Effective Field Theory

The Wilsonian renormalization group reconceived every QFT as an effective field theory (EFT): a description valid below some energy scale, built from all operators consistent with the symmetries and organized by an expansion in $E/\Lambda$. This page collects the EFT toolkit — top-down and bottom-up construction, power counting, matching and running — and the canonical examples. EFT is arguably the single most important conceptual framework in modern QFT, turning "non-renormalizable" from an insult into a systematic predictive method.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

The core idea

Physics separates by scale: to describe hydrogen you do not need quarks, and to compute $\beta$-decay you do not need the $W$ boson explicitly. An EFT makes this precise. Given a separation of scales $E\ll \Lambda$, write the most general local Lagrangian for the light degrees of freedom, consistent with the symmetries, ordered by operator dimension:

$$\boxed{{\mathcal{L}}_{\text{eff}}={\mathcal{L}}_{\text{renormalizable}}+\sum _i\frac{c_i}{{\Lambda }^{d_i-4}}{O}_i,[O_i]=d_i>4.}$$

The higher-dimension (irrelevant) operators are suppressed by powers of $E/\Lambda$, so at low energy only finitely many matter to any desired accuracy — the theory is predictive despite being non-renormalizable. The Wilsonian insight is that this is not an approximation to something better; it is how all QFT works.

Top-down vs. bottom-up

Top-down (matching). The full high-energy theory is known; integrate out the heavy fields and match onto the EFT by demanding the two theories give the same low-energy observables at the boundary scale $\mu \sim \Lambda$ (Wilsonian matching). The coefficients $c_i$ are then computed from the underlying theory.

Bottom-up. The high-energy theory is unknown; write down all allowed operators with $c_i$ as free parameters to be measured. Deviations of the $c_i$ from zero probe the scale $\Lambda$ of new physics. This is the logic of the Standard Model EFT (SMEFT): extend the Standard Model with all dimension-6 operators and constrain them with data.

The archetype: Fermi theory

Weak $\beta$-decay below the $W$ mass is the paradigmatic EFT. Integrating out the heavy $W$ from the electroweak theory reduces its exchange to a contact four-fermion interaction:

$$\frac{g^2}{m_W^2-q^2}\stackrel{q^2\ll m_W^2}{\to }\frac{g^2}{m_W^2}\equiv \frac{G_F}{\sqrt{2}}\times (\text{const}),{\mathcal{L}}_{\text{Fermi}}=-\frac{G_F}{\sqrt{2}}(\bar{\psi }{\gamma }^{\mu }\psi )(\bar{\psi }{\gamma }_{\mu }\psi ).$$

The four-fermion operator has dimension 6, so $G_F\sim 1/m_W^2$ has negative mass dimension — Fermi theory is non-renormalizable, exactly as expected for an EFT. Its breakdown at $\sqrt{s}\sim m_W$ (amplitudes grow like $G_{F}s$ and violate unitarity) predicted the scale of the $W$ before it was discovered. This is the EFT logic in its purest form: the cutoff of the effective theory is the mass of the new physics.

Chiral perturbation theory

The low-energy EFT of QCD is written not in quarks and gluons (strongly coupled, confined) but in the pions — the pseudo-Goldstone bosons of spontaneously broken chiral symmetry. Chiral perturbation theory organizes their interactions as an expansion in $p/{\Lambda }_{\chi }$ with ${\Lambda }_{\chi }\sim 1$ GeV, governed entirely by the symmetry-breaking pattern. It computes low-energy pion physics that perturbative QCD cannot touch — a striking demonstration that an EFT's degrees of freedom need not be those of the underlying theory.

Matching and running

An EFT calculation spanning scales combines two operations:

1. Matching at each heavy threshold $\mu \sim M$: fix the EFT coefficients so full and effective theories agree (a one-time boundary condition).
2. Running the coefficients with their own RG equations between thresholds, resumming the large logarithms $\ln (M/E)$.

The decoupling theorem guarantees this works — heavy particles leave only renormalizations plus $1/M$-suppressed operators. This match-and-run tower (electroweak → Fermi → chiral, or SM → SMEFT) is how precision predictions are made across widely separated scales.

Why EFT is the modern viewpoint

• Every QFT is an EFT, including the Standard Model (an EFT below the Planck or some new-physics scale) and even general relativity (a perfectly good EFT of gravity below $M_{\text{Planck}}$, despite being non-renormalizable).
• Renormalizability is demystified: it is just the statement that the leading terms dominate at low energy (Wilsonian).
• New physics is parametrized model-independently by higher-dimension operators.

Summary

• An EFT describes physics below a scale $\Lambda$ using all symmetry-allowed operators, ordered by $E/\Lambda$.
• Top-down: integrate out heavy fields and match; bottom-up: measure the coefficients (SMEFT).
• Archetypes: Fermi theory (integrating out $W$), chiral perturbation theory (pions from QCD).
• Match-and-run spans scales; every QFT is an EFT.

Where this leads

• The RG foundation: Wilsonian RG and EFT.
• Gravity as an EFT: QFT in curved spacetime.
• The scale-invariant fixed points EFT flows between: conformal field theory.

References

• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 12; Vol. 2, Ch. 19.
• Georgi, Ann. Rev. Nucl. Part. Sci. 43, 209 (1993).
• Manohar, Introduction to Effective Field Theories (Les Houches, 2017).
• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 12.