= Solution
An \b[<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> as
$$
\mathcal L_{\mathrm{EFT}}=\mathcal L_{d\leq4}+\sum_{d>4}\sum_i\frac{C_i^{(d)}(\mu)}{M^{d-4}}\mathcal O_i^{(d)}.
$$
Here $\mathcal O_i^{(d)}$ 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> $\varphi$ and a heavy real <scalar field> $S$, with
$$
\mathcal L_{\mathrm{UV}}=\frac12(\partial\varphi)^2-\frac{m^2}{2}\varphi^2-\frac\lambda{4!}\varphi^4
+\frac12(\partial S)^2-\frac{M^2}{2}S^2-\frac a2 S\varphi^2,\qquad M\gg m,E.
$$
The coupling $a$ has mass dimension one. The light-field symmetry is $\varphi\mapsto-\varphi$. For $m^2\geq0$ 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 $S=\varphi=0$, with weak enough couplings for the tree approximation.
At tree level the heavy <equation of motion> is $(M^2+\Box)S=-a\varphi^2/2$. Substitute its solution back into the action, including both its quadratic and source terms, to obtain
$$
\Delta\mathcal L_{\mathrm{eff}}=\frac{a^2}{8}\varphi^2\frac1{M^2+\Box}\varphi^2
=\frac{a^2}{8M^2}\varphi^4-\frac{a^2}{8M^4}\varphi^2\Box\varphi^2+\frac{a^2}{8M^6}\varphi^2\Box^2\varphi^2+\cdots.
$$
This is a <derivative expansion> valid for small momentum transfers, not an exact local replacement near the pole. The first term gives \b[$\lambda_{\mathrm{eff}}=\lambda-3a^2/M^2$] in the $-\lambda_{\mathrm{eff}}\varphi^4/4!$ convention. After <integration by parts>, the next term is $+a^2(\partial_\mu\varphi^2)(\partial^\mu\varphi^2)/(8M^4)$; it is a dimension-six local operator. Writing $a=g_*M$ puts its coefficient in the usual $g_*^2/M^2$ 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 $\varphi\varphi\to\varphi\varphi$. With $s,t,u$ the <Mandelstam variables>, the full theory has three heavy-exchange channels:
$$
\mathcal M_{\mathrm{UV}}=-\lambda+a^2\left(\frac1{M^2-s}+\frac1{M^2-t}+\frac1{M^2-u}\right).
$$
For $|s|,|t|,|u|\ll M^2$,
$$
\mathcal M_{\mathrm{UV}}=-\lambda+\frac{3a^2}{M^2}+\frac{a^2(s+t+u)}{M^4}
+\frac{a^2(s^2+t^2+u^2)}{M^6}+\cdots.
$$
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.
\b[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.
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