Conformally Coupled Scalar EFT

• Conformally Coupled Scalar EFT is a framework that integrates scalar fields with nonminimal curvature couplings to uphold local Weyl invariance in theories of gravity, cosmology, and CFTs.
• Its formulation utilizes higher curvature tensors and invariant actions to ensure second-order field equations, thereby avoiding Ostrogradsky instabilities and ghost states.
• This EFT framework enables precise predictions for operator spectra, boundary/defect dynamics, and inflationary models, bridging gravitational physics with advanced quantum field techniques.

A conformally coupled scalar effective field theory (EFT) describes scalar fields whose dynamics and interactions are constructed to be compatible with local conformal (Weyl) symmetry. This framework synthesizes methods from gravitational theory, particle physics, and statistical field theory to encode the consequences of conformal invariance and its breaking, extending from the formulation of unique curvature couplings in gravity and cosmology to the systematic construction of boundary and defect theories in conformal field theory (CFT). The structure of conformally coupled scalar EFTs plays a central role in gravitational physics, inflationary cosmology, the paper of critical phenomena, the theory of conformal defects and boundaries, and in precision bootstrap programs for CFT operator spectra.

1. Geometric Construction and Higher Curvature Conformal Couplings

Conformally coupled scalar EFTs generalize the minimal coupling of scalar fields to gravity by introducing specific nonminimal curvature couplings that render the action invariant under local Weyl rescalings: $$g_{\mu\nu} \rightarrow e^{2\omega(x)} g_{\mu\nu}, \quad \phi \rightarrow e^{s\omega(x)}\phi,$$ with $s$ the conformal weight of the scalar. A systematic procedure constructs conformally invariant couplings to higher curvature invariants by introducing a four-rank tensor $S_{\alpha\beta\gamma\delta}$ built from the Riemann tensor and derivatives of $\phi$, possessing Riemann symmetries (except for the Bianchi identity) and transforming covariantly under Weyl rescalings. The action for the $k$-th order (Euler density) term takes the schematic form: $$I^{(k)} = \int d^d x \sqrt{-g} \, \phi^{m_k} \, \delta^{b_1\ldots b_{2k}}_{a_1\ldots a_{2k}} S^{a_1 a_2}_{b_1 b_2} \ldots S^{a_{2k-1} a_{2k}}_{b_{2k-1} b_{2k}},$$ where $m_k$ is fixed to ensure integrality of exponents, and the index contraction involves generalized Kronecker deltas. Under Weyl rescalings, $$S_{\alpha\beta\gamma\delta}\rightarrow e^{2(s-1)\omega}S_{\alpha\beta\gamma\delta}$$, with $s=1-D/(2k)$ in $D$ spacetime dimensions (Oliva et al., 2011). This construction extends in a mathematically controlled fashion to higher-order (Lovelock-type) curvature terms, ensuring that both the equations of motion and stress-energy tensor remain of second order—a crucial property to avoid Ostrogradsky instabilities and ghosts in higher-derivative gravity.

2. Dynamics, Field Equations, and Spectral Applications

The conformal coupling ensures that the field equations derived from such actions (including gravity, scalar, and higher-derivative sectors) retain second-order dynamics even for higher curvature or higher-derivative operators. For example, in the presence of quadratic or higher-order curvature interactions, as in quadratic gravity, the equations for static black holes force the Ricci scalar to a constant value, leading to solutions matching Schwarzschild–(A)dS at infinity even when the scalar “hair” is nontrivial (Caceres et al., 2020). These analytic solutions provide benchmarks for exploring the interplay of conformal matter and higher curvature corrections, and the associated field content—when viewed as an EFT—can capture the operator spectrum of Wilson–Fisher fixed points and more exotic CFTs by organizing anomalous dimensions and operator mixing matrices to high loop order (Henriksson et al., 16 Jul 2025).

The conformally coupled scalar field is also critical in CFT boundary and defect theories, where the EFT encodes the response of heavy operators to the presence of boundaries or impurities. The universal features of defect/boundary EFTs can be derived by integrating out modes associated with the transverse size of the defect, resulting in an effective action

$$S_{\mathrm{BEFT}}(\Delta) = \frac{1}{\Delta^{d-1}}\int d^{d-1}x \sqrt{g} \left( -\mathcal{E} + \cdots \right),$$

where $\Delta$ is the transverse displacement, and $\mathcal{E}$ is the Casimir energy density, subject to non-positivity and convexity bounds from unitarity (Diatlyk et al., 3 Jun 2024).

