lorentz tests with short-range gravity

Lorentz Tests with Short-Range Gravity

Indiana University, Bloomington IU Center for Exploration of Energy and Matter (CEEM)

Josh Long

IUPUI – IUB

Indiana University Collaborative Research Grant

Outline

Parameterization

Experimental approach

Expected LV signals

Limits on LV coefficients (d=6, 8)

Outlook - optimization

Motivation and existing (non-LV) limits

Experimental challenges

r

m1 m2

mB

Yukawa Interaction

a = strength relative to gravity

Power Law

m1 m2

m=0

m=0

r0 = experimental scale

set limits on bn for n = 2 - 5

Parameterization

/1 2( ) 1 rm mU r G e

r

a

101 2( ) 1

n

n

rm mU r G

r rb

rangeBm c

Search for Lorentz Violation

Source:

http://www.physics.indiana.edu/~kostelec/mov.html

Test for sidereal variation in force signal: Standard Model Extension (SME)

Q. G. Bailey and V. A. Kostelecký, PRD 74 045001 (2006)

V. A. Kostelecký, PRD 69 105009 (2004)

Q. G. Bailey, V. A. Kostelecký, R. Xu, PRD 91 022006 (2015)

V. A. Kostelecký and M. Mewes, PLB 766 137 (2017)

GR LVL L L

ˆˆ ˆ( , , )LVL f s q k

3( )

4,even

ˆ ,...dd

d

s s

Modified field equations

Solution in linearized, non-relativistic limit:

(d = mass dimension)

Expanded to gravitational sector

talks by Q. Bailey (M Th 9:00), J. Tasson (T F 9:00),

M. Mewes (W 9:00), …

𝛿𝑈 𝒓 ∼ 𝐺𝑁𝑚

𝑟𝑑−3𝑑

Short Range Limits and Predictions

Experimental limits:

Theoretical predictions:

Limits still allow forces 1 million times stronger than gravity at 5 microns

Moduli, dilatons: new particles motivated by string models

Vacuum energy: prediction from new field which also keeps cosmological constant small

“Large” extra dimensions

Stanford: A. Geraci et al., PRD 78 022002 (2008)

Casimir: Y.-J. Chen et al., PRL 116 221102 (2016)

Eot-Wash: D. Kapner et al., PRL 98 021101 (2007)

HUST: W.-H. Tan et al., PRL 116 131101 (2016)

Irvine: J. Hoskins et al., PRD 32 3084 (1985)

Torsion Osc: JCL et al., Nature 421 922 (2003)

Irvine, HUST, Eot-Wash, = torsion pendulum experiments

Stanford, IUPUI: MEMS-type experiment

Theory: S. Dimopoulos, A. Geraci, PRD 68 124021 (2003)

Challenge: scaling and backgrounds

m1, 1 m2, 2

~ 2r 3 2

41 2 1 21 22 2

(4 )~

(2 ) 4

Gm m G rF G r

r r

r = 100 m F ≈ 10-17 N

1 = 2 = 20 g/cm3, r = 10 cm F ≈ 10-5 N

Electrostatic:

Magnetic (contaminant):

Casimir:

e0V 2

r2 FE ~

ħc

r 4 FC ~

FM ~

r 4

0 1 2 1 2ˆ ˆ[ 3( )( )]r r

Shielding fails (modes penetrate) below D ~ P ~ 100 nm

D

6

HUST short-range experiment

W.-H. Tan, et al., PRL 116 131101 (2016).

Tungsten ( =19) test and source masses: ~ 200 m thick

Minimum gap: 295 m

BeCu membrane shield (not shown): 30 m

Drive frequency: 0.26 mHz

Signal frequency: 2.1 mHz

DtY = 3.4 × 10-17 Nm (a = 0.1, = 1 mm)

DtN = 0.7 × 10-17 Nm

Limits: Scenarios with a ≥ 1 excluded

at 95% CL for ≥ 59 m

rotation

axis x

-y

9cm

Experimental Approach

Source and Detector Oscillators Shield for Background Suppression

~ 5 cm

Planar Geometry - null for 1/r2

Resonant detector with source mass driven on resonance

1 kHz operational frequency - simple, stiff vibration isolation

Stiff conducting shield for background suppression

Double-rectangular torsional detector: high Q, low thermal noise

Central Apparatus

Scale: 1 cm3

detector mass

shield

source mass PZT bimorph

transducer amp box

tilt stage

vibration isolation stacks

Figure: Bryan Christie (www.bryanchristie.com) for Scientific American (August 2000)

