CPT and DECOHERENCE in QUANTUM GRAVITY N E

CPT and DECOHERENCE in QUANTUM GRAVITY N. E. Mavromatos King’s College London Physics Department MRTN-CT 2006 -035863 DISCRETE 08 SYMPOSIUM ON PROSPECTS IN THE PHYSICS OF DISCRETE SYMMETRIES IFIC – Valencia (Spain), December 11 -16 2008 23/05/2007 KAON '07 N. MAVROMATOS

OUTLINE v (i) Theoretical motivation for CPT Violation (CPTV) : Lorentz violation (LV): microscopic & cosmological v v Briefly (ii) Quantum Gravity Foam (QGF) (Decoherence) This talk v Te. V photon astrophysics LV tests Precision tests of QGF-CPTV of smokinggun-evidence type: neutral mesons factories – entangled states: EPR correlations modified (ω-effect) Disentangling (i) from (ii) ω-effect as discriminant of space-time foam models This talk Towards microscopic models from (non -critical) strings & Order of magnitude estimates of expected effects v Neutrino Tests of QG decoherence Damping factors in flavour Oscillation Probabilities – suppressed though by neutrino mass differences This talk DISCRETE 08, IFIC (Valencia) , December 08 This talk N. E. MAVROMATOS 2

OUTLINE v (i) Theoretical motivation for CPT Violation (CPTV) : Lorentz violation (LV): microscopic & cosmological Briefly v v Te. V photon astrophysics LV tests Precision tests of QGF-CPTV of smokinggun-evidence type: neutral mesons factories – entangled states: EPR correlations modified (ω-effect) (ii) Quantum Gravity Foam (QGF) Disentangling (i) from (ii) IMPORTANT : QUANTUM GRAVITY (Decoherence ) ω-effect as discriminant of space-time DECOHERENCE CURRENT BOUNDS & This talk foam models MICROSCOPIC BLACK HOLES AT LHC v This talk v Neutrino Tests of QG decoherence Towards microscopic models from (non Damping factors in flavour Oscillation -critical) strings & Details of microscopic model matter– asuppressed lot though by Probabilities Order of magnitude estimates of expected neutrino mass differences effects before concusions are reached in excluding This talk large extra dimensional models by such decoherence studies… DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS This talk 3

Generic Theory Issues CPT SYMMETRY: § (1) Lorentz Invariance, (2) Locality , (3) Unitarity l Theorem proven for FLAT space times (Jost, Luders, Pauli, Bell, Greenberg ) v Why CPT Violation? § Quantum Gravity (QG) Models violating Lorentz and/or Quantum Coherence: (I) Space-time foam: QG as “Environment” v Decoherence, CPT Ill defined (Wald 1979) (II) Standard Model Extension: Lorentz Violation in Hamiltonian H: CPT well defined but non-commuting with H (III) Loop QG/space-time background independent; Non-linearly Deformed Special Relativities : Quantum version not fully understood… DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 4

CPT THEOREM DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 5

CPT THEOREM DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 6

CPT THEOREM DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 7

CPT THEOREM DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 8

CPT THEOREM J. A. Wheeler 10 -35 m Space-time Foam DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 9

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 10

Arguments in favour of holographic picture : Path Integral over non-trivial BH topologies decays with time, general, in leaving only trivial (unitary) topology contributions. In(Maldacena, space-times Hawking) with Horizons Arguments against resolution of issue: issue (e. g. De Sitter cosmology…) (i) not rigorous proof though over space-time measure. (ii) Entanglement entropy (Srednicki, Einhorn, Brustein, Yarom ). (iii) Also, Space-time foam may be of different type, e. g. due to stochastic space-time point-like defects crossing brane worlds (Dparticle foam) …. (Ellis, NM, Nanopoulos, Sarkar). Hence possible non-trivial decoherence effects in effective theories. Worth checking experimentally…. CPTV issues DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 11

Arguments in favour of holographic picture : Path Integral over non-trivial BH topologies decays with time, general, in leaving only trivial (unitary) topology contributions. In(Maldacena, space-times Hawking) with Horizons Arguments against resolution of issue: issue (e. g. De Sitter cosmology…) (i) not rigorous proof though over space-time measure. (ii) Entanglement entropy (Srednicki, Einhorn, Brustein, Yarom ). (iii) Also, Space-time foam may be of different type, e. g. due to stochastic space-time point-like defects crossing brane worlds (Dparticle foam) …. (Ellis, NM, Nanopoulos, Sarkar). Hence possible non-trivial decoherence effects in effective theories. Worth checking experimentally…. CPTV issues DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 12

Arguments in favour of holographic picture : Path Integral over non-trivial BH topologies decays with time, general, in leaving only trivial (unitary) topology contributions. In(Maldacena, space-times Hawking) with Horizons Arguments against resolution of issue: issue (e. g. De Sitter cosmology…) (i) not rigorous proof though over space-time measure. (ii) Entanglement entropy (Srednicki, Einhorn, Brustein, Yarom ). (iii) Also, Space-time foam may be of different type, e. g. due to stochastic space-time point-like defects crossing brane worlds (Dparticle foam) …. (Ellis, NM, Nanopoulos, Sarkar). Hence possible non-trivial decoherence effects in effective theories. Worth checking experimentally…. CPTV issues DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 13

Arguments in favour of holographic picture : Path Integral over non-trivial BH topologies decays with time, general, in leaving only trivial (unitary) topology contributions. In(Maldacena, space-times Hawking) with Horizons Arguments against resolution of issue: issue (e. g. De Sitter cosmology…) (i) not rigorous proof though over space-time measure. (ii) Entanglement entropy (Srednicki, Einhorn, Brustein, Yarom ). (iii) Also, Space-time foam may be of different type, e. g. due to stochastic space-time point-like defects crossing brane worlds (Dparticle foam) …. (Ellis, NM, Nanopoulos, Sarkar). Hence possible non-trivial decoherence effects in effective theories. Worth checking experimentally…. CPTV issues DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 14

In general, in space-times with Horizons (e. g. De Sitter cosmology…) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 15

COSMOLOGICAL MOTIVATION FOR CPT VIOLATION? Supernova and CMB Data (2006) Baryon oscillations, Large Galactic Surveys & other data (2008) Evidence for : Dark Matter(23%) Dark Energy (73%) Ordinary matter (4%) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 16

