Introduction to the Standard Model of elementary particle physics

Transcription

1 Introduction to the Standard Model of elementary particle physics Anders Ryd May 31, 2011 Abstract This short compendium will try to explain our current understanding of the microscopic universe. This is known as the Standard Model of elementary particle physics. The Standard Model of elementary particles explains what the fundamental constituents of matter is, what the forces are and how they act on matter. 1 Introduction The Standard Model of particle physics is a bit of a misnomer; in fact it is a very well tested theory. Calculations based on the Standard Model has been tested in some cases to 1 part in 10 billion. So far all experimental data is consistent with the Standard Model. At the heart, the Standard Model explains what matter is, what the forces are, and where mass comes from. This is a rather impressive scope! In the next few pages we will dissect the different components of the Standard Model to understand what it tells us. First, we discuss what matter is. Then we move on to the forces, and last mass is discussed. As will be explained the source of mass is yet not experimentally verified and there are some hints here that the Standard Model does not contain the ultimate explanation of how the universe works. The Standard Model of particle physics was developed in the late 1960 s and early 1970 s. At this point many aspects of the model were theoretical predictions. Basically all of these predictions have now been experimentally verified. 2 Matter Over 2500 years ago the Greek speculated that all of the objects around us were made from a small number of indivisible particles, called atoms. They did not have the ability - or interest - to experimentally verify this. Skipping about 2300 year forward, in the eighteenth and nineteenth century science had advanced far enough that chemists had started to identify these elementary building blocks - the atoms. About 100 different atoms were found. By studying their chemical properties it was noted that they fell into a pattern and they were organized in the periodic table of the elements as shown in Fig. 1. In a famous experiment by Hans Geiger, Ernest Marsden, and Ernest Rutherford in 1909 it was shown that the atom consisted of a massive nucleus surrounded by electrons. A model of the atom was developed where the nucleus is built from protons and neutrons that are very 1

2 Figure 1: The periodic table of elements. closely bound. The protons have a positive electric charge, while the neutron is electrically neutral. The nucleus is surrounded by a cloud of electrons. The electrons are negatively charged and balance the charge from the protons to make an atom electrically neutral. This lead to a fairly simple picture; there were electrons, protons, and neutrons. All the known atoms were built up from these three basic building blocks. However, in the years after the second world war experiments revealed that other particle could be created in high energy collisions of particles. By the 1960 s well over 100 new particle had been observed. All of them were short lived and promptly decayed. However, we were again back at a picture that had 100 s of elementary particles. It was suggested that all these particles could be built from more fundamental constituents, known as quarks. It was postulated that there was an up quark with an electric charge equal to 2/3 of the proton and a down quark with an electric charge equal to 1/3 of the proton. The proton would then consist of two up quarks and one down quark while the neutron consists of one up quark and two down quarks. In this model one should also be able to build a particle that consists of three up quarks and has an electric charge that is twice that of the proton. In fact, such a particle were observed (it is known as the ++ ). A pictorial view of the structure of matter is shown in Fig. 2. 2

3 Figure 2: Matter as you zoom in more and more until you get to resolve the quarks in the protons and neutrons. Again we have ended up with a fairly simple picture of the mater; we have the up and down quarks, there is the electron, and also an electrically neutral partner to the electron called the neutrino. (Note that the difference in electrical charge between the down quark and the up quark is the same as the difference in electrical charge between the electron and neutrino. This will be relevant later.) This simple picture got slightly more complicated when it was discovered that each of these 4 particles (the up, down, electron, and neutrino) each had 2 heavier partners. These heavier partners basically have the same properties, except that their mass is larger. In Fig. 3 is a summary of these different particles. In the world around us that we can see an touch, all matter is made up from atoms that in turn are built from the up and down quarks plus electrons. However, if you look out in the universe the picture is a bit different. We will come back to this later. 2.1 Antimatter For each of the matter particles discussed in the previous section there exists a corresponding antiparticle. The antiparticle has the same mass as the original particle, but other properties 3

