Search for Lorentz violation in short-range gravity
aa r X i v : . [ h e p - e x ] A p r Search for Lorentz violation in short-range gravity
J.C. Long and V. Alan Kosteleck´y
Physics Department, Indiana University, Bloomington, IN 47405, U.S.A.
Abstract
A search for sidereal variations in the force between two planar tungsten oscillators separated byabout 80 µ m sets the first experimental limits on Lorentz violation involving quadratic couplingsof the Riemann curvature, consistent with no effect at the level of 10 − m . U ( r ) generated by a source of mass density ρ ( r ) acquires an extra perturbative term withfour spatial derivatives, − ~ ∇ U = 4 πG N ρ + ( k eff ) jklm ∂ j ∂ k ∂ l ∂ m U, (1)where ( k eff ) jklm are effective coefficients with dimensions of squared length that can betaken as constant on the scale of the solar system [7]. The extra term violates rotationsymmetry and hence Lorentz invariance. It is the general leading-order term in a naturalperturbative expansion because a term with three derivatives is excluded by Newton’s thirdlaw [5]. The presence of four derivatives implies corrections to the Newton force that areinverse quartic and hence appear only at short range. The rotation violation implies effectsin the laboratory depending on orientation and also on sidereal time due to the rotation ofthe Earth, thereby ensuring that the resulting experimental signals are distinct from thoseassociated with conventional Yukawa or inverse-power corrections. The extra term offers a2 etector Mount Transducer Probe Source Mass PZT Bimorph Tuning Block Source Mount Detector to Amplifier Torsion Axis 1 cm
FIG. 1: Schematic of the Indiana short-range experiment. general description of dominant noncentral short-range corrections to Newton gravity arisingfrom an underlying unified theory.Here, we present new data acquired in March 2012 from a short-range experiment [8–10]located in Bloomington, IN. We use these data to search for sidereal variations involving non-central inverse-quartic corrections to Newton’s law, obtaining first constraints on quadraticLorentz-violating curvature couplings at the level of 10 − m . We also extend the analysisto incorporate the 2002 dataset obtained with the apparatus located in Boulder, CO [9].Note that existing searches for pure-gravity local Lorentz violation within this frameworkhave been restricted to the context of a Lorentz-violating inverse- square law [11–18]. A fewother short-range experiments [19–22] may have potential sensitivity to the modifications(1), while some experiments optimized for nonperturbative corrections to Newton’s law couldconceivably be adjusted to study perturbative effects [23–26]. Note also that constraints onforces with various inverse-power laws have appeared in the literature [27], but only in thecontext of Lorentz-invariant effects.The design and operation of the experiment is described elsewhere [8–10]. Here, wesummarize briefly the basic features. Each of the two test masses is a planar tungstenoscillator of approximate thickness 250 µ m, separated by a gap of about 80 µ m, arranged asshown in Fig. 1. A stiff conducting shield is placed between them to suppress electrostatic3nd acoustic backgrounds. The planar geometry concentrates as much mass as possibleat the scale of interest while being nominally null with respect to inverse-square forces,thereby suppressing the Newton background relative to new short-range effects. The force-sensitive ‘detector’ mass is driven by the force-generating ‘source’ mass at a resonance near1 kHz. Vibration isolation is a key requirement for this setup, and operation at 1 kHz ischosen because at this frequency a comparatively simple passive vibration-isolation systemcan be used. The entire apparatus is enclosed in a vacuum chamber and operated at 10 − torr to minimize the acoustic coupling. Detector oscillations are read out via capacitivetransducer probes coupled to a sensitive differential amplifier, with the signal fed to a lock-in amplifier referenced by the same waveform used to drive the source mass. This design hasproved