Transcript Black Hole

The Fundamental Forces
1. Costituents of Matter
2. Fundamental Forces
3. Particle Detectors
4. Symmetries and Conservation Laws
5. Relativistic Kinematics
6. The Static Quark Model
7. The Weak Interaction
8. Introduction to the Standard Model
9. CP Violation in the Standard Model (N. Neri)
1
The concept of Force in classical and quantum physics
In Classical Physics :
In Quantum Physics :
• Instantaneous action at a distance
• Field (Faraday, Maxwell)
• Represented by force lines
• 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
Electromagnetism
Guideline: explain all fundamental
phenomena (phenomena between
particles) with these interactions
4
Electromagnetism
Affects all particles with electric charge (Quarks, 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
  
1

 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:
m 2c 2
 U (r) 
U (r)  0
2
2
Interaction strength
(electric charge)
g
R
g e r / R
U (r) 
4 r

mc
Interaction range
U(r) plays the role of scattering potential in configuration space, which can be
analyzed in the (Fourier transformed) momentum space.
7
Classical and Quantum Fields
Quantum Field: photons, creation and annihilation operators
Classical Field: the role of quantum fluctuations is not important . The field can be
described as a purely classical spacetime function. As an example, one can
consider the scattering of electrons by an external e.m. field A, such as the field
generated by an heavy nucleus.







i
q
A static classical field would have the form: A ( x )  A ( x )  dq A ( q ) e x
e
e

e
The scattering of an electron by the field of a heavy nucleus treated as a point
charge is called Mott Scattering.
Using the Coulomb gauge :


A0  qA  0
The potential in space and momentum space are :
 Ze

Ae ( x)    ,0,0,0 
 x


 Ze

Ae (q)    2 ,0,0,0 
q


8
Momentum space
Scattering amplitude for a particle in
a potential
Let us imagine a particle interacts with a
coupling with a potential U
u
Potential
Particle
g0
General scattering amplitude in a
(boson-mediated) potential
g
g er / R
U (r ) 
4 r
u

 i qr
g g
f (q )  g 0  U (r ) e dV   2 0 2
q m
The propagator (scattering amplitude)
associated to this potential :
Propagator
1
f (q)   2
q
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) 

q2  E 2  q 2
1
g0 g
2
2
q m
Propagator
Couplings
9
(Scattering by a potential: 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 )eiqrcos r 2 d sin  d 
2




0
0
0
0
0
 g 0  d  d  dr U (r )eiqrcos r 2 sin   2 g 0  drU (r ) r 2  d eiqrcos sin  




1
1 iqr iqr
1
iqrcos
iqrcos
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  dr U (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
10



0
 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
11
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 :
M  g0 g u ( p1' )   u( p1 ) D (k )u ( p2' )  u( p2 )
D (k ) 
1
q2
Photon
Propagator
 g
k2
And the Cross Section will have the form :
d
 M if
dq 2
2
( PS )

1
 f u  u g 2 g 0v   v i
q

J

J
'

2
( PS )
Dirac
Spinors

Phase
Space
Flux
Dirac currents
u, u , v, v
PS

12
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
d     2
  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.
13
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 )
14
 ()  i
In a Scattering Process :
Non-interacting particles
in the initial state
 ()  S  ()  S i
S
i
f
The solution of the general problem
i
d
 (t )  H I (t )  (t )
dt
 ()  i
Non-interacting particles
in the final state
t
 (t )  i  (i )  dt1 H I (t1 )  (t1 )

15
t
 (t )  i  (i )  dt1 H I (t1 )  (t1 )
t1

Can be solved by iteration. Using :
(t1 )  i  (i )  dt 2 H I (t 2 )  (t 2 )

t
 (t )  i  (i )
 dt H
1
I
(t1 )  (t1 ) 

t
 i  (i )  dt1 H I (t1 ) i  (i ) 2
t1
t
 dt  dt
1



t
t
t1
 i  (i )  dt1 H I (t1 ) i  (i ) 2

 dt  dt
1

2
H I (t1 ) H I (t 2 )  (t 2 ) 
2
H I (t1 ) H I (t 2 ) i

t2
Where the series was cut at the second
order. But one could continue like this :
(t 2 )  i  (i )  dt3 H I (t3 )  (t3 )

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 (t 2 )......H I (t n )
16
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 )  dt1 H 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 (t 2 ) 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!
17
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).
18
Lowest order of other electromagnetic processes :
e Z  e  Z


