#### Limits on $\mathit R$ from Deviations in Gravitational Force Law

This section includes limits on the size of extra dimensions from deviations in the Newtonian (1/$\mathit r{}^{2}$) gravitational force law at short distances. Deviations are parametrized by a gravitational potential of the form $\mathit V~=~−(\mathit G$ $\mathit m$ $\mathit m'/\mathit r$) [1 $+$ $\alpha$ exp($−\mathit r/R$)]. For $\delta$ toroidal extra dimensions of equal size, $\alpha$ = 8$\delta$/3. Quoted bounds are for $\delta$ = 2 unless otherwise noted.

VALUE ($\mu {\mathrm {m}}$) CL% DOCUMENT ID TECN  COMMENT
• • We do not use the following data for averages, fits, limits, etc. • •
1
 2021
Optical levitation
2
 2021
Neutron scattering
3
 2020
Torsion pendulum
$<37$ 95 4
 2020 A
Torsion pendulum
5
 2018
MICR Space accelerometer
6
 2018 A
MICR Space accelerometer
7
 2017 A
Torsion oscillator
8
 2013
Nuclei properties
9
 2011
Torsion oscillator
10
 2011
Torsion pendulum
11
 2010
Microcantilever
12
 2009
Torsion pendulum
13
 2008
Microcantilever
14
 2008
Newton's constant
15
 2007 A
Torsion oscillator
$<37$ 95 16
 2007
Torsion pendulum
$<47$ 95 17
 2007
Torsion pendulum
18
 2005
Microcantilever
$<130$ 95 19
 2004
Torsion pendulum
20
 2003
Microcantilever
${ {}\lesssim{} }\text{ 200}$ 95 21
 2003
Microcantilever
$<190$ 95 22
 2001
Torsion pendulum
23
 1985
Torsion pendulum
 1 BLAKEMORE 2021 obtain constraints on non-Newtonian forces with strengths $\vert \alpha \vert$ ${ {}\gtrsim{} }$ $10^{8}$ and length scales $\mathit R$ $>$ 10 ${{\mathit \mu}}$m. See their Fig. 4 for more details including comparison with previous searches.
 2 HEACOCK 2021 obtain constraints on non-Newtonian forces with strengths $10^{18}{ {}\lesssim{} }$ $\vert {{\mathit \alpha}}\vert { {}\lesssim{} }$ $10^{25}$ and length scales $\mathit R$ $≅$ $0.02 - 10$ nm. See their Figure 3 for more details. This improves the results of HADDOCK 2018. These constraints do not place limits on the size of extra flat dimensions.
 3 LEE 2020 search for new forces probing a range of $\vert \alpha \vert$ $≅$ and length scales $\mathit R$ $≅$ $7 - 90$ $\mu$m. For $\delta$ = 1 the bound on $\mathit R$ is 30 $\mu$m. See their Fig. 5 for details on the bound.
 4 TAN 2020A search for new forces probing a range of $\vert \alpha \vert$ $≅$ $4 \times 10^{-3} - 1 \times 10^{2}$ and length scales $\mathit R$ $≅$ $40 - 350$ $\mu$m. See their Fig. 6 for details on the bound.
 5 BERGE 2018 uses results from the MICROSCOPE experiment to obtain constraints on non-Newtonian forces with strengths $10^{-11}{ {}\lesssim{} }$ $\vert {{\mathit \alpha}}\vert { {}\lesssim{} }$ $10^{-7}$ and length scales $\mathit R$ ${ {}\gtrsim{} }10^{5}$ m. See their Figure 1 for more details. These constraints do not place limits on the size of extra flat dimensions.
 6 FAYET 2018A uses results from the MICROSCOPE experiment to obtain constraints on an EP-violating force possibly arising from a new U(1) gauge boson. For $\mathit R$ ${ {}\gtrsim{} }10^{7}$ m the limits are $\vert {{\mathit \alpha}}\vert$ ${ {}\lesssim{} }$ a few $10^{-13}$ to a few $10^{-11}$ depending on the coupling, corresponding to $\vert {{\mathit \epsilon}}\vert$ ${ {}\lesssim{} }$ $10^{-24}$ for the coupling of the new spin-1 or spin-0 mediator. These constraints do not place limits on the size of extra flat dimensions. This extends the results of FAYET 2018.
 7 KLIMCHITSKAYA 2017A uses an experiment that measures the difference of Casimir forces to obtain bounds on non-Newtonian forces with strengths $\vert \alpha \vert$ $≅$ $10^{5} - 10^{17}$ and length scales $\mathit R$ = $0.03 - 10$ $\mu$m. See their Fig. 3. These constraints do not place limits on the size of extra flat dimensions.
 8 XU 2013 obtain constraints on non-Newtonian forces with strengths $\vert \alpha \vert$ $≅$ $10^{34} - 10^{36}$ and length scales $\mathit R$ $≅$ $1 - 10$ fm. See their Fig. 4 for more details. These constraints do not place limits on the size of extra flat dimensions.
