Fact-checked by Grok 2 weeks ago

Effective action

In quantum field theory, the effective action is a fundamental functional that generalizes the classical action by incorporating all quantum corrections, serving as the generating functional for one-particle irreducible (1PI) correlation functions.^[1] Introduced by Julian Schwinger in 1954,^[2] it is defined through the Legendre transform of the connected generating functional W[J]: \Gamma[\phi_c] = W[J] - \int d^4x \, J(x) \phi_c(x), where \phi_c(x) is the vacuum expectation value of the field induced by the external source J(x).^[3] Varying the effective action with respect to the classical field \phi_c yields the quantum effective field equations, which encapsulate the full dynamics of the quantized theory.^[1] The effective action arises from the path integral formulation of the generating functional Z[J] = e^{iW[J]}. In the limit of constant background fields and vanishing sources, the effective potential V_{\text{eff}}(\phi_c), derived from the effective action, includes quantum loop corrections, such as the one-loop Coleman-Weinberg potential.^[3] It is stationary at the physical vacuum expectation value, where \delta \Gamma / \delta \phi_c(x) = -J(x) = 0. Although the standard effective action depends on gauge choices, Vilkovisky's formulation ensures gauge invariance in theories with symmetries.^[1] The effective action plays a central role in renormalization, rendering 1PI Green's functions finite by absorbing ultraviolet divergences into counterterms.^[1] It underpins effective field theories (EFTs), which describe low-energy physics by integrating out high-energy degrees of freedom, as in the Wilsonian approach. Applications include Yang-Mills gauge theories for strong interactions, effective descriptions in quantum gravity and cosmology, and phenomena in condensed matter physics like superconductivity. In non-perturbative contexts, such as Schwinger pair production in strong fields, the effective action computes exact quantum effects like vacuum decay.^[4]

Fundamentals

Definition and Motivation

In quantum field theory, the effective action \Gamma[\phi] is defined as the Legendre transform of the connected generating functional W[J], given by

\Gamma[\phi] = W[J] - \int d^4x \, J(x) \phi(x),

where the background field \phi(x) is the expectation value \phi(x) = \frac{\delta W[J]}{\delta J(x)} evaluated at the source J that yields the desired \phi, and the inverse relation is J(x) = \frac{\delta \Gamma[\phi]}{\delta \phi(x)}.^[5] This functional encapsulates the quantum-corrected action for a classical background field \phi after integrating out all quantum fluctuations in the path integral formulation Z[J] = \int \mathcal{D}\tilde{\phi} \exp\left(i S[\tilde{\phi}] + i \int J \tilde{\phi}\right), with W[J] = -i \ln Z[J] (in units where \hbar = 1).^[6] Physically, \Gamma[\phi] generates all one-particle-irreducible (1PI) correlation functions via functional derivatives, providing a concise summary of the theory's quantum dynamics.^[5] The effective action incorporates all quantum corrections beyond the classical action, serving as the generator of 1PI Green's functions that resum diagrams in perturbation theory. A key feature is that varying the effective action with respect to the background field yields the quantum effective field equations \frac{\delta \Gamma}{\delta \phi(x)} = -J(x), which encapsulate the full dynamics of the theory when the source vanishes J=0, determining the physical vacuum and possible symmetry breaking. This structure ensures that symmetries of the underlying theory constrain \Gamma[\phi], preserving physical consistency.^[6] A representative example is provided by scalar field theory, where the effective action \Gamma[\phi] for a constant background \phi yields the effective potential V_{\text{eff}}(\phi) = -\Gamma[\phi]/\text{Vol}, incorporating vacuum energy shifts and interaction corrections that determine the theory's ground state and stability at low energies.^[5]

Relation to Path Integrals

In the path integral formulation of quantum field theory, the generating functional Z[J] serves as the foundational object for computing correlation functions and is expressed as

