The one-loop spectral action at $\mathcal{O}(\mathcal{R}^{2})$ in the Weyl basis is [Alfyorov:2026ff]

where $z\equiv\Box/\Lambda^{2}$, the normalized form factors satisfy $\hat{F}_{i}(0)=1$, and the coefficients are [Alfyorov:2026ff]

Linearizing the total action $S=S_{\mathrm{EH}}+\Gamma^{(1)}$ around flat space $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and decomposing via Barnes–Rivers spin projectors yields [Stelle:1976gc, vanDam:1970vg] the dressed propagator denominators

for the spin-2 (transverse-traceless) and spin-0 (scalar trace) sectors, respectively. Both satisfy $\Pi(0)=1$, recovering GR in the infrared. At conformal coupling $\xi=1/6$, the scalar sector decouples identically: $\Pi_{\mathrm{s}}\equiv 1$.

For a static point source $T_{00}=M\,\delta^{(3)}(\vec{x})$, the linearized field equations in Fourier space yield the temporal and spatial metric potentials

The potentials factorize through Newton kernels

with $K_{\Phi}(0)=K_{\Psi}(0)=1$ (Newtonian limit). The coefficients $4/3$, $2/3$, and $\pm 1/3$ are fixed by the spin decomposition and are universal for all quadratic-gravity theories [Stelle:1976gc, Edholm:2016].

In the local Yukawa approximation ($\hat{F}_{i}\approx 1$, valid for $k^{2}\ll\Lambda^{2}$), the propagator denominators reduce to $\Pi_{\mathrm{TT}}\approx 1+(13/60)z$ and $\Pi_{\mathrm{s}}\approx 1+6(\xi-1/6)^{2}z$, yielding the partial-fraction decomposition

where the effective masses are

The inverse Fourier transform, via the Gradshteyn–Ryzhik identity

gives the central result:

The corresponding spatial potential (from $K_{\Psi}$) is

Note the different coefficients: the spin-2 contribution enters with $2/3$ (not $4/3$) and the spin-0 with $-1/3$ (not $+1/3$), reflecting the different projections of the graviton propagator onto $g_{00}$ and $g_{ij}$.

At conformal coupling, the scalar term is absent:

The Yukawa amplitudes $-4/3$ (spin-2) and $+1/3$ (spin-0) are fixed by the spin structure of the graviton propagator and are independent of $\Lambda$, $\xi$, or any coupling constant. This is a key distinction from generic Yukawa-modified gravity scenarios, where the coupling strengths are free parameters.

1. 1.

   Finite at the origin. The potential ratio (12) satisfies $V(0)/V_{\mathrm{N}}(0)=1-4/3+1/3=0$: the Newtonian $1/r$ singularity is cancelled and the physical potential $V(r)\to-GM(4m_{2}/3-m_{0}/3)$ is finite as $r\to 0$.

2. 2.

   Newtonian recovery. Both Yukawa terms decay exponentially, so $V(r)\to V_{\mathrm{N}}(r)$ for $r\gg 1/\Lambda$.

3. 3.

   Mass ratio. At minimal coupling $\xi=0$: $m_{2}/m_{0}=\sqrt{10/13}\approx 0.877$, fixed by the SM spectrum.

4. 4.

   Scalar decoupling. At $\xi=1/6$, the scalar mass diverges and the potential reduces to (14). The Newtonian singularity reappears: $V(r)/V_{\mathrm{N}}(r)\to-1/3$ as $r\to 0$.

The potential ratio is plotted in Fig. 1 for several values of $\xi$.

The Eddington–Robertson–Schiff parameter $\gamma$ measures the ratio of the spatial and temporal metric potentials:

In GR, $\gamma=1$ identically. Since $\Phi_{\mathrm{N}}=\Psi_{\mathrm{N}}$ in the Newtonian limit, the ratio of Eqs. (13) and (12) gives

For $r\gg 1/m_{2}$ (which includes all solar-system distances for any viable $\Lambda$), both exponentials are negligible and $\gamma\to 1$ with corrections

The leading correction is positive, spin-2 dominated, and exponentially suppressed.

At $r=0$ with $\xi\neq 1/6$:

At minimal coupling ($\xi=0$), $\gamma(0)\approx 1.098$; at conformal coupling ($\xi=1/6$), $\gamma(0)=-1$, reflecting the absent scalar cancellation.

For $\Lambda$ satisfying the experimental bounds derived below, the departure from GR is negligible at all solar-system distances. The effective PPN parameters are listed in Table 1. The preferred-frame parameters $\alpha_{1}=\alpha_{2}=\alpha_{3}=0$ and the conservation-law parameters $\zeta_{i}=0$ follow from the diffeomorphism invariance of the spectral action [Will:2014, Chamseddine:1996zu].

The most precise tests of the gravitational inverse-square law at sub-millimeter distances are torsion-balance experiments by the Eöt-Wash group [Lee:2020, Kapner:2007, Adelberger:2009]. The most recent result [Lee:2020] excludes Yukawa deviations with $|\alpha|\geq 1$ at ranges $\lambda\geq 38.6\,\mu\mathrm{m}$ (95% CL).

The SCT spin-2 Yukawa has coupling $|\alpha_{1}|=4/3>1$, so the exclusion curve is crossed at a shorter range than the $|\alpha|=1$ boundary. From the published exclusion contour at $|\alpha|=4/3$:

giving

This is the primary laboratory bound on the SCT spectral scale. The corresponding physical length is $1/\Lambda=76.8\,\mu\mathrm{m}$, and the effective masses are $m_{2}\approx 5.5\times 10^{-3}\,\mathrm{eV}$ and $m_{0}\approx 6.3\times 10^{-3}\,\mathrm{eV}$ (at $\xi=0$). The scalar Yukawa ($|\alpha_{2}|=1/3$) produces a weaker constraint because $|\alpha_{2}|<1$ lies below the current exclusion curve at all explored ranges.

