Elementary ParticlesFundamental forces in Nature

Ch 43

Finer Structure observed

As the momentum of a particle increases, its wavelength decreases, providing details of smaller and smaller structures:Cf: the Heisenberg microscope

1) Deep Inelastic Scattering (similar to Rutherford scattering); seeing smaller details

"for his pioneering studies of electron scattering in atomic nuclei and for his thereby achieved discoveries concerning the structure of the nucleons"

The Nobel Prize in Physics 1961

Robert Hofstadter

λ (20 GeV) ~ 10-16 m

2) With additional kinetic energy more massive particles can be produced: particle physics = high energy physics

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.

High-Energy Particles and Accelerators

The Nobel Prize in Physics 1939

"for the invention and development of the cyclotron and for results obtained with it"

Ernest Lawrence

The Nobel Prize in Chemistry 1951

"for the chemistry oftransuranium elements"

Inventor of the synchrotronEdwin McMillan

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

Cyclotron Frequency

qBm

mqBrrT ππ 2/

2speed

distance===

Stability: Lorentz force = centripetal force

Revolving time:

Frequency

rmvqvB

2=

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 high-energy 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

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.

Particle Exchange

Visualization of interactions using Feynman diagrams

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

Maximum distance travelled (Range)

Particle Exchange

2~mc

t hΔ

h≈ΔΔ Et

mctcx h~Δ≈Δ

ElectromagnetismGravitation

Infinite rangem = 0 Strong force

Weak forceFinite range

m ≠ 0

IntermezzoWave equations, quantum fields

Schrödinger equationfree partcile non-relativistictime-dependent

( ) ( )txt

itxxm

,,2 2

22Ψ

∂∂

=Ψ∂∂

− hh

xip

∂∂

−= hˆt

iE∂∂

= hˆ

( ) ( )txEtxm

p ,,2ˆ 2

Ψ=Ψ

Operators

Relativistic analog for the energy ( ) ( ) ( )txEtxcmtxpc ,ˆ,,ˆ 24222 Ψ=Ψ+Ψ

( ) ( ) ( )txt

txcmtxx

c ,,, 2

2242

2

222 Ψ

∂∂

−=Ψ+Ψ∂∂

− hhOr (use operators):

Klein-Gordon equation: valid for spinless massive particles

“Similar”relativistic wave equationfor particles with spin

( ) ( ) ( )txt

itxmctxx

cii

i ,,, 2 Ψ∂∂

=Ψ+Ψ∂∂

− ∑ hh βα for “spinor”wave functions

Dirac equation: valid for massive particles with spin

IntermezzoInteractions via virtual particles

Ψ=Ψ∂∂

−Ψ∇ 2

42

2

2

22 1

h

cmtc

Klein-Gordon equation (rewrite and 3-dimensional)

0=mMassless 012

2

22 =Ψ

∂∂

−Ψ∇tc

This is the classical wave equation for electromagnetism:Photons are the (virtual) partciles mediating the force

Static problem: 01 22

2 =⎟⎠⎞

⎜⎝⎛ Ψ

=Ψ∇drdr

drd

rSolution:

re

0

2

4πε−

=Ψ

πmm =Mass Solution:r

egrr '/

2−

−=Ψ with: cmr

π

h='

Concept of the Yukawa potentialπ-mesons mediate the nuclear force(“residual strong force”)

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.

Particle Exchange

Yukawa predicted a particle thatwould mediate the strong forcesin 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 Hideki Yukawa

The Nobel Prize in Physics 1949

"for his prediction of the existence of mesons on the basis oftheoretical work on nuclear forces"

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:

Particle Exchange

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.

(Note, mesons not the true carriers gluons)

four known forcesrelative strengths for two protons in a nucleus, and their field particles

Particle Exchange

Intermezzo

Relativistic quantum fields and antiparticles( ) ( ) ( )tx

ttxcmtx

xc ,,, 2

2242

2

222 Ψ

∂∂

−=Ψ+Ψ∂∂

− hhKlein-Gordon equation:

For every solution (E, p) ( ) ( )⎥⎦⎤

⎢⎣⎡ −⋅=Ψ tiExpiNtx p

rr

hexp,

There is also a solution: ( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ +⋅−=Ψ=Ψ tiExpiNtxtx p

rr

hexp,,~ **

Corresponding to negative energy and momentum -p 24222 mccmcpEE p −≤+−=−=

Note: Dirac equation more elegant: four solutions found : two with positive energy, two with negative energyFor each spin= ½ and spin = -½ The Nobel Prize in Physics 1933

"for the discovery of new productive forms of atomic theory"

Problem with Klein-Gordon: positive-definite probability not guaranteed

Negative probability Anti-particle

Intermezzo

The Dirac SeaQuestion; What are those negative energy states ?

Vacuum:All the negative energy states are filled

Pair creation

Pauli principleFermi-energy levelChoice of zero-level for energy

A positron is a hole in the electron sea

cf: semi-conductors

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.

Particles and Antiparticles

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 particlesBut with different quantum numbers

Isospin I = ½ , MI = + ½ for protonMI = - ½ for neutron

Importance of symmetry in particle physics

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 numberis 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

Particle Interactions and Conservation Laws

Conservation of energy, momentum, and angular momentum

Noether theorems:

Conservation laws ↔ Fundamental symmetries in nature

Emmy Noether

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 Interactions and Conservation Laws

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).

Particle Classification

Weak force

Strong force

Hadron decay Weak force

Particle Classification

Bosons

Bosons

Fermions

Fermions

BE-FDstatistics

Almost all of the particles that have been discovered are unstable.

Weak decay: lifetimes ~ 10-13 sElectromagnetic: ~ 10-16 s Strong decay: ~ 10-23 s.

Particle Stability and Resonances

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.

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)

Strange Particles? Charm? Toward a New Model

Particle classifications

Quantum numbers, symmetries, and methods of “Group theory”: SU(3), SU(2), etc.

Meson octet Baryon decuplet

Murray Gell-Mann

The Nobel Prize in Physics 1969"for his contributions and discoveries

concerning the classification of elementary particles and their interactions"

Prediction of the Ω- partcile;observation after two years

quark compositions for some baryons and mesons:

Quarks

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.

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.

Quarks

Only three ?Have we missedthe fourth becauseof high mass ?

Three generations – Three families

Note: weak decay between families

Heavier familiesare unstable

Cros

s se

ctio

n

energy (GeV)

Z0 decays inquark pairs (no top quarks!)lepton pairs

e+e−, μ+μ−, τ+τ−

neutrino pairs

Lifetime1/τ = Γ withΓ = Σ ΓiSum over all decay channels

Only thee families, it seems

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–antiquarkconfigurations are seen. Color also ensures that the exclusion principle is still valid.

Color

The need for an additional quantum number (satisfy Pauli principle)

Baryons and mesons do not have color (white)

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.

Quantum Chromodynamics (QCD)Quark Confinement

shortdistance

largedistance

rTr

cU s0color 3

4+−=

hα

T0 ≈ 0.9 GeV/fm

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.

Quantum Chromodynamics (QCD)

What about the mesons and the nuclear binding ?

Manifestation, residual effect of QCD gluon forces

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.

The “Standard Model”: Quantum Chromodynamics (QCD) and 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 themasses of the W± and Z: m ≈ 80 or 90 GeV/c2 ≈ 102 GeV/c2.

Compare to Yukawa’s theory and analysis

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.”

Grand Unified Theories