Vibration isolation stacks: Brass disks connected by fine wires; soft springs which attenuate at ~1010 at 1 kHz (reason for using 1 kHz)

Readout: capacitive transducer and lock-in amplifier referenced by source drive frequency

Vacuum system: 10-7 torr

Interaction Region

10 m stretched Cu

membrane shield

(shorter ranges

possible)

detector mass

front rectangle

(retracted)

source mass

(retracted)

Thinner shield

60 m thick sapphire plate replaced by 10 m stretched copper membrane

Compliance ~5x better than needed to suppress estimated electrostatic force

Minimum gap reduced from 105 m (2003) to 40 m.

Central Apparatus

~50 cm

Inverted micrometer stages for

full XYZ positioning

Torque rods for micrometer stage control

Vacuum

system

base

plate

Readout – replaced with differential design

• Sensitive to ≈

10 fm thermal

oscillations

• Interleave on

resonance, off

resonance runs

• Typical

session: 8hrs

with 50% duty

cycle

Sensitivity: increase Q and statistics, decrease T

)]/exp(1)][/exp(1)[/)(exp(2)( 2 a dsddsY tttdAGtF

• Signal

Force on detector due to Yukawa interaction with source:

• Thermal Noise

t

kTDFT

4

Q

mD

~ 3 x 10-15 N (for a = 1, = 50 m)

~ 3 x 10-15 N (300 K, Q = 5 x104, 1 day average)

~ 7 x 10-17 N (4 K, Q = 5 x105, 1 day average)

sensitivity

10-13 g

Force Measurement Data – March 2012

19 hours on-resonance data collected over 3 days with interleaved diagnostic data

On-resonance: Detector thermal motion and amplifier noise

Off-resonance: amplifier noise

On Resonance Off Resonance

Force Measurement Data - Detail

off-resonance

on-resonance

Von – Voff = 0.93 ± 0.74 V (1)

Net Signal:

F = 4.0 ± 3.2 fN

Force:

Detector – probe force from ~ nV scale “ground” fluctuations on detector mass

Possible Source:

Upper: 1 day integration time, 50 micron gap, 300 K

Lower: 1 day integration time, 50 micron gap, 4.2 K, factor 50 Q improvement

Current Limits (2) and Projected Sensitivity

Present gap ~ 100 microns; need flatter, more level elements

Search for Lorentz Violation

Source:

http://www.physics.indiana.edu/~kostelec/mov.html

Test for sidereal variation in force signal: Standard Model Extension (SME)

Q. G. Bailey and V. A. Kostelecký, PRD 74 045001 (2006)

V. A. Kostelecký, PRD 69 105009 (2004)

Q. G. Bailey, V. A. Kostelecký, R. Xu, PRD 91 022006 (2015)

V. A. Kostelecký and M. Mewes, PLB 766 137 (2017)

GR LVL L L

ˆˆ ˆ( , , )LVL f s q k

3( )

4,even

ˆ ,...dd

d

s s

Modified field equations

Solution in non-relativistic limit:

3( ) ~ N

dd

G mU r

r

(d = mass dimension)

Expanded to gravitational sector

(talks by Q. Bailey, M Th 9:00, …)

y (east)

x

(south)

(zenith)

z

q

f

Ms

Mt

r

SME gravitational potential correction

( )lab

3( ) ( , ) N dN s t

jm jmddjm

G M MU Y k

r q f

r

SME lab frame

d ≥ 4, even = mass dimension of LV operator

j = d - 2 or d - 4

m = -j, …, j

= Newtonian, spherical coefficients of Lorentz

violation in the SME standard lab frame

( )labN djmk

⇒ Dependence on orientation in lab, sidereal time

d FORCE correction

(1/r d-2 )

# of coefficients

(4d-10)

k units

(lengthd-4)

4 1/r2 6 1

6 1/r4 14 length2

8 1/r6 22 length4

.

.

. V. A. Kostelecký and M. Mewes, PLB 766 137 (2017)

→Test with large gravity signal (“g”)

(superseded by absence of

gravitational Cerenkov radiation)

Mt

[1] Q. G. Bailey, V. A. Kostelecký, R. Xu, PRD 91 022006 (2015).

