Effective-Theory Architecture Atlas

An effective theory should be chosen from the physical problem, not from a familiar acronym. Start with the observable and state, identify the active degrees of freedom and scale hierarchy, specify how symmetries act, and only then choose the power counting, matching data, and uncertainty model. This chapter is organized around that decision: a common architecture map comes first, followed by bounded cards for Goldstone, few-body, heavy-particle, nonrelativistic, electroweak, gravitational, hydrodynamic, and open-system theories.

The cards explain construction and domain. They do not reproduce coefficient catalogs, current fits, process phenomenology, or full nonequilibrium dynamics; each of those is linked to the relevant subject page.

Enter through the physical hierarchy

At finite accuracy, an EFT is predictive only after more has been fixed than its field names. A useful architecture card records nine connected choices:

Card field	Question it must answer
Degrees of freedom	Which fields, collective variables, modes, or density-matrix variables remain dynamical?
Hierarchy	Which ratios are small, and over what kinematic or state domain?
Symmetry realization	Which symmetries constrain the theory, and are they linear, nonlinear, gauge, spacetime, or contour symmetries?
Kinematics and locality	Is the expansion relativistic, heavy-particle, nonrelativistic, light-cone, derivative, background-field, or temporally nonlocal?
Power counting	Which operators, loops, and iterations contribute at each declared order?
Matching or input	Which coefficients come from a parent theory, data, symmetry, or nonperturbative calculation?
Observables	Which quantities can the EFT predict in its domain?
Uncertainty and breakdown	What is omitted, how is its size tested, and what observation ends the expansion?
Application route	Where are coefficients, phenomenology, dynamics, or evidence developed in detail?

When a local action description applies, the finite-order structure may be written schematically as

$$\mathcal L_{\mathrm{EFT}}^{[N]} =\sum_{\nu\leq N}\sum_i C_i^{(\nu)}(\mu)\,\mathcal O_i^{(\nu)}(\mu) +O(\epsilon^{N+1}).$$

Here $\epsilon$ is the architecture-specific expansion parameter and $\nu$ is its homogeneous order; neither is necessarily canonical dimension. For hydrodynamic or open-system theories, the object being expanded may instead be a constitutive relation, a doubled-field action, or an influence kernel. Identifying the degrees of freedom, fixing a power counting, and determining a finite set of inputs are logically distinct steps, as emphasized in Manohar 2018, §§ 3.1–3.3, pp. 11–15, Open PDF and Burgess 2007, § 3.3, pp. 32–34, Open PDF.

Two formulations belong to the same architecture when a controlled field or basis transformation preserves the state, expansion, and observables. They belong to different architectures when the transformation changes the active modes, symmetry realization, locality or contour structure, or homogeneous counting.

Check your preparation

The overview has no mandatory prerequisite. Use these tasks to find the shortest repair route before entering a specialized card.

Can you…	If yes	If unsure, repair here
name a dimensionless expansion parameter and a finite-order remainder?	Begin with the architecture map.	Effective Field Theory as a Controlled Expansion
distinguish matching from RG evolution?	Enter heavy-particle, nonrelativistic, electroweak, or gravity routes.	Integrating Out Heavy Fields
distinguish an operator list from an independent basis?	Enter the SMEFT/HEFT route.	From Operator Lists to Independent Bases
decide when a momentum region requires a homogeneous EFT mode?	Enter the nonrelativistic route or compare with SCET.	Modes, Virtualities, and EFT Scale Separation
identify when shallow poles promote repeated interactions?	Enter the nuclear and few-body route.	Nonperturbative Iteration, Shallow Scales, and Power-Counting Consistency
work with nonlinear symmetry coordinates?	Enter the chiral or HEFT routes.	Cosets and Nonlinear Realizations
distinguish an in–in contour from an in–out amplitude?	Enter the open-system route; it also helps with fluctuating hydrodynamics.	Closed-Time-Path Grammar

The gravity card additionally uses Levi–Civita Connections, Geodesics, and Riemann Curvature. The hydrodynamic card uses Current Sources and Generating Functionals.

Choose a route

First encounter. Read A Map of Effective-Theory Architectures after the controlled-expansion repair above. The exit is a completed card and a defensible rejection of at least one superficially plausible alternative.

