Scope a first research project

A good first project makes one decision with one inspectable result. It is small enough to finish, but sharp enough to return yes, no, or inconclusive under rules written before the calculation. This guide turns the heavy-exchange comparison from the preceding lessons into such a project.

Required background. Bring the matched comparison and proposed discriminating test. The exact residuals are derived in Reproduce and validate a result; this page uses them to design a project rather than deriving them again.

Ask for a decision, not a topic

“Study effective field theory” is a topic. “Check a low-energy expansion” is a task, but it still does not say what the result will decide. Use this grammar:

For one object in one declared regime, does one comparison meet one numerical or logical criterion, so that I can make one next decision, while not claiming the nearest stronger conclusion?

Every phrase has work to do:

• The object fixes what is actually calculated.
• The regime prevents a result from migrating beyond its assumptions.
• The comparison names the alternatives or approximation orders.
• The criterion makes failure possible.
• The decision explains why the output matters.
• The nearest nonclaim blocks the most tempting overinterpretation.

Write the criterion before the main calculation. Transparent methods and analysis plans make deviations visible and reduce the temptation to redefine success after seeing the answer National Academies 2019, Appendix C, pp. 209–220. For an exploratory project, predeclaration does not forbid learning; it separates the planned test from follow-up questions generated by the result.

Shrink until one output can decide

Use this decision tree before adding calculations:

Does the question enable one named decision?
├─ No → name the decision or stop.
└─ Yes
   Can one observable or theorem target answer it?
   ├─ No → choose the most discriminating target; defer the rest.
   └─ Yes
      Can one table, figure, bound, or derivation expose the answer?
      ├─ No → define a pilot with one inspectable output.
      └─ Yes
         Is one principal uncertainty or failure mode left?
         ├─ No → split the project at the first independent uncertainty.
         └─ Yes
            Can success, revision, and stopping be decided in advance?
            ├─ No → sharpen the criterion before computing.
            └─ Yes → write the brief and begin.

When the tree says “defer,” write the deferred question under the nearest nonclaim. Do not let it re-enter as an unannounced second objective.

Learn only the minimum background

Background is sufficient when you can define the object, justify the test, and interpret its failure. It is not the number of papers read. For each required item, demand a demonstration:

Need	Minimum source	Demonstration before starting
Physical meaning	Heavy-scalar matching	Explain why $x=q^2/M^2$ controls the local expansion and why the pole is excluded.
Exact comparison	Residual derivation	Derive or verify $\epsilon_{\mathrm{LO}}=\lvert x\rvert$ and $\epsilon_{\mathrm{NLO}}=\lvert x\rvert^2$.
Numerical discipline	Numerical reproducibility	State numeric type, input order, tolerance, and an independent exact check.

Stop adding prerequisites when all three demonstrations work. Further reading may improve interpretation, but it should not silently enlarge the project.

Make the method capable of failing

Describe a method as inputs, transformations, outputs, and decision rules—not as “write code and inspect the result.” A falsifiable computational test includes:

1. an ordered input set and a domain boundary;
2. formulas or algorithms fixed before execution;
3. one output with named columns and precision;
4. numerical and scientific checks with tolerances;
5. interpretations for pass, fail, and inconclusive outcomes; and
6. a rule for changes made after the first run.

Clear, specific descriptions of inputs, computation, and environment are necessary for another researcher to repeat a result National Academies 2019, ch. 4, pp. 55–70. The test is scientific only if a failed check changes the conclusion rather than being dismissed as inconvenient.

A filled heavy-exchange project

This is an instructional project. Its analytic answer is known, so it trains question design, controlled comparison, and preservation; it is not offered as a new physics result.

Question, decision, and nearest nonclaim

Question. For one normalized tree-level heavy-scalar exchange channel,

$$F(x)=\frac{1}{1-x}, \qquad F_{\mathrm{LO}}(x)=1, \qquad F_{\mathrm{NLO}}(x)=1+x,$$

do the computed LO and NLO relative errors follow the predicted $x$ and $x^2$ scaling at the predeclared points

$$x\in\{0.02,0.05,0.10,0.20\}?$$

Decision enabled. If the implementation recovers both scaling laws within tolerance and passes an exact rational check, use it as the verified starting point for a later, separately scoped EFT calculation. Otherwise, correct or reject this implementation before adding physical complexity.