DARK ENERGY& Cosmological CPTV? KNOW VERY LITTLE ABOUT IT… v EMBARASSING SITUATION 74% OF THE UNIVERSE BUDGET CONSISTS OF UNKNOWN SUBSTANCE v v v Certainly of Quantum Gravitational origin v If cosmological constant (de Sitter), then quantization of field theories not fully understood due to cosmic horizon CPT invariance? Could be: § a Cosmological Constant § Quintessence (scalar field relaxing to minimum of its potential) § Something else…Extra dimensions, colliding brane worlds etc. DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 17

DARK ENERGY& Cosmological CPTV? KNOW VERY LITTLE ABOUT IT… v EMBARASSING SITUATION 74% OF THE UNIVERSE BUDGET CONSISTS OF UNKNOWN SUBSTANCE v v v Certainly of Quantum Gravitational origin v If cosmological constant (de Sitter), then quantization of field theories not fully understood due to cosmic horizon CPT invariance? Could be: § a Cosmological Constant § Quintessence (scalar field relaxing to minimum of its potential) § Something else…Extra dimensions, colliding brane worlds etc. DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 18

DARK ENERGY& Cosmological CPTV? KNOW VERY LITTLE ABOUT IT… v EMBARASSING SITUATION 74% OF THE UNIVERSE BUDGET CONSISTS OF UNKNOWN SUBSTANCE v v v Certainly of Quantum Gravitational origin v If cosmological constant (de Sitter), then quantization of field theories not fully understood due to cosmic horizon CPT invariance? Could be: § a Cosmological Constant § Quintessence (scalar field relaxing to minimum of its potential) § Something else…Extra Outer Horizon (live ``inside’’ black hole): dimensions, colliding brane Unstable, indicates expansion worlds etc. DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 19

DARK ENERGY& Cosmological CPTV? KNOW VERY LITTLE ABOUT IT… v EMBARASSING SITUATION 74% OF THE UNIVERSE BUDGET CONSISTS OF UNKNOWN SUBSTANCE v v Could be: § a Cosmological Constant § Quintessence (scalar field relaxing to minimum of its potential) § Something else…Extra dimensions, colliding brane worlds etc. DISCRETE 08, IFIC (Valencia) , December 08 v Certainly of Quantum Gravitational origin v If cosmological constant (de Sitter), then quantization of field theories not fully understood due to cosmic horizon CPT invariance? Global (Cosmological FRW solution) N. E. MAVROMATOS 20

DARK ENERGY& Cosmological CPTV? KNOW VERY LITTLE ABOUT IT… v EMBARASSING SITUATION 74% OF THE UNIVERSE BUDGET CONSISTS OF UNKNOWN SUBSTANCE v v v Certainly of Quantum Gravitational origin v If cosmological constant (de Sitter), then quantization of field theories not fully understood due to cosmic horizon CPT invariance? Could be: § a Cosmological Constant § Quintessence (scalar field Global (Cosmological FRW solution) relaxing to minimum of its potential) § Something else…Extra dimensions, colliding brane horizon: Cosmological (global) de. Sitter worlds etc. DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 21

Space-Time Foam & Intrinsic CPT Violation DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 22

CPT symmetry without CPT invariance ? DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 23

CPT symmetry without CPT invariance ? DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 24

Stochastic Light-Cone Fluctuations CPT may also be violated in such stochastic models DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 25

Caution on CPTV & Lorentz Violation v CPT Operator well defined but NON-Commuting with Hamiltonian [H , Θ ] 0 § Lorentz & CPT Violation in the Hamiltonian l l Neutral Mesons & Factories, Atomic Physics, Anti-matter factories, Neutrinos, … Modified Dispersion Relations (GRB, neutrino oscillations, synchrotron radiation…) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 26

Caution on CPTV & Lorentz Violation v CPT Operator well defined but NON-Commuting with Hamiltonian [H , Θ ] 0 § Lorentz & CPT Violation in the Hamiltonian l l v. CAUTION: Neutral Mesons & Factories, Atomic Physics, Anti-matter factories, Neutrinos, … Modified Dispersion Relations (GRB, neutrino oscillations, synchrotron radiation…) DISCRETE 08, IFIC (Valencia) , December 08 LV does not necessarily imply CPTV e. g. Standard Model Extension, Non-commutative Geometry field theories N. E. MAVROMATOS 27

STANDARD MODEL EXTENSION DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 28

Non-commutative effective field theories CPT invariant SME type field theory (Q. E. D. ) - only even number of indices appear in effective nonrenormalisable terms. (Carroll et al. hep-th/0105082) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 29

Non-commutative effective field theories CPT invariant SME type field theory (Q. E. D. ) - only even number of indices appear in effective nonrenormalisable terms. (Carroll et al. hep-th/0105082) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 30

STANDARD MODEL EXTENSION DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 31

Lorentz Violation & Anti-Hydrogen v Trapped Molecules: NB: Sensitivity in b 3 that rivals astrophysical or atomic-physics bounds can only be attained if spectral resolution of 1 m. Hz is achieved. Not feasible at present in anti-H factories DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 32

Tests of Lorentz Violation in Neutral Kaons DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 33

EXPERIMENTAL BOUNDS DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 34

DETECTING CPT VIOLATION (CPTV) Complex Phenomenology No single figure of merit v Neutral Mesons: K, B, (unique (? ) QG induced decoherence tests) mesonfactories entangled states K± charged-meson decays v Antihydrogen (precision spectroscopic tests on free & trapped molecules search forbidden transitions) v DISCRETE 08, IFIC (Valencia) , December 08 Low-energy atomic Physics Experiments v Ultra – Cold Neutrons v Neutrino Physics v Terrestrial & Extraterrestrial tests of Lorentz Invariance - search for modified dispersion relations of matter probes: GRB, AGN photons, Crab nebula synchrotron radiation, Flares…. v N. E. MAVROMATOS 35

Order of Magnitude Estimates DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 36

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 37

Order of Magnitude Estimates DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 38

Order of Magnitude Estimates DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 39

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 40

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP Adler-Horwitz decoherent evolution model (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 41

