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Introduction lecture
1.
2.
3.
4.
5.
6.
7.
Introduction
Angular momentum and spin
Anti-matter
Width - Lifetime
Experimental spectra
Cross section
Luminosity
8. Interactions : introduction
9. Electromagnetism
10. Strong interaction
11. Weak interaction
Few concepts, first glance in the
particles’ world
Overview of the 3 interactions
(detailed in the next lectures)
12. Symmetry : what is it ?
13. Relationship between conservation and symmetries
14. Continuous symmetries
15. Discrete symmetries : P, C, T
16. Conservation law : summary
MH Schune, Kiev, Feb 2008
Symmetries in particle physics
1
Warning …
This lecture :
•
is going to be too fast
•
will probably repeat many things you already know
•
Will hopefully say with my words some concepts which will prove to be useful in the
following lectures
MH Schune, Kiev, Feb 2008
2
Few concepts, first
glance in the
particles’ world
MH Schune, Kiev, Feb 2008
3
1. Introduction
A bit of history
•
•
•
•
•
1897 (Thompson)
electron discovery
1912 (Rutherford)
proton discovery
~1930 (Pauli/Fermi)
neutrino νe hypothesis
1958 (Reines-Cowan) : experimental evidence
1932 (Chadwick)
neutron discovery
1932 (Anderson)
positron discovery :e+ 1st antiparticle
In the ’30s one knows : e- p n and νe
the electromagnetic force and the γ
Framework to try to explain the forces between p, then between p and n⇒ Strong interaction
Observation of unstable particles in cosmic rays + β decays
When the muon was discovered the physicist I.
Rabi said :
Many particles :
Increasing mass
•1937 μ→eνeνμ
•
⇒ Weak interaction
Discovered in 1962
•1947 π→ μνμ
•1947 strange particles : K Λ
It remains in fact a very good question…..
1955 (Chamberlain, Segre, Wiegand, Ypsilantis) antiproton discovery
MH Schune, Kiev, Feb 2008
4
1950
Larger statistics
Accelerators era :
More and more
hadrons …
•1948 first π produced
Control on the particles
•1950 π0 discovery
•1954 K+ K0 , Σ production …
•1964 Ω discovery
Greek : 4 elements : earth, air, fire, water
100
China : 5 elements : earth, wood, metal, fire and water
India (3000a.C) : 5 elements :space, air, fire, water and
Philosophers + Alchemists add : ether, mercury, sulphur, salt
Chemistry
Partic
les “Z
o o”
Number of particles
85 elements
4
1661
Boyle defines chemistry
1789
Lavoisier makes the list of 33 elements
1868
Mendeleïev classify the elements
1914
85 elements are known
e,p
0
1500
1800 1900
MH Schune, Kiev, Feb 2008
years
5
1964 Zweig-GellMann-Neeman : theoretical introduction of quarks
’70 SLAC
Deep inelastic scattering experiments
Experimental evidence of quarks
p≡(uud)
n ≡(udd)
Strange particles (Λ,K…): quark s
’90 HERA
…. New particles..
•1974 (Richter/Ting) J/ψ discovery (cc̅) : c quark
•1975 (Perl)
discovery of the τ lepton
•1977 (Ledermann) Υ discovery (bb̅ bound state) : b quark
•1995 (CDF/DØ coll.) t quark
•2000 (Donut) : ντ
MH Schune, Kiev, Feb 2008
6
100
3 quarks
4 leptons
Particl
es “Zo
o
Number of
particles
”
85 elements
chemistry
4
12 : 6 quarks
6 leptons
e,p
0
1500
years
1800 1900
12 matter particles to explain all known particles !
Hadrons : any particle which
undergoes the strong interaction
q
q
q
(Nucleon : neutron and proton)
q q
Baryons half integer spin
(ex: p =(uud) )
Mesons integer spin
(ex: π = (ud) )
Leptons : Any particle which does not undergo the strong interaction (e,μ,τ) (νe,νυ,ντ)
MH Schune, Kiev, Feb 2008
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Interaction particles
•
Classical mechanics : interactions = force field
•
Modern physics :
interactions = interaction particles which are field quanta
1973 Observation at CERN of the “weak neutral currents”
Interactions between neutrinos Æ “Z0” ?
1976 Standard Model. Electroweak unification
They are vectors of the weak interaction, their masss
⇒ γ, Z0 W+are predicted
1983 Observation at CERN of the Z0 and W+- bosons
1989 « Mass production» of Z0 at LEP at CERN
1996 Production of W+W- pairs at LEP at CERN
MH Schune, Kiev, Feb 2008
8
The elementary particles of the Standard Model
matter :
3 families
interactions :
W±, Z0 (weak)
NB : the gravitational force is
extremely weak in the particles
world ⇒ not discussed here
g : gluon (strong)
γ (electromagnetic)
All tests of the SM have been successful up to now ! But it does
not answer to many fundamental questions. For example :
• Why 3 families ?
