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

[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 anyone 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,

 

[Orodruin]

where the sum taken over neutrinos flavors

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

Do anyone know how did he drive this ?

Just expand the quotient for small $m_i$.

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 ..,

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

 

[Safinaz]

Hi, thanks for replying:

Just expand the quotient for small $m_i$.

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

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

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 ..

 

[Orodruin]

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

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

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 ..

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 $i$.