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 : • Instantaneous action at a distance • Field (Faraday, Maxwell)
F
k r
2 Inverse square law In Quantum Physics : • Exchange of Quanta 2
Classical and Quantum concepts of Force: an analogy
Let us consider two particles at a separationdistance r .
r .
If a source particle emits a quantum that reaches the other particle, the change in momentum will be: And since:
c
t
r r
p
p
t
r c r
We have:
F
p
t
k r
2 A concept of force based on the exchage of a force carrier. In a naive representation: 3
Fundamental Forces of Nature
Gravity Elettromagnetism Strong Nuclear Force Weak Nuclear Force 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):
e
2
c
1 137
dyne erg cm cm 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 Propagator Final state Interaction range estimate by using the static Klein-Gordon equation: 2
U
(
r
)
m
2
c
2
U
(
r
) 2 0 Interaction strength (electric charge)
g R
mc U
(
r
)
g
4
e
r
/
R r
Interaction range U(r) plays the role of scattering potential in configuration space. In the (Fourier transformed) momentum space…… 7
Momentum space Scattering amplitude for a particle in a potential Let us imagine a particle interacts with a coupling with a potential U
u
Particle
u g
0 Propagator
g
Potential
U
(
r
)
g
4
e
r
/
R r f
(
q
)
g
0
U
(
r
)
e i
q r
dV
q
2
g
0
g m
2
f
(
q
) 1
q
2 Photon Propagator Or, more precisely, including also energy :
q
2
E
2
q
2
f
(
q
)
q
2
g
0
g m
2 Momentum space
U
(
r
)
g
4
e
r
/
R r
Configuration space Scattering amplitude for a particle in a (boson-mediated) potential
f
(
q
) 1
q
2
m
2
g
0
g
Propagator Couplings
q
q
(the details)
f
(
q
)
g
0
U
(
r
)
e i
q r
dV
q
2
g
0
g m
2 in this section
f
(
q
)
g
0
U
(
r
)
e
i
q r
g
0 0 2
d
0 0
dr dV
g
0
U
(
r
)
e
iqr
cos
r
2
d
U
(
r
)
e
iqr
cos
r
2
sin
2
g
0
sin
d
0
dr U
(
r
)
r
2 0
d
0
d
sin
e
iqr
cos
1
iqr
0
d e
iqr
cos
1
iqr
e
iqr
e
iqr
1
qr e
iqr
cos
sin
2 sin
qr
4
g
0
g q g
0 0
dr U
(
r
)
r
2 sin
qr qr
0
dr e
r
/
R e iqr
e
iqr
2
i
4
g
0
q
0
dr U
(
r
) sin
qr
4
q g
0 0
dr g
4
r g
0
g
2
iq
0
dr
e
(
iq
1 /
R
)
r
e
(
iq
1 /
R
)
r
e
r
/
R r
sin
qr g
0
g
2
i q
(
e
(
iq
1 /
iq
1 /
R
)
r R
)
e
(
iq
1 /
R
)
r
(
iq
1 /
R
) 0
g
0
g
2
iq
(
iq
1 /
R
)
e
(
iq
1 /
R
)
r
(
iq
1 / (
iq R
)(
iq
1 1 / /
R
)
e
(
iq
1 /
R
)
r R
) 0 9
f
(
q
)
g
0
g
2
iq
(
iq
1 /
R
)
e iqr e
r
/
R
(
i
2
q
2 (
iq
1 /
R
)
e
iqr e
r
/
R
1 /
R
2 ) 0
g
0
g
2
iq
(
iq
1 /
R
) (
iq
1 /
R
)
q
2 1
R
2
g
0
g
2
iq
2
iq q
2
m
2
q
2
g
0
g m
2
R
mc
In the chosen system of units 10
Scattering amplitude and cross section Let us imagine the interaction of two Dirac (charged) particles A typical matrix element for this process will have the form :
M
g
0
g u
(
p
1 ' )
u
(
p
1 )
D
(
k
)
u
(
p
2 ' )
u
(
p
2 )
u
Particle
u g
0 Propagator
g v
Particle
v f
(
q
) 1
q
2 Photon Propagator And the Cross Section will have the form :
d
dq
2
M if
2 (
PS
)
u
u g
1
q
2
g
0
v
v
2 (
PS
)
J
J
' Dirac currents Phase Space Flux
PS
Dirac Spinors
u
,
u
,
v
,
v
11
e
Scattering Rutherford
e
Electrons in initial and final states 1
q
2 Intermediate virtual photon
The Feynman Diagrams
time
e
e e
e
e
e
d
dq
2
q
2 2
q