+ antiparticles
• Why several interactions with very different intensities ?
•Origin of the masses of the particles (ad-hoc Higgs boson) which
has not been seen yet …
•Mass hierarchy
•The neutrinos masses : mν ≲ qq eV and mν ≠ 0
Charged leptons
MH Schune, Kiev, Feb 2008
quarks
9
The mass
Defined by : m2c4=E2-p2c2
Invariant length of the Energy-momentum 4-vector
With c=1 E, p and m are expressed using the same unity (GeV/MeV ….)
– When p =0 ⇒ E = mc2
– When v increases ⇒ E2 et p2c2 increase but their difference remains constant
– m is a Lorentz invariant
New particles production:
Mass/energy
It is not “divisibility” !
Since c is large
small mass
=
Large energy
mass
energy
energy
mass
A particle is a lump of energy
MH Schune, Kiev, Feb 2008
10
Quantum mechanics Relativity
Quantum field theory
Few words on quantum field theory
It is our main working tool for particles physics in the Standard Model and beyond
It comes from the marriage between quantum mechanics and relativity
E = mc2
(High energy physics)
mass/energy
New particles production
p= h/λ
(physics on extremely small scales)
Particle/Wave
Study the matter
h = 6.62606876(52) 10−34 Js
= = h / 2π
MH Schune, Kiev, Feb 2008
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2. Angular momentum and spin
Angular momentum
Classical mechanics :
G G G
L=r∧ p
3 components :
• can be measured with
infinite precision
• can have all values
Quantum Mechanics :
G
same definition, with the operators R and P (notation L or L )
•The algebra of the components of L : [Li,Lj] = i εijkLk ; εijk= 0, +1, -1
according to ijk. One also has : L²=Li²+Lj²+Lk² ; [L²,Li]=0
•2 independent operators (usually : L2 et Lz )
2=
(2 useful quantum numbers)
=
•quantification :
•L2 : ℓ(ℓ+1) ħ² ; ℓ is an integer
•Lz: mħ with m = -ℓ, -ℓ+1, …, -1, 0, 1,…, ℓ-1, ℓ −=
−2=
• Addition of 2 angular momenta :
j1, j 2 ; m1, m2 =
j1 + j 2
J
∑ ∑
J = j1 − j 2 M =− J
J,M J,M j1, j 2 ; m1, m2
MH Schune, Kiev, Feb 2008
ClebschGordan(CG)
coefficients 12
Spin
• The spin is the intrinsic kinetic momentum of a particle.
• it can be half-integer
• It determines the behavior of a given particle.
• Few examples of experimental evidences for the spin :
• Fine structure of the atoms spectral lines : each line is made of several components
very close in frequency
• “Abnormal » Zeeman effect : Each spectral line is divided in a given number of
equidistant lines when the atom is in an uniform magnetic field. «Anomaly» : the atoms
of Z odd (ex. Hydrogen) divide into an even number of sub-level. In fact the number of
levels is 2A+1 Æ proof of half integer kinetic momentum !
• The spin has no classical equivalent. Trying to explain it saying that the particle rotates on its
own axis does not work.
e,p,n have very different characteristics (charge/ mass/interaction) but they
have the same spin : ½
MH Schune, Kiev, Feb 2008
13
•
The spin obeys the same laws as the other kinetic momenta :
– Algebra similar as the L one
– S2 can have the values s ( s + 1) = 2 (s can be half integer)
– And Sz : m=
with m = − s, − s + 1,... − 1, 0,1,...., s − 1, s
– One can add a spin with
• An other spin (S = S1 ⊕ S2)
• With an total angular momentum (J = L ⊕ S)
A particle can have any angular momentum L but its spin S is fixed
integer spin (Bosons)
Half integer spin (Fermions)
spin 0
spin 1
spin 1/2
spin 3/2
Elementary
-
Vectors of the
interactions
quarks, leptons
-
Composite
pseudo-scalar
mesons (p,K..)
Vector mesons (ρ,K*) some baryons (octet)
MH Schune, Kiev, Feb 2008
some baryons
(decuplet)
14
spin/statistics theorem (Pauli 1940)
Pauli’s exclusion principle : two particles of half integer spin (fermions)
cannot be simultaneously in the same quantum state
Pauli’s principle
anti-symmetry of the wave function by the
exchange of 2 particles (for the fermions)
Bohr and Pauli
For 2 particles one in the state ψα, the other one in the state ψβ, one can write :
ψ (1,2) =
1
ψ α (1)ψ β ( 2) + ψ β (1)ψ α ( 2) )
(
2
Symmetric (bosons)
ψ (1,2) =
1
ψ α (1)ψ β ( 2) − ψ β (1)ψ α ( 2) )
(
2
anti-symmetric (fermions)
If 2 fermions are in the same state (α= β) their wave function is 0 ! This problem does not exist
for bosons which can occupy the same state (ex. supra- conductors).
