Universal Scalaron EFT

• Universal Scalaron EFT is an effective framework that isolates a propagating scalar degree (the scalaron) emerging from quadratic curvature corrections in gravity.
• It systematically constructs invariant operators up to six-derivative order using geometric matching, Legendre transforms, and frame universality principles.
• The framework bridges inflationary predictions, renormalization group flow, and parity-violating gravitational waves, offering insights for cosmology and quantum gravity.

A universal scalaron effective field theory (EFT) provides a systematically defined framework for capturing the dynamics of a single scalar degree of freedom—the "scalaron"—arising in diverse extensions of gravity. This concept is anchored by several structural, geometric, and field-theoretic principles, leading to powerful phenomenological predictions spanning inflationary cosmology, infrared gravity, and quantum corrections. Scalaron EFTs arise from higher-curvature theories, quantum anomalies, nonminimal couplings, and RG flows, with a central role for frame universality—ensuring physical predictions are independent of the choice of conformal frame. This article delineates the construction, operator structure, matching principles, and cosmological roles of universal scalaron EFTs, integrating state-of-the-art results and methodology.

1. Scalaron Emergence: Foundations and Field-Theoretic Context

The scalaron is a propagating scalar degree of freedom universally emerging in four-dimensional extensions of general relativity whenever the action contains quadratic curvature terms or nonminimal scalar couplings. For theories of the form $S=\int \sqrt{-g}[R + \alpha R^2 + \dots ]$, a Legendre transformation and Weyl rescaling isolate the scalaron in the Einstein frame, yielding %%%%1%%%% where $V(\phi)$ has universal Starobinsky-type form $$V(\phi) \sim (1-e^{-\sqrt{2/3} \phi/M_{\rm Pl}})^2/(4\alpha)$$ (Brax et al., 2023, Arapoğlu et al., 29 Dec 2025). Nonminimal scalar couplings $F(\phi)R$ and integrating out heavy fields also map onto an effective $R^2$ term, further ensuring scalaron universality (Arapoğlu et al., 29 Dec 2025, Li et al., 2024).

Quadratic and higher-curvature gravity admits a covariant Hubbard-Stratonovich/Legendre lift, algebraic auxiliary elimination (including Palatini connections), and Jordan-Einstein frame Weyl rescalings—all commutative at the action level up to boundary terms (Arapoğlu et al., 29 Dec 2025). This result ensures that the scalaron EFT is independent of the sequence of algebraic manipulations, preserving frame universality and physical equivalence.

2. Operator Basis: Scalar-Tensor EFT to Sixth Derivative Order

Scalaron EFTs are systematically constructed by enumerating diffeomorphism-invariant operators involving the metric $g_{\mu\nu}$ and the scalaron $\phi$, classified by derivative order and parity (Babichev et al., 15 Dec 2025). Weinberg's four-derivative basis includes:

• $(\nabla\phi)^4$,
• curvature invariants $$R_{\alpha\beta\gamma\delta} R^{\alpha\beta\gamma\delta} - 4 R_{\alpha\beta} R^{\alpha\beta} + R^2$$ (Gauss-Bonnet),
• Pontryagin density $$\epsilon^{\alpha\beta\gamma\delta} R_{\alpha\beta}{}^{\zeta\eta}R_{\gamma\delta\zeta\eta}$$.

At six derivatives, the minimal on-shell basis comprises (Babichev et al., 15 Dec 2025):

Parity-Even Operators	Parity-Odd Operators
$R_{αβγδ}R^{αβ}{}_{ζη}R^{γδζη}$	$\epsilon_{ζηθκ}R_{αβγδ}R^{αβζη}R^{γδθκ}$
$R_{αβγδ}R^{αβγδ}(\nabla\phi)^2$	$\epsilon_{γδζη}R_{αβ}{}^{γδ}R^{αβζη}(\nabla\phi)^2$
$$R_{αβγδ}\nabla^\alpha\phi\nabla^\gamma\phi\nabla^\delta\nabla^\beta\phi$$	$$\epsilon_{γδζη}R_{αβ}{}^{γδ}\nabla^\alpha\phi\nabla^\zeta\phi\nabla^\eta\nabla^\beta\phi$$
$(\nabla\phi)^6$
$$(\nabla\phi)^2 \nabla_\gamma\nabla_\beta\phi\nabla^\gamma\nabla^\beta\phi$$

Each term is multiplied by an arbitrary function of $\phi$ and suppressed by appropriate powers of the cutoff $\Lambda$. Redundancy elimination via Bianchi identities, integration by parts, and leading-order EOMs yields a minimal, physical operator set (Babichev et al., 15 Dec 2025).

3. Geometric One-Loop Universal Matching and EFT Construction

Integrating out heavy fields in scalar-tensor theories at one loop proceeds via geometric master formulas relying on the field-space metric $G_{ij}(\varphi)$ and its curvature (Li et al., 2024). The one-loop effective action is

$$\mathcal L_\text{EFT} = \frac{1}{2} Z(\phi) (\partial_\mu\phi)^2 - V_\text{eff}(\phi) + \frac{1}{16\pi^2}\left[ - \frac{1}{12 M^2} W_{;\phi A}W_{;\phi A} (\square\phi)^2 + \frac{1}{24 M^4} (W_{;\phi A}W_{;\phi A})^2 (\partial_\mu\phi\partial^\mu\phi)^2 \right] + \mathcal O(M^{-4}),$$

where $Z(\phi)$ and $V_\text{eff}(\phi)$ are determined by covariant derivatives of the potential and the heavy mass $M$ (Li et al., 2024). A non-derivative field redefinition can trade potential couplings for a nontrivial field-space metric, simplifying all matching to geometric data.

