Physics Notes: The Standard Model

Lecture 2: Jan 18, 2020 Back to PHY30

Quantum Chromodynamics

Basic Intuitive Picture. QCD is similar to QED, the theory of electrons and photons. QED is also the quantum mechanical theory of the Coulomb force. If applied to atoms where one does not worry about the motion or the nature of the nucleus (other than it’s charge), then QED is also the theory of atoms. In the same sense QCD is the theory of quarks and gluons and the things that they make. The things quarks and gluons make are called hadrons.

Hadrons include protons and neutrons which each contain three quarks. Hadrons also include mesons which are made of quark/anti-quark pairs. Hadrons also full of gluons. Gluons are electrically neutral stuff that binds the quarks together. [Most of the mass of hadrons comes from the binding of quarks by gluons.]

Reminder About the Mathematics of Spin

The mathematics of spin is really group theory. The group describing spin is really the group describing rotations of space. The theory of spin is also the theory of symmetries of physics with respect to rotations of space.

We are going to approach spin from the theory of angular momentum. We described this using commutation relations.

[or more generally

]

This was worked out in the previous quarter. We broke the symmetry in our mathematical description by focusing on the z component. We can’t measure non-commuting observables exactly at the same time due to the uncertainty principle.

We found that L_z was quantized and the allowed values differed by . Because spin values must be symmetric about 0, there are two possibilities. Either the spin values are integer multiples of , or they are offset by . Any other shift of the spin values would not be symmetric about 0.

The important point for the current discussion is that the quantization gives rise to spin multiplets. The simplest is the multiplet for the electron.

Bosons have integer spin. The first one has spin 0 – only one state. There are particles with spin 1, which have 3 states {-1, 0, +1}. The photon is a spin 1 particle [But 0 is not a spin state for the photon because it is massless]. There are other spin 1 particles.

The next interesting family has spin 3/2, which has 4 states. There are nuclei with spin 3/2. The next boson has spin 2.

When we speak of an object of spin l, then l denotes the highest value of spin along the z-axis.

The number of independent spin states is 2l+1.

If you have more than one particle, you can combine their spins into a composite spin.

What kind of angular momentum values can you make from two electrons (ignoring h-bar for now).

The maximum value occurs when both spins are both +1/2, totaling to 1. There is also a minimum that occurs when both spins are -1/2, totaling to -1. There are two ways for the spins to total to 0. So we have total spin states {-1, 0, +1}. How many states are there for a spin 1 particles? Ans: 3.

So our combinations of spin ½ particles yield 4 states.

We have an extra total spin 0 state when compared to a spin 1 particle. We need to re-express that last two states to make a spin 0 state and a spin 1 state. The key here is that we are talking about spin along the z-axis, not total spin. We can take sums and differences of the last two states above.

The symmetric combination with a + sign

Has total spin 1, but spin 0 along the z-axis. Along with the first two states, they form the three states of a spin 1 particle.

The anti-symmetric state

Has total spin 0 and obviously spin 0 along the z-axis. This state corresponds to a spin 0 particle with no angular momentum.

Now what happens if you put together 3 spin ½ particles.

How many states are there? Ans: 8, because each particle can be spin up or spin down.

How many value are there for the spin along the z-axis? Ans: 4= 2(3/2)+1 .

Two of the 8 states yield the +3/2 and -3/2 spin states. The other 6 states yield either spin ½ or spin -½ . There is no way to make 0.

In the lecture Professor Susskind says that there are two distinct ways to make spin ½, but I count 3. .

The real goal here is to talk about Isospin. Isospin is a property of particles that resembles spin. Isospin does not come from rotation in our normal 3 dimensions. Instead, think of it as spin in a separate imaginary space. It behaves mathematically just as spin does.

Isospin arises as part of the behavior of quarks.

The natural mass scale for hadrons is 100’s of MeVs, corresponding to the binding energy of quarks.

The lightest(stable) hadrons are made from up and down quarks, which are very light. To a good approximation we can pretend that up and down quarks have the same mass (basically 0 compared to the binding energy).

One can imagine a symmetry involving swapping all up quarks for down quarks. [We also have to ignore the charge difference.] We can map this on to the model of combinations of spin ½ particles.

Aside: Where does “Iso” come from? It comes from Isotope.

Isospin dates back to Heisenberg, clearly predating the concept of quarks. It is not a precise symmetry of nature because it ignores the mass differences and charge differences of the quarks. Electric forces in the nucleus only play a significant role when the nucleus is big, with many protons. The mass difference between the up and down quarks is insignificant.

One quark by itself can’t be observed (we will see why later). The simplest object built from quarks that we can observe is made up of 3 quarks – the neutron and the proton. This works just like 3 spins.

You can make a spin 3/2 combination and you can also make a spin ½ combination in 2 ways [I still count 3].

Let’s focus first on the spin ½ combination. It has two states, corresponding to the proton and the neutron.

Proton – duu Neutron- udd

Imagine that the proton is made of 3 quarks at specific locations

But we remember that quarks are fermions, so we have to antisymmetrize the up quarks.