Goldstone and few-body systems. Use the map and nonlinear-realization repair, then read Chiral Effective Theory and Nonlinear Symmetry. If shallow scattering scales force promoted iteration, continue to Nuclear and Few-Body EFT Architecture; the chiral card is helpful there but not a hard dependency.

Heavy and nonrelativistic systems. After the map, the Dirac field, and heavy-field matching, read Heavy-Particle EFT and HQET Architecture. Continue to NRQED, NRQCD, and Potential EFT Architecture only when a heavy pair introduces the additional momentum and kinetic-energy scales.

Electroweak or gravitational deformations. For electroweak symmetry realized linearly or nonlinearly, combine the map with operator-basis symmetry constraints and enter SMEFT and HEFT: Architecture and Domain. For a derivative expansion constrained by diffeomorphism invariance, combine the map, EFT power counting, and curvature conventions before Effective Field Theory of Gravity: Architecture and Power Counting.

Conserved slow variables or reduced dynamics. Combine the map with source-defined Ward identities for Hydrodynamic Effective-Theory Architecture. If the system is obtained by tracing an environment, begin instead from closed-time-path grammar and read Open-System Effective-Theory Architecture and Consistency Conditions; hydrodynamics is helpful context, not a requirement.

These are dependency-aware reading routes, not a ranking of the theories. A focused reader may begin at a specialized card once its hard preparation is in place.

What distinguishes nearby architectures

Architecture selection turns on the origin of the expansion, not on overlapping particle content.

Nearby choices	Decisive question	Architectural consequence
Chiral EFT versus nuclear/few-body EFT	Are derivative and explicit-breaking scales sufficient, or do shallow poles force selected interactions to be iterated?	The latter changes the homogeneous order and the regulator-independence test.
HQET versus NRQED/NRQCD	Is there one heavy source with residual momentum, or a slow heavy pair with $\lvert\mathbf p\rvert\sim Mv$ and $E\sim Mv^2$?	The pair requires distinct momentum and kinetic-energy scales and, eventually, potential modes.
SMEFT versus HEFT	Is the light Higgs organized with a linear electroweak doublet and an analytic canonical-dimension expansion, or independently of nonlinear Goldstone coordinates?	The symmetry realization and counting determine different allowed expansions; one is not merely a basis change of the other.
Hydrodynamic versus open-system EFT	Are conserved densities the autonomous slow variables, or has an environment been traced out?	The second problem requires doubled variables, normalization and positivity constraints, and possibly memory kernels.

This comparison also gives a failure test. If the proposed EFT cannot state which side of its decisive question applies, its name does not yet define a controlled approximation.

Exact chapter guide

1. A Map of Effective-Theory Architectures. Compares all selected frameworks using the common card and teaches selection from hierarchy, fields, symmetry, counting, matching, and state. It requires the controlled-expansion page; truncation diagnostics, heavy-field matching, and basis construction are useful depth. Continue to the card selected by the map.

2. Chiral Effective Theory and Nonlinear Symmetry. Builds the Goldstone-field architecture from nonlinear transformations, explicit-breaking spurions, and derivative counting. The map and coset construction are hard preparation. Nuclear-force promotion is deferred to the next card, while pion and hadron phenomenology belongs to the gauge-theory volume.

3. Nuclear and Few-Body EFT Architecture. Organizes large scattering lengths, promoted interactions, few-body forces, regulator independence, and order-by-order uncertainty. Its hard entry is nonperturbative iteration; the chiral card is recommended when pions are active. Nuclear phenomenology, strong dynamics, finite-volume evidence, and many-body applications remain with their subject volumes.

4. Heavy-Particle EFT and HQET Architecture. Separates $p^\mu=Mv^\mu+k^\mu$, derives the projected heavy field and inverse-mass expansion, and exposes reparameterization constraints. It requires the map, the Dirac field, and heavy-field matching. A slow heavy pair is handed to the nonrelativistic card.

5. NRQED, NRQCD, and Potential EFT Architecture. Separates mass, soft momentum, kinetic energy, and potential or ultrasoft modes using velocity counting. It follows the heavy-particle card, mode construction, and matching. Named QED/QCD coefficients and bound-state phenomenology belong downstream.

6. SMEFT and HEFT: Architecture and Domain. Chooses between linear and nonlinear electroweak realizations, states flavor and input assumptions, and connects counting to matching and truncation. It requires the map and symmetry constraints on operator bases. Complete bases, global fits, coefficient limits, and precision observables belong to the Standard Model treatment.

