Physics Notes: The Standard Model

Lecture 5: Feb 8, 2010 Back to PHY30

Before Class Questions

What holds neutrons apart?

They are fermions and can’t be in the same states

Deep inside a neutron star it is an open question if neutrons are distinct, or perhaps the quarks are moving freely.

White dwarves are held against gravity by the electric charge.

White Dwarf -> Neutron Star is a supernova.

When fermionic pressure gets too big in a neutron star the result is a black hole.

What is the connection between neutrino mass and oscillation between types?

If neutrino had no mass, then they would move at the speed of light and their local clock would be stopped. There would be no way to oscillate. Having mass then enables oscillation between the neutrino types. A follow up question was about the determination of the mass of the neutrino. The answer was that in principle one could do it by carefully measuring the velocities of the products of neutron decay and then deduce the missing momentum and energy. Then you could solve for the mass of the neutrino. The measurements, while possible, are difficult.

Why is Trace(G) = 0 required for generators of a group representation?

This is a review question covered in Class 4. Prof Susskind went on illustrate symmetry with spin. Take which is a Trace=0 matrix.

We expect that the values of spin should be symmetric. Trace=0 matrices have the property that their Eigen are symmetric [I think this may not be completely correct. We know that the sum of the Eigen values equals the trace, which doesn’t require symmetric Eigen values. For 2x2 matrix it does], which matches our expectation that for every spin up state their should be a corresponding spin down state. This also re-makes the point that the generators are Hermitian operators corresponding to measurables, the Eigen values of the generators being the possible measurement values.

Gauge Theory

Example – Maxwell’s Equations

There are 3 components of the electric field + 3 components of the magnetic field, or we could use the 4 vector potential . is the electrostatic potential.

This produces waves. The waves have a direction and a polarization

Sources and sinks for these are charges and currents.

Field lines end on charges.

, known as Gauss’s law. The flux through a closed surface is equal to the charge inside.

The Essence of a Gauge Field

If the gauge field is weakly coupled, then it will be like the electromagnetic field.

For the moment pretend that Baryon # is a “charge”. Then this would generate forces like electric charge. There would be the equivalent of photons.

Of course there is no such field.

Quark color is a conserved charge.

Gauge fields always have conserved quantities associated with symmetries.

In QCD we will call the field known as a gluon field.

The field transforms under SU(3).

is a symmetry operation [I assume the meaning here is that if we rotate the colors, that the physics remains the same]

The representation used for particles is the “3” representation and for anti-particles we use the “” representation.

The number of generators is 8.

What is ? It has the same symmetry structure as , an anti-quark quark pair. There would be 9 generators except that we can remove because it doesn’t change under our symmetry.

Gluons do carry color charge, unlike the photon because contains which would represent anti-red + blue.

Gluons can emit gluons too!

The result of this interaction is that instead of having spread out field lines between quarks, they form bundles known as flux tubes.

Gauge Theory Summary

Symmetries <-> Charge conservation

You have a coupling constant which is the amplitude for the emission or absorption of a field boson.

The electric charge “e” of the electron is the coupling constant for emission or absorption of a photon. is then the probability. Prof Susskind added the phrase, “when you stop the electron” and gave the example of the electron hitting a screen having probability of emitting a flash. [I find myself a bit confused about coupling constants and the events or time rate at which it is applied. An accelerated charge will emit photons. Maybe that’s what’s going on.][Followup – I asked the question in the next class. The idea is that the electron is continuously emitting and absorbing virtual photons. There is about 1/137 chance of there being one. If you hit/accelerate the electron you interfere with the re-absorption and may give it enough energy for the photon to not be re-absorbed]

The fine structure constant and is a pretty weak coupling.

The name “Strong” for “Strong Interactions” pre-dated QCD. Why were they considered strong?

First – get units straight.

so

Characteristics of things governed by .

Transit time across atomic diameter ~

Electron orbit time for hydrogen atom (apparently velocity of electron in first orbital of the Bohr model is !) is .

What is the decay time for the excited hydrogen atom?

All related by the fine structure constant.

What about Hadrons?

Hadron diameter is on the order of

Transit Time (for light):

Orbital time:

Decay of excited state:

Why aren’t these spread out like they were for the electron process? The reason is that the equivalent of the fine structure constant is much closer to 1.

[The rates appear to be more limited by the speed of light constraints than by the coupling constant]

The rate of these processes is whey they are called “Strong”.

The Weak Interaction

Events governed by the Weak interaction are slow – even slower than atomic events.

A good example is neutron decay:

The half-life is about 12 minutes (Wikipedia says 15). One reason is that there is very little excess energy. However, if you take a process with favorable energies

Then the decay time is about 10ns, which is still very slow.

Chart of quarks and leptons used to illustrate the Weak interaction:

Now Build a Gauge Theory of the Weak Interaction

The gauge bosons look like (transform like) anti-quark/quark pairs.

appear to express the difference between the pairs of particles in a cell, where you combine the anti-particle of one with the particle of the other.

It is simply an empirical fact that operate on both quark pairs and lepton pairs.

We can draw some Feynman diagrams to show the possibilities:

Similar diagrams are valid for quarks too:

Take an example decomposition :

Note: the most probable outcome is actually an anti-muon-neutrino/muon pair, but the electron pair can happen. Not clear why the muon is favored.

If the energy is insufficient to create a , then the process can still happen, but the lifetime of the has to be short. [My take: If the lifetime is short enough then the wavelength and frequency of the is not well defined which implies that the energy/mass of the is not well defined either. The lower the available energy, the shorter the lifetime, and I would guess, the lower the probability. Much like tunneling]

Example: Muon decay

And a last example of Neutron decay:

Thefor the Weak interaction is similar in magnitude to the fine structure constant. This leaves open the question of why the time scales are slower. [My guess is that the large mass of the slows things down]