Z[J] = \int \mathcal{D}\phi \, \exp\left( i \left( S[\phi] + \int J \phi \right) \right),

where S[\phi] is the classical action, \phi represents the field configurations, and J is an external source coupled to the field. This integral over all possible field paths encodes the quantum dynamics, with Z[J] generating connected correlation functions via functional derivatives with respect to J. To derive the effective action \Gamma[\phi], one employs the background field method by splitting the field into a fixed background configuration \phi and quantum fluctuations \eta, such that the total field is \phi + \eta. The path integral then becomes Z[J] = \int \mathcal{D}\eta \, \exp\left( i S[\phi + \eta] + i \int J (\phi + \eta) \right), and integrating out the fluctuations \eta yields \exp\left( i \Gamma[\phi] \right), where \Gamma[\phi] = S[\phi] + \Gamma_{\rm int}[\phi] and \Gamma_{\rm int}[\phi] encapsulates all quantum corrections from the loops of \eta. This conceptual process effectively sums the contributions of all quantum modes around the background, providing a non-perturbative expression for the quantum-corrected action. For constant background fields, the effective action simplifies to \Gamma[\phi] = V_{\rm eff}(\phi) \cdot {\rm Vol}, where V_{\rm eff}(\phi) is the quantum effective potential and Vol denotes the spacetime volume; minimizing \Gamma[\phi] in this case determines the vacuum structure, incorporating radiative corrections to the classical potential. The effective action \Gamma[\phi] satisfies the quantum equation of motion \frac{\delta \Gamma}{\delta \phi} = -J, which generalizes the classical Euler-Lagrange equations by linking the source J to the background field at the quantum level.

Generating Functionals

Effective Action as Legendre Transform

In quantum field theory, the effective action \Gamma[\phi] is constructed as the Legendre transform of the connected generating functional W[J], which itself derives from the path integral generating functional Z[J] via W[J] = -i \ln Z[J]. This transformation shifts the description from external sources J to the vacuum expectation values \phi = \langle \phi \rangle, providing a functional whose stationary points yield the equations of motion for the full quantum theory. The explicit form is given by

\Gamma[\phi] = W[J] - \int d^4x \, J(x) \phi(x),

where the source J and field \phi are conjugate variables satisfying \phi(x) = \frac{\delta W[J]}{\delta J(x)} and J(x) = \frac{\delta \Gamma[\phi]}{\delta \phi(x)}.^[3]^[7] The Legendre transform is invertible under the condition that W[J] is convex in J, a property that holds in the Euclidean formulation where W corresponds to the free energy and ensures a bijective mapping between J and \phi. This convexity guarantees the existence of a unique inverse transformation back to W[J], allowing the effective action to fully encode the connected correlation functions through its functional derivatives. In the tree-level or semiclassical approximation, where quantum fluctuations are neglected, the effective action simplifies to the classical action \Gamma_0[\phi] = S[\phi], as the source J enforces the classical field equation \frac{\delta S}{\delta \phi} = -J.^[7] Beyond the classical limit, the effective action admits a loop expansion in powers of \hbar: \Gamma[\phi] = \Gamma_0[\phi] + \hbar \Gamma_1[\phi] + \mathcal{O}(\hbar^2), where higher-order terms incorporate quantum corrections. The one-loop contribution \Gamma_1[\phi] arises from Gaussian fluctuations around the background field and takes the form \Gamma_1[\phi] = \frac{i}{2} \mathrm{Tr} \ln \left( \frac{\delta^2 S[\phi]}{\delta \phi \delta \phi} \right), reflecting the functional determinant in the path integral evaluation. This expansion systematically resums diagrams into vertex functions, with \Gamma[\phi] generating one-particle-irreducible (1PI) graphs.^[3] A key relation in this framework connects the second functional derivatives: the Hessian \Gamma^{(2)}[\phi] = \frac{\delta^2 \Gamma[\phi]}{\delta \phi \delta \phi} is the inverse of W^{(2)}[J] = \frac{\delta^2 W[J]}{\delta J \delta J}, i.e.,

\Gamma^{(2)}[\phi] = \left( W^{(2)}[J] \right)^{-1}.

