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arXiv:1207.2442v1 [gr-qc] 10 Jul 2012
Torsion-balance tests of the weak equivalence
principle
T A Wagner, S Schlamminger‡, J H Gundlach and E G
Adelberger
Center for Experimental Nuclear Physics and Astrophysics, Box 354290, University
of Washington, Seattle, WA 98195-4290
E-mail: eric@npl.washington.edu
Abstract.
principle and then review recent torsion-balance results that compare the differential
accelerations of beryllium-aluminum and beryllium-titanium test body pairs with
precisions at the part in 1013level. We discuss some implications of these results
for the gravitational properties of antimatter and dark matter, and speculate about
the prospects for further improvements in experimental sensitivity.
We briefly summarize motivations for testing the weak equivalence
PACS numbers: 04.80.-y, 04.80.Cc, 12.38.Qk
Submitted to: Class. Quantum Grav.
‡ current address: National Institute of Standards and Technology, Gaithersburg, Maryland 20899,
USA
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Torsion-balance tests of the weak equivalence principle
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1. General framework
The weak equivalence principle (WEP) states that in a uniform gravitational field all
objects, regardless of their composition, fall with precisely the same acceleration. In
Newtonian terms, the principle asserts the exact identity of inertial mass mi(the mass
appearing in Newton’s second law) and gravitational mass mg(the mass appearing in
Newton’s law of gravity). The WEP implicitly assumes that the falling objects are bound
by non-gravitational forces. The strong equivalence principle extends the universality
of free fall to objects (such as astronomical bodies) where the effects of gravitational
binding energy cannot be neglected.
WEP tests were traditionally interpreted in Newtonian terms, i.e. as searches for
possible departures from exact equality of mg/mifor objects 1 and 2 as specified by the
E¨ otv¨ os parameter
η1,2=
a1− a2
(a1+ a2)/2=
(mg/mi)1− (mg/mi)2
[(mg/mi)1+ (mg/mi)2]/2, (1)
where a is the measured free-fall acceleration. In this case, the properties and location of
the attractor toward which the objects were falling was irrelevant. For technical reasons
described below, the classic experiments at Princeton[1] and Moscow[2] used the sun as
the attractor.
However, as emphasized by Fischbach[3], it is appropriate to view WEP tests as
probes for possible new Yukawa interactions, potentially much weaker than gravity,
that would be essentially undetectable by other means. In this case, we ascribe any
violation of the WEP to a previously unknown Yukawa interaction arising from quantum
exchange of new bosons that couple to vector or scalar charges of the test bodies and
attractor. Vector or scalar boson exchange forces of quantum field theories produce a
spin-independent potential between test body i and attractor A of the form
VOBE(r) = ∓˜ g2
4πr
where ˜ q is a fermion’s scalar or vector dimensionless charge, ˜ g is a coupling constant,
and λ = ¯ h/(mbc) is the range of the force mediated by bosons of mass mb. The − and
+ signs apply to scalar and vector interactions, respectively. The total potential can be
written in a form appropriate for WEP tests as
˜ qi˜ qA
exp(−r/λ) ,(2)
Vi,A= VG+ VOBE= VG(r)
?
1 + ˜ α
?˜ q
µ
?
i
?˜ q
µ
?
A
exp(−r/λ)
?
, (3)
where the dimensionless ratio (˜ q/µ) is an object’s charge per atomic mass unit (u), and
the dimensionless Yukawa strength parameter
˜ α = ±˜ g2/(4πGu2) . (4)
In this case
η1,2= ˜ α
??˜ q
µ
?
1
−
?˜ q
µ
?
2
? ?˜ q
µ
?
A
?