This can be generalized for a larger system of particles.
MH Schune, Kiev, Feb 2008
15
Helicity
JG
• Particle of spin S
•
•
G
Axis orientation in the momentum direction n
JG
Helicity :
JG JG
JG p
JG JG
JG G JG G JG
G
Λ = n ⋅ S with n= JG
Λ = n ⋅ J because p ⋅ L = p ⋅ r ∧ p = 0
p
(
•
Eigenvalues −s ≤ λ ≤ s
•
if mass=0
)
2s + 1 values
only 2 eigenvalues : ±s
s
Right-handed particle
s
Left-handed particle
The helicity is invariant under rotation (scalar product of 2 vectors).
MH Schune, Kiev, Feb 2008
16
3. Anti-matter
•
•
In a magnetic field the number of particles which turn right = number of particles which turn
left
Hypothesis : electron and proton
wrong !
Anderson 1932
MH Schune, Kiev, Feb 2008
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•
•
•
•
•
The radius of curvature is smaller above
the plate. The particle is slow down in the
lead
Æ the particle in incoming from the bottom
The magnetic field direction is known
Æ positive charge
From the density of the drops one can
measure the ionizing power of the particle
Æ minimum ionizing particle
Similar ionizing power before and after the
plate
Æ same particle on the 2 sides
Curvature measurement after the lead :
particle of ~23MeV
Æ it is not a non-relativistic proton
because he would have lost all its energy
after ~5mm (a track of ~5 cm is
observed)
Lead
plate
Momentum
direction
1 cm
Particle of positive electric charge and with a mass much smaller than the proton mass
(< 20 me) : the positron
MH Schune, Kiev, Feb 2008
18
Matter and anti-matter particles are produced in the interaction of particles with matter
+
π and π
−
particles having the
same mass, spin…
but
Opposite electric
charge
opposite curvature
in a magnetic field
MH Schune, Kiev, Feb 2008
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4. Width and lifetime
Lifetime : the exponential law
Instable particles and nuclei : number of decays per unit of time
(ΔN/ΔT) proportional to the number of particles/nuclei (N)
ΔN= cte × N × Δt ⇒ exponential law
N0
τ
N(t) = N0e-t/τ
N0/e
Mean lifetime (defined in the particle
rest frame)
The majority of the particles are instable
τ from 10-23 sec (resonances)
to ~10+3 sec (neutron)
t
The probability for a radioactive nucleus to decay during a time interval t, does not
depend on the fact that the nucleus has just been produced or exists since a time T :
⎡Survival probability ⎤ ⎡ Survival probability
⎢after the time T + t ⎥ = ⎢ after the time T
⎣
⎦ ⎣
⎤ ⎡Survival probability ⎤
⎥ × ⎢ after the time t
⎥
⎦ ⎣
⎦
MH Schune, Kiev, Feb 2008
ea+b = ea × eb
20
Few important examples of different lifetimes
•
Stable particles : γ,e,p,ν Æ the only ones !
proton stability τ(p)> ~1032 ans
•
particles with long lifetimes :
n → p + e- + ν̅e
μ− → e- + ν̅e + νμ
π+→μ+ νμ (mainly)
K+
•
particle with short lifetimes :
D+
B+
Δ++→N π
τ =6.13 10+2 sec, β decay
τ = 2.2 10-6 sec, cosmic rays
τ =2.6 10-8 sec
τ =1.2 10-8 sec
τ =1.04 10-12 sec
τ =1.6 10-12 sec
τ ~ 10-23 sec
MH Schune, Kiev, Feb 2008
21
particles which can be directly detected
•
•
The lifetimes are given in the particle rest
frame
What we see is the lifetime in the
laboratory rest frame
« Event display » of the BELLE experiment
(e+e- Æ BB̅, ECM=10.58 GeV)
Æ one should take into account the
relativistic time dilation
Æ In real life one measures lengths in the
detector
L=
βγ ×
Boost ×
cτ
lifetime
K+
π-
•
Some particles are seen as stable in the
detectors.