3. Spontaneous and Explicit Breaking of Conformal Symmetry

The spontaneous breaking of conformal symmetry is modeled by a scale or time-dependent vacuum expectation value (VEV) of a scalar primary operator, such as

$$\langle \Phi \rangle = \begin{cases} f^\Delta & \text{(constant VEV; Poincaré breaking)} \ C^\Delta/(-t)^\Delta & \text{(time-dependent VEV; pseudo-conformal universe)} \end{cases}$$

where $f$ or $C$ parameterizes the breaking scale. Fluctuations around this background organize as an EFT expansion in inverse powers of the VEV (or symmetry-breaking scale), with the Lagrangian expanded in conformally invariant building blocks, \begin{align*} \mathcal{L}_0 &= \Phi^{D/\Delta}, \ \mathcal{L}_2 &= \Phi^{(D-2)/\Delta} \frac{(\partial\Phi)^2}{\Phi^2}, \ \mathcal{L}_4,\,\mathcal{L}_4' &:\quad \text{(various higher-derivative invariants)}, \end{align*} implying that correlators exhibit a controlled expansion valid at long distances (or late times, in the pseudo-conformal scenario). For defect CFTs and broken conformal symmetry on space-like surfaces, this results in universal predictions for correlation functions of heavy operators, accessible via Mellin–Barnes representations and in-in formalism for quantum fields (Hinterbichler et al., 2022).

Explicit symmetry breaking, for instance via mass terms or relevant curvature couplings, can be incorporated systematically—yielding, for example, the dilaton EFTs for near-conformal gauge theories. Here, both the spontaneous and explicit breaking terms determine scaling relations and operator dimensions, with corrections directly accessible to comparison with lattice simulations and numerical bootstrap data (Appelquist et al., 2017, Appelquist et al., 2018, Fodor et al., 2019).

4. Boundaries, Defects, and Universal CFT Data

The effective field theory for CFTs with conformal boundaries or defects is constructed by considering a pair of closely spaced boundaries/defects and integrating out heavy modes with mass proportional to $1/\Delta$, where $\Delta$ is their separation. The leading effective action yields universal formulas for the boundary structure constants at high scaling dimension,

$$B(\Delta) \sim \frac{2^{(d-1)/2}}{\sqrt{d\pi}} \frac{[(d-1)\mathcal{E}S_{d-1}]^{(d+1)/(2d)}}{\Delta^{(2d+1)/(2d)}} \exp \left[-d\left(\frac{\Delta}{d-1}\right)^{(d-1)/d} (\mathcal{E} S_{d-1})^{1/d}\right],$$

with $\mathcal{E}$ again the Casimir energy density and $S_{d-1}$ the area of the unit $(d-1)$-sphere (Diatlyk et al., 3 Jun 2024). In conformally invariant free theories, this matches known results for Dirichlet or Neumann boundary conditions, with the corresponding $\mathcal{E}$ directly related to the negative thermal free energy density. Reflection positivity and unitarity impose $\mathcal{E} \geq 0$, leading to nontrivial bounds on defect CFT data.

The thermal effective theory for CFTs is subsumed as a special case of the boundary EFT, and the formalism extends naturally to periodic or thermal boundary conditions, with the Casimir energy matching (up to geometric pre-factors) the free energy on $S^1 \times S^{d-1}$.

5. Cosmology, Gravitational Physics, and Black Hole Solutions

Conformally coupled scalar EFTs are extensively employed in cosmology and gravitational physics. A prototypical application in inflationary cosmology requires the inflaton to be conformally coupled ($\xi = 1/6$), ensuring the absence of causal pathologies (light-cone propagation by massive fields) and enabling novel “superinflation” regimes with possible blue tensor spectra for primordial gravitational waves (Faraoni, 2013).

