Questions: Electroweak Precision Measurements

Question 1

The Z boson mass (91.1876 +/- 0.0021 GeV) and width (2.4952 +/- 0.0023 GeV) were measured with extraordinary precision at LEP. The total width depends on the number of light neutrino species N_nu. How does this measurement work?

• A. Each neutrino species is directly detected and counted
• B. The total Z width Gamma_Z = Gamma_had + 3*Gamma_l + N_nu*Gamma_nu, where Gamma_had and Gamma_l are measured from visible decays; the invisible width Gamma_inv = Gamma_Z - Gamma_had - 3*Gamma_l then determines N_nu = Gamma_inv/Gamma_nu, where Gamma_nu is the Standard Model prediction for one neutrino species
• C. The Z lineshape is narrower for fewer neutrinos because there are fewer decay channels
• D. Neutrinos produce missing energy events that are directly counted

Answer: (B)

LEP measured the Z lineshape (cross section vs. center-of-mass energy) with extreme precision. The visible decays (hadrons and charged leptons) determine Gamma_had and Gamma_l. Subtracting these from the total width gives the invisible width. Dividing by the Standard Model prediction for Z -> nu nu-bar gives N_nu = 2.984 +/- 0.008, consistent with exactly 3 and ruling out a fourth light neutrino. This is one of the most precise counting experiments in physics.

Question 2

Before the top quark was discovered in 1995 at the Tevatron, electroweak precision data from LEP and SLD predicted its mass to be approximately 170-180 GeV. How could virtual particles that had never been directly observed be 'weighed'?

Model answer: Radiative corrections from virtual top quarks in loop diagrams affect electroweak observables. The top quark appears in W and Z self-energy corrections, and its contribution depends on m_t^2 (quadratic sensitivity, unlike the Higgs which enters logarithmically). For example, the rho parameter rho = M_W^2/(M_Z^2 * cos^2(theta_W)) receives a correction delta-rho proportional to G_F * m_t^2. By measuring M_W, M_Z, sin^2(theta_eff), and other observables to high precision, and computing the loop corrections as functions of m_t and m_H, a global fit determines the preferred values of m_t and m_H. The prediction m_t = 173 +/- 13 GeV (before discovery) was confirmed when the top quark was found at 173.1 GeV.

This predictive success demonstrated that the Standard Model is not just a tree-level theory but a precision quantum field theory whose loop structure is quantitatively correct. It also showed the power of indirect constraints: virtual particles modify measurable quantities through quantum corrections.

Question 3

The W boson mass is one of the most important electroweak precision observables. At tree level, M_W = M_Z * cos(theta_W). Loop corrections shift M_W by several hundred MeV. These corrections are dominated by the top quark and, to a lesser extent, the Higgs boson.

• T. True
• F. False

Answer: True

At tree level, the relationship M_W = M_Z * cos(theta_W) is exact, giving M_W approximately 79.8 GeV. Radiative corrections, dominated by top quark loops (delta M_W ~ +290 MeV from the top) and Higgs boson loops (delta M_W ~ -40 MeV for m_H = 125 GeV), shift this to M_W approximately 80.36 GeV. The current experimental world average (80.3692 +/- 0.0133 GeV) agrees with the Standard Model prediction at the level of a few MeV. Any discrepancy would be evidence for new physics contributing to the loop corrections. A 2022 CDF measurement claiming M_W = 80.4335 +/- 0.0094 GeV was in tension with the SM but has not been confirmed by other experiments.