•
Example a pion (cτ = 7.8m) :
if Eπ = 20 GeV Æ γ= 20/mπ = 142.9 ;
β = 0.999975
γ
Β0 → Κ*0 γ
K+ π -
Æ L = 1114.3m
particles which can be directly detected in the detector : n, γ ,e, p, μ, π±, K±
MH Schune, Kiev, Feb 2008
22
Width
•
The uncertainty principle from Heisenberg for an unstable particle is :
Heisenberg :
ΔE Δt ~ =
Δmc2 = Γc2
τ
By definition :
Γc ≡
2
Uncertainty on the mass (width Γ)
due to τ
=
The faster the decay, the larger the uncertainty of on m
τ
Stable particle ↔ well defined mass state
197 × 10 −15
−22
=c = 197 MeV × 1fm ; = =
=
6.582
10
MeV.s
8
3.10
Measuring widths, one is able to
have information on very small
lifetimes. This is the way one can
have information on a phenomenon
extremely fast (the fastest in
Nature?...) : a particle with a lifetime
of 10-23 sec)
Decay
mc2
τ
Γc2
Κ∗0→ K- π+
892 MeV
1.3 10-23 s
51 MeV
π0→ γ γ
135 MeV
8.4 10-17 s
8 eV
Ds → φπ+
1969 MeV
0.5 10-12 s
10-3 eV
Measurable
width
Measurable lifetimes
MH Schune, Kiev, Feb 2008
23
Breit-Wigner
•
Schrödinger equation (free particle with energy E0):
∂ψ
i=
= Hψ = E0ψ
∂t
⇒ ψ = ae
(approximate computations)
i
− E0 t
=
−i
c2
m0 t
=
2
(particle rest frame E 0 =m0 c )
⇒ ψ = ae
– stable particle :
ψ (t ) = ψ (0) = a0
2
2
2
c2 ⎛
Γ⎞
– unstable particle :
− i ⎜ m0 − i ⎟t
2⎠
=⎝
⇒ ψ (t ) = a0e
⇒ a = a0e
−
t
2τ
⇒| ψ (t ) |2 = |ψ (0) |2 e −t / τ
We want the probability to find a state of energy E
A(E ) =
1
2π
+∞
i
Et
=
∫ ψ (t )e dt ∝
0
1
(E − m c )
2
0
Γc 2
+i
2
1
2
⇒ A ∝
Γ ≡ half maximum
width
0.5
Probability = |A|2
(E − m c )
2
0
2
+ Γ 2c 4 / 4
MH Schune, Kiev, Feb 2008
E-m0=-Γ/2
m0
E-m0=+Γ/2
E
24
Several possible final states (decay modes/channels) :
⇒ branching ratios (BRi) : probability to obtain a final state i (Σi BRi=1)
partial width Γi (definition) : BRi=Γi/Γ
Example:
Relation between lifetime, partial widths and branching ratios : Λ→pπ in 64 % of the cases
Λ→nπ0 in 36 % of the cases
τ=
= 1 = BRi
= 2
2
c Γ c Γi
Relation for 2 particles with :
Part. a ; decay mode 1
Part. b ; decay mode 2
BR1 Γ1 τ a
=
BR2 Γ 2 τ b
Example : Z0 partial widths
(PDG2002)
MH Schune, Kiev, Feb 2008
25
5. Experimental spectra
experimental spectrum K-π+ :
•
Search for a K- and a π+ in the detector and computation of the invariant mass
Fitted by a Breit-Wigner Γ =51MeV
This is a K* !
BaBar
892 MeV
« combinatorial » background
MH Schune, Kiev, Feb 2008
26
π0 experimental spectrum :
2 γ reconstruction and computation of the invariant mass.
PDG Æ
τ = 8.4 x10-17 s
Fit by a gaussian
Γ= 8 eV
σ ~ 7MeV
?
Detector resolution
effect
« combinatorial » background
135 MeV
MH Schune, Kiev, Feb 2008
27
Ds experimental spectrum : (Ds → φπ+ and φ → π+π-)
Γ ~ 10-3 eV
PDG Æ τ = 500 x10-15 s
But one sees >> 10-3 eV
Fit by a gaussian σ ~ 10MeV
Detector resolution
effect
D
Ds
⇒ One measures directly
«long » lifetimes not through
widths
1969 MeV
« combinatorial » background
MH Schune, Kiev, Feb 2008
28
π
τ(Ds) :
L
Measurement of the Ds :lifetime
L ⋅ m
t =
p
Ds
φ
t : proper time
Experiment CLEO : τ(Ds) = 486.3±15.0±5.0 fs
MH Schune, Kiev, Feb 2008
29
6. Cross section
dNint
= n1v1n2σ
dtdV
•
σ is thus defined by :
– The number of interactions per unit of volume and time the physics
processes are « hidden » in this term and depend upon the beam energy)
– The beam energy (usually v1~c !)
