For scattering of relativistic electrons on fixed nucleus:

Diagrams describing scattering of electron and positron

Very near to experimental value, more accurate form ↔ full apparatus of QED

Experimental tests of QED:

1) Magnetic moment of electron

Experiment: 1.001 159 652 187(4) μB

QED: 1.001 159 652 307(110) μB

2) Magnetic moment of muon

Experiment: 1.0011659160(6) eħ/2mμ

QED: 1.0011659200(20) eħ/2mμ

Feynman diagrams for calculation of electron magnetic moment – (a) – basic the lowest diagram

3) Hyperfine hydrogen structure

4) Positronium properties

= dynamic theory of quarks and gluons describing color strong interaction

Similarity with quantum electrodynamics:

1) QED - interaction of charges by „massless“ photons

QCD – interaction of color charges by „massless“ gluons

2) QED – gauge theory – comutative symmetry group UQ(1)

QCD – gauge theory – noncomutative color symmetry group SUC(3)

Diferencies with quantum electrodynamics:

- gluons mediate interaction and they have also color charge → gluons color interact together
- both combination of quark with color and anti-quark with anticolor and combination of three quarks with three different colors are colorless

Strong interaction bonds quarks to colorless hadrons and creates nuclear force mediated by meson exchange.

Asymptotic freedom – magnitude of color forces increases with decreasing of distance and with increasing of transferred momenta (energy) → for high energies quarks start to be free particles → perturbation approximation can be used for high energies.

Low energies → necessity of nonperturbative theory - quarks bonded to hadrons → the larger distance of quarks the larger interaction → impossibility of quarks → confinment

Sufficient energy → creation of quark and antiquark pairs → new hadron.

Even higher energies → produced quarks end in colorless bounded states → production of hadron jets

Strong nuclear force between hadrons → residual color force of Van der Waals type

Experimental evidence for validity of QCD:

1) Non observed of free quarks

2) Results of scattering experiments for very high energies (dependency of cross section on transferred momentum)

3) Properties of hadron jet production

Reconstruction of two jets in DELPHI experiment

Unification of electromagnetic and weak interaction

(Electroweak interaction description)

It does not create bounded particle states – it realizes only by decay

The known manifestation of weak interaction – beta decay:

Very small value of coupling constant. Very short range 10-18 m

Conception of mediate gauge bosons → finding of theory of weak interaction description with renormalization similar to QED and QCD.

Weak intensity of interaction and its short range given by large mass of gauge bosons

Description by Feynman diagrams:

Feynman diagram of beta decay

Basic vertexes of weak interaction

Example of Feynman diagram for neutral and charged currents:

proton

proton

hadrons

hadrons

Confirmation of assumption of such theory of electroweak interaction:

Existence of W+, W-, Z0 gauge bosonswith masses ~ 80 and 90 GeV

Existence of neutral charges given by Z0 boson

Confirmed at CERN

Increasing of mass given by Higgs mechanism – existence of Higgs boson

Neutrino interactions – clean weak interactions

Production and decay of W+ boson observed by Delphi experimenton LEP2 accelerator at CERN

Standard model of matter and interactions

Particles and interactions of standard model:

I. Particles of matter – fermions and antifermions (s=1/2):

1) three lepton families (e, νe), ( μ, νμ), ( τ, ντ)

2) three quark families (d, u), (s, c), (b, t)and their antiparticles

II. Particles of interactions – gauge bosons (s=1):

1) electroweak boson with m0 = 0 (photon γ)

2) three electroweak bosons with m ≠ 0 (W+, W-, Z0)

3) eight colored gluons

III. Higgs bosons (s=0)

Interactions:

1) Electromagnetic of photon interaction

2) Interaction of bosons W+, W-, Z0

3) Strong interaction of gluons with gluons and quarks

Charges of single interactions:

Strong – color (red, green, blue)

Electromagnetic – electric charge

Weak – flavor 6 types (u, d, s, c, b, t, for quarks e, νe, μ, νμ, τ, ντ for leptons

Interaction intensity α

unified

SU(5)

weak SU(2)

electromagnetic U(1)

Energy E [GeV]

Higgs boson – Higgs mechanisms – gives mass to originally mass less gauge bosons W+, W-, Z0

Gauge symmetries → coupling constants of interactions changes with transferred momentum:

Electroweak interaction: coupling constant is increasing

Strong interaction: coupling constant is decreasing

Intensity of interactions become equal for energy 1019 GeV

Describes very accurately almost all experimental measurements at micro world

Coupling constants become equal for high transfer momenta ( high energies)

Way above standard model – Grand unification

Extreme success of standard model. Anyhow there are reasons, why go above:

I)Big number of parameters in standard model (masses of leptons, quarks, gauge bosons, Higgs, different mixing parameters)

II)Existence of many symmetries between particles and interactions of standard model (for example symmetry between quark and lepton families).

III)Noninclusion of gravitation – forth fundamental interaction.