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 42

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 43

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 44

D-particle Foam Models Bulk closed string DISCRETE 08, IFIC (Valencia) , December 08 ELLIS, NM, WESTMUCKETT N. E. MAVROMATOS 45

D-particle Recoil & LIV models DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 46

D-particle Recoil & LIV models String World-sheet torn apart non-conformal Process, Liouville string DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 47

D-particle Recoil & LIV models DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 48

A (non-critical) string theory time Arrow Ellis, NM Nanopoulos Non-equilibrium Strings (non-critical), due to e. g. cosmically catastrophic events in Early Universe, for instance brane worlds collisions: World-sheet conformal Invariance is disturbed Central charge of world. Sheet theory ``runs’’ To a minimal value Zamolodchikov’s C-theorem An H-theorem for CFT Change in degrees of Freedom (i. e. entropy) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 49

A (non-critical) string theory time Arrow Ellis, NM Nanopoulos Non-equilibrium Strings (non-critical), due to e. g. cosmically catastrophic events in Early Universe, for instance brane worlds collisions: World-sheet conformal Invariance is disturbed Central charge of world. Sheet theory ``runs’’ To a minimal value Zamolodchikov’s C-theorem An H-theorem for CFT Change in degrees of Freedom (i. e. entropy) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 50

A (non-critical) string theory time Arrow Ellis, NM Nanopoulos Non-equilibrium Strings (non-critical), due to e. g. cosmically catastrophic events in Early Universe, for instance brane worlds collisions: World-sheet conformal Invariance is disturbed Central charge of world. Sheet theory ``runs’’ To a minimal value Zamolodchikov’s C-theorem An H-theorem for CFT Change in degrees of Freedom (i. e. entropy) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 51

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 52

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 53

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 54

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 55

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 56

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 57

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 58

Order of Magnitude Estimates (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ: Decoherence damping exp (-D t), D = γ (Δm 2)/E The parameters σ and γ depend on microscopic model and are suppressed by (powers of) the string scale MString DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 59

Complex Phenomenology of CPTV v CPT Operator well defined but NON-Commuting with Hamiltonian [H , Θ ] 0 § Lorentz & CPT Violation in the Hamiltonian l l Neutral Mesons & Factories, Atomic Physics, Anti-matter factories, Neutrinos, … Modified Dispersion Relations (GRB, neutrino oscillations, synchrotron radiation, Te. V AGN…) DISCRETE 08, IFIC (Valencia) , December 08 v CPT Operator ill defined (Wald), intrinsic violation, modified concept of antiparticle § Decoherence CPTV Tests l Neutral Mesons: K, B & factories (novel effects in entangled states : (perturbatively) modified EPR correlations) l l l Ultracold Neutrons Neutrinos (highest sensitivity) Light-Cone fluctuations (GRB, Gravity-Wave Interferometers, neutrino oscillations) N. E. MAVROMATOS 60

Complex Phenomenology of CPTV v CPT Operator well defined but NON-Commuting with Hamiltonian [H , Θ ] 0 § Lorentz & CPT Violation in the Hamiltonian l l Neutral Mesons & Factories, Atomic Physics, Anti-matter factories, Neutrinos, … Modified Dispersion Relations (GRB, neutrino oscillations, synchrotron radiation, Te. V AGN…) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 61

Multi-messenger observations of the Cosmos cosmic accelerator Us protons E>1019 e. V ( 10 Mpc ) neutrinos gammas ( z < 1 ) protons E<1019 e. V protons/nuclei: Deviated by magnetic fields, Absorbed by radiation field (GZK) photons: Absorbed by dust & radiation field (CMB) neutrinos: Difficult to detect ⇒ DISCRETE Three 08, “astronomies” De. Naurois 2008 62 IFIC (Valencia) , December possible. . . 08 N. E. MAVROMATOS

VHE Experimental World Today TIBET MILAGRO MAGIC STACEE TIBET ARRAY ARGO-YBJ MILAGRO STACEE CACTUS TACTIC PACT Canary Islands GRAPES TACTIC HESS CANGAROO 08, IFIC (Valencia) , December 08 M. DISCRETE MARTINEZ N. E. MAVROMATOS 63

The MAGIC Collaboration (Major Atmospheric Gamma-ray Imaging Cherenkov Telescope ) Observation of Flares from AGN Mk 501 Red-shift: z=0. 034 DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 64

The MAGIC ``Effect’’ DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 65

Possible Interpetations v Delays of more energetic photons are a result of AGN source Physics (SSC mechanism) MAGIC Coll. Ap. J 669, 862 (2007) : SSC I Bednarek & Wagner ar. Xive: 0804. 0619 : SSCII v Delays of more energetic photons occur in propagation due to new fundamental Physics (e. g. refractive index in vacuo due to Quantum Gravity space-time Foam Effects…) MAGIC Coll & Ellis, NM, Nanopoulos, Sakharov, Sarkisyan [ar. Xive: 0708. 2889] (individual photon analysis – reconstruct peak of flare by assuming modified dispersion relations for photons, linearly or quadratically suppressed by the QG scale) SOURCE MECHANISM BIGGEST THEORETICAL UNCERTAINTY AT PRESENT…. DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 66

Quantum-Gravity Induced Modified Dispersion for Photons Modified dispersion due to QG induced space-time (metric) distortions (c=1 units): DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 67

MAGIC Results (ECF Method): Quadratic Linear 95% CL DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 68

A Stringy Model of Space -Time Foam Ellis, NM, Nanopoulos Open strings on D 3 -brane world represent electrically neutral matter or radiation, interacting via splitting/capture with D-particles (electric charge conservation). D-particle foam medium transparent to (charged) Electrons no modified dispersion for them Photons or electrically neutral probes feel the effects of D-particle foam Modified Dispersion for them…. DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 69

A Stringy Model of Space -Time Foam Ellis, NM, Nanopoulos Open strings on D 3 -brane world represent electrically neutral matter or radiation, interacting via splitting/capture with D-particles (electric charge conservation). D-particle foam medium transparent to (charged) Electrons no modified dispersion for them Photons or electrically neutral probes feel the effects of D-particle foam Modified Dispersion for them…. NON-UNIVERSAL ACTION OF D-PARTICLE FOAM ON MATTER & RADIATION DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 70