4 Figure 3: The simplified periodic table of quarks and leptons. like the electric charge is the opposite for the antiparticle. The antiparticle of the electron is called the positron. The positron has the same mass as the electron, but the opposite electric charge. When the same matter and antimatter particles get in contact with each other they can annihilate each other and turn into pure energy (photons). One of the mysterious still of the universe is why it seems to be made up mostly of matter 1 and there is not yet any evidence for antimatter out in the universe. 2.2 Problems 1. Calculate the number of atoms in one kilogram of iron. 2. Estimate the number of protons in the universe. Assume that average galaxy is made from 10 billion stars and that there are about 10 billion galaxies in the universe. 1 Of course what we call matter vs. antimatter is an arbitrary choice. It is just more convenient to call the stuff that we are made from matter as supposed to antimatter. However, if there were a planet somewhere in the universe that was made from antimatter the inhabitants would certainly call that antimatter. Is there a way to tell if they are made from antimatter before we make contact with them? 4

5 3. Verify the the electric charge of the proton and neutron is correct based on the charges of their quark constituents. Consider the sun to be an average star with a mass of about kg. 3 Forces We are familiar with many forces from our everyday life. The gravitational force keeps us on the ground, the friction force keeps us from sliding of roads when we drive, the contact force that prevents us from slipping into the ground, the electrostatic force between electrically charged objects, the magnetic force that makes the compass point to the north, etc. Similar to how the Standard Model explained how all matter is built out of a few basic constituents, the standard model also simplifies the description of forces. That is all forces except for gravity. Gravity is not part of the Standard Model. The electric force and magnetism appeared to be different forces, but through the work of James Clark Maxwell and others in the nineteenth century it was realized that the electric and magnetic force had the same origin: there was one electromagnetic force. With a common description it is said that the electric and magnetic forces were unified. Most other forces, such as friction, contact forces etc. are just manifestations of the electromagnetic force on small scales. The equations that govern the electromagnetic fields allowed for a solution that explained light. Light was electromagnetic waves that travel by the speed of light (in vacuum). Hence, besides unifying the electromagnetic force, Maxwell explained what light was. In classical physics the electric field is defined as the force a test charge feels. The electric field of an electrically charged particle fills all of space and generates an electrical potential that falls as 1/r, where r is the distance from the electric charge. In the Standard Model a slightly different picture is taken. However, mathematically you can show that the two descriptions give the same observable effects. In this picture the force comes from exchanging a photon, see Fig. 4. In this picture two electrically charged particles are traveling towards each other, one emits a photon and change direction. The photon is absorbed by the other particle within a very short time and it also gets a change in direction. To illustrate how this works, think of a two people standing in boats and tossing a bowling ball back and forth, see Fig. 5. When Alex throws the ball to Jenny, Alex will recoil against the ball and start moving backwards. When Jenny receives the ball she will absorb the momentum and start moving to the right. The net effect of this is equivalent to a repulsive force that pushes the boats away. In the Standard Model the electromagnetic force is described by the exchange of photons. We say that the photons couple to the electric charge. This is consistent with the classic picture, where the force is proportional to the electric charge. An electrically neutral object does not feel the electromagnetic force. 5

6 Figure 4: Two electrically charged particles exchange a photon. 3.1 Weak force In the Standard Model there are more forces than the electromagnetic force. The weak force is mediated by two types of particles, the electrically neutral Z 0 and the electrically charged W ±. The Z 0 is very similar to the photon but there is one very important difference. The Z 0 is not massless, like the photon. In fact, the Z 0 is very heavy on the scale of fundamental particles. Only the top quark is heavier. The mass of the Z 0 is about the same as 100 protons, or one Zirconium (Zr) atom. The fact that the Z 0 is massive has a profound effect. The electric potential around an electrically charged particle was 1/r. For a massive particle this changes to e mrc/ h /r Which means that for large distances the potential due to the weak force is exponentially suppressed. (If m = 0 as for the photon you recover the 1/r form.) But for small distances where mrc/ h << 1 the exponential is approximately one and there is no exponential suppression. The upshot of this is that the weak for is about as strong as the electromagnetic force at short distances, but for longer distances the weak force is much weaker. The other two particles, the W + and W, are also massive. They are just slightly lighter than the Z 0. Again, this has the effect that the range of the force is short due to the exponential suppression. However, the fact that they are charged has another important effect. Consider the case we had in Fig. 4. If we replaced the photon here with an W + or W it would not work - the electric charge would not be conserved. So the type of interactions the W ± generate are different. Instead of just giving the particles a kick, the particles involved change type. Consider the example in Fig. 6. In this example the initial d quark (with charge 1/3) emits a W and becomes a u quark (with charge +2/3). The W 6