1 +r
λ
?
exp(−r/λ) .(5)
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Torsion-balance tests of the weak equivalence principle
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For electrically neutral bodies consisting of atoms with proton and neutron numbers Z
and N, respectively, a general vector charge-to-mass ratio can be parameterized as
(˜ q/µ) = (Z/µ)cos˜ψ + (N/µ)sin˜ψ with tan˜ψ ≡
˜ qn
˜ qe+ ˜ qp
, (6)
where˜ψ is an unknown parameter that ranges between −π/2 and +π/2. It is easy to see
that any vector interaction must violate the equivalence principle because particles and
anti-particles have opposite vector charges. Less dramatically, different atoms composed
of ordinary matter must also have different vector (˜ q/µ) ratios because the vector charge
of an atom is the sum of the charges of its ingredients, but the atom’s mass is less than
the mass of its ingredients because of binding energy.
Note that the ˜ q of any given atom vanishes for some value of˜ψ. This implies that
to make a comprehensive and unbiased test of the WEP it is necessary to
(i) test with 2 different test-body composition dipoles falling toward 2 different
attractors to avoid accidental cancellations of the charges of the test body dipole
or the attractor, and
(ii) use attractors with the smallest practical distance from the test bodies to cover a
wide span of Yukawa ranges λ.
The situation for scalar charges is considerably more complicated because scalar
charges are neither conserved nor Lorentz invariant (the charge density rather than
the charge itself is a Lorentz scalar) and binding energy along with virtual fermion-
antifermion loops carry scalar charges. Furthermore, a scalar interaction can couple to
Tµ
massless scalar fields, the Tµ
indistinguishable from normal gravity.) Therefore, detailed field-theoretic calculations
are required to compute scalar charges of neutral atoms. Nevertheless, one expects the
WEP component of a general scalar charge-to-mass ratio will be roughly described by
equation 6 as well.
A particularly interesting scalar interaction arises from dilaton exchange, where the
dilaton is the scalar partner of the massless graviton that is inherent in string theories.
Kaplan and Wise[4] found that the force generated by low-mass dilatons is dominated
by coupling to the gluon field strength, which gives a force between nucleons that is
∼ 103times stronger than gravity with only a small, 0.3%, WEP-violating component.
A long-range dilaton field with such couplings is clearly ruled out by many experiments.
It is, therefore, usually assumed that the dilaton has a finite mass which gives its force
a small range, allowing it to evade the experimental bounds. We will return to this
issue in Sec. 4.2. Donoghue and Damour[5, 6] made related calculations but took an
entirely different point of view. They computed the WEP-violating effects but allowed
the various dilaton-coupling terms to be free parameters, but implicitly assumed that
composition-independent coupling to the gluon field was dominant. They suggested
that WEP experiments be analyzed to set bounds on bilinear combinations of dg, de
(couplings to gluon and electromagnetic field strength) and dme, dˆ mand dδm(couplings
µ, the trace of the energy-momentum tensor i.e. effectively to mass. (Note that, for
µterm has no experimental significance because it would
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Torsion-balance tests of the weak equivalence principle
4
to the masses of the electron and to the average and difference of the up and down quark
masses). The dominant WEP-violating effects are expected to arise from deand dˆ m.
2. Principles of torsion-balance tests of the WEP
The remarkable sensitivity of torsion-balance null tests of the WEP results from two
properties of such instruments:
(i) A freely hanging torsion balance responds only to a difference in the directions
of the external force vectors on the test bodies and not on their magnitudes[7]
(see figure 1). This allows instruments with tolerances at the 10−5level to make
measurements with a precision of a part in 1013. In fact, current experiments are
limited by gravity gradients which, when coupled to imperfections in the geometry
of the torsion pendulum, also give a difference in the directions of the forces on the
test bodies[8]. The highest precisions have been obtained by uniformly rotating
the balance with respect to the attractor, giving a WEP-violating signal that is a
sinusiodal function of the rotation angle. The classic WEP tests of the Princeton[1]
and Moscow[2] groups employed the sun as the attractor and let the earth’s rotation
provide the smooth rotation of the instrument.
