# A New Torsion Balance for the Search of Long-range Interactions Coupling to Baryon and Lepton Numbers

AA New Torsion Balance for the Search ofLong-range Interactions Coupling to Baryon andLepton Numbers

Ramanath Cowsik, Dawson Huth, Tsitsi Madziwa-Nussinov

McDonnell Center for the Space Sciences at Washington University in St. Louis,St. Louis MOE-mail: [email protected]

December 2020

Abstract.

We have developed a torsion balance with a sensitivity about ten timesbetter than those of previously operating balances for the study of long range forcescoupling to baryon and lepton numbers. We present here the details of the designand expected characteristics of this balance. Operation of this balance for a year willalso result in improved bounds on long range interactions of dark matter violatingEinstein’s equivalence principle.

1. Introduction

For over a century experimental eﬀorts have probed Einstein’s General Relativity (GR)theory; ﬁnding agreement with theoretical expectations at every turn [1]. However, thepresence of dark matter and seeming incompatibility with the quantum theory of theStandard Model of Particle Physics (SM) leaves hollow a uniﬁed view of the universe.To address this, numerous theories have been proposed to bring GR and SM into asingle framework at the cost of violating Einstein’s Equivalence Principle (EEP) andintroducing new particles or forces [2, 3], thus motivating experiments to search for aviolation of the EEP. Also, after decades of direct and indirect dark matter searchesrevealing nothing [4, 5] EEP experiments are yet another route to gain insight into thenature of dark matter [6]Some of the best modern tests of the EEP constrain the E¨otv¨os parameter η whichdescribes the validity of the universality of free fall (UFF). Terrestrial experiments usingrotating torsion balances have placed upper bounds on composition-dependent forcesin terms of the E¨otv¨os parameter η Be − Ti = (0 . ± . × − [7] and η Be − Al =( − . ± . × − [8]. More recently torsion balance tests of the EEP have usedchiral test masses probing violations of gravitational parity [9, 10] reporting η left − right =[ − . ± . ± . × − [11]. The ﬁrst results from the MICROSCOPEspace based mission have reported η Ti − Pt = [ − ± ± × − [12]. a r X i v : . [ g r- q c ] J a n New Torsion Balance for the Search of Long-range Interactions η in the regime of the Strong Equivalence Principle(SEP), where contributions from massive self-gravitating test bodies can no longer beneglected. A. M. Archibald et al. [13] analyzed timing observations of the pulses fromthe pulsar over a six-year period showing that the relative accelerations of the whitedwarfs and the neutron star varied by no more than a fraction ∼ . × − of theirmean accelerations or η SEP ∼ . × − .The possible existence of a ’dark/hidden sector’ of particles which are neutral tothe forces of the SM has been a leading motivation for Beyond Standard Model (BSM)physics searches and EP experiments are capable of probing the parameter spaces ofthese models [6]. Despite the sensitive bounds on η shown above plenty of untouchedparameter space exists for future EEP violation searches to explore in the context ofBSM physics and constrain properties of new particles of interest such as dark photonsand Weakly Interacting Massive Particles (WIMPs) [14, 15].The work presented here covers a recent ’pilot experiment’ of the operation of along-period torsion balance instrument sensitive to long-range forces coupling to baryonand lepton numbers. We discuss the subsequent instrument upgrades and the design ofa new torsion balance developed to enhance our instrument’s sensitivity to these forces.We conclude with the expected response of the new balance should the EEP be violatedand prospects for placing lower bounds on η .

2. Pilot Experiment

The design of the pilot long-period torsion balance shown in Fig. 1 follows theclassic design concepts developed by Dicke [16], Braginsky [17], and the more recentworks of the E¨ot-Wash group [8] and Zhu et al. [11]. The balance bob has four-foldazimuthal symmetry with 14.33 g test masses composed of Al and SiO . This symmetrysigniﬁcantly reduces the bob’s coupling to gravitational gradients. The composition ofthe test masses were chosen to give large diﬀerences in baryon number per amu ( B/µ ),lepton number per amu (

L/µ ), and (( B − L ) /µ ) thereby enhancing the bob’s sensitivityto equivalence principle (EP) violating forces which couple to these charges [18]. Valuesfor these parameters with respect to this pilot experiment, the E¨ot-Wash group balance,and the new balance we have constructed can be found in Table 1. The compositiondipole generated by these charge diﬀerences is subject to the gravitational ﬁeld of theSun and that of the dark matter halo centered about our Galactic Center. It is expectedthat any EP violating forces associated with those gravitational ﬁelds will exert a torqueon the balance bob with a period of the length of the diurnal or sidereal day, respectively.Data acquisition with this instrument started on December 22, 2017 and continued untilJune 10, 2018 producing ∼

115 continuous days of useful data used in analysis. Thebalance bob’s long natural period combined with the low frequency of the diurnal orsidereal signal cause long strings of uninterrupted data acquisition to be critical for thesuccess of this instrument. This pilot experiment shows that collecting data of this kind

New Torsion Balance for the Search of Long-range Interactions

3. Noise Diagnostics and Remediation

We have studied the eﬀects of the variations of Earth’s magnetic ﬁeld, absolutetemperature and temperature gradient ﬂuctuations, atmospheric pressure changes, andthe instrument’s support structure response to ambient seismic and thermal noise havebeen studied. In each noise study a length of ∼ New Torsion Balance for the Search of Long-range Interactions R xy ( τ ) = 1 T (cid:90) T − T ( x ( t + τ ) − µ x )( y ( t ) − µ y ) σ x σ y dt, (1)where R XY is the normalized correlation coeﬃcient between two sets of data x and yof length T calculated for some time lag τ between x and y. These data sets havemeans µ and standard deviations σ . This analysis showed that correlations with theEarth’s magnetic ﬁeld and variations in temperature gradients are the most signiﬁcantnoise contributions with normalized correlation coeﬃcients of 0.4 and 0.3 respectively.Pressure variations yielded coeﬃcients at the level of 0.01 and the coeﬃcients for absolutetemperature were also very low. Plots of the correlation coeﬃcients as functions of timelag for the largest contributors are shown in Fig. 3.Based on these studies we added magnetic shielding around the chamber anddeveloped a water circulation system for evening out thermal gradients. For magnetic New Torsion Balance for the Search of Long-range Interactions