7 Figure 5: Two people standing on a boat and throw an object between them. This will cause the two to drift apart. in turn splits into an electron and the anti-neutrino. This is an example of a beta decay. If the initial d quark was in an neutron, this would generate the decay n p + e ν e. 3.2 Strong force Besides the electromagnetic and weak forces, there is one more force in the Standard Model: the strong force. This is the force that is responsible for binding the quarks together to form the protons and neutrons. If you consider the proton as constituting two up quarks and one down quark it is easily seen that some new force is need to bind these particles together, the electric repulsion would otherwise make the quarks fly apart. There are several very peculiar properties of the strong force. We will discuss some of them here. Like the photon is the mediator of the electromagnetic force, the gluons are the mediators of the strong force. Similar to the exchange of a photon between two charged particle, a gluon can be exchanged between two quarks. This is illustrated in Fig. 7. However, the gluons do not couple to the electric charge, rather they couple a different type of charge called the color. In fact there are three such charges, called red, green, and blue. Like for the electric charge where we have positive and negative charges the three color charges also have corresponding anti-charges: anti-red, anti-green, and anti-blue. But even more surprising is the gluons. Note that the photon was electrically neutral, while the gluons carry one color charge (red, green, or blue) and one anti-charge (anti-red, anti-green, or anti-blue). So if we now look at the diagram again where two quarks exchange gluons and we add the color to the diagram, as shown in Fig. 8, we have to conserve color in each vertex of the diagram. Since the gluons carry color, gluons also couple to gluons. This is illustrated in Fig. 9. This leads to a much more complicated behavior of the strong interaction. In Fig. 10 are two electric charges and two color charges (quarks). Notice that the electric field lines just spread out over all of space, while the color field lines forms a long string, or flux tube. The reason that the field lines for the strong field bundle up is that there is a force between the 7

8 Figure 6: A neutron can decay to a proton and an electron anti-neutrino pair. This is know as beta-decay in nuclear physics. This illustration shows how one d quark is transformed to a u quark when a W is exchanged. gluons as they them self have a color charge. For the electric field the force falls as 1/r 2 with distance. While for the strong field the force is constant. Hence the work needed to separate two electric charges is finite, while for the two quarks you would need an infinite amount of energy. The property that you can not separate two quarks is known as confinement. The fact that a quark can never exist free means that it is hard to establish properties of the quarks, such as their masses. Hence quarks are always observed bound together by the strong force. There are two different types of particles that are made from quarks. One type is known as mesons. These consists of a quark and an anti-quark pair. The quark and anti-quark carry opposite color, e.g. red and anti-red. The other type of matter made from quarks is the baryons. Baryons are made from three quarks, like for example the protons and neutrons. The three quarks, e.g. uud in a proton, each carry a different color. If the color is indicated by r, g, b as subscripts you would for example have u r u g d b. We said earlier that free quarks could not exist, why can we have three quarks with three different colors? In some sense this is like when you mix red, green, and blue light you get white like on a TV screen, in the same sense the three quarks that form a proton has no net color charge. 3.3 Summary of forces To summarize this, the Standard Model includes three forces. The electromagnetic force, mediated by the photon, the weak force, mediated by the Z 0 and W ± and the strong force, mediated by the gluons. In the Standard Model it is shown that the electromagnetic and weak force is actually of the same origin. The strong force is mathematically very similar, but at this point not unified with the electroweak force. Gravity is not included in the Standard Model. 8

