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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 $\Gamma =(2M_i{)}^{-1}{\sum }_{\mathrm{spins}}\int d{\Phi }_f|\mathcal{M}{|}^2$, 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.

The Standard Model assigns the same gauge coupling to each lepton generation in the weak charged current. Different decay rates still arise from masses and phase space. Ratios of leptonic pion decay rates cancel the common hadronic decay constant and quark mixing coefficient, isolating universality and helicity suppression. Electromagnetic radiative corrections must be included for precision comparisons to inclusive measured decay rates.

The derivative convention uses hypercharge Y with $Q=T_3+Y$. For each lepton family $l=e,\mu ,\tau$, the left-handed fields form an SU(2) doublet $L_l=({\nu }_l,l{)}_L^T$, with $Y=-1/2$. Their weak isospin values are $T_3=+1/2,-1/2$ and their electric charges are $0,-1$. The right-handed charged lepton $l_R$ is an SU(2) singlet, with $T_3=0$, $Y=-1$ and charge $-1$. The minimal model used here has no right-handed neutrino. A sterile right-handed neutrino, if added as a singlet with $Y=0$, would have no charged weak current interaction. Antiparticles carry the conjugate representations and opposite electric charges.

Expanding ${\bar{L}}_{l}i{\gamma }^{\mu }{D}_{\mu }{L}_l$ supplies $-g{\bar{L}}_l{\gamma }^{\mu }{A}_{\mu }^a{\sigma }_a{L}_l/2$. The off-diagonal entries of $A^a{\sigma }_a$ are $\sqrt{2}{W}^{+}$ and $\sqrt{2}{W}^{-}$. With the chiral projector $P_L=(1-{\gamma }^5)/2$, one has ${\bar{\nu }}_L{\gamma }^{\mu }{l}_L=\bar{\nu }{\gamma }^{\mu }{P}_{L}l$. Thus the two W boson interactions areFor quarks, separate rotations diagonalize the up-type and down-type mass terms. If their left-handed rotations are $U_{uL}$ and $U_{dL}$, the charged interaction contains the CKM matrix $V=U_{uL}^{†}{U}_{dL}$:There is an implicit sum over color charge. This quark mixing permits transitions between different families, rather than just the paired fields in a single doublet.

Let $J^{\mu }={\sum }_l{\bar{\nu }}_l{\gamma }^{\mu }(1-{\gamma }^5)l+J_{\mathrm{had}}^{\mu }$ and $\kappa =g/(2\sqrt{2})$. At momenta much smaller than $m_W$, neglect derivatives in the heavy W boson equation. Its mass and source terms can be completed to a square:This explicitly performs integrating out a charged vector boson. The resulting Fermi interaction isThe omitted terms contain derivatives and are relatively suppressed by momentum squared divided by $m_W^2$.

For the leptonic pion decay, Lorentz covariance restricts a vacuum-to-spin-zero current matrix element to a constant times $p^{\alpha }$. In the rest frame, rotations eliminate the spatial components. Under parity, the temporal vector current is even, whereas the vacuum is even and the pion is odd, so its temporal matrix element equals its negative and vanishes. The temporal axial current is odd and is allowed. This proves the vacuum-to-pseudoscalar current selection rule, rather than merely assuming an axial current. Define the pion decay constant byThen the vector-minus-axial charged weak current has matrix element $-i{f}_{\pi }{V}_{ud}{p}^{\alpha }$, up to an irrelevant overall phase convention. Consequently the scattering amplitude for the decay can be writtenUse $p=k+q$, the momentum-space Dirac equation ${\bar{u}}_e/k=m_e{\bar{u}}_e$, and $/q{v}_{\nu }=0$ for a massless neutrino. Since $/q(1-{\gamma }^5)=(1+{\gamma }^5)/q$, the neutrino contribution vanishes, leavingThe fermion spin sum is, using the supplied gamma matrix trace identities,Here $(k+q{)}^2=m_{\pi }^2$, $k^2=m_e^2$, and $q^2=0$. The mass term in the trace vanishes; anticommuting ${\gamma }^5$ reduces the remaining trace to twice $\mathrm{tr}(/k/q)$. It follows thatUsing two-body decay phase space, with final momentum magnitude $(m_{\pi }^2-m_e^2)/(2m_{\pi })$, gives the leptonic pseudoscalar decay widthThus the requested $m_e^2(m_{\pi }^2-m_e^2)$ factor is already present in the squared amplitude, and the phase space supplies one more threshold factor.

The physical explanation is helicity suppression. In the pion rest frame the two momenta are opposite. The massless antineutrino has positive helicity; zero total angular momentum would require the electron to have positive helicity too. The charged weak current instead creates a left-chiral electron, whose helicity is negative when its mass vanishes. A nonzero electron mass admits the required opposite helicity with an amplitude proportional to $m_e$. The decay amplitude vanishes for $m_e=0$, and the rate is suppressed by $m_e^2$.

Pion beta decay changes a charged pion into a neutral pion and a lepton pair through the weak charged current. The small charged-neutral pion mass gap severely restricts phase space, giving an approximately fifth-power dependence on that gap. It does not have the charged-lepton mass suppression of direct leptonic pion decay.

Cecil Frank Powell's 1950 lecture illustrates the successive decays of pions and muons through their tracks and explains why the nearly fixed secondary muon range, allowing for range straggling, supports a single neutral decay partner. He describes its identification as a neutrino tentatively. The neutral particle itself leaves no recorded track. He contrasts this decay with the nuclear reactions caused by stopped negative pions, connecting the latter with the nuclear force.^[ref]

Here is a modern special relativity derivation for the dominant leptonic pion decay, assuming a free pion at rest, a massless muon neutrino, and no emitted photon. Write $m_{\pi }$ and $m_{\mu }$ for the pion and muon masses. By conservation of momentum the daughter momenta are opposite with equal magnitude $p$. By conservation of energy and the energy–momentum relation,Substitute $E_{\mu }=m_{\pi }{c}^2-pc$, square, and cancel $p^2{c}^2$. This givesThe resulting $p$ is fixed, although the muon's direction is not set by the incoming stopping track. Thus the kink angle relative to that earlier track is not the angle between the two daughters: in the pion rest frame, the daughters are back to back. A neutral-partner arrow in a diagram represents inferred momentum, not an observed string of grains.

For an original calculation, adopt rounded input values $m_{\pi }{c}^2=139.57\mathrm{MeV}$ and $m_{\mu }{c}^2=105.66\mathrm{MeV}$. They give $pc=29.79\mathrm{MeV}$, $T_{\mu }=4.119\mathrm{MeV}$ and $E_{\nu }=29.79\mathrm{MeV}$. Their sum satisfies $T_{\mu }+E_{\nu }=(m_{\pi }-m_{\mu })c^2=33.91\mathrm{MeV}$. These are rounded calculation inputs and outputs, not a fit to the 1947 plates.

At fixed material and particle identity, a fixed initial kinetic energy suggests a narrow particle range distribution broadened by range straggling and measurement uncertainty. The synthetic example convolves a single chosen range with a normal distribution; a second example superposes two such chosen ranges. It explains why repeated comparable stopping distances can be informative, without pretending to extract a historical spectrum.

The decay equations do not identify the neutral partner as a neutrino on their own: a massless photon would satisfy the same energy and momentum equations. Nor does any isolated kink establish the pion's identity. The inference gains force from calibrated particle range, track grain density, repeated decay patterns and independent particle properties. Historical particle names should not obscure the distinction: a muon is a lepton, whereas a pion is a meson.