+)=m

Ch 43
Elementary Particles
Fundamental forces in Nature
Finer Structure observed
As the momentum of a particle increases, its wavelength
decreases, providing details of smaller and smaller structures:
Cf: the Heisenberg microscope
λ (20 GeV) ~ 10-16 m
1) Deep Inelastic Scattering (similar to Rutherford scattering); seeing smaller details
2) With additional kinetic energy more massive particles can be produced: particle physics =
high energy physics
The Nobel Prize in Physics 1961
"for his pioneering studies of
electron scattering in atomic nuclei
and for his thereby achieved discoveries
concerning the structure of the nucleons"
Robert Hofstadter
High-Energy Particles and Accelerators
Cyclotron/Synchrotron
Charged particles are maintained in
near-circular paths by magnets, while
an electric field accelerates them
repeatedly. The voltage is alternated
so that the particles are accelerated
each time they traverse the gap.
The Nobel Prize in Chemistry 1951
The Nobel Prize in Physics 1939
"for the chemistry of
transuranium elements"
"for the invention and development
of the cyclotron and for results
obtained with it"
Ernest Lawrence
Inventor of the synchrotron
Edwin McMillan
Cyclotron Frequency
mv 2
Stability: Lorentz force = centripetal force
qvB =
r
distance
2πr
2πm
Revolving time:
=
=
T=
speed
qBr / m qB
Frequency
A small cyclotron of maximum radius R = 0.25 m accelerates protons in a
1.7-T magnetic field.
Calculate
(a) the frequency needed for the applied alternating voltage
(b) the kinetic energy of protons when they leave the cyclotron
Small cyclotrons; non-relativistic motion
Large Hadron Collider
The maximum possible energy is obtained from an accelerator
when two counter-rotating beams of particles collide head-on.
Fermilab (r= 1 km) is able to obtain 1.8 TeV in proton–antiproton
collisions;
The Large Hadron Collider (LHC, r=4.3 km) will reach energies of
14 TeV.
Particles at relativistic speeds
Determine the energy required to accelerate a proton in a highenergy accelerator
(a) from rest to v = 0.900c and
(b) from v = 0.900c to v = 0.999c.
(c) What is the kinetic energy achieved by the proton in each
case?
The last bit requires most of the energy
Cf: the problem of space travel
Particle Exchange
The electromagnetic force acts over a distance – direct contact
is not necessary. How does that work?
Because of wave–particle duality, we can regard the
electromagnetic force between charged particles as due to:
1. an electromagnetic field, or
2. an exchange of photons.
Visualization of interactions using Feynman diagrams
Particle Exchange
The photon is emitted by one electron and absorbed by the other; it is
never visible and is called a virtual photon. The photon carries the
electromagnetic force.
Originally, the strong force was thought to be carried by mesons. The
mesons have nonzero mass, which is what limits the range of the force, as
conservation of energy can only be violated for a short time.
Virtual particle limited energy
Limited lifetime
Δt ~
h
mc 2
Maximum distance travelled (Range)
Electromagnetism
Gravitation
Æ Infinite range
m=0
ΔtΔE ≈ h
Δx ≈ cΔt ~
Strong force
Weak force
Æ Finite range
h
mc
m≠0
Intermezzo
Schrödinger equation
free partcile
non-relativistic
time-dependent
Wave equations, quantum fields
h2 ∂2
∂
(
)
,
x
t
i
Ψ ( x, t )
−
h
Ψ
=
2
2m ∂x
∂t
∂
∂x
∂
Eˆ = ih
∂t
pˆ 2
Ψ ( x , t ) = EΨ ( x , t )
2m
c 2 pˆ 2 Ψ ( x, t ) + m 2c 4 Ψ ( x, t ) = Eˆ 2 Ψ ( x, t )
Relativistic analog for the energy
Or (use operators):
pˆ = −ih
Operators
2 2
−c h
∂2
∂x
2
Ψ ( x, t ) + m c Ψ ( x, t ) = − h
2 4
2
∂2
∂t
2
Ψ ( x, t )
Klein-Gordon equation: valid for spinless massive particles
“Similar”
relativistic wave equation
for particles with spin
− i hc ∑ α i
i
∂
∂
Ψ ( x, t ) + βmc 2 Ψ ( x, t ) = ih Ψ ( x, t )
∂x
∂t
Dirac equation: valid for massive particles with spin
for “spinor”
wave functions
Intermezzo
Interactions via virtual particles
Klein-Gordon equation
(rewrite and 3-dimensional)
Massless
2
∇ Ψ−
Concept of the Yukawa potential
h
1 ∂2
2
c ∂t
∇ 2Ψ =
Solution:
m 2c 4
Ψ=
− e2
Ψ=
4πε 0 r
m = mπ
π-mesons mediate the nuclear force
(“residual strong force”)
c ∂t
2
∇ Ψ−
Static problem:
Mass
2
2
m=0
Solution:
1 ∂2
2
2
Ψ
Ψ=0
This is the classical wave equation
for electromagnetism:
Photons are the (virtual) partciles
mediating the force
1 d ⎛ 2 dΨ ⎞
r
⎟=0
2 dr ⎜
⎝ dr ⎠
r
e−r / r '
Ψ = −g
r
2
with:
r'=
h
mπ c
Particle Exchange
The mass of the meson can be calculated,
assuming the range, d, is limited by the
uncertainty principle:
For d = 1.5 x 10-15 m, this gives 130
MeV.