A Stringy Model of Space -Time Foam Ellis, NM, Nanopoulos Open strings on D 3 -brane world represent electrically neutral matter or radiation, interacting via splitting/capture with D-particles (electric charge conservation). D-particle foam medium transparent to (charged) Electrons no modified dispersion for them Photons or electrically neutral probes feel the effects of D-particle foam Modified Dispersion for them…. NON-UNIVERSAL ACTION OF D-PARTICLE FOAM ON MATTER & RADIATION DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 71

Stringy Uncertainties & the Capture Process Ellis, NM, Nanopoulos ar. Xiv: 0804. 3566 During Capture: intermediate String stretching between D-particle and D 3 -brane is Created. It acquires N internal Oscillator excitations & Grows in size & oscillates from Zero to a maximum length by absorbing incident photon Energy p 0 : Minimise right-hand-size w. r. t. L. End of intermediate string on D 3 -brane Moves with speed of light in vacuo c=1 Hence TIME DELAY (causality) during Capture: DELAY IS INDEPENDENT OF PHOTON POLARIZATION, HENCE NO BIREFRINGENCE…. BIREFRINGENCE DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 72

Stringy Uncertainties & the MAGIC Effect v v v D-foam: transparent to electrons D-foam captures photons & re-emits them Time Delay (Causal) in each Capture: v v Independent of photon polarization (no Birefringence) Total Delay from emission of photons till observation over a distance D (assume n* defects per string length): Effectively modified Dispersion relation for photons due to induced metric distortion G 0 i ~ p 0 REPRODUCE 4± 1 MINUTE DELAY OF MAGIC from Mk 501 (redshift z=0. 034) For n* =O(1) & Ms ~ 1018 Ge. V, consistently with Crab Nebula & other Astrophysical constraints on modified dispersion relations…… DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 73

Stringy Uncertainties & the MAGIC Effect v v v D-foam: transparent to electrons D-foam captures photons & re-emits them Time Delay (Causal) in each Capture: v v Independent of photon polarization (no Birefringence) COMPATIBLE WITH STRING UNCERTAINTY Total Delay from emission of photons PRINCIPLES: till observation over a distance D (assume n* defects Δ t Δ x ≥ α ’ , Δ p Δ x ≥ 1 + α ’ (Δ p ) 2 + … per string length): (α’ = Regge slope = Square of minimum string length scale) Effectively modified Dispersion relation for photons due to induced metric distortion G 0 i ~ p 0 REPRODUCE 4± 1 MINUTE DELAY OF MAGIC from Mk 501 (redshift z=0. 034) For n* =O(1) & Ms ~ 1018 Ge. V, consistently with Crab Nebula & other Astrophysical constraints on modified dispersion relations…… DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 74

Complex Phenomenology of CPTV v CPT Operator well defined but NON-Commuting with Hamiltonian [H , Θ ] 0 § Lorentz & CPT Violation in the Hamiltonian l l Neutral Mesons & Factories, Atomic Physics, Anti-matter factories, Neutrinos, … Modified Dispersion Relations (GRB, neutrino oscillations, synchrotron radiation…) DISCRETE 08, IFIC (Valencia) , December 08 v CPT Operator ill defined (Wald), intrinsic violation, modified concept of antiparticle § Decoherence CPTV Tests l Neutral Mesons: K, B & factories (novel effects in entangled states : (perturbatively) modified EPR correlations) this talk l l l Ultracold Neutrons Neutrinos (highest sensitivity) Light-Cone fluctuations (GRB, Gravity-Wave Interferometers, neutrino oscillations) N. E. MAVROMATOS 75

Complex Phenomenology of CPTV v CPT Operator well defined but NON-Commuting with Hamiltonian [H , Θ ] 0 § Lorentz & CPT Violation in the Hamiltonian l l Neutral Mesons & Factories, Atomic Physics, Anti-matter factories, Neutrinos, … Modified Dispersion Relations (GRB, neutrino oscillations, synchrotron radiation…) DISCRETE 08, IFIC (Valencia) , December 08 v CPT Operator ill defined (Wald), intrinsic violation, modified concept of antiparticle § Decoherence CPTV Tests l Neutral Mesons: K, B & factories (novel effects in entangled states : (perturbatively) modified EPR correlations) this talk l l l Ultracold Neutrons Neutrinos (highest sensitivity) Light-Cone fluctuations (GRB, Gravity-Wave Interferometers, neutrino oscillations) N. E. MAVROMATOS 76

QUANTUM GRAVITY DECOHERENCE & CPTV NEUTRAL MESON PHENOMENOLOGY 23/05/2007 KAON '07 N. MAVROMATOS

QG DECOHERENCE IN NEUTRAL KAONS: SINGLE STATES DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 78

Decoherence vs CPTV in QM DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 79

Neutral Kaon Asymmetries DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 80

Neutral Kaon Asymmetries DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 81

Neutral Kaon Asymmetries Effects of α, β, γ decoherence parameters DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 82

Decoherence vs QM effects DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 83

Indicative Bounds DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 84

Neutral Kaon Entangled States v Complete Positivity Decoherence matrix Different parametrization of (Benatti-Floreanini) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 85

Neutral Kaon Entangled States v Complete Positivity Decoherence matrix Different parametrization of (Benatti-Floreanini) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 86

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 87

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 88

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 89

Order of Magnitude Estimates However there are models with inverse energy dependence, e. g. (i) Adler’s Lindblad model for Energy-driven QG Decoherence in two level systems (hep-th/0005220): decoherence Lindblad operator propotional to Hamiltonian Decoherence damping exp(-D t) , Decoherence Parameter estimate: D = (Δm 2)2/E 2 MP (ii) Stochastic models of foam in brane/string theory (D-particle recoil models (below)) Decoherence Parameters estimates depend on details of foam, e. g. distribution of recoil velocities of populations of D-parfticle defects in space time (Sarkar, NM) : (a) Gaussian D-particle recoil velocity distribution, spread σ : Decoherence damping in oscillations among two-level systems : exp (-D t 2), D = σ2(Δm 2)2/E 2 (b) Cauchy-Lorentz D-particle recoil velocity distribution, parameter γ : Decoherence damping exp (-D t), D = γ (Δm 2)/E DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 90

