LORENTZ AND CPT VIOLATIONS FROM CHERN-SIMONS MODIFICATION OF QED. Alexander A. Andrianov. Instituto Nazionale di Fisica Nucleare Sezione di Bologna

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1 LORENTZ AND CPT VIOLATIONS FROM CHERN-SIMONS MODIFICATION OF QED Alexander A. Andrianov Instituto Nazionale di Fisica Nucleare Sezione di Bologna Paola Giacconi & R. S. Dipartimento di Fisica Università di Bologna JHEP 02 (2002) 030

2 0. INTRODUCTION 1. THE MAXWELL - CHERN - SIMONS ( MCS ) FREE RADIATION FIELD 2. FREE DIRAC SPINOR FIELDS IN THE UNIFORM AXIAL - VECTOR BACKGROUND 3. THE PHYSICAL ULTRAVIOLET CUT OFF AND THE RADIATIVELY INDUCED CHERN SIMONS VERTEX 4. CONCLUSIONS Brazil - August 2002

3 0. INTRODUCTION Lorentz symmetry breaking at the M Planck scale : ST RIN G & BRAN E T HEORIES N ON COMMUT AT IVE FIELD T HEORIES Induced low-energy very small deviation from the Lorentz covariant Standard Model : CPT violating modification of electrodynamics S.M. Carroll, G.B. Field & R. Jackiw (1990) Lorentz symmetry breaking from U(1) Anomaly A.A. Andrianov & R. Soldati (1995) CPT & Lorentz spontaneous breaking D. Colladay & V.A. Kostelecky (1998) explicit parity even Lorentz symmetry breaking S.R. Coleman & S.L. Glashow (1999)

4 Lorentz non-covariant Lagrange models : Power counting renormalizable SU(3) SU(2) U(1) gauge invariant Privileged Inertial Reference Frames ( PIRFs ) : Cosmic Microwave Background ( CMB ) isotropic tiny anisotropies in laboratory experiments translation invariance within PIRFs Lorentz symmetry breaking by constant 4-vectors Constant four-vectors η α & b µ very small within PIRFs owing to experimental bounds η α & b µ may have cosmological origin : T ORSION -like b µ = ɛ µνρσ T νρσ 0 QUIN T ESSEN CE-like η α = α Φ 0

5 Lorentz non-covariant CPT-even perturbations B 2 (1 + ɛ) B 2 velocity of light υ light = c 1 + ɛ c maximal attainable velocity of a material body υ body c υ light < υ body vacuum Čerenkov radiation primary cosmic p up to ev ɛ < Evading the GZK Cutoff? ev from 50 Mpc K. Greisen; G. Zatsepin & V. Kuz min (1966) Dispersion relation of the a -type particle E a = c 2 a p 2 + m 2 ac 4 a υ a c a Cosmic Microwave Photon scattering p + γ CMB (1232) p + γ CMB p + π NO GZK Cutoff c > c p > c π

6 1. MAXWELL CHERN SIMONS FREE PHOTONS MCS free Lagrange density in the Axial Gauge L MCS = 1 4 F νλ (x)f νλ (x) 1 2 η αa β (x) F αβ (x) B(x)η α A α (x) F αβ (x) (1/2)ɛ αβρσ F ρσ (x) B(x) is the auxiliary field η α Chern-Simons & Axial Gauge constant vector Going in the momentum representation A ν (x) d 4 k (2π) 3/2 Ãν (k) exp{ik x} Euler Lagrange equations of motion ( k 2 g νσ iɛ µρνσ η µ k ρ ) Ãσ = 0 η σ Ã σ (k) = k σ Ã σ (k) = 0