    Z
Bremsstrahlung

2
 3 Z2
2
 3 Z2

Z 
 Z  e e Z


    Z
Pair Production


Z 
19
d
 M if
2
dq
Cross Section :
R   NT 
2
( PS )


f u  u e
1
ev  v i
2
q
2
( PS )

  1029 cm2  105 barn
(typical cross section of electromagnetic processes)
Reaction rate
Number of targets
Incident flux
t
Lifetimes and their relation to scattering processes
  h
Total amplitude
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
20
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 
0  1 0 0 

g ( x)    
0 0  1 0 


0
0
0

1


Far away from sources
of mass/energy (in a
flat spacetime)
Energy-Momentum Tensor
Metric Tensor
G  G (   g )
21
Gravity and Electromagnetism at the
particle scale (the two classical theories)
Gmm
ee
?? 2
2
r
r
e2
(4.8 10 10 ) 2
dyne cm cm
1
  

c 1.054 10 27  3 1010 erg s cm
137
s
G  6.674 10
11
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
mm ?? 
kg 2
kg 2
r2
r
2.12 1015
(1.67 1027kg) 2 ?? 
2
kg
1
5.9  10 39 
137
….gravity weaker by many
orders of magnitude
22
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 (in all regimes). 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
dyne cm cm
1


c 1.054 10 27  3 1010 erg s cm
137
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
23
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 :
This neutron star is about to become a black hole
2GM
rs 
2
c
Schwarzschild radius
Generally a macroscopic bodies has dimensions much bigger than its Schwaszschild radius
An object whose radius is smaller than the Schwarzschild radius is called a Black Hole
2GM
 2.95 km
2
c
2GM

 8.87 mm
c2
Sun: rs 
Some notewhorty Schwarzschild radii :
Earth: rs
24
Schwarzschild Black Hole (static and spherically symmetric)
2GM
rs 
2
c
 
M
4 3
 rs
3
2GM
 2.95 km
2
c
2GM

 8.87 mm
2
c
Sun: rs 
Earth: rs
3Mc 6
1

 2
3
3
4 8G M
M
Smaller Black Holes have higher density
What would be the density of the Sun as
a Black Hole?
3c 6
 

3
2
32 G M
Similar to the nuclear
matter density
25
Flying away from the Schwarzschild Radius :
A Black Hole
A semiclassical calculation (do not really replace a fully
general-relativistc treatment)
rs
The kinetic energy of a massive body escaping from rs to infinity :
GMm
  1 mc  mc 
 mc 2
r
2
2
GM
GM 2GM
r
 2  2
2
(  1) c v / 2
v
The «classical» nonsense
(but v<<c)
1 2 GMm
mc 
0
2
r
WRONG !
Incidentally gives the correct result for the Schwarzschild radius.

26
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
c 5
 31019 GeV
2G
The concept of a Planck scale:
2. Gravity on particles = Electromagnetism on particles (as shown before)
27
Can a high-energy heavy ion collider form a Black Hole?
M  200 GeV
E  6.5 TeV
M  200 GeV
rs
E  6.5 TeV
Even assuming that all the mass gets all the energy in a single unit
(which is unrealistic because the collision happens at the level of the elementary
constituents)
2Gm 2  6.6  1011  400  1.7  1027
 41
rs  2 

10
m
16
c
9  10

1034
19
C 


5

10
m
27
8
mc 400  1.7  10  3  10
Not enough mass!
28
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
29
The Planck scale in Astrophysics :
How powerful can be an astrophysical object ?
• Gamma Ray Bursts (energy emitted in e.m. form)
• Active Galactic Nuclei
L 1052 1053 erg s 1
L 1048 erg s 1 (for a long time)
45
1
• Supernovae L 10 erg s
for the first 100 s
43
1
• Supernovae L 10 erg s for the first month
Planck Power : (Planck Energy) / (Planck Time)
Absolute maximum of energy generation in a radius R
EP
c 5
WP 

