# Mean Lifetimes & Branching Raitos

1. Nov 14, 2007

### genloz

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 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:
$$\Gamma_{overall} = \frac{1}{\tau}$$
$$Branching Ratio= \frac{\Gamma_{partial}}{\Gamma_{overall}}$$

3. The attempt at a solution
$$\tau = ((5*10^{-6})+(50*10^{-3})+(20*10^{-6}))/3$$
$$\tau = 1.66*10^{-2}$$

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

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?

2. Nov 16, 2007

### 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?

3. Nov 17, 2007

### genloz

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