Matter-gravity interactions in the SME

Transcription

1 Matter-gravity interactions in the SME Jay D. Tasson Carleton College

2 outline gravitationally coupled matter motivation path to experimental analysis establishing the fluctuations experiments constraining torsion with Lorentz violation Kostelecký, JT, PRD 11; PRL 09 Kostelecký, Russell, JT, PRL 08

3 sensitivities with ordinary matter Data Tables for Lorentz and CPT violation Kostelecký, Russell Rev. Mod. Phys. 11

4 sensitivities with ordinary matter ~1/2 of leading ordinary matter & light possibilities explored other sectors nearly unexplored use gravitational experiments to get more! Data Tables for Lorentz and CPT violation Kostelecký, Russell Rev. Mod. Phys. 11

5 gravitationally coupled fermions 1 L fermion = 1 2 ieμ a ψ(γ a c νλ e νa e λ b γb e ν e νa...) D μ ψ ψ(m+a μ e μ a γa +...)ψ coefficients fields for Lorentz violation particle-species dependent spacetime dependent additional coefficients for LV Idea : new gravitational couplings provide new LV sensitivity access a μ coefficient challenging to observe in flat spacetime 1) Kostelecký PRD 04

6 gravity (no Lorentz violation) vierbein general Lorentz a,b,c, links Minkowski tangent space to each spacetime point incorporates spinors weak-field limit metric fluctuation

7 perturbative proceed exactly long expressions lots of hard (impossible?) work not needed proceed perturbatively easier what should we keep? linearize gravity expand coefficients about the vev (watch notation!)

8 order counting gravity x LV vev is really our interest c μν h αβ O(1, 1) number of LV vevs number of metric fluctuations what s in the fluctuations? what order is c μν a λ h αβ 7 aδ

9 order counting gravity X LV vev is really our interest c μν h αβ O(1, 1) number of LV vevs number of metric fluctuations what s in the fluctuations? what order is c μν a λ h αβ 7 aδ O(3, 2)

10 order counting later we ll also use post-newtonian counting for order in velocity PNO(n) what is the smallest possible post-newtonian order of c μν a λ h αβ 7 aδ

11 path to experimental analysis effective quantum field theory modified Newton s laws

12 path to experimental analysis L fermion H Relativistic expand to desired order in LV and gravity Euler-Lagrange eq. relativistic quantum experiments

13 path to experimental analysis L fermion H Relativistic expand to desired order in LV and gravity Euler-Lagrange eq. relativistic quantum experiments

14 Dirac equation L fermion Euler Lagrange Dirac Eq. found: Dirac Eq. needed: LV/gravity perturbations i(γ 0 + C 0 ) 0 ψ =[ iγ j j +...]ψ iγ 0 0 ψ =[ iγ j j +...]ψ Problem: non-standard time derivatives caused by gravity Lorentz violation solutions gravity -- define curved scalar product 1 LV in flat space -- redefine fermion field 2 questions Are the solutions equivalent? Which should be used for gravity and LV? 1) Parker PRD 80 2) Bluhm, Kostelecký, Russell PRD 98

15 curved-product / Parker method Dirac Eq. remove problematic term via left multiplication by identify hamiltonian continuity Eq. probability density curved product i(γ 0 + C 0 ) 0 ψ =[ iγ j j +...]ψ γ 0 (1 C 0 γ 0 ) 0 [ψ (1 + γ 0 C 0 )ψ]+ j [ψ γ 0 (γ j + C j )ψ] =0 hψ 1, ψ 2 i P = curvature and/or LV in the time direction alter the probability density Z H ψ d 3 x ψ 1 (1 + γ0 C 0 )ψ 2

16 field-redefinition method L fermion perform field redefinition ψ =(1 1 2 γ0 C 0 )χ L fermion Dirac Eq. i 0 χ =[ iγ 0 (γ 0 γ j 1 2 C0 γ 0 γ j γ0 γ j C 0 ) j +...]χ hamiltonian H χ 6= H ψ Z hψ 1, ψ 2 i f = d 3 x ψ 1 ψ 2 scalar product remains flat interpretation: the perturbation is an effective inverse vierbein γ μ + C μ = E μ aγ a transform to different fermion on a space with E 0 a = δ 0 a

17 comparison physical equivalence hamiltonians typically have different forms equivalent for time-independent perturbation theory E (1) = hψ (0),H (1) ψ ψ(0) i f = hψ (0),H χ (1) ψ (0) i f curved-product advantages - simpler hamiltonian - easier to obtain field-redefinition advantages - easy match with classical theory - ease of working consistently at a given order - vierbein interpretation

