Scope, conventions, and status

A scientific claim is useful only when a reader can tell what it says, where it applies, and what supports it. This lesson turns a traced source claim into a bounded evaluation without compressing assumptions, evidence, confidence, or the state of a broader question into one word such as “established.”

Required background. Complete Literature, sources, and claim tracing so that the statement being evaluated is attached to its actual source and context. The heavy-scalar matching example provides helpful technical background, but the derivation below is self-contained.

Evaluate a scientific claim

Use separate fields because each answers a different question:

1. Claim type: Is this a definition, theorem, controlled derivation, numerical result, measurement, inference, or proposal?
2. System and kinematic scope: Which fields, states, observables, dimensions, and ranges of energy or momentum are included?
3. Assumptions: Which dynamical, perturbative, regularity, boundary, or scale-separation conditions make the reasoning valid?
4. Conventions: Which metric, Fourier transform, action, amplitude, and normalization choices affect the displayed formulas?
5. Evidence basis and independence: What calculations, data, or comparisons support the claim, and which failure modes do they genuinely test separately?
6. Confidence: How strongly is the claim supported within its stated scope, and what could change that judgment?
7. Publication or current-use disposition: Is a paper corrected or withdrawn, a dataset superseded, or a software version obsolete? Record this only when it bears on use of the claim.
8. Broader question state: Does the result settle the surrounding scientific question, only one special case, or neither?
9. Evidence cutoff: Through what date, dataset release, or version was mutable evidence checked?

These fields are not a score. High confidence in a narrow tree-level identity does not imply that a broader decoupling problem is closed, and an old publication date does not make a derivation false.

Derive the bounded heavy-scalar claim

Consider two real scalar fields in four spacetime dimensions: a light field $\phi$ of mass $m$ and a heavy field $H$ of mass $M$, with $M\gg m$. The relevant Lorentzian Lagrangian is

$$\mathcal L = \frac12(\partial\phi)^2-\frac12m^2\phi^2 +\frac12(\partial H)^2-\frac12M^2H^2 -\frac g2 H\phi^2 .$$

We ask only for tree-level processes whose channel invariants satisfy $\lvert q^2\rvert\ll M^2$ and remain away from the heavy-particle pole. Integrating the heavy kinetic term by parts gives

$$\mathcal L_H = -\frac12H(\Box+M^2)H-\frac g2H\phi^2 .$$

Completing the square, or equivalently solving the classical equation

$$(\Box+M^2)H=-\frac g2\phi^2 ,$$

produces the exact tree-level nonlocal contribution

$$\Delta\mathcal L_{\mathrm{tree}} = \frac{g^2}{8}\, \phi^2\frac{1}{\Box+M^2}\phi^2 .$$

For fields whose Fourier support obeys $\lvert q^2\rvert/M^2<1$, the inverse operator has a convergent geometric expansion mode by mode:

$$\begin{aligned} \Delta\mathcal L_{\mathrm{EFT}} &= \frac{g^2}{8M^2}\phi^4 -\frac{g^2}{8M^4}\phi^2\Box(\phi^2) +\mathcal O\!\left(\frac{g^2\phi^2\Box^2\phi^2}{M^6}\right). \end{aligned}$$

Thus heavy exchange is represented by local light-field operators ordered by powers of $q^2/M^2\sim E^2/M^2$. This is the low-energy organization developed in effective-field-theory treatments; it does not require the heavy propagator to disappear from the underlying theory (Weinberg 1979, pp. 327–340; Georgi 1993, pp. 209–252).

The error is explicit for one exchange channel. With $x=q^2/M^2$,

$$\mathcal M_{\mathrm{full}}(q^2) = \frac{g^2}{M^2-q^2} = \frac{g^2}{M^2} \left(1+x+x^2+\cdots\right).$$

In four dimensions $[g]=1$, so every term in this amplitude is dimensionless. This is a quick check on both the contact coefficient and the derivative expansion.

