Transcript Document

The Fundamental Forces
1. Costituents of Matter
2. Fundamental Forces
3. Particle Detectors (N. Neri)
4. Experimental highlights (N. Neri)
5. Symmetries and Conservation Laws
6. Relativistic Kinematics
7. The Static Quark Model
8. The Weak Interaction
9. Introduction to the Standard Model
10. CP Violation in the Standard Model (N. Neri)
1
The concept of Force
In Classical Physics :
In Quantum Physics :
• Instantaneous action at a distance
• Field (Faraday, Maxwell)
• Exchange of Quanta
k
F 2
r
Inverse square law
2
Classical and Quantum concepts of Force: an analogy
Let us consider two particles at a
separationdistance r
.
If a source particle emits a quantum that reaches the
other particle, the change in momentum will be:
And since:
We have:
c t  r
r
.
r p  
p  c

t r r
p
k
F 
 2
t
r
A concept of force based on the exchage of a
force carrier.
In a naive representation:
3
Fundamental Forces of Nature
Gravity
Strong Nuclear Force
Weak Nuclear Force
Elettromagnetism
Guideline: explain all fundamental
phenomena (phenomena between
particles) with these interactions
4
Electromagnetism
Affects all particles with electric charge (Quarkl, Leptons, W)
Responsible of the bound between
charged particles, e.g. atomic stability
Coupling constant: the electric charge
Range of the force: infinite
Classical theory: Maxwell Equations (1861)

 F  J

  F    F   F  0
F: Electromagnetic Field Tensor
J: 4-current
5
Quantum Electrodynamics (QED) is the quantum relativistic theory of
electromagnetic interactions. Its story begins with the Dirac Equation (1928) and
goes on to its formulation as a gauge field theory as well as the study of its
renormalizability (Bethe, Feynman, Tomonaga, Schwinger, Dyson 1956).
F. Dyson showed the equivalence between the method of Feynman diagrams and
the operatorial method of Tomonaga and Schwinger, making commonplace the
use of Feynman diagrams for the description of fundamental interactions.
A Feynman Diagram is a pictorial representation of a fundamental physical
process that corresponds in a rigorous way to a mathematical expression.
The pictorial representation is – however – more intuitive.

The basic structure of the electromagnetic interaction (CGS):
e2
1
 
c 137
 dyne cm cm 
  

erg
cm


Fine structure
constant
It determines the intensity of the coupling at vertices
of electromagnetic Feynman diagrams

e
e
6
The Feynman Diagram and the (bosonic) Propagator:


• Does not correspond to any physical process
• If interpreted as a physical process, it would violate E-p conservation law
• Two (or more) vertices diagrams have physical meaning
time
e
e
The concept of exchange of
quanta (represented by the
propagator) is the analog of
the classical concept of a force
field between two charges
Initial
state
Final
state
Propagator
Interaction range estimate by using the static
Klein-Gordon equation:
g er / R
U (r ) 
4 r
m2c 2
 U (r ) 
U (r )  0
2
2
Interaction strength
(electric charge)
g
R

mc
Interaction range
U(r) plays the role of scattering potential in configuration space. In the (Fourier
transformed) momentum space……
7
Momentum space
u
Scattering amplitude for a particle in
a potential
Let us imagine a particle interacts with a
coupling with a potential U

 i qr
g g
f (q )  g 0  U (r ) e dV   2 0 2
q m
Or, more precisely, including also energy :

q2  E 2  q 2
Scattering amplitude for a particle
in a (boson-mediated) potential
Propagator
Potential
Particle
g0
g er / R
U (r ) 
4 r
g
u
f (q) 
1
q2
Photon
Propagator
g g
f (q)  2 0 2
q m
g er / R
U (r ) 
4 r
Momentum space
Configuration space
f (q) 
1
g0 g
2
2
q m
Propagator
Couplings
(the details)

 i qr
g g
f (q )  g 0  U (r ) e dV   2 0 2
q m

q q
in this section

 i qr
f (q )  g 0 U (r )e dV  g 0 U (r )eiqr cos r 2 d sin d 
2




0
0
0
0
0
 g 0  d  d  dr U (r )eiqr cos r 2 sin  2 g 0  drU (r ) r 2  d eiqr cos sin 




