New experimental limits on non-Newtonian forces in the micrometer-range
NNew experimental limits on non-Newtonian forces in the micrometer-range
A. O. Sushkov ∗ Yale University, Department of Physics, P.O. Box 208120, New Haven CT 06520-8120, USA
W. J. Kim † Dept. of Physics, Seattle University, 901 12th Avenue, Seattle, WA 98122, USA
D. A. R. Dalvit ‡ Theoretical Division MS B213, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
S. K. Lamoreaux § Yale University, Department of Physics, P.O. Box 208120, New Haven, CT 06520-8120, USA (Dated: August 15, 2011)We report measurements of the short-range forces between two macroscopic gold-coated platesusing a torsion pendulum. The force is measured for separations between 0.7 µ m and 7 µ m, and iswell described by a combination of the Casimir force, including the finite-temperature correction,and an electrostatic force due to patch potentials on the plate surfaces. We use our data to placeconstraints on the Yukawa-type “new” forces predicted by theories with extra dimensions. Weestablish a new best bound for force ranges 0.4 µ m to 4 µ m, and, for forces mediated by gaugebosons propagating in (4 + n ) dimensions and coupling to the baryon number, extract a (4 + n )-dimensional Planck scale lower limit of M ∗ >
70 TeV.
It is remarkable that two of the greatest successes on20th century physics, General Relativity and the Stan-dard Model, appear to be fundamentally incompatible.Intense effort is devoted to searching for a frameworkthat connects gravity to the rest of physics, and stringtheory, or M-theory, is a candidate. There is still a num-ber of outstanding problems, two of the most seriousones are the gauge hierarchy problem and the cosmo-logical constant problem. Theoretical approaches haveincluded proposals incorporating n extra spatial dimen-sions [1], predicting deviations from Newtonian gravityat sub-millimeter length scales. The rationale is to bringdown the Planck scale from M P = 10 GeV in 4 di-mensions to the electroweak scale M ∗ ≈ n )dimensions, thereby addressing the gauge hierarchy prob-lem. In addition, in this scenario, gauge bosons thatpropagate in the bulk of the n extra dimensions but cou-ple to the Standard Model baryon number, can mediateforces that are a factor of ≈ M ∗ /M N ) ≈ strongerthan gravity, here M N ≈ ≈ M ∗ /M P , diluted, exactly like the gravitational inter-action, by the bulk (4 + n )-dimensional volume [2].A large amount of experimental work has been doneto search for such forces in a wide range of distancescales [3]. The Yukawa potential due to new interactionsis typically taken to modify the gravitational inverse-square law: V ( r ) = − G m m r (cid:16) αe − r/λ (cid:17) , (1)where G is the gravitational constant, and the new in- teraction parameters are the strength α and the range λ . The strength α is constrained to be below unity for λ > µ m [4], but at shorter ranges the experimen-tal limits are not as stringent [5–8]. The measurementsat short ranges are complicated by the presence of theCasimir force [9, 10], as well as the electrostatic forcesdue to surface patch potentials [11, 12]. See [13] for a re-cent overview of tests of gravity at sub-millimeter ranges.Recent measurements of the attractive force betweentwo gold-coated flat and spherical plates for separationsbetween 0.7 µ m and 7 µ m have improved our understand-ing of the Casimir and the electrostatic patch forces inthis separation range, and detected the thermal Casimirforce [14]. We now use these measurements to place lim-its on new interactions in the micron range.Our apparatus, which has been more fully describedin [14] comprises a torsion pendulum suspended inside avacuum chamber (pressure 5 × − torr) by a tungstenwire of 25 µ m diameter and 2.5 cm length. The force tobe measured is between the two glass plates, each coatedwith a 700 ˚A (optically thick) layer of gold evaporatedon top of a 100 ˚A-thick layer of titanium. One is a flatplate mounted on one side of the pendulum, the otheris a spherical lens (radius of curvature R = 15 . d . The attractive force between the plates createsa torque on the pendulum body, which is counteractedby a pair of “compensator” electrodes on the opposite a r X i v : . [ qu a n t - ph ] A ug end of the pendulum. The voltage that has to be ap-plied to the compensator electrodes to keep the pendu-lum stationary is proportional to the force between theCasimir plates, with the calibration coefficient extractedfrom the measurements of the electrostatic force betweenthe plates. Further details of the measurement techniquecan be found in Ref. [14].The total force between the plates can be written as: F = F Casimir + F electric + F gravity + F new . (2)The gravitational (Newtonian) force between the plates, F gravity , is very nearly a constant ( ≈
20 pN) in the stud-ied range of separations, and is neglected in the anal-ysis. F new is the hypothetical new force, arising fromthe Yukawa potential in Eq. (1). The Casimir forcebetween the spherical lens and the planar plate is cal-culated in the proximity force approximation (valid for d (cid:28) R ) as F Casimir = 2 πRE
Casimir , where E Casimir is theCasimir interaction energy per unit area between two flatparallel plates separated by a distance d . The latter iscomputed using the Lifshitz formalism with temperature T = 300 K, and the gold optical permittivity data [17],extrapolated to zero frequency using the Drude modelwith parameters ω p = 7 .
