Agravity: Dimensionless Quadratic Gravity

• Agravity is a dimensionless gravity theory built from R² and Rμν² terms that achieves perturbative renormalizability and dynamically generates the Planck scale.
• It replaces the Einstein–Hilbert term with a four-derivative action, yielding a massless graviton alongside a ghost-like massive spin-2 component known as the anti-graviton.
• The framework underpins classically scale-invariant models of inflation and high-energy scattering, with its renormalization group behavior offering insights into ultraviolet physics.

Agravity, short for “adimensional gravity,” denotes a class of gravity theories in which the fundamental Lagrangian contains no intrinsic mass or length scale, all couplings are dimensionless, and perturbative renormalizability is obtained by replacing the Einstein–Hilbert two-derivative graviton kinetic term with a four-derivative one built from quadratic curvature invariants. In the formulation introduced in “Agravity” (Salvio et al., 2014), the gravitational sector is based on $R^2$ and $R_{\mu\nu}^2$ terms with dimensionless couplings $f_0$ and $f_2$, the Planck scale can arise dynamically at quantum level, inflation is generic because Einstein-frame flatness is controlled by beta functions, and the higher-derivative spin-2 sector can be interpreted as “gravity minus an anti-graviton.” Subsequent work studies the ultraviolet fate of the theory, its embedding into classically scale-invariant particle models, and its ultra-Planckian scattering behavior (Salvio et al., 2017).

1. Foundational definition and action

The defining agravity action is

$$S=\int d^4x\,\sqrt{|g|}\left[ \frac{R^2}{6f_0^2} +\frac{\frac13 R^2-R_{\mu\nu}^2}{f_2^2} +L_{\rm SM}^{\rm adim} +L_{\rm BSM}^{\rm adim} \right],$$

with $f_0$ and $f_2$ dimensionless. In the same construction, the Standard Model sector is written without explicit mass terms,

$L_{\rm SM}^{\rm adim} = -\frac{F_{\mu\nu}^2}{4g^2} +\bar\psi i\slashed{D}\psi +|D_\mu H|^2 -(yH\psi\psi+\text{h.c.}) -\lambda_H |H|^4 -\xi_H |H|^2 R,$

and a generic beyond-the-Standard-Model scalar can be added through

$$L_{\rm BSM}^{\rm adim} = |D_\mu S|^2 -\lambda_S |S|^4 +\lambda_{HS}|S|^2|H|^2 -\xi_S |S|^2 R.$$

The Einstein–Hilbert term $-M_{\rm Pl}^2R/16\pi$ and the Higgs mass term are absent at the fundamental level, so all scales must be generated dynamically (Salvio et al., 2014).

The renormalizability claim is tied to the propagator of the higher-derivative graviton. Once a Planck mass is induced, the spin-2 propagator takes the schematic form

$R_{\mu\nu}^2$0

with

$R_{\mu\nu}^2$1

This is the sense in which the theory was described as “gravity minus an anti-graviton”: the usual massless graviton is accompanied by a massive spin-2 ghost-like state, while the $R_{\mu\nu}^2$2 term supplies a scalar graviton (Salvio et al., 2014).

A closely related Jordan-frame presentation appears in classically scale-invariant model-building, where the agravity terms are combined with scalar non-minimal couplings,

$R_{\mu\nu}^2$3

This makes explicit that agravity is not an add-on to an already defined Einstein theory; it is the gravitational sector appropriate to a classically scale-invariant, dimensionless ultraviolet completion (Farzinnia et al., 2015).

2. Renormalization-group structure and ultraviolet behavior

A central structural feature of agravity is the asymmetry between the two gravitational couplings. At one loop,

$R_{\mu\nu}^2$4

so $R_{\mu\nu}^2$5 is asymptotically free, whereas

$R_{\mu\nu}^2$6

which is positive for small physical $R_{\mu\nu}^2$7. The conformal mode of the graviton may be isolated through

$R_{\mu\nu}^2$8

for which the $R_{\mu\nu}^2$9 term becomes

$f_0$0

This identifies $f_0$1 as the self-coupling of the conformal mode (Salvio et al., 2017).

The main ultraviolet result of “Agravity up to infinite energy” is that the non-asymptotic freedom of $f_0$2 does not by itself force a Landau pole at finite energy. When $f_0$3 becomes large, the conformal mode decouples from the rest of the theory provided that all scalars become asymptotically conformally coupled,

$f_0$4

and all other couplings approach ultraviolet fixed points. In that regime agravity flows to conformal gravity at infinite energy,

$f_0$5

rather than encountering a finite-energy inconsistency (Salvio et al., 2017).

