TutorialFrom textbook to notebook — worked examples for each kinematic limit
The Framework page states what the physics is. This page shows you how to get there from what you already know. Each section bridges a specific textbook derivation to the computation you will perform in the lab notebooks.
Work through the exercises in order. Each builds on the previous one. By the end, you will have reproduced — by hand and numerically — the three kinematic limits of the mediator propagator and the statistical tools needed to set constraints.
Textbook Map
The following table connects each topic to the textbook sections where it is developed. You are not expected to read all of these cover to cover; the specific sections listed contain exactly the material you need.
| Topic | Primary Reference | Sections | Prepares you for |
|---|---|---|---|
| Propagator intuition | Griffiths, Introduction to Elementary Particles | §1.5, §6.1–6.3 | All labs |
| Yukawa potential from Fourier transform | Schwartz, QFT and the Standard Model | §2.2, §3.4 | Lab 1 |
| Hydrogen-like wavefunctions, |ψ(0)|² | Griffiths, Introduction to Quantum Mechanics | Ch. 4 (hydrogen atom) | Lab 2 |
| EFT matching: heavy mediator → contact operator | Schwartz, QFT and the Standard Model | §31.1–31.3 | Lab 2 |
| Breit-Wigner, Z⁰ line shape, partial widths | Thomson, Modern Particle Physics | Ch. 15 | Lab 3 |
| Likelihood, χ², confidence intervals | Cowan, Statistical Data Analysis | Ch. 7, 9, 10 | Labs 1–4 |
| Exclusion limits, Feldman–Cousins | Feldman and Cousins, Phys. Rev. D 57, 3873 (1998) | Full paper | Labs 1, 4 |
Schwartz §31 on effective field theories is graduate-level material. You do not need the full machinery. What you need is one specific move: when the mediator mass is much larger than the momentum transfer, the propagator collapses to a constant and the interaction becomes local. The worked example below isolates exactly that step.
Tutorial 1: Propagator → Yukawa Potential
Textbook basis: Griffiths EP §6.1–6.3; Schwartz §3.4
Prepares you for: Lab 1
The problem
You know the propagator in momentum space: M(q) ~ g²/(q² + mϕ²). You need the potential in position space. This requires a three-dimensional Fourier transform.
Step 1: Set up the integral
The static potential is:
Switch to spherical coordinates in momentum space. Let θ be the angle between q and r. The volume element is d³q = q² sinθ dq dθ dφ, and the azimuthal integration gives 2π.
Perform the angular integration over θ. You should find that ∫₀π exp(iqr cosθ) sinθ dθ = 2 sin(qr)/(qr). This is a standard result — verify it by substituting u = cosθ.
Step 2: Reduce to a radial integral
After the angular integration:
Substitute u = qr. Show that the integral becomes (1/r²) ∫₀∞ du · u sin(u) / (u² + (mϕr)²). Look up the standard result: ∫₀∞ du · u sin(u)/(u² + a²) = (π/2) e⁻ᵃ. You do not need contour integration. This integral is tabulated in Gradshteyn and Ryzhik §3.723 and can be verified numerically.
Step 3: Assemble the result
Verify two limits:
• Set mϕ = 0. You should recover the Coulomb potential V(r) = −g²/(4πr). Massless mediator → infinite range.
• Compute the screening length λ = 1/mϕ for mϕ = 0.2 meV. You should get λ ≈ 1 mm. This is the range at which the Yukawa exponential suppresses the force to 1/e of the Coulomb value.
Numerical check: Open the Lab 1 notebook template. Plot V(r) for mϕ = 0 (Coulomb) and mϕ = 0.2 meV (Yukawa) on the same axes for r ∈ [10 μm, 10 mm]. Verify that the curves diverge at r ~ λ.
Many textbooks perform this integral using contour integration in the complex q-plane (closing in the upper half-plane, picking up the pole at q = imϕ). That method is elegant but requires residue calculus that not all students have seen. The substitution method above gives the same answer and requires only a tabulated integral. If you do know contour methods, use them as a cross-check, not as the primary derivation.
Tutorial 2: Heavy Mediator → Contact Operator → Energy Shift
Textbook basis: Griffiths QM Ch. 4; Schwartz §31.1–31.3
Prepares you for: Lab 2
The problem
A new scalar mediator with mass mϕ couples to electrons with strength g. If mϕ is much larger than the typical momentum transfer in positronium (q ~ αem me ≈ 3.7 keV), the propagator simplifies and the interaction becomes local. You need to compute the resulting energy shift of the ground-state hyperfine splitting.
Step 1: Expand the propagator
The propagator at momentum transfer q is:
When q²/mϕ² ≪ 1, expand: ≈ (1/mϕ²)[1 − q²/mϕ² + ...]. Keep only the leading term.