1
1 iqr iqr
1
iqr cos
iqr cos
0 d sin e   iqr 0 d e   iqr e  e  qr 2sin qr



4 g 0
sin qr 4 g 0
g r / R
 4 g 0  drU (r ) r 2

dr
U
(
r
)
sin
qr

dr
e r sin qr 


qr
q 0
q 0 4 r
0




iqr
g0 g
 e iqr g 0 g
r / R e
( iq1/ R ) r
( iq1/ R ) r

dr
e

dr
e

e



q 0
2i
2iq 0

g0 g e
g 0 g   (iq 1/ R)e
e
 (iq 1/ R)e




2i q (iq 1/ R)  (iq 1/ R) 0 2iq 
 (iq 1/ R)(iq 1/ R)
( iq1/ R ) r
( iq1/ R ) r
( iq1/ R ) r
( iq1/ R ) r



0
9

g 0 g   (iq 1/ R)e e  (iq 1/ R)e
f (q ) 

2iq 
 (i 2 q 2 1/ R 2 )
iqr r / R

iqr r / R
e


g 0 g (iq 1/ R)  (iq 1/ R)

 
1
 0 2iq
q2  2
R
g 0 g 2iq
g0 g

2 2
2
2
2iq q  m
q m
R

mc
In the chosen system of units
10
Scattering amplitude and cross
section
u
v
Propagator
Particle
Particle
g0
Let us imagine the interaction of two Dirac
(charged) particles
g
u
v
f (q) 
A typical matrix element for this process will have the
form :
1
q2
Photon
Propagator
M  g0 g u ( p1' )   u( p1 ) D (k )u ( p2' )  u( p2 )
And the Cross Section will have the form :
d
 M if
2
dq
2
( PS )

2
1
( PS )
 u   u g 2 g 0v   v
q

J

J
'

Dirac currents
Phase
Space
Flux
PS

Dirac
Spinors
u , u , v, v
11
e
e

1
 2
q

Scattering
Rutherford
The Feynman
Diagrams
Electrons in
initial and final
states
time
Intermediate
virtual photon
2

e
e e  e e
e
   2
d
  2   4
2
dq
q  q
Rutherford Scattering
Well defined initial and final states
The simplest Feynman Diagram given the initial and final states («tree level»).
The diagram contains two vertices where the coupling constants appear
The diagram REPRESENTS the exchange of a virtual particle (the photon)
between the charged particles that are the sources of the electromagnetic field
12
A taste of the S-Matrix expansion (and Feynman Diagrams)
In a Theory of Interacting Quantum Fields
L  L0  LI
1
L0  N [ (i     m)  ( A ) ( A ) ]
2

LI  N[ e   A  ]
Normal Product
Free Fermion field
with mass m
Free E.M. field
The evolution of the system in the
Interaction Picture is described by:
Fermion current
interacting with the
electromagnetic field
d
i
(t )  H I (t ) (t )
dt
H I (t )  eiH0 (t t0 ) HeiH0 (t t0 )
13
 ()  i
In a Scattering Process :
Non-interacting particles
in the initial state
 ()  S  ()  S i
S
i
f
The solution of the general problem
d
i
(t )  H I (t ) (t )
dt
 ()  i
Non-interacting particles
in the final state
t
(t )  i  (i)  dt1H I (t1 ) (t1 )

14
t
(t )  i  (i)  dt1H I (t1 ) (t1 )

t
Can be solved by iteration :
1 (t )  i  (i)  dt1H I (t1 ) i

t
t


 2 (t )  i  (i )  dt1 H I (t1 ) i  (i )  dt2 H I (t 2 ) 1 (t ) 
t
 i  (i )  dt1 H I (t1 ) i  (i ) 2

t1
t
 dt  dt
1

2
H I (t 2 ) H I (t1 ) i

Power series expansion (Dyson Expansion) of the Scattering Matrix (power series
in the Interaction Hamiltonian. Or power series in the interaction coupling constant
S 


n
(

i
)

 dt1
n 0

t n1
t1
 dt .....  dt
2

n
H I (t1 ) H I (t2 )......H I (tn )

15
Feynman diagrams are a pictorial representation of this kind of perturbative series
To every term of the series a diagram is associated following precise formal rules
t
S  i  (i)  dt1H I (t1 ) i  (i) 2