18 path to experimental analysis L fermion L fermion H Relativistic expand to desired order in LV and gravity field redefinition Euler-Lagrange eq. relativistic quantum experiments

19 H rel = relativistic hamiltonian H (0,0) i(h jk + h 00 η jk )γ0 γ k j + ih j0 j 1 2 mh 00 γ T αβγ ² αβγδ γ 5 γ 0 γ δ j h 0k ² jkl γ 5 γ 0 γ l i j h j i[ j h 00 + k h jk ]γ 0 γ j + a 0 me 0 +2ic (j0) j [mc 00 ie j j ]γ 0 f j j γ 0 γ 5 +[a j + i(c 00 η jk + c jk ) k ]γ 0 γ j +[ b 0 2id (j0) j ]γ 5 +[ih 0j +2g j(k0) k ]γ j [b j + i(d jk k + d 00 j ) 1 2 mgkl0 ² jkl ]γ 5 γ 0 γ j ( 1 2 Hkl ² jkl + md j0 )γ 5 γ j i² jlm (g l00 η km glmk ) k γ 5 γ j + ih (1,1) j j0 1 2 mh(1,1) γ i(h(1,1) + h (1,1) jk 00 η jk )γ 0 γ k j + ea 7 0 a j h h j0 + 7eaj 1 2 a j h 00 1 i 2 ak h jk γ 0 γ j h + e 7 b0 + b j i h j0 γ 5 h 7ebj b j h i bk h jk γ 0 γ 5 γ k + ih 7ec00 η jk + ec 7 kj c 00 ³h 00 η jk h jk +2c (l0) h l0 η jk c k0 h j0 4 1 c j0 h k0 2 1 ³ c lj h 00 η l k + hl k c kl h l i j γ 0 γ k j h 2i 7ec (j0) + c(k0) h jk + c jk i h h k0 j m 7ec c 00 h 00 +2c (j0) hj0i γ 0 h 7ed 2i (j0) + d(k0) h jk + d (jk) i 7ed00 h k0 γ 5 j + ih d 00 h 00 +2d (k0) hk0i γ 0 γ 5 γ j j 7edkj +ih d k0 h j0 1 4 d j0 h k0 1 2 d 00 h jk 1 ³ 2 d lj h 00 η l k + hl k d kl h l i j γ 0 γ 5 γ k j 7ed j0 +mh ηjk 1 2 dj0 h jk d kj h j0i γ 5 γ k 4 1 imdj0 h k0 ² jkl γ l h i 7ej e 0 h j0 2 1 i e j h 00 ek h jk γ 0 j me me j h j me 0 h j0 γ0 γ j h 7efj f 0 h j0 1 i 2 f j h 00 fk h jk γ 0 γ 5 j h + 2eg 7 k(j0) 4 1 g l00 hl0 η jk g l(j0) h l k 2g k(l0) hl j +2g k(jl) hl0i γ k j h + i 7eg l00 jk η 1 eg 7 klj 1 g k00 ³ h η jl jl h 1 2 g n00 hln η jk 1 4 gjk0 h l0 +2g l(n0) h n0 η jk 1 8 gkl0 h j gklj h mig jk0 hk0 γ 0 γ j mgjk0 h l0 ² jkl γ gkln h j n 1 2 gknj h l i n ² klm γ 5 γ m j 1 h i 2 m 7eg jk0 km0 j + g h m + gjkm h m0 ² jkl γ 0 γ 5 γ l 1 h 7eH jk jm k H h 2 n 1 i 2 Hjk h 00 ² jkl γ 5 γ l h 7eHj0 i Hk0 h jk + H jk h k0i γ j ih 18 h 00 h 00 η jk h l0 hl0 η jk 1 4 h 00 h jk 8 3 i h jl hl k γ 0 γ k j ih k0 h jk h j m 18 h 00 h h j0 hj0i γ 0

20 path to experimental analysis L fermion L fermion H Relativistic H NonRel expand to desired order in LV and gravity field redefinition Euler-Lagrange eq. relativistic quantum experiments Foldy-Wouthuysen expn (for us, drop spin including torsion) inspection non-relativistic quantum experiments L Classical ẍ μ h μν variation But it still has fluctuations ec μν in it. What should we do about that? 7