3

2 4

ˆˆ ( , )( , ) ( )

jjj

N

k TRg T G d r

Rr r

r r r r

ˆwhere ( ) / , R r r r r

eff eff

105ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( , ) ( ) 45( )2

j k l m n j k l

j klmn klmnk T k R R R R R k R R R R

Acceleration of test mass (at r) due to source mass density ( ) r

[2] J. C. Long, V. A. Kostelecký, PRD 91 092003 (2015).

Example: SME d=6 force (cartesian)

eff eff eff

9 ˆ ˆ ˆ ˆ ˆ( ) 30( ) 18( )2

j k l m k

klkl jklm jkllk R k R R R k R

y (east)

x

(south)

(zenith)

z

Ms R

SME lab frame

= cartesian coefficients of Lorentz violation in the SME standard lab frame eff( ) jklmk

ˆ jR = projection of unit vector along R in jth direction

r r

y

x

z

q

f

m1

m2

R

Angular dependence of lab frame force

1 2

4ˆ ˆ ˆ( , , )jklm jklmN

z iklm z x y z

G m mF k R R R

R

• equal contributions of

F > 0, F < 0

over half-sphere

(each component)

(Fz only)

Γz(θ,f)

f

q = π/2

q = 0

(zenith)

H. Meyer, IU

y

x

q

f

m1

m2

R

Lab frame point-plane force

• fundamental dependence = 1/d2

max max

11 4 2

0 0

( , )( , ) sin cos

jklmjklm jklmNz

z N jklm z

G mF G m k dm d d

R d

qq

q q q q

z

• Fz vanishes for infinite area plate (q, f cover half-sphere)

d

• optimal distance? (small d and but small solid angle)

• Not (Newtonian) “edge effect,” where F > 0 but does not depend on d unless any (finite)

plate edges are aligned

H. Meyer, IU

Calculation of the Fitting Function

[1] V. A. Kostelecký and M. Mewes, PRD 66 056005 (2002)

• Transform to sun-centered frame [1]:

= sidereal frequency, c = colatitude = 0.89,

ignore boost

eff eff( ) ( ) ( )jJ kK lL mM

jklm JKLMk T R R R R k

d FORCE correction

(1/r d-2 )

# of coefficients

(4d-10)

k units

(lengthd-4)

# signals

(2d-3)

4 1/r2 6 1 5

6 1/r4 14 length2 9

8 1/r6 22 length4 13

2 experiments:

4d-6

( > # of coefficients)

y

x

z

q

f

m1

m2

R

Angular, time dependence of z-force

eff( ) 1

XXYYk

colatitude = 0.89 (IUB)

q = 0.89

Γz(θ,f)

f

q = π/2

q = 0

(zenith)


[1] V. A. Kostelecký and M. Mewes, PRD 66 056005 (2002).

• Transform to sun-centered frame [1]:

• Detector has distributed mass:

x = mode shape from finite element model [2]

= sidereal frequency, c = colatitude = 0.89,

ignore boost

31( ) ( ) ( , )z

PF T d r F T

zx r r

[2] H Yan, et al., Class. Quantum Grav. 31 205007 (2014)

Need Fz only

eff eff( ) ( ) ( )jJ kK lL mM

jklm JKLMk T R R R R k

C, S functions of test mass geometry,

• Force:

eff( )JKLMk (sun frame)


4

01

sin( ) cos( )p m mm

F C S m T C m T

0 eff eff eff[ (1.8 2.3)( ) (1.8 2.3)( ) (3.6 4.7)( )

XXXX YYYY XXYYC k k k

0 eff eff eff[ 1.8( ) 1.8( ) 3.6( )


eff eff13.5( ) 13.5( ) ] nN

XXZZ YYZZk k

eff eff(13.5 7.5)( ) (13.5 7.5)( ) ] nN

XXZZ YYZZk k

eff eff eff[5.0( ) 3.6( ) 12.2( )

XXXZ YYYZ ZZZXS k k k

eff eff eff14.1( ) 3.6( ) 5.0( ) ] nN

ZZZY XXYZ YYXZk k k

eff eff eff[3.6( ) 5.0( ) 14.1( )

XXXZ YYYZ ZZZXC k k k

eff eff eff12.2( ) 5.0( ) 3.6( ) ] nN

ZZZY XXYZ YYXZk k k

→ Limits on 𝑘 eff ≈force sensitivity(noise)