To every term in
the S expansion a
diagram can be
drawn, following
precise formal
rules (outside of
the goal of this
course)
t1
t
 dt  dt
1

2
H I (t2 ) H I (t1 ) i

+

Fundamental (“tree level”)
 2
First order in Perturbation Theory
While Feynman diagrams are NOT a picture of the real physical process (just a
representation of a mathematical expression) they can give a lot of grasp on the
physics at work. After all, Quantum Mechanics is just a representation!
16
Perturbation Theory: a few more ideas
The occurrence probability of :
ee  ee
It can by calculated by summing up the
amplitudes due to various diagrams:
P (ee  ee) 
2
=
+

Fundamental (“tree level”)
+
 2
+ ……
+
 2
 2
First order in Perturbation Theory
Higher-order terms in the expansion, which are negligible if the
coupling constant is small. Which is the case of QED.
The graphs have constituent lines (electrons) exchanging force carriers ( photons).
17
Lowest order of other electromagnetic processes :
e Z  e  Z


    Z
Bremsstrahlung

2
 3 Z2

Z 
 Z  e e Z



    Z 2  3 Z 2
Pair Production

Z 
18
d
 M if
2
dq
Cross Section :
R   NT 
2
( PS )

2
1
( PS )
 u   u e 2 ev   v
q

  1029 cm2  105 barn
(typical cross section of electromagnetic processes)
Reaction rate
Lifetimes:
Total amplitude
h

Number of targets
Incident flux
t
  h
Branching ratio of different final states
   1  2  ..... n  ( B1  B2  .... Bn ) 
Partial amplitudes to different final states
B1  B2  .... Bn  1
Decay processes are represented by the same
kind of diagrams that are used to describe
scattering processes. The lifetime has a similar
dependence on the coupling constants
Electromagnetic processes:
  1018 s
1

   2
19
Gravity
Concerns all forms of energy of the Universe (mass included)
Responsible of bounds between macroscopic bodies
2   4 G 
Classical field theory (Newton,
1687) for the masses
Gravitational potential
Mass density
“Geometrized” spacetime field theory (Einstein, 1915)
General Relativity
The Principle of Equivalence between inertial mass
(inertia to a force) and gravitational mass (gravitational
charge) made it possible to consider gravity as a
property of the spacetime background
Einstein Tensor
Cosmological Costant
G  g   
8 G
T
c4
1 0 0
0  1 0

0 0  1

0 0 0
0
0 
0

 1
Far away from sources
of mass/energy (in a
flat spacetime)
g ( x)  
Energy-Momentum Tensor
Metric Tensor
G  G (  g )
20
Gravity and Electromagnetism at the
particle scale (the two classical theories)
Gmm
ee
?? 2
2
r
r
e2
(4.8 1010 ) 2
dynecm cm
1
 


c 1.05410 27  3 1010 erg s cm
137
s
11
G  6.67410
Same dependence
on distance
Fine structure constant
m

m
6.671011 
c
11
N m 2  6.67410

kg
1.051034 s kg 2 1.051034 kg 2 3 108
2.121015
G
c
2
kg
Gravity constant (written in a way to show h and c
Now let us compare:
Need to choose charges
and masses. For the case
of two protons
2.121015 m m

2.121015
c ?? 2 c
m m ?? 
kg 2
kg 2
r2
r
2.121015
(1.671027 kg) 2 ?? 
2
kg
1
5.9  10 39 
137
….gravity weaker by many
orders of magnitude
21
Gravity is normally negligible at the atomic and subatomic level.
But not at the Planck Mass:
2.121015
2
M P ?? 
2
kg
M P  1019 GeV / c 2 1019 1.781027 kg 1.78108 kg
2.121015
(1.78108 kg) 2  
2
kg
The Planck Mass can be defined as the mass that an elementary particle should have so that its
gravitational interactions would be similar in strength to that of other interactions (electromagnetic,strong).
Currently we have no valid quantum theory of gravity. If however such a theory exists, perhaps it could
have a structure similar to QED:
Electromagnetism
e
Photon
Spin1
e
Gravity
R
Charge
 
e2
(4.8 1010 ) 2
dynecm cm
1


 27
10
cm
c 1.05410  3 10
137
erg s
s
Electromagnetism
M G
M G
Graviton
Spin 2
R
Energy
Two adimensional
constants (at the
mass and charge
of the proton)
GM 2 2.121015
M 2 (kg 2 )

