Recent breakthrough and outlook in constraining the nonNewtonian gravity and axionlike particles from Casimir physics
Eur. Phys. J. C
Recent breakthrough and outlook in constraining the nonNewtonian gravity and axionlike particles from Casimir physics
G. L. Klimchitskaya 0 1
0 Institute of Physics, Nanotechnology and Telecommunications, Peter the Great Saint Petersburg Polytechnic University , Saint Petersburg 195251 , Russia
1 Central Astronomical Observatory at Pulkovo of the Russian Academy of Sciences , Saint Petersburg 196140 , Russia
The strongest constraints on the Yukawatype corrections to Newton's gravitational law and on the coupling constants of axionlike particles to nucleons, following from recently performed experiments of Casimir physics, are presented. Specifically, the constraints obtained from measurements of the lateral and normal Casimir forces between sinusoidally corrugated surfaces, and from the isoelectronic experiment are considered, and the ranges of their greatest strength are refined. Minor modifications in the experimental setups are proposed which allow for strengthening the resultant constraints up to an order of magnitude. The comparison with some weaker constraints derived in the Casimir regime is also made.

It is well known that many extensions of the Standard
Model predict the existence of light scalar particles [1,2]. An
exchange of one such particle between two atoms results in
the Yukawatype correction to Newton’s gravitational
potential [3]. The same correction arises in the multidimensional
unification theories with a lowenergy compactification scale
[4,5]. It should be stressed that within the micrometer and
submicrometer interaction ranges such kinds of corrections
are consistent with all available experimental data even if they
exceed the Newtonian gravity by the orders of magnitude
[3]. However, the gravitational experiments of
Cavendishtype [6,7], measurements of the Casimir force [8,9], and
experiments on neutron scattering [10] allow one to constrain
parameters of the Yukawatype interaction in micrometer and
submicrometer ranges.
An important object of the Standard Model and its
generalizations is the pseudoscalar particle axion introduced in
[11,12] for exclusion of large electric dipole moment of a
neutron and strong CP violation in QCD. More recently,
axions and various axionlike particles have been actively
discussed as the most probable constituents of dark
matter [1,13]. Although an exchange of one axion between two
nucleons results in the spindependent potential, which
averages to zero after a summation over the volumes of two
macroscopic bodies, an exchange of two axionlike particles
interacting with nucleons via the pseudoscalar Lagrangian
leads to the spinindependent force [14]. Using this fact, the
coupling constants of axionlike particles to nucleons have
been constrained from the results of Cavendishtype
experiments [15,16] and measurements of the Casimir–Polder and
Casimir forces [17–20].
Recently, a considerable strengthening of constraints on
the strength of Yukawa interaction in the wide range from
40 nm to 8 µm was achieved in the socalled isoelectronic
(Casimirless) experiment, where the contribution of the
Casimir force was nullified [21]. The obtained constraints
are up to a factor of 1000 stronger than the previously
known ones. The experimental data of [21] were also used
to strengthen the constraints on the axiontonucleon
coupling constants. The stronger up to a factor of 60 constraints
have been obtained over the wide range of axion masses from
1.7 meV to 0.9 eV [22]. This corresponds to the wavelength
of axionlike particles from 1.2 × 10−4 to 2.2 × 10−7 m,
respectively.
This paper summarizes the strongest constraints on the
Yukawatype corrections to Newtonian gravity and on
the coupling constants of axionlike particles to nucleons
obtained so far from the Casimir physics. It is demonstrated
that by minor modifications of the already performed
experiments with retained sensitivity and other basic
characteristics it is possible to find even stronger constraints.
Specifically, the experiment [23,24] is considered on measuring
the lateral Casimir force between two aligned sinusoidally
corrugated Aucoated surfaces of a sphere and a plate. It is
shown that with appropriately increased corrugation
amplitudes and decreased periods the constraints of [8] on the
parameters of Yukawatype correction to Newtonian gravity
can be strengthened by up to a factor of 10. The same
modification made in the experiment [25,26] on measuring the
normal Casimir force between sinusoidally corrugated
surfaces also allow one to strengthen the constraints [9] by up
to an order of magnitude. Furthermore, it is shown that with
increased thickness of Au and Si sectors of the structured
disc in the isoelectronic experiment [21] the obtained
constraints on the Yukawatype interaction can be strengthened
by up to a factor of 3 over the interaction range from 500 nm
to 1.2 µm.
