Screened Scalars in the Laboratory

A screened scalar cannot be tested by placing a universal Yukawa curve over laboratory data. Its mass, coupling, and source charge depend on the local field, which in turn depends on the chamber walls, vacuum density, source and detector geometry, ambient boundary value, and sometimes the field history. The prediction is therefore a nonlinear boundary-value problem attached to a particular apparatus.

Required background. Weak-Field Gravity supplies the Newtonian background and force observable; Probability and Statistical Inference supplies likelihood and nuisance-parameter reasoning. Helpful background. Classical Field Theory supplies variational field equations, boundary data, and stability tests.

Density-dependent scalar profiles

Consider an Einstein-frame scalar with matter coupled to $A^2(\phi)g_{\mu\nu}$. In a static, weak-field laboratory with nonrelativistic density $\rho(\mathbf x)$,

$$\nabla^2\phi = V_{,\phi}(\phi)+\rho(\mathbf x)A_{,\phi}(\phi) \equiv V_{{\rm eff},\phi}(\phi;\rho).$$

Here $\rho$ is the conserved nonrelativistic rest-mass density used in the Einstein-frame effective potential $V_{\rm eff}=V+\rho A$; pressure and relativistic matter corrections are neglected. Other density conventions move factors of $A$ into the source. A chameleon example has

$$V(\phi)=\frac{\Lambda^{4+n}}{\phi^n}, \qquad A(\phi)\simeq1+\frac{\beta\phi}{M_{\rm Pl}},$$

with $\Lambda\gt0$, $n\gt0$, $\beta\gt0$, $\phi\gt0$, and $\lvert\beta\phi/M_{\rm Pl}\rvert\ll1$ for the linearized coupling. Setting $V_{{\rm eff},\phi}=0$ gives

$$\phi_{\min}=\left(\frac{n\Lambda^{4+n}M_{\rm Pl}}{\beta\rho}\right)^{1/(n+1)}, \qquad m_{\rm eff}^2=\frac{n(n+1)\Lambda^{4+n}}{\phi_{\min}^{n+2}}.$$

Increasing density lowers the minimum and raises the mass: $m_{\rm eff}\propto\rho^{(n+2)/[2(n+1)]}$. A denser region therefore has a shorter response length. These are homogeneous-medium equilibrium results; a finite chamber need not reach that minimum. A symmetron example instead suppresses its coupling in dense regions,

$$V(\phi)=-\frac{\mu^2}{2}\phi^2+\frac{\lambda_s}{4}\phi^4, \qquad A(\phi)\simeq1+\frac{\phi^2}{2M^2}.$$

Here $M_{\rm Pl}=(8\pi G)^{-1/2}$ is the reduced Planck mass, $\mu,M\gt0$, and $\lambda_s\gt0$. The quadratic coefficient becomes $(\rho/M^2-\mu^2)/2$. Above $\rho=\mu^2M^2$ the homogeneous stable minimum is $\phi=0$, where $d\ln A/d\phi$ vanishes; below it the two homogeneous minima are $\phi=\pm\sqrt{(\mu^2-\rho/M^2)/\lambda_s}$. This shows explicitly that mass screening and coupling screening are different mechanisms Burrage and Sakstein 2018, Eqs. (2.7)–(2.15) and §2.2.

A test body feels a scalar acceleration schematically proportional to $-\nabla\ln A(\phi)$. But only the unscreened fraction of an extended body may source or respond to the field. A thin-shell formula derived for an isolated sphere in a homogeneous environment is an asymptotic check; it is not a substitute for solving a patterned source, shield, wall, and detector. The field must cross dense walls whose thickness may exceed its local Compton wavelength, enter a low-density chamber whose size may prevent it from reaching its vacuum minimum, and adjust around each body.

The apparatus is the boundary condition

A well-posed laboratory calculation specifies:

• the full chamber and wall domains, material densities, coatings, shield, source, sensor, and vacuum-gas profile;
• either a far-field value at an exterior boundary or a sufficiently extended exterior material domain;
• continuity conditions for $\phi$ and its normal derivative when the action has no singular surface term;
• the branch or initial continuation used when $V_{\rm eff}$ has multiple minima; and
• the force observable, obtained from the detector response or a stress integral rather than a field value at one point.

The chamber-size and wall-density dependence is not a small correction. Burrage and Sakstein 2018, §3.4 and Eq. (3.35) explains why the field may fail to reach its density minimum in a finite vacuum and why nonspherical geometries require numerical treatment.

