Quarks, Gluons, and the Big Bang

Maurice Barnhill

 Next section Previous section End of section Index of notes Syllabus Announcements
 Last revised 1999/03/12

Class Notes V: Conservation Laws

Outline

1. Background
1. Definition of Conservation
2. Particles and Antiparticles
2. Conserved Quantities
1. Momentum - related to uniformity of physics in space
2. Energy
1. Related to uniformity of physics in time
2. Related by special relativity to rest masses
3. Set of examples that indicate that masses always decrease in decays
3. Charge
1. Found experimentally
2. Illustrative set of examples
4. Angular momentum
1. Further discussion to come under Quantum Mechanics
1. Integer and half-integer spins
2. Related to uniformity of physics under rotations
3. Complicated in table by existence of orbital angular momentum
4. Rule about number of 1/2 - integer spins
5. A few more quantities will be discussed in later sections of the notes
3. Particle categories
1. Leptons and Lepton number
1. Antiparticles have negative lepton numbers
2. Decays that display law
2. Baryons and Baryon number
1. Antiparticles have negative baryon numbers
2. Defined as number of protons in ultimate decay list
3. Examples
3. Everything else: Mesons
1. Mesons: matter
2. Mesons: force carriers

Background

"Conserved" is one of the many words that have somewhat different meanings in physics than they do in general English. In physics, a quantity that is conserved does not change its value with time, even if other variables in the situation are changing immensely. Usually constants are not called "conserved quantities" unless there is some kind of general rule that requires that they not change, but the important part of the definition is the lack of change itself. Conserved quantities can be very useful, since once you measure them you know their value forever. Often, the most important properties of a system or the important part of the motion of a system can be predicted from conservation laws alone. In such cases you can avoid making -- or even understanding how to make -- complicated calculations to determine all the details of the system's change with time.

Some quantities which are not strictly constant nevertheless change slowly or change only with a very low probability. If during a major change in a system, such as a decay of a particle, there is a general rule which requires some quantities to change rarely, the rule is often called a "selection rule." A selection rule ensures that a quantity is almost conserved. It may also ensure that a quantity is completely conserved, in which case it may be called a selection rule or a conservation law more or less at random.

While studying Special Relativity we saw that particles and antiparticles exist, and that particle/antiparticle pairs may be viewed as having the same mass and opposite charges. Many other quantities can also have opposite values in particles and antiparticles, and when such quantities are conserved they are especially useful.

In the rest of this section we will discuss conserved and nearly-conserved properties of elementary particles. In many case we will infer the conservation law by looking at the decay processes in the particle table. I have extracted a few particles to make a smaller table for exploring these relations. You should look at the full table and check whether the conservation laws are obeyed by the decay processes listed there. There are too many decays in the full table to check all of them, but it is a good idea to check a few of the decay processes to make sure that you understand the rules.

Conserved Quantities

Energy and Momentum

Elementary Particles, like all objects in physics, are subject to the conservation of energy and momentum. Momentum is the product of the mass of the particle and its velocity; the correct mass is the mass in the reference system being used, not the rest mass. Since momentum involves the velocity, and the velocity contains a statement of the direction of movement, momentum has a direction. The total momentum of an isolated system does not change with time, even though the parts of the system may exchange momenta with each other. Theoretically, the conservation of momentum is related to the fact that the laws of physics are the same in all locations of the Universe.

The sum of the rest-mass energy and the kinetic energy of a particle is given by E = sqrt(p2 c2 + m0 c4). The sum of the energies of all the parts of an isolated system does not change with time, even though the parts may exchange energy with each other. Theoretically, the conservation of energy is related to the fact that the laws of physics are the same at all times.

Energy and momentum conservation are vital to determining experimentally what reactions take place among elementary particles and how the various particles decay, so knowing that they exist is very important. However, neither is directly useful in describing the character or the types of particles, so we will not make much use of either in the rest of the notes.