(ii) Although an actual torsion oscillator has many modes (twist, pendulum, bounce,
wobble, etc.) the frequency of the twist mode (∼mHz) lies well below that of all the
other modes (∼Hz). Therefore the other modes can be damped before their energy
Figure 1. Operating principle of the E¨ otv¨ os torsion balance. This idealized balance
consists of two test bodies attached to a rigid, massless frame that is supported by a
perfectly flexible torsion fibre. F1and F2denote the external forces on the test bodies.
The torque about the fibre axis is Tz= (F1× F2· r12)/|F1+ F2|. The signal is the
change in Tzwhen the instrument is rotated about the fibre axis so that the component
of r12along the direction of F1× F2changes sign.
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Torsion-balance tests of the weak equivalence principle
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has had much chance to leak into the twist mode, so that the torsion ocillator
effectively has a single mode that can operate close to the thermal limit. The
thermal torque noise at frequency f in a single-mode torsional oscillator with fibre
torsional constant κ and quality factor Q has a power spectral density (see [9])
τ(f)2= 4kBTκ/(2πfQ) ; (7)
where it is assumed that the damping is dominated by internal friction in the
suspension fibre. (The related case of a rotating 2-dimensional oscillator is discussed
in [10].)
3. Modern experimental tests and their results
The instrument rotation scheme of the classic Princeton and Moscow experiments, while
very smooth, had two main disadvantages:
(i) The 24 hour signal period posed serious problems. Most noise sources increase as the
frequency decreases (as 1/f for fibre damping and 1/f2for several other sources).
Furthermore many possible systematic effects have a 24 h period (temperature,
vibration, power fluctuations, etc.).
(ii) The solar attractor rendered the experiments completely insensitive to Yukawa
forces with ranges less than 1011m.
To avoid these limitations, the E¨ ot-Wash group developed a series of torsion balances
equipped with uniformly-rotating turntables[7, 8, 11, 12, 13, 14]. This allowed the earth
to be used as the attractor and placed the signal at the turntable’s rotation frequency
(∼mHz for our apparatus). The centrifugal force due to earth’s rotation pushes a torsion
pendulum in the Northern Hemisphere toward the south. This force is balanced against
a horizontal component of gravity, which in Seattle, Washington at a latitude of 47.7◦N
is 1.68 cm s−2, giving a maximum horizontal acceleration three times greater than that
toward the sun.
The requirements on the constancy of the turntable rotation rate[8], as well as the
alignment of its rotation axis with the suspension fibre, are quite severe. Suppose that
the turntable rotation rate ωttis not completely constant so that
ωtt(t) = ωc+
N
?
n=1
ωne−inωct. (8)
This will induce a twist angle θ of the torsion pendulum
θ(t) =
N
?
n=1
−inωc
0− (nωc)2ωne−inωct,
ω2
(9)
where pendulum damping has been neglected and ω0is the frequency of free torsional
oscillations. The n = 1 term will generate a spurious WEP signal that must be cancelled
by combining data with 2 opposite orientations of the composition dipole in the rotating
balance.
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Torsion-balance tests of the weak equivalence principle
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The choice of ωtt involves competing considerations of thermal and other low-
frequency torque noises and the noise in the twist readout system. The response of
a damped torsion oscillator to a torque of magnitude T varying at a frequency ωsis
θ(ωs) =T
κ
?
where κ is the torsional spring constant. Running on resonance (ωs = ω0) is only
sensible when the θ readout noise is completely dominant. Otherwise, the signal-to-noise
ratio is optimized by a compromise between the thermal torque noise (which falls with
increasing ωs) and noise from imperfect turntable rotation (which rises as ωsincreases).