c
 1039
2
c
kg
c
Gravity
22
The Planck scale
The Schwarzschild Radius : the radius of a sphere such that, if all the mass of an object
is compressed within that sphere, the escape speed from the surface of the sphere would
equal the speed of light (wikipedia).
Every massive object has a Schwarzschild radius :
Escape velocity :
This neutron star is about to become a black hole
2Gm
c
r
Speed of light
2Gm
rs  2
c
Schwarzschild radius
An object whose radius is smaller than the Schwarzschild radius is called a Black Hole
Some notewhorty Schwarzschild radii :
2Gm
 2.95 km
c2
2Gm
rs  2  8.87 m m
c
Sun: rs 
Earth:
23
The Compton Wavelength : instrinsic quantal space scale associated to a particle
C 

mc
The concept of a Planck scale:
1. Schwarzschild Radius = Compton Wavelength
2Gm

rs  2 
 C
c
mc
mP 
EP  m P c 2 
c
2G
c5
 31019 GeV
2G
The concept of a Planck scale:
2. Gravity on particles = Electromagnetism on particles (as shown before)
24
The three fundamental constants
of the Universe
G  6.7 1011 m 3 kg 1 s 2
c  3.0 108 m s 1
  1.11034
Js
G
35

1
.
6

10
m
3
c
What is the (only) way to form a
length with these constants ?
lP 
What is the (only) way to form a
mass with these constants ?
MP 
c
 2.2 108 kg
G
tP 
G
 44

5
.
4

10
s
5
c
What is the (only) way to form a
time with these constants ?
One then has a Planck energy
..and a Planck temperature
EP  M P c 2  2.0109 J  1.21019 GeV
TP  EP / k  1.41032 K
25
In General Relativity :
actually: Energy (not just matter)
A notewhorty application of General
Relativity (with some assumptions
regarding the matter distribution of
the Universe)
Cosmology
The weak equivalence principle, also known as the universality of free fall or the Galilean equivalence
principle can be stated in many ways.
The strong EP includes (astronomic) bodies with gravitational binding energy (e.g., 1.74 solar-mass pulsar
PSR J1903+0327, 15.3% of whose separated mass is absent as gravitational binding energy).
The weak EP assumes falling bodies are bound by non-gravitational forces only. Either way,
The trajectory of a point mass in a gravitational field depends only on its initial position and
velocity, and is independent of its composition and structure.
26
Electromagnetic radiator

2
 t2  A  0
• Two polarization states
4-vector
• Photon: spin 1
Graravitational radiator
Flat spacetime
g     h 
curvature
In the linearized (weak field) theory, far away from the
source (and in the De Donder gauge):

2

 t2 h   0
• Four polarization states
• Graviton: spin 2
Trace reverse h
(tensor-like)
• Electromagnetic waves discovered in 1886 (Hertz).
• Gravity waves not yet detected.
27
Gravity wave source candidates :
• Systems whose mass distribution that changes rapidly in time.
• High masses, small times. Black-holes, Neutron Stars merging. Supernovae.
• Mass variation not having a spherical symmetry
1993 Hulse & Taylor measured the orbital decrease
rate (7 mm/day) of the binary pulsar PSR B1913+16.
This energy loss is in agreement with the prediction of
General Relativity  indirect evidence for the emission
of Gravity Waves.
Siccome esiste un segno solo della massa
(diversamente dalle cariche!), il momento piu’
basso e’ il 4-polo
Effect of a
gravity wave: a
space
deformation with
two polarization
states :
http://demonstrations.wolfram.com/GravitationalWavePolarizationAndTestParticles/
28
The VIRGO Interferometer (Cascina, Pisa) for the detection of gravitational waves
29
Weak Nuclear Force
Affects Quarks and Leptons (carriers of a “weak charge”)
Generally, the Weak Nuclear process is dwarfed by Electromagnetic or Strong
Nuclear processes.
Weak Nuclear processes are commonplace whenever:
• Conservation laws are violated (conserved in Strong or EM interactions)
• Neutral particles and/or particles with no Strong Nuclear interaction intervene
n  p  e   e
Neutron Beta Decay
  900 s
Let us get familiar with why some process just cannot take place
n  p    NO
Violates E conservation
n  e    NO Violates conservation of baryon and lepton numbers
n  p   NO Violates electric charge conservation
The number of Baryons and Leptrons cannot change arbitrarily:
 Proton stability
30
“Specific” particles:
• The Photon. Its presence is indicative of the Electromagnetic Interaction.
• The Neutrino. This particle interacts only weakly.
• W,Z. Appear only in Weak Interactions.
 e  p  n  e
Antineutrino absorption
Are there Weak Interactions without neutrinos? Yes!
  p  
It takes place through the Weak Interaction because it violates
the Strangeness quantum number
10
  2.6 10
(d , u )
The decay diagram
W
(u, u, d )
(u, d , s)