4 2 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
L
0
L I L
0
N
[
(
i
m
)
1 2 (
A
) (
A
) ]
L
I
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 dt
(
t
)
H I
(
t
)
(
t
)
H I
(
t
)
e iH
0 (
t
t
0 )
He
iH
0 (
t
t
0 ) 13
In a Scattering Process : Non-interacting particles in the initial state
(
)
i
(
)
S
(
)
S i S
i f
(
)
i
Non-interacting particles in the final state The solution of the general problem
d i dt
(
t
)
H I
(
t
)
(
t
)
(
t
)
i
(
i
)
t
dt
1
H I
(
t
1
)
(
t
1
)
14
(
t
)
i
(
i
)
t
dt
1
H I
(
t
1
)
(
t
1
)
Can be solved by iteration : 1
(
t
)
i
(
i
)
t
dt
1
H I
(
t
1
)
i
2
(
t
)
i
i
(
i
)
t
dt
1
H I
(
t
1
)
i
(
i
)
t
dt
1
H I
(
t
1
)
i
(
i
)
t
dt
2
H I
(
t
2
)
1
(
t
)
(
i
)
2
t
dt
1
t
1
dt
2
H I
(
t
2
)
H I
(
t
1
)
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
0
(
i
)
n
dt
1
t
1
dt
2
.....
t
n
1
dt n H I
(
t
1
)
H I
(
t
2
)......
H I
(
t n
)
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
S
i
(
i
)
t
dt
1
H I
(
t
1
)
i
(
i
)
2
t
dt
1
t
1
dt
2
H I
(
t
2
)
H I
(
t
1
)
i
To every term in the S expansion a diagram can be drawn, following precise formal rules (outside of the goal of this course)
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 : It can by calculated by summing up the amplitudes due to various diagrams:
e
e
e
e
P
(
e
e
e
e
)
+ …… 2 = + + +
Fundamental (“tree level”)
2
2 First order in Perturbation Theory
2 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
Bremsstrahlung
Z
2 3
Z
2
Z
Z
e
e
Z
Pair Production
Z
2 3
Z
2
Z
18
Cross Section :
R
N T
d
dq
2
M if
2 (
PS
)
u
u
1
e q
2
ev
v
2 (
PS
) 10 29
cm
2 10 5
barn
(typical cross section of electromagnetic processes) Reaction rate Number of targets Incident flux t
Lifetimes:
h
Total amplitude Branching ratio of different final states
h
1 2 .....
n
(
B
1
B
2 ....
B n
) 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 Partial amplitudes to different final states
B
1
B
2 ....
B n
1 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 Classical field theory (Newton, 1687) for the masses 2 4
G
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 Far away from sources of mass/energy (in a flat spacetime)
g
(
x
) 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Einstein Tensor Cosmological Costant Energy-Momentum Tensor Metric Tensor
G
g
8
G c
4
T
G
G
(
g
) 20
Gravity and Electromagnetism at the particle scale (the two classical theories)
G m m r
2
e
2
c
( 4 .
8 10 10 ) 2 1 .
054 10 27 3 10 10
dyne cm cm
erg s cm s
1 137 ??
e r
2
e
Same dependence on distance Fine structure constant
G
6 .
674 10 11
G
2 .
12 10 15
kg
2
c N m m kg
2 6 .
674 10 11 1 .
05 10 34
s m kg
2 6 .
67 10 11 1 .
05 10 34 Gravity constant (written in a way to show h and c
kg
2
c
3 10 8 Now let us compare: 2 .
12 10 15
kg
2
m m r
2
c
??
r
2
c
2 .
12 10 15
mm kg
2 ??
Need to choose charges and masses. For the case of two protons 2 .
12 10 15 ( 1 .
67 10 27
kg
) 2
kg
2 5 .
9 10 39 1 137 ??
….gravity weaker by many orders of magnitude 21
Gravity is normally negligible at the atomic and subatomic level.
But not at the Planck Mass:
M P
10 19
GeV
/
c
2 10 19 1 .
78 10 27
kg
1 .
78 10 8
kg
2 .
12 10 15
kg
2
M P
2 ??
2 .
12 10 15 ( 1 .