tP
2G
c5
c5

 2.51059 erg s 1
G G 2
Physical meaning : Mass entirely converted into energy in a time equal to the
light crossing the gravitational radius of the object.
Does not depend on microphysical scales.
30
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.
31
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 (in de Donder and Transverse Traceless 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.
32
The Gravitational Multipole Expansion
The potential
generated by this
mass distribution at x :
Rewrite :
The origin is the
center of mass
Intention : expand as a power series in r/x. Use trick:
(Legendre poly for abs(x) and abs(z) less than one.
Obtain :
Dipole term goes to
zero because of
definition of the center
of mass ! Obtain :
33
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.
Since (differently from electric charges) one
has only one sign of the mass, the lowest
moment is the 4-pole.
Effect of a
gravity wave: a
space
deformation with
two polarization
states :
http://demonstrations.wolfram.com/GravitationalWavePolarizationAndTestParticles/
34
Detectors for gravitaty waves : Auriga, Nautilus, Explorer, LIGO, VIRGO…
The VIRGO Interferometer (Cascina, Pisa) for the detection of gravitational waves
35
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 Leptons cannot change arbitrarily:
 Proton stability
36
“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
d
u
s
u
d
u
d
u

p
37
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
38
An estimate of the Weak Coupling Constant
  n   
(Weak)
1


  1010 s
2
weak
0    
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
39
The Range of the Weak
Nuclear Interaction: 10-18 m
Compton Wavelength argument :
Weak Interactions Propagator :
R

c 197 MeV fm
3



2

10
fm
2
mc mc
90 GeV
g2
f (q)  2
q  M W2 Z
Low energies
q2 << M2WZ
g2
f (q) 
 GF
2
MWZ
u
d
g
1
M W2 Z
g
e
The Fermi constant of the low energy Weak Interaction.
An effective interaction of the form:
J weak
GF
e
'
J weak
2
g
LFermi  GF J  J '  2 J  J '
MW Z
40
The Fermi Weak Coupling Constant
GF  1.2 10
It is often quoted as:
What is actually meant is:
Using the usual expression:
One finds:
5
GeV
2
GF
2 g2

(c ) 3 8M W2
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
41
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
42
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) :
43
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
44
     
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):
Interaction
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. Not discussed in this course.
45
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)    
46

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
47
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
9.3% BR
Tau decay
Feynman
diagram
(Lifetime 5x10-13 sec)
48
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
N l  N e  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»
49
Examples of lepton number conservation:
Muon decay :
Neutron decay :
Please note that :
These processes are ok :
These processes are not ok!
In the Standard Model there
are no processes that mix
letpons from different
generations
50
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
A «strong»
 s  g  1 coupling constant !
2
s
51
The gluon
m0
However, the gluon has a very short range
Confinement: range limited to 10-15 m
R

mc
s 1
Two independent
polarizazion states
The six color charges (sources of the strong nuclear field):
Color Confinement : only
color singlet states are
possible as observed
particles.
Antiquarks carry anticolor
Quarks carry color
Those are states with no
color, invariant under the
symmetry group
transformation of Strong
Interactions.
52
Color neutrality: colorless
states (color singlet states):
1
( r r  bb  g g )
3
Meson color wavefunction
1
( rgb  rbg  gbr  grb  brg  bgr )
6
Baryon color wavefunction
The strong force is mediated by 8 gluons
rg
rb
r
gb
gr
br
bg
gs
1
rr  gg 
2
1
rr  gg  2bb 
6
b
 gs gs  s
rb
b
gs
Gluons are
colored !
r
Gluons, being colored, carry the strong charge themselves.
53
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
54
55
Gluon – Gluon Interactions
56
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
57
How the Strong Nuclear force is different :
Increasing the distance between
constituents makes e.m. energy go down
Increasing the distance between
constituents makes strong energy go up
These particles are mostly
pions (and Kaons)
58
A few aspects of the Strong Force (QCD, Quantum Chromodynamics)
3-gluon vertex
(typical of non abelian gauge theories)
Color lines
representation
The inside of a hadron
(a lively place!)
Strong flux tube
Gluon force lines
Compare with
electric dipole:
59
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
p T  300 MeV
60
Strong Interaction at work : Hadronic Physics
Cross Sections
Lifetimes
R   NT
  h
≈100 MeV
10-23 s
  1026 cm2  102 barn
10 13 cm
 23
t

10
s
10
3 10 cm / s
Hadron crossing time
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 2    Ei     Pi 
 i
  i 
61
Important
Fundamental Interactions do not generally take
place one at at time (while one of them is often
dominant)
D mesons decay to K mesons.
They are Weak Interaction processes.
However, Strong Interaction corrections
might be present (actually: they are
always present if quarks are involved)
FSI: Final State Interactions
K meson decaying weakly to pions.
FSI due to Strong Interactions
62
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.
63
The running of the coupling constant and the vacuum
Let us take the case – pour fixer les idées – of Electromagnetism (and QED)
Now, it is well known that the classical theories has some infinity problems
related to the concept of a bare charge :