s
u
d
s
u
d
d
u
u

p
31
The importance of Weak Interactions: the pp cycle in the Sun :
99,77%
p + p  d+ e+ + e
84,7%
0,23%
p + e - + p  d + e
~210-5 %
d + p  3He +
13,8%
13,78%
7Be
3He
+ 4He 7Be + 
+ e-  7Li + e
3He+3He+2p
7Li
+ p ->+
7Be
8B
0,02%
+ p  8B + 
 8Be*+ e+ +e 3
++
He+p+e
e
2
The pp cycle is responsible for ~98% of the energy generation in the Sun
32
An estimate of the Weak Coupling Constant
  n   
(Weak)
1


0    
  1010 s
2
weak
e

c
 weak
10
4


10

1010
 weak
1
(Electromagnetic)
2
18
  1019 s

g2

c
   2
Weak charge
Weak Interaction Carriers and Propagator
W±
80.4 GeV/c2
Spin 1
Z0
91.2 GeV/c2
Spin 1
u
u
g
g
W
d
e

d
g

e
  e    e
 e p  n e

g
W
Z0
g
e
e

g
e
e
33
The Range of the Weak
Nuclear Interaction: 10-18 m
Compton Wavelength argument :
Weak Interactions Propagator :
R

c 197MeV fm
3



2

10
fm
2
mc mc
90GeV
g2
f ( q)  2
2
q  MWZ
Low energies
q2 << M2WZ
g2
f (q) 
 GF
2
M WZ
u
d
g
1
2
M WZ
e
g
The Fermi constant of the low energy Weak Interaction.
An effective interaction of the form:
J weak
GF
e
'
J weak
LFermi
2
g
 GF J  J '  2 J  J '
M WZ
34
The Fermi Weak Coupling Constant
5
GF  1.2 10
It is often quoted as:
What is actually meant is:
Using the usual expression:
One finds:
GeV
2
GF
2 g2

(c)3 8MW2
GF
5
2

1.2

10
GeV
( c )3
c 197 MeV  fm
GF  8.9 105 MeV  fm3
As an example, the cross section for the process
 e e  e e
2 GF2 me2
 46
2
e 

88

10
cm
 4
35
Two fundamental types of Weak processes:
• Charged Weak Currents: W exchange (a charged carrier W+ and W-)
• Weak Neutral Currents: Z exchange (a neutral carrier, Z0)
e q  e q
Photon-mediated
e q  e q
Z-mediated
e u   e d
W-mediated
e    e  
Z-mediated
36
Charged Currents Weak Interactions: Nuclear Beta Decay
A(Z , N )  A(Z 1, N 1)  e   e
n  p  e   e
d  u  e   e
(at the nuclear level)
(at the free neutron level)
(at the fundamental constituents level)
Charged Currents Weak Interactions: Antineutrino Scattering
 e  p  e  n
(at the free proton level)
 e  u  e  d
(at the fundamental constituents level)
At the fundamental level,
weak processes involve
Quarks and Leptons (as
well as weak carriers) :
37
Neutrino classification. The Lepton families.
While it is easy to distinguish between an electron and a positron (because of
the opposite electric charge), this is not so trivial for neutrinos.
One possibility is the dinamical distinction based on the lepton (electron) that
is produced together with the (anti)neutrino.
Definition of the ELECTRON NEUTRINO: this is the neutrino being emitted
together with the positron in the process:
A(Z , N )  A(Z 1, N 1)  e   e

While the ELETRON ANTINEUTRINO is the one being emitted in the process:
A(Z , N )  A(Z 1, N 1)  e   e
ELECTRON NEUTRINO and ELECTRON ANTINEUTRINO NEUTRINO are
associated to ELECTRON and ANTIELECTRON
38
     
Muons and Muon Neutrinos
These neutrinos are different from the ones emitted in beta
decays. In turn, they are a Neutrino and an Antineutrino.
      
