Last week we introduced our toy particle theory.
There were three types of particle: A, B and C. Each had a quantum field that consisted of a pair of Creation and Annihilation Operators (CAOs).
A = A+ + A-
B = B+ + B-
C = C+ + C-
We allowed a single interaction potential, gABC, where g is the coupling constant that defines the strength of the interaction and has to be determined experimentally.
Then we showed that we could expand the ABC product of fields into eight terms consisting of three CAOs. But when we acted on a state with a single A particle, |a>, only two of those terms survived, because the B- and C- terms annihilated the |a> state to zero.
ABC |a>
= A+B+C+ |a>
+ A-B+C+ |a>
= |aabc> + |bc>
We’re now going to use this result to calculate the probability of an A particle decaying into a B and C particle.
We start off with an “A” particle, wait a while, then find that it decays into a B and C pair. In the Standard Model, the allowed decay modes are determined by conservation of mass, charge, spin etc. Here, we’ll just assume that this particular decay is possible.
Call the initial state with the “A” particle |a>. Call the final state with the B and C particles |bc>. The decay from |a> into |bc> is due to the interaction of the A, B and C fields. i.e. It is due to the gABC interaction potential. I’ll look at why this is in a bit more detail in the comments.
So we start with the initial state
|a>
and we act on it with the interaction potential gABC to give the result
gABC|a>
Remember, |a> is just a probability distribution (PD). gABC is an operator that changes |a> into another PD. We want to find the probability that this resulting PD is the same as the |bc> PD. So we take the overlap:
< bc | gABC | a >
This is what we want to calculate. Once we have this probability, that an “A” particle changes into a B and C particle, then it is a standard calculation to turn this into the lifetime for an “A” particle. (Actually we want the square of this overlap, but as usual I’m ignoring that.)
We already know what gABC |a> gives:
gABC |a> =
g |aabc> + g |bc>
We now take the overlap.
< bc | gABC | a > =
< bc | g |aabc> + < bc | g |bc>
g is just a number, we can move it around as we please.
< bc | gABC | a > =
g < bc | aabc> + g < bc | bc>
The first term gives zero:
< bc | aabc> = 0 (remenber, only identical states have an overlap)
So only the final term survives.
< bc | gABC | a > =
g < bc | bc>
But the overlap of any state with itself is always the number one. The probability of any state, given that we know it’s in that state, is one, < bc | bc> = 1. So we’re left with the result g.
< bc | gABC | a > = g
If you have the maths, and you feel up to it, you can find a more complete version of this calculation here: http://www.damtp.cam.ac.uk/user/tong/qft/three.pdf, section 3.2.1.
Now it may seem as if we haven’t really achieved very much. We put in a single number “g”, and we got exactly the same single number back out again. We constructed a whole bunch of permutations of fields, only one of which turned out to do anything, and the thing that it did was to return the number one.
But we’ve achieved more than first seems apparent.
First, if we already know the value of g, as we do when g is the charge on the electron, then we can calculate our decay probability and particle lifetime directly. Alternatively, we can compare the lifetime that we get based on g, with the experimental lifetime of |a> -> |bc> to tell us the value of g. Now that we have the number “g”, we can use it in further calculations, such as the scattering probabilities that we’ll do next week.
Second, we’ve determined that particle decays are governed by the strength of the interaction. The stronger the interaction, the higher the probability that the particle will decay and so the shorter the lifetime of the particle. This is confirmed experimentally. The strong force typically has decay times ~10^-23s. Whereas the electromagnetic interaction, which is about 1% of the strength of the strong interaction, typically has decay times ~10^-16s. The weak interaction, the weakest of the three, typically has decay times ~10^-13s.
Third, we don’t need to know what an “A” particle actually is in order to calculate it’s decay time. The only information we’ve fed into this calculation is the interaction potential. This is true for real world particles too.
For example, muon decay is a two stage process. First the muon decays into a muon neutrino and a W- particle. Then the W- decays into an electron and an electron anti-neutrino. Both the theoretical and the measured lifetimes are 2.2 10^-6 seconds. Yet, as we discussed in an earlier post, QFT has no idea what any of these particles actually are. However it does know all their measurable properties, or at least, the properties of their probability distributions, and that’s good enough to do the calculation.
Once again I feel I have to point out how remarkable this is. We’re treating particles as “black boxes”. We know nothing about what is inside. All we know are the externally visible properties. Yet this is still sufficient to predict it’s lifetime to decay into some completely different particles.
Next week we’ll look at particle scattering, followed by Feynman diagrams the week after. And that will probably be the end of the series.
Platitude of the Day 