# Torsion-balance tests of the weak equivalence principle

**ABSTRACT** We briefly summarize motivations for testing the weak equivalence 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 $10^{13}$ level. 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.

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- A M Nobili, D M Lucchesi, M T Crosta, M Shao, S G Turyshev, R Peron, G Catastini, A Anselmi, G Zavattini[Show abstract] [Hide abstract]

**ABSTRACT:**Through the contributions of Galileo, Newton, and Einstein, we recall the universality of free fall (UFF), the weak equivalence principle (WEP), and the strong equivalence principle (SEP), in order to stress that general relativity requires all test masses to be equally accelerated in a gravitational field; that is, it requires UFF and WEP to hold. The possibility of testing this crucial fact with null, highly sensitive experiments makes these the most powerful tests of the theory. Following Schiff, we derive the gravitational redshift from the WEP and special relativity and show that, as long as clocks are affected by a gravitating body like normal matter, measurement of the redshift is a test of UFF/WEP but cannot compete with direct null tests. A new measurement of the gravitational redshift based on free-falling cold atoms and an absolute gravimeter is not competitive either. Finally, we compare UFF/WEP experiments using macroscopic masses as test bodies in one case and cold atoms in the other. We conclude that there is no difference in the nature of the test and that the merit of any such experiment rests on the accuracy it can achieve and on the physical differences between the elements it can test, macroscopic proof masses being superior in both respects. V C 2013 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4798583]American Journal of Physics 07/2013; 81(527). · 0.78 Impact Factor - SourceAvailable from: Mitsuhiro Kimura
##### Article: Prospects for measuring the gravitational free-fall of antihydrogen with emulsion detectors

AEgIS Collaboration, S. Aghion, O. Ahlén, C. Amsler, A. Ariga, T. Ariga, A. S. Belov, G. Bonomi, P. Bräunig, J. Bremer, [......], P. Scampoli, A. Sosa, J. Storey, M. A. Subieta Vasquez, M. Spacek, G. Testera, D. Trezzi, R. Vaccarone, C. P. Welsch, S. Zavatarelli[Show abstract] [Hide abstract]

**ABSTRACT:**The main goal of the AEgIS experiment at CERN is to test the weak equivalence principle for antimatter. AEgIS will measure the free-fall of an antihydrogen beam traversing a moir\'e deflectometer. The goal is to determine the gravitational acceleration g for antihydrogen with an initial relative accuracy of 1% by using an emulsion detector combined with a silicon micro-strip detector to measure the time of flight. Nuclear emulsions can measure the annihilation vertex of antihydrogen atoms with a precision of about 1 - 2 microns r.m.s. We present here results for emulsion detectors operated in vacuum using low energy antiprotons from the CERN antiproton decelerator. We compare with Monte Carlo simulations, and discuss the impact on the AEgIS project.Journal of Instrumentation 06/2013; 8(08). · 1.66 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Motivated by the seesaw mechanism for neutrinos which naturally generates small neutrino masses, we explore how a small grand-unified-theory-scale mixing between the standard model Higgs boson and an otherwise massless hidden sector scalar can naturally generate a small mass and vacuum expectation value for the new scalar which produces a false vacuum energy density contribution comparable to that of the observed dark energy dominating the current expansion of the Universe. This provides a simple and natural mechanism for producing the correct scale for dark energy, even if it does not address the long-standing question of why much larger dark energy contributions are not produced from the visible sector. The new scalar produces no discernible signatures in existing terrestrial experiments so that one may have to rely on other cosmological tests of this idea.Physical Review Letters 08/2013; 111(6):061802. · 7.73 Impact Factor

Page 1

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

2

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

3

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˜ψwithtan˜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

Page 4

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.

Page 5

Torsion-balance tests of the weak equivalence principle

5

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

6

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

Page 7

Torsion-balance tests of the weak equivalence principle

7

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

Page 8

Torsion-balance tests of the weak equivalence principle

8

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.

Page 9

Torsion-balance tests of the weak equivalence principle

9

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.

Page 10

Torsion-balance tests of the weak equivalence principle

10

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

Page 11

Torsion-balance tests of the weak equivalence principle

11

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

Page 12

Torsion-balance tests of the weak equivalence principle

12

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

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Torsion-balance tests of the weak equivalence principle

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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.