Yukawa and Fifth-Force Searches

A Yukawa search turns the mass and charge distribution of an apparatus into a force template labeled by a range $\lambda$ and a coupling strength. The template is useful precisely because it is operational: it predicts a force or torque for a specified geometry. It is not, by itself, a complete theory, and a composition-independent result cannot automatically be applied to a composition-dependent mediator.

Required background. Weak-Field Gravity supplies the Newtonian limit and potential-to-force map; Probability and Statistical Inference supplies nuisance-parameter and interval reasoning. Helpful background. Classical Field Theory supplies the massive Green function and matter-source language.

Massive exchange and its force law

For pointlike, universally coupled bodies, write

$$V(r)=-\frac{Gm_1m_2}{r} \left(1+\alpha e^{-r/\lambda}\right), \qquad F_r=-\frac{Gm_1m_2}{r^2} \left[1+\alpha\left(1+\frac{r}{\lambda}\right)e^{-r/\lambda}\right].$$

The radial force is negative for attraction. Differentiating the potential supplies the factor $1+r/\lambda$; omitting it is a common error. In units with $\hbar=c=1$, a mediator of mass $m_\phi$ has $\lambda=m_\phi^{-1}$. Thus the exponential has a dimensionless argument, $\alpha$ is dimensionless, and $Gm_1m_2/r$ has energy dimensions.

A nonuniversal mediator couples to charges $Q_A$, not simply to inertial masses. For an attractive scalar exchange with potential $-g^2Q_AQ_Be^{-r/\lambda}/(4\pi r)$, a convenient pair-dependent coordinate is

$$\alpha_{AB} = \frac{g^2}{4\pi G} \frac{Q_A}{m_A}\frac{Q_B}{m_B},$$

where the normalization and dimensions of $g$ and $Q$ must be stated. With dimensionless $Q$ and dimensionless $g$ in natural units, the displayed ratio is dimensionless because $Gm_Am_B$ is dimensionless. Vector exchange between like charges is repulsive and has the opposite sign in this potential convention; it corresponds to negative $\alpha_{AB}$. Changing test material changes the charge-to-mass ratio and hence the signal. A bound on a universal $\alpha$ is therefore not automatically a bound on baryon number, neutron excess, lepton number, or another charge direction. The Yukawa parameterization and its domain as a single-mediator approximation are reviewed in Adelberger, Heckel, and Nelson 2003, §§II–III.

The limits clarify the physics. At $r\gg\lambda$, the new term is exponentially small. At $r\ll\lambda$, it approaches a rescaling of the inverse-square force, so an experiment with only a narrow span of small separations is degenerate with its absolute force calibration. At $\alpha=0$, the Newtonian result is recovered for every $\lambda$. These are checks on both analytic formulas and numerical kernels.

Finite bodies are part of the prediction

For density fields $\rho_A$ and charge-per-mass fields $q_A$, the additional interaction is

$$U_{\rm Y} = -G\alpha_0 \int_{V_1}d^3x\int_{V_2}d^3x'\, \rho_1(\mathbf x)q_1(\mathbf x) \rho_2(\mathbf x')q_2(\mathbf x') \frac{e^{-R/\lambda}}{R}, \qquad R=\lvert\mathbf x-\mathbf x'\rvert.$$

Force and torque follow by differentiating with respect to the measured translation or rotation. The integral must include finite thicknesses, gaps, coatings, patterned holes, glue layers, roughness when relevant, and metrology uncertainty. Lee and collaborators used a Fourier–Bessel treatment of a patterned torsion balance rather than a point-mass substitution; see Lee et al. 2020, Eq. (1), Fig. 1, and fit equation on p. 4.