The proposed modifications in measurements of the lateral
Casimir force [23,24], normal Casimir force [25,26], and in
the isoelectronic experiment [21] are also used to derive the
prospective constraints on the coupling constants of
axionlike particles to nucleons. According to the results obtained,
the modified experiments will give the possibility to obtain
up to factors of 2.4, 1.7, and 1.7 stronger constraints,
respectively, than those found in [20,22] from the original
measurement data.
The paper is organized as follows. In Sect. 2 the strongest
constraints on the Yukawatype corrections to Newtonian
gravity within the micrometer and submicrometer interaction
range (both already obtained and prospective) are discussed.
Section 3 presents similar results for the axiontonucleon
coupling constant in the range of axion masses from 0.1 meV
to 20 eV. In Sect. 4 the reader will find the conclusions and
discussion.
Throughout the paper units with h¯ = c = 1 are used.
2 Constraints on the Yukawatype corrections to
Newtonian gravity
The Yukawatype correction to Newton’s gravitational
potential between two point masses m1 and m2 placed at the points
r1 and r2 is usually parametrized as [3]
V Yu(r12) = −α Gm1m2 e−r12/λ,
r12
where α is the dimensionless interaction constant, G is
the Newtonian gravitational constant, and r12 = r12 =
r1 − r2. If the Yukawa interaction arises due to exchange
of a scalar particle with mass M between two atoms with
masses m1 and m2, the quantity λ = 1/M has a meaning of
the Compton wavelength of this scalar particle. If the
correction (1) arises from multidimensional physics, then λ is the
characteristic size of a compact manifold generated by the
extra dimensions.
It is well known that in the nanometer interaction range
the strongest constraints on α given by Casimir physics
follow [8] from measurements of the lateral Casimir force
between two aligned sinusoidally corrugated Aucoated
surfaces of a sphere and a plate [23,24]. The sphere was made
of polystyrene of density ρs = 1.06 g/cm3 and uniformly
coated with a layer of Cr of density ρCr = 7.14 g/cm3 and
thickness Cr = 10 nm and then with a layer of Au of density
ρAu = 19.28 g/cm3 and thickness (Asu) = 50 nm. The
external radius of the sphere was R = 97.0 µm. The
longitudinal sinusoidal corrugations covering the region of the sphere
nearest to the plate have had an amplitude A2 = 13.7 nm and
a period = 574.7 nm. The corrugated plate was made of
hard epoxy with density ρ p = 1.08 g/cm3 and coated with a
layer of Au of thickness (Apu) = 300 nm. The sinusoidal
corrugations on the plate have had A1 = 85.4 nm and the same
period as on a sphere (the latter is a condition for obtaining
a nonzero lateral Casimir force).
The lateral Yukawa force in the experimental
configuration described above was found in [8] by the pairwise
summation of potentials (1) over the volumes of interacting bodies
with subsequent negative differentiation with respect to the
phase shift ϕ between corrugations on a sphere and a plate
FpYsu,,cloart(a, ϕ) = 8π 3Gαλ3 lat(λ) e−a/λ
Here, a is the separation distance between the zero levels
of corrugations on a sphere and a plate, In(z) is the Bessel
function of imaginary argument, and the following notations
are introduced:
lat(λ) = ρAu − (ρAu − ρ p)e− (Apu)/λ × ρAu (R, λ)
In the experiment [23,24], the lateral Casimir force was
independently measured as a function of the phase shift ϕ
between corrugations over the range of separations a from
120 to 190 nm. At each separation ai the measured
maximum amplitude of the lateral Casimir force was achieved at
some phase shift ϕi and found in agreement with
theoretical predictions of the exact theory within the limits of the
experimental errors i Flat. The latter were obtained at the
95% confidence level. This means that the Yukawa force (2)
satisfies the inequality
Fig. 1 Constraints on the parameters of Yukawatype correction to
Newton’s gravitational law obtained in [10] from the experiment on
neutron scattering (line 1), in [8] from measuring the lateral Casimir force
[23,24] (line 2), in [9] from measuring the normal Casimir force [25,26]
(line 3), and in [21] from the isoelectronic Casimirless experiment (line
4). The longdashed and shortdashed lines show the prospective
constraints obtained in this work (see the text for further discussion). The
regions of the plane below each line are allowed and above each line
are excluded
FpYsu,,cloart(ai , ϕi ) ≤
In Fig. 1 the constraints on α and λ following from (4)
are reproduced from Fig. 1 of [8] by the solid line 2. Here
and below the regions of the (λ, α)plane above a line are
excluded and below a line are allowed by the results of
respective experiment. For comparison purposes, line 1 in Fig. 1
shows the strongest constraints on α and λ following from
the experiment on neutron scattering [10]. It can be seen that
the constraints of line 2 become stronger only for λ > 9 nm.