– The number of particles per unit of volume in the beam (n1)
– The number of particles per unit of volume in the target (n2)
•
For a i → f transition :
1
dσ =
F
•
•
Phase
space
σ : [L]2
1 barn =
∑
int
d 3 pk
f | T | i (2π ) δ (Q f − Qi )∏
3
(
2
π
)
2 Ek
k =1
2
4
4
n
Flux factor
10-24
cm2
MH Schune, Kiev, Feb 2008
30
7. Luminosity
Instantaneous luminosity
dN
= L ⋅σ
dt
Cross section
Number of
interactions /s
dNint
= n1v1n2σ
dtdV
dN1,2/dV=n1,2
luminosity
cm-2 sec-1
k bunches
kfN+ N−
L=
4π sx sy
f (=c/circumference) frequency
N+ : number of electrons in a bunch
N- : number of positrons in a bunch
y
e+
x2
ez
− 2
1
2s x
ρ ± ( x, y ) =
e
e
2π sx sy
−
y2
2s 2y
x
MH Schune, Kiev, Feb 2008
31
charge
⎡C ⎤
I (e) = ⎢ ⎥
⎣s⎦
An example : PEP-2
time
I (e ) = N (e ) × qe × N
Circumference
2200 m
I(e-)
0.75 A
I(e+)
2.16 A
Npaquets
2 x 1658
N(e-)/bunch
2.1 1010
N(e+)/bunch
6.0 1010
Beams size
sx=150 μm, sy=5 μm
L=
e
bunches
c
×
Lcirc
kfN+ N−
4π sx sy
⇒ L=3 1033 cm-2 s-1
Macroscopic quantity →
relates the microscopic
world (σ) to a number of
events
dN
= L ⋅σ
dt
MH Schune, Kiev, Feb 2008
32
Integrated luminosity
• Product of the luminosity of a characteristic time (1 year .. , experiment lifetime …)
Lint = ∫ L dt
cm −2
( )
or barn−1 b−1
•PEP-2 example L=3 1033 cm-2 s-1 1 year (~ 107 seconds)
• Lint = 3 1040 cm-2 = 30 fb-1
1 b = 10-24 cm2
• N = σ Lint
1fb = 10-15 b
• production cross section of the l’Υ(4s) : ~1.1 nb
1fb-1= 1039 cm-2
⇒ 33 106 Υ(4s) produced by year by the PEP-2 machine
Lint takes into account the machine operation : convenient !
MH Schune, Kiev, Feb 2008
33
Summary
Elementary particles :
3 families of fermions + vector bosons of the
interactions
(+ anti-matter)
leptons, hadrons, mesons, baryons
Kinetic momentum :
angular (integer) or intrinsic (spin: integer or half-integer)
eigenvalues of J2 and Jz: kinetic momentum is quantified
addition of angular momenta
identical particles; Pauli’s exclusion principle
Few properties :
helicity: Λ = n.S
Width: Γ= h̷/τ ↔ lifetime
Breit-Wigner, branching ratios
stable particles, can be directly detected
cross section
Luminosity: instantaneous and integrated
MH Schune, Kiev, Feb 2008
34
Overview of the 3
interactions (detailed
in the next lectures)
MH Schune, Kiev, Feb 2008
35
8. Interactions : introduction
Classical physics :
«modern» physics:
The particle P1 creates around it a
force field. If one introduces the
particle P2 in this field it undergoes
the force.
P1 and P2 exchange a field quantum;
the interaction boson
P1
P2
Electrostatic example :
P1
JG JG
F E P2
q1
r
The heavier the ball, the
more difficult it will be to
throw it far away
q2
JG
JG
kq1 JJG
F = q 2 E (r ) = q 2 2 ur
r
Interaction vector
Range of the interaction ∝1/mass
of the vector
MH Schune, Kiev, Feb 2008
36
Range of an interaction
•
Creation and exchange of an interaction particle
⇒ violation of the energy conservation principle during a limited time
Δt ≈
•
=
=
=
ΔE mc 2
Heisenberg principle
During Δt the particle can travel R =c Δt
=c
R=
mc 2
Range → « reduced » wave length (Compton)
with ħc ≅ 197.3 MeV fm
Example : an interaction particle with m = 200 MeV ⇔ R = 1 fm
MH Schune, Kiev, Feb 2008
37
Shape of the interaction potential
Klein-Gordon equation for a spin 0 particle :
E 2 = p 2c 2 + m 2c 4
∂ 2ψ
2
( i = ) 2 = ( i = ) c 2∇2ψ + m2c 4ψ
∂t
2 4
∂ 2ψ
m
c
− 2 = −c 2∇ 2ψ + 2 ψ
∂t
=
2 2
2
∂
ψ
m
c
1
⇒ ∇ 2ψ − 2 ψ − 2 2 = 0
=
c ∂t
2
∂
∂t
p = − i =∇
E = i=
operators
(one only deals with
stationary states)
2 2
m
c
∇ 2ψ − 2 ψ = 0
=
1 d ⎛ 2 dU (r ) ⎞ m 2c 2
=
r
U (r )
In spherical symmetry : ψ=U(r) and ΔU(r)=∇2U(r)= 2
⎜
⎟
2
r dr ⎝
dr ⎠
=
if m≠0 :
g 2 −r / R
U (r ) = − e
r
=
R=
mc
if m=0 :
r>0
Yukawa potential
g coupling constant
ΔU (r ) = 0
U (r ) = −
1 q1q2
4πε 0 r
r>0
qi = charge
In this case the Yukawa potential is
equivalent
MH Schune, Kiev,
Feb 2008 to the Coulomb one
38
Force
Vector
Lifetime (order
of magnitude)
Strong
Relative intensity
(order of magnitude)
1
Gluons
10-24 s
electromagnetic
10-2
Photon
10-19 - 10-20 s
Weak
10-5
W and Z0
10-16 - 10+3 s
Gravitation
10-40
Graviton
???