Complex Phenomenology of CPTV v CPT Operator well defined but NON-Commuting with Hamiltonian [H , Θ ] 0 § Lorentz & CPT Violation in the Hamiltonian l l Neutral Mesons & Factories, Atomic Physics, Anti-matter factories, Neutrinos, … Modified Dispersion Relations (GRB, neutrino oscillations, synchrotron radiation…) DISCRETE 08, IFIC (Valencia) , December 08 v CPT Operator ill defined (Wald), intrinsic violation, modified concept of antiparticle § Decoherence CPTV Tests l Neutral Mesons: K, B & factories (novel effects in entangled states : (perturbatively) modified EPR correlations) this talk l l l Ultracold Neutrons Neutrinos (highest sensitivity) Light-Cone fluctuations (GRB, Gravity-Wave Interferometers, neutrino oscillations) N. E. MAVROMATOS 91

Complex Phenomenology of CPTV v CPT Operator well defined but NON-Commuting with Hamiltonian [H , Θ ] 0 § Lorentz & CPT Violation in the Hamiltonian l l Neutral Mesons & Factories, Atomic Physics, Anti-matter factories, Neutrinos, … Modified Dispersion Relations (GRB, neutrino oscillations, synchrotron radiation…) DISCRETE 08, IFIC (Valencia) , December 08 v CPT Operator ill defined (Wald), intrinsic violation, modified concept of antiparticle § Decoherence CPTV Tests l Neutral Mesons: K, B & factories (novel effects in entangled states : (perturbatively) modified EPR correlations) this talk l l l Ultracold Neutrons Neutrinos (highest sensitivity) Light-Cone fluctuations (GRB, Gravity-Wave Interferometers, neutrino oscillations) N. E. MAVROMATOS 92

Entangled States: CPT & EPR correlations v Novel (genuine) two body effects: § If CPT not-well defined modification of EPR correlations (ω-effect) Unique effect in Entangled states of mesons !! Characteristic of ill-defined nature of intrinsic CPT Violation (e. g. due to decoherence) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 93

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 94

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 95

CPTV & EPR-correlations modification DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 96

CPTV & EPR-correlations modification DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 97

CPTV & EPR-correlations modification CPTV KLKL, ωKSKS terms originate from Φ-particle , hence same dependence on centre-of-mass energy s. Interference proportional to real part of amplitude, exhibits peak at the resonance…. DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 98

CPTV & EPR-correlations modification KSKS terms from C=+ background no dependence on centre-of-mass energy s. Real part of Breit-Wigner amplitude Vanishes at top of resonance, Interference of C=+ with C=-- background, vanishes at top of the resonance, opposite signature on either side…. . DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 99

CPTV & EPR-correlations modification DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 100

CPTV & EPR-correlations modification DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 101

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 102

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 103

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 104

B-systems, ω-effect & demise of flavour-tagging Alvarez, Bernabeu NM, Nebot, Papavassiliou Kaon systems have increased sensitivity to ω-effects due to the decay channel π+π-. v B-systems do not have such a “good’’ channel but have the advantage of statistics Interesting limits of ω-effects there v v Flavour tagging: Knowledge that one of the two-mesons in a meson factory decays at a given time through flavour-specific “channel’’ Unambiguously determine the flavour of the other meson at the same time. Not True if intrinsic CPTV – ω-effect present : Theoretical limitation (“demise’’) of flavour tagging DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 105

B-systems, ω-effect & demise of flavour-tagging DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 106

B-systems, ω-effect & demise of flavour-tagging CP parameter CPTV parameter (QM) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 107

Equa. L-Sign di-lepton charge asymmetry Δt dependence ALVAREZ, BERNABEU, NEBOT v Interesting tests of the ω-effect can be performed by looking at the equal-sign di-lepton decay channels DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 108

Equa. L-Sign di-lepton charge asymmetry Δt dependence ALVAREZ, BERNABEU, NEBOT v Interesting tests of the ω-effect can be performed by looking at the equal-sign di-lepton decay channels DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 109

Equa. L-Sign di-lepton charge asymmetry Δt dependence ALVAREZ, BERNABEU, NEBOT v Interesting tests of the ω-effect can be performed by looking at the equal-sign di-lepton decay channels DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 110

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 111

Δt observable Expt. DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 112

CURRENT EXPERIMENTAL LIMITS Δt observable Expt. DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 113

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 114

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 115

D-particle Foam & Entangled States v If CPT Operator well-defined as operator, even if CPT is broken in the Hamiltonian… (e. g. Lorentz violating models) KSKL DISCRETE 08, IFIC (Valencia) , December 08 Φ N. E. MAVROMATOS KSKL 116

EPR Modifications & D-particle Foam v If CPT Operator ill-defined, unique consequences in modifications of EPR correlations of entangled states of neutral mesons in meson factories (Φ-, B-factories) Bernabeu, NM, Papavassiliou, Nebot, Alvarez, Sarben Sarkar Φ KSKL IF CPT ILL-DEFINED (e. g. D-particle Foam) Induced metric due to capture/recoil, stochastic fluctuations, Decoherence DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 117

EPR Modifications & D-particle Foam v CPT Operator ill-defined, unique consequences in modifications of EPR correlations of entangled states of neutral mesons in meson factories (Φ -, B-factories) Bernabeu, NM, Papavassiliou, Nebot, Alvarez, Sarben Sarkar KSKL Φ KSKS , KLKL IF CPT ILL-DEFINED (e. g. flavour violating (FV) D-particle Foam) KSKL KSKS , KLKL Induced metric due to capture/recoil, stochastic fluctuations, Decoherence Estimates of such D-particle foam effects in neutral mesons complicated due to strong QCD effects present in such composite neutral particles DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 118