Note that slightly stronger constraints than those shown by
line 2 were obtained [27] from measurements of the Casimir
force between two crossed cylinders [28]. The experiment
[28], however, suffers from several uncertainties (see [29,30]
for details), which make the deduced results not enough
reliable.
Now let us show that the experiment [23,24] on
measuring the lateral Casimir force has a good chance for obtaining
stronger constraints at the expense of only minor
modification of the parameters of a setup. For this purpose, the same
corrugation amplitude on a plate is preserved, but the
amplitude on a sphere is increased up to A2 = 25 nm. The
respective increase in the thickness of an Au coating on a sphere
up to (Asu) = 70 nm is made. The corrugation period on a
sphere and a plate is decreased down to = 300 nm. The
sphere radius R = 100 µm is chosen almost the same as in the
already performed experiment. Computations of the
prospective constraints were done by using Eqs. (2)–(4). Within the
interaction range λ < 7 nm the strongest constraints on α
follow at a1 = 120 nm. At this separation the value of the
experimental error 1 Flat = 10 pN has been used in
agreement with already performed experiment [23,24]. In a similar
way, within the interaction ranges 7 nm < λ < 18 nm and
λ > 18 nm the strongest constraints on α are obtainable at
a2 = 125 nm and a3 = 135 nm, where 2 Flat = 4.5 pN and
3 Flat = 2.5 pN, respectively [23,24].
The resulting prospective constraints are shown by the
longdashed line in Fig. 1. It is seen that they become stronger
than the constraints of line 1, following from the experiments
on neutron scattering, for λ > 5 nm.
Another experiment, used for constraining the
Yukawatype interaction, measured the normal Casimir force between
sinusoidally corrugated surfaces of a sphere and a plate
[25,26]. In this experiment, the polystyrene sphere was
coated with a layer of Cr of thickness Cr = 10 nm, then
with a layer of Al of thickness Al = 20 nm, and finally with
a layer of Au of thickness (Asu) = 110 nm. The sphere radius
was R = 99.6 µm. The parameters of sinusoidal
corrugations on the sphere were A2 = 14.6 nm and = 570.5 nm.
The corrugated plate was made of hard epoxy and coated
with a layer of Au of thickness (Apu) = 300 nm. The
corrugations on the plate had the same period as on a sphere
and an amplitude of A1 = 40.2 nm [25,26]. An opposed to
experiment on measuring the lateral Casimir force [23,24],
in this experiment the axes of corrugations on a sphere and
a plate should not necessarily be parallel.
For the sake of simplicity, however, here the case of
parallel axes of corrugations is considered. In this case the normal
Yukawa force in the experimental configuration is given by
[9]
nor(λ) = ρ Au − (ρ Au − ρ p)e− (Apu)/λ
Al −
The theoretical predictions for the normal Casimir force
calculated using the scattering theory have been confirmed
experimentally within the experimental errors i Fnor. This
means that any additional normal force should satisfy the
inequality
FpYsu,,cnoorr(ai ) ≤
The constraints on the parameters of Yukawa interaction
obtained from Eqs. (5)–(7) are shown by line 3 in Fig. 1
reproduced from Fig. 4 of [9]. As is seen in Fig. 1, the
constraints of line 3 become stronger than the constraints of
line 2 for λ > 11 nm. Thus, line 2 presents the strongest
constraints within only a very narrow interaction interval of
2 nm width. At the same time, the improved experiment
proposed here on measuring the lateral Casimir force (the
longdashed line in Fig. 1) leads to the strongest constraints up
to λ = 18 nm (the intersection between the longdashed line
and line 3). As a result, the constraints of the longdashed
line are stronger than those of line 2 by up to a factor of 10
within the interaction range from 5 to 18 nm. The maximum
strengthening holds at λ = 9 nm at the intersection of lines 1
and 2.