Lederman, Schwartz, Steinberger
Experiment (1962)
Use of a muon antineutrinos
beam from kaon decays in flight
(at Brookhaven):
Muon Neutrino
Muon
Electron (muon) neutrinos produce
electrons (muons) when brought to interact
with matter.
  p   n
YES
  p  e n
NO
Lepton masses are well known. Neutrino masses are a non-trivial subject (Neutrino
Oscillations) but in general they are not zero.
39
How to build a Neutrino Beam ?
Schematics of an example: the CERN (to Gran Sasso) beam :
• Production of particles
• Selection of particles (energy, type)
• Kaon and pions decay to muons
• Muon decays
p+C
 (interactions) 
, K  (decay in flight)    
40

The third Lepton: the Tau.
Its mass is 1.78 GeV.
Its associate neutrino is the Neutrino τ.
Discovered in 1977 at SLAC (Stanford, California).
Detection reaction (in e+e- collisions)
e  e        e   X
X: undetected particles (neutrinos!).
This reaction featured a threshold at 3.56 GeV. With hindsight, this is twice the tau
lepton mass !
Further analysis revealed that the reactions
actually taking place where of the type:
e e       e    e    
Tau neutrino interactions where subsequently
discovered in 2002 (DoNUT experiment)
 three fundamental leptons and neutrinos
41
Donut experiment in a
nutshell: discovery of
the tau neutrino !
While the electron is stable, muon and tau neutrinos decay (weakly) :
   e   e   
         
   e   e   
      
(Lifetime 2x10-6 sec)
17.4% BR
17.8% BR
(Lifetime 5x10-13 sec)
9.3% BR
42
Lepton Numbers:
Ne  N (e )  N (e )  N ( e )  N ( e )
Electron Lepton Number
N  N (  )  N (  )  N (  )  N (  )
Muon Lepton Number
N  N (  )  N (  )  N ( )  N ( )
Tau Lepton Number
To the best of our knowledge, these three numbersa are conserved in all
interactions (with the exception of NEUTRINO OSCILLATIONS). As a
consequence, the decay :
 e
is not seen to take place. The TOTAL LEPTON NUMBER (the sum of all three
numbers) is conserved in all known interactions
Nl  Ne  N  N
A fundamental property of Weak Interactions: all leptons and associated neutrinos
behave exactly in the same way, when mass differences are taken into account.
«Weak Interactions Universality»
43
Strong Nuclear Force
Affects the Quarks constituting the hadrons
Responsible of hadron stability (baryons, mesons)
Quarks have a strong charge (color)
Mediated by GLUONS
Force strength ?
Strong decay
Electromagnetic decay
K  p  0 (1385)    0
 1023 s
0 (1192)   
 1019 s
 strong
1018
2


10

1023
s
A «strong»
 g 1 coupling constant !
2
s
44
The gluon
m0
However, the gluon has a very short range
Confinement: range limitated to 10-15 m
R

mc
s 1
Two independent
polarizazion states
The six color charges (sources of the strong nuclear field):
Color neutrality: colorless
states (color singlet states):
1
( r r  bb  g g )
3
Antiquarks carry anticolor
1
( grb gbr bgr rgb rbg  brg)
6
Quarks carry color
45
A few color singlets:
The pion:
3    u r d r  ub d b  u g d g
The proton:
3 p  ub ur d g  u g ub d r  ur u g d b  udu s  duu
s
The strong force is mediated by 8 gluons
rg
rb
r
gb
gr
br
1
rr  gg 
2
bg
gs
1
rr  gg  2bb 
6
b
 g s g s  s
rb
b
gs
Gluons are colored !
r
Gluons, being colored, carry the strong charge themselves.
46
Quark – Gluon Interactions
Interaction between Quarks is mediated
by Gluons
S
Gluons carry the Quarks color
(by contrast: the photon DOES NOT
carry the electron charge)
Color charge is conserved in Strong
Interactions
Gluon
propagator
Color lines are continuous
47
Asymptotic Freedom and Confinement
s 
low q
2
1
The two Strong Interaction regimes
Running coupling constant
ln (q /  )
2
2
  270 MeV
high q 2
The coupling constant is small (adimensional: <<1)
The perturbative expansions converges rapidly
The coupling constant is big
The perturbative expansion has problems
High distances and confinement regimes
Vs (r )  
4 s
 kr
3 r
Phenomenological potential
Confinement part
“Coulombian” part, one gluon-exchange
Two Quarks
Q
Q
Moving apart
Q
Q
The energy stored increases up to the creation of a Quark-Antiquark pair
Q
Q
Q
Q
Q
Q
48
A few aspects of the Strong Force (QCD, Quantum Chromodynamics)
3-gluon vertex
(typical of non abelian gauge theories)
Color lines
representation
anti  b
g
b
anti  g
The inside of a hadron
Strong flux tube
Gluon force lines
Compare with
electric dipole:
49
The Fragmentation (Hadronization) Process
First step: a fundamental process described by a
Feynman diagram
(however, the quarks live inside a proton)
Second step: hadronization.
The final state gets enriched by
many particles (pions…) extracted
from vacuum as the quarks get
farther apart