EPR Modifications & D-particle Foam v CPT Operator ill-defined, unique consequences in modifications of EPR correlations of entangled states of neutral mesons in meson factories (Φ -, B-factories) Bernabeu, NM, Papavassiliou, Nebot, Alvarez, Sarben Sarkar KSKL Φ KSKS , KLKL IF CPT ILL-DEFINED (e. g. flavour violating (FV) D-particle Foam) KSKL KSKS , KLKL Induced metric due to capture/recoil, stochastic fluctuations, Decoherence Estimates of such D-particle foam effects in neutral mesons complicated due to strong QCD effects present in such composite neutral particles DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 119

v Neutral mesons no longer indistinguishable particles, initial entangled state: Φ KSKL KSKS , KLKL IF CPT ILL-DEFINED (e. g. flavour violating (FV) D-particle Foam) If QCD effects, sub-structure in neutral mesons ignored, and D-foam acts as if they were structureless particles, then for MQG ~ 1018 Ge. V (MAGIC) the estimate for ω: | ω | ~ 10 -4 |ζ|, for 1 > |ζ| > 10 -2 (natural) Not far from sensitivity of upgraded meson factories ( e. g. DAFNE 2) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 120

v Neutral mesons no longer indistinguishable particles, initial entangled state: Φ KSKL KSKS , KLKL IF CPT ILL-DEFINED (e. g. flavour violating (FV) D-particle Foam) If QCD effects, sub-structure in neutral mesons ignored, and D-foam acts as if they were structureless particles, then for MQG ~ 1018 Ge. V (MAGIC) the estimate for ω: | ω | ~ 10 -4 |ζ|, for 1 > |ζ| > 10 -2 (natural) Not far from sensitivity of upgraded meson factories ( e. g. DAFNE 2) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 121

v Neutral mesons no longer indistinguishable particles, initial entangled state: Φ KSKL KSKS , KLKL IF CPT ILL-DEFINED (e. g. flavour violating (FV) D-particle Foam) If QCD effects, sub-structure in neutral mesons ignored, and D-foam acts as if they were structureless particles, then for MQG ~ 1018 Ge. V (MAGIC) the estimate for ω: | ω | ~ 10 -4 |ζ|, for 1 > |ζ| > 10 -2 (natural) Not far from sensitivity of upgraded meson factories ( e. g. DAFNE 2) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 122

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 123

NB: Problem direction D-particle Recoil & the “Flavour” of recoil Not all particle species interact the same way with D-particles dependence e. g. electric charge symmetries should be preserved, hence LIV …. + electrically-charged excitations cannot split and attach to neutral D-particles…. Stochastically Neutrinos (or neutral mesons) are good candidates… flct. Environment But there may be flavour oscillations during the capture/recoil process, i. e. wave Decoherence, -function of recoiling string might differ by a phase from incident one…. CPTV ill defined… In statistical populations of D-particles, one might have isotropic situations, with << ui >> = 0, but stochastically fluctuating << ui ui >> 0. For slow recoiling heavy D-particles the resulting Hamiltonian, expressing interactions of neutrinos (or “flavoured” particles, including oscillating neutral mesons), reads: DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 124

NB: Problem direction D-particle Recoil & the “Flavour” of recoil Not all particle species interact the same way with D-particles dependence e. g. electric charge symmetries should be preserved, hence LIV …. + electrically-charged excitations cannot split and attach to neutral D-particles…. Stochastically Neutrinos (or neutral mesons) are good candidates… flct. Environment But there may be flavour oscillations during the capture/recoil process, i. e. wave Decoherence, -function of recoiling string might differ by a phase from incident one…. CPTV ill defined… In statistical populations of D-particles, one might have isotropic situations, with << ui >> = 0, but stochastically fluctuating << ui ui >> 0. For slow recoiling heavy D-particles the resulting Hamiltonian, expressing interactions of neutrinos (or “flavoured” particles, including oscillating neutral mesons), reads: DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 125

D-particle recoil and entangled Meson States v Apply non-degenerate perturbation theory to construct “gravitationally dressed’’ states from DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 126

D-particle recoil and entangled Meson States v Apply non-degenerate perturbation theory to construct “gravitationally dressed’’ states from DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 127

D-particle recoil and entangled Meson States v Apply non-degenerate perturbation theory to construct “gravitationally dressed’’ states from DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 128

D-particle recoil and entangled Meson States v Apply non-degenerate perturbation theory to construct “gravitationally dressed’’ states from DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 129

D-particle recoil and entangled Meson States v Apply non-degenerate perturbation theory to construct “gravitationally dressed’’ states from Similarly for the dressed state is obtained by DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 130

D-particle recoil and entangled Meson States v Apply non-degenerate perturbation theory to construct “gravitationally dressed’’ states from Similarly for the dressed state is obtained by DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 131

D-particle recoil and entangled Meson States v Apply non-degenerate perturbation theory to construct “gravitationally dressed’’ states from Similarly for the dressed state is obtained by DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 132

D-particle recoil and entangled Meson States v Apply non-degenerate perturbation theory to construct “gravitationally dressed’’ states from Similarly for the dressed state is obtained by DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 133

D-particle recoil and entangled Meson States v Apply non-degenerate perturbation theory to construct “gravitationally dressed’’ states from Similarly for the dressed state is obtained by Prediction of -like effects in entangled states …. ( Bernabeu, NM, Papavassiliou) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 134

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 135

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 136

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 137

ω-effect as discriminant of space-time foam models Bernabeu, NM, Sarben Sarkar ω-effect not generic, generic depends on details of foam Initially dressed states depend on form of interaction Hamiltonian HI (non-degenerate) perturbation theory determine existence of ω-effects (I) D-foam: features: direction of k violates Lorentz symmetry, symmetry flavour non conservation non-trivial ω-effect (II) Quantum Gravity Foam as “thermal Bath’’ (Garay) no ω-effect v Bath frequency “atom’’ (matter) frequency DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 138

ω-effect as discriminant of space-time foam models Bernabeu, NM, Sarben Sarkar ω-effect not generic, generic depends on details of foam Initially dressed states depend on form of interaction Hamiltonian HI (non-degenerate) perturbation theory determine existence of ω-effects (I) D-foam: features: direction of k violates Lorentz symmetry, symmetry flavour non conservation non-trivial ω-effect (II) Quantum Gravity Foam as “thermal Bath’’ (Garay) no ω-effect v Bath frequency “atom’’ (matter) frequency DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 139