Charge

The charge of a particle controls the size of an electric force exerted on the particle. If the size of the charge doubles, so does the size of an electric force exerted by a given outside charge. The charge of a proton, e, is measured to be 1.60 x 10 -19 Coulombs. All directly observable particles have a charge that is zero or an integer timese. Quarks and antiquarks have charges which may be positive or negative and have magnitudes of e / 3 or 2 e / 3. Hence charge is quantized, that is, it occurs in quantities that are integers or simple fractions times a base size of e. As long as magnetic monopoles are not found, there is no theoretical understanding of charge being quantized.

We can look at the small data table to see whether charge is conserved or not. In the line governing the mu-, the mu decays into an electron and two neutrinos. In compact form, mu- --> e- + antineutrino + neutrino. The muon has charge -1, and elsewhere in the table you can find that the charge of the electron is -1 and of each of the neutrinos is 0. Hence the sum of the charges on the left of the decay is -1 and the sum of the charges on the right is -1. The charge does not change during the decay. Continuing, the mu+ decay has total charge of +1 on both sides; the decay of the pi0 has total charge 0 on both sides, and so forth through all the decays. In each case the total charge is the same before and after decay, and we conclude that experimentally charge is conserved in all of these decays. The same is true of all other reactions ever observed in the laboratory.

Angular Momentum

"Angular momentum" or "spin" is a quantity which measures how quickly an object is rotating or revolving about a fixed line called an axis. Classically, it is given by the product  m r v  of the mass of the object, its distance from the axis, and its velocity perpendicular to the line between it and the axis. The conservation of angular momentum is related to the fact that the laws of physics don't depend on the orientation of your coordinate system.

When an object is rotating internally, its angular momentum is usually called "spin." When it is necessary to specifically state that an angular momentum is not due to internal rotation, it is called "orbital angular momentum." Since in particle physics we deal mostly with spins, I will normally use the shorter word "spin" to refer to angular momentum.

Quantum Mechanics puts some complicated restrictions on the behavior of spins, which we will talk about in the Quantum Mechanics section of the notes. For the moment, we need to know only two things: The first is that spin always comes as an integer or half an integer times a fixed number called h-bar, Planck's constant over 2 pi. The second fact can be seen by looking at the rules for combining two or more spins. For the lower values of spin, the combinations are

 0 + n = n 1/2 + 1/2 = 0 or 1 1/2 + 1/2 + 1/2 = 1/2 (two different ways) or 3/2 1/2 + 1 = 1/2 or 3/2 1 + 1 = 0 or 1 or 2 1/2 + 2 = 3/2 or 5/2 1 + 2 = 1 or 2 or 3

Notice that more than one spin can result from the addition of two spins. We will see how in the Quantum Mechanics section. The addition rules are complicated, but a useful generalization can be made, and it is our second useful fact about spins. If we add an odd number of half-integer spins, the result is a half-integer spin. If we addan even number of half-integer spins, the result is an integer spin. The number of integer spins added together does not affect this rule, only the number of half-integer spins.

Since spin is conserved during a particle decay, the spins of the decay products must add up to be the spin of the decaying particle. Almost. To complicate matters the decay products may leave the point of decay with orbital angular momenta as well as spins, and it is the total which must be conserved. However, all orbital angular momenta are an integer times h-bar and cannot change the total angular momentum from half-integer to integer or vice versa. Therefore, in the decay of an integer-spin particle, there are zero or another even number of half-integer-spin decay products, and in the decay of a half-integer-spin particle, there are an odd number of half-integer-spin products. So the number of half-integer-spin particles in a decay is conserved, if pairs of half-integer-spin particles are ignored.

Other Conserved Quantities

There are several more conserved quantities. Lepton number and baryon number will be discussed next in conjunction with the classification of particles into leptons, baryons, and mesons. Some other quantities, such as strangeness, charm, top, and bottom, can't be understood completely until we have surmised the existence of quarks. Quantities called hypercharge and isospin will come up as we exhibit the groupings of particles that lead to the surmise of the existence of quarks. All but isospin act basically the same way that charge does; isospin mimics angular momentum.