7 MCS polarizations: 2D projector to η ν and k µ e µν g µν η k D (ηµ k ν + η ν k µ ) + k2 D ηµ η ν + η2 D kµ k ν D (η k) 2 η 2 k 2 e µν η ν = e µν k ν = 0 e µ λ eλν = e µν e µ µ = 2 Linear polarization real vectors e µν = a=1,2 e (a) µ e (a) ν g µν e (a) µ e (b) ν = δ ab Chiral polarization complex vectors ε (L) µ 1 2 ε (R) µ 1 2 e µν = a=l,r ( ) e (1) µ + ie (2) µ ( ) e (1) µ ie (2) µ { ε (a) µ ε (a) ν + ε (a) ν } ε (a) µ

8 ONLY CHIRAL POLARIZATIONS ARE THE EIGENSTATES OF THE MCS KINETIC OPERATOR ( k 2 g σ ν ( k 2 g σ ν iɛ ν µρσ ) η µ k ρ ε (L) σ = iɛ ν µρσ ) η µ k ρ ε (R) σ = ( k 2 ) D ε (L) ν ( k 2 + ) D ε (R) ν LET US SEARCH FOUR REAL SOLUTIONS OF THE QUARTIC ON SHELL CONDITION (k 2 ) 2 D = (k 2 ) 2 (η k) 2 + η 2 k 2 = 0 MCS CHIRAL POLARIZATIONS DO NOT EXACTLY COINCIDE WITH MAXWELL S CIRCULAR POLARIZATIONS

9 pure space-like general solution η µ = (0, η ) k0± 2 = k η 2 ± η k 2 cos 2 ϕ η 2 ω±( 2 k, η ) LEFT ( + ) and RIGHT ( ) deformed hyperboloyds cos ϕ k η k η Ã µ (k) = a=l,r ε (a) µ (k)f a (k) F L (k) = f L ( k)δ[k 2 0 ω + 2 ( k, η)] F R (k) = f R ( k)δ[k 2 0 ω 2 ( k, η)] f L ( k) and f R ( k) are regular functions on the support of the δ distributions ( deformed hyperboloyds )

10 monochromatic plane wave solutions are possible only with a definite chiral polarization { A (L) µ, (t, x) = f Lε (L) k µ exp i k x itω + ( } k, η) { A (R) µ, (t, x) = f Rε (R) k µ exp i k x itω ( } k, η) + c.c. + c.c. WAVE PACKETS ARE SLOWLY SPLITTED INTO LEFT ( + ) AND RIGHT ( ) WAVE PACKETS MOVING WITH DIFFERENT GROUP VELOCITIES υ ± = k ω ± ( k, η) ± η k η ω ± ( k, η) ( 2 k η) 2 + η 4 υ ± < 1

11 * VACUUM BIREFRINGENCE * Group Velocity Difference between Left and Right Wave Packets υ υ R υ L Vanishes at HIGH frequencies lim υ = 0 k Grows at LOW frequencies lim υ = 1 k 0 TIME DELAY of LEFT HANDED WAVES WITH RESPECT TO RIGHT HANDED ONES IS CALLED VACUUM BIREFRINGENCE

12 * UPPER BOUND * EXPERIMENTAL ABSENCE OF THIS EFFECT IN RADIO-ASTRONOMY OBSERVATIONS OF DISTANT QUASARS AND RADIO GALAXIES η ev B. Nodland & J. P. Ralston (1997) J. F. C. Wardle, R. A. Perley & M. E. Cohen (1997)

13 pure time-like general solution η µ = (η 0, 0) ω ± ( k, η 0 ) = k ( k ± η 0 ) WAVE PACKETS ARE SUPERLUMINAL LEFT ( + ) AND RIGHT ( ) WAVE PACKETS MOVE WITH TACHIONIC GROUP VELOCITIES υ ± = k ω ± ( k, η 0 ) ( 1 ± η 0 2 k ) υ ± > 1 ω ( k, η 0 ) becomes IMAGINARY when k < η 0 UNPHYSICAL SOLUTIONS