Keeping only the contact term gives the exact relative error

$$\frac{\lvert\mathcal M_{\mathrm{full}}-\mathcal M_0\rvert} {\lvert\mathcal M_{\mathrm{full}}\rvert} =\lvert x\rvert ,$$

while also keeping the first derivative correction gives $\lvert x\rvert^2$. For identical-particle scattering, this expansion must hold in every contributing channel: $\lvert s\rvert,\lvert t\rvert,\lvert u\rvert\ll M^2$.

Check the sign and translate the convention

Under the site’s metric and amplitude conventions, the $H\phi\phi$ vertex is $-ig$ and the heavy propagator is $i/(q^2-M^2+i0)$. One channel therefore gives

$$\begin{aligned} i\mathcal M &=(-ig)^2\frac{i}{q^2-M^2+i0}\\ &=\frac{ig^2}{M^2-q^2-i0}, \end{aligned}$$

which fixes the positive leading coefficient in the low-energy amplitude. The local term $+g^2\phi^4/(8M^2)$ has four-point rule

$$i(4!)\frac{g^2}{8M^2} =\frac{3ig^2}{M^2}.$$

That factor of three equals the leading $s$-, $t$-, and $u$-channel contributions. This comparison checks both the sign and the identical-field combinatorics.

In Euclidean momentum the same propagator kernel is

$$\frac{1}{M^2+q_E^2} = \frac{1}{M^2} \left(1-\frac{q_E^2}{M^2}+\cdots\right).$$

Analytic continuation gives $q_E^2=-q^2$, recovering the Lorentzian series $M^{-2}(1+q^2/M^2+\cdots)$. The apparently opposite momentum-correction sign before this translation is representational; a mismatch after translating both the momentum and action conventions is a substantive error.

Filled claim sheet

Claim. For $\lvert q^2\rvert\ll M^2$, tree-level exchange of the heavy scalar in the theory above is reproduced by a sequence of local operators involving only $\phi$. Channel by channel, the first omitted amplitude correction after the contact interaction is bounded at order $E^2/M^2$; cancellations among channels or symmetries can make the combined correction smaller.

Claim type. Controlled analytic derivation: a geometric expansion of a known tree propagator.

System and kinematic scope. Two real scalars in four-dimensional relativistic quantum field theory; amplitudes with light external states; tree level; all relevant exchange-channel invariants small compared with $M^2$; no kinematics near the heavy pole.

Assumptions. $M$ is nonzero and parametrically larger than the external scales; the coupling and field content are those displayed above; the heavy field enters quadratically in the part integrated out; the same low-energy expansion is valid throughout the momentum support being approximated.

Conventions. Lorentzian metric $(+---)$, Fourier modes for which $\Box\mapsto-q^2$, interaction $-gH\phi^2/2$, and $S=1+iT$ amplitude conventions. The Euclidean translation uses $q_E^2=-q^2$.

Evidence basis. First, completing the square gives the exact tree-level nonlocal action. Second, the exchange diagram fixes the pole, sign, and channel structure. Third, the exact geometric-series remainder quantifies the truncation error. The action and diagram checks expose different algebraic mistakes, but they share the same Lagrangian and tree-level assumptions; they are corroborating calculations, not fully independent physical evidence.

Confidence. High for this algebraic statement inside its declared domain. Confidence would fall if the proposed application sampled $\lvert q^2\rvert\sim M^2$, required heavy on-shell production, or relied on uncalculated loop effects. This is not a statistical confidence level.

Publication or current-use disposition. Not applicable to the mathematical validity of this instructional derivation. The cited publisher records establish technical and historical context; no claim here depends on a mutable dataset or software release.

Broader question state. The example resolves how this particular tree-level propagator is expanded at low energy. General decoupling, loop matching, running, and possible nondecoupling effects are outside the result.

Evidence cutoff. The source identities and publisher information cited here were checked through 2026-08-13. The algebraic claim itself is not a survey of research activity through that date.