21 Lorentz-symmetry breaking explicit Lorentz violation is a predetermined property of the spacetime inconsistent with Riemannian geometry Bianchi identities require D μ T μν =0 spontaneous LV arises dynamically consistent with geometry possible in numerous underlying theories: string theory, quantum gravity expand about vev t λμν... = t λμν... + et 7 λμν... vev corresponds to Minkowski matter coefficients fluctuations about the vacuum

22 matter sector constant coefficients satisfy the Bianchi identities to 3 rd post-newtonian order models which produce constant coefficients are minimally coupled stop here for since sensitivities can be achieved c μν nonminimal coupling generic analysis yields fluctuations at lower post- Newtonian order consider for a μ since constant part provides no signals upon investigating spontaneous breaking we find Minkowski-spacetime coefficients a μ = a μ αaν h μν 1 4 αa μh ν ν metric fluctuation g μν = η μν + h μν characterize nonminimal couplings in dynamical theories

23 countershaded Lorentz violation a μ = a μ αaν h μν 1 4 αa μh ν ν unobservable shift in fermion phase observable effects via gravity coupling a μ for matter is unobservable in flat-spacetime tests observable effects are suppressed by the gravitational field a μ a μ could be large (~ 1eV) relative to existing matter-sector bounds c.f. GeV b μ < 10 30

24 classical results h 00 = 2Gm r 1+(c S ) m (as eff ) ẍ j = 1 2 j h 00 +(c T ) j k k h m T α(a T eff ) 0 j h S and T denote composite coefficients for source and test respectively modified metric & particle equation of motion experimental hooks particle-species dependence time dependence

25 species dependence species dependence c T μν= P w=p,n,e N w m w m T c w μν (a eff ) T μ= P w=p,n,e N w (a eff ) w μ combining data from multiple experiments independent p, n sensitivity = experiments with charged matter independent e sensitivity = mass fraction of species w in test body

26 sample Newtonian results

27 time dependence standard frame for reporting SME bounds V =10 4 boost and rotation of test annual & sidereal variations ~x 2g αa T ẑ 2gV αa X sin(ωt )ẑ 2 5 gv L αa X sin(ωt + ψ)ŷ

28 experiments time and species dependent equations of motion imply signals in: lab tests gravimeter Weak Equivalence Principle (WEP) space-based WEP exotic tests charged matter antimatter higher-generation matter solar-system tests laser ranging perihelion precession light-travel/clock tests time delay Doppler shift red shift... Kostelecký, Tasson PRD 11

29 lab tests acceleration of a test particle T ~x 2 1 m gv α(a T eff ) Xsin(ΩT )ẑ+ gv (c T ) TX sin 2χ sin (ΩT )ˆx annual variations monitor acceleration of of one particle over time gravimeter monitor relative behavior of of particles -- WEP (Weak (Weak Equivalence Principle) test frequency and phase distinguish from other effects

30 media coverage

31 lab tests acceleration of a test particle T ~x 2 5m gv Lα(a T eff ) Xsin(ωT + ψ)ŷ V L 10 2 V sidereal variations

32 gravimeters ³ ~F = m T g 2gα a T T + mt a S m S T measure local gravitational acceleration example stretched spring ³ 2gα ~ a T ~v + mt~ a S ~v m S F = kx F k search for changes in stretch with time

33 gravimeters ³ ~F = m T g 2gα a T T + mt a S m S T ³ 2gα ~ a T ~v + mt~ a S ~v m S measure local gravitational acceleration more sensitive superconducting levitation acceleration sensitivity at parts in notice change in g over fraction of centimeter

34 free-fall WEP (Weak Equivalence Principle) tests ³ ~F = m T g 2gα a T T + mt a S m S T ³ 2gα ~ a T ~v + mt~ a S ~v m S compare gravitational acceleration for different bodies x(t) drop objects from rest and search for change in relative position

35 free-fall WEP (Weak Equivalence Principle) tests ³ ~F = m T g 2gα a T T + mt a S m S T ³ 2gα ~ a T ~v + mt~ a S ~v m S compare gravitational acceleration for different bodies Bremen drop tower 120m free fall

36 force-comparison WEP tests torque on a composition dipole (or higher multipole) sample: dipole torsion pendulum equation of motion I θ +2Iγ θ + hκθ = i ω 2 1 R r 0 m cos η 4 cos2 2χ(c A (TZ) cb (TZ) )cos(ω pt + ωt + ΩT )... periodic torque frequencies are linear combinations of: turntable freq. sidereal freq. annual freq.

