E+e- collison Resonance peaks cross-section partial widths

[binbagsss]

Question:

The Breit-Wigner cross-section for a resonance R is $\sigma_{i \to f} =12\pi\frac{\Gamma_{R\to i} \Gamma_{R\to f}}{(s-M^{2})^{2}+M^{2}_{R}\Gamma^{2}_{R total}}$ [1],

where $s$ is the com energy squared, $M_{R}$ is the mass of the resonance , $\Gamma_{R total}$ is the total width of the particle, and $\Gamma_{R\to i,f}$ is the partial width for the decay of the particle into the initial and final state particles respectively.

Assuming that $\Gamma_{R \to e+e-}=\Gamma_{R \to \mu+\mu-}=\Gamma_{R \to \tau+\tau-}$ use the plot (attached) of the cross-section near the peak to obtain $\Gamma_{R total}, \Gamma_{R \to \mu+\mu-}, \Gamma_{R \to hadrons}$ and the mass of the resonance?

(The cross-section for $e+e- \to \mu+\mu-$ has been multiplied by ten in the figure).

Solution:

I'm fine with attaining these figures:

$M_{R}=91.2 Gev$
$\sigma_{hadrons}=39.5 nb$
$\sigma_{\mu+\mu-}=2.0nb$

I understand that the formula reduces to $\sigma_{i \to f} =12\pi\frac{\Gamma_{R\to i} \Gamma_{R\to f}}{M^{2}_{R}\Gamma^{2}_{R total}}$ around the peak.

I don't understand:

1) Why $\Gamma_{total}=(92.5-90.0)=2.5GeV$

So i see this has been read of from the hadron peak, but I thought that this figure corresponds to $\Gamma_{R \to hadrons}$.

I think I'm still struggling with these definitions of initial widths, final width and total width. I know that the product has many decay routes, and $\Gamma_{total}=\Sigma\Gamma_{i}$ . I'm unsure of what is defined as an initial state and what is defined as an final state.

Maybe when I understand these better I'll understand why $2.5GeV$ is not $\Gamma_{R \to hadrons}$.

2) Why $\sigma_{\mu+\mu-}=\frac{12\pi\Gamma^{2}_{\mu+\mu-}}{M^{2}\Gamma_{total}^{2}}$.

As in I don't understand why $\Gamma_{R \to i}= \Gamma_{R \to f}$ here.

So when computing $\sigma_{hadrons}$, $\Gamma_{R \to i}= \Gamma_{\mu+\mu-}$ and $\Gamma_{R \to f}= \Gamma_{hadrons}$ Again I'm unsure.

Any one who can help shed some light on this, greatly appreciated !

 

[mfb]

There is no attachment which makes your questions hard to understand.

1) The width of the peak is always the total width because it comes from the denominator only (the numerator does not depend on s).

2) This is the peak value? See where the cross-section gets maximal, plug it into the formula.

$\Gamma_{R \to i}= \Gamma_{R \to f}$ for muons because it got defined above (the process has the same width for electrons and muons).

 

[binbagsss]

Apologies, attached here.
(can no longer edit).

 

[binbagsss]

bump.

 

[binbagsss]

[QUOTE="mfb, post: 5050415, member: 405866] (the process has the same width for electrons and muons).[/QUOTE]

I don't know how you know which of electrons or muons is the initial/final decay mode?
What's meant by these terms? I can't find a clear definition on the internet, thanks appreciate it.

 

[mfb]

I don't know how you know which of electrons or muons is the initial/final decay mode?

You have electron/positron collisions -> ee is your initial state.
You consider the decay to muons -> µµ is your final state.

Decay and production give the same factor, it does not matter which one is which.

 

[binbagsss]

You have electron/positron collisions -> ee is your initial state.
You consider the decay to muons -> µµ is your final state.

Decay and production give the same factor, it does not matter which one is which.

Ah thanks . so for $\sigma_{hadrons}, \Gamma_{i}=\Gamma_{e+e-}$ and $\Gamma_{f}=\Gamma_{hadrons}$, (For$\Gamma_{i}$ we've just used $\Gamma_{e+e-}=\Gamma_{\mu+ \mu-}$

I'm still confused with $\Gamma_{total}=\Sigma \Gamma_{i}$
Isn't the only initial state e+e- here?

The solution gives it as $\Gamma_{total}=\Gamma_{hadrons}+\Gamma_{u+u-}+\Gamma_{\tau+ \tau-}+\Gamma_{unseen decays}$

Thanks.

 

[mfb]

I'm still confused with $\Gamma_{total}=\Sigma \Gamma_{i}$
Isn't the only initial state e+e- here?

That should be the sum over all partial decay widths.

 

[binbagsss]

That should be the sum over all partial decay widths.

Ah thanks. Apologies it had $\Gamma_{total}=3\Gamma_{\mu+ \mu-}+\Gamma_{hadrons}+\Gamma_{unseen}$
So its took $\Gamma_{e+e-}$ into account - I don't understand why, this isn't a decay route?

I'm still unsure why you read $\Gamma_{total}$ from the hadron peak, why not the $\mu+ \mu-$ peak say?

 

[binbagsss]

So $\Gamma_{total}$ is the total width of the resonance peak,
But there are two resononances - one from the $\mu+\mu-$ and one from the $e+e-$ , how do you know which to consider?

 

[mfb]

e+ e- is a possible decay.

You can get the total width from all peaks. Shouldn't matter which one you take (you can even use multiple together). Hadrons are more frequent.
What do you mean with two resonances?