This inverse correspondence identifies \Gamma^{(2)}[\phi] as the full quantum propagator in the background \phi, while W^{(2)}[J] yields the connected two-point function.^[7]

Vertex Functions and 1PI Diagrams

The vertex functions in quantum field theory are defined as the successive functional derivatives of the effective action \Gamma[\phi] with respect to the classical background field \phi. Specifically, the n-point vertex function is given by

\Gamma^{(n)}[\phi](x_1, \dots, x_n) = \left. \frac{\delta^n \Gamma[\phi]}{\delta \phi(x_1) \cdots \delta \phi(x_n)} \right|_{\phi \text{ fixed}},

where the derivatives are evaluated at a constant or slowly varying background field \phi. These vertex functions represent the coefficients in the local expansion of the effective Lagrangian \mathcal{L}_\text{eff} extracted from \Gamma[\phi] = \int d^4x \, \mathcal{L}_\text{eff}(\phi, \partial\phi, \partial^2\phi, \dots), capturing the quantum-corrected interactions among n field quanta while incorporating all one-particle-irreducible (1PI) contributions.^[8] Diagrammatically, the effective action \Gamma[\phi] is expressed as the sum of all connected 1PI Feynman diagrams, with external legs sourced by the background field \phi and no disconnected components allowed. A 1PI diagram is one that cannot be divided into two disjoint subdiagrams by removing a single internal propagator, ensuring the structure encodes only irreducible quantum corrections. This expansion organizes the quantum effects hierarchically by the number of loops, with each term in the loop expansion contributing additively to \Gamma[\phi].^[9] The proper vertices \Gamma^{(n)} correspond precisely to the amputated versions of these 1PI diagrams, where external propagators attached to the \phi legs are removed. Amputation isolates the irreducible interaction kernels, facilitating the construction of full scattering amplitudes by attaching propagators and summing over tree-level diagrams built from these vertices. This property underpins the additivity in the loop expansion, as higher-loop 1PI graphs contribute independently without redundant resummations of reducible parts.^[8] In scalar \phi^4 theory with interaction \frac{\lambda}{4!} \phi^4, the four-point vertex function \Gamma^{(4)} illustrates this structure. At tree level, it consists of the single vertex -i\lambda. The one-loop correction includes three 1PI diagrams: the s-, t-, and u-channel bubbles, each formed by a single loop propagator connecting pairs of external legs via two \lambda vertices. Higher-order terms, such as two-loop contributions, incorporate diagrams like the sunset topology (a self-energy insertion on an internal line of a one-loop graph), along with crossed ladders and other irreducible structures, systematically building the quantum-corrected interaction.^[10]

Properties

Symmetries and Ward Identities

In quantum field theory, symmetries of the classical action S[\phi] impose corresponding constraints on the effective action \Gamma[\phi], ensuring that the quantum theory respects the underlying invariance structure in the absence of anomalies. Specifically, if the classical action is invariant under an infinitesimal transformation \delta \phi = \epsilon K[\phi], where K[\phi] is the generator of the symmetry (e.g., a rotation or shift), then the effective action satisfies \int d^dx \frac{\delta \Gamma[\phi]}{\delta \phi(x)} K[\phi](x) = 0. This condition arises because the path integral measure and regularization preserve the symmetry, leading to the same functional form for \Gamma[\phi] as for S[\phi] at the level of the background field \phi. For global symmetries like O(N) invariance, this invariance holds order by order in perturbation theory, constraining the form of vertex functions derived from \Gamma.^[11] Ward-Takahashi identities provide the quantum generalization of these symmetry constraints, relating different n-point vertex functions \Gamma^{(n)} obtained from functional derivatives of \Gamma[\phi]. In the anomaly-free case, these identities take the form of a contraction of \Gamma^{(n+1)} with the symmetry current or generator equaling zero, such as \partial_\mu \Gamma^\mu_{(n+1)} = 0 for conserved currents, ensuring that the structure of the effective action mirrors the classical one. These relations are derived by considering infinitesimal transformations in the generating functional and translating them to the 1PI sector, preserving current conservation and forbidding certain interaction terms. For instance, in quantum electrodynamics (QED), U(1) gauge symmetry leads to the Ward-Takahashi identity for the photon two-point function, k_\mu \Gamma^{\mu\nu}(k) = 0, implying transversality of the photon self-energy and protecting the massless pole.^[12]^[13] In quantum chromodynamics (QCD), chiral symmetries constrain the effective action for pions, which emerge as pseudo-Goldstone bosons from spontaneous symmetry breaking. The SU(2)_L × SU(2)_R chiral Ward identities dictate that the pion decay constant and axial-vector couplings satisfy relations like the Goldberger-Treiman relation, limiting the form of low-energy interactions in the effective Lagrangian to terms invariant under chiral rotations. These identities ensure that pion scattering amplitudes vanish in the soft-pion limit, as derived from the generating functional. For non-Abelian gauge theories, such as QCD's SU(3)_c, the Ward-Takahashi identities generalize to Slavnov-Taylor identities, which incorporate ghost fields and BRST symmetry to maintain gauge invariance. These relate all Green functions, including vertices involving gluons, and enforce transversality conditions like D_{\mu\nu}^{-1}(k) k^\nu = 0 for the gluon propagator in the effective action.^[14]^[15]