The same analysis gives a technically useful reformulation: agravity = conformal gravity + two extra conformally coupled scalars. With an $f_0$6 doublet $f_0$7, the Weyl-breaking piece may be rewritten as

$f_0$8

This equivalence organizes the renormalization group without denominators and clarifies why the conformal mode behaves like a strongly coupled scalar quartic interaction in the ultraviolet (Salvio et al., 2017).

3. Dynamical generation of the Planck scale and naturalness claims

Agravity was introduced together with the claim that the Planck scale and flat space can emerge dynamically. If a scalar $f_0$9 acquires a vacuum expectation value through dimensional transmutation and couples non-minimally to curvature, then in the simplest case

$f_2$0

In the Einstein frame the scalar potential takes the form

$f_2$1

and for an adimensional scalar potential $f_2$2 with $f_2$3,

$f_2$4

The stationary condition is

$f_2$5

To obtain both the Planck scale and vanishing vacuum energy, the stronger conditions

$f_2$6

are imposed at the Planck scale (Salvio et al., 2014).

Within this framework, the weak scale is generated through portals and non-minimal couplings rather than explicit mass parameters. In a classically scale-invariant extension containing a complex singlet $f_2$7 and three right-handed Majorana neutrinos, the singlet develops a Coleman–Weinberg-generated vacuum expectation value $f_2$8, the Higgs mass parameter is induced as

$f_2$9

and the effective reduced Planck mass arises from the vacuum value of

$$S=\int d^4x\,\sqrt{|g|}\left[ \frac{R^2}{6f_0^2} +\frac{\frac13 R^2-R_{\mu\nu}^2}{f_2^2} +L_{\rm SM}^{\rm adim} +L_{\rm BSM}^{\rm adim} \right],$$0

The same paper stresses that all scales are free from mutual quadratic destabilization because they are induced by dimensionless couplings and vacuum expectation values rather than by explicit mass inputs (Farzinnia et al., 2015).

The naturalness claim is correspondingly strong. The original agravity paper argues that quadratically divergent corrections are absent by dimensional analysis, and that the electroweak hierarchy can be generated with small gravitational couplings, estimating $$S=\int d^4x\,\sqrt{|g|}\left[ \frac{R^2}{6f_0^2} +\frac{\frac13 R^2-R_{\mu\nu}^2}{f_2^2} +L_{\rm SM}^{\rm adim} +L_{\rm BSM}^{\rm adim} \right],$$1 as sufficient for gravitational corrections to generate the observed hierarchy. The later ultraviolet analysis identifies two viable regimes for maintaining small Higgs-mass corrections: $$S=\int d^4x\,\sqrt{|g|}\left[ \frac{R^2}{6f_0^2} +\frac{\frac13 R^2-R_{\mu\nu}^2}{f_2^2} +L_{\rm SM}^{\rm adim} +L_{\rm BSM}^{\rm adim} \right],$$2 at the Planck scale, or $$S=\int d^4x\,\sqrt{|g|}\left[ \frac{R^2}{6f_0^2} +\frac{\frac13 R^2-R_{\mu\nu}^2}{f_2^2} +L_{\rm SM}^{\rm adim} +L_{\rm BSM}^{\rm adim} \right],$$3 with the scalar component of the graviton above the Planck scale, in both cases with $$S=\int d^4x\,\sqrt{|g|}\left[ \frac{R^2}{6f_0^2} +\frac{\frac13 R^2-R_{\mu\nu}^2}{f_2^2} +L_{\rm SM}^{\rm adim} +L_{\rm BSM}^{\rm adim} \right],$$4 controlling the dominant correction [(Salvio et al., 2014); (Salvio et al., 2017)]. A plausible implication is that agravity’s naturalness program is inseparable from its specific RG behavior; it is not merely the statement that higher derivatives improve power counting.