ω-effect as discriminant of space-time foam models Bernabeu, NM, Sarben Sarkar ω-effect not generic, generic depends on details of foam Initially dressed states depend on form of interaction Hamiltonian HI (non-degenerate) perturbation theory determine existence of ω-effects (I) D-foam: features: direction of k violates Lorentz symmetry, symmetry flavour non conservation non-trivial ω-effect (II) Quantum Gravity Foam as “thermal Bath’’ (Garay) no ω-effect v Bath frequency “atom’’ (matter) frequency DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 140

ω-effect as discriminant of space-time foam models Bernabeu, NM, Sarben Sarkar ω-effect not generic, generic depends on details of foam Initially dressed states depend on form of interaction Hamiltonian HI (non-degenerate) perturbation theory determine existence of ω-effects (I) D-foam: features: direction of k violates Lorentz symmetry, symmetry flavour non conservation non-trivial ω-effect (II) Quantum Gravity Foam as “thermal Bath’’ (Garay) no ω-effect v Bath frequency “atom’’ (matter) frequency DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 141

PRECISION T, CP & CPT TESTS WITH CHARGED KAONS 23/05/2007 KAON '07 N. MAVROMATOS

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 143

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 144

DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 145

QG DECOHERENCE & NEUTRINOS 23/05/2007 KAON '07 N. MAVROMATOS

QG Decoherence & Neutrinos v Stochastic (quantum) metric fluctuations in Dirac or Majorana Hamiltonian for neutrinos affect oscillation probabilities by damping exponential factors – characteristic of decoherence v Quantum Gravitational MSW effect v Precise form of neutrino energy dependence of damping factors linked to specific model of foam DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 147

QG Decoherence & Neutrinos v Stochastic (quantum) metric fluctuations in Dirac or Majorana Hamiltonian for neutrinos affect oscillation probabilities by damping exponential factors – characteristic of decoherence v Quantum Gravitational MSW effect v Precise form of neutrino energy dependence of damping factors linked to specific model of foam DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 148

QG Decoherence & Neutrinos v Stochastic (quantum) metric fluctuations in Dirac or Majorana Hamiltonian for neutrinos affect oscillation probabilities by damping exponential factors – characteristic of decoherence v Quantum Gravitational MSW effect v Precise form of neutrino energy dependence of damping factors linked to specific model of foam DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 149

Stochastic QG metric fluctuations Consider Dirac or Majorana (two-flavour) Hamiltonian with mixing , in such a metric background, with equation of motion: Oscillation Probability DISCRETE 08, IFIC (Valencia) , December 08 Flavour states N. E. MAVROMATOS Mass eigenstates 150

Stochastic QG metric fluctuations Oscillation Probability Two kinds of foam examined: (i) Gaussian distributions NM, Sarkar , Alexandre, Farakos, Pasipoularides (ii) Cauchy-Lorentz In D-particle foam model, hoi n= ui involves recoil velocity distribution of Dparticle populations. NB: damping suppressed by neutrino mass differences DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 151

Stochastic QG metric fluctuations Oscillation Probability Two kinds of foam examined: (i) Gaussian distributions (ii) Cauchy-Lorentz In D-particle foam model, hoi n= ui involves recoil velocity distribution of Dparticle populations. NB: damping suppressed by neutrino mass differences DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 152

Quantum Gravitational MSW Effect ncbh Black Hole density in foam Neutrino density matrix evolution is of Lindblad decoherence type: Barenboim, NM, Sarkar, Waldron-Lauda DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 153

Quantum Gravitational MSW Effect ncbh Black Hole density in foam Neutrino density matrix evolution is of Lindblad decoherence type: CPT Violating (time irreversible, CP symmetric) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 154

Quantum Gravitational MSW Effect OSCILLATION PROBABILITY: Damping suppressed by neutrino-flavour MSW-coupling differences DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 155

Quantum Gravitational MSW Effect OSCILLATION PROBABILITY: Damping suppressed by neutrino-flavour MSW-coupling differences DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 156

Quantum Gravitational MSW Effect OSCILLATION PROBABILITY: Damping suppressed by neutrino-flavour MSW-coupling differences DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 157

Quantum Gravitational MSW Effect OSCILLATION PROBABILITY: Damping suppressed by neutrino-flavour MSW-coupling differences DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 158

Current Experimental Bounds Lindblad-type decoherence Damping: Fogli, Lisi, Marrone, Montanino, Palazzo t = L (Oscillation length, (c=1) Including LSND & Kam. Land Data: Best fits Beyond Lindblad: stochastic metric Fluctuations damping: Barenboim, NM, Sarkar, Waldron-Lauda DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 159

Current Experimental Bounds Lindblad-type decoherence Damping: Fogli, Lisi, Marrone, Montanino, Palazzo t = L (Oscillation length, (c=1) Including LSND & Kam. Land Data: Best fits Beyond Lindblad: stochastic metric Fluctuations damping: SOME EXPONENTS NON ZERO, NOT ALL… DISCRETE 08, IFIC (Valencia) , December 08 Barenboim, NM, Sarkar, Waldron-Lauda N. E. MAVROMATOS 160

Current Experimental Bounds Including LSND & Kam. Land Data: Barenboim, NM, Sarkar, Waldron-Lauda t = L (Oscillation length, (c=1) Beyond Lindblad: stochastic metric Fluctuations damping (Best Fits): DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 161

Current Experimental Bounds Including LSND & Kam. Land Data: Barenboim, NM, Sarkar, Waldron-Lauda Lindblad type Decoherence Beyond Lindblad: stochastic metric Fluctuations damping (Best Fits): t = L (Oscillation length, (c=1) NB: Lindblad-type damping Also induced by uncertainties in energy of neutrino beams Ohlsson, Jacobson Probably too Large to be QG Effect…. LSND+ Kam. Land DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 162

Current Experimental Bounds Including LSND & Kam. Land Data: Barenboim, NM, Sarkar, Waldron-Lauda Lindblad type Decoherence Beyond Lindblad: stochastic metric Fluctuations damping (Best Fits): t = L (Oscillation length, (c=1) NB: Lindblad-type damping Also induced by uncertainties in energy of neutrino beams Ohlsson, Jacobson LSND+ Kam. Land DISCRETE 08, IFIC (Valencia) , December 08 Probably too Large to be QG Effect…. But in that case all exponents must be non zero Hence…. still unresolved N. E. MAVROMATOS 163