Nearest nonclaim. Success does not prove the Appelquist–Carazzone theorem, validate loop matching, cover all exchange channels, establish an experimental constraint, or show that a truncated local expansion remains accurate near $|x|=1$. The original theorem has broader hypotheses and content Appelquist and Carazzone 1975, pp. 2856–2861; the toy propagator expansion is treated in a modern EFT setting by Manohar 2018, §5.4, eq. (5.20), preprint p. 37, PDF.

Method and acceptance threshold

For each $x$ in the listed order, compute

$$\epsilon_{\mathrm{LO}}(x) =\frac{|F-F_{\mathrm{LO}}|}{|F|}, \qquad \epsilon_{\mathrm{NLO}}(x) =\frac{|F-F_{\mathrm{NLO}}|}{|F|}.$$

The analytic predictions are

$$\epsilon_{\mathrm{LO}}(x)=x, \qquad \epsilon_{\mathrm{NLO}}(x)=x^2 \quad (x>0).$$

Use binary64 arithmetic without intermediate rounding. The project passes only if all of the following hold:

$$\begin{aligned} \max_x|\epsilon_{\mathrm{LO}}(x)-x|&\leq5\times10^{-13},\\ \max_x|\epsilon_{\mathrm{NLO}}(x)-x^2|&\leq5\times10^{-13},\\ \max_x|(1-x)F(x)-1|&\leq5\times10^{-15}. \end{aligned}$$

Also require $\epsilon_{\mathrm{NLO}}<\epsilon_{\mathrm{LO}}$ at every point. The tolerances are fixed for this well-conditioned deterministic calculation; they are not estimates of uncertainty in a realistic EFT prediction.

Independently repeat the calculation with exact rationals

$$x\in\left\{\frac1{50},\frac1{20},\frac1{10},\frac15\right\}.$$

This check changes the numerical failure mode: exact rational arithmetic has no binary rounding. It still shares the same mathematical specification, so it does not test whether the toy model represents a different physical system.

One inspectable output

Produce one machine-readable table with the columns x, exact, lo, nlo, relative_lo, and relative_nlo. The expected values are:

$x$	$F(x)$	$F_{\mathrm{LO}}$	$F_{\mathrm{NLO}}$	$\epsilon_{\mathrm{LO}}$	$\epsilon_{\mathrm{NLO}}$
$0.02$	$1.020408163265$	$1$	$1.02$	$0.02$	$0.0004$
$0.05$	$1.052631578947$	$1$	$1.05$	$0.05$	$0.0025$
$0.10$	$1.111111111111$	$1$	$1.10$	$0.10$	$0.0100$
$0.20$	$1.250000000000$	$1$	$1.20$	$0.20$	$0.0400$

Compute the acceptance statistics from unrounded values, not from the displayed decimals. A reader should be able to decide the project from this table plus the three threshold values; a decorative plot is unnecessary.

Risks and decision rules

Risk	Early signal	Decision
Wrong sign or denominator	$F$ decreases for positive $x$, or $(1-x)F\neq1$	Stop interpretation; repair the defining formula and rerun every row.
Wrong approximation order	NLO does not improve on LO, or its error scales as $x$	Check that $F_{\mathrm{NLO}}=1+x$; do not weaken the threshold.
Rounding hides failure	Results depend on displayed precision	Recompute from full-precision values and preserve both raw and displayed tables.
Scope expands toward the pole	A point above $0.20$ is added to “make the result stronger”	Reject the addition here; make breakdown behavior a new project.
Toy result is overinterpreted	The conclusion mentions loops, all heavy fields, or data	Replace it with the nearest nonclaim and keep the conclusion at one-channel tree level.

Proceed. All numerical thresholds and the exact-rational check pass; preserve the result and hand it forward.

Revise once. A localized transcription or implementation error is identified. Record the failed output and the change, then rerun the entire four-point test without changing inputs or tolerances.

Stop successfully. The table and decision are complete. Do not add more $x$ values merely because the calculation is easy.

Stop inconclusively. The formulas, input identity, or arithmetic route cannot be reconciled. Preserve the discrepancy and do not use the implementation as a basis for a more complicated calculation.

Stop at the domain boundary. This project ends at $x=0.20$. Any study that moves toward $x=1$, where the exact function has a pole, or toward the geometric-series boundary $|x|=1$ more generally, where a low-order local approximation deteriorates, requires a new question, criterion, and interpretation.

Preservation and handoff

Keep a compact project folder containing:

• the completed brief;
• a plain-text calculation script or formula sheet;
• the exact ordered inputs and unrounded output table;
• the runtime and numeric type, exact command, warnings, and file checksums;
• the separate exact-rational calculation; and
• a short README stating the result, decision, nonclaim, and how to repeat the test.

