I'm attempting to calculate the branching fraction of a particular Kaon decay, namley $K^{+}\rightarrow{\pi^{+}\pi^{0}}$. I know what the branching fraction equation is, namely:

$$ BR=\frac{\Gamma_j}{\Gamma} $$

Where $\Gamma=1/\tau$. Now, I have been given $\Gamma_{j}$ as $1.2\times{10^{-8}}\,\mathrm{eV}$, and $\tau$ as $1.2\times{10^{-8}}\,\mathrm{s}$, rather this is stated as the mean lifetime of the $K^+$ species. Putting this all together I get a branching fraction of $1.44\times10^{-16}\,\mathrm{eV}{\mathrm{s}}$.

Surely this is way too small to be a viable branching fraction...? Usually it is quoted as a percentage so I was expecting something like 0.2...?

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    $\begingroup$ DarthPlagueis: "the branching fraction equation [... where] $\Gamma = 1/\tau$." -- It bears being pointed out that the relation between the (full) decay width $\Gamma$ and the mean life duration $\tau$ is instead: $$\Gamma = \frac{\hbar}{\tau}.$$ "$\tau$ [...] the mean lifetime" -- Note that the PDG is presenting the relevant duration values as "mean life". $\endgroup$ – user12262 Oct 18 '15 at 21:30

Energies are equivalent to the (sometimes angular) frequencies of the photons which have those energies via $E = h f.$

The dimensionless value you are looking for is probably your current value divided by $\hbar,$ but it strongly depends on how the $\Gamma_j$ in units of $\text{eV}$ was being calculated. (You see $h$ when people are quoting optical spectra because they care about real frequencies; you see $\hbar$ when people are using units which set $\hbar = 1.$)

  • $\begingroup$ Hmmmm, interesting. I now get a branching fraction of 0.218, I wonder why this was not explicit in the question? Should I have known this...? $\endgroup$ – DarthPlagueis Oct 18 '15 at 19:56
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    $\begingroup$ Particle physicists use $\hbar = 6.58 \times 10^{-22}\,\mathrm{MeV \cdot s}$ almost exclusively for these things. The Particle Physics Booklet doesn't even quote $h$ in natural units. @Darth If this is a course on particle physics, then either you should have known or the problem is intended to teach this convention. For reference, the PDB quotes that branching ratio at 20.66%. $\endgroup$ – dmckee --- ex-moderator kitten Oct 18 '15 at 20:38

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