Λ≈10 fN

10 nN/m2≈ 10−6m2

C, S eff( )i ii

k

Data – time series

• 21 hrs of data accumulated over 3 days in March 2012

• On-resonance (signal) data accumulated in 14 minute sets (off-resonance, diagnostic data in

between)

• T0 = 2000 vernal equinox

PRD 80 016002 (2009)

Atom interferometer data for small g:

• Include old data: 22 hrs accumulated over 5 days in August 2002

(Boulder, CO: colatitude = 0.872 [Bloomington: 0.887]; same orientation in lab)

2002 2012

• Signal estimate: discrete Fourier transforms:

2

( )cosm i ii

c F T m TN

2

( )sinm i ii

s F T m TN

Ti, F(Ti) = points from data plot

Measured signals

• Fourier amplitudes (1 stat errors):

Measured signals

• Continuous transforms

(fN, ± 1):

18

eff eff1

[( ) ] [( ) ]JKLM i JKLMi

P k p k

2 2eff theory measured ( , )[( ) ] exp (( , ) ( , ) ) / 2

i ii JKLM i i i i C Sp k C S C S

eff eff eff[( ) ] [( ) ] (all other )XXXX JKLMP k P k d k

• Probability for all k̅eff

• Data are Gaussian

• Estimate of individual k̅eff

Constraints on k̅eff

Systematic Errors (2012)

(m) (m)

18

eff eff1

[( ) ] [( ) ]JKLM i JKLMi

P k p k

2 2eff theory measured ( , )[( ) ] exp (( , ) ( , ) ) / 2

i ii JKLM i i i i C Sp k C S C S

eff eff eff[( ) ] [( ) ] (all other )XXXX JKLMP k P k d k

• Probability for all k̅eff

• Data are Gaussian

• Estimate of individual k̅eff

Independent coefficient values

(2, units 10-5 m2)

Constraints on k̅eff

eff effmax min

1[( ) ] [( ) ] ( )JKLM JKLMP k P k x d x

x x

D DD D

• Include systematic uncertainties (average over 1000 configs.)

e.g., overlap Dx:

eff eff14.0( ) 14.0( ) ] nNm

XXZZ YYZZk k

0 eff eff eff[4.4( ) 4.4( ) 8.7( )


eff eff eff[ 5.3( ) 1.7( ) 5.3( ) ]nNm

YYYZ ZZZY XXYZS k k k

eff eff eff[ 5.3( ) 1.7( ) 5.3( ) ]nNm

XXXZ ZZZX YYXZC k k k

• Fourier components (fNm, ± 1) [1]: • SME theoretical torque

Combined analysis: HUST short-range experiment

[1] C.-G. Shao, et al., PRL 117 071102 (2016)

→ Limits on 𝑘 eff ≈torque sensitivity

Λ≈10 aNm

10 nNm/m2≈ 10−9m2

Combined analysis: constraints on k̅eff


(2, units 10-9 m2)

C.-G. Shao, et al., PRL 117 071102 (2016)


(2, units 10-9 m2) [1]

[1] C.-G. Shao, et al., PRL 117 071102 (2016)

N(6)eff eff43

5Re 2 3

7 XXXZ XYYZk k k

Combined analysis: constraints on [2] (6)N

jmk

[2] V. A. Kostelecký and M. Mewes, PLB 766 137 (2017)

No overlap

with vacuum

coefficients (or

G-wave

constraints)

Combined analysis: constraints on fundamental (6) (6)

1 2,k k

Constraints (2, units 10-9 m2) on 59 independent coefficients k1, one at a time

C.-G. Shao, et al., PRL 117 071102 (2016)

Constraints (2, units 10-9 m2) on 72 independent coefficients k2, one at a time

Combined analysis: constraints on fundamental (6) (6)

1 2,k k

C.-G. Shao, et al., PRL 117 071102 (2016)

y (east)

x

(south)

z

q

f Ms

Mt

r

d=8 analysis (spherical)