Name the nearest stronger nonclaim

The nearest tempting overstatement is:

Every heavy particle decouples in every quantum field theory, with all effects suppressed by $E^2/M^2$.

The calculation above does not establish that statement. It does not prove the Appelquist–Carazzone theorem, whose hypotheses and renormalization argument cover a much broader class of theories (Appelquist and Carazzone 1975, pp. 2856–2861). It also does not analyze loop matching, logarithmic running, operator mixing, couplings that scale with $M$, symmetry-breaking effects, anomalies, thresholds, or known nondecoupling regimes. Even when a local expansion exists, symmetries can change the first allowed power correction.

The strongest warranted sentence is the bounded claim in the filled sheet. Naming the nearest stronger nonclaim prevents a correct toy calculation from being used as evidence for a theorem it has not proved.

Copyable claim-evaluation template

Claim:

Claim type:

System and kinematic scope:

Assumptions:

Conventions and translation:

Evidence basis:

Independence of checks:

Confidence within scope:
What would change it:

Publication or current-use disposition, if relevant:

Broader question state:

Evidence cutoff:

Strongest warranted statement:

Nearest stronger statement not warranted:

Keep “not applicable” distinct from “not checked.” The first says a field does not bear on the claim; the second identifies unfinished work.

Exercises

Quantify rather than say “small”

Suppose one channel has $\lvert q^2\rvert/M^2=0.04$. Find the exact relative error of the contact-only approximation and of the approximation that also retains the first derivative correction.

Solution

For the geometric series, the contact-only relative error is exactly $\lvert x\rvert=0.04$, or $4\%$. Retaining $1+x$ leaves relative error $\lvert x\rvert^2=0.0016$, or $0.16\%$. These numbers apply to this channel and tree-level propagator; they do not include loops or errors from other kinematic regions.

Translate the derivative sign

A Euclidean calculation reports

$$\frac{1}{M^2+q_E^2} =M^{-2}-q_E^2M^{-4}+\mathcal O(q_E^4M^{-6}).$$

Does the minus sign contradict the positive $q^2/M^2$ correction in the Lorentzian amplitude?

Solution

No. Under analytic continuation $q_E^2=-q^2$, so $-q_E^2M^{-4}=+q^2M^{-4}$. After the momentum convention is translated, both expressions give

$$\frac{1}{M^2-q^2} =M^{-2}+q^2M^{-4}+\mathcal O(q^4M^{-6}).$$

Comparing the unconverted symbols would create a false sign disagreement.

Classify evidence independence

You verify the contact coefficient by completing the square, by drawing the exchange diagram, and by evaluating the same propagator numerically at ten low momenta. Are these three independent demonstrations that heavy particles generally decouple?

Solution

No. Completing the square and the diagram provide useful cross-checks because they are vulnerable to different sign and combinatorial mistakes, but both use the same Lagrangian and tree approximation. Ten numerical evaluations of the same geometric formula test implementation and truncation behavior, not new dynamics. Together they strongly check the bounded tree-level claim; none supplies independent evidence for a general all-orders decoupling theorem or for a different theory.

Next: reproduce and validate

The completed sheet identifies what a reproduction must hold fixed and which checks can fail independently. Continue to Reproduce and validate a result to turn the bounded claim into an executable comparison with declared tolerances and failure conditions.

References

• Appelquist, Thomas, and J. Carazzone. “Infrared Singularities and Massive Fields.” Physical Review D 11 (1975): 2856–2861. https://doi.org/10.1103/PhysRevD.11.2856.
• Georgi, Howard. “Effective Field Theory.” Annual Review of Nuclear and Particle Science 43 (1993): 209–252. https://doi.org/10.1146/annurev.ns.43.120193.001233.
• Weinberg, Steven. “Phenomenological Lagrangians.” Physica A: Statistical Mechanics and its Applications 96, nos. 1–2 (1979): 327–340. https://doi.org/10.1016/0378-4371(79)90223-1.