For the strong, electromagnetic and
gravitational interactions these orders
of magnitudes can be obtained
comparing the binding energy of 2
protons separated by ~1fm
The intensity of the interactions dictates the particles lifetimes and their interaction cross
sections.
MH Schune, Kiev, Feb 2008
39
9 electromagnetism (QED)
•
•
•
Between charged particles
Vector of the interaction : the photon (γ)
One Feynman graph for QED:
e-
e-
R. Feynman
√α
γ (virtual)
1/q2
√α
e-
et
electrons exchanging a photon
or
An e- which emits a γ and moves
back. The γ is absorbed by an
other e- whose direction is modified
MH Schune, Kiev, Feb 2008
40
Feynman graph
•
•
•
A powerful « graphical » method to display the interaction in perturbations theory (each
diagram is a term in the perturbation series)
Each graph is equivalent to « a number »
→ computation of the matrix elements and of the transition probabilities
Vector boson of the interaction
particle
•
•
•
Horizontal axis : the time
Lines are particles which propagate in space-time
The
represent the vertices «location» of the interaction (where there is quantum
number conservation)
e-(k’)
e-(k)
Feynman rules :
External lines: fields
(spinors, vectors, …)
Vertex: √α factor in the matrix element
« interaction intensity »
γ (q)
Propagator:
μ- (p)
t
μ- (p’)
factor igυν /(q2-m2) (depends also on
spin …)
MH Schune, Kiev, Feb 2008
41
α=
e2
4πε 0 =c
=
Virtual particles
1
137
Example QED : e+e- symmetric collision in the rest frame
e-
a
Re
s
e
l
c
rti
a
lp
√α
√α γ
a
Re
e+
In the rest frame :
Ee + + Ee − = Eγ
JJG JJG JJG
p+ + p− = pγ
s
e
l
tic
r
a
lp
mγ2 = 2me2 + 2Ee + Ee − − 2 p+ p− cosθ
It can be interpreted as :
JJG JJG JJG G
p+ + p− = pγ = 0
θ = π ⇒ mγ2 = 2me2 + 2Ee + Ee − + 2 p+ p−
incompatible with mγ = 0
The γ is « off-shell »
e-
√α
γ
Or
Creation of a massive virtual
photon during a « short » time
the γ can only exist virtually thanks to ΔE.Δt ≈ ħ
2 γ production going in opposite directions
Virtual e-
e+
Violation of the energymomentum conservation law
√α
→ energy-momentum conservation
γ
MH Schune, Kiev, Feb 2008
42
e+e- Æ e+e- interaction
√α
e-
e-
e-
√α
√α
e-
~α
+
e+
e+
√α
e+
γ exchange between an e+
and an e-
e- √α
√α e-
e-
√α
e-
e+
√α
√α
√α
√α
√α
e+
e+
e-
√α
+
e+
e+
e+e- pair annihilation in γ and γ
conversion in an e+ and an e-
√α
√α
+…
~α2
e+
exchange of 2 γ between an e+ and an e-
√α
√α
e-
~α3
+…
√α
e+
α small (1/137) : one can develop in
perturbation series
MH Schune, Kiev, Feb 2008
43
The way we see the electron and the photon is modified
electron :
photon :
e-
ee-
The electron emits and absorbs all the time
virtual γ, it can be seen as :
γ
eγ
e-
e-
e-
e-
ee-
eγ
e-
e-
γ
e+
e-
γ
ee-
…
e-
=> Theoretical (α « running ») and experimental (g-2)
consequences
MH Schune, Kiev, Feb 2008
44
Theoretical consequences of the way we see the electron
α (μ2 )
α (Q 2 ) =
The electron is surrounded by e+e- pairs with the
e+ preferentially oriented towards the e-
1−
→ Screening of the electron charge
α (μ2 )
3π
⎛ Q2 ⎞
log ⎜ 2 ⎟
⎝μ ⎠
-
-
-
- -+ - +
+
+
