Hyperbolic form factors for Yukawa interactions, and applications to the Earth

This paper introduces the hyperbolic form factor as a bilateral Laplace transform of density distributions to model Yukawa interactions mediated by massive particles, applying it to Earth's density profile to derive significantly relaxed coupling limits for new spin-0 and spin-1 mediators with masses around $10^{-12}$ eV/$c^2$.

The Gist

Imagine the Earth not just as a solid rock, but as a giant, layered cake. Now, imagine there is a new, invisible "ghost force" trying to push or pull on objects near this cake. This force is different from gravity because it has a limited range—it gets weaker very quickly as you move away, kind of like how the smell of a perfume fades as you walk out of the room.

This paper is about figuring out exactly how strong this "ghost force" feels when you are standing on a satellite orbiting the Earth, and how the Earth's internal layers (the crust, the mantle, the core) change that feeling.

Here is the breakdown of the paper's ideas using simple analogies:

1. The "Ghost Force" and the "Smell" Analogy

Scientists are looking for a new force that might explain things like dark matter or why the universe is expanding so fast. This force is carried by a particle called a "mediator."

• The Mediator: Think of this particle like a messenger. If the messenger is heavy, they get tired quickly and can only travel a short distance (short range). If they are very light, they can travel far (long range).
• The Range ($\lambda$): This is the "smell radius." If the force has a short range, only the Earth's outer skin (the crust) matters. If it has a long range, the whole Earth (including the heavy core) matters.

2. The Problem: The Earth is Messy

The Earth isn't a perfect, uniform ball of clay. It has a dense iron core, a rocky mantle, and a thin crust.

• The Old Way: To calculate how this ghost force interacts with the Earth, scientists usually had to build a complex "5-layer cake" model (Core, Mantle, Crust, etc.) and do heavy math for every single layer. It was like trying to calculate the wind resistance of a car by measuring every single bolt and screw.
• The New Idea: Pierre Fayet (the author) asks: Can we find a simple mathematical "shortcut" that gives us the same answer without the headache?

3. The "Hyperbolic Form Factor": The Earth's "Ghost Signature"

The paper introduces a mathematical tool called the Hyperbolic Form Factor.

• The Metaphor: Imagine the Earth is a speaker playing a song. The "Ordinary Form Factor" is the song played normally (like a standard radio wave). The "Hyperbolic Form Factor" is what happens if you play the song in a weird, hyper-accelerated time machine.
• Why it matters: This "weird song" tells us exactly how the Earth's density changes the strength of the ghost force. It acts like a volume knob.
  • If the Earth were a uniform ball of clay, the volume knob would be set to a specific number.
  • Because the Earth is dense in the middle and light on the outside, the volume knob is turned up. The force feels stronger than expected because the outer layers are "closer" to the satellite than the center is.

4. The "Effective Density": A Magic Average

The paper defines a concept called Effective Density.

• The Analogy: Imagine you are trying to guess how heavy a suitcase is by lifting it from different angles. If you lift it from the handle, it feels heavy. If you lift it from the bottom, it feels lighter.
• The "Effective Density" is a "magic average" that changes depending on how far away you are.
  • Far away (Long range): The Earth looks like a uniform ball. The magic average is the Earth's total average density.
  • Close up (Short range): The force only "sees" the outer shell. The magic average drops to match the density of the crust.
• The paper shows that this "magic average" smoothly slides from the Earth's center density to the surface density as you get closer.

5. The Big Discovery: Simple Shapes Work Best!

This is the most surprising part of the paper. The author tested two very simple, almost "silly" density models against the complex 5-layer Earth model:

1. The "1/R" Model: Imagine the Earth's density gets stronger the closer you get to the center, following a simple curve ($1/r$).
   • Result: This simple curve matched the complex Earth model surprisingly well for most distances.
2. The "Super-Mix" Model: The author took the "1/R" curve and mixed it with a straight-line curve (where density drops linearly from center to surface).
   • Result: This mix ($\rho' = \frac{5}{4} - \frac{r}{R} + \frac{R}{3r}$) was astonishingly accurate. It matched the complex 5-layer Earth model to within 0.7% for almost all practical distances!

Why is this cool? It means we don't need a supercomputer to model the Earth's core to test for new physics. A simple algebraic formula is good enough.