The rotating torsion balance used for the recent E¨ ot-Wash test of the WEP[13, 14] is
ω2
0
(ω2
0− ω2
s)2+ (ω2
0/Q)2,(10)
Figure 2. Simplified scale drawing of the E¨ ot-Wash WEP torsion balance.
depicted in figure 2. An air-bearing turntable driven by an eddy-current motor provided
a highly uniform rotation rate. A laser autocollimator measured the twist of the torsion
pendulum[15]. Additional sensors on the apparatus measured temperature, vacuum
pressure, and tilts. Feedback to the tilt sensors aligned the rotation axis with local
vertical by controlling thermal-expansion legs that supported the turntable[12]. The
balance was surrounded by passive thermal and magnetic shields. Large masses placed
nearby compensated the leading static environmental gravity gradients by more than
two orders of magnitude. An ion pump maintained the vacuum chamber at a pressure of
< 10−4Pa. The apparatus is located within a temperature-stabilized foam box inside a
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Torsion-balance tests of the weak equivalence principle
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temperature-controlled room. The pendulum’s twist angle and 27 other environmental
sensors were recorded every ≈ 3s by a data acquisition system. The recorded twist
angle was passed through a digital notch filter to remove the pendulum oscillation, then
separated into Fourier components by fitting the time series from two complete turntable
rotations with sines and cosines of harmonics of the turntable angle, plus a 2nd-order
polynomial drift.
Figure 3. [Colour online] Torsion pendulum used in the recent E¨ ot-Wash WEP test.
An Al frame holds 4 mirrors and supports 8 barrel-shaped test bodies, 4 of which are Be
and 4 are Ti or Al. The structure underneath the pendulum allows the pendulum to be
parked to prevent damage when the apparatus is serviced and catches the pendulum if
a small earthquake should break the suspension fibre. The tungsten fibre is just visible
at the top.
The torsion pendulum used for measurements with Be-Ti and Be-Al test body pairs,
shown in Figure 3, was supported by a 1.07 m long, 20 µm thick tungsten fibre. The
pendulum’s design, with 4-fold azimuthal and up-down symmetries, reduces systematic
effects by minimizing the coupling to gravity gradients and by allowing for four different
orientations of the pendulum with respect to the turntable rotor. The gravitational
multipole framework described in [8] was used to suppress couplings to environmental
gravity gradient fields that fall off more slowly than r−6, with the exception of the
primary four-fold symmetry of the pendulum that gave a weak signal at the fourth
harmonic of the turntable rotation frequency. This was readily distinguished from a
WEP-violation whose signal is at the turntable rotation frequency.
The test bodies, which comprise 40 g of the pendulum’s 70 g mass, all have
identical masses and outside dimensions to suppress systematic effects.
removable, which allowed us to use two different composition dipoles and to rearrange
test bodies to invert the composition dipole on the pendulum frame. This last strategy
canceled systematic effects that followed the pendulum frame rather than the test
They are
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Torsion-balance tests of the weak equivalence principle
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Table 1. Charge-to-mass ratios of selected test body materials. Z, N and B are the
atomic, neutron and baryon numbers, respectively. Qˆ mand Qeare the dilaton charge-
to-mass ratios associated with the average light quark mass and the electrostatic field
strength, respectively [6].
BeTiAlPt
Z/µ
N/µ
B/µ
Qˆ m
Qe
0.44384
0.55480
0.99865
0.07526
0.00072
0.45961
0.54147
1.00107
0.08267
0.00228
0.48181
0.51887
1.00068
0.08076
0.00174
0.39983
0.60032
1.00015
0.08526
0.00428
bodies themselves. The pendulum is coated with ≈ 300 nm of gold and is surrounded
by a gold-coated electrostatic shield to minimize electrical effects from work-function
variations. The test-body materials were selected for their scientific impact and for
practical concerns such as mechanical stability and freedom from magnetic impurities.
Table 1 summarizes the charges of some test body materials.
Figure 4 shows the power spectral density of the observed twist signal and
demonstrates that the instrument operates close to the thermal limit. Figure 5
Figure 4. [Colour online] Power spectral density of the twist signal. The upper [blue]
histogram shows WEP data taken with ωtt/ω0= 2/3. The curve is the thermal noise
predicted by equation 7 for a room-temperature oscillator with Q = 6000. The peaks
at integer multiples of ωttarise from reproduceable variations in ωtt(see equation 9).