+
++
+
e-
-
+
+
- -
+
+
+
+
+
- -
-
-
-
Vacuum polarization
+
Test charge
Measured charge of the e-
Measurement of the electron charge with a
test charge : the closer we are, the larger
the charge we see
high energy
low energy
1/137
distance
MH Schune, Kiev, Feb 2008
45
(g-2) : Experimental evidence of the vacuum polarisation
Gyro-magnetic ratio g
• The magnetic moment associated associated to the angular momentum of the electron
JG
n
r
•
e
n : unity vector
v : electron speed
S : surface
I : intensity = charge / time
v
JG
JG
JG
JG
e
e
2
μ =I S n=
πr n =
( mvr ) n Angular
2π r
2m
momentum
v
μ = μB A
with
G
μ = μB L
G
e=
μB =
2m
=A
Bohr magneton
Intrinsic magnetic momentum :
JG
JG
μ = g μB S
Dirac : for spin ½ point-like
particles : g=2
spin
gyro-magnetic spin ratio
MH Schune, Kiev, Feb 2008
46
The value of g is modified by :
+…
« e-»
One defines a =
« e-»
g −2 g
α
1
= −1=
+ ... ≈
2
2
2π
800
a=0.00115965241 ± 0.00000000020
experiment (10-11 precision )
a=0.00115965238 ±0.00000000026
theory (α3)
MH Schune, Kiev, Feb 2008
47
10 Strong interaction
Yukawa :
Yukawa’s pontential
g 2 −r / R
U (r ) = − e
r
=
R=
mc
σ forte
g2 ⇔
r>0 , g coupling constant . Analogy with QED :
⎛ =c ⎞
~ 30mb ~ π ⎜
2 ⎟
m
c
⎝ π ⎠
2
e2
4πε 0
Geometrical size
R
π
nucleon-nucleon scattering
⇒mπ~200 MeV
Yukawa (1934) : range of the interaction ~1fm (short distance interaction between the
nucleons) due to the existence of field quantum with a mass ~200 MeV
p
π discovery in 1947
√αstrong
p
n
π
MH Schune, Kiev, Feb 2008
√αstrong
n
α strong
g2
=
=c
48
strong interaction : quarks and gluons (QCD « Quantum ChromoDynamics »)
•
•
Colour = charge of the strong interaction
Only quarks feel the strong interaction (leptons are not colored)
But:
As for QED :
q
q
g
QCD
QED
3 colours
One charge
Gluons :coloured
Æ Coupling between
3 and 4 gluons
Photon : neutral
Æ no coupling between
photons
g
q
q
g
γ
g
γ
γ
Asymptotic freedom : αs decreases when the
energy increases (or the distance decreases)
MH Schune, Kiev, Feb 2008
49
11. Weak interaction
•
The particles lifetimes are very different :
–
–
–
–
–
–
Γ∝
Δ++→p π
Σ0→Λγ
π0→γγ
Σ→nπ
π→μν
n→pνe
1
τ
~ 10-23 sec
~610-20 sec
~10-16 sec
~10-10 sec
~10-8 sec
~15 minutes
strong
Electromagnetic
weak
2
∝ M ∝ ~ coupling constant
τ ( Δ → nπ ) 10 sec ⎛ αW ⎞
= −10
=⎜
⎟
τ ( Σ → nπ ) 10 sec ⎝ α s ⎠
−23
2
2
⇒ αw~10-6
~same phase space
MH Schune, Kiev, Feb 2008
50
•
All fermions (leptons+quarks) carry a « weak charge» :
Neutral Vertex
Charged Vertex
l+
νl
Z0
W-
→ short range
l-
l-
MW~MZ~100 GeV
leptons
Charged Vertex
q’(-2/3)
Neutral Vertex
W-
Z0
q
q
q(-1/3)
quarks
MH Schune, Kiev, Feb 2008
51
Interactions : summary
•
•
The interactions are mediated by vector bosons
interaction range ∝ 1/mass
Feynman graph = display of a matrix element of the transition in the perturbations series
framework
Virtual particles (off-shell particles during a short time)
•
•
QED: electric charge, γ, vacuum polarisation, α ↗ with energy
QCD: colour, gluons (self-interaction), αs ↘ with energy (asymptotic freedom)