Convexity and Jensen's Inequality

The effective action \Gamma[\phi] in quantum field theory is a convex functional of the classical field configuration \phi. This means that for any two field configurations \phi_1 and \phi_2, and for $0 \leq \lambda \leq 1,

\Gamma[\lambda \phi_1 + (1-\lambda) \phi_2] \leq \lambda \Gamma[\phi_1] + (1-\lambda) \Gamma[\phi_2],

with equality holding only if \Gamma is linear along the line connecting \phi_1 and \phi_2.^[5] The origin of this convexity lies in the Legendre transform relating \Gamma[\phi] to the generating functional of connected Green's functions W[J], which is concave in the source J.^[16] For the effective potential V_\mathrm{eff}(\phi), defined as the value of \Gamma[\phi] per unit volume for constant \phi, convexity implies that V_\mathrm{eff}(\phi) \geq V_\mathrm{eff}(\phi_\mathrm{cl}) for any \phi, where \phi_\mathrm{cl} is the configuration minimizing V_\mathrm{eff}; this ensures that the vacuum energy density at the true vacuum is the minimum value of the effective potential, bounding V_\mathrm{eff}(\phi) from below for other \phi. Physically, convexity guarantees the positive-definiteness of the second functional derivative \Gamma^{(2)}[\phi_\mathrm{cl}] at minima, ensuring positive mass-squared values and a spectral gap in the excitation spectrum; apparent violations of positivity signal instabilities, such as tachyonic instabilities in the potential.^[5] In theories exhibiting spontaneous symmetry breaking, tree-level approximations to the effective potential often display flat directions connecting degenerate minima, but the full quantum convexity enforces curvature along these directions, resolving the degeneracy through loop corrections.^[17]

Computation Methods

Perturbative Approaches

In perturbative quantum field theory, the effective action Γ[φ] admits a loop expansion in powers of the loop-counting parameter ℏ, expressed as Γ[φ] = S[φ] + ∑_{l=1}^∞ ℏ^l Γ_l[φ], where S[φ] is the classical action at tree level and each Γ_l[φ] arises from the sum of all one-particle irreducible (1PI) diagrams with l loops evaluated in the background of the mean field φ.^[18] This expansion organizes quantum corrections systematically, with higher-order terms suppressed in the weak-coupling regime, and relies on the 1PI structure to generate connected vertex functions from its functional derivatives.^[18] The one-loop contribution Γ_1[φ], which dominates low-order corrections, takes the form

\Gamma_1[\phi] = \frac{i}{2} \operatorname{Tr} \ln \left( \frac{\delta^2 S}{\delta \phi^2} \right),

