# Physics - Semileptonic decays

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### Semileptonic decay estimator

This script takes two mesons and calculates the branching fraction of semileptonic decay (via $$e\nu$$ emission) from one meson to the other, taking the mass, quark contents and lifetime of the mesons into account. The expression for the partial width of a semileptonic decay of a pseudoscalar meson, $$P$$, to another psuedoscalar meson, $$Q$$, and leptons $$e\nu$$ is given by:

$\Gamma(P\to Qe\nu) = \frac{G_F^2|V_{qq}|^2\Delta^5}{30\pi^3}$

where:

$\Delta = \frac{M_P^2-M_Q^2}{2M_P}$

and $$|V_{qq}|$$ is the relevant CKM matrix element.

This is then turned into a branching fraction by taking the ratio with the total width:

$\mathcal{B}(P\to Qe\nu) = \frac{\Gamma(P\to Qe\nu)}{\Gamma(P)} = \frac{\Gamma(P\to Qe\nu)\tau_P}{\hbar}$

where $$\tau_P$$ is the lifetime of the meson, $$P$$.

Process Phase space (MeV) Width (MeV) Branching fraction PDG value