6. The Real-World Application: The MICROSCOPE Satellite

The paper applies this to the MICROSCOPE experiment, a satellite that dropped two different metal cylinders (Titanium and Platinum) in space to see if they fell at the exact same speed.

• The Test: If the "ghost force" exists, and if it couples differently to Titanium and Platinum, the cylinders would fall at slightly different speeds, violating Einstein's Equivalence Principle.
• The Result: The satellite found no difference (which is great for Einstein!).
• The New Limits: Using the new "simple formula" for the Earth's shape, the authors recalculated how strong this ghost force could be before we would have seen it.
  • They found that for certain "heavy" mediators (short range), the force could be 34 times stronger than previously thought and we still wouldn't have noticed it!
  • This pushes the boundaries of what we know about the universe, telling us exactly how "weak" this new force must be.

Summary

The paper is a masterclass in simplification. It takes a messy, complex problem (calculating forces inside a layered Earth) and shows that a few elegant mathematical tricks (hyperbolic functions and simple density curves) can replace years of complex modeling.

The Takeaway: The Earth is complex, but when it comes to detecting new, weak forces, the Earth behaves almost exactly like a simple, smooth mathematical curve. This allows scientists to set much tighter and more accurate limits on the existence of new "dark" forces in our universe.

Technical Summary

1. Problem Statement

The paper addresses the theoretical challenge of calculating the Yukawa potential induced by an extended body (specifically the Earth) for a new, finite-range interaction mediated by a light boson (mass $m = 1/\lambda$).

• Context: New physics beyond the Standard Model often predicts "dark sector" forces mediated by light spin-0 or spin-1 bosons (e.g., dark photons or $B-L$ gauge bosons). These forces would manifest as deviations from the Equivalence Principle (EP).
• The Challenge: Experiments like MICROSCOPE test the EP by measuring the differential acceleration of test masses (Titanium and Platinum) in Earth orbit. To set precise limits on the coupling strength of these new forces, one must accurately calculate the potential generated by the Earth's non-uniform density distribution.
• The Gap: Standard calculations often assume a homogeneous sphere or rely on complex multi-shell numerical models. There is a need for robust, analytic expressions for the "hyperbolic form factor" that accounts for the Earth's radial density profile $\rho(r)$, particularly for mediators with ranges $\lambda$ comparable to or smaller than the Earth's radius ($R$).

2. Methodology

The author develops a mathematical framework based on bilateral Laplace transforms and analytic continuation to define and compute hyperbolic form factors.

• Definition of Hyperbolic Form Factor ($\Phi(x)$):
  The paper defines the hyperbolic form factor for a spherically symmetric density $\rho(r)$ as the expectation value:
  $$\Phi(x) = \langle \cosh(\vec{k} \cdot \vec{r}) \rangle = \left\langle \frac{\sinh(kr)}{kr} \right\rangle$$
  where $x = kR = R/\lambda$, $k$ is the mediator mass, and $R$ is the Earth's radius.
  This is related to the ordinary (Fourier) form factor $f(k) = \langle \sin(kr)/kr \rangle$ via a duality transformation $k \to \pm ik$ (or $x^2 \to -x^2$).

• Potential Calculation:
  The external Yukawa potential $V(r)$ for a sphere of total charge $Q$ is expressed as:
  $$V(r) = \frac{Q e^{-kr}}{4\pi r} \Phi(x)$$
  The factor $\Phi(x)$ corrects the point-source potential to account for the extended density distribution.

• Effective Density ($\bar{\rho}(x)$):
  The author introduces an effective density $\bar{\rho}(x)$ such that the actual non-uniform sphere generates the same potential as a homogeneous sphere with density $\bar{\rho}(x)$. It is defined as:
  $$\bar{\rho}(x) = \rho_0 \frac{\Phi(x)}{\phi(x)}$$
  where $\phi(x)$ is the form factor for a homogeneous sphere. The paper proves that for decreasing density profiles ($d\rho/dr < 0$), $\bar{\rho}(x)$ decreases from the average density $\rho_0$ (at $x=0$) to the surface density $\rho(R)$ (at large $x$).

• Inversion Formula:
  A key theoretical contribution is an inversion formula allowing the recovery of the density distribution $\rho(r)$ from the analytic continuation of the form factor $\Phi(ix)$:
  $$\rho(r) = \rho_0 \frac{2R}{3\pi r} \int_0^\infty \Phi(ix) \sin\left(\frac{x r}{R}\right) x \, dx$$
  This establishes a direct link between the Laplace transform of the density and the physical potential.