By modifying parameters of corrugations, it is possible to
strengthen the constraints of line 3 from measurements of the
normal Casimir force between corrugated surfaces. Here, an
increase of the corrugation amplitudes on a sphere and a plate
up to A2 = 25 nm and A1 = 85 nm, respectively, is proposed.
The corrugation period is decreased to = 300 nm. All the
other parameters remain as presented above in accordance
with [25,26].
The prospective constraints on the Yukawatype
corrections to Newtonian gravity, which could be obtained from
measurements of the normal Casimir force between
corrugated surfaces with increased amplitudes and decreased
periods of corrugations, can be found by substitution of (5) and
(6) in (7). The strongest constraints were obtained at a1 =
127 nm where the experimental error was 1 Fnor = 0.94 pN
[9,25,26]. The derived constraints are shown by the
shortdashed line in Fig. 1. They become stronger than the
constraints of line 1 for λ > 7.5 nm.
The recently performed isoelectronic experiment is an
important breakthrough in the field. It allowed significant
strengthening of the constraints on Yukawa interaction in the
micrometer and submicrometer interaction ranges [21]. As
discussed in Sect. 1, in the isoelectronic experiment the
contribution of the Casimir force is nullified. This is achieved by
making the difference force measurement between a smooth
Aucoated sphere and either Au or Si sectors of the
structured disc. A sapphire sphere of density ρsap = 4.1 g/cm3
was coated with a layer of Cr of thickness Cr = 10 nm and
then with a layer of Au of thickness (Asu) = 250 nm. The
resulting sphere radius was R = 149.3 µm. The structured
disc consisted of Au and Si sectors (ρSi = 2.33 g/cm3) of
thickness D = 2.1 µm and was coated with overlayer of
Cr of thickness Cr = 10 nm and Au of (Apu) = 150 nm
thickness. These overlayers nullify the difference of Casimir
forces between a Aucoated sphere and Au and Si sectors of
the disc. They do not contribute to the difference of Yukawa
forces between a sphere and the sectors.
The difference of Yukawatype forces in the experimental
configuration of [21] is given by [21,31,32]
FpYsu,,dnioffr(a) = −4π 2Gαλ3 Re−a/λ(ρAu − ρSi)
× (1 − e−D/λ)[ρAu + (ρCr − ρAu)e− (Asu)/λ
+ (ρsap − ρCr)e−( (Asu)+ Cr)/λ].
Here, a is the separation distance between the sphere and
Au and Ni sectors of the disc. The experimentally measured
separation between the two test bodies is given by
z = a −
In the experiment [21] no differential force was observed
within the minimum detectable force Fmin(a). This means
that the difference of Yukawatype forces (8) satisfies the
inequality
FpYsu,,dnioffr(a) ≤ Fmin(a).
The numerical analysis of (8) and (10) results [21] in line 4
in Fig. 1 which is presented under the same number in Fig. 2
over a wider interaction range extended to larger values of λ.
It is seen that the constraints of line 4 become stronger than
the constraints of line 3, derived from measuring the normal
Casimir force between corrugated test bodies, for λ > 31 nm.
Approximately the same left border of the region, where the
constraints of line 4 are the strongest ones, is given by
experiments measuring the effective Casimir pressure between two
parallel plates by means of micromachined oscillator [33,34].
The latter constraints are not shown in Fig. 1 because they
are slightly weaker than those shown by line 3. Similarly,
the constraints following from measurements of the Casimir
force between the smooth surfaces of a sphere and a plate by
means of an atomic force microscope [35] are even weaker
[30] and do not determine the left border of the region where
line 4 indicates the strongest constraints obtained so far.