37 force-comparison WEP tests torque on a composition dipole (or higher multipole)

38 force comparison WEP tests compare gravitational force for different bodies U. Washington torsion pendulum differential gravitational force sensitivity of parts in

39 experimental sensitivities α(a eff ) w T, α(a eff) w J, α(a eff) w T, α(a eff) w J, m w c w TT m n c n TJ m w c w TT m n c n TJ Experiment actual feasible future future torsion pendulum falling corner cube atom interferometry superconducting gravimeter drop tower balloon drop bouncing masses w = p, n, e; J = X, Y, Z; sensitivities in GeV sensitivities are to various combinations of above coefficients iactual = achieved in current analysis, feasible = attainable with existing data / operating experiments u future = attainable in planned experiments

40 experimental sensitivities α(a eff ) w T, α(a eff) w J, α(a eff) w T, α(a eff) w J, m w c w TT m n c n TJ m w c w TT m n c n TJ Experiment actual feasible future future torsion pendulum falling corner cube atom interferometry multiple experiments 10 5 needed 10 5 for for maximum number of of sensitivities superconducting gravimeter drop tower other coefficients - yield yield additional - effects balloon drop bouncing masses w = p, n, e; J = X, Y, Z; sensitivities in GeV sensitivities are to various combinations of above coefficients iactual = achieved in current analysis, feasible = attainable with existing data / operating experiments u future = attainable in planned experiments

41 space-based E.P. tests long free-fall times improved sensitivity = differential acceleration sensitivity test a/a MicroSCOPE Galileo Galilei STEP crude sensitivity estimates based on a/a =10 18 c n JJ no sum c n (TJ) α(a eff ) w J GeV α(a eff ) w T,mw c w TT GeV w = p, n, e; J = X, Y, Z

42 exotic tests variations of above tests involving experimentally challenging matter charged matter separate proton and electron coefficients theoretically interesting -- bumblebee electrodynamics higher-generation matter few existing bounds antimatter separate CPT even and odd coefficients CPT odd L = 1 2 (m N w m w c w TT )v2 gz(m + N w m w c w TT +2αN w (a eff ) w T ) m i,eff m g,eff differing gravitational response for matter and antimatter

43 a toy model for antimatter gravity L = 1 2 (m N w m w c w TT )v2 gz(m + N w m w c w TT +2αN w (a eff ) w T ) m i,eff m g,eff Isotropic Parachute Model (IPM) Kostelecký, Tasson PRD mw c w TT = α(a eff) w T Matter m i,eff = m g,eff a=g m i,eff 6= m g,eff ā=g(1 4mw N w 3m cw TT )

44 interpretation of c μν coefficients found that fluctuations in c μν are not needed to PNO(3) did the coordinate shift found same Einstein equation (and hence metric) as the gravity sector after yesterdays construction!

45

46 Questions? (before I talk about torsion)

47 effective Lorentz violation E Looks like Lorentz violation

48 effective Lorentz violation E Looks like Lorentz violation

49 effective Lorentz violation E I m not so sure Looks like Lorentz violation

50 Irreducible decomposition: T αμν = 1 3 (g αμt ν g αν T μ ) ² μναβ A β + M αμν T μ g αβ T αβμ, A μ 1 6 ²αβγμ T αβγ, M αμν 1 3 (T αμν + T μαν + T μ g αν ) 1 (μ ν). 3 There are independent components Minimal torsion coupling D μ ψ μ ψ ab iωμ σ ab ψ ω ab μ =[termsinvolving μ e b α ]+ 1 2 (T λμν T μνλ T νμλ )e λa e νb Flat-space limit: L 3 4 A μψγ 5 γ μ ψ, L SME b μ ψγ 5 γ μ ψ, Related ideas on axial component of constant torsion: Lämmerzahl, PLA vol. 228, p. 223 (1997) (from Hughes-Drever); Shapiro, Phys. Rep. vol 357, p. 113 (2002) (from b μ bounds)

51 Nonminimal torsion couplings up to mass dimension 5 L T 1 2 i ψγ μ μ ψ m ψψ +ξ (4) 1 T μ ψγ μ ψ + ξ (4) 2 T μ ψγ 5 γ μ ψ + ξ (4) 3 A μ ψγ μ ψ + ξ (4) 4 A μ ψγ 5 γ μ ψ iξ(5) 1 T μ ψ μ ψ ξ(5) 2 T μ ψγ5 μ ψ iξ(5) 3 Aμ ψ μ ψ ξ(5) 4 Aμ ψγ5 μ ψ iξ(5) 5 M λ μν ψ λ σ μν ψ iξ(5) 6 T μ ψ ν σ μν ψ iξ(5) 7 A μ ψ ν σ μν ψ iξ(5) 8 ²λκμν T λ ψ κ σ μν ψ iξ(5) 9 ²λκμν A λ ψ κ σ μν ψ. We see: -- 4 terms of mass dimension independent terms of mass dimension 5 -- the mixed component of torsion in one term -- coupling constants ξ for each term

