Charged-current chirality and the Fermi limit

The charged interaction connects the upper and lower members of a left-chiral doublet. Writing the charged vector fields with a square-root-of-two normalization yields a vertex coefficient g divided by square root of two when the current contains chiral fields. In a four-component notation the left projector is one half of one minus gamma-five. Moving that projector into a V-minus-A current therefore introduces another explicit factor one half at each vertex.

The underlying statement is about chirality, not a universal prohibition on a massive particle of one helicity. A relativistic massive spinor contains a suppressed component of the opposite chirality, and boosts can reverse its helicity. For effectively massless particles the familiar helicity selection emerges, with the corresponding antiparticle assignments. Quark charged currents also contain a unitary mixing matrix after mass diagonalization; a mixing entry multiplies the amplitude, while its modulus squared enters a simple rate comparison only after other factors are controlled.

For conserved leading currents at momentum much smaller than the W mass, the propagator denominator can be expanded in momentum squared over mass squared. Combining the two g-over-square-root-of-two vertices and the two left projectors gives the conventional Fermi coefficient. In muon decay, one may write the operator as a product of two V-minus-A currents with coefficient minus G-Fermi divided by square root of two. Equivalent Fierz-reordered forms give the same amplitude when fermion labels and signs are handled consistently.

Using the tree-level W mass expresses G-Fermi directly through the vacuum scale. The constant has inverse-energy-squared units, as required for a dimension-six operator built from four fermion fields. The leading neglected propagator terms are suppressed by momentum squared over W mass squared, not merely by a smaller gauge coupling. Near the W pole the expansion fails, and its width and complete gauge-consistent amplitude become relevant. For precision matching, electroweak loops and the definition of the measured Fermi constant must also be specified.

For a massless final electron and neutrinos at leading order, the muon width scales as G-Fermi squared times the fifth power of the muon mass, divided by a fixed three-body phase-space factor. The fifth power follows from dimensions: the squared coefficient contributes energy to the minus four, so a width needs five powers of the available mass. Physical lifetime extraction additionally includes the electron mass and inclusive electromagnetic corrections.

The same weak coefficient appears in beta decay, but hadronic vector and axial matrix elements replace free elementary currents. Nuclear structure, radiative corrections and phase space cannot be inferred from G-Fermi alone. Likewise, using charged-current universality as a precision test requires consistent input definitions across processes. The effective theory is useful precisely because it separates a short-distance coefficient from long-distance matrix elements and a controlled momentum expansion, rather than asserting all weak rates are identical.