7. Effective Field Theory of Gravity: Architecture and Power Counting. Organizes the diffeomorphism-invariant derivative expansion below the gravitational cutoff and carefully distinguishes the four-dimensional pure-gravity one- and two-loop statements. It requires the map, power counting, and curvature conventions. Curved-background applications and ultraviolet quantum-gravity proposals are separate destinations.

8. Hydrodynamic Effective-Theory Architecture. Starts from conserved slow variables, constitutive relations, frame choice, derivative counting, and fluctuation constraints. It requires the map and source-defined currents; closed-time-path grammar is recommended. Full transport and nonequilibrium dynamics belong to the thermal and nonequilibrium volume.

9. Open-System Effective-Theory Architecture and Consistency Conditions. Tests a bounded doubled-field kernel for normalization, Hermiticity, dissipation, noise, and the assumptions behind a Markovian limit. Closed-time-path grammar is hard preparation and hydrodynamics is useful context. Environment tracing, general influence actions, memory dynamics, and Lindblad applications are developed downstream.

A chapter-scale selection test

The central decision can be run as a short sequence:

1. Declare the observable, state, kinematic window, and desired accuracy.
2. List the modes or collective variables that can become on shell or remain slow in that window.
3. State every hierarchy as a dimensionless ratio and identify the limit in which it vanishes.
4. Specify the symmetry realization, locality or contour structure, and field normalization.
5. Derive a homogeneous counting for operators, loops, and any required iterations.
6. Identify matching data or empirical inputs, then keep running distinct from matching.
7. Name the invariant observable check, the truncation diagnostic, and the first breakdown signal.
8. Route coefficients, dynamics, phenomenology, and evidence to their application pages.

A proposal fails this test if it supplies only an operator list, a cutoff, or a model name. It also fails if an omitted mode produces a retained-order singularity, if renormalization generates an uncounted operator, or if resummation promotes terms without a new consistency analysis.

Review the chapter

Use the following checks to test whether the architecture choice is physically complete and whether nearby alternatives have been rejected for stated reasons.

Check	Successful response	Repair
Classify a heavy-light meson and a near-threshold heavy pair.	Uses residual-momentum counting for the first and separate $Mv$ and $Mv^2$ scales for the second; does not classify by the word “heavy” alone.	Revisit the heavy-particle and nonrelativistic cards.
Reject one of SMEFT or HEFT for a stated light spectrum.	Names the symmetry realization and Higgs-coordinate assumption that fails, then states the counting consequence.	Revisit the architecture map and SMEFT/HEFT card.
Diagnose a proposed fluid EFT with no frame or derivative order.	Identifies the missing constitutive convention and explains why transport data cannot yet be compared.	Revisit the hydrodynamic card.
Test a local Markovian open-system approximation.	States the scale separation behind short memory and checks normalization and positivity in the declared domain.	Revisit closed-time-path grammar and the open-system card.
Transfer the card to an unfamiliar low-energy theory.	Supplies all nine card fields, identifies at least one competing architecture, and gives a falsifiable breakdown condition.	Return to the map and the controlled-expansion page.

Passing these checks means being able to choose and reject architectures for stated reasons. It does not certify a phenomenological calculation or replace the specialist validation developed in another volume.

Connections to applications

The Standard Model and gauge-theory volume develops named coefficients, processes, fits, and particle or nuclear phenomenology. Nonperturbative QFT develops strong dynamics and bound-state mechanisms, while Lattice and Hamiltonian Methods develops numerical continuum and finite-volume evidence. Thermal and Nonequilibrium QFT develops full hydrodynamic and open-system dynamics, and Many-Body and Quantum Matter treats finite-density and emergent applications.

QFT in Curved Spacetime develops gravitational EFT on curved backgrounds; Holography and Quantum Gravity treats ultraviolet and quantum-gravity interpretations. Within this volume, Naturalness, Scale Sensitivity, and Emergence asks a different question: how threshold sensitivity, symmetry protection, tuning measures, and effective variables should be interpreted once an architecture has already been chosen.

References

• Burgess, C. P. (2007), “Introduction to Effective Field Theory,” Annual Review of Nuclear and Particle Science 57, 329–362. DOI. Open PDF.
• Manohar, A. V. (2018), Introduction to Effective Field Theories, Les Houches lecture notes, arXiv:1804.05863 [hep-ph]. arXiv. Open PDF.