From Fig. 1 one can also see that the proposed
modification of an experiment on measuring the normal Casimir
force between corrugated surfaces of a sphere and a plate
would lead to the strongest constraints up to λ = 36 nm
where the shortdashed line intersects line 4. By and large
the constraints of the shortdashed line are up to an order of
magnitude stronger than the constraints of line 3. The
maximum strengthening holds at λ = 11 nm.
For comparison purposes, in Fig. 2 the constraints on
Yukawatype corrections to Newtonian gravity following
from the previous Casimirless experiment [36] (line 5),
from recent experiment on measuring the difference of lateral
forces [37] (line 6), from measuring the Casimir force using
Fig. 2 Constraints on the parameters of Yukawatype corrections to
Newton’s gravitational law obtained in the recent [21] and previous [36]
isoelectronic experiments (lines 4 and 5, respectively), from measuring
the difference of lateral forces [37] (line 6), from the torsion pendulum
experiment [38] (line 7), and from the Cavendishtype experiments [39,
40] (line 8). The dasheddotted line shows the prospective constraints
obtained in this work (see the text for further discussion). The regions of
the plane below each line are allowed and above each line are excluded
the torsion pendulum [38] (line 7), and from the
Cavendishtype experiments [39,40] (line 8) are also shown. As is seen
in Fig. 2, the isoelectronic experiment (line 4) provides the
strongest constraints over a wide interaction range λ ≤ 8 µm,
and narrows the region of λ where the gravitational
constraints have been considered as the strongest ones. It is seen
also that the isoelectronic experiment alone provides up to a
factor of 1000 stronger constraints than several other
experiments using different laboratory setups.
Here, the possibility to further strengthen the constraints
on nonNewtonian gravity following from the isoelectronic
experiment [21] is proposed. For this purpose, we suggest to
increase the thickness of Au and Si sectors up to D = 10 µm.
The resulting strengthening of the obtained constraints is
determined by the single factor in Eq. (8) containing the
quantity exp(−D/λ). These constraints are shown by the
dasheddotted line in Fig. 2. As a result, the prospective
isoelectronic experiment would present the strongest constraints
in the wider interaction range 31 nm < λ < 12 µm. The
largest strengthening up to a factor of 3 will be achieved at
λ = 8 µm.
3 Constraints on the coupling constants of axionlike particles to nucleons The experiments of Casimir physics discussed above allow for constraining the coupling constants of axionlike particles
to a proton and a neutron if an interaction via the pseudoscalar
Lagrangian is assumed. In this case the spinindependent
effective potential between two nucleons at the points r1 and
r2 is caused by the exchange of two axions [14,41,42]
In this equation, the coupling constants between axion and
proton (k, l = p) and neutron (k, l = n) are notated gak
and gal , m and ma are the mean nucleon and axion masses,
respectively, and K1(z) is the modified Bessel function of the
second kind.
Let us begin with an experiment on measuring the lateral
Casimir force between corrugated surfaces on a sphere and
a plate [23,24] briefly discussed in Sect. 2. The additional
lateral force due to twoaxion exchange in the experimental
configuration was found in [20]. At the phase shift ϕ = π/2
between corrugations the amplitude of this force is equal to
[20]
CAu + (CCr − CAu)
× e−2mau (Asu) − CCre−2mau( (Asu)+ Cr) .
Here, mH is the mass of atomic hydrogen and the coefficient
C for any material k (k = Au, Cr etc.) is defined as
ga2p Zk ga2n Nk
where Zk and Nk are the number of protons and the mean
number of neutrons in an atom or a molecule of the
material under consideration, μk = mk /mH, and mk is the mean
atomic (molecular) mass. Specifically, ZAu/μAu = 0.40422,
ZCr/μCr = 0.46518, NAu/μAu = 0.60378, and NCr/μCr =
0.54379 [3]. Note that polystyrene and hard epoxy contribute
negligibly little to the force due to twoaxion exchange, and
these contributions are omitted in (12).
The constraints on the gan and gap have been found by
substitution of the force amplitude (12) in (4) in place of
FpYsu,,cloart (see Fig. 2 of [20]). They are shown by line 1 in
Fig. 3 under a condition gan = gap. As in the case of Yukawa
interaction, the regions above each line in Fig. 3 are excluded
and below each line are allowed.
