Density Screening: Phenomenological Profile and Thin-Shell Replacement

Revision note (v13.0). This document is not a first-principles derivation, and the v12 file name is kept only so that links remain valid. The v12 derivations of the spatial factor $S_r$ and of the density factor $S_\rho$ are withdrawn. The profile $S_\rho=1-\tanh(\rho/\rho_c)$ survives only as a phenomenological interpolation used in the old forecasts. Corrected here: the factor-of-two inconsistency between $\tfrac12(1-\tanh)$ and $1-\tanh$, the position of the profile centre, the value of $S_\rho$ for platinum ($2.3\times10^{-17}$ at 21,450 kg/m³), and the overlap radius implied by $\rho_c=1.1\times10^{3}$ kg/m³ ($7.1\times10^{-11}$ m). Replaced by thin-shell screening of the scalar sector, in the screened scalar sector. See the architecture section of the main document.

1. Status

Nothing in this document is derived from first principles. The v12 version presented a "semi-microscopic EFT derivation" of the spatial factor $S_r(r)=e^{-r/\lambda_c}$ and a "disciplined EFT closure" of the density factor $S_\rho(\rho)$. In v13.0:

2. The Phenomenological Profile

The profile used in the v12 forecasts is

Properties, with $x=\rho/\rho_c$:

• $S_\rho(0)=1$. The profile is normalised to unity in vacuum by definition.
• $S_\rho(\rho_c)=1-\tanh(1)=0.238$.
• $S_\rho=\tfrac12$ at $x=\operatorname{artanh}(\tfrac12)=0.549$, that is at $\rho=604$ kg/m³.
• For $x\gg1$, $S_\rho\to2e^{-2x}$.

The tanh is centred at zero, not at $\rho_c$. Since $\tanh(\rho/\rho_c)$ is an odd function about $\rho=0$, the profile is the right half of a sigmoid centred at zero density. The parameter $\rho_c$ sets the width of the fall-off. It is not the density at which a transition occurs, and the description of $\rho_c$ as a transition or critical density in v12 is withdrawn. A transition centred at $\rho_c$ would need a different function, such as $\tfrac12[1-\tanh((\rho-\rho_c)/w)]$, with an extra parameter $w$.

2.1. Values for solid materials

Approximate room-temperature densities are used.

Platinum. For $\rho=21{,}450$ kg/m³, $\rho/\rho_c=19.5$ and $S_\rho=2/(e^{39}+1)=2.3\times10^{-17}$. A value of $S_\rho$ for platinum evaluated at $8{,}900$ kg/m³ appeared in earlier material. That density is not platinum. It is close to the densities of copper (about 8,960 kg/m³) and nickel (about 8,900 kg/m³). At 8,900 kg/m³ the profile gives $1.9\times10^{-7}$, which overstates the platinum value by about ten orders of magnitude.

Consequence. With this profile a composition signal from solid test masses is suppressed to an unmeasurable level. That follows from the chosen exponential fall-off and is not a derived result. It is one reason the profile is replaced by thin-shell screening, where the suppression depends on the shell thickness and not on an exponential of the local density.

3. The Withdrawn Overlap Derivation and the Factor of Two

The v12 argument defined an overlap probability

and set the screening factor proportional to the probability of not being screened:

It then used $S_\rho=1-\tanh(\rho/\rho_c)$ elsewhere, stating that the factor $\tfrac12$ is "absorbed" by redefining $\delta(Z,A)\to2\delta(Z,A)$. The two forms differ by exactly a factor of two at every density, and the absorption is not a consistent redefinition, for these reasons.

1. The derived form gives $S_\rho(0)=\tfrac12$. It would mean that half of the coupling is screened in vacuum, which contradicts the use of an unscreened coupling in vacuum in every forecast.
2. The assumed overlap probability is unphysical at zero density. $P_{\rm overlap}(0)=\tfrac12$, but an overlap probability must vanish as $\rho\to0$. The form was chosen to produce the tanh and was not derived from a model of the medium.
3. The absorption changes the meaning of $C$, and v12 stated it in the wrong direction. Absorbing the factor into $\delta$ changes the coefficient $C$ by a factor of two, so $C=0.03$ in one convention is a different number in the other. To rewrite $\delta\cdot\tfrac12(1-\tanh)$ as $\delta_{\rm eff}\cdot(1-\tanh)$ one needs $\delta_{\rm eff}=\delta/2$, whereas v12 wrote $\delta\to2\delta$. The v12 statement that the quoted coefficient $2.36\times10^{-7}$ "already incorporates this factor of two" was never shown, and that coefficient is itself withdrawn (the correct value is $C\varepsilon=4.13\times10^{-5}$ at $C=0.03$).