4. Design of the New Balance

We are interested in detecting long-range forces which couple to baryon and leptonnumbers so choosing test body materials which maximize these quantities is critical toa sensitive measurement of η . To this end a new balance was designed incorporatingCu and ultra-high molecular weight polyethylene (UHMWP). We ensured UHMWP isvacuum compatible by placing a sample with high surface area in a vacuum chamberseparate from our instrument. The sample was pumped down to ∼ − torr withina day and reached a few parts in 10 − torr after a few days giving a suﬃcient levelof vacuum for our experiment without a prolonged outgassing period. Compared tothe materials of the pilot balance this choice increases ∆( B/µ ) by over an order ofmagnitude; similarly ∆(

L/µ ) and ∆( B − L/µ ) are enhanced by a factor of ∼ − B/µ ) and (

L/µ ) of the Sun relative to the surroundinginterstellar medium and strong dipole moments of our balance will allow us to sensitivelyprobe the coupling strengths for baryon-baryon and lepton-lepton interactions.The balance is designed with a ring-shaped geometry where each semicircle hasthe same mass and the ﬁrst and second order moments. The copper semicircle iscomprised of four 90 ◦ arcs with two arcs stacked vertically on each half semicircleseparated by ∼ . ◦ arcs joined at the ends.The UHMWP semicircle is also covered with Cu foil to prevent the accumulation ofpatch charges and the entire assembly is joined with conductive epoxy. The higherazimuthal symmetry reduces couplings to gravitational gradients which induce spurioustorques on the balance and also removes any preferential direction it may be deﬂectedby radiometric ﬂow compared to four-fold symmetric designs. We design for a mass of New Torsion Balance for the Search of Long-range Interactions

Characteristics E¨ot-Wash [7, 8] Pilot CurrentMaterials Be-Ti Al-SiO Cu-C H Total Mass (g) 70 72 530Tine Length/Radius (cm) 2.01 25 25Moment of Inertia (g cm ) 3 . × . × . × Fiber Length (m) 1.07 1.67 1.67Fiber Section 20 µ m dia. 18 µ m dia. 15 × µ m Torsion Constant (dyne cm rad − ) 2 . × − . × − . × − Natural Period (s) 798 12800 12100Signal Torque (dyne cm) 3 . × − . × − . × − Expected Deﬂection (rad) 1 . × − . × − . × − Nyquist Torque (dyne cm) 3 . × − . × − . × − SNR ( τ S /τ Ny ) 1 .

24 7 .

99 21 . B/µ ) 2 . × − . × − . × − ∆( L/µ ) 1 . × − . × − . × − ∆( B − L/µ ) 1 . × − . × − . × − Charge Composition of Sun (

B/µ ) (

L/µ ) ( B − L/µ )0.9943 0.8493 0.1450

Table 1: Comparison of characteristics of our prototype instrument, proposed, and theE¨ot-Wash instrument. Values for expected deﬂection, signal torque, and Nyquist torqueassume an EP violation at the level of η ∼ − and an observation time of 10 s. Thecharge characteristics of the Sun are shown for referenceA thin tungsten ﬁber is used to suspend the balance because of tungsten’s hightensile strength and large Q factor. The power spectral density of data from the pilotexperiment is shown in Fig. 4 with a Lorentzian ﬁt parameterized by a Q of 800 whichis limited by the length of the data set. Tungsten has been shown to oﬀer a Q factorup to 6000 by the E¨ot-Wash group [7]. A data set much longer than the 100 days ofobservations that we had is needed to establish such a high value of Q for our balancewith a natural period of ∼ ,

500 seconds. A ﬁber of rectangular cross-section is usedrather than a circular section. Circular section ﬁbers with radius r have a torsionconstant proportional to r while a rectangular section is proportional to a · b where a is the width and b is the thickness of the tungsten strip. The rectangular geometry New Torsion Balance for the Search of Long-range Interactions ∼ . × − Hz.allows for a suﬃciently large cross-sectional area to support the more massive balancewhile only increasing the torsion constant of the ﬁber by a factor of ∼ . × − rad in the balance position for a given signal amplitude while increasing theSNR of the balance with respect to the Nyquist thermal noise in the suspension ﬁberby almost a factor of 3 compared to the pilot balance and a factor of ∼

17 compared tothe E¨ot-Wash balance.

New Torsion Balance for the Search of Long-range Interactions

5. Closing Remarks

Terrestrial experiments still hold much promise for measuring violations of the EEP togreater precision and the various motivations for new physics beyond the StandardModel fuel these eﬀorts. In this work we have described past and current workssearching for EP violations including a pilot experiment of a long-period torsion balanceinstrument. The lessons learned from this prototype have given us insight into whereimprovements can be made for better environmental isolation and a more sensitivetorsion balance. Many of these changes have been implemented or are actively beingdeveloped and a measurement of η at the level of 10 − or lower can be made.

6. Acknowledgements

We would like to thank the NSF for initial funding of this project, followed by fundingfrom the McDonnell Center for the Space Sciences. We also recognize the earliercontributions of Michael Abercrombie, Adam Archibald, Maneesh Jeyakumar, NadathurKrishnan, and Kasey Wagoner towards this eﬀort.

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