(8)lab

5( ) ( , ) NN s t

jm jm

jm

G M MU Y k

r q f r

(8) lab ( ) (8)( )im TN j N

jm mm jmm

k e d k c

*( )( ) ( 1)m

jm j mk k

6

061

( ) sin( ) cos( )s tLV m m

m

GM MdF U c s m T c m T

r

r

1 13 4,1 14 4,1 15 6,1 16 6,1Re Im Re Imc k k k ka a a a

1 14 4,1 13 4,1 16 6,1 15 6,1Re Im Re Ims k k k ka a a a



5 21 6,5 22 6,5Re Imc k ka a

5 22 6,5 21 6,5Re Ims k ka a

0 1 4,0 2 6,0c k ka a





6 11 6,6 12 6,6Re Imc k ka a

6 12 6,6 11 6,6Re Ims k ka a

a1-a22 = f q,f,c

y (east)

x

(south)

z

q

f Ms

Mt

r f

q = π/2

q = 0

(zenith)

a2

DC (m=0, k60) 61,k 2 3 4 5 6a15

a16

a5

a6

a19

a20

a9

a10

a21

a22

a11

a12

Angular dependence of force ( in sun frame) (8)Njmk

( j = 6 terms)

colatitude = 0.89 (IUB)

q = 0.89

dMt

Point-plane, time dependence of z-force

y

x

q

f

Ms R

z

d

Force on Mt from

5 mm × 5 mm plate, d = 200 m

scaled by 1/R6

dMt

Point-plane, time dependence of z-force

y

x

q

f

Ms R

z

d

Force on Mt from

5 mm × 5 mm plate, d = 200 m

scaled by 1/R6

q = 0.89

d = 8 fitting function

1 2 6

( , , )ii t sG dV dV

r

a q f c

Abs(avg) ≈ .01 N/m4

→ Limits on 𝑘𝑗𝑚 ≈noise

Λ ≈10 fN

.01 N/m4≈ 10−12m4

F k

Abs(avg) ≈ 10-5 Nm/m4

→ Limits on 𝑘𝑗𝑚 ≈noise

Λ ≈10 aNm

10−5Nm/m4≈ 10−12m4

kt HUST:

Measured signals

• HUST: > 2000 hrs of data accumulated from December 2014 to August 2015

• Averaged over attractor rotational period DT = 3846 s

• Include data from 8 f0 (below) and 16 f0 [1] for two independent sets

HUST 8 f0 time series and Fourier transform IU, HUST Fourier components (± 2 stat)

[1] C.-G. Shao, et al., PRL 117 071102 (2016)

≈ 10fN ≈ 10aNm


(2, units 10-13 m4)

Combined analysis: constraints on (preliminary) (8)N

jmk


• smaller test mass gap r (sensitivity to F ~ 1/r n ≥ 4)

• smaller solid-angle W = A/r2 “acceptance”

(avoid angular averaging of F to 0)

Basic strategies conflict:

A

r

W

f

q = π/2

q = 0

(zenith)

4max max 4,46, 4, ~ ~ id j m F Y e f

Df

→ want Df ≈ /2

• Increase sensitivity with “more”

(periodic) masses

Outlook – optimization: HUST LV pendulum

(poster by Y. Chen, talk by C. Shao, F 2:30)

C.-G. Shao, et al., PRD 94 104061 (2016)

• Strips ≈ 1 x 2 x 20 mm

• Gap r ≈ 0.4 → 1mm

Df

Df ≈ 140→ 90

• LV torque relative to

torque at r = 0.4 mm

Gap r (mm)


• smaller test mass gap r (sensitivity to F ~ 1/r n ≥ 4)

• smaller solid-angle W = A/r2 “acceptance”

(avoid angular averaging of F to 0)

Basic strategies conflict:

A

r

W

f

q = π/2

q = 0

(zenith)

4max max 4,46, 4, ~ ~ id j m F Y e f

Df

→ want Df ≈ /2

• Increase sensitivity with “more”

(periodic) masses

• “antisymmetric” arrangement

Outlook – optimization: HUST LV pendulum

(poster by Y. Chen, talk by C. Shao, F 2:30)

C.-G. Shao, et al., PRD 94 104061 (2016)

• LV torque enhancement relative to

no-strip design

Summary

• Consistent with ~ fN, aNm sensitivity of the experiments and no evidence of

sidereal effects

• 131 constraints on fundamental coefficients:

• Short-range gravity experiments: First independent constraints on 14 d = 6

nonrelativistic coefficients: k̅eff < 10-9 m2

• Preliminary independent constraints on 22 d = 8 nonrelativistic coefficients:

(6) (6) 9 2

1 2, 10 mk k

N(8) 12 410 mjmk

• Experiments optimized for LV: improvement by 1-2 orders of magnitude (d = 6)

lorentz tests with short-range gravity

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