The small peak at ωtt/2 is caused by the turntable leveling system that recomputed
the tilt every two turntable rotations[12]. The lower [green] histogram displays data
taken with the turntable stationary and the pendulum resting on a support to show
the readout noise. The low-frequency readout noise is ascribed to thermal fluctuations.
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Figure 5.
configurations of the pendulum. The final result is in the difference between the means
of the two configurations (shown as solid lines).
Data collected in the Ti-Be (first 4 runs) and Be-Ti (last 2 runs)
summarizes the Be-Ti composition dipole measurements. Each data point represents
about two weeks of data, with daily reversals of the pendulum orientation with respect
to the turntable rotor. A linear drift was removed to correct for slow environmental
variations (the drift correction was insignificant compared to the statistical errors). The
difference in the mean values for each configuration contains the signal. The offset from
zero is due to systematic effects that follow the orientation of the pendulum frame.
Approximately 75 days of data were collected using the Be-Ti test bodies and 110
days using the Be-Al test bodies. Systematic investigations were performed each time
the vacuum system was pumped out and then repeated after the measurements were
completed to ensure that the systematic effects had not changed.
Several environmental conditions are known to produce effects that can mimic a
WEP-violating signal. Tilts of the rotation axis with respect to local vertical, coupling of
the pendulum to gravity gradients, temperature fluctuations and gradients and magnetic
fields all produce such effects. The systematic errors associated with these effects were
measured following the strategy described in detail in [8, 12, 13]. Each “driving term”
was deliberately exaggerated and its effect on the WEP-violating signal was measured;
this signal was then scaled to the driving term observed in the actual WEP data. Gravity
gradients were measured with a specially designed gradiometer pendulum that could
be configured to give sensitivity to a particular mulitpole component of the gradient;
this information had been used to design the gradient compensators shown in figure 2.
Systematic errors from gravity gradients were measured by rotating the compensators
by 180◦about the vertical axis, so that instead of canceling the ambient gradient they
effectively doubled it. The ratio of the twist signals with the WEP and gradiometer
pendulums in the two compensator positions determined the effects of gravity gradients
on the WEP pendulum; this was used to correct the WEP signal.
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It is well known that small tilts of the apparatus induce a twist in the fibre because
of tiny asymmetries in the upper fibre attachment point.
sensors placed above the upper attachment of the fibre and beneath the pendulum
(see figure 2) measured the turntable tilt. A feedback loop locked the turntable rotation
axis with a precision of a few nanoradians to local vertical as determined by the upper
tilt sensor. However, the lower tilt sensor revealed that the direction of local vertical at
the pendulum position differed from that at the upper sensor by ∼ 50 nrad. Since the
fibre axis is determined by local level at the pendulum site, corrections were needed to
account for this gradient in the down direction. The tilt-induced twist was measured
by purposely tilting the apparatus by a measured amount. The resulting feed-through
of a small tilt into pendulum twist was typically around 5%, but varied from mirror to
mirror. Corrections for tilt were applied to obtain the final result.
Temperature gradients and magnetic effects were primarily minimized by multi-
stage passive shielding. The magnetic systematic uncertainty was found by removing
the outermost mu-metal shield (which normally reduced the ambient laboratory field to
≈ 2.5 × 10−6T) and measuring the effect on the twist signal when a strong permanent
magnet was placed outside the vacuum vessel. In the absence of any shielding the
magnet’s field at the pendulum would have been ≈ 1.7×10−4T. Data were taken with
both the north and south poles pointing toward the pendulum. The pendulum twist
did not significantly change when the magnet orientation was reversed. The magnetic
systematic error was computed by scaling this upper limit on the twist change by the
ratio of the normal to enhanced fields inside the outermost shield.
The effect of temperature gradients was measured by placing large temperature-
controlled copper plates next to the apparatus and measuring the pendulum signal as a
function of the applied temperature gradient. Temperature gradients of up to 15 K/m
were applied, while in normal operation the apparatus saw a gradient of ∼ 44 mK/m.