obtained by integrating out quadratic fluctuations around the background field φ in the path integral representation.^[18] Higher loops Γ_l[φ] for l ≥ 2 involve increasingly complex 1PI graphs, such as those from four-point interactions in scalar theories, and can be computed diagrammatically using Feynman rules adapted to the background field.^[18] The background field method provides a practical framework for evaluating this loop expansion, particularly in gauge theories where gauge invariance must be maintained. Here, the total field is decomposed as φ_total = φ + η, with φ as the fixed background and η the quantum fluctuation; the effective action is then generated by the 1PI diagrams for η, with vertices derived from the action expanded around φ, ensuring the result is gauge-invariant under background transformations.^[19] This approach simplifies multi-loop calculations by avoiding explicit gauge fixing for the background and directly yields covariant counterterms upon renormalization.^[19] Dyson-Schwinger equations offer an alternative perturbative route to the effective action, manifesting as functional differential equations satisfied by Γ[φ]. These stem from the path integral identity δΓ/δφ_i = J_i, linking the mean field φ to the external source J, and can be solved iteratively starting from the classical action S[φ] by inserting perturbative expansions of the propagators and vertices.^[20] In practice, this iteration generates the loop series through recursive relations among 1PI functions, with consistency checked via Ward identities for symmetries like gauge invariance.^[20] Renormalization is essential to render the perturbative effective action finite, achieved by adding local counterterms to S[φ] that absorb divergences in the loop integrals of Γ_l[φ]. The renormalized effective action Γ_ren[φ] then features scale-dependent vertices whose running is governed by β-functions, extracted from the renormalization group equation applied to these vertices— for instance, the β-function for a quartic coupling λ arises from the divergent part of the four-point 1PI function at two loops.^[18] This procedure ensures multiplicative renormalizability, with the β-functions quantifying how couplings evolve under changes in the renormalization scale, preserving the structure of the loop expansion.^[19]

Non-Perturbative Techniques

Non-perturbative techniques for computing the effective action address regimes where coupling strengths are strong or expansions in small parameters fail, providing exact or approximate solutions through functional flows, numerical discretizations, or semiclassical approximations. These methods preserve key properties such as symmetries during computation, ensuring the resulting effective action respects underlying invariances of the theory. Unlike perturbative expansions, they capture phenomena like confinement and dynamical mass generation directly from the path integral formulation. The functional renormalization group (FRG) offers an exact flow equation for the scale-dependent effective action \Gamma_k, where k acts as an infrared cutoff suppressing low-momentum fluctuations. The Wetterich equation governs this evolution:

\partial_t \Gamma_k = \frac{1}{2} \mathrm{Tr} \left[ (\Gamma_k^{(2)} + R_k)^{-1} \partial_t R_k \right],

^[21] with t = \ln(k/\Lambda) (where \Lambda is the ultraviolet cutoff), \Gamma_k^{(2)} the second functional derivative of \Gamma_k, and R_k a regulator function chosen to optimize convergence, such as the optimized sharp cutoff R_k = (k^2 - p^2) \theta(k^2 - p^2).^[21] This equation integrates out fluctuations shell by shell from ultraviolet to infrared scales, yielding the full effective action \Gamma = \lim_{k \to 0} \Gamma_k. Practical implementations truncate the functional dependence, often using derivative expansions or ansätze for \Gamma_k, to solve the infinite-dimensional partial differential equation numerically.^[21] Lattice simulations discretize the spacetime continuum into a finite grid, transforming the path integral into a multidimensional integral amenable to Monte Carlo methods for estimating the partition function and deriving the effective action. In quantum chromodynamics (QCD), for instance, the Yang-Mills action is formulated on a hypercubic lattice with spacings a, and configurations are generated via importance sampling algorithms like hybrid Monte Carlo to compute expectation values of operators. The effective action for glueball states, which are gluon bound states, is extracted from correlation functions of loop operators, revealing non-perturbative spectra such as the lowest scalar glueball mass around 1.7 GeV in pure gauge SU(3). These simulations handle strong-coupling dynamics exactly within the lattice regularization, though continuum extrapolation requires careful control of finite-volume and discretization artifacts. Instanton methods employ semiclassical approximations to evaluate non-perturbative contributions from field configurations mediating tunneling between vacua, particularly in theories with degenerate ground states. Instantons are solutions to the Euclidean equations of motion with finite action S_{\mathrm{inst}}, contributing to the effective action as \Gamma_{\mathrm{inst}} \sim \exp(-S_{\mathrm{inst}}/\hbar), where the prefactor arises from collective coordinates and fluctuations around the instanton path. In quantum mechanics or field theories like QCD, these saddle-point evaluations capture vacuum tunneling rates and topological effects, such as the \theta-dependence in the effective potential, beyond perturbative vacuum stability. Variational principles construct approximate effective actions by optimizing trial forms subject to exact constraints, such as those from Schwinger-Dyson equations, which are functional differential equations enforcing the stationarity of \Gamma under field variations. For a trial \Gamma[\phi], the variation \delta \Gamma / \delta \phi = 0 must satisfy the Schwinger-Dyson relations derived from the path integral, like \langle \delta S / \delta \phi \rangle = 0 in the presence of sources. This approach, often using ansätze with adjustable parameters (e.g., Gaussian or mean-field forms), minimizes violations of these identities, providing bounds or approximations validated by the convexity of \Gamma. Seminal applications include deriving effective potentials in scalar theories where the trial action incorporates resummed propagators.

