= Decay amplitude
{title2=$\mathcal M(i\to f)$}
The invariant decay amplitude is the transition matrix element for one incoming particle and a specified on-shell final state, with the overall momentum delta function factored out. For covariantly normalized states, write the connected <S-matrix> element as $i(2\pi)^4\delta^4(p_i-p_f)\mathcal M(i\to f)$. Its overall phase does not affect the <decay width>. The rate is $\Gamma=(2M_i)^{-1}\sum_{\rm spins}\int d\Phi_f|\mathcal M|^2$, with the initial spin average included when the initial state is unpolarized. For a spin-zero parent and an angle-independent two-body amplitude, the <two-body Lorentz-invariant phase space> gives $\Gamma=|\mathbf k|\sum|\mathcal M|^2/(8\pi M_i^2)$. In <leptonic pion decay>, contracting the axial hadronic matrix element with the chiral <weak charged current> makes this amplitude proportional to the charged-lepton mass, proving <helicity suppression>.
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