 
For instance, a classical potential would not be
 ( x )  Q  ( x  x0 )

defined at the point of the source
x0
m  m0 
 2
1  2
1  2 1
d
x
E

m

d
x
E

d
x
E 
0



2
2 IN
2 OUT
Classical self-energy of an electron
2
a
1 r
1  e 
2
2


 m0   
4

r
dr

4

r
dr 

2 0 3 
2 a  4 r 2 
2
8 2  2
 m0 
18
a

0
e2
r dr 
2
4
2

1
dr
a 4 r 2 
a
OUT
IN
6
2
8 2 2 5 e
1 16 2 2 a
e
  a 
 m0 
 


En extended classical model of
90
8a
20 9
a 8a
the electron
e2
e2
 m0 

20a 8a
 m0 
64
Just using the external energy term would imply an electron
radius a of ~10-14 cm (classical electron radius).
The particle must have at least this size or its e.m. energy would
exceed its total energy!
In disagreement with experiments on the structure of the electron
e2
m
8a
In QED the model is different. The electron mass in free space is affected by the
presence of virtual particles around the bare electron :
Electron self-energy graph
These effects will generate an electron mass of the form
f (r )
m  m0 
1
f (r )dr

2
The usual electric field squared when r >> λc of the e
A shielded E2 when r ~ λc of the electron
The electric field in the proximity of the electron is reduced by the virtual Quantum
Field fluctuation effects
65
In QED the electron mass in free space is affected by the presence of virtual
particles around the bare electron, giving a correction to the mass :
 3

1
m  m0 1 
ln
 ...
 2 am0

This is now only logarithmically divergent .
When a «realistic» value is used for a (10-18 m)
one gets m = 1.04 m0
In QED the fundamental paradoxes appear at a
much smaller scale
The same set of fundamental interactions (the same kind of Feynman diagrams)
affect the charge of the electron and give rise to the running of the coupling
constant :
When at distances comparable – or
smaller – than the electron Compton
wavelength, the coupling constant
gets corrected
e2
1
1
 

c 127 137
66
Here is the high-energy (short-distance) behaviour of the coupling
constant. The probe that we are using is penetrating inside the
virtual e+e- pairs and is starting to see the bare charge
Here is our normal
«everyday life»
coupling constant and
electron charge
67
68
On the other hand, in a quantum field theory (like QED), infinities needs to be
dealt with and properly subtracted. This procedure is called renormalization.
Conceptually, in symbols ,
for a quantum field theory
perturbative expansion :

e0  e f ( x , t )  e
2
0
The bare charge is just a parameter of the
theory. It is not observable
2
Infinite divergent
quantity
Observed
charge
This is called renormalization. The extent to which this is possible (at all orders
in perturbation theory) defines whether the theory is renormalizable.
Dissatisfied with all of this ?
Think twice :
The measurement of the electron's g-2 is
the most precisely determined quantity in
science. It has recently been measured to
3 parts in 1013 and its value calculated in
QED from a summation of 12,672
Feynman diagrams ! (10.th order in alpha)
69
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
Quantum Gravity Epoch
Grand Unification Epoch
(Strong+Electroweak)
Electroweak Epoch
Everyday Universe
70
g2
f (q)  2
q  M W2 Z
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
MWZ
MWZ

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
71
The (predictable) surprise Test on Fundamental Interactions
Gravity
Carrier
Electro
magnetism
Weak
Nuclear
Strong
Nuclear
Graviton
Photon
W,Z
8 gluons
Spin
2
1
1
1
Mass
0
0
82, 91 GeV
0
Range

Energy

Electric q
1018 m
Weak q
1015 m
Color
10 39
1/ 137
10 5
0.1  1
1029 cm 2
1042 cm 2
1027 cm 2
10 19 s
10 8 s
10 23 s
Source
Coupling Constant
(proton)
1 GeV Cross Section
Lifetime for decay
72
The Fundamental Interactions
Carrier
Gravity
Electro
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
Energy
Electric charge
Weak charge
Color
10-39
1/137
10-5
0.1 - 1
Coupling Constant
(proton)
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
73
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 (at the energy
of the 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 q 2 )  s  1 (low q 2 )
74