This formalism unifies weak-field, Starobinsky (Arbuzov et al., 2020), and RG-improved scalaron EFTs (Brax et al., 2023), and captures the universal matter coupling exponent $\beta=1/\sqrt{6}$ generated by the Legendre/Weyl map.

4. Renormalization Group Flow and IR Physics

The universal scalaron EFT incorporates running couplings $\alpha(\mu)$ and $\Lambda(\mu)$ associated with $R^2$ and cosmological constant terms, respectively. As particles are integrated out at decreasing scale $\mu$, beta functions for the scalaron mass and vacuum energy are dictated by the decoupling theorem and captured by one-loop RG equations (Brax et al., 2023):

$$\mu\,{d \alpha(\mu) \over d\mu} = -{7\alpha^2 \over 8\pi^2 M_{\rm Pl}^4} \mathrm{STr}[M^4(\mu)], \qquad m_\phi^2(\mu) = {1 \over 6\alpha(\mu)},$$

where STr denotes a supertrace over active field masses. At electron threshold $\mu=m_e$, matching conditions link scalaron mass and the observed vacuum energy, confining the IR mass window to $$3.8 \times 10^{-3}\,\rm eV \lesssim m_\phi \lesssim 0.1\,\rm eV$$ (Brax et al., 2023). This RG structure ties Starobinsky inflation to late-universe acceleration within a single-field scalaron paradigm.

5. Frame Universality and Nonminimal Coupling Matching

A fundamental theorem asserts that Hubbard–Stratonovich lifts, algebraic elimination of auxiliaries, and Jordan-Einstein Weyl rescalings commute up to boundary terms, ensuring frame-independent identification of the propagating scalaron (Arapoğlu et al., 29 Dec 2025). This property guarantees that integrating out heavy fields, performing field redefinitions, or switching conformal frames yields the same bulk scalaron EFT.

Universal matching formulas relate generic nonminimal scalar couplings $F(\phi)R$ to $R^2$ terms via

$$\alpha'_{\rm NMC} = \frac{1}{2} F_a (M^{-2})^{ab} F_b,$$

yielding a unique scalaron sector with positive-definite kinetic term and predictive inflationary observables (Arapoğlu et al., 29 Dec 2025). This universality applies also to Palatini gravity and extensions.

6. Parity-Violating Extensions and Phenomenological Signatures

Parity-odd six-derivative operators in the universal scalaron EFT, such as $\epsilon_{ζηθκ}R_{αβγδ}R^{αβζη}R^{γδθκ}$, induce chirality in gravitational-wave propagation and cosmic birefringence in CMB polarization (Babichev et al., 15 Dec 2025). Observable effects include different dispersion relations for tensor helicities and possible detection in stochastic gravitational wave backgrounds or CMB tensor-mode polarization. Such signatures sharply probe strong-curvature, quantum, or stringy corrections to gravity.

7. Scalar Gravitational Waves: Anomaly-Induced EFT

Quantum trace anomalies in massless or nearly massless fields mandate a universal infrared scalaron extension of general relativity, generating a dynamical conformalon field $\varphi$ that couples via anomaly-induced terms (Mottola, 2016):

$$S_{\rm eff}[g,\varphi] = \frac{1}{16\pi G} \int \sqrt{-g}(R - 2\Lambda) + S_{\rm anom}[g;\varphi],$$

where $S_{\rm anom}$ is constructed from the Weyl tensor square, Gauss-Bonnet density, and additional conformal invariants. Linearized analysis reveals massless scalar gravitational wave ("breathing" mode) solutions with positive energy and monopole coupling, potentially detectable in neutron star mergers due to QCD trace anomaly sources (Mottola, 2016). The anomaly-induced scalaron propagates over macroscopic distances, but couples only to anomalous terms, not directly to classical energy-momentum.

8. Inflationary and Cosmological Implications

Universal scalaron EFTs generically recover Starobinsky inflation, yielding firm predictions for CMB observables (Brax et al., 2023, Arapoğlu et al., 29 Dec 2025, Arbuzov et al., 2020):

$$n_s \simeq 1 - {2 \over N_*}, \qquad r \simeq {12 \over N_*^2}.$$

With $N_*\in[50,60]$, $r\simeq(3.3-4.8)\times10^{-3}$ and $n_s\simeq(0.960-0.967)$—directly testable by upcoming CMB surveys. In single-field attractor scenarios, isocurvature modes are exponentially suppressed, and non-Gaussianity constraints serve as falsifiable tests of the universal scalaron framework (Arapoğlu et al., 29 Dec 2025).

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Universal scalaron EFTs provide a technically robust, phenomenologically precise framework for extending general relativity via higher-curvature gravity, quantum anomalies, and geometric field theory principles. Operator enumeration, one-loop matching, RG running, parity violation, and frame universality converge to a predictive scalar-tensor theory eminently suited for probing inflation, late-universe acceleration, strong-curvature physics, and quantum gravitational effects (Babichev et al., 15 Dec 2025, Arapoğlu et al., 29 Dec 2025, Brax et al., 2023, Arbuzov et al., 2020, Li et al., 2024, Mottola, 2016).