Capstones

Use these project blueprints when a single lesson exercise is too narrow and a new research question is still too open. Each blueprint ends in one bounded calculation or reproduction, with preparation, decisive checks, and a sentence that limits the conclusion.

Availability note. No scored or independently reviewed capstone assessment is currently offered; the blueprints below are self-directed research practice, not formal assessments or certifications.

Choose one project, one result, and one stopping rule

Choose by the work you want to practice, not by the number of topics a project mentions.

Choose	Best fit and work product
Scalar action to scattering rate	Connect formulations to a measurable process; derive a normalized tree-level rate from the action and four-point correlator.
Ward identity in Compton scattering	Make a diagram or sign error fail visibly; produce a two-diagram QED amplitude with polarization-replacement tests.
Heavy-field matching and running	Carry short-distance information across scales; produce a matched prediction with scale cancellation and a truncation check.
Infrared-safe observable and uncertainty	Combine real and virtual terms; produce a finite inclusive result with numerical and theoretical uncertainties separated.
Reproduce a lattice scalar fixture	Test a computational claim; produce a clean rerun, continuum fit, correlated uncertainty, and independent analytic check.

Before beginning, write three sentences:

1. Primary claim: the one result you will calculate, including its regime and requested accuracy.
2. Independent check: a comparison that does not reuse the decisive step of the main derivation or implementation.
3. Stopping rule: the first failed identity, uncontrolled limit, or missing input that will make you report a bounded partial result instead of extending the claim.

Adding another loop order, observable, field, or numerical method is not a substitute for finishing those three sentences.

Project blueprint 1: scalar action to scattering rate

Question. Starting from real scalar $\lambda\phi^4$ theory, can you connect the local action to the connected four-point function, amputate it with LSZ, and obtain a correctly normalized $2\to2$ tree-level rate?

Concrete deliverable. Write a short calculation that derives the field equation and free propagator, obtains the leading connected four-point correlator with its combinatorial factor, extracts the invariant amplitude, and converts it into a center-of-mass differential or total cross section. Include the identical-final-state factor and state every normalization used.

Preparation. Use classical fields, actions, and local dynamics, functional integrals and correlators, perturbative expansion and Feynman rules, and LSZ and tree amplitudes. The compressed QFT refresher is useful if you have done this before but cannot reconstruct one link in the chain.

Minimum required steps.

1. State the metric, Fourier, state, field, and one-particle normalizations.
2. Vary the action, invert the quadratic kernel, and identify the pole and residue of the two-point function.
3. Derive the leading connected four-point function rather than quoting a vertex rule without its contraction count.
4. Apply LSZ and show how external pole residues and field normalizations cancel from the observable.
5. Insert flux, two-body phase space, and the identical-particle factor to obtain the rate.

Decisive checks. Verify mass dimensions at every stage, crossing symmetry of the scalar amplitude, positivity of the final rate, and invariance after a consistent rescaling of the interpolating field. Recompute the contact factor once by Wick contractions and once from the Feynman rule.

Scope and failure boundary. The result is a tree-level prediction for stable asymptotic particles in the perturbative regime. Ordinary LSZ does not cover a confining theory, an unstable external state, or a background without suitable asymptotic states. A large loop correction or a violated partial-wave bound ends the stated accuracy.

Optional extension. Add one-loop bubble contributions and check the imaginary part against the two-particle cut, or replace the contact interaction by exchange of a heavy scalar and compare with blueprint 3.

Project blueprint 2: Ward identity in Compton scattering

Question. Can you derive the two tree diagrams for $e^-\gamma\to e^-\gamma$ and show that gauge-representative dependence cancels only when their signs, fermion ordering, and propagator momenta are consistent?

Concrete deliverable. Produce the tensor amplitude $\mathcal M^{\mu\nu}$, demonstrate both external-photon Ward checks, and compute either one fixed-polarization rate or the unpolarized differential cross section in a stated frame. Include a one-page table that maps each algebraic cancellation to the assumption it uses.

