After correcting for QED effects, including initial-state radiation, the measured \(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \mu^{+} \mu^{-}\)and \(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow\) hadrons cross sections at the peak of the Z resonance give $$ \sigma^{0}\left(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow z \rightarrow \mu^{+} \mu^{-}\right)=1.9993 \mathrm{nb} \quad \text { and } \quad \sigma^{0}\left(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow z \rightarrow \text { hadrons }\right)=41.476 \mathrm{nb} . $$ (a) Assuming lepton universality, determine \(\Gamma_{\ell t}\) and \(\Gamma_{\text {hadrons }}\). (b) Hence, using the measured value of \(\Gamma_{2}=2.4952 \pm 0.0023 \mathrm{GeV}\) and the theoretical value of \(\Gamma_{v v}\) given by Equation (15.41), obtain an estimate of the number of light neutrino flavours.

The Z resonance occurs when the energy of colliding particles, like electrons and positrons, matches the Z boson's mass. The Z boson is an essential mediator of the weak force and has a mass of about 91.2 GeV. When electrons and positrons collide at this energy, they can produce a Z boson, which then decays into other particles like muons and hadrons. This resonance is crucial for studying properties such as cross sections and decay widths.
In our example, we see different cross sections measured at the peak of the Z resonance. These values indicate how frequently different decay processes, like \[ \mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \mu^{+} \mu^{-} \] and \[ \mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \text{hadrons} \], occur. By analyzing these cross sections, we can infer parameters like decay widths and confirm theoretical predictions.

Decay widths describe how quickly a particle decays into other particles. For the Z boson, the total decay width \[ \Gamma_{Z} \] means the sum of all partial decay widths, reflecting every possible decay channel.
To find individual decay widths such as \[ \Gamma_{\ell} \] for leptons, and \[ \Gamma_{\text{hadrons}} \] for hadrons, we use measured cross sections. For instance, cross section \[ \sigma^{0}\left( \mathrm{e}^{+} \mathrm{e}^{-} \rightarrow z \rightarrow \mu^{+} \mu^{-} \right) \] is related to \[ \Gamma_{\ell} \] and helps us determine it.
By assuming lepton universality, we take \[ \Gamma_{\ell} \] for channels involving electrons, muons, or taus to be equal. Using known relationships and measured values, we can solve for these widths.

Lepton universality is a principle stating that all leptons (e.g., electrons, muons, and tau particles) behave identically except for differences due to their masses. This assumption simplifies calculations in particle physics. For instance, we assume \[ \Gamma_{e} = \Gamma_{\mu} = \Gamma_{\tau} = \Gamma_{\ell} \].
When examining the Z boson's decays, lepton universality implies that decay widths into different lepton pairs are equal. This allows us to aggregate data from decays into different leptons and use them collectively to derive several useful parameters. For example, the cross section for \[ \mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \mu^{+} \mu^{-} \] helps us solve for \[ \Gamma_{\ell} \], making our calculations more straightforward.

Estimating the number of light neutrino flavours involves analyzing the Z boson's invisible decay width, \[ \Gamma_{\text{inv}} \], parts of which are due to neutrino decays. Using the measured total decay width \[ \Gamma_{Z} \], and subtracting the calculated contributions from leptonic and hadronic decays, we find \[ \Gamma_{\text{inv}} \].
This invisible width is then related to the number of neutrino flavours, \[ N_{u} \], using the theoretical decay width per neutrino, \[ \Gamma_{u u} \]. By solving the equation \[ \Gamma_{\text{inv}} = N_{u} \Gamma_{u u} \], we estimate \[ N_{u} \], which should align with our understanding that there are three families of neutrinos: electron, muon, and tau neutrinos.
This estimation technique provides vital confirmation of the Standard Model, emphasizing the role of neutral Z boson decays in studying neutrino families.