The maximum twist signal change in the temperature test was scaled to temperature
gradients seen in normal data and assigned equally to systematic uncertainties in the
north and west signals.
Table 2 summarizes the lab-fixed systematic effects in the Be-Ti measurement.
When astronomical objects were viewed as the attractors, their additional signal
Dual-axis electronic tilt
Table 2. Error budget for the lab-fixed Be-Ti differential accelerations. Corrections
were applied for gravitational gradients and tilt, only upper limits were obtained on
the magnetic and temperature effects. All uncertainties are 1σ.
Uncertainty source∆aN,Be−Ti(10−15m s−2)∆aW,Be−Ti(10−15m s−2)
Statistical
Gravity gradients
Tilt
Magnetic
Temperature gradients
3.3 ± 2.5
1.6 ± 0.2
1.2 ± 0.6
0 ± 0.3
0 ± 1.7
−2.4 ± 2.4
0.3 ± 1.7
−0.2 ± 0.7
0 ± 0.3
0 ± 1.7
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Table 3. Differential accelerations in the lab-fixed frame (∆aNand ∆aW) and toward
the sun and galactic center (∆a⊙ and ∆ag). The E¨ otv¨ os parameters, η⊕, η⊙ and
ηDM,were calculated using the horizontal gravitational accelerations of earth, sun and
galactic dark matter, 0.0168m/s2, 5.9×10−3m/s2and 5×10−11m/s2[16], respectively.
Uncertainties are 1σ with systematic and statistical uncertainties added in quadrature.
Be-TiBe-Al
∆aN
∆aW
∆a⊙
∆ag
η⊕
η⊙
ηDM
(10−15m s−2)
(10−15m s−2)
(10−15m s−2)
(10−15m s−2)
(10−13)
(10−13)
(10−5)
0.6 ± 3.1
−2.5 ± 3.5
−1.8 ± 2.8
−2.1 ± 3.1
0.3 ± 1.8
−3.1 ± 4.7
−4.2 ± 6.2
−1.2 ± 2.2
0.2 ± 2.4
−3.1 ± 2.4
−1.2 ± 2.6
−0.7 ± 1.3
−5.2 ± 4.0
−2.4 ± 5.2
modulation reduced the systematic uncertainties so that those results were dominated
by the statistical uncertainty in contrast to the lab-fixed results where the statistical
and systematic uncertainties were comparable.
The basic results from the E¨ ot-Wash Be-Ti and Be-Al WEP tests are summarized
in table 3.
4. Some implications of the results for new Yukawa interactions
4.1. Results
The properties of our terrestrial attractor allow the lab-fixed Be-Ti and Be-Al results
in table 3 to constrain exotic Yukawa interactions with ranges down to 1 m. A torsion
balance located on a flat, level region would have essentially no sensitivity for forces with
λ ≤ rearth(see [7]). However, the E¨ ot-Wash laboratory is located on a hillside above
a deep lake, with the pendulum only 0.75 m from a wall excavated from the hillside.
The complex regional topography plus details of the laboratory environment enormously
enhance the sensitivity for Yukawa interactions with λ < 107m. But determining the
sensitivity for such forces is a challenging undertaking because one needs to compute
the horizontal component of a Yukawa force from a complicated object. We estimated
this strength using geophysical models extending from the detailed local topography to
regional geology[17, 18] to the gross structure of the earth[19, 20, 21]. The left panel in
figure 6 shows 95% CL limits on the magnitude of the Yukawa strength ˜ α as a function
of range λ assuming a charge ˜ q = N = B − L, where B and L are the baryon and
lepton numbers, respectively. This is a particularly interesting charge because B −L is
conserved in grand unified theories. The bump in Figure 6 at λ ∼ 105m comes from an
east-west density asymmetry in the subduction zone for the Juan de Fuca plate. As λ
increases beyond 60 km, which is approximately the depth of the subduction zone, the
supporting mantle quickly reduces the asymmetry to maintain hydrostatic equilibrium.