Applications

Renormalization Group Flow

In the renormalization group (RG) framework, the effective action \Gamma_\Lambda[\phi] is defined at a momentum cutoff \Lambda, incorporating fluctuations up to that scale, and its evolution under RG transformations is obtained by integrating out high-momentum modes between \Lambda and a lower scale \Lambda' < \Lambda. This flow describes how the theory changes under scale transformations, with the effective action at the infrared scale \Lambda \to 0 yielding the full quantum effective action \Gamma[\phi]. The process aligns with Wilson's momentum-shell integration, where infinitesimal changes in the cutoff lead to a differential equation governing the scale dependence of \Gamma_\Lambda. The scale invariance properties of the effective action are captured by the Callan-Symanzik equation, which relates the dependence of \Gamma on the renormalization scale \mu to anomalous dimensions and \beta-functions, ensuring consistency under rescaling. In the context of RG flows, fixed points occur where the effective action \Gamma_*[\phi] becomes scale-invariant, meaning its form remains unchanged under RG transformations. A prominent example is the Wilson-Fisher fixed point in O(N) scalar theories near four dimensions, obtained via \epsilon-expansion around the Gaussian fixed point, where interactions lead to non-trivial scaling dimensions for the fields and vertices. The \beta-functions, which dictate the running of couplings g as \frac{d g}{d \ln \mu} = \beta(g), can be derived directly from the vertices of the effective action \Gamma^{(n)}, incorporating anomalous dimensions that arise from the wave function renormalization and higher-point correlations. These \beta-functions determine the flow trajectories in coupling space, with infrared fixed points corresponding to conformal field theories. Additionally, the convexity of the effective potential, a key property of \Gamma[\phi], is preserved throughout the RG flow, ensuring thermodynamic stability at all scales. The decoupling theorem states that heavy particles with masses m \gg \mu decouple from the low-energy effective action for light fields, rendering \Gamma for light degrees of freedom independent of physics above the heavy scale, with effects appearing only as local operators suppressed by powers of $1/m. This theorem underpins the separation of scales in quantum field theories, allowing the effective action to focus on relevant low-energy dynamics.

Effective Field Theories

In effective field theories (EFTs), the effective action \Gamma_\text{EFT}[\phi] encapsulates the low-energy physics below a characteristic scale \Lambda by organizing interactions into a systematic expansion of operators consistent with the underlying symmetries. The leading terms typically include the canonical kinetic term and potential for light fields \phi, augmented by higher-dimensional operators O_i suppressed by inverse powers of \Lambda:

\Gamma_\text{EFT}[\phi] = \int d^4 x \left[ \frac{1}{2} (\partial \phi)^2 - V(\phi) + \sum_i c_i O_i \right],

