Wilsonian matching and power counting

Work in four-dimensional Euclidean spacetime with a real light field $\phi$, a heavy field $H$, and interaction $gH\phi^2/2$. Both fields have mass dimension one, so $g$ has dimension one. The heavy quadratic operator is $A=M^2-\partial^2$, whose momentum eigenvalue is positive $M^2+Q^2$. Complete the square in $H$ with source $J=g\phi^2/2$. Gaussian integration produces a light-field term with a minus sign, because the completed square subtracts half of $JA^{-1}J$.

The determinant is independent of $\phi$ in this deliberately simple Gaussian model and only changes the vacuum normalization. Extra interactions involving $H$ would make further terms possible. The negative quartic contribution here does not assert that an isolated light potential is globally stable: the full model must contain appropriate stabilizing interactions and be used around its stated vacuum. Euclidean action coefficients also should not be copied into Minkowski scattering rules without translating conventions.

In a channel with Euclidean transfer $Q$, expand the positive kernel $g^2/(M^2+Q^2)$ as a geometric series. This isolates a dimensionless coefficient $g^2/M^2$ followed by corrections in $Q^2/M^2$. In position space the first term multiplies $\phi^4$, a dimension-four operator. The next can be written using derivatives on $\phi^2$ and has dimension six, so its coefficient has mass dimension minus two. Integration by parts can change its appearance but cannot change a matched observable.

Every relevant external invariant must be small compared with the heavy scale. Low total energy alone is insufficient if a different channel carries a large transfer or if another nearby threshold was omitted. A geometric expansion at small spacelike momentum has an explicitly calculable remainder in this example. More general effective theories estimate the next allowed order using power counting, symmetries, and coefficient sizes. An unexpectedly enhanced coefficient is information to investigate, rather than permission to claim a universal tiny error.

Matching fixes coefficients at a chosen scale by equating the same low-energy quantities in two descriptions. Renormalization-group running then transports those coefficients between scales within the effective theory. When light loops occur on both sides, common infrared contributions must cancel in the matching difference; otherwise long-distance physics would be incorrectly counted as a new short-distance coefficient. Operators related by field redefinitions can trade coefficients, so the chosen basis and renormalization prescription belong in a reproducible result.

Our Gaussian elimination is exact for the heavy field in the stated action, but evaluating the remaining light-field theory at tree level is an additional approximation. Report separately which derivative terms are retained, which loop orders are evaluated, and how measured inputs were fixed. This bookkeeping makes the effective theory useful for precision work: increasing one order has a stated purpose and test. It also exposes the boundary where a heavy degree of freedom must be restored rather than approximated by more local terms.