CPT Symmetry, Quantum Gravity and Entangled Neutral Kaons Antonio Di Domenico Dipartimento Di Fisica, Sapienza Università Di Roma and INFN Sezione Di Roma, Italy

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CPT symmetry, Quantum Gravity and entangled neutral kaons Antonio Di Domenico Dipartimento di Fisica, Sapienza Università di Roma and INFN sezione di Roma, Italy

Fourteenth Marcel Grossmann Meeting - MG14

University of Rome "La Sapienza" - Rome, July 12-18, 2015

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International photos of the meeting available here Meals Organizing Hotels Local Organizing International Organizing Committee chair: Remo Ruffini, University of Rome and ICRANet Participants International International Coordinating Committee chair: Robert Jantzen, Villanova University Coordination Local Organizing Committee chair: Massimo Bianchi, University of Rome "Tor Vergata" Preliminary List Scientific Program The Fourteenth Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Contacts Relativity, Gravitation, and Relativistic Field Theory will take place at the University of Rome Sapienza July 12 - 18, Plenary Invited Contact us Speakers 2015, celebrating the 100th anniversary of the Einstein equations as well as the International Year of Light under the aegis of the United Nations. Plenary Program Links For the first time, in addition to the main meeting in Rome, a series of satellite meetings to MG14 will take place. Chairpersons of The registration to MG14 will also cover participation in one additional satellite meeting. ICRA Homepage Parallel Sessions Preregistration will take place Sunday July 12 from 10am to 6pm at the Department of Physics, and the official ICRANet Homepage Parallel Sessions opening of the meeting Monday morning, July 13. During the six day conference a wide variety of topics will be discussed in the morning plenary sessions beginning with mathematical topics on Monday, quantum aspects of Sponsors Parallel Sessions by gravity on Tuesday, precision tests of general relativity on Wednesday, relativistic astrophysics on Thursday, Day cosmology and astroparticle physics on Friday and the latest scientific news and the history of physics Saturday. There will be five plenary lectures each morning and up to twenty parallel sessions in the four weekday afternoons Global Program excluding Wednesday, which is left free for a trip and the evening conference banquet.

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Contact email: mg14[AT]icra.it CPT: introduction The three discrete symmetries of QM, C (charge conjugation: q ! -q), P (parity: x ! -x), and T (time reversal: t ! -t) are known to be violated in nature both singly and in pairs. Only CPT appears to be an exact symmetry of nature.

CPT theorem : R. Jost

J. Schwinger (1957) G. Lüders (1951) (1954)

W. Pauli J. Bell (1952) (1955)

Exact CPT invariance holds for any quantum field theory (like the Standard Model) formulated on flat space-time which assumes: (1) Lorentz invariance (2) Locality (3) Unitarity (i.e. conservation of probability). Testing the validity of the CPT symmetry probes the most fundamental assumptions of our present understanding of particles and their interactions.

A. Di Domenico 14th Marcel Grossmann meeting July 12-18, 2015 Rome 2

CPT: introduction The three discrete symmetries of QM, C (charge conjugation: q ! -q), P (parity: x ! -x), and T (time reversal: t ! -t) are known to be violated in nature both singly and in pairs. Only CPT appears to be an exact symmetry of nature.

Intuitive justification of CPT symmetry [1]: For an even-dimensional space => reflection of all axes is equivalent to a rotation e.g. in 2-dim. space: reflection of 2 axes = rotation of π around the origin

In 4-dimensional pseudo-euclidean space-time PT reflection is NOT equivalent to a rotation. Time coordinate is not exactly equivalent to space coordinate. Charge conjugation is also needed to change sign to e.g. 4-vector current jµ. (or axial 4-v). CPT (and not PT) is equivalent to a rotation in the 4-dimensional space-time [1] Khriplovich, I.B., Lamoreaux, S.K.: CP Violation Without Strangeness.

A. Di Domenico 14th Marcel Grossmann meeting July 12-18, 2015 Rome

CPT: introduction Extension of CPT theorem to a theory of quantum gravity far from obvious. (e.g. CPT violation appears in several QG models) huge effort in the last decades to study and shed light on QG phenomenology ⇒ Phenomenological CPTV parameters to be constrained by experiments

Consequences of CPT symmetry: equality of masses, lifetimes, |q| and |µ| of a particle and its anti-particle. Neutral meson systems offer unique possibilities to test CPT invariance; e.g. taking as figure of merit the fractional difference between the masses of a particle and its anti-particle:

−18 neutral K system m 0 − m 0 mK <10 K K

−14 m 0 m 0 m 10 neutral B system B − B B <

proton- anti-proton −8 mp −mp mp <10 Other interesting CPT tests: e.g. the study of anti-hydrogen atoms, etc..