Look into the future: Potential of J-PARC, CNGS NM, Sakharov, Meregaglia, Rubbia, Sarkar JPARC CNGS DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 164

Look into the future: Potential of J-PARC, CNGS DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 165

Look into the future: High Energy Neutrinos Morgan, Winstanley, Anchordoqui, Goldberg, Hooper, Subir Sarkar, Weiler, Halzen… Much higher sensitivities from high energy neutrinos AMANDA, ICE CUBE, ASTROPHYSICAL/COSMOLOGICAL NEUTRINOS (e. g. if neutrinos with energies close to 1020 e. V from GRB At redshifts z > 1 are observed …) DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 166

Look into the future: High Energy Neutrinos Morgan, Winstanley, Anchordoqui, Goldberg, Hooper, Subir Sarkar, Weiler, Halzen… Much higher sensitivities from high energy neutrinos AMANDA, ICE CUBE, ASTROPHYSICAL/COSMOLOGICAL NEUTRINOS (e. g. if neutrinos with energies close to 1020 e. V from GRB At redshifts z > 1 are observed …) Recently: Interesting scenarios of Lindblad decoherence with SOME damping exponents having energy dependence E-4 proposed for reconciling LSND (anti-ν) & MINIBOONE Farzan, Smirnov, Schwetz, DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 167

Look into the future: High Energy Neutrinos Morgan, Winstanley, Anchordoqui, Goldberg, Hooper, Sarkar, Weiler, Halzen… Much higher sensitivities from high energy neutrinos Global Best Fit AMANDA, ICE CUBE, ASTROPHYSICAL/COSMOLOGICAL NEUTRINOS (e. g. if neutrinos with energies close to 1020 e. V from GRB At redshifts z > 1 are observed …) Recently: Interesting scenarios of Lindblad decoherence with SOME damping exponents having energy dependence E-4 proposed for reconciling LSND (anti-ν) & MINIBOONE Farzan, Smirnov, Schwetz, Microscopic QG models ? DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 168

Look into the future: High Energy Neutrinos Morgan, Winstanley, Anchordoqui, Goldberg, Hooper, Sarkar, Weiler, Halzen… Much higher sensitivities from high energy neutrinos Global Best Fit AMANDA, ICE CUBE, ASTROPHYSICAL/COSMOLOGICAL NEUTRINOS (e. g. if neutrinos with energies close to 1020 e. V from GRB At redshifts z > 1 are observed …) Recently: Interesting scenarios of Lindblad decoherence with SOME damping exponents having energy dependence E-4 proposed for reconciling LSND (anti-ν) & MINIBOONE Farzan, Smirnov, Schwetz, Microscopic QG models ? e. g. in our stochastic fluctuating metric models E-4 scaling is obtained in Gaussian model with σ2 ~ k-2 DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 169

QG Decoherence & LHC Black Holes v In string theory or extra-dimensional models (in general) QG mass scale may not be Planck scale but much smaller : MQG << MPlanck v Theoretically MQG could be as low as a few Te. V. In such a case, collision of energetic particles at LHC could produce microscopic Black Holes at LHC, which will evaporate quickly. v If there is decoherence in such models, then, can the above tests determine the magnitude of MQG beforehand? v Highly model dependent issue… DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 170

QG Decoherence & LHC Black Holes Models of Decoherence: (i) Parameters = O(E 2/MQG) (e. g. D-particle recoil LV foam ) current bounds: Kaons MQG > 1019 Ge. V Photons MQG > 1018 Ge. V, Neutrinos MQG > 1026 Ge. V No low MQG mass scale allowed… v (ii) Parameters = O((Δm 2)2/E 2 MQG) (e. g. Adler’s model of decoherence, stochastic D-particle LI models…) Low MQG mass models (even Te. V) allowed by current measurements of decoherence… compatible with LHC BH observations…. DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 171

CONCLUSIONS v We know very little about QG so experimental searches & tests of various theoretical models will definitely help in putting us on the right course … v (Intrinsic) CPT &/or Lorentz Violation & decoherence might characterise QG models… v There may be `smoking-gun’ experiments for intrinsic CPTV & QG Decoherence , unique in entangled states of mesons (ω-effect best signature, if present though…. However, not generic effect, depends on model of QG foam…) foam… v The magnitude of such effects is highly model dependent, may not be far from sensitivity of immediate-future facilities. v QG Decoherence effects in neutrino oscillations yield damping signatures, but those are suppressed by (powers of) neutrino mass differences. Difficult to detect … Nevertheless future (high energy neutrinos) looks promising… DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 172

CONCLUSIONS v We know very little about QG so experimental searches & tests of various theoretical models will definitely help in putting us on the right course … v (Intrinsic) CPT &/or Lorentz Violation & decoherence might characterise QG models… Microscopic Time Irreversibility…. v There may be `smoking-gun’ experiments for intrinsic CPTV & QG Decoherence , unique in entangled states of mesons (ω-effect best signature, if present though…. However, not generic effect, depends on model of QG foam…) foam… v The magnitude of such effects is highly model dependent, may not be far from sensitivity of immediate-future facilities. v QG Decoherence effects in neutrino oscillations yield damping signatures, but those are suppressed by (powers of) neutrino mass differences. Difficult to detect … Nevertheless future (high energy neutrinos) looks promising… DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 173

CONCLUSIONS v We know very little about QG so experimental searches & tests of various theoretical models will definitely help in putting us on the right course … v (Intrinsic) CPT &/or Lorentz Violation & decoherence might characterise QG models… Microscopic Time Irreversibility…. v There may be `smoking-gun’ experiments for intrinsic CPTV & QG Decoherence , unique in entangled states of mesons (ω-effect best signature, if present though…. However, not generic effect, depends on model of QG foam…) foam… v The magnitude of such effects is highly model dependent, may not be far from sensitivity of immediate-future facilities. v QG Decoherence effects in neutrino oscillations yield damping signatures, but those are suppressed by (powers of) neutrino mass differences. Difficult to detect … Nevertheless future (high energy neutrinos) looks promising… DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 174

Further Questions… DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 175

Further Questions… DISCRETE 08, IFIC (Valencia) , December 08 N. E. MAVROMATOS 176