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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
Introduction
Interaction – term describing possibility of energy and momentum exchange or possibility of creation
and anihilation of particles
The known interactions: 1) Gravitation 2) Electromagnetic 3) Strong 4) 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
E   m02c4  p 2c2
Virtual particle – temporarily existed particle, it is not valid relation (they exist thanks Heisenberg
uncertainty principle):
E   m02c4  p 2c2
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?
Interactions and their character
Four known interactions:
Interaction
intermediate boson
interaction constant
range
Gravitation
graviton
2·10-39
Weak
W+ W- Z0
7·10-14 *)
10-18 m
Electromagnetic
γ
7·10-3
infinite
Strong
8 gluons
1
10-15 m
infinite
*) Effective value given by large masses of W+, W- a Z0 bosons
Interaction intensity α
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
strong SU(3)
unified
SU(5)
weak SU(2)
electromagnetic U(1)
Range of interaction depends on mediate
particle mass
Magnitude of coupling constant on their
properties (also mass)
Energy E [GeV]
Leveling of coupling constants for high
transferred momentum (high energies)
Example of graphical representation of exchange
interaction nature during inelastic electron scattering on
proton with charm creation using Feynman diagram
Symmetries
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)
A) 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) Symmetry of 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-symmetry t → -t.
Their combination:
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:
 

r  r  r





dr
 dr
p  m  p  m
 p
dt
dt
Pseudovector (axial vector) transformation during mirroring:
  
  



L  r  p  L  r  p   r   p   L
1) Decay of mesons K+ and K-:
Meson spins: I=0, orbital moments of π meson systems: l = 0 → parity after decay is given by
intrinsic parities of π mesons (are pseudoscalars with parity П(π+) = П(π -) = П(π0) = –1)
Two possible decays for K+:
K+ → π+ + π0
K+ → π + + π+ + π -
→
→
П = П(π+)∙П(π0) = 1
П = П(π+)∙П(π+) ∙П(π -) = -1
for K-:
K- → π- + π0
K- → π + + π - + π -
→
→
П = П(π -)∙П(π0) = 1
П = П(π+)∙П(π -) ∙П(π -) = -1
Because П(K+) = П(K-) = -1 → decay to two π does not conserved parity
2) Asymmetry of electron emission direction during beta decay against spin direction - firstly for
60Co – C.S.Wu 1957:
60Co
60Co
→ 60Ni + e- + anti-νe
Polarization by strong magnetic field → enhanced
emission of electrons to opposite direction against
magnetic field (spin) direction
decay
60Co
decay
at the mirror
3) Only left-handed neutrina exist (helicity –1) and righthanded antineutrina (helicity +1) → only P transformation
→ left-handed neutrino to right-handed neutrino
Occurs only for weak interactions → very small effects
→ world in the mirror differs from normal world only
a little
Orientation of nucleus spin and electron
momentum at the normal world and at mirror
world
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
Simultaneously C and P transformation → lefthanded neutrino transforms to right-handed
antineutrino → CP symmetry conserves
Neutrino
at the mirror
Orientation of neutrino spin and momentum
at normal world and mirror world
Violation of CP symmetry:
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
Quantum electrodynamics
Electromagnetic interaction description:
Macroworld - Maxwell theory of electromagnetic field → classic electrodynamics – description by

fields:


A
E  grad  
B

rot
A
t





B
Satisfy Maxwell equations:
I. series:
div
E

rotE  
40
t

(at vacuum)



E 

rotB  0  j   0
div
B
0
II. series:


t














A

A

A
 grad 
Gauge invariance:
t
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:
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
Elementary QED vertex
(straight lines with arrow
represent electron, wave line
photon)
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 
A ~ e
2
40 c
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:
d
Z 2 2 c 

dq 2
q 4c2
2
For scattering of relativistic electrons on fixed
2
nucleus:
d 4  Z2 2 c 

dq
2

4 2
qc
cos 2
Diagrams describing scattering of electron and positron
2
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
3) Hyperfine hydrogen structure
4) Positronium properties
Feynman diagrams for calculation of electron
magnetic moment – (a) – basic the lowest
diagram
Quantum chromodynamics (QCD)
= 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:
1) gluons mediate interaction and they have also color charge → gluons color interact together
2) 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:
n  p  e   e
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:
Basic vertexes of weak interaction
Feynman diagram of beta decay
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 bosons with 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
experiment on 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
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
Interaction intensity α
Describes very accurately almost all experimental measurements at micro world
strong SU(3)
unified
SU(5)
weak SU(2)
electromagnetic U(1)
Energy E [GeV]
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 charge quantization is great mystery in the frame of standard model.
2) Existence of symmetries between quark and lepton families:
 sa 
For each lepton family quarks family
 e   u a   
    b a 
     a       a 
     a 
in three colors exists.
e

d
c
     
   t 
 

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:
e  d  X  uu
 e d  Y  ud
→ 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
Supersymmetric theories
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:
E
 2
2
m
c
Vgrav (r)  G N
 GN  
r
r
where GN is Newton gravitation constant (GN = 6.67·10-11m3kg-1 s-2 = 6.71·10-39 ħc(GeV/c2)-2)
From Heisenberg uncertainty principle:
r
 c c


p pc E
2
Inconsiderable influence of gravitation interaction is in the case Vgrav ~ E and then:
2
Vgrav
E
 2
2
GN
c 
E

 GN

 E  2   E
c
 c 
c 
 
E
Correspondent energetic scale of non-negligence of gravitation interaction:
2
2
GN
c
c
E
E
 E  2   E   2  

 1.49 1038 (GeV/c 2 ) 2
39
2 2
c
G N 6.7110 c(GeV/c )
c 
c 
and then E  1019 GeV, corresponding size scale ~10-35 m (Planck scale):
r
c

E
c
c 2
c
GN

G N
6.67  10 11 m 3 kg 1s 2  1.054  10 34 kgm 2 s 2

 5  10 34 m
3
3
8

1
c
3  10 ms


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 spacetime → 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.