 9 BEZERRA 2011 obtain constraints on non-Newtonian forces with strengths $10^{11}{ {}\lesssim{} }$ $\vert {{\mathit \alpha}}\vert { {}\lesssim{} }$ $10^{18}$ and length scales $\mathit R$ = $30 - 1260$ nm. See their Fig. 2 for more details. These constraints do not place limits on the size of extra flat dimensions.
 10 SUSHKOV 2011 obtain improved limits on non-Newtonian forces with strengths $10^{7}{ {}\lesssim{} }$ $\vert {{\mathit \alpha}}\vert$ ${ {}\lesssim{} }$ $10^{11}$ and length scales 0.4 ${{\mathit \mu}}$m $<$ ${{\mathit R}}$ $<$ 4 ${{\mathit \mu}}$m (95$\%$ CL). See their Fig. 2. These bounds do not place limits on the size of extra flat dimensions. However, a model dependent bound of ${{\mathit M}_{{{*}}}}$ $>$ 70 TeV is obtained assuming gauge bosons that couple to baryon number also propagate in (4 + ${{\mathit \delta}}$) dimensions.
 11 BEZERRA 2010 obtain improved constraints on non-Newtonian forces with strengths $10^{19}{ {}\lesssim{} }$ $\vert \alpha \vert { {}\lesssim{} }$ $10^{29}$ and length scales $\mathit R$ = $1.6 - 14$ nm (95$\%$ CL). See their Fig.$~$1. This bound does not place limits on the size of extra flat dimensions.
 12 MASUDA 2009 obtain improved constraints on non-Newtonian forces with strengths $10^{9}{ {}\lesssim{} }\vert \alpha \vert { {}\lesssim{} }10^{11}$ and length scales $\mathit R$ = $1.0 - 2.9$ $\mu$m (95$\%$ CL). See their Fig.$~$3. This bound does not place limits on the size of extra flat dimensions.
 13 GERACI 2008 obtain improved constraints on non-Newtonian forces with strengths $\vert \alpha \vert$ $>$ 14,000 and length scales $\mathit R$ = $5 - 15$ $\mu {\mathrm {m}}$. See their Fig. 9. This bound does not place limits on the size of extra flat dimensions.
 14 TRENKEL 2008 uses two independent measurements of Newton's constant $\mathit G$ to constrain new forces with strength $\vert \alpha \vert ≅10^{-4}$ and length scales $\mathit R$ = $0.02 - 1$ m. See their Fig. 1. This bound does not place limits on the size of extra flat dimensions.
 15 DECCA 2007A search for new forces and obtain bounds in the region with strengths $\vert \alpha \vert$ $≅$ $10^{13} - 10^{18}$ and length scales $\mathit R$ = $20 - 86$ nm. See their Fig. 6. This bound does not place limits on the size of extra flat dimensions.
 16 KAPNER 2007 search for new forces, probing a range of $\vert \alpha \vert$ $≅$ $10^{-3} - 10^{5}$ and length scales $\mathit R$ $≅$ $10 - 1000$ $\mu$m. For $\delta$ = 1 the bound on $\mathit R$ is 44 $\mu$m. For $\delta$ = 2, the bound is expressed in terms of ${{\mathit M}_{{{*}}}}$, here translated to a bound on the radius. See their Fig. 6 for details on the bound.
 17 TU 2007 search for new forces probing a range of $\vert \alpha \vert$ $≅$ $10^{-1} - 10^{5}$ and length scales $\mathit R$ $≅$ $20 - 1000$ $\mu$m. For $\delta$ = 1 the bound on $\mathit R$ is 53 $\mu$m. See their Fig. 3 for details on the bound.
 18 SMULLIN 2005 search for new forces, and obtain bounds in the region with strengths $\alpha$ $\simeq{}$ $10^{3} - 10^{8}$ and length scales ${{\mathit R}}$ = $6 - 20$ ${{\mathit \mu}}$m. See their Figs.$~$1 and 16 for details on the bound. This work does not place limits on the size of extra flat dimensions.
 19 HOYLE 2004 search for new forces, probing $\alpha$ down to $10^{-2}$ and distances down to 10$\mu$m. Quoted bound on $\mathit R$ is for $\delta$ = 2. For $\delta$ = 1, bound goes to 160 $\mu$m. See their Fig. 34 for details on the bound.
 20 CHIAVERINI 2003 search for new forces, probing $\alpha$ above $10^{4}$ and $\lambda$ down to 3$\mu$m, finding no signal. See their Fig.$~$4 for details on the bound. This bound does not place limits on the size of extra flat dimensions.
 21 LONG 2003 search for new forces, probing $\alpha$ down to 3, and distances down to about 10$\mu$m. See their Fig.$~$4 for details on the bound.
 22 HOYLE 2001 search for new forces, probing $\alpha$ down to $10^{-2}$ and distances down to 20$\mu$m. See their Fig.$~$4 for details on the bound. The quoted bound is for $\alpha$ ${}\geq{}$ 3.
 23 HOSKINS 1985 search for new forces, probing distances down to 4$~$mm. See their Fig.$~$13 for details on the bound. This bound does not place limits on the size of extra flat dimensions.
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