effective in suppressing all background forces to the extent that only thermal noise isobserved, arising from dissipation in the detector mass. The output of the lock-in amplifierconstitutes the raw data. These data are converted to force readings by comparison withthe detector thermal noise, the scale of which is determined using the equipartition theorem[8]. Following data collection in 2002, this experiment set the strongest limits on unobservedforces of nature between 10 and 100 µ m [9]. The apparatus has since been optimized toexplore gaps below 50 µ m, and operation at the thermal noise limit has recently beendemonstrated [10].Measuring the coefficients ( k eff ) jklm in Eq. (1) is the goal of the present analysis. Thecoefficients are totally symmetric, implying 15 independent observables for Lorentz violation.Following standard convention, we extract values of these observables in the canonical Sun-centered frame [3, 28], with Z axis along the direction of the Earth’s rotation and X axispointing towards the vernal equinox. As the Earth rotates, the coefficients measured in thelaboratory vary with sidereal time T . The Earth’s boost β ⊕ ≃ − can be neglected here.The transformation from the Sun-centered frame ( X, Y, Z ) to the laboratory frame ( x, y, z )therefore involves a time-dependent rotation R jJ ( T ) [5] that depends on the Earth’s siderealfrequency ω ⊕ ≃ π/ (23 h 56 min) and the colatitude χ of the laboratory, which is 0.887 inBloomington and 0.872 in Boulder. The laboratory coefficients ( k eff ) jklm ( T ) are thus relatedto the coefficients ( k eff ) JKLM in the Sun-centered frame by( k eff ) jklm ( T ) = R jJ R kK R lL R mM ( k eff ) JKLM . (2)The cartesian components g j ( r , T ) of the modified gravitational acceleration at position4 and at sidereal time T contain the conventional Newton acceleration along with an inverse-quartic correction term, g j ( r , T ) = − G N Z d r ′ ρ ( r ′ ) b R j | r − r ′ | + k j ( b R , T ) | r − r ′ | ! . (3)Here, b R = ( r − r ′ ) / | r − r ′ | , while k j ( b R , T ) = ( k eff ) klmn ˆ R j ˆ R k ˆ R l ˆ R m ˆ R n − k eff ) klmm ˆ R j ˆ R k ˆ R l + ( k eff ) klkl ˆ R j − k eff ) jklm ˆ R k ˆ R l ˆ R m + 18( k eff ) jkll ˆ R k (4)controls the inverse-quartic force correction, which varies with direction ˆ R and sidereal time T . Note that the T dependence is oscillatory and includes components up to the fourthharmonic of ω ⊕ .The detector is a constrained mechanical oscillator with distributed mass. The modalamplitude at any point in the detector mass is strongly dominated by vertical motion. Thisis particularly true near the thermal noise limit, where the amplitudes are of order 1 pm[10]. The experiment is thus sensitive predominantly to the z component F p of the effectiveforce at the location of the capacitive probe, which can be written as F p ( T ) = 1 d Z D d r ξ ( r ) F z ( r , T ) . (5)Here, ξ ( r ) is the detector mode-shape function, which is the amplitude of the displacementof the detector at point r when undergoing free oscillations in the relevant mode of interest,and the displacement d is the oscillation amplitude of the detector at the location of theprobe. These quantities are derived from a finite-element model of the detector mass andhave the same arbitrary normalization. The integration is taken over the volume D of thedetector over which the force is applied.For the purposes of the present analysis, Eq. (5) is evaluated by Monte-Carlo integration,using the z component F z ( r ) of the force (3) expressed in terms of the coefficients ( k eff ) JKLM in the Sun-centered frame along with the geometrical parameters listed in Table II of Ref.