Separation of scales and local expansions

Use natural units with ħ and c equal to one, so masses, energies, and inverse lengths share one unit. Suppose a process has typical external momentum Q and an omitted heavy excitation has mass M. A local derivative expansion requires Q well below the relevant heavy singularity. With several masses or external invariants, every ratio entering the expansion must be controlled; one small center-of-mass energy does not automatically make a large momentum transfer harmless.

A heavy propagator illustrates the mechanism. Far below its pole, expand its momentum dependence as a geometric series. A constant term becomes a contact interaction, while successive powers of momentum become derivatives acting on local field products. The expansion does not remove the heavy contribution completely: its couplings and mass appear in the contact coefficients. Several distinct microscopic theories can generate the same first few coefficients, so a low-energy fit usually does not reconstruct a unique heavy spectrum.

A light field can go on shell within the low-energy domain and produces poles or branch cuts that no finite polynomial in momentum can reproduce. It should therefore remain dynamical when those processes matter. Light loops can generate logarithms and imaginary parts in the effective theory even though the short-distance matching coefficients are local. Locality of the Wilson expansion does not mean that every predicted amplitude is an analytic polynomial of external momentum.

A derivative expansion is also different from simply deleting high Fourier modes of the retained fields. In a mass-independent renormalization scheme, EFT loop integrals need not have a literal momentum cutoff at M. Matching and counterterms account consistently for short-distance sensitivity. The physical claim is that finitely many operators determine predictions through a chosen order at low external scales. If a previously omitted state becomes nearly on shell, it may need to be introduced explicitly rather than absorbed into more coefficients.

When a ratio is small, a low-order description can still fail a demanding precision goal. Conversely, an omitted contribution can be unimportant at one observable's accuracy but become the leading contribution when a symmetry suppresses lower orders. A practical specification names the observable, the retained orders in every expansion, and the expected first omitted terms. This also separates truncation error from experimental error and from uncertainty in fitted coefficients.

Approaching a heavy threshold makes successive derivatives less suppressed and eventually invalidates the local expansion. A large coefficient can signal a lower effective breakdown scale, an enhanced coupling, or a fine-tuned low-energy structure requiring reorganized counting. It does not follow that every coefficient must be one. The geometric example provides an unusually transparent diagnostic because its exact answer is known; in applications without a known full theory, one must test convergence and justify coefficient-size assumptions independently.