= Solution
For the <Pole forward-backward asymmetry for charged leptons>, <Integration>[integrating] the normalized distribution over the two hemispheres cancels the even term in the numerator. Since $\int_{-1}^{1}(1+z^2)dz=8/3$ and $\int_0^1z\,dz-\int_{-1}^0z\,dz=1$,
$$
\boxed{A_{FB}=\frac34\left(\frac{L^2-R^2}{L^2+R^2}\right)^2
=\frac34\left[\frac{1-4\sin^2\theta_W}{1-4\sin^2\theta_W+8\sin^4\theta_W}\right]^2.}
$$
Within this single-$Z$, tree-level model, a nonzero asymmetry requires unequal left/right <Gauge coupling>[coupling] magnitudes, demonstrating <parity violation> in the vertices. For $g_L=g_R$ the odd angular term vanishes and $A_{FB}=0$; equal magnitudes with opposite signs also give zero. This inference is process-specific: a forward-backward asymmetry is not universally a parity-odd observable, since reversing both beam and outgoing <momenta> leaves their relative <Angle> unchanged. Purely electromagnetic <Bhabha scattering> is a concrete parity-conserving process with a forward-peaked, asymmetric distribution.
For the supplied $\sin^2\theta_W=0.23153$,
$$
\boxed{A_{FB}^{\mathrm{pred}}=0.01620.}
$$
The measurement differs by $0.0188-0.01620=0.00260$, or $0.79$ times its quoted <Measurement uncertainty>[uncertainty] of $0.0033$, so it is consistent with the prediction. Statistical fluctuations alone preclude exact equality; <photon> exchange, <initial-state radiation> and <Electroweak interaction>[electroweak] <radiative corrections> also modify the idealized pure-resonance prediction and provide physical reasons for small differences.
Back to article page
