FrameworkTheory reference for all four labs
This page contains the theoretical backbone shared by all four labs. The physics here is standard; the pedagogical contribution is organising three experiments as limits of one amplitude.
1. The Mediator Propagator
The tree-level amplitude for exchange of a mediator of mass mϕ between two particles is
where g is the coupling strength and q is the four-momentum transfer. The denominator (q² − mϕ²) is universal — it encodes that the mediator has a definite mass. Spin affects the numerator and selection rules but not the pole structure.
This framework treats the denominator as the organising principle. Labs 1–2 use a scalar mediator (simplest numerator). Lab 3 uses the Z⁰ (vector). The synthesis in Lab 4 asks you to evaluate what this difference means for unification onto a common parameter plane.
Three kinematic limits of this single expression produce three distinct experimental signatures:
| Limit | Condition | Result | Lab |
|---|---|---|---|
| Static | |q| ≪ mϕ | Yukawa potential V(r) ∝ e−mϕr/r | Lab 1 |
| Contact | |q| ~ αme ≪ mϕ | Local operator ∝ δ³(r) | Lab 2 |
| On-shell | q² → mϕ² | Resonance (Breit-Wigner peak) | Lab 3 |
2. Derivation A: Propagator → Yukawa Potential
The static potential in position space follows from Fourier-transforming the amplitude:
Evaluate in spherical coordinates. The angular integration gives sin(qr)/(qr). The remaining radial integral is:
Substituting u = qr and recognising the standard integral gives:
No contour integration is required. The integral is evaluated by substitution and recognition of a tabulated result.
Verify that setting mϕ = 0 recovers the Coulomb potential. Interpret: a massless mediator gives an infinite-range force; a massive mediator gives an exponentially screened force with range λ = 1/mϕ (natural units) or λ = ℏ/(mϕc).
Compute: λ = 1 mm corresponds to mϕ ≈ 0.2 meV. λ = 10 μm corresponds to mϕ ≈ 20 meV.
3. Derivation B: Propagator → Contact Operator
When mϕ ≫ αme, expand the propagator:
The leading term is q-independent: a contact interaction. In position space this corresponds to δV(r) ∝ (g²/mϕ²) · δ³(r). Since |ψ(0)|² ∝ (meα)³ for positronium:
This is the same propagator as in Derivation A, at different kinematics. Both limits follow from expanding M(q) in powers of q²/mϕ².
Verify dimensionally. Estimate: for δ(ΔE) = 1 MHz, what is the implied g²/mϕ²?
4. Derivation C: Propagator Pole → Breit-Wigner
When q² → mϕ², the width Γ regularises the pole:
For the Z⁰:
The invisible width Γinv = ΓZ − Γhad − 3Γlep counts channels with no visible final state.
Show that the peak cross section determines ΓeeΓf, and that the FWHM gives ΓZ. Verify these are sufficient to extract Γinv given external input on Γee.
5. Statistical Methods
5.1 Maximum Likelihood Estimation
Given data {xi} and model parameters θ, the likelihood is L(θ) = ∏i p(xi | θ). For Gaussian measurements, maximising the log-likelihood is equivalent to minimising:
Parameter uncertainties: Δχ² = 1 (one parameter, 68% CL) or Δχ² = 2.30 (two parameters, 68% CL). A Python template is provided in framework/likelihood_utils.py.
5.2 Likelihood-Ratio Test
To compare nested models:
Under H₀, Λ follows a χ² distribution with Δndf degrees of freedom. For Lab 1: H₀ has 1 parameter (G), H₁ has 3 (G, α, λ), so Δndf = 2. Λ > 5.99 rejects H₀ at 95% CL.
5.3 Exclusion Contours
Construct 95% CL contours using Feldman–Cousins or profile likelihood. All exclusion plots must state the confidence level and method. See Feldman & Cousins, Phys. Rev. D 57, 3873 (1998).
If you have not previously performed likelihood-ratio tests, work through the scaffolded exercises in framework/likelihood_utils.py before starting Lab 1.
6. Notation and Conventions
| Symbol | Meaning | Notes |
|---|---|---|
| g | Mediator coupling strength | Dimensionless |
| mϕ | Mediator mass | In eV unless stated otherwise |
| λ = ℏ/(mϕc) | Compton wavelength | Range of the Yukawa force |
| α | Yukawa coupling relative to gravity (Lab 1) | Not fine-structure constant in this context |
| αem | Fine-structure constant ≈ 1/137 | Used in Labs 2–3 |
| q | Four-momentum transfer | In the static limit, |q| is three-momentum |
| κ | Torsion constant | Lab 1 only |
| ΓZ, Γinv | Z⁰ total and invisible widths | Lab 3 only |
| Λ | Likelihood-ratio test statistic | Not to be confused with Λ as energy scale |
α appears in two roles: as the Yukawa coupling in Lab 1 and as the fine-structure constant in Labs 2–3. Context must disambiguate. In the synthesis (Lab 4), use g² and mϕ exclusively.