Two hadrons collide

Partons in hadrons collide as described by Structure
Functions (we assume, perturbatively).

Scattered parton emits a shower of quarks and gluons

Hadronization


Partons pick up color matching partner from sea
of virtual quarks and gluons
We can then observe these hadrons or their decays
pT  300 MeV
50
R   NT
Cross Sections
  h
Lifetimes
  1026 cm2  102 barn
1013 cm
 23
t

10
s
10
3 10 cm / s
Hadron crossing time
≈100 MeV
10-23 s
Hadron lifetime measurements:
• Invariant mass reconstruction
• Intrinsic width is determined
• Use of the Uncertainty Principle
Invariant Mass: relativistic invariant
which is equal to the total mass in
the center-of-mass system
2
  
2 
M    Ei     Pi 
 i
  i 
2
51
Unification of Forces
Unification of Forces:
a constant concept in
the development of
Physics
The Coupling
Constants are not
really constant
GUT hypotesis: the
unification of all forces
at high energies
The exact value of the grand unification energy (if grand unification is
indeed realized in nature) depends on the precise physics present at
shorter distance scales not yet explored by experiments. If one assumes
supersymmetry, it is at around 1016 GeV.
52
Interactions unified at high energies.
Interactions are different at lower energies because of symmetry breaking
If true, all of this has taken place during the history of the Universe
Careful:
this is a theoretical
(fascinating) hypotesis
53
g2
f ( q)  2
2
q  MWZ
The most recent case: Electroweak Unification
Guideline: Electromagnetic and Weak Forces as manifestations of a unique
Force at q2>104 GeV2, with only one coupling constant e.
At low energies, the symmetry is broken.
The presence of Weak Neutral Currents was required based on the
rinomalizability of the theory. So, the diagrams of the interactions look like:
e
g
W
e
e
e
e
e


W
W
e
W

W
e
g2
e2
 GF  2
2
M WZ
M WZ

W
Z0

Weak Neutral Currents allow the
renormalization process, provided
there is the right connection between
the coupling constants
g
e
g
e
W
Z0
g
g


e
g
W

g e
54
The Fundamental Interactions
Gravity
Elettro
magnetism
Weak
Nuclear
Strong
Nuclear
Graviton
Photon
W,Z
8 Gluons
Spin
2
1
1
1
Mass
0
0
82,91 GeV
0
Range
∞
∞
10-18 m
10-15 m
Source
Mass
Electric charge
Weak charge
Color
Coupling Constant
(proton)
10-39
1/137
10-5
1
1 GeV Cross Section
10-29 cm2
10-42 cm2
10-27 cm2
Lifetime for decay
10-19 s
10-8 s
10-23 s
55
The Coupling Costants
Gravity (proton mass)
GM 2 2.121015
M 2 (kg 2 )
39


c

10
c
kg 2
c
E.M. (proton charge)
e2
(4.8 1010 ) 2
dyne cm cm
1
 

c 1.0541027  3 1010 erg s cm
137
s
Weak (proton mass)
GF
2
5
2
2
5
m
c

1
.
2

10
GeV
m
c

10
p
c 3 p
Strong (proton mass)
 s 1 (high q2 )  s 1 (low q2 )
56