Weinberg angle and neutral-current couplings

Define sine and cosine of the weak angle as g-prime and g divided by their positive combined magnitude. The photon is sine times W-three plus cosine times B; the orthogonal Z is cosine times W-three minus sine times B. Substitution into the gauge interaction gives the photon coefficient g sine T-three plus g-prime cosine Y. The equality of these two coefficients makes the result the elementary coupling times electric charge.

The orthogonal combination has coefficient g over cosine multiplying T-three minus sine squared times electric charge. This formula acts on each chiral field in its own representation. A right-chiral singlet has no T-three but still couples to Z through its nonzero charge. Consequently the word neutral does not mean electrically neutral particles only, nor does it mean identical interactions for the two chiralities. The neutrino and charged lepton therefore probe different combinations of the same gauge parameters.

For a Dirac fermion write the Z interaction with an overall g over cosine and coefficients g-left and g-right multiplying the chiral projectors. Their values are T-three minus Q sine squared and minus Q sine squared, respectively, where T-three is that of the left-chiral partner. Rewriting the projectors gives vector coefficient g-left plus g-right and axial coefficient g-left minus g-right, but introduces an overall one half.

A paper that writes an overall g over twice cosine therefore lists vector and axial numbers different from a paper that keeps g over cosine and divides its vector and axial numbers by two. The predictions agree when the entire normalization is translated. For effectively massless fermions, the parity asymmetry formed from the two squared chiral couplings is also expressible using vector and axial couplings. It depends on their relative weights rather than the common overall gauge factor.

Define the fermion asymmetry parameter as left-coupling squared minus right-coupling squared divided by their sum, for real leading couplings and negligible fermion masses. At an ideal isolated Z pole, the forward-backward asymmetry for electron–positron annihilation into a fermion pair is three quarters times the electron and final-fermion asymmetry parameters. Photon interference, radiation, acceptance, finite masses, and loop form factors require corrections before comparing this formula with a measured number.

Beyond tree level, an on-shell angle defined from W and Z masses, a running subtraction-scheme angle, and a fermion-specific effective angle extracted from a vertex need not coincide. The differences are calculable within a specified scheme and order. One cannot insert a fitted effective angle into every bare tree identity and interpret the residual as new physics. Neutral-current tests become powerful by comparing consistently calculated observables, with input correlations and common theoretical uncertainties retained.