52 Minimal SME in Minkowski space L SME = 1 2 i ψγ μ μ ψ m ψψ a μ ψγ μ ψ b μ ψγ5 γ μ ψ 1 2 H μν ψσ μν ψ 1 2 ic μν ψγ μ ν ψ 1 2 id μν ψγ 5 γ μ ν ψ 1 2 ie μ ψ μ ψ f μ ψγ 5 μ ψ 1 4 ig μνλ ψσ μν λ ψ Colladay and Kostelecký, PRD vol. 58, (1998) Match between fixed torsion and minimal SME b μ mg μ (A) = (ξ (4) 2 2mξ (5) 8 )T μ (ξ (4) 4 2mξ (5) 9 )A μ, g μνα (M) = 2ξ (5) 5 M αμν.

53 To identify constraints on torsion, need to identify frame in which T is constant Cases i) galactic or cosmologically sourced -- T constant on scale of solar system -- modulations from seasonal revolution of Earth -- modulations from rotation of Earth and apparatus ii) Sun sourced -- no effect from revolution of Earth about Sun -- modulations from rotation of Earth and apparatus iii) Earth sourced -- no effect from rotation of Earth -- modulations from rotation of apparatus

54 University of Washington torsion-pendulum experiment New CP-Violation and Preferred-Frame Tests with Polarized Electrons, Physical Review Letters 97, (2006) Eöt-Wash Group

55 Sensitivities achieved First bounds on 11 mixed-symmetry components First bounds on all 4 trace components Best bounds on all 4 axial-vector components (minimal-coupling case) A T < GeV, A X < GeV, A Y < GeV, A Z < GeV. Quantity Sensitivity Quantity Sensitivity Source ξ (4) 2 T T GeV ξ (5) 8 T T ξ (4) 2 T X GeV ξ (5) 8 T X S ξ (4) 2 T Y GeV ξ (5) 8 T Y S ξ (4) 2 T Z GeV ξ (5) 8 T Z S,E ξ (4) 4 A T GeV ξ (5) 9 A T ξ (4) 4 A X GeV ξ (5) 9 A X S ξ (4) 4 A Y GeV ξ (5) 9 A Y S ξ (4) 4 A Z GeV ξ (5) 9 A Z S,E ξ (5) 5 M TTX ξ (5) 5 M TTY ξ (5) 5 M TTZ ξ (5) 5 M XXZ ξ (5) 5 M YYX ξ (5) 5 M ZZY ξ (5) 5 M TXY S,E ξ (5) 5 M TYZ S ξ (5) 5 M TZX S ξ (5) 5 M XY Z ξ (5) 5 M YZX Kostelecký, Russell, Tasson, Phys. Rev. Lett. 100, (2008)

56 Sensitivities achieved First bounds on 11 mixed-symmetry components First bounds on all 4 trace components Best bounds on all 4 axial-vector components (minimal-coupling case) A T < GeV, A X < GeV, A Y < GeV, A Z < GeV. Quantity Sensitivity Quantity Sensitivity Source ξ (4) 2 T T GeV ξ (5) 8 T T ξ (4) 2 T X GeV ξ (5) 8 T X S ξ (4) 2 T Y GeV ξ (5) 8 T Y S ξ (4) 2 T Z GeV ξ (5) 8 T Z S,E ξ (4) 4 A T GeV ξ (5) 9 A T ξ (4) 4 A X GeV ξ (5) 9 A X S ξ (4) 4 A Y GeV ξ (5) 9 A Y S ξ (4) 4 A Z GeV ξ (5) 9 A Z S,E ξ (5) 5 M TTX ξ (5) 5 M TTY ξ (5) 5 M TTZ ξ (5) 5 M XXZ ξ (5) 5 M YYX ξ (5) 5 M ZZY ξ (5) 5 M TXY S,E ξ (5) 5 M TYZ S ξ (5) 5 M TZX S ξ (5) 5 M XY Z ξ (5) 5 M YZX Further improvements since our paper! Kostelecký, Russell, Tasson, Phys. Rev. Lett. 100, (2008)