Lab 3: Z⁰ Resonance, Invisible Width, and Pole PhysicsOn-shell limit of the mediator propagator
Learning Objectives
- Extract the Z⁰ mass, total width, and partial widths from LEP line-shape data.
- Determine the invisible width Γinv and infer the number of light neutrino generations.
- Understand how a propagator pole governs a resonance cross section.
- Quantify both statistical and systematic uncertainties on extracted parameters.
- Prepare the Z⁰ results for comparison with Labs 1–2 in the Synthesis.
Three-Step Protocol
| Step | For this lab |
|---|---|
| Entrance Session | Present the Breit-Wigner formalism, explain how the propagator pole produces a resonance, and describe the invisible-width method for neutrino counting. Explain the connection to the propagator structure in Labs 1–2. |
| Active Lab | Fit the LEP line-shape data. Extract mZ, ΓZ, σpeak. Compute Γinv and Nν. Check fit quality as you go — plot residuals during the fit, not after. |
| Findings Session | Present your extracted Z⁰ parameters with uncertainties, comparison to published LEP values, and the systematic discussion. Identify what remains unresolved. |
Background
The Z⁰ Resonance as a Propagator Pole
The Z⁰ production cross section near the pole is described by a Breit-Wigner form arising directly from the propagator evaluated at q² ≈ mZ²:
This is the on-shell limit of the propagator structure introduced in the Framework: when q² approaches m², the denominator nearly vanishes and the amplitude peaks. The width ΓZ encodes all decay channels. The invisible width Γinv = ΓZ − Γhad − 3Γlep counts channels producing no visible final state.
Neutrino Counting and BSM Sensitivity
The Standard Model predicts Γinv = Nν · Γν, where Γν is the single-neutrino partial width. The LEP result Nν = 2.984 ± 0.008 is consistent with three generations. Any additional light species coupling to the Z increases Γinv.
Labs 1–2 probed the propagator far from the pole (|q| ≪ mϕ). Lab 3 probes it at the pole (q² = mZ²). The same denominator structure that produces Yukawa screening at long distances produces a Breit-Wigner resonance at the mass shell. The invisible width teaches a complementary lesson: couplings determine both the strength of the potential (Labs 1–2) and the rate of decay (Lab 3).
Data Analysis Programme
You work with published LEP line-shape data (hadronic cross section vs centre-of-mass energy √s).
- Fit the Breit-Wigner line shape to extract mZ, ΓZ, and σpeak.
- Compute Γhad from the peak cross section and known Γee.
- Extract Γinv = ΓZ − Γhad − 3Γlep.
- Determine Nν with uncertainty and compare with the SM prediction.
- Compute the 95% CL upper limit on additional invisible decay width ΔΓinv.
Statistical and Systematic Effects
Fit Quality
- Perform χ² minimisation using the full covariance matrix of the LEP data points.
- Report fit χ²/ndf and check goodness of fit.
- Extract parameter uncertainties from the fit covariance matrix (not from parameter scans).
- Compute the correlation coefficient between mZ and ΓZ and discuss its origin.
Systematic Effects
| Systematic Effect | Estimation Method | Impact |
|---|---|---|
| Beam energy calibration | From published LEP energy working group | Dominant systematic on mZ |
| Initial-state radiation (ISR) | Radiator function convolution | Modifies line shape; must be included in fit |
| γ–Z interference | Include interference term in fit model | Asymmetric distortion of peak |
| Selection efficiency vs √s | Parametrize from published LEP efficiencies | Affects σpeak extraction |
| Γee input value | Compare CODATA value vs LEP-internal determination | Propagates directly into Γhad |
Your lab notes should include a table comparing your extracted Z⁰ parameters with published LEP/CODATA values, including both statistical and systematic uncertainties on each parameter.
Lab Notes
Your analysis notes should contain: Breit-Wigner fit results with covariance matrix, Γinv extraction, Nν determination, systematic discussion, and the parameter-comparison table. These feed into your report and the pole-inference panel in Lab 4.