78 10 8
kg
) 2
kg
2 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
Charge
Photon Spin1 e R
Gravity
M G Graviton Spin 2 M G
Energy
R
e
2
c
( 1 .
054 4 .
8 10 10 10 27 ) 2 3 10 10
dyne cm cm
erg s cm s
1 137 Electromagnetism Two adimensional constants (at the mass and charge of the proton)
GM
2
c
2 .
12 10 15
kg
2
c M
2 (
kg
2 )
c
10 39 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 : This neutron star is about to become a black hole Escape velocity :
2
Gm
c r
Speed of light
r
s
2
Gm c
2 Schwarzschild radius An object whose radius is smaller than the Schwarzschild radius is called a
Black Hole
Some notewhorty Schwarzschild radii :
Sun: Earth:
r s r s
2
Gm c
2 2 .
95 2
Gm
8 .
87
c
2
km mm
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
r
s
2
Gm c
2
mc
C
m
P
c
2
G E
P
m
P
c
2
c
5 2
G
3 10 19
GeV
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
c
3 .
0 10 8
m
3
kg
1
s
2
m s
1 1 .
1 10 34
J s
What is the (only) way to form a
length
with these constants ?
What is the (only) way to form a
mass
with these constants ?
What is the (only) way to form a
time
with these constants ?
t l P
M
P
G
c
3
1 .
6
10
35
m
c G
2 .
2
10
8
kg
P
G
c
5
5 .
4
10
44
s
One then has a Planck energy
E
P
M
P
c
2
2 .
0
10
9
J
1 .
2
10
19
GeV
..and a Planck temperature
T
P
E
P
/
k
1 .
4
10
32
K
25
In General Relativity : A notewhorty application of General Relativity (with some assumptions regarding the matter distribution of the Universe) actually: Energy (not just matter) 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
t
2
A
0 4-vector • Two polarization states • Photon: spin 1 Graravitational radiator
g
h
Flat spacetime curvature In the linearized (weak field) theory, far away from the source (and in the De Donder gauge): 2
t
2
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
(
u
,
d
,
s
) It takes place through the Weak Interaction because it violates the Strangeness quantum number 2 .
6 10 10
s
(
u
,
u
,
d
) (
d
,
u
) The decay diagram
s d u W
d u u d u
p
31
The importance of Weak Interactions: the pp cycle in the Sun :
p + p 99,77%
d+ e
+
+
e
p + e
-
0,23% + p
d +
e
84,7% 13,78%
7
Be + e
7
d + p
3
He +
13,8%
3
He +
4
He
7
Be +
Li +
e
~2
10 -5 % 0,02%
7
Be + p
8
B +
3
He+
3
He
+2p
7
Li + p ->
+
8
B
8
Be*+ e
+
2
+
e 3
He+p
+e
+
+
e 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 2
weak
10 10
s
weak
10 18 10 10 10 4 0
e
2
c weak
(Electromagnetic)
g
2
c
10 19
s
1 2 Weak charge Weak Interaction Carriers and Propagator W ± 80.4 GeV/c 2 Spin 1
e p
n e
Z 0 91.2 GeV/c 2 Spin 1
u g u e
g g W
W
d
e
e g d e
e
e
e
g g Z
0
e
33
The Range of the Weak Nuclear Interaction: 10 -18 m Compton Wavelength argument :
R
mc
c mc
2 197
MeV
90
GeV fm
2 10 3
fm
Weak Interactions Propagator :
f
(
q
)
g
2 2
M W Z
G F u
e g d g
1
M
2
W Z e
f
(
q
)
q
2
g
2 2
M W Z
Low energies q 2 << M 2 WZ The Fermi constant of the low energy Weak Interaction. An effective interaction of the form:
J weak J
'
weak G F L Fermi
G F J
J
'
g
2 2
M W Z J
J
' 34
The Fermi Weak Coupling Constant
It is often quoted as:
G F
What is actually meant is:
(
G
F
c
)
3 5
GeV
2 5
GeV
2 Using the usual expression: One finds:
G
F
c
197
MeV
fm
5 3 As an example, the cross section for the process
e e
e e
e
2
G F
2 4
m e
2
88
10
46
cm
2
G F
(
c
) 3 2
g
8
M W
2 2 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, Z 0 )
e q
e q
Photon-mediated
e q
Z-mediated
e q
e u
e
W-mediated
d
e
Z-mediated
e
36
Charged Currents Weak Interactions: Nuclear Beta Decay
)
1,
N e
e
(at the nuclear level)
n e
e
(at the free neutron level)
d e
e
(at the fundamental constituents level) Charged Currents Weak Interactions: Antineutrino Scattering
e
p
e
n
(at the free proton level)
e 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: ) 1,
N e
e
While the ELETRON ANTINEUTRINO is the one being emitted in the process: ) 1,
N 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.
