Weak Interaction: e+e- to Mu+Mu- | Probability & Exceptions

[Soff]

$$e^+ e^- \rightarrow \mu^+ \mu^-$$

The reaction above can take place in two different ways:
1) The electromagnetic interaction by exchanging a $$\gamma$$
2) The weak interaction by exchanging a $$Z^0$$

Can somebody explain which way is more probable and why.
I'm not sure but I guess that the first way is more probable because the photon has zero mass and the $$Z^0$$ is a rather heavy particle. Is that always true or are there any exceptions?

 

[Meir Achuz]

In terms of the cm energy W,
the photon amplitude ~alpha/W^2.
The Z exchange ampllitude ~alpha/(W^2+\M_Z^2) .
This means the photon exchange dominates until W~M.
The Weak Interaction is called weak, because it was first discovered at low energies.

 

[Soff]

Thanks for your reply!
I think I have now understood the reason why the gamma-exchange is prefered. Only at high energy is the weak interaction almost equally probable as the gamma-exchange. Are weak interactions most likely observed in high energy reactions?

 

[AlphaNumeric]

Yes, when you get the collision energies which are comparable to that of the rest mass of the weak bosons (so >80GeV ish) then you have enough energy to start producing them with relative ease. Below that you don't get much effect.

The reaction you asked about is a very nice QED reaction to learn about because it's tree level process is only a single diagram and you can general your workings to both taus and light quarks. Colliding electrons and positrons allows for many quark bound states to be investigated and pretty good calculations of the mass and charge of new quarks.

It's done in great detail in Chapter 5 of Peskin & Schroder's 'An Introduction to QFT'. They go through the physical explanation, the QED calculation, non-relativistic limit and then how to generalise it to other products and the physical measurements which vindicate such notions.

If you are learning QFT and don't have a copy of that book, buy it. It's a brilliant textbook. £40 or so but well worth it.

 

[Meir Achuz]

Thanks for your reply!
I think I have now understood the reason why the gamma-exchange is prefered. Only at high energy is the weak interaction almost equally probable as the gamma-exchange. Are weak interactions most likely observed in high energy reactions?

Although the weak interaction is weak at low energy, that is where it is most often observed. All of beta decay is due to the weak interaction.