4. Inflationary constructions and scale-invariant model building

Inflation is described as generic in agravity because the Einstein-frame potential is nearly flat whenever the underlying Jordan-frame theory is adimensional. In the original formulation the slow-roll parameters are given by beta-function combinations, and when the inflaton is identified with the field that generates the Planck scale—the “Higgs of gravity”—the resulting quadratic effective potential yields

$$S=\int d^4x\,\sqrt{|g|}\left[ \frac{R^2}{6f_0^2} +\frac{\frac13 R^2-R_{\mu\nu}^2}{f_2^2} +L_{\rm SM}^{\rm adim} +L_{\rm BSM}^{\rm adim} \right],$$5

for $$S=\int d^4x\,\sqrt{|g|}\left[ \frac{R^2}{6f_0^2} +\frac{\frac13 R^2-R_{\mu\nu}^2}{f_2^2} +L_{\rm SM}^{\rm adim} +L_{\rm BSM}^{\rm adim} \right],$$6 $$S=\int d^4x\,\sqrt{|g|}\left[ \frac{R^2}{6f_0^2} +\frac{\frac13 R^2-R_{\mu\nu}^2}{f_2^2} +L_{\rm SM}^{\rm adim} +L_{\rm BSM}^{\rm adim} \right],$$7-folds (Salvio et al., 2014).

A more general treatment in “Dynamically Induced Planck Scale and Inflation” embeds this mechanism into full agravity. After introducing the scalar graviton degree of freedom, inflation becomes effectively multifield, but the classical dynamics converges toward an attractor trajectory. Two limiting regimes are emphasized. If the Planckion dominates, agravity reproduces the quadratic-inflation result $$S=\int d^4x\,\sqrt{|g|}\left[ \frac{R^2}{6f_0^2} +\frac{\frac13 R^2-R_{\mu\nu}^2}{f_2^2} +L_{\rm SM}^{\rm adim} +L_{\rm BSM}^{\rm adim} \right],$$8; if the scalar graviton dominates, it reproduces Starobinsky inflation with

$$S=\int d^4x\,\sqrt{|g|}\left[ \frac{R^2}{6f_0^2} +\frac{\frac13 R^2-R_{\mu\nu}^2}{f_2^2} +L_{\rm SM}^{\rm adim} +L_{\rm BSM}^{\rm adim} \right],$$9

The full agravity prediction interpolates continuously between these limits,

$f_0$0

while keeping $f_0$1 (Kannike et al., 2015).

A more explicit particle-physics realization is given by the minimal classically scale-invariant and $f_0$2-symmetric extension of the Standard Model containing one additional complex gauge singlet scalar and three right-handed Majorana neutrinos. In that model the pseudo-Nambu–Goldstone boson of approximate scale symmetry is identified with the inflaton. Along the tree-level flat direction,

$f_0$3

and the canonically normalized inflaton $f_0$4 acquires a Coleman–Weinberg-type potential with

$f_0$5

The mass spectrum entering $f_0$6 includes the agravity states,

$f_0$7

and the inflaton mass is radiatively generated as

$f_0$8

In this realization, small $f_0$9 approaches quadratic chaotic inflation, while larger $f_2$0 flattens the potential; the paper states that the Planck data favor $f_2$1, $f_2$2, and $f_2$3, and that small-field inflation is the phenomenologically viable branch (Farzinnia et al., 2015, Farzinnia, 2015).

These inflationary results are model-dependent rather than universal statements about every agravity theory. What remains common across the constructions is the same gravitational sector, the absence of explicit mass scales, the dynamical induction of $f_2$4, and the use of RG-controlled flatness to support slow roll.

5. Ultra-Planckian scattering and recent amplitude results

Recent work extends agravity from model-building and RG analysis to explicit trans-Planckian scattering amplitudes. In “Ultra-Planckian Quark and Gluon Scattering in Agravity,” tree-level graviton-mediated QCD processes

$f_2$5

were computed from the agravity action

$f_2$6

The paper reports that all corresponding differential cross sections scale as

$f_2$7

in the ultra-Planckian regime, that positivity is process- and angle-dependent in channels such as $f_2$8, and that forward singularities are infrared effects from massless graviton exchange, regularized by finite quark masses where appropriate (Cunha et al., 26 May 2025).