Yukawa predicted a particle that
would mediate the strong forces
in the bonding of a nucleus: M ~ 100 MeV
(Yukawa assumed: d = 2 fm)
Later is was found: m(π+)=m(π-)=140 MeV/c2
m(π0)=135 MeV/c2
The Nobel Prize in Physics 1949
Hideki Yukawa
"for his prediction of the existence
of mesons on the basis of
theoretical work on nuclear forces"
Particle Exchange
Strong force: The meson was soon discovered, and is called the pi
meson, or pion, with the symbol π.
Pions are created in interactions in particle accelerators. Here are
two examples:
(Note, mesons not the true carriers Æ gluons)
The weak nuclear force is also carried by particles; they are called
the W+, W-, and Z0. They have been directly observed in
interactions.
A carrier for the gravitational force, called the graviton, has been
proposed, but there is as yet no theory that will accommodate it.
Particle Exchange
four known forces
relative strengths for two protons in a nucleus, and their field particles
Intermezzo
Relativistic quantum fields and antiparticles
Klein-Gordon equation:
For every solution (E, p)
There is also a solution:
2 2
−c h
∂2
∂x
2
Ψ ( x, t ) + m c Ψ ( x, t ) = − h
2 4
(
2
∂2
∂t
2
Ψ ( x, t )
)
⎡i r r
⎤
Ψ ( x, t ) = N exp ⎢ p ⋅ x − iE p t ⎥
⎦
⎣h
~
⎡i r r
⎤
Ψ ( x, t ) = Ψ * ( x, t ) = N * exp ⎢ − p ⋅ x + iE p t ⎥
⎣h
⎦
(
Corresponding to negative energy and momentum -p
)
E = − E p = − p 2c 2 + m 2c 4 ≤ −mc 2
Problem with Klein-Gordon: positive-definite probability not guaranteed
Negative probability
Note: Dirac equation more elegant: four solutions found :
two with positive energy, two with negative energy
For each spin= ½ and spin = -½
Anti-particle
The Nobel Prize in Physics 1933
"for the discovery of new
productive forms of atomic theory"
Intermezzo
Question; What are those negative energy states ?
Vacuum:
All the negative energy states are filled
Pauli principle
Fermi-energy level
Choice of zero-level for energy
The Dirac Sea
Pair creation
A positron is a hole in the electron sea
cf: semi-conductors
Particles and Antiparticles
The positron is the same as the
electron, except for having the
opposite charge (and lepton
number).
Every type of particle has its own
antiparticle, with the same mass
and most with the opposite
quantum number.
A few particles, such as the photon
and the π0, are their own
antiparticles, as all the relevant
quantum numbers are zero for
them.
bubble chamber photograph
incoming antiproton and a proton
(not seen) that results in the creation
of several different particles and
antiparticles.
Concept of Particle Physics: Isospin
- Protons and neutrons undergo the same nuclear force
- No need to make a distinction between the two
- There is just a two-valuedness of the same particle
Define protons and neutrons as identical particles
But with different quantum numbers
Isospin
I = ½ , MI = + ½ for proton
MI = - ½ for neutron
Importance of symmetry in particle physics
Particle Interactions
and Conservation Laws
In the study of particle interactions, it was found
that certain interactions did not occur, even
though they conserve energy and charge, such as:
A new conservation law was proposed: the
conservation of baryon number. Baryon number
is a generalization of nucleon number to include
more exotic particles.
Particle Interactions and Conservation Laws
Baryon Number:
B = +1; protons, neutrons,
B = -1; anti-protons, anti-neutrons
B = 0 : electrons, photons, neutrino’s (all leptons and mesons)
Conservation of Baryon number: principle of physics
Leptons :
- Electron
- Muon (about 200 times more massive)
- Tau (about 3000 electron masses)
Conservation of Lepton numbers; Le, Lμ, Lτ
Conservation of energy, momentum, and angular momentum
Noether theorems:
Conservation laws ↔ Fundamental symmetries in nature
Emmy Noether
Particle Interactions
and Conservation Laws
This accounts for the following decays (weak interaction):
Decays that have an unequal mix of e-type and μ-type leptons are
not allowed.
Neutrino-oscillations seem to suggest that this is not always true
Particle Interactions and Conservation Laws
Which of the following decay schemes
is possible for muon decay?
(a)
(b)
(c)
All of these particles have Lτ = 0.
Particle Classification
Gauge bosons are the particles that
mediate the forces.
• Leptons interact weakly and (if
charged) electromagnetically, but not
strongly.
• Hadrons interact strongly; there are
two types of hadrons, baryons (B = 1)
and mesons (B = 0).