[8]. Note that the source amplitude for the 2012 dataset was 22 . ± . µ m and the averagegap was 77 . ± µ m. The experiment is performed on resonance, so the Monte-Carlo5lgorithm computes the Fourier amplitude of Eq. (5) averaged over a complete cycle of thesource-mass oscillation, taking into account the measured source-mass curvature and modeshape. The result can be expressed as a Fourier series in the sidereal time T , F p ( T ) = C + X m =1 S mω sin( mω ⊕ T ) + C mω cos( mω ⊕ T ) . (6)The Fourier amplitudes in this expression are linear combinations of the coefficients( k eff ) JKLM . Their weights are functions of the source and detector mass geometry and thelaboratory colatitude. Using approximately 500 million random pairs of points for each testmass suffices to resolve all harmonics. Systematic errors from the dimensions and positionsof the test masses [8] can be determined at this stage, by computing the mean and standarddeviation of a population of Fourier amplitudes generated with a spread of geometries basedon metrology errors. For the 2002 data, the systematic error on the weights ranges fromabout 10% to 75%. For the 2012 data, it ranges from 15% to 50% on the most resolvableterms, while a few poorly resolved ones have systematic errors in excess of 100%. Most ofthe systematic error is due to the uncertainty in the average gap, with a smaller contributionfrom the source amplitude.All 15 independent components of ( k eff ) JKLM appear in the Fourier series (6), althoughno single amplitude contains all of them. The transformation (2) predicts some simplerelations among the amplitudes, each of which is satisfied by the results of the numericalintegration. Performing the numerical integration for a hypothetical geometry with anaverage gap an order of magnitude larger than the largest dimension of either mass producesa result agreeing to within a few percent with the analytical expression for point masses ofthe same mass and separation. This limiting case confirms that some contributions from( k eff ) JKLM are resolvable only due to the planar geometry.Figure 2 displays the force data acquired during the runs in 2012 and in 2002 as a functionof the sidereal time T measured in seconds from T = 0, which is taken to be the 2000 vernalequinox. The force data were collected at a 1 Hz rate in 14.4-minute sets (2012 run) andin 12-minute sets (2002 run), with comparable intervals between each set during whichdiagnostic data were taken to monitor the experiment for gain and frequency drifts. Eachdata point represents the mean of a 14.4- or 12-minute set. Each error bar shown is the 1 σ standard deviation of the mean, including both the statistical uncertainty and the systematicerrors associated with the force calibration. The 2002 force calibration and parameters are6 .590x10 -400-300-200-1000100200300400 Time (dd:hh:mm) F o rce (f N ) Time (s)
Mar 2012
FIG. 2: Data from the Indiana short-range experiment. given by Eq. (2) and Table 1 of Ref. [8]. The 2012 parameters are unchanged except that themechanical quality factor was 22479 ±
64, the resonance frequency was 1191.32 ± ± × − m / . The calibration uncertainties forthe 2002 and 2012 data increase the errors by about 1% and 2%, respectively.Figure 2 represents a finite time series of force data with uneven time distribution. Toanalyze the data for Lorentz violation, we adopt a well-established procedure [13]. The idealmeasure of each harmonic signal component is the corresponding Fourier amplitude in Eq.(6). Each of these nine amplitudes, k = 1 , . . .
9, can be estimated by the discrete Fouriertransform ˜ d k = N P j f ( T j ) a k ( T j ), where N is the total number of force-data points plotted7
012 data 2012 data 2002 data 2002 dataMode ˜ d k ˜ D k ˜ d k ˜ D k C − . ± . − . ± . − . ± . . ± . S ω − . ± . − . ± . . ± . . ± . C ω . ± . . ± . − . ± . − . ± . S ω − . ± . − . ± . − . ± . − . ± . C ω − . ± . − . ± . − . ± . − . ± . S ω − . ± . − . ± . . ± . . ± . C ω − . ± . − . ± . − . ± . − . ± . S ω − . ± . . ± . . ± . . ± . C ω . ± . − . ± . . ± . . ± . in Fig. 2, f ( T j ) are the values of the force at each time T j , and a k ( T j ) is either sin( ω k T j ) orcos( ω k T j ) with ω k = mω ⊕ , m = 0 , , , ,
