Lab 2: Positronium Hyperfine Splitting and BSM Contact CorrectionsBound-state limit of the mediator propagator
Learning Objectives
- Calculate the QED prediction for positronium ground-state HFS to leading order.
- Derive the BSM correction δ(ΔE) ∝ (g²/mϕ²)|ψ(0)|² from the propagator in the bound-state limit.
- Compare the theoretical prediction with published experimental values.
- Extract constraints on g²/mϕ² from the current theory–experiment discrepancy.
- Quantify theoretical and experimental uncertainties and their propagation into BSM limits.
Three-Step Protocol
| Step | For this lab |
|---|---|
| Entrance Session | Present the positronium HFS physics, the connection to |ψ(0)|², and how the EFT contact operator arises from the same propagator as Lab 1. Explain the theory–experiment comparison strategy. |
| Active Lab | Work through the numerical programme: compute the wavefunction at the origin, evaluate ΔEhfs, derive the BSM correction, and construct exclusion contours. Estimate orders of magnitude as you go. |
| Findings Session | Present your exclusion contour in (g, mϕ) space, the error budget, and the sensitivity analysis. Discuss whether the constraint is theory-limited or experiment-limited. |
Background
Positronium Ground-State HFS
The ground-state hyperfine splitting arises from the spin-spin interaction of the electron-positron pair at zero separation. The leading-order QED result is:
Higher-order QED corrections bring the prediction to approximately 203.391 GHz. The experimental value is approximately 203.389 GHz, with both uncertainties at the ~1 MHz level. Reproduce the leading-order result and understand the structure of the first radiative correction (order α⁵) without deriving it.
Representative values — verify current values before your report.
BSM Correction from a New Scalar Mediator
A new scalar mediator coupling to electrons with strength g modifies the contact interaction. The propagator evaluated at bound-state momentum transfer q ~ αem me reduces to −1/mϕ² when mϕ ≫ αem me. The resulting local operator modifies only |ψ(0)|² terms — only S-wave states are affected:
See Framework, §3 for the full derivation scaffold.
This is the same propagator as in Lab 1, evaluated in a different kinematic regime. In Lab 1, |q| ≪ mϕ gives a Yukawa tail. Here, |q| ~ αem me ≪ mϕ gives a contact operator. Verify explicitly that both limits emerge from expanding M(q) in the same small parameter q²/mϕ².
Numerical Programme
- Compute the positronium ground-state wavefunction at the origin from the Coulomb solution.
- Evaluate ΔEhfs at leading order and compare with the known QED result.
- Compute δ(ΔE) as a function of mϕ and g for a scalar mediator.
- Using the current ΔEtheory − ΔEexpt discrepancy as an upper bound, derive exclusion contours in (g, mϕ) space.
- Place these contours on the same parameter plane as the Lab 1 Yukawa results (after appropriate axis mapping).
Statistical and Systematic Effects
Error Budget
The BSM constraint comes from the difference between theory and experiment:
where σcombined includes:
| Uncertainty Source | Origin | Treatment |
|---|---|---|
| QED theory truncation | Unknown higher-order α corrections | Estimate from last known term; assign as systematic |
| Experimental statistical uncertainty | Finite measurement precision | Propagate directly |
| Experimental systematic | Apparatus limitations | Taken from published error budget |
| Hadronic vacuum polarisation | Non-perturbative QCD | Assign systematic from literature range |
Your lab notes should include an uncertainty decomposition table separating theoretical systematics from experimental uncertainties. State whether the BSM constraint is theory-limited or experiment-limited.
Sensitivity Analysis
Produce a sensitivity curve showing how the exclusion contour in (g, mϕ) space shifts as σcombined is varied by factors of 0.5, 1, 2, and 5. This reveals which improvement — theory or experiment — would have the greatest impact on BSM reach.
Lab Notes
Your analysis notes should contain: leading-order HFS derivation, BSM correction calculation, exclusion contour in (g, mϕ) space, full error budget, and sensitivity analysis. These feed into your report and the unified exclusion plot in Lab 4.