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The neutral kaon: a two-level quantum system 0 Since the first observation of a K0 (V- K 0 K particle) in 1947, several phenomena ππ observed and several tests performed: S=+1 S=-1 • strangeness oscillations s d s d • regeneration • CP violation • Direct CP violation € € € • precise CPT tests • … € € One of the most intriguing physical systems in Nature. T. D. Lee € € Neutral K mesons are a unique physical system which appears to be created by nature to demonstrate, in the most impressive manner, a number of spectacular phenomena...... If the K mesons did not exist, they should have been invented “on purpose” in order to teach students the principles of quantum mechanics. Lev B. Okun

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The neutral kaon system: introduction

The time evolution of a two-component state vector Ψ = a K 0 + b K 0 0 0 in the { K , K } space is given by ∂ K 0 K 0 (Wigner-Weisskopf approximation): i Ψ(t) = HΨ(t) ππ d s d ∂t s H is the effective hamiltonian (non-hermitian), decomposed into a Hermitian € € € € € part (mass matrix M) and an anti-Hermitian part (i/2 decay matrix Γ) : € € i € $ m11 m12 ' i $ Γ11 Γ€12 ' € H = M − Γ = & ) − & ) 2 % m21 m22 ( 2% Γ21 Γ22 ( Diagonalizing the effective Hamiltonian: eigenstates eigenvalues 1 K 1 K 0 1 K 0 i S,L = [( +εS,L ) ± ( −εS,L ) ] λ €= m − Γ 2 1+ εS,L S,L S,L 2 S,L ( ) |K1,2> are −iλS,L t 1 K t = e K 0 CP=±1 states S,L ( ) S,L ( ) = [ K1,2 +εS,L K2,1 ] (1+ εS,L ) τS ~ 90 ps τL ~ 51 ns € ∗ -3 KL →ππ violates CP KS KL ≅εS +εL ≠ 0 small CP impurity ~2 x 10

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€

€ € CPT violation: standard picture

CP violation: T violation: ε H − H −iℑM − ℑΓ 2 S ,L = ε ±δ ε = 12 21 = 12 12 2(λS − λL ) Δm +iΔΓ / 2

CPT violation:

m 0 − m 0 − i 2 Γ 0 − Γ 0 H11 − H22 1 ( K K ) ( )( K K ) δ = = 2(λS − λL ) 2 Δm + iΔΓ/2

• δ ≠ 0 implies CPT violation Δm = mL − mS , ΔΓ = ΓS − ΓL • ε ≠ 0 implies T violation −15 • ε ≠ 0€ or δ ≠ 0 implies CP violation Δm = 3.5 ×10 GeV 2 m 7 10−15 GeV (with a phase convention ℑ Γ 12 = 0 ) ΔΓ ≈ ΓS ≈ Δ = ×

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€ neutral kaons vs other oscillating meson systems

Δm <Γ> ΔΓ/2 (GeV) (GeV) (GeV) (GeV)

K0 0.5 3x10-15 3x10-15 3x10-15

D0 1.9 6x10-15 2x10-12 1x10-14

0 -13 -13 -15 B d 5.3 3x10 4x10 O(10 ) (SM prediction)

0 -11 -13 -14 B s 5.4 1x10 4x10 3x10

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“Standard” CPT tests

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CPT test at CPLEAR

e+ Test of CPT in the time evolution of K0 K0 neutral kaons using the semileptonic π- τ=0 τ asymmetry ν

⎧ R+ (τ ) −αR− (τ ) R− (τ ) −αR+ (τ ) ⎪Aδ (τ ) = + R ( ) R ( ) R ( ) R ( ) ⎪ + τ +α − τ − τ +α + τ ⎪ 0 +(−) −(+) ⎨R+(−) (τ ) = R (K t=0 → (e π v)t=τ ) ⎪ 0 −(+) +(−) R ( ) R K t 0 (e v) ⎪ −(+) τ = ( = → π t=τ ) ⎪ ⎩α =1+ 4ℜε L

Aδ (τ >> τ S ) = 8ℜδ ℜδ = (0.30 ± 0.33 ± 0.06) × 10-3 CPLEAR PLB444 (1998) 52

A. Di Domenico 14th Marcel Grossmann meeting July 12-18, 2015 Rome

The Bell-Steinberger relationship

(1965) J. Bell J. Steinberger

Unitarity constraint: K = aS KS + aL KL d 2 2 ⎛ K t ⎞ a f T K a f T K ⎜− ( ) ⎟ = ∑ S S + L L ⎝ dt ⎠t=0 f Sum over all possible decay products yields two trivial relations: (sum over few decay products for kaons; 2 many for B and D mesons => not easy to evaluate) ΓS,L = ∑ f T KS,L f and a not trivial one, i.e. the B-S relationship: All observables quantities ∗ ∑ f T KS f T KL K K 2 i f L S = (ℜε + ℑδ) = ∗ i(λS − λL )