14 2. CPT ODD FREE DIRAC SPINOR FIELDS CPT ODD free Lagrange density of Dirac spinors L spinor = ψ(x) (iγ µ µ m γ µ b µ γ 5 ) ψ(x) b µ constant axial-vector ψ(x)γ µ b µ γ 5 ψ(x) breaks Lorentz and CPT momentum space CPT ODD Dirac equation ψ(x) d 4 p (2π) 3/2 ψ(p) exp{ip x} (γ µ p µ m γ µ b µ γ 5 ) ψ(p) = 0

15 Diagonalization of Dirac s operator ψ(p) A B φ(p) A p 2 + b 2 m (b p + mb µ γ µ ) γ 5 B γ ν p ν + m + b ν γ ν γ 5 quartic scalar free field equation [ (p 2 + b 2 m 2) 2 + 4b 2 m 2 4(b p) 2] φ(p) = 0 on shell condition for CPT odd spinors ( p 2 + b 2 m 2) 2 + 4b 2 m 2 4(b p) 2 = 0

16 pure space-like general solution b µ = (0, b ) p 2 0± = p 2 + b 2 + m 2 ± 2 b p 2 cos 2 θ + m 2 cos θ p b p b pure time-like general solution b µ = (b 0, 0) p 2 0± = ( p ± b 0 ) 2 + m 2 (+) POSITIVE HELICITY particles NEGATIVE HELICITY anti-particles ( ) NEGATIVE HELICITY particles POSITIVE HELICITY anti-particles

17 (+) 1P STATES pure space-like case b µ = (0, b ) p 2 + p 2 0+ p 2 = b 2 + m b p 2 cos 2 θ + m 2 > 0 pure time-like case b µ = (b 0, 0) p 2 + p 2 0+ p 2 = b m b 0 p > 0 (+) 1P energy-momentum eigenstates PHYSICAL STATES p 2 + > 0 p R 3

18 ( ) 1P STATES pure space-like case b µ = (0, b ) p 2 p 2 0 p 2 > 0 p b < 1 2 b 2 m 2 pure time-like case b µ = (b 0, 0) p 2 p 2 0 p 2 > 0 p < b m 2 2 b 0 ( ) 1P states separated by THE PHYSICAL LIGHT-CONE BORDER p 0 = p = b 2 + m 2 2 b 0 sgn(p 0 ) b cos θ b µ

19 ( ) 1P energy-momentum eigenstates PHYSICAL STATES p 2 0 p b 2 + m 2 2 b 0 sgn(p 0 ) b cos θ ( ) 1P energy-momentum eigenstates UNPHYSICAL STATES p 2 < 0 p > b 2 + m 2 2 b 0 sgn(p 0 ) b cos θ

20 GROUP VELOCITIES OF SPINOR WAVE PACKETS υ ± < 1 b 2 > 0 p R 3 υ ± < 1 b µ = (0, b ) p R 3 CPT ODD FREE SPINOR QUANTIZATION CONSISTENT AND CAUSAL b 2 > 0 b µ = (0, b) ( ) UNPHYSICAL STATES at very high momenta

21 EXPERIMENTAL LIMITS spatial components b constrained by PENNING TRAPS EXP H. G. Dehmelt et al. (1999) b < ev HYDROGEN MASERS EXP D. F. Phillips et al. (2001) b < ev SPIN POLARIZED MATTER EXP B. R. Heckel et al. (2002) b < ev temporal component b 0 constrained by PRECISION DETERMINATION OF m e Particle Data Group (2000) b 0 < 10 2 ev LORENTZ and CPT Breaking owing to TINY b 0 NOT EXCLUDED AT PRESENT

22 3. RADIATIVELY INDUCED CS ACTION CPT ODD QED Lagrange density L = L MCS + L spinor + e ψ(x)γ µ ψ(x)a µ (x) e is the electron charge L MCS = 1 4 F νλ (x)f νλ (x) 1 2 η αa β (x) F αβ (x) B(x)η α A α (x) L spinor = ψ(x) (iγ µ µ m γ µ b µ γ 5 ) ψ(x) To be definite without loss of generality η µ = (0, η ) b µ = (b 0 > 0, 0)

