Leptonic pion decay (${\pi }^{+}\to {\ell }^{+}{\nu }_{\ell }$)

The pion decay constant gives $⟨0|J_A^{\mu }|\pi (p)⟩\propto f_{\pi }{p}^{\mu }$. Contracting the momentum with the leptonic weak charged current and applying the external Dirac equations makes the amplitude proportional to the charged-lepton mass. This is an explicit example of helicity suppression.

A charged pion decays to a charged antilepton, its associated neutrino, and a photon. Inner bremsstrahlung from charged external states interferes with structure-dependent radiation. An experimentally resolved photon needs an energy cut: the real-radiation rate alone diverges in the soft limit, whereas a suitably inclusive combination with virtual radiative corrections is finite.^[ref]

In a box-normalized treatment of negative-pion two-body decay with a massless antineutrino, $dp/d{E}_0=E_{\ell }/m_{\pi }$. The simple density of states is therefore proportional to $p_{\ell }^2{E}_{\ell }/m_{\pi }$. 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 $m_{\ell }^2{p}_{\ell }^2$. A Lorentz-invariant phase-space measure distributes normalization factors differently but gives the same observable ratio when the amplitude is normalized consistently.

Lorentz covariance allows only a multiple of $p^{\mu }$ for a vacuum-to-spin-zero one-particle current. In its rest frame only the temporal component can remain. A parity-even temporal vector current between a parity-even vacuum and parity-odd particle changes sign, forcing that component to vanish. A temporal axial current is parity-odd and is allowed. This explains why the pion decay constant enters the axial, rather than vector, part of the weak charged current.

At tree level with universal weak charged current coupling and massless neutrinos, the leptonic pseudoscalar decay width gives the displayed ratio. The pion decay constant, Fermi constant and common quark mixing coefficient cancel. The charged-lepton mass factor is helicity suppression: a spin-zero parent cannot emit the two massless helicities selected by the chiral weak interaction. The ratio tests this suppression and lepton universality; scalar or pseudoscalar four-fermion interactions can modify it. Radiative corrections must be included for a precision comparison.

In the convention $⟨0|J^{\mu }|P(p)⟩=-i\sqrt{2}{F}_P{p}^{\mu }$, the Fermi interaction gives $\mathcal{M}=G_F{F}_P{p}_{\mu }{\bar{u}}_{\ell }{\gamma }^{\mu }(1-{\gamma }_5)v_{\nu }$. With a massless antineutrino, the external Dirac equations reduce this to $G_F{F}_P{m}_{\ell }{\bar{u}}_{\ell }(1-{\gamma }_5)v_{\nu }$. The spin sum is $8G_F^2{F}_P^2{m}_{\ell }^{2}k\cdot q=4G_F^2{F}_P^2{m}_{\ell }^2(M_P^2-m_{\ell }^2)$. The two-body Lorentz-invariant phase space has integrated value $(1-m_{\ell }^2/M_P^2)/(8\pi )$, and the initial flux factor is $1/(2M_P)$, giving the displayed width. If the decay constant is instead defined without the square root of two, $f_P=\sqrt{2}{F}_P$ and the denominator is $8\pi$. A weak mixing coefficient multiplies the amplitude and its modulus squared multiplies the width.