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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
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)
Calculation of the Fitting Function
[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)
Calculation of the Fitting Function
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( )
XXXX YYYY XXYYC k k k
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( )
XXXX YYYY XXYYC k k k
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
Independent coefficient values
(2, units 10-9 m2)
C.-G. Shao, et al., PRL 117 071102 (2016)
Independent coefficient values
(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
3 17 4,3 18 4,3 19 6,3 20 6,3Re Im Re Imc k k k ka a a a
3 18 4,3 17 4,3 20 6,3 19 6,3Re 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
2 3 4,2 4 4,2 5 6,2 6 6,2Re Im Re Imc k k k ka a a a
2 4 4,2 3 4,2 6 6,2 5 6,2Re Im Re Ims k k k ka a a a
4 7 4,4 8 4,4 9 6,4 10 6,4Re Im Re Imc k k k ka a a a
4 8 4,4 7 4,4 10 6,4 9 6,4Re Im Re Ims k k k ka a a 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
Independent coefficient values
(2, units 10-13 m4)
Combined analysis: constraints on (preliminary) (8)N
jmk
Outlook - optimization
• 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)
Outlook - optimization
• 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)