In gravitational contexts, nonminimal and conformal couplings enable analytic constructions of black hole solutions, including neutral and charged (dyonic) topological black holes with scalar “hair” and higher curvature corrections (Caceres et al., 2020), as well as rotating generalizations (e.g., rotating Bekenstein–Bocharova–Myers–Bekenstein (BBMB) black holes (Astorino, 2014)). The geometric structure often forces the Ricci scalar to a fixed value, aligning the asymptotic solution with standard general relativity even in the presence of nontrivial scalar configurations.

Scalar field configurations with conformal couplings can drastically alter the gravitational “charge” of matter via spontaneous symmetry breaking, leading, for example, to repulsive gravity and bouncing cosmological solutions when the effective gravitational constant becomes negative in a broken vacuum (Antunes et al., 2017).

6. Classification and Uniqueness in the Space of Scalar EFTs

Within the broader classification of Lorentz-invariant local scalar EFTs, conformally coupled scalar EFTs occupy a non-exceptional sector. Theories are characterized by four parameters: number of derivatives per interaction ($\rho$), the scaling of the amplitude under a soft limit ($\sigma$), the leading valency ($v$), and the spacetime dimension ($d$). Conformally coupled EFTs typically exhibit “trivial” soft behavior ($\sigma$ equal to or slightly above $\rho$), as opposed to “exceptional” theories, such as the nonlinear sigma model, Dirac–Born–Infeld (DBI), and special Galileon, whose amplitude structures are fully fixed by enhanced soft symmetry ($\sigma = \rho + 1$) (Cheung et al., 2016).

For derivatively coupled scalars, requiring exact conformal invariance is a stronger constraint than mere scale invariance. For instance, conformal invariance uniquely fixes the DBI theory (quartic $O(p^4)$ vertex in $D=0$) and special Galileon ($O(p^6)$ in $D=-2$) as the only solutions compatible with vanishing trace of the energy-momentum tensor, verified both at the level of Lagrangian improvements and on-shell via conformal Ward identities and double Adler zeroes in dilaton soft limits (Cheung et al., 2020).

7. Multi-loop Renormalization, Operator Spectra, and Applications

The paper of conformally coupled scalar EFTs at multi-loop order provides critical input for perturbative and nonperturbative analyses in CFTs. Efficient renormalization techniques, such as the improved $R^*$ method employing small-momentum asymptotic expansions, enable determinantion of anomalous dimension matrices for primary composite operators up to high loop order, with application to Ising, $O(n)$, and hypercubic CFTs (Henriksson et al., 16 Jul 2025).

Diagonalization of these matrices after fixing to the IR critical point gives scaling dimensions,

$$\Delta = \Delta_\text{classical} + \text{eigenvalues}(\Gamma),$$

which can be used as seed or benchmark data for numerical conformal bootstrap studies. The quantitative precision achieved for operator scaling dimensions in, e.g., the three-dimensional Ising or $O(3)$ CFTs, is essential for refining gap assumptions and guiding the isolation of the allowed operator space.

Summary Table: Conformally Coupled Scalar EFT—Key Themes

Feature	Description	Example Reference
Higher curvature conformal couplings	Four-index tensor construction, second-order EOM	(Oliva et al., 2011, Caceres et al., 2020)
Spontaneous conformal breaking	EFT in inverse–VEV expansion, defect/early universe applications	(Hinterbichler et al., 2022)
Boundary/defect structure	Effective action for boundaries/defects, universal asymptotics	(Diatlyk et al., 3 Jun 2024)
Cosmological implications	Superinflation, blue tensor spectrum, repulsive gravity	(Faraoni, 2013, Antunes et al., 2017)
EFT classification	Periodic table: $(\rho,\sigma,v,d)$, conformal invariants	(Cheung et al., 2016, Cheung et al., 2020)
Multi-loop operator spectrum	Renormalization, anomalous dimensions, bootstrap seed data	(Henriksson et al., 16 Jul 2025)

The conformally coupled scalar effective field theory framework thus provides a versatile and technically rigorous tool for capturing the consequences of conformal invariance and its breaking across quantum field theory, gravity, cosmology, and critical phenomena. The mathematical structure, emphasis on second-order dynamics, and the link with operator spectra render it foundational both in fundamental theory and in precision applications to experiment and numerical simulation.