Breit Wigner for photon intermediates

[bayners123]

Hey!

I'm hoping someone can help me understand a basic problem I'm having with understanding the BW formula:

$$\sigma(i,j) = \frac{\pi}{k^2} \frac{\Gamma_i \Gamma_j}{(E - E_0)^2 + \Gamma}$$

In this, $E_0$ is the "characteristic rest mass energy of the resonance." I thought this meant the rest mass of the intermediate particle for the resonance, but in the case of a photon how does this work?

Then I tried to consider the products, but surely the energy of the final system, and therefore it's rest mass, depends on the energy of the incident particles?

The specific reaction I'm trying to understand is $e^- + e^+ \rightarrow \gamma \rightarrow \mu^- + \mu^+$


Thanks for any help!

 

[Bill_K]

What's wrong with the obvious, E₀ = 0?

 

[Vanadium 50]

The Breit-Wigner is not appropriate for that reaction. It would be appropriate if you were going through a massive resonance, like a phi.

 

[bayners123]

The Breit-Wigner is not appropriate for that reaction. It would be appropriate if you were going through a massive resonance, like a phi.

Gotcha. Turns out that the resonances I was looking at are due to $b\bar{b}^\star = \Upsilon^\star$ intermediates that decay to $\Upsilon$s. For anyone else who stumbles on this, these are the resonances at ~10GeV.

 

[Bill_K]

Well you may be seeing another resonance, but I still think the Breit-Wigner form (or a limiting case of it) is entirely appropriate for a reaction like this. I'm Looking at Halzen and Martin, who analyze e⁺ e^- → μ⁺ μ^-, specifically for the interference effect caused by neutral currents. That is, the intermediate particle can either be γ or Z. They find the invariant matrix elements (ignoring nonessential factors)

M_γ = e²/s
M_Z = g²/(s - M_Z² + iM_ZΓ_Z)

where s is the Mandelstam s. The latter is clearly Breit-Wigner, and the former a limiting case of it.