9 Figure 7: Two quarks exchange a gluon in the strong interaction. 3.4 Problems 1. As discussed above, the range of the weak force is suppressed by the exponential term e mrc/ h. By setting mrc/ h = 1 we can estimate the range of the weak force. Using the mass of the Z boson estimate the range and compare to the size of a typical atom. 4 Mass and the Higgs There is one slight issue. In order to mathematically write down the Standard Model one need to assume that all particles are massless. This obviously is not consistent with reality. But thanks to a beautiful mathematical trick call spontaneous symmetry breaking this can be repaired. Instead of just postulating that the particles in the Standard Model has mass one can add the Higgs to the existing particles and forces. The Higgs particle couples to the different particles with different strengths and this coupling gives the particles in the Standard Model mass. All particles, 6 quarks, 3 leptons, and 3 neutrinos, have been found in particle physics experiments. However the Higgs has so far eluded discovery after more than 40 years of searches since it was postulated in the late 60 s. The Standard Model makes many predictions about the Higgs, but it don t tell us the Higgs mass. It is likely that the Higgs particle has not been discovered as it is just to heavy to be produced in our experiments (so far). 9

10 Figure 8: In this figure the quarks and gluon has been labeled by the color to more carefully indicate how the strong force works. 5 Outlook The Standard Model has been incredibly successful. All experiments done so far has been consistent with the predictions of the Standard Model. However, there are a few hints that there are something beyond the Standard model. A few of these are mentioned below. 5.1 Dark Matter and Dark Energy When astronomers study galaxies they can look at the rotation curves of the stars. This looks at the velocity of the stars in galaxies as a function of the radius from the center of the galaxy. It has been observed since the 1930 s that the velocity is consistent with much more mass in the galaxy than what you see from the light emitted by stars. Using gravitational lensing where light is bent by the mass of galaxies and clusters of galaxies, astronomers have made made maps of the mass distributions and seen that there is much more mass than what you expect from the observed number of stars. These observations point to a large fraction of dark matter in the universe. The term dark is used to mean that it does not emit light. That is it does not couple to the photon - it is electrically neutral. Even more strange are the observations from the cosmic microwave background studies. In these studies small temperature variations from the time when the universe was only about 300,000 years old are used to probe the early universe. These observation tells us that the universe is full of a substance known as dark energy. The dark energy providing a negative pressure that is driving an expansion of the universe. 10

11 Figure 9: This diagram shows an example of how gluons can couple to each other and generate very complicated diagrams. All of these, and other, observations tells us that the universe is made from about 4% of regular matter that we have here on earth. Dark energy makes up about 23% and mostly the universe is made from dark energy, 73%. So the Standard Model is only useful to explain about 4% of the stuff in the universe. 5.2 The Higgs Mass Another problem with the Standard Model is Higgs itself. Though it does an remarkable job at giving mass to all the other particles in the Standard Model, its own mass suffers from a theoretical problem known as fine tuning. It turns out that in order to give the Higgs a Figure 10: Illustration of how the electric and color fields differs. The electric field spreads out all over space, while the color field tends to stick together (like glue) and form a bundle. 11

12 mass that makes the theory consistent one has to tune a parameter in the model to about 36 significant digits. This is what is meant by fine tuning. This seems exceedingly unlikely so theorists have thought of ways to solve this problem. The most popular idea is known as super symmetry. Super symmetry adds more particles and solves the problem with fine tuning. But then there should be many more particles, and so far none of them has been seen. Again it is likely that they are heavy and has not been seen yet in experiments. It is clear from all the experiments done so far that the Standard Model is an excellent model for the physics we have explored so far. This is similar to classical mechanics. It works very well for macroscopic objects in our everyday life. But if you start to look at objects that travel at speeds close to the speed of light you will need to use special relativity which provides a correct description for objects that travel at very high speeds. Similar if you look at very small objects you need to use quantum mechanics. As experiments in particle physics probe higher and higher energies it is expected that one day we will run into a crack in the Standard Model and it will give us an hint about the new physics beyond the Standard Model. 12