54 eV, γ = 0 .
051 eV [14].The electrostatic force is given by the expression F electric = π(cid:15) R (cid:20) ( V − V m ) d + V rms d (cid:21) , (3)where (cid:15) is the permittivity of free space, V is thecomputer-controlled bias voltage applied between theplates, and the “minimizing potential” offset V m is due tothe contact potential difference of approximately 20 mVbetween the two plates, caused by the several soldercontacts around the electrical loop connecting the twoplates. Our measurements show that the minimizing po-tential V m ( d ) is nearly independent of separation in the0.7 µ m ≤ d ≤ µ m range (average variation is 0.2 mV). V rms is a parameter characterizing the magnitude of thevoltage fluctuations across the plates’ surfaces, giving riseto a patch-potential electrostatic force given by the sec-ond term in brackets. Such voltage patches are alwayspresent even on chemically inert metal surfaces preparedin an ultra-clean environment [18, 19], and can be causedby spatial changes in surface crystalline structure, surfacestresses, and adsorbed impurities or oxides. The exactform of the electrostatic patch force is determined by thepatch voltage size distribution spectrum on the plates[11], and in particular by the relationship between threelength scales: the typical patch size λ , the plate separa-tion d , and the “effective interaction length” r eff = √ Rd .In the limit d (cid:28) λ (cid:28) r eff the patch force is well de-scribed by π(cid:15) RV rms /d [12].A further correction is needed to account for fluctu-ations in plate separation d [20]. The sources of thesefluctuations are surface roughness of the plates, and pen-dulum fluctuations, caused, for example, by apparatus vibrations. In addition to radius of curvature measure-ments, surface roughness measurements were performedwith the Micromap TM-570 interferometric microscope,yielding an rms roughness of S q ≈
10 nm for the curvedplate, and S q ≈ d were measured by connecting aninductor in parallel with the Casimir plates, and monitor-ing the resonance frequency of the resulting LC-circuit;rms fluctuations of < ∼
40 nm were recorded. In addi-tion, statistical error of ±
10 nm in determination of d contributes in quadrature to the fluctuations mentionedabove. We take the total rms plate separation fluctuationof δ = (40 ±
20) nm. From the Taylor expansion of theCasimir force about the mean plate separation, we de-duce that a correction term F (cid:48)(cid:48) C δ / d . In addition, since the same correction ex-ists for the electrostatic force, the plate separation d ex-tracted from the electrostatic calibration was correctedby a factor 1 + ( δ/d ) , and the electrostatic patch force V rms /d was corrected by the same factor.The data are well described by the Drude model, usingthe distance correction derived from auxiliary measure-ments as described above (no free parameters), togetherwith a least-squares fit for two parameters, which are V rms and an overall force offset. Given that only twowell-understood fitting parameters are needed to fullydescribe our data, which spans more that an order ofmagnitude in distance and more than two orders of mag-nitude in force, we are confident that, together with a1 /d patch potential force, the finite temperature Drudemodel provides the correct explanation of the Casimirforce between Au surfaces. The reduced χ of the fit is1.04. Therefore we can set bounds on additional forcesthat might be present, at a level of confidence based onthe statistical fluctuations in the difference between thedata and the corrected model. The force data, groupedinto distance bins and averaged, are shown in Figure 1,together with the best-fit line (red), and the Casimir force(dashed blue line). The difference between the red andblue curves is due to the patch potential 1 /d force. Thefit residuals are shown in the inset.According to Eq. (2), these residuals can be used toplace a limit on the hypothetical “new” force F new be-tween the plates. Integrating over the two gold- andtitanium-coated plates gives the following approximateexpression for the force: F new =4 π GRαλ e − d/λ [ ρ Au + ( ρ Ti − ρ Au ) e − d Au /λ +( ρ g − ρ Ti ) e − ( d Au + d Ti ) /λ ] , (4)where ρ Au = 19 g/cm is the gold density, d Au = 700 ˚A isthe gold layer thickness, ρ Ti = 4 . is the Ti den-sity, d Ti = 100 ˚A is the titanium layer thickness, and ρ g = 2 . is the substrate glass density. This F o r ce ( p N ) plate separation ( m)