Constraints on vector interactions coupled to other charges can be inferred from the
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Figure 6. The left panel shows 95% CL upper bounds on the strength of a vector
Yukawa interaction coupled to ˜ q = B − L. The curve labeled EW shows the limit
extracted from the lab-fixed results in table 3. Curves labeled Princeton, Moscow,
EW94, and EW99 are extracted from [1], [2], [8] and [11], respectively. The two
LLR constraints are derived from lunar laser-ranging results[22] for the earth-moon
differential acceleration toward the sun (right curve) and the inverse-square law
violation obtained from anomalous precession of the lunar orbit (left curve). The
right panel shows how the constraints on an infinite-range interaction depend on˜ψ the
parameter that describes the interaction charge. The combined E¨ ot-Wash result from
Be-Ti and Be-Al attracted toward the earth and toward the sun is indicated by EW.
The Moscow result[2] used a Al-Pt dipole attracted to the sun; the left pole arises when
the sun’s charge is zero, while the pole on the right occurs where the charge difference
of the test bodies vanishes.
right panel in figure 6, which displays 95% CL limits on |˜ α| as a function of˜ψ for an
infinite-ranged interaction. Since any single pair of test bodies (or source) has a value of
˜ψ for which its charge difference (or charge) vanishes, two different pairs of test bodies
and two different sources must be used to obtain limits for all values of˜ψ.
Figure 7 shows an example of WEP bounds on scalar interactions, the Donoghue-
Damour[6] scenario for WEP violation by massless dilatons. Their predicted WEP-
violating effects are dominated by couplings to the average light quark mass and the
electromagnetic field strength via the “dilaton coefficients” Dˆ m and De, respectively.
Our 95% CL limits in the Dˆ m-Deparameter space demonstrate that the effects of a
massless dilaton must be suppressed by a factor of at least ∼ 1010. (The individual
95 %CL constraints on Dˆ mand Deare (−0.3 ± 3.2)× 10−10and (+1.7± 10.3)× 10−10,
respectively.) This suggests that the dilaton must have a finite mass so that its short-
range force was not detected in WEP experiments. In this case, inverse-square law tests,
which probe the dominant composition-independent coupling to the gluon strength, set a
conservative lower limit of 3.5 meV on the dilaton mass[23, 4]. This lower limit becomes
13 meV in the standard model if the string scale is set to the Planck scale[4].
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Figure 7. [Colour online] 95% CL constraints on couplings of a long-range dilaton
field. We use our Be-Ti and Be-Al results plus the Moscow Al-Pt limit[2] to constrain
the dominant dilaton couplings Deand Dˆ min the Donoghue-Damour framework. For
standard dilaton couplings these coeffcients would be of order unity.
4.2. Some implications of the results
The impressive recent technical progress[24, 25] toward trapping antihydrogen,¯H, has
revived interest in probing the gravitational properties of antimatter by testing the
suggestion that antimatter could fall with an acceleration perceptibly different from
g[26]. It is worth asking how plausible this is, especially considering the extraordinary
technical difficulties involved in measuring the freefall acceleration of antihydrogen.
In field theory terms, if antihydrogen were to fall with an acceleration different from
hydrogen it could occur if and only if there were a vector interaction that coupled
to Z. But the WEP results summarized above set extremely strong upper limits on
such vector interactions. To be explicit, what should one expect if one could do a
hydrogen-antihydrogen freefall comparison at the location of the E¨ ot-Wash WEP torsion
balance? To answer this, we used our geophysical earth model to calculate the ratio, as
a function of λ, of the vertical to horizontal Yukawa forces at our site. Then, using our
constraint on˜ψ = 0 vector interactions, we computed the upper bound on the vertical
component of ∆a¯H−H/g. The results, shown in figure 8, indicate that any anomalous
gravitational acceleration must be extremely small, well below the sensitivity of current
technology. One might object that our˜ψ = 0 assumption is unwarranted because it
assumes that antineutrons and neutrons should fall with identical accelerations. Indeed
it is, so we also computed the upper bound on ∆a¯H−H/g for the values of˜ψ that gave
the weakest constraint at each value of λ; the results are also shown in figure 8. Had we
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Figure 8.
freefall accelerations of hydrogen and antihydrogen. The region above the solid line is
excluded for all values of ˜ q. The region above the dashed line is excluded if we assume
that ˜ q = Z. The corresponding constraints for neutrons and antineutrons are almost
the same as those for hydrogen and antihydrogen.
