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Integrating out a field means performing its path integral while retaining the other fields as backgrounds. The result is generally a nonlocal quantum effective action; for a field much heavier than all external momenta, a derivative expansion turns it into local effective interactions.

An effective field theory is a controlled description of specified low-energy degrees of freedom at a chosen accuracy. It does not require knowing all physics at arbitrarily short distances. If new particles or strong dynamics enter at a scale $M$, processes with characteristic energy and momentum transfers $E\ll M$ can be described using the light fields and interactions consistent with their symmetries. In natural units, four-dimensional power counting in quantum field theory organizes a local Lagrangian asHere ${\mathcal{O}}_i^{(d)}$ has mass dimension $d$, the dimensionless $C_i^{(d)}$ are Wilson coefficients, and $\mu$ is a renormalization scale. In a relativistic vacuum with canonical fields, a typical insertion of a dimension-$d$ interaction contributes an additional power $(E/M{)}^{d-4}$, multiplied by its couplings and any light mass ratios. Additional expansions, such as a loop expansion, must also be specified. Symmetry or on-shell identities can postpone a particular observable's first correction.

The theory is useful when its retained states and expansion parameters adequately describe the experiment. It ceases to be a reliable truncated local description near an omitted particle's production threshold, near a heavy propagator pole, or where the retained dynamics become too strongly coupled for an assumed perturbative expansion. A light particle cannot be removed merely because one wants fewer variables: its propagation can produce nonanalytic momentum dependence that must be represented by retained light fields. Heavy-field decoupling can shift renormalizable masses and couplings as well as generate suppressed interactions; those low-energy parameters must be measured or matched, not assumed unchanged. The ultraviolet cutoff is an organizational scale, while a calculation may use a regulator other than a hard momentum cutoff.

Construction starts by identifying the light particles, the hierarchy of scales, and the exact or approximate symmetries relevant to the problem. Form all allowed local field operators through the desired order in power counting in quantum field theory. Choose an operator basis in effective field theory: remove equivalent terms by integration by parts, algebraic identities and allowed field redefinitions. Operators proportional to lower-order equations of motion can be redundant operators for on-shell amplitudes, provided coefficients are consistently transformed. This reduces bookkeeping without imposing additional physics assumptions.

If an ultraviolet completion is known, determine the Wilson coefficients by matching in effective field theory: calculate low-energy amplitudes or appropriate correlation functions in both descriptions using the same infrared conventions, and adjust coefficients so they agree to the chosen order. Equivalently, integrating out a field performs its path integral while retaining the light fields as backgrounds. Heavy propagators have an analytic expansion below their singularities, producing a derivative expansion; heavy loops also generate local terms and logarithms in their coefficients. Without a specified ultraviolet completion, the coefficients are parameters to be constrained by data.

An effective field theory remains predictive even when it contains nonrenormalizable interactions. At each fixed order in energy and loops there are finitely many required coefficients and counterterms. Renormalization absorbs divergences into that order's allowed operators. The renormalization group evolves the Wilson coefficients between matching and measurement scales, compensating scale dependence in matrix elements and, when appropriate, resumming large logarithms. The truncation error in effective field theory is estimated from the first omitted orders, under a stated coupling-size assumption; it is separate from parameter uncertainty and cannot be inferred merely by writing down infinitely many terms.

A concrete example is heavy scalar exchange in effective field theory. Take a light real scalar field $\phi$ and a heavy real scalar field $S$, withThe coupling $a$ has mass dimension one. The light-field symmetry is $\phi \mapsto -\phi$. For $m^2\ge 0$ and $\lambda >3a^2/M^2$, the displayed scalar potential is bounded below: completing the square in $S$ leaves a positive light quartic. Thus the example can be treated as a stable theory around $S=\phi =0$, with weak enough couplings for the tree approximation.

At tree level the heavy equation of motion is $(M^2+\square )S=-a{\phi }^2/2$. Substitute its solution back into the action, including both its quadratic and source terms, to obtainThis is a derivative expansion valid for small momentum transfers, not an exact local replacement near the pole. The first term gives ${\lambda }_{\mathrm{eff}}=\lambda -3a^2/M^2$ in the $-{\lambda }_{\mathrm{eff}}{\phi }^4/4!$ convention. After integration by parts, the next term is $+a^2({\partial }_{\mu }{\phi }^2)({\partial }^{\mu }{\phi }^2)/(8M^4)$; it is a dimension-six local operator. Writing $a=g_{\ast }M$ puts its coefficient in the usual $g_{\ast }^2/M^2$ form. These coefficients are a tree-level matching in effective field theory result.

One can directly check the matching through the on-shell scattering amplitude for $\phi \phi \to \phi \phi$. With $s,t,u$ the Mandelstam variables, the full theory has three heavy-exchange channels:For $|s|,|t|,|u|\ll M^2$,The effective field theory reproduces these terms in order: a shifted quartic, then local derivative interactions. Since $s+t+u=4m^2$ on shell, the first derivative correction is a light-mass-dependent constant; it vanishes for $m=0$. This illustrates why an operator basis in effective field theory can trade some derivative operators for mass-dependent or higher-field interactions using field redefinitions. For massless external particles, the first nonconstant correction in this four-point tree amplitude starts at the following order.

The example exhibits the central logic: keep the light field, encode virtual heavy exchange in matched local coefficients, and control the error by expanding in momentum divided by the heavy scale. No heavy particle is actually produced in the domain of the approximation. Near $s=M^2$, the full propagator is resonant and the truncated effective field theory fails; retaining $S$ or adopting a different description is then necessary. Light loops are computed within the effective field theory, while higher-order matching supplies the corresponding heavy corrections.

A gauge boson couples to the conserved current of a charged fermion throughwith the appropriate chiral projector when the gauge representation is chiral. At momentum transfer $|q^2|\ll M_X^2$, its propagator can be expanded aswhere the longitudinal term drops for a conserved current. The operation of integrating out a field, applied to $X_{\mu }$, gives the local four-fermion interactionup to the normalization used for $J^{\mu }$. Hence the effective Fermi coupling scales as $G\sim g_X^2/M_X^2$. For Standard Model charged currents, the conventional normalization is

Let $K_M=-{\partial }^2+M^2$ and let ${\Delta }_M=K_M^{-1}$ be its Euclidean quantum field theory propagator. With the source-sign convention suited to the expression in the question, definewhere $N_0$ makes $Z_0[0]=1$. Since inserting $\chi (x)$ is equivalent to acting with $-\delta /\delta J(x)$, the integral over $\chi$ gives

Put $\rho ={\phi }^{†}\phi$. The heavy-field equation of motion isThis solution describes the low-energy sector without an independent incoming heavy-particle wave. It is valid when all external momentum invariants and light masses squared are small compared with $M^2$.

After an integration by parts, the terms involving the heavy field are $-\frac{1}{2}\varphi (\square +M^2)\varphi -g\rho \varphi$. Substituting the solution into both terms, rather than the source term alone, givesHence, with the question's sign convention for the local quartic interaction,This is integrating out a field followed by a derivative expansion. The negative quartic means attraction. It also makes the truncated potential unbounded at large fields; minimizing the original potential over $\varphi$ reveals the same negative quartic. Thus neither displayed model, without extra stabilizing interactions, supplies a globally stable nonperturbative vacuum. The matching describes perturbative low-energy scattering about the local vacuum. The simple equation-of-motion matching fixes the leading tree-level coefficient; precision quantum matching requires consistent loop renormalization and the additional effective operators at the requested accuracy.