Resolution. The profile is normalised by definition to $S_\rho(0)=1$, so the form is $1-\tanh(\rho/\rho_c)$. No factor $\tfrac12$ enters and no redefinition of $\delta$ is made. The coefficient $C$ is defined once, in the species couplings of the Level 1 action. The statistical-field-theory derivation of the tanh is withdrawn.

4. The Withdrawn Spatial Factor $S_r$

The v12 text derived $S_r=e^{-r/\lambda_c}$ from the decay of the coherence of a "vacuum state" with an assumed decoherence rate $\Gamma\propto\alpha_{\rm EME}\mathcal{N}$ and a propagation time $t=r/c$, giving $\lambda_c=\hbar c/(\alpha_{\rm EME}E_{\rm ZPF})$. It stated that $\Gamma$ was "calculated from the QFT self-energy diagram". No such calculation exists in the corpus. The step from a decoherence rate to an exponential range factor was assumed, and $\alpha_{\rm EME}$ was not shown to be dimensionless or equal to one. The universal $S_r$ with fixed $\lambda_c=1\ \mu$m is withdrawn.

The thermal wavelength $\hbar c/k_BT=7.63\ \mu$m at 300 K may be cited as a possible origin of a micrometre scale. The electron mass cancels in that expression, so it is not a derivation. The replacement is the environment-dependent range

with $m_{\rm eff}(\rho)$ obtained from the effective potential of the scalar sector (see the screened scalar sector and the quantum-mechanical foundation).

5. The Overlap Radius Implied by $\rho_c$

The v12 text related the critical density to an overlap radius through

and then used the measured benchmark $\rho_c=1.1\times10^{3}$ kg/m³ to fix $r_{\rm overlap}$. Inverting with $m_p=1.6726\times10^{-27}$ kg:

The mean spacing of protons at this density is $(m_p/\rho_c)^{1/3}=1.15\times10^{-10}$ m.

This radius is atomic in size (1.35 Bohr radii). It is 185 times larger than the $3.86\times10^{-13}$ m used in v12 as the size of the "microscopic QVP cloud" (the electron reduced Compton wavelength). If clouds of radius $3.86\times10^{-13}$ m began to overlap at one per cloud volume, the density would be $3m_p/(4\pi(3.86\times10^{-13}\ {\rm m})^3)=6.9\times10^{9}$ kg/m³, about $6\times10^{6}$ times $\rho_c$. The v12 reading of $\rho_c$ as the onset of overlap of the stated clouds is therefore inconsistent with its own cloud size. Neither length is derived. At v13.0, $\rho_c$ is a fitted width parameter of an interpolation, with no mechanism that places it near 1.1×10³ kg/m³.

6. Replacement: Thin-Shell Screening

In the v13.0 scalar sector, dense bodies are screened by the thin-shell mechanism of chameleon-type models, which replaces the ad hoc density factor. The following is a summary. The derivation, the conditions and the open calculations are in Sections 3, 4 and 8 of the screened scalar sector.

• A body of radius $R$ sources the scalar only from a shell of thickness $\Delta R$ near its surface. The external scalar force is reduced by about $3\Delta R/R$ relative to an unscreened body of the same mass, and the effective coupling of a screened body is $\beta_{\rm eff}\approx3(\Delta R/R)\,\beta$.
• The screening depends on the body's gravitational potential and on the field value in the surrounding medium, and so on the environment, not on a single critical density.
• The range of the force is the environment-dependent $\lambda(\rho)$ from the effective mass, and it is long in a vacuum chamber.
• Low-density bodies and single atoms are expected to be unscreened. That is why atom interferometry near a low-density source mass is a test of this class of model.
• The screening and range factor $f$ for a laboratory source has not been computed. The simplest range estimate is $f\approx e^{-r/\lambda(\rho)}$.
• The Earth and the Sun must satisfy thin-shell conditions that follow from MICROSCOPE and Cassini. For the legacy benchmark the Earth requirement is $3\,\Delta R_\oplus/R_\oplus\lesssim1.9\times10^{-7}$. A first-pass estimate for uniform spheres ($n=1$, $\Lambda=2.4$ meV, $\beta_0=0.053$) gives $1.2\times10^{-7}$ at a galactic ambient density of $1.7\times10^{-21}$ kg/m³ and $5.1\times10^{-8}$ at an interplanetary density of $10^{-20}$ kg/m³, so the benchmark passes by a factor of about 1.5 to 4. The Sun gives $3\,\Delta R_\odot/R_\odot\approx4\times10^{-11}$. The full calculation is open. The numbers are in section 4 of the screened scalar sector.

The tanh profile is retained only so that the v12 plots can be reproduced. It must not be quoted as a derived suppression.

7. Summary of Corrections