An infinite-slab check is especially useful. Two uniform slabs of densities $\rho_1,\rho_2$, thicknesses $t_1,t_2$, and vacuum gap $s$ have Yukawa pressure

$$\frac{F_{\rm Y}}{A} = -2\pi G\alpha\rho_1\rho_2\lambda^2 e^{-s/\lambda} \left(1-e^{-t_1/\lambda}\right) \left(1-e^{-t_2/\lambda}\right).$$

One can derive this result in three steps. For two layers a normal distance $d$ apart, the transverse kernel integrates to

$$\int_0^\infty 2\pi R_\perp\,dR_\perp\, \frac{e^{-\sqrt{R_\perp^2+d^2}/\lambda}}{\sqrt{R_\perp^2+d^2}} =2\pi\lambda e^{-d/\lambda}.$$

Integrating each layer depth gives two factors $\lambda(1-e^{-t_i/\lambda})$. Finally $F/A=-\partial(U/A)/\partial s$ removes one factor of $\lambda$, leaving the displayed pressure. This calculation assumes infinite lateral extent and an additive linear interaction.

The sign is attractive for $\alpha\gt0$. When either slab is much thinner than $\lambda$, its factor becomes $t_i/\lambda$, correctly making the response proportional to its areal mass. When both are thick, only material within a few ranges of each surface contributes. A finite-element or direct-quadrature calculation should reproduce this result as the lateral dimensions grow.

From a kernel to an experimental constraint

For each trial $\lambda$, precompute a unit-coupling template $\mathbf T_{\rm Y}(\lambda)$ from the measured geometry. A linear forward model can then be written

$$\mathbf d = \mathbf T_{\rm N} +\alpha\,\mathbf T_{\rm Y}(\lambda) +A_{\rm e}\mathbf T_{\rm e} +A_{\rm m}\mathbf T_{\rm m} +\mathbf J\delta\boldsymbol\zeta +\boldsymbol\epsilon.$$

Here $\mathbf d$ may contain force or torque harmonics at many separations; $\mathbf T_{\rm e}$ and $\mathbf T_{\rm m}$ are electrostatic and magnetic controls; and $\delta\boldsymbol\zeta$ represents alignment and separation errors. The likelihood, covariance, prior measure, and interval construction determine the reported constraint. A prior uniform in $\alpha$ differs from one uniform in $\log\lvert\alpha\rvert$, especially near zero. Composition-dependent analyses must share material-assay nuisance parameters across runs rather than refitting unrelated effective couplings.

Three independent checks should precede interpretation:

1. reproduce the Newtonian geometric signal and its calibration;
2. recover a blinded Yukawa injection at several $\lambda$ values, including one near a body thickness and one near an edge dimension; and
3. compare the same extended geometry using direct volume quadrature and an independent spectral or multipole method.

The single-Yukawa description breaks down for multiple mediators, nonlinear screening, environment-dependent masses or charges, and strong coupling that invalidates the linear field equation. Those cases require their own boundary-value problem, not a relabeled exclusion curve.

Exercises

Derive the thin-slab scaling of the pressure when $t_1,t_2\ll\lambda$, and explain what apparatus parameter becomes degenerate with $\alpha$ if every measured separation also satisfies $s\ll\lambda$.

Solution

Expand each thickness factor as $1-e^{-t_i/\lambda}\simeq t_i/\lambda$. Substitution gives $F_{\rm Y}/A\simeq-2\pi G\alpha\rho_1\rho_2t_1t_2e^{-s/\lambda}$, so the result depends on the two areal mass densities. If $s\ll\lambda$, then $e^{-s/\lambda}\simeq1-s/\lambda$ changes only weakly across the data. The leading Yukawa contribution is therefore nearly a constant rescaling of the calibrated gravitational force. Absolute torque scale, density, or thickness uncertainty can mimic $\alpha$ unless the experiment spans separations comparable to $\lambda$ or supplies an independent null geometry.

References

• Adelberger, E. G., Heckel, B. R., and Nelson, A. E., “Tests of the Gravitational Inverse-Square Law,” Annual Review of Nuclear and Particle Science 53 (2003) 77–121, DOI.
• Lee, J. G., Adelberger, E. G., Cook, T. S., Fleischer, S. M., and Heckel, B. R., “New Test of the Gravitational $1/r^2$ Law at Separations down to 52 Microns,” Physical Review Letters 124 (2020) 101101, DOI.

Further reading

• Murata, J., and Tanaka, S., “A Review of Short-Range Gravity Experiments in the LHC Era,” Classical and Quantum Gravity 32 (2015) 033001, DOI.