Physics Notes: The Standard Model

Lecture 4: Feb 1, 2010 Back to PHY30

Opened with question about Lagrangian for QED. I missed the beginning but the question seemed to be about modeling currents.

If we remember our Dirac matrices where the alphas correspond to space and beta to time, the beta matrix being I, then we get

and

The first part being charge density and the second part being “current” in different directions.

Note: these terms have U(1) symmetry – charge conservation

Suppose that we have a particle that has charge 2e – a “dielectron” with field .

Terms in our Lagrangian would have to look like

For charge conservation, but then would need to transform differently than . We will need

and more generally , where n is the charge.

For the photon, we have n=0, so we under our U(1) symmetry we have .

Now lets try an example with the Neutron.

Let N be the field for the Neutron, P for Proton, for Electron, and for the neutrino. Then we could have a term like:

which annihilates a Neutron, creates a Proton, creates an Electron and creates an anti-neutrino (or destroys a neutrino leaving a hole).

Commercial during class: Please buy Susskind’s book “The Black Hole Book”. In fact, I did order it and read it.

More Group Theory

We need to understand some group theory to apply rotations to states of particles. For discrete states like spin (u or d), or color (r, b or g), a rotation mixes up the amplitudes while maintaining the magnitude of the state vector.

For spin ½ we use 2x2 matrices to encode the rotations. For N distinct

states we would use NxN matrices.

Reminder: Unitary implies that , which specifies 4 equations of the U elements. We also have =1, which removes one more parameter and moves us from the Unitary group to the Special Unitary group.

Actually SU(2) is not really the rotation group. It has a 2->1 correspondence where both U and –U correspond to the same rotation. Not really important for the discussion.

Generators of a Group

For continuous groups represented as matrices, one can pick a set of elements infinitesimally close to from which all group members can be constructed by repeated application. We can describe this situation as a parameterized difference from I

Then we have

If we ignore the term which will scale away as , then

or , meaning that T is Hermitian

But Hermitian operators are our observables. You can think of these generators as vectors that represent small rotations. A rotation about an arbitrary axis can be formed as a linear combination of the generators.

We still have the rule, so Trace(T) = 0

These generators are angular momentum operators for SU(2). Conserved quantities are the generators.

On to SU(3)

3x3 matrices -> 18 total parameters

removes 9 of the parameters by specifying 9 independent equations, removes one more, leaving 8.

Our group members then should be , j=1..8, a linear combination of the generators.

Combining Systems

By this we mean making a system from known components, like two electrons. In this case we could have 3 states, spin -1, spin 0, and spin +1 based on the spins aligning down, misaligning, or aligning up. Once we have a spin 0 state, then the state is invariant with respect to a rotation. This is an important notion - that by picking the right basis states we can create subgroups, which are closed under transform.

Now consider quarks

or we could equivalently mix fields

It gets a bit fuzzy here, but will hopefully be clarified later.

The matrix representation for this transform is called the “3”.

For anti-quarks the representation is called the “” representation and it is the complex conjugate of the “3” representation. This is the same as the electron symmetry using for electrons and positrons.

The generators of “” are the negation of the generators for “3”.

A Two Quark System

Suppose we make a 2 quark system. This gives us 9 states (rr, rb, rg, br, bb, bg, gr, gb, gg) that have to be mixed together by our abstract rotation.

In combining the corresponding representations we will use the symbol to avoid confusion with multiplication

In our two quark system, the representations combine as

, where is the symmetric case and 6 is the anti-symmetric case. (It appears that these cases are sub-groups of the product groups, when applied to the right basis set of states, more later …)

The important case is the case. For a quark anti-quark pair, we have

The 1 case is called a singlet and is interpreted as dropping color. This can be seen as , which are transformed into each other, remaining colorless.

If we then combine another quark with our quark-quark system we will have

which takes us back to 1 or 8 again, (or to the cases). The 1 representation is again a singlet which is colorless and finite energy. The 8 sub-group will turn out to be a set of infinite energy states.

The gluon, which carries color charge, will need to transform like .

A postulate of QCD: All free particles of nature are colorless. They are the singlets.

There are only two such ways to combine quarks out of which other particles can be composed.

: Mesons (quark+anti-quark)

: Baryons (three quarks)

Can we make particles out of gluons? If we combine two, then our group product

The “1” is a singlet and is colorless. The name for this particle is a “glueball”.

Mesons, baryons, and the glueball are all color singlets.

The relationship between color and charge

A pair has 0 electrical charge.

If we go through all the cases for 3 quarks we have possible charge values of , so it turns out that colorless -> integer charge.

What’s Really Going On With Representations?

Note: The doesn’t really go to 1 or 63. It goes to 1, 8, 8, 10, 10, 27.

Suppose we have two particles with spin ½. Then our field vector could be written

If we make an appropriate basis change to

Then we can see that under rotation, the first element is not transformed into the others. It is a singlet. The other 3 elements are mixed. This is the “3” sub-group.

Homework: find representations of SU(3). Doesn’t work in all dimensions.

I’m not sure what the point of this next bit is – my notes are unclear:

where e is the e-Permutation symbol, or

For , we need 64 basis states and we would need a 64x64 element matrix to represent the transformation. However, with a suitable choice of basis states the matrix decomposes into a block diagonal form with blocks of size 1, 8, 8, 10, 10, 27. This means that the states that are linear combination of the basis states associated with a block are closed. Some of these blocks correspond to infinite energy.

Prof Susskind attributed Chromodynamics to someone whose name appeared to begin with N, but I didn’t catch it.

Confinement

Another Postulate: gluons play the role of photons in QED

The 8 gluons correspond to the 8 conserved quantities, which are the 8 generators of SU(3). They are conserved just like angular momentum.

Quarks are surrounded by gluon fields (of 8 types).

Analogy to electric fields where a particle pair + - has less energy when the particles are close together and + + has more energy. A net charge in a region -> more energy.

If the electric charge were larger, then it would become very difficult to have a net charge in a region and electrons would be confined in pairs with positrons.

If the color charge is large, then we won’t be able to pull particles apart.

Gluons are different from photons in that they carry color. They are “charged”. Unlike photons, they can interact.

The important new behavior is that unlike electric field lines that spread out all over the place, the gluon lines of flux attract each other, causing them to form bundles between charges.

The quarks may be pulled apart by simply hitting one of them hard. If the moving quark runs out of kinetic energy, then it is pulled back. Another possibility is that the “string” breaks with the formation of another quark/anti-quark pair. This process can continue, producing a series of particles.

QCD is called a gauge theory. In such theories, there are conserved quantities like electric charge or color based on underlying symmetries.

Side comment about lines of flux: If we postulate that lines of flux have to end on particles, then the speed of light limit says that particles can’t disappear because you would either have unterminated flux lines after the particle disappeared, or you could take advantage of the disappearance of the flux lines to signal faster than the speed of light.

Where are we?

We have a lot of the pieces.

We have electric charge, spin and color. We need more on SU(2) and the Higgs particle.