Muon Neutrino Muon Lederman, Schwartz, Steinberger Experiment (1962) Use of a muon antineutrinos beam from kaon decays in flight (at Brookhaven):
p
n
YES Electron (muon) neutrinos produce electrons (muons) when brought to interact with matter.
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
(Lifetime 2x10 -6 sec)
e
e
17 .
4 % 17 .
8 %
BR BR
(Lifetime 5x10 -13 sec) 9 .
3 %
BR
42
Lepton Numbers
:
N e N
N e
( )
)
N
e
N
e N
(
)
N
(
)
N
(
)
N
(
)
N
N
(
)
N
(
)
N
N
Electron Lepton Number Muon Lepton Number 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» 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 ?
K
Strong decay
p
0 ( 1385 ) 10 23
s
0
strong
10 18 10 23 10 2 Electromagnetic decay 0 ( 1192 ) 10 19
s
s
g s
2 1 A «strong» coupling constant !
44
The gluon
m
0 However, the gluon has a very short range
R
m c
Confinement: range limitated to 10 -15 m
s
1 Two independent polarizazion states The six color charges (sources of the strong nuclear field): Quarks carry color Antiquarks carry anticolor Color neutrality: colorless states (color singlet states): 1 3 (
r r
b b
g g
) 1 6 (
grb
gbr
bgr
rgb
rbg
brg
) 45
A few color singlets: The pion: The proton:
3
u
r
d
r
u
b
d
b
u
g
d
g
3
p
u b u r d g
u g u b d r
u r u g d b
udu s
duu s
The strong force is mediated by 8 gluons
r g r b g b g r b r b g
1 2
r r
g g
1 6
r r
g g
2
b b
Gluons are colored !
r
g
s
b b g
s r b r
g
s
Gluons, being colored, carry the strong charge themselves.
g
s
s
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
low q
2
s
ln (
q
2 1 / 2 ) 270
MeV
The two Strong Interaction regimes Running coupling constant
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
V s
(
r
) 4 3
s r
kr
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 b g 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
p T
300
MeV
50
Cross Sections Lifetimes
R
N T
h
≈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
M
2
i E i
2
i
P i
2 10 26
cm
2 10 2
barn t
10 13 3 10 10
cm cm
/
s
10 23
s
Hadron crossing time 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 10 16 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
The most recent case: Electroweak Unification
f
(
q
)
q
2
g
2 2
M W Z
Guideline: Electromagnetic and Weak Forces as manifestations of a unique Force at q 2 >10 4 GeV 2 , 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
e e
e
e
g e
g
W
W
W
e
W
e W
g
e e
g e
Z
0
Z
0
W
g g W
W
Weak Neutral Currents allow the renormalization process, provided there is the right connection between the coupling constants
g
2
M
2
W Z
G F
e
2
M
2
W Z g
e
54
The Fundamental Interactions
Gravity Elettro magnetism Weak Nuclear Strong Nuclear
Spin Mass Range Source Coupling Constant (proton) 1 GeV Cross Section Lifetime for decay Graviton 2 0 ∞ Mass 10 -39 Photon 1 0 ∞ Electric charge 1/137 W,Z 1 82,91 GeV 10 -18 m Weak charge 10 -5 8 Gluons 1 0 10 -15 m Color 1 10 -29 cm 2 10 -19 s 10 -42 cm 2 10 -8 s 10 -27 cm 2 10 -23 s 55
The Coupling Costants
Gravity (proton mass) E.M. (proton charge) Weak (proton mass) Strong (proton mass)
GM
2
c
2 .
12 10 15
kg
2
c M
2 (
kg
2 )
c
10 39
e
2
c
( 4 .
8 10 1 .
054 10 27 10 ) 2 3 10 10
dyne cm erg s cm s cm
1 137
G F
3
m p c
2 1 .
2 10 5
GeV
2
m p c
2 10 5
s
1 (
high q
2 )
s
1 (
low q
2 ) 56