The proton and the neutron are symmetric with each other and form an isospin doublet. These objects are also spin ½.

There is another object with isospin 3/2 and spin 3/2.

Uuu and ddd

These are a little more massive than the proton or the neutron. They are known as the delta D particles. There must be two more states given that these are spin 3/2 particles. These states have the same quark content as the proton and neutron, but have total isospin 3/2 and “z” component isospin of ½ .

All four of these objects have nearly the same mass, but they are different in mass from the neutron and proton.

m_np»940MeV

m_D=1200MeV

The delta can decay into a proton or neutron and a meson (quark/antiquark pair).

The isospin ¾ D particle can decay into a proton and a meson. The delta has a mass of 1200MeV, we subtract the mass of the proton leaving about 260MeV, which is larger than the mass of the meson (pion) which is about 140Mev, which leaves some kinetic energy left over for the pion to fly away. This is a p⁺ with charge +1 and the proton had charge +1, matching the charge of the delta. Delta particle have a very short lifetime, approximately 10^-26 seconds.

There is still something wrong. The uuu delta particles with spin 3/2 requires that we have all three quarks in the same state and isospin. This would seem to require that all the quarks are in the same state even though they are fermions. This was the first clue for color, which would distinguish the three up quarks.

Color values are red green and blue (red white and blue has gone away, but used to be used). Color does not affect charge or mass. We can now list the quarks as the product of the quark type u, d, c, s, t, b and the color R,G,B yielding 18 different states.

The three quarks could have differed by position or momentum, but this was ruled out. [It wasn’t clear why.]

Spin was discovered in helium because it had two electrons apparently in the same orbital state. They needed another quantum number. We have the same situation here.

Nambu is the one who recognized that the Delta situation required another degree of freedom to distinguish the three quarks.

So the spin 3/2 D particle is .

The “total” color of all observed particles is neutral. The D_3/4 particle was important because it created the maximum conflict.

Unless you have something to set a direction, then there is no difference between spin up and spin down. Isospin works the same way. Color does too.

A real proton would be a superposition of all different legal color assignments for the three quarks.

BTW - Chromo in Quantum Chromo Dynamics stands for color.

We have the analog of electrons – the quarks. Now we need the glue that holds them together – like a photon or the electrostatic field.

The analog of the photon is the gluon. They are also spin 1, and massless like photons. Like electrons, quarks emit and absorb gluons.

Gluons have a bit more character than photons, which have only polarization. Almost all the behavior of a photon can be shown in one diagram. Out of this one can build forces. You could flip these lines around to indicate positrons going back in time. This one diagram can be used to build all the important interactions. One way to think of the photon, for bookkeeping purposes, is as an electron positron pair. We would then redraw the diagram with the photon replaced by an electron positron pair.

There is no content in this other than to keep track of the bookkeeping for charge and other properties. This isn’t too useful for QED, but is helpful for QCD.

Remember that quarks can be red green or blue. We can pick a basis of red, green and blue for a quark and make a column vector out of it. A quark and an antiquark would have column vectors like:

A gluon in some respects behaves as a combination of a quark and an antiquark. If we label a quark by a color, then we would label a gluon by a pair of a color and an anti color, naively yielding 9 gluon labels. There is a subtlety which reduces this to 8, but we will come to that later.

The sum of the diagonals doesn’t represent an independent quantum state, but we will ignore that for now.

Using this notion, we can draw the quark gluon interaction where a red quark comes in and a green quark goes out as.

The gluon type is . This is the analog of the basic vertex of QED in QCD.

There is another building block that doesn’t exist for electrons and photons. Suppose a gluon is moving along. We will start with a gluon.

What we see here is that a gluon can interact with gluons. By using the gluon=quark/antiquark pair fiction, we can easily find all the possible diagrams.

Now we come to forces. The gluon behaves like the photons. We can draw a diagram for the interaction between a quark and an antiquark.

If we draw a few extra lines along the sides, we can turn this into a gluon-gluon interaction illustrating the exchange of a gluon between gluons. This is really new and represents forces between gluons. Quarks can bind together to make hadrons. This diagram allows up to bind gluons together without any quarks. Even a single gluon propagating along would emit and absorb its own gluons.

Gluons are massless in a special sense. Suppose we have a small object of mass m and we have something dangling off of it. If you move the small mass with a small force, then the effective mass is the sum of the mass of the small object and the dangling object. If you apply a large high frequency force, then the mass would just be m. So what is the mass? It seems to depend on the applied force. The mass of objects depends on the frequency of the force. In the same sense the mass of a quark is frequency dependent. In the context of a hadron, if you slowly moved a quark, the whole hadron would drag along. If you applied a very high frequency force, then you would see the normal value we ascribe to the quark mass.

The parameters we describe particles with depend on the frequencies and wavelengths at which we measure them. We call this phenomenon running constants.

Q: How did people figure out that quarks had fractional charges?

A: By the time I (Susskind) came into physics, quarks already existed. There were clues and inconsistencies. Confinement was one important fact. It took about 10 years to settle it.