The Standard Model at the \(Z\) pole
Experiments at the Large Electron–Positron collider (LEP) at CERN were decisive for precision studies of the electroweak interactions.
(i) Consider the decay \(Z\to f\bar{f}\). Neglect the masses of the fermions.
a) Show that the total polarization of the fermion pair can be found by measuring the asymmetry \[A_{LR}^f = \frac{\Gamma(Z\to f_L\bar{f}_R)-\Gamma(Z\to f_R\bar{f}_L)}{\Gamma(Z\to f_L\bar{f}_R)+\Gamma(Z\to f_R\bar{f}_L)}.\]
b) Express \(A_{LR}^f\) in terms of the electric charge \(Q_f\), the third component of the weak isospin \(T_3^f\) of the fermion and \(\sin^2\theta_w\) in the Standard Model.
c) Compute \(A_{LR}^f\) for the charged leptons, the neutrinos and the quarks.
(ii) Define the forward–backward asymmetry of the process \(e^+e^-\to f\bar{f}\) as \[A_{FB}^f = \frac{\int\limits_0^1 d\cos\theta\frac{d\sigma}{d\cos\theta} - \int\limits_{-1}^0 d\cos\theta\frac{d\sigma}{d\cos\theta}}{\int\limits_0^1 d\cos\theta\frac{d\sigma}{d\cos\theta} + \int\limits_{-1}^0 d\cos\theta\frac{d\sigma}{d\cos\theta}}.\]
d) Find \(A_{FB}^f\) as a function of \(\sqrt{s}\). Plot it.
e) Show that at \(\sqrt{s}=m_Z\) the asymmetries \(A_{LR}^f\) and \(A_{FB}^f\) are related by \[A_{FB}^f = \frac{3}{4}A_{LR}^e A_{LR}^f.\]
f) Show that at \(\sqrt{s}=m_Z\) \[\sigma(e^+e^-\to f\bar{f}) = \frac{12\pi}{m_Z^2}\frac{\Gamma(Z\to e^+e^-)\Gamma(Z\to f\bar{f})}{\Gamma^2(Z\to \text{ all})}.\]
(iii) Discuss how \(\sigma(e^+e^-\to f\bar{f})\) and \(\Gamma(Z\to \text{ all})\) would change if there were a fourth type of neutrino.