Lab 4: SynthesisUnification, gap analysis, and critical evaluation
Learning Objectives
- Map the results of Labs 1–2 onto a common (g², mϕ) parameter plane.
- Determine what Lab 3 contributes to the synthesis — and what it cannot.
- Produce a unified exclusion plot with clearly stated confidence levels.
- Identify the gap: parameter-space regions where none of the three experiments has sensitivity.
- Critically evaluate whether the propagator-unification narrative holds up.
Three-Step Protocol
| Step | For this lab |
|---|---|
| Entrance Session | Present the parameter-mapping strategy: how (α, λ) from Lab 1 and (g, mϕ) from Lab 2 translate to common coordinates. Explain the interpretive challenge for Lab 3 (vector vs scalar). State your plan for the gap analysis. |
| Active Lab | Perform the mapping calculations. Build the unified exclusion plot. Test both competing hypotheses. Draft the gap analysis. This session is primarily computational and interpretive. |
| Findings Session | Present the unified exclusion plot, the hypothesis comparison, and the gap analysis. Discuss which aspects of the propagator-unification concept survived contact with data and which did not. |
The Mapping Problem
Labs 1–3 each constrain physics beyond the Standard Model, but they report results in different variables. Before you can combine them, you must translate each measurement into common coordinates.
Lab 1 → (g², mϕ)
The torsion-balance exclusion curve lives in (α, λ) space. The mapping is:
This translation is direct: both α and λ have unambiguous physical meaning for a scalar mediator.
Lab 2 → (g², mϕ)
The positronium exclusion contour enters directly from the EFT matching. The BSM correction δ(ΔE) ~ (g²/mϕ²) me³ αem³ already contains the right variables.
Lab 3 → The Interpretive Step
The Z⁰ analysis extracts (mZ, ΓZ, Γinv). These constrain additional species coupling to the Z, but the Z⁰ is a vector boson — not the same hypothetical scalar mediator probed in Labs 1–2.
The Z⁰ results cannot be placed on the (g², mϕ) plane for a scalar mediator without an explicit model link. You have two honest options:
Option A: Present Lab 3 as a separate “pole inference panel” — showing how (M, Γ) encode coupling strength and hidden channels — alongside the Labs 1–2 exclusion plot. This is the cleaner interpretation.
Option B: Assume a specific model where a new scalar also contributes to invisible Z decays (e.g. via Z → ϕϕ if mϕ < mZ/2), derive the constraint on g from ΔΓinv, and plot it. This requires explicit model assumptions that must be stated.
Whichever you choose, you must justify it. Asserting that all three experiments constrain “the same mediator” without stating the assumptions is not acceptable.
The Unified Exclusion Plot
Produce a single figure with:
- Horizontal axis: mediator mass mϕ (logarithmic), from ~10⁻⁶ eV to ~10² GeV.
- Vertical axis: coupling strength g² (logarithmic).
- Lab 1 exclusion region (low mϕ, large Compton wavelength).
- Lab 2 exclusion region (intermediate mϕ, sensitivity through |ψ(0)|²).
- Lab 3 contribution — either as exclusion band (Option B) or as an inset panel (Option A).
- Published limits (Eöt-Wash, LEP, collider searches) for comparison.
- A secondary axis showing Compton wavelength λ = ℏ/(mϕc).
- All excluded regions labelled with confidence level (95% CL) and method.
The plot must show each experiment’s reach into mediator-mass space — which extends far beyond its operating energy scale. If your plot suggests each experiment probes only its native scale, it is wrong.
Gap Analysis
Identify the region of (g², mϕ) space where none of the three experiments — nor published limits — provides constraints. Then address:
- What mediator masses and couplings fall in the gap?
- What physical reason explains the gap?
- What measurement geometry, energy scale, or experimental technique would be needed to reach it?
Competing Hypotheses
Hypothesis A: Unified Mediator
A single scalar mediator (g, mϕ) explains the Yukawa deviation in gravity, the HFS shift in positronium, and is consistent with collider constraints. Test whether a common parameter region exists.
Hypothesis B: Regime-Specific Physics
Each regime requires independent effective operators. No single mediator spans centimetres to 100 GeV. Evaluate whether the data favour scale separation.
Compare the two hypotheses quantitatively — using likelihood ratio, Δχ², or Bayes factor. State which is preferred by the data.
Critical Evaluation
This is the intellectually hardest part of the capstone. In your findings session and later in your seminar, you should address:
Does gravitational metrology realistically probe high-energy physics — or is this primarily a precision-engineering field? Does the propagator-unification concept reflect genuine physical unity, or is it a pedagogical convenience imposed on experiments that are fundamentally independent?
Present arguments on both sides with specific, quantitative support from your own results. State your own position and justify it. The gap analysis should inform your argument.
Lab Notes
Your synthesis notes should contain: parameter mapping calculations, the unified exclusion plot, hypothesis comparison, and a draft of the critical evaluation. These form the core of your report and the culmination of your seminar presentation.
Archive
After assessment, your final synthesis plot and a one-paragraph summary of your gap-analysis conclusion will be deposited (with your permission) in the archive/ directory. Over time, this becomes a visible record of how different students’ analysis choices produce different exclusion contours from the same underlying physics. That variation is itself a teaching tool.