•
Weak: concerns all fermions, W±,Z0
•
MH Schune, Kiev, Feb 2008
52
Symmetries in
particle physics
MH Schune, Kiev, Feb 2008
53
12. What is a symmetry ?
•
A physics law is symmetric wrt a transformation if the related equation does not change
under this transformation
G
JG
d r
F =m 2
dt
2
example:
symmetric wrt time reversal t→-t
z
An object is not symmetric “by chance”, the
existence of a symmetry gives information about
the object
Various types of symmetries:
• Geometrical symmetries : rotations, translations, t→-t
• Internal symmetries (related to Quantum mechanics): isospin transformation, Charge
MH Schune, Kiev, Feb 2008
54
13 Relationship between conservation laws and symmetries
In classical mechanics :
symmetry principle ↔ non observable quantity ↔ invariance
Momentum conservation law
Absolute position non
Invariance under
⇔
⇔
observable
translation
Absolute direction
Angular momentum
Invariance under
⇔
⇔
observable
conservation
rotation
E. Noether
=> Noether’s theorem :
For any continuous symmetry for a given system corresponds a
conservation law for this system.
In quantum mechanics :
- Noether’s theorem also works :
invariance/symmetry ⇔ conservation law
- symmetry operator T: unitary (T+T = 1, Q’ = T Q T+)
- a given operator T (which can be an observable) is a symmetry operator for H (= does not
change H) if it commutes with H ([H,T]=0)
=> The associated quantum numbers
are
conserved
( selection rules )
MH Schune,
Kiev,
Feb 2008
55
14. continuous symmetries
Continuous symmetry : additive quantum number (conserved)
- space-time symmetry (translation, rotation)
For a unitary transformation Tα one can write Tα = exp(- iαQ)
Q is called the transformation’s generator
# of generators = # of parameters in the transformation (eg : 3 generators for the rotation)
The momentum operators are the generators of the translation
- internal symmetry (gauge symmetry : EM) : if global : quantum number
conservation (eg baryonic one) ; if local : « appearance » of a vector field (the
photon) see later…
MH Schune, Kiev, Feb 2008
56
15. discrete symmetries : multiplicative quantum numbers
G
G
G
P ψ ( r ) = ψ ( −r ) = ε ψ ( r )
G
G
G
G
2
2
x’ P ψ (r ) = ε ψ (r ) = P ψ ( −r ) = ψ (r )
G
G
⇒ ε = ±1 ψ + (r ) et ψ − (r )
parity P:
z
x
P
y
y’
z’
Examples:
Intrinsic parity of a particle :
JP : Spinparity
(the spin is invariant under P)
G
G
Pr = − r
G JG G JG
P(r ⋅ p ) = r ⋅ p
G JG G JG
P(r ∧ p ) = r ∧ p
G JG G
G JG G
P[(r ∧ p ) ⋅ r '] = −(r ∧ p ) ⋅ r '
spin 0
spin 1
vector (parity -1)
scalar (parity +1)
pseudovector (parity +1)
pseudoscalar (parity -1)
0+
parity +1
scalar
0-
parity -1
Pseudo-scalar
1+
parity +1
Pseudo-vector
1-
parity -1
vector
Parity for a two-particles system:
Intrinsic parity of the 2 components + parity of the angular momentum L (the part related to
the spin is invariant under P)
Ptot=P1 P2 (-1)L
MH Schune, Kiev, Feb 2008
57
Charge conjugation (C)
•
particle ↔ antiparticle
•
+
−
+
C
π
→
π
≠
±
π
Not defined in some cases :
In particular, the fermions are not C-eigenstates
•
C can be defined for a neutral boson or for a particle-antiparticle system
C π
(γ, Z0, π0, ρ, η, …):
•
0
= ηC π
η2 =1
0
C π
C
0
= ± π
0
Example: π0 →2 γ decay
Cγ=-1 ⇒ Cπ0 =1
Experimentally :
(
BR (π
) < 3.1⋅ 10
→ γγ )
BR π 0 → γγγ
Time reversal (T)
t → −t
G G
⇒r =r
JG
JG
p = −p
G
G
L = −L
0
t=t1
−8
90 % CL
C is conserved by the elm interaction
t=-t1
t=0
t=0
t=t1
t=-t1
t
complicated in Quantum Mechanics
-t
MH Schune, Kiev, Feb 2008
58
16. Conservation laws : summary
•
•
Preserved by all interactions :
– energy and momentum
– Total angular momentum
– electric charge
– Baryonic number
– Leptonic numbers
Practically the difference between
N(quarks) and N(anti-quarks)
Apart from the weak interaction, all other interactions preserve :
– The number of quarks of each type (u,d,s,c,b,t)
– Charge conjugation C
– Parity P
The flavour :
– Time reversal T
strangeness Ns − Ns
MH Schune, Kiev, Feb 2008
charm
Nc − Nc
beauty
Nb − Nb
59
We now have with us all what we need to travel in the particles’ world
MH Schune, Kiev, Feb 2008
60