IV)Experimental evidences:

1) Existence of baryon asymmetry at Universe

2) Evidence of neutrino oscilation existence

3) Existence of nonbaryonic dark matter at Universe

Grand unified theory

Laws not explained by standard model:

1) Origin of electric charge quantization:

Quantization of angular momentum in ħ/2 units – it results from properties of symmetry groups, which result to angular momentum conservation law ( are noncommutative – nonabelian).

Quantization of charge in e/3 unit dos not result from properties of symmetry groups, which lead to charge conservation (is commutative).

Nature of electric chargequantization is great mystery in the frame of standard model.

2) Existence of symmetries between quark and lepton families:

For each lepton family quarks family

in three colors exists.

Proposal of their solution in the frame of Grand unification:

Assumption A: symmetry groups of standard model are part of high degree noncomutative symmetry group → source of electric charge quantization

Assumption B: single quarks of different colors and leptons in corresponding families are only different states of one particle (for example ured, ublue, ugreen, νe or dred, dblue, dgreen, e-)

Given assumptions → weak interaction between leptons mediated by W±, Z0 bosons and strong interaction between quarks mediated by color gluons are different manifestations of one fundamental interaction.

Intensity of interaction connected with electric charge increases with transferred momentum (energy). Intensity of interaction connected with color charge decreases with transferred momentum (energy)→ for high energy ( ~ 1015 GeV) magnitude of these forces become equal.

Grand unification theory → searching of nonabelian symmetry group containing standart model groups, which unifies quarks and leptons to one family (multiplet).

Intermedial bosons mediate transition between particles → existence of gauge bosons, which change quarks to leptons and vice versa → X, Y bosons (leptoquarks) - MX,Y ≈ 1015 GeV,

Feynman vertices for interaction of leptoquarks. Other may be obtained by changing of particles by antiparticles (change of arrow orientation)

Leptoquark charges: QX = -4/3e a QY = -1/3e

Their change to both antilepton – antiquark and also quark pair, diagrams are above or for example:

→ nonconservation of baryon and lepton numbers → change of nucleons to leptons → proton decay: through virtual X, Y boson:

p = uud → e+

energy and momentum conservation laws → more then one particle is created → decays p → e+π0, p → e+π+π– and similar.

Examples of Feynman diagrams of proton and bounded neutron decays

Proton decay was searched also by big Cherenkov detector Kamiokande (Japan). Picture of installed photomultipliers

Very large MXY → long lifetime of proton τp > 1031 years. Depends on concrete form of theory (used symmetry groups). Experiment τp > 5·1032 years.

Implications for Universe origin: Inflation during interaction split, baryon asymmetry of Universe

To date restriction of symmetries on transformation of similar types of particles:

1) rotation → change of electron spin projection

2) rotation in isospin space → changes: p → n, π -→ π0 → π+ ...

3) change of quark to lepton

Supersymmetric theories:

Searching of symmetries, which make possible transformation of bosons to fermions → supersymmetric (SUSY) symmetries.

Theories invariant against such transformations → supersymmetric theories.

These theories lead to doubling of number of fundamental particles → each has its supersymmetric partner:

boson → supersymmetric fermion (fotino, gravition, gluino, …)

fermion → supersymmetric boson (s-quark, s-lepton)

They were not observed up to now – if they exist, their observation is waited in near future.

Supergravitation, superstrings:

For close distances (high energies) gravitation starts to be significant:

where GNis Newton gravitation constant (GN = 6.67·10-11m3kg-1 s-2 = 6.71·10-39 ħc(GeV/c2)-2)

From Heisenberg uncertainty principle:

Inconsiderable influence of gravitation interaction is in the caseVgrav ~ Eand then:

Correspondent energetic scale of non-negligence of gravitation interaction:

and then E 1019 GeV, corresponding size scale ~10-35 m (Planck scale):

Near to the scale of Grand Unification → description of fundamental interaction on this scale must include gravitation.

Problems with construction of quantum theory of gravitation ↔ divergences while cross sections are calculated ↔ renormalization is not working for Einstein General Theory of Relativity.

Supersymmetric theories → better behavior of divergences. Supersymmetric theories cover gravitation → supergravitation.

Even the best from these theories are not without divergences.

Nature of divergences:point like character of particles → interaction in accurate point of space-time → zero uncertainty in position → from Heisenberg uncertainty principle infinite uncertainty of transferred momentum.

Removal of divergences: transition to finite particle sizes (~10-35 m) → interaction vertexes are not accurately localized → finite inaccuracy of transferred momentum → divergences disappear.

Theories describing particles as very small linear objects – string theories.

Description of interaction by diagrams - string diagrams.

Application of perturbation theory (depends on string coupling constant magnitude):

interaction

time

Interaction of couple of strings:

Process is described by diagrams with different numbers of loops – the more loops the smaller influence (diagram without loops is dominant – virtual string pairs)

Two extra dimensions to spacetime compactified into sphere shape (taken from book B. Greene: The Elegant Universe)

Introduction more then four dimensional description of space-time (10 - 11), part of dimensions is compactified it takes effect only on ultra subatomic level

Geometry of compactified dimensions determines basic properties of particles (masses, charges)

Elimination of divergences – relation between black holes and elementary particles

Many (six) of different string theories all are part of common M-theory.

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