4. For this part of the analysis, we treat the 2012and 2002 results as separate datasets. The nine components ˜ d k extracted from the 2012dataset and from the 2002 dataset are listed in the second and fourth columns of Table I.The uncertainties are determined by propagating the errors of the time-series data in Fig. 2.The uncertainties can also be estimated by computing the Fourier transforms at severalfrequencies above and below the signal frequency and calculating the root mean square ofthe values obtained. The former method is slightly more pessimistic and is adopted here.For a finite time series, the Fourier components overlap. The overlap can be quantifiedby a correlation covariance matrix cov( a k , a k ′ ) = (2 /N ) P j a k ( T j ) a k ′ ( T j ). The covariancematrix relates the amplitudes ˜ D k for continuous data to the amplitudes ˜ d k for discrete dataaccording to ˜ d k = P k ′ cov( a k , a k ′ ) ˜ D k ′ . The nine continuous amplitudes ˜ D k can be obtainedby applying the inverse matrix cov − . For the 2012 and 2002 datasets, the results of thiscalculation are also displayed in the third and fifth columns of Table I. The ˜ D k can be takento represent the measured values of the force components. These values largely are consistentwith zero within the quoted errors, which include the small calibration systematics along withstatistical errors. The modes at 3 ω appear to display resolved signals at this stage. However,the associated coefficient weights are tiny, so these force components become swamped by8 oefficient 2012 value 2002 value Combined(10 − m ) (10 − m ) (10 − m )( k eff ) XXXX . ± . . ± . . ± . k eff ) Y Y Y Y . ± . − . ± . . ± . k eff ) ZZZZ . ± . − . ± . . ± . k eff ) XXXY . ± . . ± . − . ± . k eff ) XXXZ − . ± . − . ± . − . ± . k eff ) Y Y Y X − . ± . − . ± . − . ± . k eff ) Y Y Y Z . ± . − . ± . . ± . k eff ) ZZZX − . ± . − . ± . − . ± . k eff ) ZZZY . ± . − . ± . . ± . k eff ) XXY Y . ± . − . ± . . ± . k eff ) XXZZ . ± . − . ± . . ± . k eff ) Y Y ZZ . ± . − . ± . . ± . k eff ) XXY Z − . ± . . ± . − . ± . k eff ) Y Y XZ . ± . . ± . . ± . k eff ) ZZXY − . ± . − . ± . − . ± . σ ) from the 2012, 2002, and combined datasets, with all othercoefficients vanishing. position systematics in the final results below.Individual measurements of the independent components of ( k eff ) JKLM can be extractedfrom a global probability distribution formed using the values of the nine continuous am-plitudes ˜ D k and their errors. Each measured amplitude can be assigned a correspondingprobability distribution p k = p k (( k eff ) JKLM ) that is a function of the 15 independent com-ponents of ( k eff ) JKLM . The p k are assumed to be gaussian with means µ k and standarddeviations σ k . The global probability distribution P = P (( k eff ) JKLM ) of interest is then theproduct of the individual p k , taking the form P = P exp " − X k =1 ( ˜ D k − µ k ) σ k . (7)In this expression, P is an arbitrary normalization. The predicted signal µ k =9 k (( k eff ) JKLM ) for the k th amplitude is determined from Eqs. (5) and (6), and the vari-ance σ k includes all statistical and systematic errors.An independent measurement of any one chosen component of ( k eff ) JKLM can in principlebe obtained by integrating the global probability distribution P over all other components.The result is a distribution involving the chosen component with a single mean and standarddeviation, which constitute the estimated component measurement and its error. However,the 2012 dataset alone contains only nine signal components, which is insufficient to con-strain independently each of the 15 degrees of freedom in ( k eff ) JKLM . Following standardpractice in the field [3], we can obtain maximum-sensitivity constraints on each componentof ( k eff ) JKLM in turn by integrating the global probability distribution with the other 14 de-grees of freedom set to zero. The resulting measurements and 2 σ errors on each independentcomponent of ( k eff ) JKLM are displayed in the first two columns of Table II. Note that thefirst column reveals our choice for the 15 independent components of ( k eff ) JKLM . Note alsothat the sensitivity of the apparatus to the coefficients ( k eff ) JKLM can be crudely estimatedas the ratio of the thermal-noise force at the location of the probe ( ∼
