# I $\mu~\to e~ \gamma$ decay width and neutrino propagator

1. Feb 4, 2017

### Safinaz

Hi all,

I'm studying $\mu \to e~ \gamma$ decay from cheng & Lie' book " gauge theory of elementary particles ". In Equation (13.84), he wrote the neutrino propagator
$\sum_i \Big ( \frac{U^{*}_{ei} U_{\mu i}}{(p+k)^2-m_i^2} \Big),$
(where the sum taken over neutrinos flavors) in the form:

$\sum_i U^{*}_{ei} U_{\mu i} \Big ( \frac{1}{(p+k)^2} + \frac{m_i^2}{[(p+k)^2]^2} + ...... \Big)$
$= \sum_i \frac{U^{*}_{ei} U_{\mu i} m_i^2}{[(p+k)^2]^2} + ........$

Do any one know how did he drive this ? Then he write that:

the leading term vanishes, $\sum_i U^{*}_{ei} U_{\mu i}$, reflecting the GIM cancellation mechanism.

I can't get this statement ..

Thanks,

2. Feb 4, 2017

### Orodruin

Staff Emeritus
No, the sum should be taken over the neutrino masses. The flavour eigenstates do not have definite masses.

Just expand the quotient for small $m_i$.

It follows directly from the unitarity of $U$. It is the $\mu$-$e$ element of $U U^\dagger = 1$.

3. Feb 4, 2017

### Safinaz

Can you please write the general form, like for instance taylor expansion, it seems i'm not so good in math !

Actually this still not clear for me , $\sum_i U^{*}_{ei} U_{\mu i}$ is multiplied by the first term $\frac{1}{(p+k)^2}$ as well as the second term $\frac{m_i^2}{[(p+k)^2]^2}$, so why the first one which has been ignored .. also what's GIM mechanism and he says due to this mechanism this first leading term vanishes ..

4. Feb 4, 2017

### Orodruin

Staff Emeritus
$$\frac{1}{1+x} = 1 - x + x^2 + \ldots$$

The first term $\frac{1}{(p+k)^2}$ is independent of $i$ and can be taken out of the sum. The second term is not independent of $i$.