Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 45 / 3 / Solution

The product of leptonic weak charged currents in the Fermi interaction contains four-fermion cross terms such asThis produces ${\mu }^{-}\to e^{-}+{\bar{\nu }}_e+{\nu }_{\mu }$, and the same interaction permits neutrino-electron scattering and inverse muon decay. Pure electromagnetic interactions have no neutrino coupling and preserve the separate charged-lepton species, so cannot produce these processes by themselves. The factor $1-{\gamma }_5=2P_L$ fixes the chiral structure of the weak charged current.

The pion is a pseudoscalar with negative parity, while the strong-interaction vacuum is parity even. Lorentz covariance restricts a vacuum-to-pion vector-current matrix element to $a{p}^{\alpha }$. In the pion rest frame only its temporal component could be nonzero, but a vector current's temporal component is parity even, whereas the pion is parity odd. The matrix element therefore equals its negative and vanishes. The temporal component of an axial current is parity odd, so its matrix element is allowed and Lorentz covariance fixes it to a constant times $p^{\alpha }$. Defining the pion decay constant in the printed normalization givesThis proves both the vector-current exclusion and the axial-current form; $F_{\pi }$ itself is a strong-interaction parameter, not fixed by symmetry.

The cross term $J_{\mathrm{leptons}}^{\alpha †}{J}_{\alpha }^{\mathrm{hadrons}}$ gives, up to an irrelevant overall phase, the decay amplitudeTake the antineutrino mass to be zero. Since $p=k+q$, ${\bar{u}}_e/k=m_e{\bar{u}}_e$, and $/q{v}_{\nu }=0$, anticommutation with ${\gamma }_5$ givesThe amplitude has helicity suppression by one power of the charged-lepton mass.

Using the external spin sums, with no initial spin average for a spin-zero pion,The middle equality follows from $(1-{\gamma }_5)/q=/q(1+{\gamma }_5)$, $(1+{\gamma }_5{)}^2=2(1+{\gamma }_5)$, and the gamma matrix trace identities. Summing a full massless neutrino spin basis introduces no factor of two: the current projector annihilates the inactive helicity.

For completeness, evaluate the two-body Lorentz-invariant phase space in the pion rest frame. Its three-momentum delta function sets $\mathbf{q}=-\mathbf{k}$, and the energy delta function is $\delta (M_{\pi }-\sqrt{|\mathbf{k}{|}^2+m_e^2}-|\mathbf{k}|)$. The radial root isCombining its derivative with $1/(4E_k|\mathbf{k}|)$ and integrating solid angle givesThe invariant squared amplitude is independent of direction, so the supplied decay formula yields the leptonic pseudoscalar decay widthThe denominator is $4\pi$ because the matrix element contains $\sqrt{2}{F}_{\pi }$. With the alternative definition $f_{\pi }=\sqrt{2}{F}_{\pi }$, the same width is $G_F^2{f}_{\pi }^2{M}_{\pi }{m}_e^2(1-m_e^2/M_{\pi }^2{)}^2/(8\pi )$. In either convention it vanishes as $m_e\to 0$.

Replacing the electron by a muon gives the same hadronic factor and weak coupling. Hence the pion electron-to-muon decay ratio isThis is a tree-level result with massless neutrinos. The cancellation of $F_{\pi }$ and $G_F$ makes the ratio a clean test of the predicted helicity suppression and electron–muon universality in the chiral weak charged current. For example, scalar or pseudoscalar four-fermion interactions could remove the charged-lepton mass factor and substantially enhance the electron mode. Radiative corrections are required for a precision experimental comparison.

With two quark generations, the charged hadronic current containsA $K^{-}$ contains $s\bar{u}$, so its annihilation into the lepton pair selects the term proportional to $\sin {\theta }_C$. Strong strangeness conservation excludes the $d$ term from its vacuum matrix element. Its amplitude is thus multiplied by $\sin {\theta }_C$ and its width by ${\sin }^2{\theta }_C$:Here $F_K$ is the pure strong-interaction decay constant in the same square-root-of-two convention. If the pion current's quark mixing coefficient is also shown explicitly, its width carries ${\cos }^2{\theta }_C$; it is already absorbed if the stated pion matrix element uses the full weak current. Thus the isolated weak-factor ratio is ${\tan }^2{\theta }_C$, while the kaon suppression relative to unit weak coupling is the requested Cabibbo suppression factor ${\sin }^2{\theta }_C$.