A. Di Domenico 14th Marcel Grossmann meeting July 12-18, 2015 Rome

–5–

“Standard” CPT test e+ measuring the time evolution of a neutral kaon beam into K0 π- semileptonic decays: -3 ℜδ = (0.30 ± 0.33 ± 0.06) × 10 τ=0 τ CPLEAR ν PLB444 (1998) 52 " f T K f T K ∗ $ )∑ S L * using the unitarity constraint 2 " K K $ f ℑδ = ℑ# L S % = ℑ) ∗ * (Bell-Steinberger relation) ) i(λS − λL ) * # % Im δ =(-0.7 ± 1.4) × 10-5 PDG fit (2014)

Γ 0 − Γ 0 ( K K ) m 0 − m 0 − i 2 Γ 0 − Γ 0 1 ( K K ) ( )( K K ) −18 δ = (10 GeV) 2 Δm + iΔΓ/2 Combining Reδ and Imδ results

Assuming Γ 0 − Γ 0 = 0 , i.e. no CPT viol. in decay: ( K K ) € € −19 at 95% c.l. mK 0 − mK 0 < 4.0 ×10 GeV −18 m 0 − m 0 10 GeV ( K K ) ( )

th A. Di Domenico 14 Marcel Grossmann meeting July 12-18, 2015Figure Rome 1: Top: allowed region at 68% and 95%12 C.L. in the ℜ(ϵ), ℑ(δ)plane.Bottom:allowed region at 68% and 95% C.L. in the ∆M,∆Γ plane.€

August 21, 2014 13:17 Entangled neutral kaon pairs

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Neutral kaons at a φ-factory Production of the vector meson φ in e+e- annihilations: K + - S,L • e e → φ σφ∼3 µb + - e φ e W = mφ = 1019.4 MeV 0 0 • BR(φ → K K ) ~ 34% KL,S • ~106 neutral kaon pairs per pb-1 produced in an antisymmetric quantum state PC -- with J = 1 1 0  0  0  0  : i = K (p) K (−p) − K (p) K (−p) 2 [ ] N     p = 110 MeV/c K p K p K p K p K = [ S ( ) L (− ) − L ( ) S (− ) ] 2 λS = 6 mm λL = 3.5 m 2 2 N = 1+ ε 1+ ε 1−ε ε ≅1 ( S )( L ) ( S L )

€

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The KLOE detector at the Frascati φ-factory DAΦNE

DAFNE collider

Integrated luminosity (KLOE)

KLOE detector

Total KLOE ∫L dt ~ 2.5 fb-1 9 (2001 - 05) → ~2.5×10 KSKL pairs

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The KLOE detector at the Frascati φ-factory DAΦNE

KLOE detector DAFNE collider

Integrated luminosity (KLOE)

Lead/scintillating fiber calorimeter drift chamber Total KLOE ∫L dt ~ 2.5 fb-1 4 m diameter × 3.3 m length 9 helium based gas mixture (2001 - 05) → ~2.5×10 KSKL pairs

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Test of Quantum Coherence

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EPR correlations in entangled neutral kaon pairs from φ

1 0 0 0 0 i = K K − K K Same final state for both kaons: f = f = π+π- 2 [ ] 1 2 π φ π )

π a.u t1 t2 π ( ) € t Δ π I( φ π π t1 t2=t1 π

EPR correlation: no simultaneous decays (Δt=0) in the same final state due to the fully destructive quantum interference Δt=t1-t2 Δt/τS

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EPR correlations in entangled neutral kaon pairs from φ

1 0 0 0 0 i = K K − K K Same final state for both kaons: f = f = π+π- 2 [ ] 1 2 π φ π )

π a.u t1 t2 π ( ) € t Δ π I( φ π π t1 t2=t1 π

EPR correlation: no simultaneous decays (Δt=0) in the same final state due to the fully destructive t/ quantum interference Δ τS Δt=|t1-t2|

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EPR correlations in entangled neutral kaon pairs from φ

1 0 0 0 0 i = K K − K K Same final state for both kaons: f = f = π+π- 2 [ ] 1 2 π φ π )