83 65 742 F o r ce r e s i du a l s ( p N ) plate separation ( m) FIG. 1: The binned experimental short-range force betweengold-coated plates. The error bars include contributions fromstatistical scatter, and uncertainties in the applied correc-tions, discussed in the text. The dashed blue line shows thetheoretical Casimir force, calculated using the Lifshitz formal-ism at 300 K, with the Drude model permittivity extrapola-tion to zero frequency. The red line shows the force, includingthe electrostatic patch potential contribution, with two freefitting parameters, as described in the text. Inset: the forceresiduals, used to place constraints on the “new” short-rangeforces. expression is a good approximation to the exact formfor the Yukawa force between the spherical lens and theflat plate provided λ , d Au , and d Ti are much less thanthe curved plate’s radius of curvature R , the flat plate’sthickness, and both plates’ diameters. These conditionsare satisfied very well in our experiment (For an exactexpression for the force F new , not subject to these as-sumptions, see [21]). The obtained 95%-confidence limitson the “new” interaction strength α at each interactionrange λ are shown in Figure 2. The figure also showslimits obtained by other experimental groups, as well assome theoretical expectations. Our experiment achievesup to a factor of 30 improvement in the limit on the in-teraction strength α for 0 . µ m < λ < µ m, comparedto previous best limits [6].Given the range of parameters α , λ that our exper-iment is most sensitive to, the most stringent limit wecan place is on the (4 + n )-dimensional Planck scale M ∗ in presence of gauge bosons that propagate in (4 + n ) di-mensions and couple to the Standard Model baryon num-ber (hatched region labeled “gauge bosons” in Fig. 2).Our data constrains the range of a hypothetical inter-action mediated by such particles (i.e. their Comptonwavelength) to be below 2 µ m, which corresponds tothe gauge particle mass of more than 0.5 eV. Assum-ing all the coupling parameters are on the order of unity,the natural scale for this mass is M ∗ /M P , which meansthat the (4 + n )-dimensional Planck scale is limited to gaugebosons Yukawamessengersdilaton s t r a n g e m o du l u s g l u o n m o du l u s A B C D E
FIG. 2: Experimental upper limits on the Yukawa forcestrength α , together with some theoretical predictions. Thearea shaded in light blue is experimentally excluded. Thecurves labeled A-E correspond to results in Refs. [7], [6],present work, [5], and [4]. The hatched area labeled “gaugebosons” is the parameter space for forces mediated by gaugebosons that propagate in (4 + n ) dimensions and couple tothe Standard Model baryon number. M ∗ >
70 TeV. This is more stringent than the astro-physical limits, based on the PSR J09052+0755 neutronstar heating from Kaluza-Klein graviton decay, for thecase of 3 or more extra dimensions [22].The authors thank Valery Yashchuk for performing thesurface roughness measurements, and Roberto Onofrioand Serge Reynaud for discussions. This work was sup-ported by the DARPA/MTO’s Casimir Effect Enhance-ment project under SPAWAR Contract No. N66001-09-1-2071. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected][1] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys.Rev. D , 086004 (1999).[2] S. Dimopoulos and A. A. Geraci, Phys. Rev. D ,124021 (2003).[3] E. Adelberger, B. Heckel, and A. Nelson, Annual Reviewof Nuclear and Particle Science , 77 (2003).[4] D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gund-lach, B. R. Heckel, C. D. Hoyle, and H. E. Swanson, Phys.Rev. Lett. , 021101 (2007).[5] A. A. Geraci, S. J. Smullin, D. M. Weld, J. Chiaverini,and A. Kapitulnik, Phys. Rev. D , 022002 (2008).[6] M. Masuda and M. Sasaki, Phys. Rev. Lett. , 171101(2009).[7] R. S. Decca, D. L´opez, H. B. Chan, E. Fischbach, D. E.Krause, and C. R. Jamell, Phys. Rev. Lett. , 240401 (2005).[8] R.S. Decca, D. L´opez, E. Fischbach, G.L. Klimchitskaya,D.E. Krause, and V.M. Mostepanenko, Phys. Rev. D ,077101 (2007).[9] H. B. G. Casimir, Proc. Kon. Nederland. Akad. Weten-sch. , 793 (1948).[10] R. Onofrio, New. J. Phys. , 237 (2006).[11] C.C. Speake and C. Trenkel, Phys. Rev. Lett. , 160403(2003).[12] W. J. Kim, A. O. Sushkov, D. A. R. Dalvit, and S. K.Lamoreaux, Phys. Rev. A , 022505 (2010).[13] I. Antoniadis, S. Baessler, M. Buchner, V.V. Fedorov, S.Hoedl, V.V. Nesvizhevsky, G. Pignol, K.V. Protasov, S.Reynaud and Yu. Sobolev, Compt. Rend. Acad. Sci. (toappear, 2011).[14] A. O. Sushkov, W. J. Kim, D. A. R. Dalvit, and S. K.Lamoreaux, Nat. Phys. , 230 (2011).[15] V. V. Yashchuk, E. M. Gullikson, M. R. Howells, S. C.Irick, A. A. MacDowell, W. R. McKinney, F. Salmassi,T. Warwick, J. P. Metz, and T. W. Tonnessen, AppliedOptics , 4833 (2006).[16] V. V. Yashchuk, A. D. Franck, S. C. Irick, M. R. How- ells, A. A. MacDowell, and W. R. McKinney, in Nano-and Micro-Metrology , edited by H. Ottevaere, P. DeWolf,and D. S. Wiersma (SPIE, Munich, Germany, 2005), vol.5858, pp. 58580A–12.[17] E. D. Palik, ed.,
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