- #1
binbagsss
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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 !
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 !
