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|Last revised 1999/03/12|
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.
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.
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.
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|
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.
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.
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.
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