= State-counting and helicity factors in pion decay
{title2=$\rho_\ell\propto p_\ell^2E_\ell/m_\pi,\quad w_\ell\propto1-\beta_\ell$}
In a box-normalized treatment of negative-<pion> two-body decay with a massless <antineutrino>, $dp/dE_0=E_\ell/m_\pi$. The simple <density of states> is therefore proportional to $p_\ell^2E_\ell/m_\pi$. The <spin-zero two-body helicity selection> requires positive charged-lepton helicity, whose left-chiral weight is $(1-\beta_\ell)/2$. Since $1-\beta_\ell=2m_\ell^2/(m_\pi E_\ell)$, the energy factors cancel in the product and give the standard relative rate proportional to $m_\ell^2p_\ell^2$. A <Lorentz-invariant phase-space measure> distributes normalization factors differently but gives the same observable ratio when the amplitude is normalized consistently.
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