where the Wilson coefficients c_i scale as $1/\Lambda^{\dim(O_i)-4} to ensure the expansion is controlled for energies E \ll \Lambda. This structure arises from the decoupling theorem, which guarantees that high-energy details manifest only through these local operators at low energies.^[22]^[23] The coefficients c_i are determined via a matching procedure, where heavy fields or modes above \Lambda are integrated out from the full ultraviolet (UV) theory's effective action \Gamma. This involves equating observables, such as correlation functions or scattering amplitudes, computed in the full theory and the EFT at the matching scale \mu \approx \Lambda, yielding tree-level or loop-level contributions to c_i. For instance, in theories with heavy particles of mass M \gg \Lambda, the leading c_i emerge from expanding the full \Gamma in powers of $1/M.^[24]^[25] A prominent example is chiral perturbation theory (ChPT), which describes pion interactions at low energies from the spontaneously broken chiral symmetry of quantum chromodynamics (QCD). The leading effective action is \Gamma_{\chi\text{PT}} \sim f_\pi^2 \Tr(\partial_\mu U \partial^\mu U^\dagger) + higher-order terms involving quark masses and derivatives, where U = \exp(i \pi^a \tau^a / f_\pi) parameterizes the Goldstone bosons, and f_\pi \approx 92 MeV sets the scale; loop corrections from this action reproduce QCD observables like pion scattering lengths.^[26] Another application is the Standard Model Effective Field Theory (SMEFT), valid above the electroweak scale \sim 246 GeV but below new physics at \Lambda \gtrsim 1 TeV, where \Gamma_\text{SMEFT} includes dimension-6 operators such as (H^\dagger i \overleftrightarrow{D}_\mu H)(\ell \gamma^\mu \ell) with coefficients encoding deviations from Standard Model predictions in processes like Higgs couplings or flavor-changing neutral currents.^[27]^[28] Power counting in EFTs organizes the expansion such that each order contributes consistently to amplitudes, while the convexity of the effective action—ensuring positive spectral densities—imposes bounds on |c_i| that maintain perturbative unitarity up to \Lambda. These positivity constraints, derived from analyticity and unitarity of the full theory, limit coefficients to avoid violations like negative scattering cross-sections, as seen in bounds on quartic Higgs interactions in SMEFT where c / \Lambda^2 < 1/(16\pi^2 v^2) with v the Higgs vacuum expectation value. Symmetries in EFTs, such as chiral ones, are often realized nonlinearly through field reparameterizations. The coefficients c_i may be evolved to lower scales via renormalization group flow for precise low-energy predictions.^[29]^[30]