Organized inputs, explicit dependencies, change tracking, and a human-readable project overview make small computational work much easier to repeat and transfer Wilson et al. 2017, Box 1. If the first run fails, keep its output beside the corrected run so the reason for the change remains inspectable.

Copyable one-page brief

Working question:
Decision this result enables:
Nearest stronger nonclaim:

Object and regime:
Minimum background:
Demonstration of each prerequisite:

Ordered inputs:
Method and approximation order:
One inspectable output:
Acceptance threshold:
Independent check and distinct failure mode:

Principal risk:
Early warning signal:
Proceed rule:
Revise rule:
Successful stopping rule:
Inconclusive stopping rule:
Domain boundary:

Files and information to preserve:
One-sentence conclusion format:
Next destination if the result passes:
Next destination if it fails:

The brief fits on one page because every field serves the decision. If a second method, observable, or conclusion needs its own threshold, split it into a continuation.

Exercises

1. Shrink a multi-project question

A proposed first project asks: “Does heavy-scalar EFT work at tree level and one loop, in all three exchange channels, and near the resonance?” Apply the decision tree and write a bounded replacement.

Solution

The proposal contains several independent targets: tree versus loop order, three kinematic channels, and low-energy versus near-pole behavior. Each adds different formulas and failure modes, so one output cannot decide the whole question.

A bounded replacement is: “For one normalized tree-level exchange channel at $x\in\{0.02,0.05,0.10,0.20\}$, do LO and NLO relative errors agree with $x$ and $x^2$ within the declared tolerance?” Its nearest nonclaim excludes loops, crossing among channels, and resonance behavior. Those become possible later projects only after this test passes.

2. Respond to a failed threshold

The first run gives $\epsilon_{\mathrm{NLO}}(0.10)=0.011$ instead of $0.010$, so the planned tolerance fails. The other rows look close. Should you increase the tolerance to $0.002$?

Solution

No. The tolerance was chosen before the run, and the analytic identity predicts $0.010$ exactly. Increasing it would redefine success after seeing the answer.

First check the input value, the definition of relative error, unrounded arithmetic, and whether NLO was entered as $1+x$. Then evaluate $(1-x)F-1$ and repeat the row with exact rationals. If a transcription error is found, preserve the failed output, make the one bounded correction, and rerun all four rows. If no cause is found, stop inconclusively rather than passing the calculation.

3. Enforce the stopping boundary

All four points pass. A student proposes adding $x=0.80$ and claims that agreement there would prove decoupling for the model. What should the project do?

Solution

Stop successfully at $x=0.20$, as declared. At $x=0.80$, the exact value is $F=5$, while LO and NLO are $1$ and $1.8$; their relative errors are $0.8$ and $0.64$. The geometric series still converges for $|x|<1$, but a low-order truncation is no longer accurate, and the pole at $x=1$ is nearby.

Studying deterioration toward the pole could be a useful new project, but it needs different accuracy questions and cannot prove the full decoupling theorem. The completed project supports only the predeclared one-channel, tree-level scaling test.

Choose the next destination

If the test passes and you want the physics needed for a less artificial matching calculation, continue through the particle and nuclear physics pathway. If the result motivates a dated literature or evidence question, carry the brief and its nonclaim into Research. If the computation should become a reusable interactive or executable study, preserve its specification, checks, inputs, runtime, and expected failure cases with the implementation.

A handoff is ready when another researcher can repeat the calculation, recover the decision, identify the nearest nonclaim, and know exactly why the project stopped.

References

• Appelquist, Thomas, and J. Carazzone. “Infrared Singularities and Massive Fields.” Physical Review D 11, no. 10 (1975): 2856–2861. DOI.

• Manohar, Aneesh V. “Introduction to Effective Field Theories.” In Effective Field Theory in Particle Physics and Cosmology: Lecture Notes of the Les Houches Summer School, Volume 108, edited by Sacha Davidson, Paolo Gambino, Mikko Laine, Matthias Neubert, and Christophe Salomon, 47–136. Oxford: Oxford University Press, 2020. DOI. Author manuscript, arXiv:1804.05863v1 (2018).

• National Academies of Sciences, Engineering, and Medicine. Reproducibility and Replicability in Science. Washington, DC: National Academies Press, 2019. DOI and open book.

• Wilson, Greg, Jennifer Bryan, Karen Cranston, Justin Kitzes, Lex Nederbragt, and Tracy K. Teal. “Good Enough Practices in Scientific Computing.” PLOS Computational Biology 13, no. 6 (2017): e1005510. DOI and open article.