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Scalar-Tensor Gravity
Yukawa Corrections
Experimental Physics
Topological Sectors

Hidden Topology → Testable Gravity: A Minimal Scalar–Tensor Module with a Yukawa Signature

Abstract

We present a minimal scalar–tensor module linking a hidden topological sector to a testable Yukawa correction to Newtonian gravity. A topological susceptibility χtop generates a scalar mass mφ; a universal Jordan-frame coupling β to matter yields a Newton–Yukawa potential with strength αy = 2β² and range λ = mφ-1. We define a falsifiable target band in (αy,λ)-space that is automatically consistent with solar-system post-Newtonian tests and not excluded by sub-millimeter torsion-balance and micro-cantilever null results, yet accessible to near-term tabletop experiments. The module is EFT-consistent, technically natural, and independent of flavor-physics sectors.

Motivation and Related Context

Light scalars coupled to matter appear in scalar–tensor gravity (e.g., Brans–Dicke), string/dilaton scenarios, and screened models such as chameleons and symmetrons. Our module differs by attributing the scalar mass to atopological susceptibility and by targeting a clean two-parameter Newton–Yukawa signature (no screening), chosen to be testable yet consistent with existing bounds.

From Hidden Topology to a Light Scalar

Let Q(x) be a hidden-sector topological density (e.g., a generalized instanton density or four-form field strength). Its susceptibility,

$$\chi_{\rm top} \equiv \int d^4x\,\langle Q(x)Q(0)\rangle$$

softly breaks the shift symmetry of the associated mode φ, inducing

$$m_\phi^2 = \frac{\chi_{\rm top}}{f_\phi^2}$$

with fφ an effective decay constant. Small mφ is technically natural: loop corrections scale as δmφ² ~ β²Λ²/(16π²) and are negligible for β≪1 and modest EFT cutoff Λ.

Universal Coupling and the Newton–Yukawa Law

Assume a universal Jordan-frame coupling. The action includes:

$$S \supset \int d^4x\sqrt{-g}\,\left[\frac{M_P^2}{2}\left(1+\frac{2\beta\,\phi}{M_P}\right)R -\frac{1}{2}(\partial\phi)^2-\frac{1}{2}m_\phi^2\phi^2\right] + S_{\rm m}[g_{\mu\nu},\Psi]$$

Linearized exchange between static sources yields the Newton–Yukawa potential:

$$V(r) = -\frac{G_N m_1 m_2}{r}\left[1+\alpha_y\,e^{-r/\lambda}\right]$$
where αy=2β² and λ=mφ-1

The fractional force deviation for separation r is:

$$\frac{\Delta F}{F_N} = \alpha_y\left(1+\frac{r}{\lambda}\right)e^{-r/\lambda}$$

Established Constraints and Testable Target

Solar-System PPN (Long Range)

For λ≪AU, Yukawa effects on time delay/deflection are exponentially suppressed. Cassini's radio-link test constrains |γPPN-1| at the 10-5 level; with e-AU/λ≈0 for our short ranges, PPN bounds are automatically satisfied and do not directly limit αy.

Laboratory Short Range (Sub-mm to mm)

Null results from torsion balances and micro-cantilevers set strong limits in the λ~10–500 μm window: Kapner et al. (Eöt-Wash, 2007), Tan et al. (HUST, 2016), and the updated Eöt-Wash millimeter-range test (2020). Chiaverini et al. (2003) provide micro-cantilever constraints near 10–100 μm. We position our target band outside these confirmed exclusions while keeping it accessible to tabletop searches at decimeter scales.

Goldilocks Target (Experiment-Facing)

Falsifiable Target Region
αy ≤ 10-4, λ ≤ 0.1–0.2 m

This region is exponentially invisible to PPN tests and not already excluded by established sub-mm limits, yet yields measurable ΔF/FN via meter-scale apparatus.

Micro-to-Macro Map and Priors

Microscopic parameters (χtop, fφ, β) map to observables by:

$$\alpha_y=2\beta^2, \quad \lambda=\frac{1}{m_\phi}=\frac{f_\phi}{\sqrt{\chi_{\rm top}}}$$

Technically natural priors take β≪1 and allow broad λ via χtop/fφ². These are agnostic to the specific UV origin of Q(x).

Radiative Stability and EFT Consistency

With β≪1, δmφ²∝β² is small, so mφ~λ-1 in the target band is stable. Laboratory momenta justify linearized EFT; screening mechanisms are unnecessary in this regime.

How to Test the Model

Desk Overlay

Plot (αy,λ) from a scan over (β,mφ) on a standard αy–λ exclusion plot. Accept model points that lie outside the published sub-mm exclusions and within the target box.

Tabletop Forecast

For a source–probe separation r in the 0.1–0.5 m range, the force equation gives the modulation amplitude. Changing r toggles the exponential, providing a clean on/off handle against systematics.

Parameter Space Visualization

Yukawa Parameter Space
(Conceptual plot - replace with digitized experimental data)
log(αy) vs log(λ)
Goldilocks Target
Range λ (m) →
← Strength αy
Blue dashed rectangle: proposed target band safe from solar-system bounds and sub-mm exclusions, yet testable with tabletop setups.

Conclusions

A hidden topological sector can endow a light scalar with a technically natural mass and a universal matter coupling that together generate a clean Newton–Yukawa signature. The resulting two-parameter space (αy,λ) contains a Goldilocks region that is (i) invisible to solar-system tests, (ii) not excluded by established sub-mm null results, and (iii) decisively testable in tabletop experiments.

References

Solar-system / PPN:
  • C. M. Will, The Confrontation between General Relativity and Experiment, Living Rev. Relativ. 17, 4 (2014).
  • B. Bertotti, L. Iess, and P. Tortora, A test of general relativity using radio links with the Cassini spacecraft, Nature 425, 374–376 (2003).
Sub-mm / short-range tests:
  • D. J. Kapner et al., Tests of the Gravitational Inverse-Square Law below the Dark-Energy Length Scale, Phys. Rev. Lett. 98, 021101 (2007).
  • J. Chiaverini et al., New Experimental Constraints on Non-Newtonian Forces below 100 Micrometers, Phys. Rev. Lett. 90, 151101 (2003).
  • W.-H. Tan et al., New Test of the Gravitational Inverse-Square Law at the Submillimeter Range, Phys. Rev. Lett. 116, 131101 (2016).
  • J. G. Lee, E. G. Adelberger, T. S. Wagner, and B. R. Heckel, New Test of the Gravitational Inverse-Square Law at Millimeter Ranges, Phys. Rev. Lett. 124, 101101 (2020).
Context: scalar-tensor, dilaton, screening:
  • C. Brans and R. H. Dicke, Mach's Principle and a Relativistic Theory of Gravitation, Phys. Rev. 124, 925–935 (1961).
  • T. Damour and A. M. Polyakov, The String Dilaton and a Least Coupling Principle, Nucl. Phys. B 423, 532–558 (1994).
  • J. Khoury and A. Weltman, Chameleon Fields: Awaiting Surprises for Tests of Gravity in Space, Phys. Rev. Lett. 93, 171104 (2004).
  • K. Hinterbichler and J. Khoury, Symmetron Fields: Screening Long-Range Forces through Local Symmetry Restoration, Phys. Rev. Lett. 104, 231301 (2010).