95% CL upper bounds on |∆aH¯ H|/g, the fractional difference in the
plotted the corresponding constraints on ∆a¯ n−n/g , a quantity that would be essentially
impossible to measure directly, the results would be essentially the same as the hydrogen-
antihydrogen bounds in figure 8.
It has been argued[26] that the existence of a scalar field could invalidate these
arguments; the scalar field would have no effect on ∆a¯H−Hbecause particles and
antiparticles have the same scalar charge, but it would contribute to the differential
accelerations of the E¨ ot-Wash test bodies.
like particles are attractive, a scalar interaction would tend to cancel a vector force.
But this cancelation must be unreasonably precise to give null results in WEP
tests with 9 different materials (ranging from Be to Pb) falling toward 3 different
attractors[2, 8, 11, 14]. Suppose the scalar charges of the materials used for these
tests differed by merely 0.1% from the vector charges in equation 6; the upper limit on
∆a¯H−H/g from a long-range vector field would still be about 1 part in 106. Reference
[27] gives a detailed discussion of the impossibility of nearly perfect scalar-vector
cancellation. Of course, our arguments rely on the CPT theorem that, to our knowledge,
has not been tested for gravity. But consider how strange it would be if, as is occasionally
suggested, antimatter fell up rather than down.
own antiparticle (such as the photon or π0) would not fall. This is excluded by many
observations.
WEP results also provide a laboratory test of the common assumption that
In fact, because scalar forces between
In that case a particle that is its
Page 15
Torsion-balance tests of the weak equivalence principle
15
gravitation is the only long-range force between dark and luminous matter. Because
almost all of the usual conclusions about dark matter rely on this assumption, finding
laboratory support for the idea has real value. The acceleration vector toward the
galaxy’s dark matter passes through our instrument’s plane of maximum sensitivity and
has an estimated magnitude of 5×10−11m s−2[16]. This acceleration can be separated
into gravitational and non-gravitational components aDM = ag
that any non-gravitational interaction with dark matter violates the WEP and search
for differential accelerations δang
we can deduce, for a given material, the magnitude of ang
parameter describing the test-body charges (see [8]). We combine the Be-Al and Be-
Ti galactic attractor results from table 3 with the 3 × 10−16m s−2upper bound on
the earth-moon differential acceleration toward the galactic center[28] extracted from
lunar laser-ranging (LLR) data to obtain an upper bound on the contribution of non-
gravitational forces to the galactic dark-matter acceleration of neutral hydrogen shown
in figure 9. Extraordinarily, for any value of˜ψ, the acceleration of hydrogen due to
non-gravitational interactions with dark matter must be less than about 5% of the total
acceleration. The bounds in figure 9 apply to any interaction whose WEP violation
is approximately described by equation 6. For example, a very similar upper bound
would arise for a dilaton-like scalar field. The WEP-violating component of the dilaton
coupling to the gluon strength is about 0.3%[4] which is very similar to the 0.25%
difference in B/µ values of Be and Ti.
DM+ ang
DM. We assume
DMfor a series of test-body pairs. From these results
DMas a function of˜ψ, the
Figure 9. Inferred 95% CL limits on the ratio of non-gravitational acceleration of
neutral hydrogen to the total acceleration toward galactic dark matter. The ratio
vanishes at˜ψ = ±90◦where the charge of hydrogen is zero. The weakest bound of
about 5% occurs near˜ψ = 45◦where ˜ q ∝ B.
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