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Q:
Why is a decay rate defined for a (weak) flavour-changing neutral current in the Standard Model?
I know that the decay rate for a $b \to s+\gamma$ transition is given in the standard model as
$
\Gamma(b\to s+\gamma) = \frac{4 G_F^2 m_b^5}{\pi^3}\left|V_{tb}V_{ts}^*C_7(\mu)\right|^2
$
where $G_F$ is the Fermi constant, $m_b$ the $b$-quark mass, $V_{tb}$ the top-quark Yukawa coupling and $V_{ts}$ the CKM matrix element.
If the top-quark Yukawa coupling is small, one can expand the factor $|V_{tb}V_{ts}^*|$ and obtain
$
\Gamma(b\to s+\gamma) \simeq \frac{G_F^2 m_b^5|C_7(\mu)|^2}{\pi^3}.
$
But in the standard model there is also a neutral, flavour-changing current given as
$
J_{NC}(b\to s+\gamma) = \frac{G_F m_b^2}{\sqrt{2}} V_{ts}^*\bar{s}\gamma_\mu(1-\gamma_5)b F_0(\mu)
$
where $F_0(\mu)$ is a short-distance function.
To define the decay rate, one has to integrate the above current over time. Doing so one obtains
$
\Gamma(b\to s+\gamma) \simeq \frac{G_F^2 m_b^5|C_7(\mu)|^2}{2\pi^3}.
$
The factor $|V_{tb}V_{ts}^*|$ is then replaced by $(F_0(0 0b46394aab
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