Meson Decay: Solving for the Ratio of Decay Rates - What is the Input Parameter?

[sunrah]

Homework Statement

$\frac{\Gamma\left( D^{*+} \rightarrow D^{0}\pi^{+}\right)}{\Gamma\left( D^{*+} \rightarrow D^{+}\pi^{0}\right)} = ?$

Homework Equations

$D^{*+} = |c\bar{d}\rangle$
$D^{*0} = |c\bar{u}\rangle$

see also Clebsch-Gordon coefficients

The Attempt at a Solution

I know that these are meson decay reactions and those look like gamma functions, but what exactly is the input parameter here? What number does the decay process deliver?

anyway

$D^{*+} \rightarrow D^{0}\pi^{+} = |c\bar{d}\rangle \rightarrow |c\bar{u}\rangle|u\bar{d}\rangle$
$= |\frac{1}{2}, \frac{1}{2}\rangle \rightarrow |\frac{1}{2}, \frac{-1}{2}\rangle|1,1\rangle$, where the numbers are isospin and its z-axis projection, e.g. $|I,I_{3}\rangle$.

now clebstch-gordon:

$|1,1\rangle |\frac{1}{2}, \frac{-1}{2}\rangle = \sum_{J =1/2} ^{3/2}|J,1/2\rangle C^{J,1/2} = \sqrt{\frac{2}{3}}\left( |1/2,1/2\rangle + |3/2,1/2\rangle\right)$

this is where i get stuck. are the CB-factors alone the input parameters of the gamma functions? thanks

 

[mark726]

.



Hello, thank you for your post. The input parameters of the gamma functions in this case are the masses of the particles involved in the decay process. The ratio of the two gamma functions gives us the relative probability of the two decay channels, which is important in understanding the dynamics of the decay. The Clebsch-Gordon coefficients that you have calculated help us determine the relative strengths of the different isospin states in the final state, but they are not the input parameters of the gamma functions. I hope this helps clarify things for you.