Compute the ratio q²/mϕ² for q = αem me ≈ 3.7 keV and mϕ = 1 MeV. You should find q²/mϕ² ≈ 1.4 × 10⁻⁵. This justifies truncating at leading order. Repeat for mϕ = 10 keV — the expansion is still valid but the correction term is no longer negligible. At what mϕ does the expansion break down?
Step 2: From momentum space to position space
A q-independent amplitude in momentum space corresponds to a δ-function in position space:
So the effective potential from the heavy mediator is:
This is the key EFT matching step: a nonlocal interaction mediated by a heavy particle becomes, at low energies, a local (contact) interaction. The mediator has been “integrated out.”
In Tutorial 1, you derived V(r) = −(g²/4πr) exp(−mϕr). Take the limit mϕr → 0 for a wavefunction that is nonzero only near the origin. The exponential → 1, but the 1/r divergence means the interaction is dominated by the shortest distances. Smearing this over the bound-state wavefunction recovers the δ-function result. Both routes give the same physics — the contact limit is the short-distance limit of the Yukawa potential.
Step 3: Compute the energy shift
First-order perturbation theory: δE = ⟨ψ|δV|ψ⟩. For a δ-function potential, this picks out the wavefunction at the origin:
Look up |ψ(0)|² for positronium. The ground state is hydrogen-like with reduced mass μ = me/2 and Bohr radius a₀’ = 2a₀:
|ψ1s(0)|² = 1/(π a₀’³) = (αem me)³ / (8π)
Compute |ψ(0)|² numerically. Then evaluate δE for g = 10⁻⁶ and mϕ = 1 MeV. Express the result in MHz. Compare to the current theory–experiment discrepancy in positronium HFS (~1 MHz). Is this coupling detectable?
Dimensional analysis check. Verify that (g²/mϕ²) · |ψ(0)|² has dimensions of energy. Work in natural units (ℏ = c = 1): g is dimensionless, mϕ has dimension [mass], |ψ(0)|² has dimension [mass]³. So δE ~ [mass]³/[mass]² = [mass] ✔. Then convert to SI and verify you get a frequency in the MHz range.
Tutorial 3: Propagator Pole → Breit-Wigner → Invisible Width
Textbook basis: Thomson, Modern Particle Physics, Ch. 15
Prepares you for: Lab 3
The problem
When the centre-of-mass energy approaches the Z⁰ mass, the propagator denominator nearly vanishes. The finite width Γ regulates the divergence and produces a resonance peak. You need to connect the propagator to the measured cross section and extract the invisible width.
Step 1: From propagator to cross section
The Z⁰ propagator near the pole is:
The cross section for e⁺e⁻ → Z⁰ → f f̄ involves |D(s)|²:
This is the Breit-Wigner form. The numerator carries the product of partial widths because the Z must couple to the initial state (e⁺e⁻, strength ∝ Γee) and the final state (f f̄, strength ∝ Γf).
Show that the peak cross section (at s = mZ²) is:
σpeak = (12π/mZ²) · Γee Γf / ΓZ²
and that the full width at half maximum is ΓZ. The peak height gives ΓeeΓf/ΓZ²; the FWHM gives ΓZ. Together with external knowledge of Γee, these two observables determine all partial widths.
Step 2: Decompose the total width
The Z⁰ decays to hadrons, charged leptons, and neutrinos. The total width is:
where Γhad and Γlep are measured from visible final states. Everything left over is the invisible width:
Using the SM prediction Γν ≈ 167 MeV and the measured Γinv ≈ 499 MeV, compute Nν. You should find Nν ≈ 2.99, consistent with three light neutrino generations.
Now compute the 95% CL upper limit on additional invisible width ΔΓinv. If the uncertainty on Γinv is ±1.5 MeV, what is the maximum ΔΓinv at 95% CL? What does this imply for the coupling of any additional species to the Z?
Numerical check. Open the Lab 3 notebook template. Using the provided LEP data, fit a Breit-Wigner to the hadronic cross section and extract mZ and ΓZ. Compare your values to the PDG: mZ = 91.1876 ± 0.0021 GeV, ΓZ = 2.4952 ± 0.0023 GeV. If your fit deviates by more than 3σ, check whether you included initial-state radiation corrections.
In Tutorials 1 and 2, you probed the propagator far from the pole: |q| ≪ mϕ. Here you probe it at the pole: q² ≈ mZ². The denominator is the same structure — (q² − m²) — but the physics is completely different. Off-shell: screening and contact operators. On-shell: resonance production and decay. Lab 4 will ask you whether these are genuinely the same physics or merely the same algebra.
Tutorial 4: Likelihood, Model Comparison, and Exclusion Contours
Textbook basis: Cowan, Statistical Data Analysis, Ch. 7, 9, 10; Feldman and Cousins (1998)
Prepares you for: Labs 1–4
The problem
You have data and two competing models. You need to decide which model fits better (model comparison) and, if neither model is clearly preferred, set upper limits on the parameters of the extended model (exclusion contours). Every lab in this capstone requires this machinery.
