Breit Wigner Formula & Positron Annihilation

[paultsui]

Hey guys!

Breit Wigner Formula describes the cross section for interactions that proceed dominantly via a intermediate particle (O*) A+B → O* → C + D:

σ = \frac{2\Pi}{k^{2}}\frac{Γ_{i}Γ_{f}}{(E-E_{o})^{2} + (Γ/2)^{2}}

A short question: Does the formula apply to situations when the intermediate particle is actually virtual?

For example, in positron electron annihilation, they form a photon which might eventually decay into another two particles. Can we calculate the resonant cross section for this process with the Breit Wigner Formula as well? If it is possible, what should we put in for E_{0}, which is supposed to be the rest mass of the intermediate particle?

 

[Dickfore]

I think that the Breit-Wigner formula applies to reactions going through the s-channel (because E describes the CM energy, which is \sqrt{s}).

The tree level diagram for e^--e⁺ annihilation (e^{-<br /> }e^{+} \rightarrow \gamma \gamma) is the following:

As you can see, this is a t-, or a u- channel process, so the formula does not apply. Also the intermediate particle is a fermion, not a photon.

 

[Bill_K]

There's a recent thread on this exact same question. I argued that the Breit-Wigner formula does apply, just put E0 = Γ = 0.

Does the formula apply to situations when the intermediate particle is actually virtual?

The intermediate particle is always virtual, how could it not be??

 

[mfb]

For processes like e^+ e^- \to \gamma \to \mu^+ \mu^-, I think you can use your formula. E₀ for a photon is 0.

 

[paultsui]

Thank you for your replies guys!

Bill_K and mfb:

I agree with you guys that this formula should apply. However, if E_{0} = 0, this would suggests that the resonance would happen when E_{CM} = 0 for all positron electron annihilation reactions.

In particular, if you like at
http://pdg.lbl.gov/2011/hadronic-xsections/hadronicrpp_page6.pdf

The graphs shows that resonance (the red lines) happen at E_CM differ from 0. Those red lines actually correspond to the production energy for the corresponding mesons. In other words, if we are allowed to apply this formula to the problem, E_{0} should not zero. So the new question would be, how should we determine E_{0}?

 

[mfb]

The resonances are usually due to particles produced from the photon, with the Z as exception. The whole energy range is far away from 0 (and has to be, as you have at least 2 electron masses), therefore you cannot see this resonance as a peak.

 

[paultsui]

Thank you for your reply mfb!

And then can we use the Breit Wigner Formula to understand the non-zero energy resonances in http://pdg.lbl.gov/2011/hadronic-xse...crpp_page6.pdf?

 

[mfb]

Should be possible, if you use the parameters for the different particles there.

J/psi, psi(2s) and the Upsilon particles are quite long-living (small Gamma), which gives them narrow peaks. In contrast, particles like Z and omega are short-living, they have broad peaks.