# Branching fraction

1. Jan 9, 2015

### Safinaz

Hi all,

I found that the branching fraction $b \to s \gamma$ decay is given by
$B \to K \gamma$ ~ 10^-4 , but now I want to know if I calculate the decay width of $b \to s \gamma$, what it should equals ? In other words I don't understand what does a branching fraction mean ..

Bests.

2. Jan 9, 2015

### Einj

The branching fraction of a certain decay $A\to B+C$ is simply defined as:

$$\mathcal{BR}(A\to B+C)=\frac{\Gamma(A\to B+C)}{\Gamma^{tot}_A},$$

where $\Gamma_A^{tot}$ is the total width of the particle A.

3. Jan 9, 2015

### Safinaz

So the branching fraction is the same as the branching ratio.

But now what is the decay width of $b \to s \gamma$, if the b→sγ branching fraction has been calculated to be
$B \to K \gamma$ ~ 10^-4 ?

Last edited: Jan 9, 2015
4. Jan 10, 2015

### Einj

You simply have to multiply the branching fraction (or ratio) by the total width of the B meson.

5. Jan 10, 2015

### Safinaz

I found in PDG that the mean life time of $B_0$ is ~ 10^-12 s, so its total decay width $\Gamma = h / 2 \pi \tau$ ~ 10^-25 GeV.s. / 10^-12 s ~ 10^-13 GeV . Which means the partial decay width of s $\gamma$ will be ~ 10^-17 GeV .

6. Jan 10, 2015

### Einj

I think your calculation is wrong. The Plank constant is $\hbar \simeq 6.58\times 10^{-16} eV\cdot s$ and I would say:

$$\Gamma=\frac{\hbar}{\tau}\simeq6.6\times10^{-4} \;eV.$$

To be fair I don't know if this is a reasonable value for the total width but I think so. Anyways, in this case you obtain the partial width to be $\sim10^{-8}\;eV$

7. Jan 10, 2015

### Safinaz

Which means the partial width ~ $10^{-17} GeV$ ..

The problem is I calculate this width by FormCalc and LoopTools and I get it much larger !

8. Jan 10, 2015

### Staff: Mentor

What is "much larger"? Factor 10? 10 orders of magnitude?
Is the result given in GeV or eV?