Now let us consider the modified setup with increased
corrugation amplitudes and decreased period as indicated in
Sect. 2. By making computations with the help of (12), (13)
and (4), where FpYsu,,cloart(a) is replaced with max Fpas,l,actor(a),
1.0
0.5
0.0
0.5
1.0
1.5 2
2.0
Fig. 3 Constraints on the coupling constants of axionlike particles
to nucleons obtained in [20] from measuring the lateral Casimir force
[23,24] (line 1), in [19] from measuring the effective Casimir pressure
[33,34] (line 2), in [22] from the isoelectronic experiment [21] (line 3),
and in [16] from the Cavendishtype experiment [15] (line 4). The
longdashed, shortdashed and dasheddotted lines show the prospective
constraints obtained in this work (see the text for further discussion). In
the inset, the range of larger axion masses is presented on an enlarged
scale. The regions of the plane below each line are allowed and above
each line are excluded
one arrives at stronger constraints on gan = gap shown by
the longdashed line in Fig. 3. They are shown also in the
inset to Fig. 3 on an enlarged scale.
In one more experiment discussed in Sect. 2 the normal
Casimir force between the sinusoidally corrugated surfaces
has been measured [25,26]. The additional normal force
arising in this experimental configuration due to twoaxion
exchange was found in [20]. It is given by
Fpas,n,coorr(a) = −
√u2 − 1
× e−2maua I0 (2mau( A1 − A2)) (1 − e−2mau (Apu) )
× CAu + (CAl − CAu)e−2mau (Asu)
+ (CCr − CAl)e−2mau( (Asu)+ Al)
For Al one has [3] ZAl/μAl = 0.48558 and NAl/μAl =
0.52304.
The constraints on gap(n) have been found by substituting
(14) in (7), where the force FpYsu,,cnoorr was replaced with Fpas,n,coorr
(see Fig. 1 of [20]). They turned out to be weaker than the
combined constraints of line 1 in Fig. 3, following from
measurements of the lateral Casimir force, and of line 2 obtained
[19] from measurements of the effective Casimir pressure by
means of micromachined oscillator [33,34]. Because of this,
the constraints of [20] are not reproduced in Fig. 3.
How
1 1
χ (r, z) = r − 2z + e−2r z r + 2z .
Note that for Si and sapphire one obtains [3] ZSi/μSi =
0.50238, NSi/μSi = 0.50628, and Zsap/μsap = 0.49422,
Nsap/μsap = 0.51412.
The constraints on the coupling constant of axionlike
particles to nucleons are recalculated here by substituting (15)
in place of  FpYsu,,dnioffr in (10) with increased thickness D
of Au and Si sectors, as proposed in Sect. 2. The obtained
constraints are shown by the dasheddotted line in Fig. 3.
For comparison purposes, line 4 reproduces constraints on
gap(n) obtained [16] from the gravitational experiment of
ever, with increased corrugation amplitudes and decreased
period, as proposed in Sect. 2, the stronger constraints can
be obtained. They are found from (7) with the above
replacement and (14). The derived constraints are shown by the
shortdashed line in Fig. 3, and on an enlarged scale in the
inset to this figure.
As is seen in Fig. 3, the proposed experiment on measuring
the normal Casimir force between corrugated surfaces (the
shortdashed line) allows one to strengthen the constraints
on gap(n) in the region of axion masses from 6 to 11 eV.
The largest strengthening by a factor of 1.7 holds for ma =
8 eV. The constraints of the longdashed line, following from
the proposed experiment on measuring the lateral Casimir
force, are stronger than those of line 2 and of the shortdashed
line for ma > 6.5 and 7.5 eV, respectively. In so doing, the
strengthening by the factors of 2 and 2.4, as compared with
line 1, are reached for ma = 10 and 15 eV, respectively.
(At the moment line 1 indicates the strongest constraints for
ma > 8 eV.)
