Book Notes / Quantum Field Theory and the Standard Model

Matthew D. Schwartz / Reading notes

Removing a field does not remove its effects

An effective theory reduces the variables needed to make predictions in a specified regime. The replacement must reproduce the relevant matrix elements or correlation functions of the more detailed theory to the stated accuracy. That requirement is called matching. Omitting an inaccessible particle is insufficient if its interactions still affect the particles being measured.
Consider a heavy scalar field φ with mass M, coupled through λφJ to the fermion bilinear J = ψ̄ψ. At tree level, its field equation is (□ + M²)φ = λJ. Solving this equation and substituting back generates the interaction (λ²/2) J(□ + M²)⁻¹J among the retained fermions.
The heavy field has disappeared from the explicit variables, but its mass and coupling remain in the interaction. The inverse differential operator also connects the current at different spacetime points. Eliminating a variable has produced a nonlocal description; a local approximation requires an additional argument.

For momentum transfer q, the heavy-field kernel is 1/(M² − q²). If |q²|/M² is small, expand it as M⁻²[1 + q²/M² + (q²/M²)² + …]. Each successive term becomes a higher-derivative interaction in position space. The local theory retains increasingly fine corrections to an interaction it cannot resolve directly.
The leading term gives λ²J²/(2M²). Its dimensions provide a quick check: in four spacetime dimensions, a fermion field has mass dimension 3/2, so J² has dimension six. The coefficient's inverse mass squared returns the Lagrangian term to dimension four. The excluded scale supplies the suppression needed for the retained interaction.
The approximation has a testable error. Write r = q²/M². Keeping the kernel through rᴺ leaves a relative kernel error of |r|ᴺ⁺¹. At |r| = 0.01, the contact term alone has 1% error; including the first momentum correction reduces it to 0.01%. “Low energy” becomes useful when the ratio and retained order are specified.
These percentages describe this propagator kernel, not automatically a scattering cross section. Other amplitude contributions can interfere or cancel, and the cross section depends on the squared total amplitude. The error of one building block does not by itself establish the relative error of the observable.

Near q² = M², the full kernel has a pole, and a few local terms cannot reproduce it. The heavy particle must be treated explicitly to describe that regime. Adding more detail to the low-momentum approximation does not remove its domain of validity.
The expansion can also fail without approaching a physical pole. At q² = −M², the exact kernel is finite, but the geometric expansion around zero no longer converges. Absence of an on-shell heavy particle does not guarantee that this particular derivative expansion is useful. Its controlling condition is the magnitude of the momentum ratio.
Momentum order and quantum-loop order are separate. The classical field-equation substitution gives the tree-level matching considered here; lowering the momentum does not establish that loop corrections are negligible. A prediction must state which terms were retained in each expansion before its accuracy can be assessed.
The value of the simpler theory is therefore conditional and precise: it reproduces selected predictions without carrying degrees of freedom that the chosen regime does not require. Matching establishes what survives; the expansion identifies what was discarded and when those omissions matter.