# Are unstable particles ever really external lines in QFT?

• A
• bjj
In summary, the Z boson can't be directly detected, so it is sometimes written as something -> Z + something, then Z-> ... but since unstable particles aren't really on their own eigenstates of the Hamiltonian, and they will eventually decay, is it more accurate to draw them always as an internal line?
bjj
TL;DR Summary
Since an unstable particle isn't an eigenstate of the Hamiltonian, and it will eventually decay, should it always be written as an internal line of a Feynman diagram?
Let's for example consider the Z boson. It can't directly be detected; so is it ever really correct to draw it as an external line on a Feynman diagram? I've seen processes involving it before be written as
something -> Z + something, then Z -> ...
but since unstable particles aren't really on their own eigenstates of the Hamiltonian, and they will eventually decay, is it more accurate to draw them always as an internal line?

A Feynman diagram describes individual, isolated, particles coming from far away, meeting at the diagram, and then receding so that they are far away again.

A short-lived particle cannot come from far away.

The factorization in the way you described, initial state --> Z and then Z--> final state, is called the narrow-width approximation, in which you replace the Breit-Wigner shaped propagator of the unstable internal particle by a delta function
$$\frac{1}{(q^2-m^2)^2+m^2\Gamma^2}\to \frac{\pi}{m\Gamma}\delta(q^2-m^2)$$

In this way the intermediate particle is set on-shell and the producton and the decay are completely factorized in the way mentioned above.
This makes calculations A LOT easier, but you are loosing spin correlations and finite-widht effects of order ##\mathcal{O}\bigl(\frac{\Gamma}{m}\bigr)##.

If this is OK for your calculation, the narrow-width approximation will usually be your method of choice.
If not, you have to do the full calculation, initial state --> final state, with your unstable particle as an internal line in your Feynman diagrams.

Thanks! This makes things easier for me to understand. I often see in papers them breaking a process up into scattering into W or Zs, then a W or Z decay, and I didn't understand why they would not just do a single diagram, but I suppose it's probably this approximation.

## 1. What is QFT?

QFT stands for Quantum Field Theory, which is a theoretical framework used to describe the behavior of particles at a subatomic level.

## 2. What are unstable particles?

Unstable particles are subatomic particles that have a short lifetime and quickly decay into other particles.

## 3. Can unstable particles be external lines in QFT?

Yes, unstable particles can be represented as external lines in QFT diagrams. However, they are typically represented as internal lines to account for their short lifetime.

## 4. Why are unstable particles represented as internal lines in QFT diagrams?

Unstable particles have a short lifetime and quickly decay into other particles, making them difficult to observe as external lines in experiments. Therefore, they are represented as internal lines to account for their contribution to the overall process.

## 5. How does QFT account for the effects of unstable particles?

QFT uses mathematical techniques, such as Feynman diagrams, to account for the effects of unstable particles. These diagrams include both internal and external lines to accurately represent the interactions between particles and their decays.

• High Energy, Nuclear, Particle Physics
Replies
4
Views
1K
• High Energy, Nuclear, Particle Physics
Replies
6
Views
2K
• High Energy, Nuclear, Particle Physics
Replies
4
Views
2K
• High Energy, Nuclear, Particle Physics
Replies
1
Views
1K
• Quantum Physics
Replies
6
Views
696
• High Energy, Nuclear, Particle Physics
Replies
11
Views
2K
• High Energy, Nuclear, Particle Physics
Replies
5
Views
2K
• Quantum Physics
Replies
134
Views
7K
• Beyond the Standard Models
Replies
2
Views
965
• Quantum Physics
Replies
31
Views
2K