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

• Soff
In summary, the reaction e^+ e^- \rightarrow \mu^+ \mu^- can occur through two different interactions, the electromagnetic interaction by exchanging a \gamma or the weak interaction by exchanging a Z^0. The electromagnetic interaction is more probable than the weak interaction at low energies, but at high energies, the two interactions become equally probable. The weak interaction is called weak because it was first discovered at low energies. However, at high enough energies, weak interactions can also be observed. This reaction is useful for studying quark bound states and can be generalized to other products. It is also well-explained in Chapter 5 of Peskin & Schroder's 'An Introduction to QFT'.

#### 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?

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.

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?

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.

Soff said:
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.

## 1. What is weak interaction?

The weak interaction is one of the four fundamental forces of nature, along with gravity, electromagnetism, and strong nuclear force. It is responsible for the decay of subatomic particles and plays a crucial role in nuclear reactions.

## 2. What is the process of e+e- to Mu+Mu- in weak interaction?

In weak interaction, e+e- (electron-positron) pairs can annihilate to form Mu+Mu- (muon-antimuon) pairs. This process involves the exchange of a W boson, which carries the weak force.

## 3. What is the probability of e+e- to Mu+Mu- in weak interaction?

The probability of this process occurring depends on the energy of the electron and positron. At low energies, the probability is very small, but at high energies, it becomes more likely.

## 4. Are there any exceptions to e+e- to Mu+Mu- in weak interaction?

Yes, there are a few exceptions to this process. For example, if the energy of the electron and positron is not high enough, they may not have enough energy to produce a W boson and thus cannot form a Mu+Mu- pair.

## 5. How does weak interaction differ from other fundamental forces?

Weak interaction is unique in that it violates the conservation of parity, meaning that it does not behave the same way if the direction of time is reversed. It also has a much shorter range than the other fundamental forces, only acting on particles within a distance of about 10^-18 meters.