23 PHYSICAL 1P ASYMPTOTIC STATES high momenta electron & positron decay e + e + + e + e + + e + + e e + e + + V. Kostelecky & R. Lehnert (2001)

24 momentum conservation p = k 1 + k 2 + k 3 kj = β j p + j j = 1, 2, 3 j p p j = 0 β 1 + β 2 + β 3 = = 0 helicity conservation p m

25 energy conservation p 0± = k (1) 0± + k(2) 0 + k(3) 0 ( p ± b 0 ) 2 + m 2 = ( k 1 ± b 0 ) 2 + m2 + ( k 2 b 0 ) 2 + m2 + ( k 3 b 0 ) 2 + m 2 decay processes take place p 2m2 b 0 Λ s

26 . STABLE 1P ASYMPTOTIC STATES p 0± = ( p ± b 0 ) 2 + m 2 p Λ s PHYSICAL UV CUTOFF Λ s PHYSICAL IN/OUT 1P STATES p 2 ± 0 p Λ s

27 1Loop induced MCS polarization tensor Π µν (k) = d 4 p i(2π) 4 tr {γµ S(p)γ ν S(p k)} Feynman s propagator of the spinor field S(p) = i A B (p 2 + b 2 m 2 + iε) 2 4 [(b p) 2 m 2 b 2 ] A p 2 + b 2 m (b p + mb µ γ µ ) γ 5 B γ ν p ν + m + b ν γ ν γ 5 UV divergencies REGULARIZATION regπ µν (k) = regπ µν even(k) + regπ µν odd (k)

28 physical UV cutoff regπ µν (k) = d p i(2π) 3 ϑ ( p 2 2 Λ ) + s lim Λ s dp 0 2π tr {γµ S(p)γ ν S(p k)} dimensional regularization regπ µν (k) = lim ω 2 d 2ω p i(2π) 2ω tr {γµ S(p)γ ν S(p k)} BOTH REGULARIZATIONS GUARANTEE MINIMAL LORENTZ SYMMETRY BREAKING OWING ONLY TO b µ = (b 0, 0)

29 1Loop induced parity odd polarization tensor regπ µν odd (k; b, m) = 4ɛµνρσ b ρ k σ regπ odd (k; b, m) INDUCED PARITY ODD EFFECTIVE ACTION BOTH REGULATORS GIVE THE SAME RESULT regπ µν odd (k; b, m) 4ɛµνρσ b ρ k σ Π odd (k = 0; m 2 /b 2 ) { = i } 2π 2 ɛµνρσ b ρ k σ 1 ϑ( b 2 m 2 ) 1 + m2 b 2 INDUCED CHERN-SIMONS LAGRANGE DENSITY L CS η α e2 2π 2 b α = 1 2 ( η αβ α)a β F { 1 ϑ( b 2 m 2 ) 1 + m2 b 2 }

30 comparison with previous different results R. Jackiw & V.A. Kostelecky (1999) M. Peréz- Victoria (1999) J.M. Chung & P. Oh (1999) M. Chaichian, W.F. Chen & R. Gonzalez Felipe (2001) J.M. Chung & B.K. Chung (2001) O.A. Battistel & G. Dallabona (2001) all those analyses contain ILLEGAL MATHEMATICAL MANIPULATIONS the vanishing result η µ = 0 S.R. Coleman & S.L. Glashow (1999) G. Bonneau (2001) NON MINIMAL Lorentz symmetry breaking regulators are tacitly assumed

31 4. CONCLUSIONS CONSISTENT QUANTIZATION η µ = (0, η ) b 2 > 0 b µ = (0, b ) N-flavours induced Chern-Simons vector η µ = 2α π N a=1 b µ a EXPERIMENTAL BOUNDS b < ev b 0 < 10 2 ev η = η (0) + η η < ev η < ev fine tuning cancellation between η (0) and η b 0 0 not yet excluded empirically