Step 1: The likelihood function
Given N measurements {xi} with known uncertainties {σi} and a model f(θ), the log-likelihood is:
Maximising ℓ is equivalent to minimising χ². The best-fit parameters θ̂ sit at the minimum of χ².
Open framework/likelihood_utils.py. Generate 20 synthetic data points from a straight line y = 2x + 1 with Gaussian noise σ = 0.5. Evaluate χ²(a, b) on a grid of slopes and intercepts. Find the minimum. Verify that Δχ² = 1 contour around the minimum contains approximately 68% of the true parameter value when you repeat the exercise 1000 times.
Step 2: Likelihood-ratio test
To compare two nested models (H₀ ⊂ H₁), compute:
Under H₀, Λ follows a χ² distribution with degrees of freedom equal to the difference in free parameters. Large Λ means H₀ is a poor fit relative to H₁.
Generate synthetic data from the Lab 1 scenario:
• H₀: V(r) = −G m₁ m₂/r (one free parameter: G)
• H₁: V(r) = −(G m₁ m₂/r)[1 + α exp(−r/λ)] (three free parameters: G, α, λ)
First generate data under H₀ (no Yukawa signal). Fit both models. Compute Λ. Since Δndf = 2, you need Λ > 5.99 to reject H₀ at 95% CL. Verify that Λ is small — the data contain no signal, so the extended model should not be significantly preferred.
Now generate data under H₁ with α = 10⁻⁴ and λ = 0.1 m. Repeat the fit and compute Λ. Is the signal detected? At what significance?
Step 3: Exclusion contours
When the data do not prefer the extended model, set upper limits. For each point (α, λ) in parameter space, ask: “Is this point consistent with the data at 95% CL?” The boundary of the region where the answer changes from yes to no is the exclusion contour.
Using your H₀ synthetic data from Exercise 4b, scan a grid in (α, λ) space. At each point, compute Δχ² = χ²(α, λ) − χ²min. The 95% CL contour for two parameters is at Δχ² = 5.99.
Plot the excluded region. Verify that the injected signal from H₁ (α = 10⁻⁴, λ = 0.1 m) falls outside the excluded region — this is consistent with the signal being undetectable in H₀ data. This is your first exclusion plot. You will produce three more in the labs and combine them in Lab 4.
The Δχ² method used above is the simplest approach. The Feldman–Cousins unified method handles the boundary between one-sided and two-sided intervals more correctly, especially near physical boundaries (e.g. α ≥ 0). For this capstone, either method is acceptable, but you must state which you used. If you want to implement Feldman–Cousins, read the original paper — it is clearly written and includes worked examples.
Full Reference List
Textbooks
- Griffiths, D. J., Introduction to Elementary Particles, 2nd ed. (Wiley, 2008). Propagators, Feynman rules, and particle phenomenology at the advanced-undergraduate level.
- Griffiths, D. J., Introduction to Quantum Mechanics, 3rd ed. (Cambridge, 2018). Hydrogen atom, perturbation theory, and bound-state wavefunctions.
- Schwartz, M. D., Quantum Field Theory and the Standard Model (Cambridge, 2014). Graduate-level QFT. Particularly §2–3 (propagators and potentials) and §31 (effective field theories).
- Thomson, M., Modern Particle Physics (Cambridge, 2013). Accessible treatment of electroweak physics, Z resonance, and LEP measurements (Ch. 15).
- Cowan, G., Statistical Data Analysis (Oxford, 1998). Compact reference for likelihood methods, hypothesis testing, and confidence intervals.
- Barlow, R., Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences (Wiley, 1989). Friendlier introduction to statistics for physicists; complements Cowan.
Key Papers
- Adelberger et al., Ann. Rev. Nucl. Part. Sci. 53, 77 (2003). Comprehensive review of inverse-square-law tests.
- Arkani-Hamed, Dimopoulos, Dvali, Phys. Lett. B 429, 263 (1998). ADD extra dimensions — theoretical motivation for sub-millimetre Yukawa searches.
- LEP Collaborations, Phys. Rep. 427, 257 (2006). Definitive Z-pole electroweak measurements.
- Karshenboim, Phys. Rev. Lett. 104, 220406 (2010). Light boson constraints from atomic precision measurements.
- Feldman and Cousins, Phys. Rev. D 57, 3873 (1998). Unified statistical approach to exclusion limits.
- Cowan et al., Eur. Phys. J. C 71, 1554 (2011). Asymptotic formulae for likelihood-ratio tests — practical computational tool.
Mathematical Tables
- Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, 8th ed. (Academic Press, 2015). The tabulated integral used in the Yukawa derivation is §3.723.