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Mean Lifetimes & Branching Raitos

  1. Nov 14, 2007 #1
    1. The problem statement, all variables and given/known data
    1. In a theory a charged particle called F- (charge=-1) exists. This particle has three decay modes that will be observed at the LHC. The mean time between F- particle creation and each kind of decay is found to be:
    Mode Mean lifetime
    Mode 1 5 microsec
    Mode 2 50 millisecond
    Mode 3 20 microsec
    What is the mean lifetime of the F particle, and what are the branching ratios for each of the three modes?

    2. Relevant equations
    None given, but my thoughts were that these were relevant:
    [tex]\Gamma_{overall} = \frac{1}{\tau}[/tex]
    [tex]Branching Ratio= \frac{\Gamma_{partial}}{\Gamma_{overall}}[/tex]

    3. The attempt at a solution
    Mean Lifetime:
    [tex]\tau = ((5*10^{-6})+(50*10^{-3})+(20*10^{-6}))/3[/tex]
    [tex]\tau = 1.66*10^{-2}[/tex]

    [tex]\Gamma_{overall} = \frac{1}{\tau}[/tex]
    [tex]\Gamma_{overall} = 60.24[/tex]

    Branching Ratio for Mode 1:
    [tex]\Gamma_{partial} = \frac{1}{\tau}[/tex]
    [tex]\Gamma_{partial} = \frac{1}{(5*10^{-6})}[/tex]
    [tex]\Gamma_{partial} = 200000[/tex]
    [tex]Branching Ratio= \frac{\Gamma_{partial}}{\Gamma_{overall}}[/tex]
    [tex]Branching Ratio= \frac{200000}{60.24} = 3320[/tex]

    But then that didn't look much like a ratio so I started to wonder if I'd made a mistake or units were incorrect or something?
     
  2. jcsd
  3. Nov 16, 2007 #2
    Okay, I had a bit of a think and came to this conclusion...
    [tex]\Gamma_{overall}=\frac{1}{\tau_{1}}+\frac{1}{\tau_{2}}+\frac{1}{\tau_{3}}[/tex]

    Therefore,

    Mode 1
    [tex]\Gamma_{partial1}=\frac{1}{5*10^{-6}}[/tex]
    [tex]\Gamma_{partial1}=200000[/tex]

    Mode 2
    [tex]\Gamma_{partial2}=\frac{1}{50*10^{-3}}[/tex]
    [tex]\Gamma_{partial2}=20[/tex]

    Mode 3
    [tex]\Gamma_{partial3}=\frac{1}{20*10^{-6}}[/tex]
    [tex]\Gamma_{partial3}=50000[/tex]


    So,
    [tex]\Gamma_{overall}=250020[/tex]

    So the branching ratios are:
    Mode 1
    [tex]\frac{\Gamma_{partial1}}{\Gamma_{overall}}=\frac{200000}{250020}[/tex]
    [tex]=0.7999[/tex]

    Mode 2
    [tex]\frac{\Gamma_{partial2}}{\Gamma_{overall}}=\frac{20}{250020}[/tex]
    [tex]=7.999*10^{-5}[/tex]

    Mode 3
    [tex]\frac{\Gamma_{partial3}}{\Gamma_{overall}}=\frac{50000}{250020}[/tex]
    [tex]=0.19998[/tex]

    Does that sound about right?
     
  4. Nov 17, 2007 #3
    I guess the width should be hbar/meanlife but I don't think that matters in this instance...
     
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