π a.u t1 t2 π ( ) € t Δ π I( φ π Interference effects are a π t1 t2=t1 π key feature of QM, “the only mystery” according to Feynman EPR correlation: => Experimental test no simultaneous decays (Δt=0) in the same final state due to the fully destructive t/ quantum interference Δ τS Δt=|t1-t2|

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+ - + - φ →KSKL→π π π π : test of quantum coherence

1 i = K 0 K 0 − K 0 K 0 2 [ ]

2 2 + − + − % + − + − 0 0 + − + − 0 0 I π π ,π π ;Δt = N π π ,π π K K Δt + π π ,π π K K Δt ( ) 2 &' ( ) ( ) € ∗ * −2ℜ π +π −,π +π − K 0K 0 Δt π +π −,π +π − K 0K 0 Δt ( ( ) ( ) )+,

€

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+ - + - φ →KSKL→π π π π : test of quantum coherence

1 i = K 0 K 0 − K 0 K 0 2 [ ]

2 2 + − + − % + − + − 0 0 + − + − 0 0 I π π ,π π ;Δt = N π π ,π π K K Δt + π π ,π π K K Δt ( ) 2 &' ( ) ( ) € ∗ , − 1−ζ ⋅ 2ℜ π +π −,π +π − K 0K 0 (Δt) π +π −,π +π − K 0K 0 (Δt) ( 00 ) ( )-.

€

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+ - + - φ →KSKL→π π π π : test of quantum coherence

1 i = K 0 K 0 − K 0 K 0 2 [ ]

Decoherence parameter: ζ = 0 → QM € 00 ζ00 =1 → total decoherence (also known as Furry's hypothesis or spontaneous factorization) [W.Furry, PR 49 (1936) 393] Bertlmann, Grimus, Hiesmayr PR D60 (1999) 114032 Bertlmann€ , Durstberger, Hiesmayr PRA 68 012111 (2003)

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+ - + - φ →KSKL→π π π π : test of quantum coherence

1 i = K 0 K 0 − K 0 K 0 2 [ ]

2 2 + − + − % + − + − 0 0 + − + − 0 0 I π π ,π π ;Δt = N π π ,π π K K Δt + π π ,π π K K Δt ( ) 2 &' ( ) ( ) € ∗ , 1 2 + −, + − K 0K 0 t + −, + − K 0K 0 t −( −ζ00 )⋅ ℜ π π π π (Δ ) π π π π (Δ ) . I(Δt) (a.u.) ( )-

Decoherence parameter: ζ = 0 → QM € 00 −6 ζ00 =1 → total decoherence ζ00 = 4 ×10 (also known as Furry's hypothesis or spontaneous factorization) [W.Furry, PR 49 (1936) 393] ζ 00 = 0 t/ Bertlmann, Grimus, Hiesmayr PR D60 (1999) 114032 Δ τS Bertlmann€ , Durstberger, Hiesmayr PRA 68 012111 (2003)

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+ - + - φ →KSKL→π π π π : test of quantum coherence • Analysed data: L=1.5 fb-1 • Fit including Δt resolution and efficiency effects + regeneration

PLB 642(2006) 315 KLOE result: Found. Phys. 40 (2010) 852 −7 ζ00 = (1.4 ± 9.5STAT ± 3.8SYST ) ×10 Observable suppressed by CP 2 2 -6 violation: |η+-| ∼ |ε| ∼ 10 2 => terms ζ00/|η+-| => high sensitivity to ζ00 € From CPLEAR data, Bertlmann et al. (PR D60 (1999) 114032) obtain:

ζ 00 = 0.4 ± 0.7 In the B-meson system, BELLE coll. (PRL 99 (2007) 131802) obtains: ζ B = 0.029 ± 0.057 00 A. Di Domenico 14th Marcel Grossmann meeting July 12-18, 2015 Rome 25

+ - + - φ →KSKL→π π π π : test of quantum coherence • Analysed data: L=1.5 fb-1 • Fit including Δt resolution and efficiency effects + regeneration

ζ 00 = 0.4 ± 0.7 In the B-meson system, BELLE coll. (PRL 99 (2007) 131802) obtains: ζ B = 0.029 ± 0.057 Best precision achievable in an entangled system 00 A. Di Domenico 14th Marcel Grossmann meeting July 12-18, 2015 Rome 26