Particle categories

Leptons and Lepton Number

Electrons get and always have gotten a lot of attention from particle physicists, since they are one of the three particles evident in atomic processes alone. Electrons do not decay, so physicists early on assumed that the total number of electrons in the universe is fixed. However, that rule is a bit too simple. Electrons have antiparticles called positrons, and you can produce positrons by the reaction   photon -> e+ + e -. Also, if an electron and positron get near each other, they can undergo a reaction e+ + e - -> 2 or 3 photons. These reactions do change the number of electrons in the universe. However, antiparticles often have quantum numbers that are opposite those of the corresponding particle, so it is appealing to count a positron as -1 electron. Then the first process has 0 electrons before the reaction and 1 - 1 = 0 electrons afterward, and the second has 1 - 1 before and 0 afterwards. Now both reactions conserve the net number of electrons, and electron conservation has been rescued.

If we say that an electron has a quantum number L = 1 and the positron has L = -1, then we can encapsulate our understanding so far by saying that L is conserved. That is just a numerical way of saying that the net number of electrons is conserved.

If the conservation of L is to be meaningful, it must be obeyed everywhere. We look in the full or the small data table and find the net result of the decay of a negative pion. Actually in some cases this decay happens directly, pi - -> e - + anti-electron-neutrino. The total L is zero before and apparently +1 after the decay. To rescue the conservation, we must assign L = -1 to the anti-neutrino, and so presumably L = +1 to the electron neutrino. [You might prefer to assign L = 1 to the pion, but then Sigma- decay would not conserve L. Check the table.] The decay pi+ -> e+ + neutrino now also conserves L, and a lot more reactions are covered.

However, we are not done yet. Look at mu - and its antiparticle mu+ . The mu - decays according to mu - -> e - + anti-electron-neutrino + muon-neutrino. L is conserved, but if we treat the mu-neutrino as being a kind of muon, so is muon number. Do we have another class of conserved particles? Or do we, perhaps, have one class, assigning mu and its neutrino L = 1 and their antiparticles L = -1?

"One class or two?" is a question that requires data in order to be sure of the answer. The data are not yet definite, unfortunately. There is now (as of late 1998) one experiment that indicates that a muon neutrino might become some other type of neutrino, conserving L but not muon number. In that case there is only one class, and we need a generic name for electrons, muons, and neutrinos. Theorists have tended to believe that there was only one class almost against the evidence, on grounds of simplicity. There is even a name, "lepton," for the class electron, muon, neutrino. The antiparticles are then called antileptons. Leptons have "lepton number" L = 1, and antileptons have L = -1.

Are there any more leptons? Find out yourself. Go through the full particle table for other particles that act like electron/electron-neutrino and muon/muon-neutrino. You have found one if you can assign L = 1 to the particles and L = -1 to the antiparticles in such a way that all the decays in the table conserve L.

Baryons and Baryon Number

Are there any more such groups? Yes, there is a group called baryons, and a corresponding conserved quantum number called B. The lightest baryon is the proton, and it is the only stable baryon. Since the neutron decays by n -> p + e - + anti-nue and the electron and neutrino are leptons, not baryons, B conservation requires that the neutron be a baryon also.

It is fairly easy to spot a baryon in the table. Suppose you are looking at a particle which might be a baryon. If it is not the proton and is a baryon, it must decay. Baryon conservation then requires a baryon among the decay products, although you may not know which of the decay products is the baryon. Let all of the decay products themselves decay (if unstable). The baryon's decay (if there is one) yields another baryon. Keep going until all the particles are stable. Among all the resulting particles there must be, net, one baryon. Since the proton is the only stable baryon, that baryon must be a proton. Hence, a particle is a baryon if and only if there is one net proton among its ultimate decay products.

Mesons

Everything else in the table besides the baryons and leptons is called a meson. Mesons have L = 0 and B = 0, and they have no net leptons or baryons in their ultimate decay products. The number of mesons is not conserved, so there is no "meson number." There are two basic classes of mesons, but we will save the details of that situation for later.

 Next section Previous section Beginning of section Index of notes Syllabus Announcements