Numerically, continue from a weak-coupling solution, solve the nonlinear elliptic equation on at least three meshes, and monitor both the field-equation residual and the integrated force. Compare with the unscreened linear solution, a one-dimensional planar limit, and the spherical thin-shell limit. Vary the outer boundary location, chamber size, wall thickness and density, vacuum density, initial guess, and solver branch. A force that moves substantially under any unphysical numerical choice is not a stable apparatus prediction.

Results and limitations through 2026-08-19

Position or test class	Evidence or result	Limitation and counterevidence
Torsion and short-range force searches	Null residuals constrain screened models after an apparatus-specific torque calculation	Patterned geometry, electrostatic shield, nonlinear source charge, and material backgrounds prevent a universal Yukawa reinterpretation
Atom and neutron probes	Source modulation and quantum spectroscopy constrain specified chameleon-like potentials in selected chambers	Atoms may be unscreened while the source is screened; surface, magnetic, optical, and chamber-field models remain shared degeneracies
Symmetron and generalized screening	Some chamber configurations have a field gradient only in a restricted mass and symmetry-breaking range	For too-long wavelengths the field cannot change across the apparatus; for too-short wavelengths the response is exponentially suppressed, giving genuine null sensitivity

The conservative position is that existing null results exclude regions only after a valid nonlinear apparatus map. A competing shortcut treats published Yukawa or chameleon contours as environment-independent constraints; evidence against that shortcut is the explicit wall, chamber-size, and source-screening dependence. The status remains model-dependent because quantum stability, strong-coupling cutoffs, thermal corrections, material response, and numerical branches need not be controlled by the same low-energy parameterization.

An adversarial independent check compares a resolved three-dimensional solution with planar, spherical thin-shell, and unscreened limits, then changes the chamber size or wall density in both calculation and experiment. A real screened response should follow the predicted environmental change; an electrostatic or thermal residual generally follows a different control.

From field solution to inference

For data $\mathbf d$ and an apparatus prediction $\mathbf s(\theta,\boldsymbol\nu)$,

$$\mathcal L(\theta,\boldsymbol\nu) \propto \exp\left[ -\frac12 \bigl(\mathbf d-\mathbf s-\mathbf b\bigr)^{\mathsf T} C^{-1} \bigl(\mathbf d-\mathbf s-\mathbf b\bigr) \right]\mathcal L_{\rm aux}(\boldsymbol\nu).$$

This expression assumes fixed covariance; include its determinant when $C$ varies with the fitted parameters. The factor $\mathcal L_{\rm aux}$ is the likelihood of independent calibration measurements, while priors multiply the complete likelihood only when forming a posterior. The theory parameters $\theta$ specify $V$ and $A$, not merely an effective $\alpha$ and $\lambda$. Nuisance variables include wall and source dimensions, densities, vacuum pressure, distance zero, field calibration, electrostatic and Casimir backgrounds, and solver error. Restrict the calculation to regions where its effective action and numerical approximation apply; a failed approximation is not an experimental exclusion. Negative evidence means the tested response is absent under those conditions, not that every screened completion is absent.

What could change the conclusion. The interpretation could change when a new calibrated dataset changes the environmental controls, a corrected three-dimensional solution moves an exclusion, an independent solver finds another physical branch, or a theory calculation changes the strong-coupling or quantum-stability domain. A tighter contour obtained from the same uncontrolled boundary approximation is not a sufficient trigger.

Exercises

A symmetron has a vacuum Compton wavelength much larger than the chamber diameter, while dense walls restore the symmetric phase. Explain why interpreting a null force as a strong coupling bound can fail.

Solution

The field must vary from its wall value to a symmetry-broken vacuum value before it can develop a gradient. If the Compton wavelength is much larger than the chamber, gradient energy makes that variation too costly, so the field may remain near the wall-imposed value throughout the apparatus. The predicted force is then small even at substantial microscopic coupling. Therefore the null measurement has little sensitivity in this regime. Solving the finite chamber and varying its size distinguish this physical loss of sensitivity from an excluded coupling; inserting only a point-particle Yukawa force would incorrectly claim a bound.

References

• Burrage, C., and Sakstein, J., “Tests of Chameleon Gravity,” Living Reviews in Relativity 21 (2018) article 1, DOI, Open PDF.

Further reading

• Hinterbichler, K., and Khoury, J., “Screening Long-Range Forces through Local Symmetry Restoration,” Physical Review Letters 104 (2010) 231301, DOI.
• Khoury, J., and Weltman, A., “Chameleon Fields: Awaiting Surprises for Tests of Gravity in Space,” Physical Review Letters 93 (2004) 171104, DOI.