10 fN) to the scale( ∼ µ N/m ) of the amplitudes in the Fourier series (6), multiplied by a suppression factorof order 10 − because the dominant contribution to the noncentral force in a parallel-plategeometry arises from edge effects [29]. This estimate matches the size of the values in thesecond column of Table II.The third column of Table II displays the values for the coefficients ( k eff ) JKLM obtainedfrom a comparable analysis of the 2002 dataset. These 2002 results are about a factor offive less sensitive than the 2012 data, a feature that can be traced to the larger average gapbetween the source and detector masses and the smaller source-mass amplitude in the 2002experiment. The final column of Table II presents the measured values of each independentcomponent taken in turn that are obtained from analyzing the combined datasets.The contents of Table II represent the first measurements of noncentral inverse-quarticcorrections to Newton gravity and hence of quadratic curvature couplings violating localLorentz invariance. The inverse-quartic dependence implies the corrections are perturbativeat squared distances greater than the coefficient values. For example, the perturbativeeffects at the apparatus scale are roughly comparable to the Newton force, while on themacroscopic scale of the laboratory the attained constraints exclude noncentral forces atabout parts in ten million. An alternative perspective can be obtained by comparing the10ength dimension associated with the coefficients ( k eff ) JKLM to the various scales set by theCompton wavelengths of elementary particles. The experiment here probes modificationsgoverned to within about an order of magnitude of the scale of the neutrino Comptonwavelength. Effects at the scales of Compton wavelengths of other particles would be smaller,reflecting the possibility that comparatively large ‘countershaded’ Lorentz violation remainsa viable possibility [30].The results reported here set a benchmark for future efforts. For example, upgradingthe apparatus used by improving the test-mass and shield flatness could reduce the averagegap by a factor of two, and refining the test-mass metrology could reduce the uncertaintyin the average gap by a factor of four. Simulations suggest these improvements would in-crease the overall sensitivity by more than an order of magnitude in the absence of newsystematics. With several months of run time, the statistical error bars could be reducedby about another order of magnitude. Moreover, other experimental groups also have thecapability of improving substantially over the results in the present work [5]. For exam-ple, the HUST experiment has recently reported sensitivities to the coefficients ( k eff ) JKLM surpassing those reported here [29]. Overall, the prospects for improved future short-rangesearches for Lorentz violation are excellent.We thank Q. Bailey, R. Decca, and R. Xu for discussions, S. Kelly for collection of the2012 data, and D. Bennett and W. Jensen for work on the Monte-Carlo code in earlierincarnations of this experiment. We are grateful to C.-G. Shao, Y.-J. Tan, W.-H. Tan, S.-Q.Yang, J. Luo, and M.E. Tobar for drawing our attention to an issue with the Monte-Carlocode used in the original version of this work. The 2012 data were taken at the IndianaUniversity Center for the Exploration of Energy and Matter. This work was supportedin part by the National Science Foundation under grant number PHY-1207656, by theDepartment of Energy under grant number DE-SC0010120, and by the Indiana UniversityCenter for Spacetime Symmetries. [1] V.A. Kosteleck´y and S. Samuel, Phys. Rev. D , 683 (1989); V.A. Kosteleck´y and R. Potting,Nucl. Phys. B , 545 (1991); Phys. Rev. D , 3923 (1995).[2] For reviews see, for example, J. Murata and S. Tanaka, arXiv:1408.3588; J. Jaeckel and A. ingwald, Ann. Rev. Nucl. Part. Sci. , 405 (2010); E.G. Adelberger, J.H. Gundlach, B.R.Heckel, S. Hoedl, and S. Schlamminger, Prog. Part. Nucl. Phys. , 102 (2009); E. Fischbachand C. Talmadge, The Search for Non-Newtonian Gravity , Springer-Verlag, 1999.[3] V.A. Kosteleck´y and N. Russell,
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