What Are the Mean Lifetime and Branching Ratios for an F- Particle?

[genloz]

Homework Statement

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?

Homework Equations

None given, but my thoughts were that these were relevant:
\Gamma_{overall} = \frac{1}{\tau}
Branching Ratio= \frac{\Gamma_{partial}}{\Gamma_{overall}}

The Attempt at a Solution

Mean Lifetime:
\tau = ((5*10^{-6})+(50*10^{-3})+(20*10^{-6}))/3
\tau = 1.66*10^{-2}

\Gamma_{overall} = \frac{1}{\tau}
\Gamma_{overall} = 60.24

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

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?

 

[genloz]

Okay, I had a bit of a think and came to this conclusion...
\Gamma_{overall}=\frac{1}{\tau_{1}}+\frac{1}{\tau_{2}}+\frac{1}{\tau_{3}}

Therefore,

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

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

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


So,
\Gamma_{overall}=250020

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

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

Mode 3
\frac{\Gamma_{partial3}}{\Gamma_{overall}}=\frac{50000}{250020}
=0.19998

Does that sound about right?

 

[genloz]

I guess the width should be hbar/meanlife but I don't think that matters in this instance...