Line 3 in Fig. 3 shows the constraints on gap(n) obtained
[22] from the recent isoelectronic Casimirless experiment
[21]. In the configuration of this experiment the difference
of additional forces due to twoaxion exchange is given by
[22]
X (z) = CAu[χ (R, z) − e−2z (Asu) χ (R −
Cavendish type [15]. As is seen in Fig. 3, the constraints of
line 3 are the strongest ones in the range of axion masses
from 1.7 meV to 0.9 eV. With increased thickness of Au
and Si sectors, stronger constraints of the dasheddotted line
could be obtained over the range of ma from 1.3 meV to
40 meV. The maximum strengthening by a factor of 1.7 holds
at ma = 1.7 meV.
4 Conclusions and discussion
In the foregoing the strongest constraints on the
Yukawatype corrections to Newtonian gravity and coupling
constants of axionlike particles to nucleons following from
Casimir physics are collected. Minor modifications in
respective experimental configurations are proposed allowing
further strengthening of the obtained constraints. Specifically,
it is shown that if one preserves the corrugation amplitude
on a plate ( A1 ≈ 85 nm), but increases it on a sphere from
A2 = 13.7 nm to A2 = 25 nm, and decreases the period
of corrugations from 574 to 300 nm, the constraints on
nonNewtonian gravity, following from measurements of the
lateral Casimir force, become stronger up to a factor of 10.
Similar modifications in the setup for measuring the normal
Casimir force between corrugated surfaces also result in up
to an order of magnitude stronger constraints. An increase of
thickness D of Au and Si sectors in the recent isoelectronic
experiment (from 2.1 to 10 µm), proposed in this paper,
results in up to a factor of 3 stronger constraints on
nonNewtonian gravity over a wide interaction range.
At the moment, the strongest constraints on the
Yukawatype corrections to Newtonian gravity for λ < 9 nm follow
from the experiments on neutron scattering [10], for 9 nm <
λ < 11 nm from measuring the lateral Casimir force, for
11 nm < λ < 31 nm from measuring the normal Casimir
force between corrugated surfaces [25,26], for 31 nm < λ <
8 µm from the isoelectronic experiment [21], and for larger
λ from the Cavendishtype experiment [39,40]. Thus, it is
shown that the constraints of [21] are the strongest ones in a
wider region of λ than is indicated in [21]. If the suggested
modifications will be implemented, the strongest constraints
for λ < 5 nm will follow from neutron scattering, for 5 nm <
λ < 10.5 nm from the proposed measurements of the lateral
Casimir force, for 10.5 nm < λ < 36 nm from the proposed
measurements of the normal Casimir force, for 36 nm < λ <
12 µm from the modified isoelectronic experiment, and for
λ > 12 µm from gravitational experiments.
The proposed modifications of the test bodies in
measurements of the lateral and normal Casimir forces, and in
the isoelectronic experiment also allow one to strengthen
constraints on the coupling constant of axionlike particles
to nucleons. At the moment, the strongest constraints for
1 µeV < ma < 1.7 meV follow [16] from the
Cavendishtype experiment [15], for 1.7 meV < ma < 0.9 eV from the
isoelectronic experiment [21,22], for 0.9 eV < ma < 8 eV
from measuring the effective Casimir pressure [19,33,34],
for ma > 8 eV from measuring the lateral Casimir force
[20,23,24]. If the suggested experiments will be realized, the
strongest constraints for 1.3 meV < ma < 0.9 eV will follow
from the isoelectronic experiment, for 0.9 eV < ma < 6 eV
from measuring the effective Casimir pressure, for 6 eV <
ma < 7.5 eV from the proposed measurements of the
normal Casimir force, and for ma > 7.5 eV from the proposed
measurements of the lateral Casimir force.
Thus, Casimir physics already resulted in strong
laboratory constraints on the nonNewtonian gravity and axionlike
particles. As shown above, the proposed minor modifications
of already performed experiments make possible obtaining
even stronger constraints. In future, more radical
improvements in the laboratory setups may be employed (for instance,
by using the test bodies with aligned nuclear spins [43] or a
large area force sensor suggested for constraining chameleon
interactions [44]).
Acknowledgements The author is grateful to R. S. Decca for sending
the numerical data for line 4 in Figs. 1 and 2 and to V. M. Mostepanenko
for useful discussions.
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