Preparation. Review fermions and spin, vector fields and gauge redundancy, symmetry, currents, and Ward identities, QED and Yang–Mills theory, and LSZ and tree amplitudes.

Minimum required steps.

1. Fix the covariant derivative, photon polarization, spinor, and gamma-matrix conventions.
2. Derive the two ordered fermion-line contributions with their internal momenta rather than importing an unlabelled formula.
3. Contract the incoming and outgoing photon indices with their momenta and use momentum conservation plus the on-shell Dirac equations.
4. Choose physical polarizations or a justified polarization sum and compute the selected rate.
5. Repeat the result after shifting either polarization by a multiple of its photon momentum.

Decisive checks. Replacing an external polarization by its momentum must give zero. Deliberately omit either diagram and confirm that the test fails. Check the electron mass dimension, the soft-photon behavior appropriate to the selected observable, and agreement between a trace calculation and at least one explicit polarization configuration.

Scope and failure boundary. This calculation tests a tree-level QED Ward identity with on-shell external states. It does not establish loop-level renormalization, anomaly cancellation, detector inclusiveness, or non-Abelian Slavnov–Taylor identities. Stop if a polarization shift changes the rate or if the kinematic limit leaves the assumptions used in the derivation.

Optional extension. Cross the amplitude into pair annihilation or add the leading soft-emission contribution and state the inclusive measurement needed for a finite prediction.

Project blueprint 3: heavy-field matching and running

Question. For a light scalar $\phi$ coupled to a heavy scalar $H$, can you replace heavy exchange by local operators, evolve the coefficients from the matching scale to a low scale, and predict an amplitude with a tested error in $Q/M$?

Concrete deliverable. Starting from the heavy-scalar model in the Core EFT lesson, derive the exact tree-level exchange amplitude, match the leading and next derivative operators near $\mu_M\simeq M$, run the retained coefficient or coefficient vector to $\mu_Q\simeq Q$, and compare the EFT prediction with the expanded full-theory result at several values of $Q/M$.

Preparation. Use perturbative expansion and Feynman rules, loops and regularization, renormalization and the renormalization group, and effective field theory and matching. For operator mixing or several scales, follow renormalization and EFT for working researchers.

Minimum required steps.

1. State the light degrees of freedom, symmetry, hierarchy, scheme, operator basis, and matching observable.
2. Expand the heavy propagator through a declared power in $Q/M$ and match all operators that contribute at that order.
3. Derive or independently verify the anomalous dimension needed for the retained light-theory operators.
4. Solve the RG equation between $\mu_M$ and $\mu_Q$ and combine the evolved coefficient with the low-energy matrix element at a consistent order.
5. Compare with the full-theory expansion and tabulate the observed error as $Q/M$ and the retained order change.

Decisive checks. Check coefficient dimensions and signs, reproduce the first omitted power in a full-theory comparison, and show cancellation of explicit and implicit renormalization-scale dependence through the calculated order. Translate the answer through one allowed finite scheme change and back.

Scope and failure boundary. The local expansion applies below the heavy threshold while $Q/M$ and the relevant couplings remain small. It cannot reproduce the heavy pole at finite order. Residual scale variation diagnoses omitted perturbative terms but is not automatically a probability interval.

Optional extension. Perform one-loop matching, include a second operator that mixes under running, or test whether a symmetry removes the nominally leading correction.

Project blueprint 4: infrared-safe observable and uncertainty

Question. Can you define one inclusive rate or event-shape bin so that its soft and collinear limits are physically meaningful, combine real and virtual terms into a finite result, and state what each uncertainty component tests?

Concrete deliverable. Reproduce one benchmark fixed-order inclusive rate or normalized event-shape bin. Provide the measurement function or cut definition, cancellation or subtraction calculation, phase-space integration, normalization, and a table separating Monte Carlo uncertainty, input covariance, residual scale dependence, and any nonperturbative or detector correction actually used.

Preparation. Begin with infrared-safe observables and synthesis and the scattering phenomenology pathway. For the calculation itself, use Real–Virtual Cancellation and Local Subtraction and Phase-Space Integration and Monte Carlo Estimators.

Minimum required steps.