Hadron decay
Weak force
Strong force
Weak force
Particle Classification
BE-FD
statistics
Bosons
Fermions
Bosons
Fermions
Particle Stability and Resonances
Almost all of the particles that have been discovered are unstable.
Weak decay: lifetimes ~ 10-13 s
Electromagnetic: ~ 10-16 s
Strong decay: ~ 10-23 s.
The lifetime of strongly decaying particles is calculated from the variation
in their effective mass using the uncertainty principle.
These resonances are often called particles.
Strange Particles? Charm?
Toward a New Model
When the K, Λ, and Σ particles were first discovered in
the early 1950s, there were mysteries associated with
them:
• They are always produced in pairs.
• They are created in a strong interaction, decay to
strongly interacting particles, but have lifetimes
characteristic of the weak interaction.
To explain this, a new quantum number, called
strangeness, S, was introduced.
Strangeness not conserved in weak interactions
Strange Particles? Charm?
Toward a New Model
Particles such as the K, Λ, and Σ have S = 1 (and
their antiparticles have S = -1); other particles have
S = 0.
The strangeness number is conserved in strong
interactions but not in weak ones; therefore, these
particles are produced in particle–antiparticle
pairs, and decay weakly.
More recently, another new quantum number
called charm was discovered to behave in the
same way.
(Later: Bottomness, Topness)
Particle classifications
Quantum numbers, symmetries, and methods of “Group theory”: SU(3), SU(2), etc.
Meson octet
Baryon decuplet
Prediction of the Ω- partcile;
observation after two years
The Nobel Prize in Physics 1969
"for his contributions and discoveries
concerning the classification of
elementary particles and their interactions"
Murray Gell-Mann
Quarks
quark compositions for some
baryons and mesons:
Due to the regularities seen in the
particle tables, as well as electron
scattering results that showed
internal structure in the proton
and neutron, a theory of quarks
was developed.
There are six different “flavors”
of quarks; each has baryon
number B = ⅓.
Hadrons are made of three
quarks; mesons are a quark–
antiquark pair.
Quarks
Table : properties of the six known quarks.
Quarks
hadrons that have been discovered
containing c, t, or b quarks.
Quarks
Truly elementary particles (having no internal structure):
quarks, the gauge bosons, and the leptons.
Three “generations” ; each has the same pattern of electric charge,
but the masses increase from generation to generation.
Three generations – Three families
Only three ?
Have we missed
the fourth because
of high mass ?
Note: weak decay between families
Heavier families
are unstable
Z0 decays in
9 quark pairs
(no top quarks!)
9 lepton pairs
ƒ e+e−, μ+μ−, τ+τ−
ƒ neutrino pairs
Lifetime
1/τ = Γ with
Γ = Σ Γi
Sum over all decay channels
Cross section
Only thee families, it seems
energy (GeV)
Color
Soon after the quark theory was proposed, it was suggested that quarks have
another property, called color, or color charge.
Unlike other quantum numbers, color takes on three values. Real particles
must be colorless; this explains why only 3-quark and quark–antiquark
configurations are seen. Color also ensures that the exclusion principle is still
valid.
The need for an additional quantum number (satisfy Pauli principle)
Baryons and mesons do not have color (white)
Quantum Chromodynamics (QCD)
Quark Confinement
The color force becomes much larger as quarks separate; quarks are
therefore never seen as individual particles, as the energy needed to
separate them is less than the energy needed to create a new quark–
antiquark pair.
Conversely, when the quarks are very close together, the force is very
small.
U color
4 α s hc
=−
+ T0 r
3 r
T0 ≈ 0.9 GeV/fm
confinement
short
distance
large
distance
Quantum Chromodynamics (QCD)
The color force becomes much larger as quarks separate;
quarks are therefore never seen as individual particles, as the
energy needed to separate them is less than the energy
needed to create a new quark–antiquark pair.
Conversely, when the quarks are very close together, the
force is very small.
What about the mesons and the nuclear binding ?
Manifestation, residual effect of QCD gluon forces
The “Standard Model”:
Quantum Chromodynamics (QCD) and gluons
These Feynman diagrams show a quark–quark interaction mediated by
a gluon; a baryon–baryon interaction mediated by a meson; and the
baryon–baryon interaction as mediated on a quark level by gluons.
time
The “Standard Model”:
Electroweak Theory
Feynman diagram for beta decay using quarks.
The Electroweak Theory
Range of weak force.
The weak nuclear force is of very short range, meaning it acts
over only a very short distance. Estimate its range using the
masses of the W± and Z: m ≈ 80 or 90 GeV/c2 ≈ 102 GeV/c2.
Compare to Yukawa’s theory and analysis
Grand Unified Theories
A Grand Unified Theory (GUT) would unite the strong,
electromagnetic, and weak forces into one. There would be (rare)
transitions that would transform quarks into leptons and vice versa.
This unification would occur at extremely high energies; at lower
energies the forces would “freeze out” into the ones we are familiar
with.
This is called “symmetry breaking.”