References

[1]
Effective Action in Quantum Field Theory | SpringerLink
The effective action lies at very heart of many deep properties of quantum field theory. It generalizes the classical action and is the generating functional.
[2]
The effective action - INIS-IAEA - International Atomic Energy Agency
Jan 2, 2025 · The concept of the effective action in quantum field theory was introduced into physics by Julian Schwinger in 1954. The effective action ...Missing: definition | Show results with:definition
[3]
[PDF] Lecture 11 The Effective Action
THE EFFECTIVE ACTION. gives the expectation value of the operator φ(x). We now define the effective action as a. functional of φc(x) as. Γ[φc(x)] ≡ W[J] −
[4]
How Quantum Field Theory Becomes “Effective” – Sean Carroll
Jun 20, 2013 · Wilsonian effective field theory gives you a systematic way of dealing with them and estimating their importance.
[5]
[PDF] Physics 215C: Quantum Field Theory
Mar 25, 2014 · 13In fact, the whole effective action Γ[φ] is a convex functional –. δ2Γ δφ(x)δφ(y) is a positive integral operator. For more on this, I ...
[6]
[PDF] QUANTUM FIELD THEORY – 230A - UCLA Physics & Astronomy
... Γ[ϕ] of Gc[J] is then defined by. Γ[ϕ] ≡ Gc[J] −. Z. d4xJ(x)ϕ(x). J=J(x;ϕ) ... effective action of the system. Why action? Here, it is useful to make a ...
[7]
https://arxiv.org/pdf/1612.00462.pdf
[8]
Renormalization Group and Critical Phenomena. II. Phase-Space ...
Nov 1, 1971 · An effective interaction is defined by integrating out the wave-packet variables with momentum of order 1, leaving unintegrated the variables ...Missing: action | Show results with:action
[9]
https://arxiv.org/pdf/hep-th/0609178
[10]
[PDF] Optical theorem and effective action: new proofs of old results in QFT
May 29, 2020 · This paper presents a new, simple proof of the optical theorem and a new proof of the effective action as generating functional of 1PI ...
[11]
[PDF] arXiv:hep-th/0609178v1 26 Sep 2006
Sep 26, 2006 · In this paper, we propose a new kind of recursion relation to obtain the 1PI Feynman diagrams for the effective action. In Sec.II, we derive the ...
[12]
[PDF] One loop radiative corrections to the translation-invariant ... - arXiv
Feb 10, 2015 · In order to evaluate the four-point function at one loop level, we have to include all the contributions that give O(λ2) and O(g4) corrections ...
[13]
[PDF] Symmetries Ward Takahashi identities and all that
Γ[~ϕ] has the same symmetry as the classical action. • Contains the full set of Ward Takahashi identities for proper Green's functions.
[14]
[PDF] 6 Symmetries in Quantum Field Theory
6.3 Ward–Takahashi identities. Closely related to the consideration of symmetries of the effective action is to consider symmetries of correlation functions ...
[15]
[0708.0971] Ward Identities for the 2PI effective action in QED - arXiv
Aug 7, 2007 · Abstract: We study the issue of symmetries and associated Ward-like identities in the context of two-particle-irreducible (2PI) functional ...
[16]
[PDF] Chiral perturbation theory to one loop
Chiral symmetry implies a set of Ward identities which link the various Green's functions and therefore interrelate the expansion coefficients. As is well-known ...
[17]
[hep-ph/0203014] An approach to solve Slavnov-Taylor identities in ...
Dec 29, 2002 · The solution restricts the ghost part of the effective action and gives predictions for the physical part of the effective action.
[18]
[PDF] arXiv:1612.00462v2 [hep-th] 31 May 2017
May 31, 2017 · The Legendre transform plays two important rôles in quantum field theory. As described in (3), it maps between the generating functional of ...
[19]
Misunderstanding that the Effective Action is Convex under Broken ...
Jun 16, 2016 · Abstract:The widespread belief that the effective action is convex and has a flat bottom under broken global symmetry is shown to be wrong.
[20]
Functional evaluation of the effective potential | Phys. Rev. D
Mar 15, 1974 · Functional evaluation of the effective potential. R. Jackiw. Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute ...
[21]
Regularization and renormalization of gauge fields - ScienceDirect
A new regularization and renormalization procedure is presented. It is particularly well suited for the treatment of gauge theories.
[22]
Effective action for composite operators | Phys. Rev. D
Oct 15, 1974 · An effective action and potential for composite operators is obtained. The formalism is used to analyze, by variational techniques, dynamical symmetry breaking.
[23]
[PDF] An Introduction to Effective Field Theory - McMaster Physics
quantum (1PI) action.15 By doing so it provides a constructive framework for defin- ing and explicitly building effective actions for a broad class of ...
[24]
[PDF] An Introduction to Effective Field Theories - ICTP – SAIFR
Jun 19, 2020 · An EFT is defined by an effective action, which in turn is completely specified by the following three ingredients: 1. Degrees of freedom. The ...
[25]
[PDF] Effective Field Theory Matching and Applications
Mar 14, 2022 · EFT matching refers to the procedure of computing the coefficients of EFT operators (including threshold corrections) in terms of the UV ...
[26]
[PDF] Lectures on Effective Field Theories
Sep 21, 2020 · As an example, consider the following effective action for a single scalar field φ: ... by differentiating Γ[φ]: hφ(x1)...φ(xn)i1PI = i. δnΓ ...
[27]
Phenomenological Lagrangians - ScienceDirect.com
Physica A: Statistical Mechanics and its Applications Volume 96, Issues 1–2, April 1979, Pages 327-340 Phenomenological Lagrangians
[28]
[2303.16922] The Standard Model effective field theory at work - arXiv
Mar 29, 2023 · In this article, after briefly presenting the salient features of the Standard Model, we review the construction of the SMEFT. We discuss ...
[29]
The standard model effective field theory at work | Rev. Mod. Phys.
Mar 19, 2024 · The standard model can be viewed as the leading term of an effective “remnant” theory (referred to as the SMEFT) of a more fundamental structure.Article Text · Standard Model Effective Field... · Low-Energy Effective Field...
[30]
[2207.14224] Positivity bounds on effective field theories with ... - arXiv
Jul 28, 2022 · This paper derives positivity bounds on EFT coefficients in theories with spontaneously broken boosts, using retarded Green's functions, and ...
[31]
Positivity in Multifield Effective Field Theories | Phys. Rev. Lett.
Sep 17, 2021 · We discuss the general method for obtaining full positivity bounds on multifield effective field theories (EFTs).Abstract · Article Text · Introduction. · General bounds from...