3. Key Contributions

A. Analytic Approximations for Earth's Density

The paper demonstrates that complex, multi-shell Earth models (like the 5-shell PREM model) can be accurately approximated by simple analytic density profiles:

1. $1/r$ Profile: A density $\rho(r) \propto 1/r$ yields a simple form factor $\Phi(x) = (\frac{\sinh(x/2)}{x/2})^2$. This is accurate to within $\sim 1\%$ for $x \le 4$ ($\lambda \ge R/4$).
2. Hybrid Profile ($\rho'$): The author proposes a composite density profile:
   $$\rho'(r) = \rho_0 \left( \frac{5}{4} - \frac{r}{R} + \frac{R}{3r} \right)$$
   This profile combines a linear decrease with a $1/r$ term. It yields a remarkably accurate analytic expression for $\Phi(x)$:
   $$\Phi'(x) = \frac{7x^2 \cosh x - 24 \cosh x + 9x \sinh x - 4x^2 + 24}{4x^4}$$
   Result: This analytic expression matches the numerical 5-shell Earth model to within 0.7% for $x \le 64$ (corresponding to interaction ranges $\lambda \ge 100$ km).

B. Power Series Expansions

The paper provides power series expansions for these form factors in terms of the even moments of the density $\langle r^{2n} \rangle$.

• For the hybrid model $\rho'(r)$, the expansion is:
  $$\Phi'(x) \approx 1 + 0.0833 x^2 + 2.73 \times 10^{-3} x^4 + \dots$$
  This is nearly identical to the expansion derived from the 5-shell model ($1 + 0.0827 x^2 + \dots$), confirming that the Earth's moment of inertia ($I \approx 0.3308 MR^2$) is the dominant factor, while fine structural details (core/mantle discontinuities) are smoothed out for finite-range forces.

C. Application to MICROSCOPE Limits

The paper recalculates the constraints on new forces based on MICROSCOPE data, incorporating the derived form factors.

• Scaling Factor: The limits on coupling constants ($g$) for a massive mediator are rescaled by a factor proportional to $1/\sqrt{\Phi(x)}$ compared to the massless case.
• Quantitative Impact: For a mediator mass $m = 10^{-12}$ eV/c$^2$ (range $\lambda \approx 200$ km), the coupling limits are relaxed by a factor of $\sim 34$ compared to the massless limit.
  • For spin-1 mediators coupled to $B-L$: $|g_{B-L}| < 3.6 \times 10^{-24}$.
  • For spin-1 mediators coupled to $B$: $|g_B| < 2.6 \times 10^{-23}$.

4. Results

• Accuracy: The analytic form factor $\Phi'(x)$ derived from the hybrid density profile $\rho'(r)$ is valid to within 0.7% up to $x=64$ ($\lambda > 100$ km). This covers the entire range of interest for current and near-future EP tests.
• Robustness: The hyperbolic form factor is shown to be insensitive to the detailed internal structure of the Earth's core. Simple profiles capture the essential physics determined by the global moment of inertia.
• Inversion: The inversion formula successfully recovers density distributions (including singular ones like point sources) from their form factors, validating the mathematical consistency of the approach.

5. Significance

• Theoretical Utility: The paper provides a rigorous mathematical tool (the hyperbolic form factor and its inversion) for analyzing finite-range interactions in astrophysical and geophysical contexts.
• Experimental Relevance: By providing highly accurate analytic approximations for the Earth's form factor, the paper removes a major source of uncertainty in interpreting MICROSCOPE and future satellite experiments. It confirms that complex Earth models are not strictly necessary for setting limits on new forces with ranges $>100$ km.
• New Physics Constraints: The results significantly update the exclusion limits for ultralight bosons. The relaxation of limits by a factor of ~34 for $m \sim 10^{-12}$ eV/c$^2$ implies that the search for these forces must probe even weaker couplings than previously estimated if the mediator has a finite range.
• Generalization: The methodology applies to any spherically symmetric body and any finite-range interaction, making it a standard reference for "dark sector" force searches involving planetary bodies.

In summary, Pierre Fayet establishes that the Earth's response to finite-range forces can be accurately modeled by simple analytic density profiles, providing a powerful and precise tool for testing the Equivalence Principle and constraining new fundamental interactions.