Tree-level matching by heavy-field elimination

In Minkowski signature (+,−,−,−), take a real light scalar phi and a real heavy scalar H of mass M, with interaction $-gH\phi^2/2$. The parameter g has mass dimension one. Include any light-sector interactions in a separate Llight. After integration by parts, the H-dependent action is a quadratic form plus a linear source. Its classical equation fixes H in terms of the source using the inverse of Box plus M squared, with the scattering boundary prescription understood.

The relevant solution describes virtual heavy response to light fields. An independent oscillating solution of the homogeneous heavy equation is not retained because the EFT is not meant to describe an external on-shell H particle. Expanding the inverse differential operator is justified when derivatives acting on the source are small compared with M. Substitution back into the action gives half the source contracted with its inverse quadratic operator, not the full linear-source contribution by itself.

If the light quartic convention is $-\lambda\phi^4/4!$, the generated positive coefficient of phi to the fourth shifts lambda downward by three g squared divided by M squared. This is a threshold correction to a dimension-four coupling, not necessarily a higher-dimensional correction. Decoupling does not mean that all heavy dependence is accompanied by higher powers of external energy: renormalizable parameters are also matched between theories.

For identical light scalars, the heavy exchange amplitude contains s, t, and u channels. With the stated interaction, its contribution is a sum of three positive fractions with denominators M squared minus the corresponding invariant. At lowest order it equals three times g squared over M squared, matching the contact contribution. Leaving out two channels would miss a factor of three and would misidentify the quartic threshold correction even if the single-channel propagator expansion were algebraically correct.

Expanding the crossed amplitude gives a two-derivative term proportional to s plus t plus u. It vanishes for the massless on-shell four-point process, although the derivative operator appears in the intermediate action. Integration by parts and the lower-order equations of motion account for this cancellation in a reduced basis. If leading light interactions are present, removing that operator can induce higher-multiplicity interactions; the cancellation is not a statement that the entire operator has no physical role in every process.

The first nonzero energy correction to this particular massless four-point amplitude is proportional to the sum of the squared invariants. Approaching any heavy-channel pole invalidates the local description. All coefficients in this lesson follow from the declared scalar toy theory and should not be promoted to universal signs or size relations for arbitrary ultraviolet completions. A different heavy spin, representation, or interaction changes the matching, while equality of the low-energy observables remains the defining test.