The Cassini spacecraft measurement of the Shapiro time delay [Bertotti:2003] constrains $|\gamma-1|<2.3\times 10^{-5}$ at $r\simeq 1\,\mathrm{AU}$. Using (17):

With $r_{\mathrm{AU}}\simeq 1.496\times 10^{11}\,\mathrm{m}$:

This is fourteen orders of magnitude weaker than the Eöt-Wash bound, reflecting the exponential suppression at astronomical distances.

The MESSENGER radioscience analysis [Verma:2014] provides $|\gamma-1|<2.5\times 10^{-5}$, yielding a comparable bound:

Casimir force measurements at sub-micrometer distances [Decca:2007, Chen:2016] require Yukawa couplings $|\alpha|\gtrsim 3000$ to be detectable, far exceeding the SCT values $|\alpha_{1}|=4/3$ and $|\alpha_{2}|=1/3$. At $r=0.1\,\mu\mathrm{m}$ with $\Lambda=2.565\times 10^{-3}\,\mathrm{eV}$, the potential deviation is only $\sim 0.3\%$, well below the experimental sensitivity. Casimir experiments provide no constraint on $\Lambda$.

Gravity Probe B [Everitt:2011], MICROSCOPE [Touboul:2022], Lunar Laser Ranging [Williams:2004], and atom interferometry experiments probe gravity at centimeter-to-orbital distances, where the SCT Yukawa corrections are suppressed by factors of $e^{-m_{2}r}\sim e^{-10^{2}}$ (atom interferometry at 1 cm) to $e^{-10^{13}}$ (Earth–Moon). These experiments provide zero effective constraint on $\Lambda$.

LLR also probes the Nordtvedt parameter $\eta_{N}=4\beta-\gamma-3$ [Williams:2004], but since $\beta$ has not been derived in the spectral action framework (it requires nonlinear field equations), this bound cannot yet be evaluated. If $\beta=1$ at leading order (as in GR), the Nordtvedt parameter vanishes automatically. (MICROSCOPE [Touboul:2022] tests the weak equivalence principle on test masses, not the Nordtvedt strong-EP parameter.)

The local Yukawa approximation replaces the full propagator denominator $\Pi_{\mathrm{TT}}(z)=1+(13/60)\,z\,\hat{F}_{1}(z)$ by its small-$z$ limit $1+(13/60)\,z$. The exact dressed propagator has a zero at $z_{0}\approx 2.41$ [Alfyorov:2026ff], yielding an exact spin-2 mass $m_{2}^{\mathrm{exact}}=\Lambda\sqrt{z_{0}}\approx 1.55\,\Lambda$, smaller than the Yukawa-approximation value $m_{2}^{\mathrm{loc}}\approx 2.15\,\Lambda$ by a factor of $1.38$. The exact Yukawa range is therefore longer than the approximation predicts, making the stated Eöt-Wash bound on $\Lambda$ conservative: the full nonlocal analysis would yield a tighter constraint.

For solar-system experiments ($r\gg 1/\Lambda$), the Yukawa corrections are exponentially suppressed regardless of the approximation, so the local and exact results are indistinguishable at current precision.

In Stelle’s local quadratic gravity [Stelle:1976gc, Stelle:1978], the potential has the same functional form (12) but with arbitrary masses $m_{2}$ and $m_{0}$ determined by two free coupling constants. The spectral action eliminates this freedom: the couplings $c_{2}=13/60$ and $3c_{1}+c_{2}=6(\xi-1/6)^{2}$ are computed from the SM particle content, yielding definite effective masses (9) and (10) as functions of $\Lambda$ alone (plus $\xi$).

The spin-2 mass $m_{2}$ corresponds to a ghost mode: the coefficient of $e^{-m_{2}r}$ in (12) is $-4/3<0$, indicating that the massive spin-2 graviton couples with wrong-sign residue. This is the well-known Ostrogradsky ghost of higher-derivative gravity [Stelle:1976gc, Woodard:2015zca]. In the spectral action framework, the dressed propagator $1/\Pi_{\mathrm{TT}}(z)$ has a pole at $z_{0}\approx 2.41$ with residue $R_{2}\approx-0.49$, approximately 50% suppressed relative to the Stelle value $R_{2}^{\mathrm{Stelle}}=-1$. The physical interpretation of this ghost (Lee–Wick field, fakeon, or dark matter candidate) is deferred to a dedicated analysis.

The modified potential (12) is determined by two inputs: the spectral scale $\Lambda$ and the Higgs non-minimal coupling $\xi$. The Yukawa amplitudes ($-4/3$, $+1/3$) are fixed by spin decomposition. The mass ratios $m_{2}/\Lambda$ and $m_{0}/\Lambda$ are fixed by the SM content. This structure means that a single measurement of the Yukawa range determines $\Lambda$ uniquely, and all other predictions follow.

The Eöt-Wash bound $\Lambda>2.565\times 10^{-3}\,\mathrm{eV}$ corresponds to a length scale $1/\Lambda<77\,\mu\mathrm{m}$, far above the Planck length $\ell_{P}\sim 10^{-35}\,\mathrm{m}$. The spectral scale is therefore not directly constrained to lie near the Planck mass by current experiments; it could be much higher. Future improvements in short-range gravity experiments, targeting $\lambda\lesssim 10\,\mu\mathrm{m}$, would push the bound to $\Lambda\gtrsim 10^{-2}\,\mathrm{eV}$.