1. Define the observable and show how its measurement function behaves when a particle becomes soft or a pair becomes collinear.
2. Display the virtual poles and the integrated or local counterterms that cancel them, using one regulator and convention consistently.
3. Integrate the finite remainder with recorded cuts, maps, tolerances, and random-stream construction.
4. Normalize the result and vary renormalization and factorization scales coherently where both occur.
5. Report numerical, parametric, perturbative, matching, and model effects separately before choosing any combination rule.

Decisive checks. Verify independence from unphysical subtraction or slicing parameters over a resolved range, reproduce a known inclusive or soft/collinear limit, repeat the integration with a different phase-space map or independent stream, and show that real–virtual poles cancel before numerical rounding obscures them.

Scope and failure boundary. The conclusion applies only to the declared observable, cuts, fixed order, scale range, and treatment of long-distance physics. Scale variation is a sensitivity test, not a universal confidence level. Do not compare directly with detector-level data unless the transfer, hadronization, and calibration assumptions are included.

Optional extension. Add logarithmic resummation and verify fixed-order expansion plus matching, or compare two infrared-safe measurement definitions that emphasize different physical scales.

Project blueprint 5: reproduce a lattice scalar fixture

Question. Can you reproduce a finite-lattice scalar pole mass in a clean environment, recover its continuum order, propagate correlated uncertainty, and distinguish successful execution from evidence for the stated numerical claim?

Concrete deliverable. Reproduce the free-scalar fixture in computational field theory onboarding at several values of $am_0$. Extract $E_a$ once from the lattice propagator and once from a periodic correlator fit, extrapolate $E_a/m_0$ to $a=0$, and provide a complete rerun packet plus a separately derived analytic reference.

Preparation. Use the numerical and reproducibility review, Lattice Regulators and Continuum Targets, Statistical Inference and Error Budgets, and Reproduce and Validate a Result.

Minimum required steps.

1. Write the finite mathematical specification, units, boundary conditions, expected $O(a^2)$ artifact, and acceptance tolerances before writing code.
2. Implement direct momentum-space and correlator-based extractions without sharing the decisive mass-extraction routine.
3. Sweep lattice spacing, temporal extent, numeric precision, fit window, and statistics; use the full stable covariance for correlated fits.
4. Fit the predicted continuum order, test plausible higher-order terms, and compare with the exact pole relation.
5. Rerun from recorded source, dependencies, command, inputs, random streams, raw outputs, and analysis choices in a clean environment.

Decisive checks. Recover the sign and coefficient of the leading free cutoff term, observe second-order convergence in its asymptotic range, inject an incorrect momentum stencil or periodic-image omission and confirm that a test fails, and verify that a diagonal-covariance fit does not silently replace the stated analysis.

Scope and failure boundary. This project verifies a free scalar regulator, implementation, and continuum-inference procedure. It is not evidence that an interacting sampler equilibrates, that a sign problem is controlled, or that the same fit model resolves excited states. Report those as new questions.

Optional extension. Add a weak interaction, Hamiltonian truncation, or a tensor-network calculation while retaining the exact free fixture and adding method-specific truncation checks.

Copyable final-report template

Use this template for any blueprint. Keep the strongest supported conclusion near the top so a reader does not have to infer it from the final plot.

project title:

question:
strongest supported conclusion:
scope and failure boundary:

theory, state, conventions, and boundary conditions:
observable, normalization, units, and kinematic regime:
preparation pages actually used:

derivation or computational method:
approximations, truncations, and first omitted terms:
inputs and their covariance:

primary result:
uncertainty components and their meanings:
decisive checks and whether each passed:
independent cross-check and shared assumptions:

source, environment, data, and rerun instructions:
unexpected failure or negative result:
stopping rule reached:

optional extension not attempted:
next exact page or research question:

A useful report remains valuable when the target calculation fails: preserve the failed check, reduce the claim, and explain which missing derivation, input, or limiting sequence would be needed next. Use Scope a First Research Project to turn that bounded gap into a new question, or return to Learning pathways to choose a specialist route for the completed result.