An analogous Abelian analysis appears in “Scattering amplitudes in dimensionless quadratic gravity coupled to QED,” which studies photon–photon, fermion–fermion, fermion–photon, scalar–fermion, scalar–photon, scalar–scalar, and annihilation channels while retaining photon–graviton interference terms. The graviton propagator again behaves as $f_2$9, and the paper verifies explicit independence of the tree-level amplitudes on the gravitational gauge-fixing parameter. Across all channels, the corresponding differential cross sections exhibit the same universal scaling

$L_{\rm SM}^{\rm adim} = -\frac{F_{\mu\nu}^2}{4g^2} +\bar\psi i\slashed{D}\psi +|D_\mu H|^2 -(yH\psi\psi+\text{h.c.}) -\lambda_H |H|^4 -\xi_H |H|^2 R,$0

while many amplitudes show strong forward or backward enhancement associated with small momentum transfer (Cunha et al., 5 Mar 2026).

These calculations do not remove the long-standing interpretational issue associated with higher-derivative ghost modes. They do, however, show that at tree level the amplitude-level behavior expected of a UV-complete dimensionless gravity theory—most notably the $L_{\rm SM}^{\rm adim} = -\frac{F_{\mu\nu}^2}{4g^2} +\bar\psi i\slashed{D}\psi +|D_\mu H|^2 -(yH\psi\psi+\text{h.c.}) -\lambda_H |H|^4 -\xi_H |H|^2 R,$1 ultra-Planckian scaling—can be realized in explicit QCD and QED processes. This suggests a shift in emphasis within the agravity literature: from formal renormalizability and inflationary model building toward concrete high-energy observables.

6. Terminological ambiguity and relation to antigravity

The literature also contains a distinct use of the word “agravity.” In “Physical Interpretation of Antigravity,” the term denotes a geodesically complete Weyl-invariant framework that includes both gravity and antigravity sectors of spacetime connected across singularities. There the effective gravitational coupling is proportional to $L_{\rm SM}^{\rm adim} = -\frac{F_{\mu\nu}^2}{4g^2} +\bar\psi i\slashed{D}\psi +|D_\mu H|^2 -(yH\psi\psi+\text{h.c.}) -\lambda_H |H|^4 -\xi_H |H|^2 R,$2, the gravity region is

$L_{\rm SM}^{\rm adim} = -\frac{F_{\mu\nu}^2}{4g^2} +\bar\psi i\slashed{D}\psi +|D_\mu H|^2 -(yH\psi\psi+\text{h.c.}) -\lambda_H |H|^4 -\xi_H |H|^2 R,$3

and the antigravity region is

$L_{\rm SM}^{\rm adim} = -\frac{F_{\mu\nu}^2}{4g^2} +\bar\psi i\slashed{D}\psi +|D_\mu H|^2 -(yH\psi\psi+\text{h.c.}) -\lambda_H |H|^4 -\xi_H |H|^2 R,$4

In that usage, “agravity” is the full framework containing both sectors, while “antigravity” refers specifically to the region with negative effective Planck mass squared (Bars et al., 2015).

This is conceptually different from the dimensionless quadratic-gravity program of (Salvio et al., 2014). In the original agravity formulation, the phrase “anti-graviton” refers to the ghost-like massive spin-2 component generated by the four-derivative propagator; it does not denote repulsive gravity. Separate antigravity literatures pursue yet other mechanisms: topologically nontrivial spacetime defects with negative active gravitational mass (Klinkhamer et al., 2018), or antimatter-gravity scenarios in which opposite-sign gravitational charges could generate virtual gravitational dipoles in the quantum vacuum (Hajdukovic, 2024). This establishes that the same lexical family—agravity, antigravity, anti-graviton—covers several non-equivalent research programs.

A common misconception is therefore to identify agravity with repulsive gravity. The data do not support that identification. In dimensionless quadratic gravity, agravity is a renormalizable, scale-free theory built from $L_{\rm SM}^{\rm adim} = -\frac{F_{\mu\nu}^2}{4g^2} +\bar\psi i\slashed{D}\psi +|D_\mu H|^2 -(yH\psi\psi+\text{h.c.}) -\lambda_H |H|^4 -\xi_H |H|^2 R,$5 and $L_{\rm SM}^{\rm adim} = -\frac{F_{\mu\nu}^2}{4g^2} +\bar\psi i\slashed{D}\psi +|D_\mu H|^2 -(yH\psi\psi+\text{h.c.}) -\lambda_H |H|^4 -\xi_H |H|^2 R,$6 terms. In the geodesically complete Weyl-invariant literature, agravity names the union of gravity and antigravity patches. In antimatter-gravity or negative-mass-defect scenarios, antigravity refers to repulsive gravitational behavior of a very different kind. The shared vocabulary reflects overlapping historical intuitions, not a single unified definition.