+ - + - φ →KSKL→π π π π : test of quantum coherence -1 CINELLI et• al. Analysed data: L=1.5 fb Cinelli et al. PHYSICAL REVIEW A 70, 022321 (2004) • Fit including Δt resolution and by the use of narrow spatial-filtering pinholes or single-mode optical fibersefficiency[6,8,9,28]. This effects operation + allowsregeneration us to select a pair of correlated k-vectors, within a mode uncertainty ⌬k. In these conditions, it may happenPLB that 642(2006) only one 315 photon in a pair passesKLOE through result the spatial: filter,Found. while Phys. the 40 other (2010) one 852 is intercepted. A similar effect can be ascribed to any frequency-filteringζ = 1.4 operation± 9.5 as well.± As3.8 a consequence,×10− a7 state is obtained00 ( that is givenSTAT by the superpositionSYST ) of the polarization-entangledObservable pairs suppressed with single-photon by CP states, in addition to the vacuum field. However, in the common case of 2 2 -6 a coincidenceviolation: experiment, |η+ the-| output∼ |ε| state∼ 10 may be considered 2 as a postselected => terms two-photon ζ00/|η+ state.-| => This high postselection sensitivity may to ζ00 € cause conceptual difficulties when performing nonlocality tests [29].From CPLEAR data, Bertlmann et al. With our(PR source D60 the (1999) adoption 114032) of postselection obtain: to eliminate the single-photon-state contribution would not be necessary FIG. 2. Bell inequalities test. The selected state is ͉⌽−͘ ͱ in principle.ζ Indeed,= 0 this.4 would± 0.7 be possible by coupling the =͑1/ 2͉͒͑H1 ,H2͘−͉V1 ,V2͒͘. full set of mode00 pairs to detectors and , in all possible In the B-meson system,A BELLEB coll. wavelength configurations, either degenerate ͑␭1 =␭2͒ or measurement times with a low uv pump power. ␲ nondegenerate(PRL͑␭ 199! ␭(2007)2͒. Of course, 131802) in this obtains: case severe limi- The experimental data given in Fig. 2 show the ជ correlation obtained by varying the angle ␪ in the range 45° tations for fullB particle detection come from the limitedBest␭ precision achievable in an entangled1 system extension ofζ the detectors= 0.029 QE’s± and0. of057 the performance of the −135°, keeping the angle ␪2 =45° fixed. The interference pat- optical components00 (mirrors, lenses, etc.). Nevertheless, an tern shows the high degree of ␲ entanglement of the source. A. Di Domenico 14th Marcel GrossmannThe meeting measured July 12-18, visibility 2015 Rome of the coincidence rate, Vജ94%, 27 ideal experiment may be conceived by which the full set of QED excited modes at any wavelength, ranging from a very gives a strong indication of the entangled nature of the state over the entire emission k cone, while the single count rates large ␭ down to ␭=␭p, can be coupled to the detectors without any geometrical or frequency constraint, i.e., without any do not show any periodic fringe behavior, as expected. In the spatial or ␭ filtering. The same idealization is possible for the same Fig. 2 the dotted line corresponds to the limit boundary source of Kwiat et al. [10]. between the quantum and the classical regimes [30], while the theoretical continuous curve expresses the ideal interferometric pattern with maximum visibility V=1. B. Bell’s inequalities test The entanglement character of the source has also been The ͉⌽−͘ Bell state has been adopted to test the violation investigated by a standard Ou-Mandel interferometric test of a Bell inequality by a standard coincidence technique and [11,33], in this case by adoption of the Bell state expressed by the experimental configuration shown in Fig. 1 [30]. Ac- by Eq. (3)(Fig. 3, inset). For this experiment the radius of cording to previous considerations, the distribution of active the iris diaphragms ͑ID͒ in Fig. 1 was set at r=0.075 cm. e.m. modes appearing in Eq. (5) and corresponding to the The experimental results shown in Fig. 3 and corresponding whole ER was spatially filtered by the annular mask. to the values ␾= ␲ and ␾=0 express a value of the interfer- The adopted angle orientations of the ␲ជ -analyzers located ence visibility VӍ88%. The full width at half maximum at the ͑1͒ and ͑2͒ sites were ͕␪ =0,␪Ј=45°͖ and ͕␪ ͑Ӎ35 ␮m͒ of the interference pattern is in good agreement A B 1 1 2 =22.5° ,␪2 =67.5°͖, together with the corresponding or- Ќ Ќ Ќ Ќ thogonal angles ͕␪1 ,␪1 ͖ and ͕␪2 ,␪2 ͖. Using these values, the standard Bell inequality parameter can be evaluated [31]:

S = ͉P͑␪1,␪2͒ − P͑␪1,␪2Ј͒ + P͑␪1Ј,␪2͒ + P͑␪1Ј,␪2Ј͉͒, ͑6͒ where Ќ Ќ Ќ Ќ ͓C͑␪1,␪2͒ + C͑␪1 ,␪2 ͒ − C͑␪1,␪2 ͒ − C͑␪1 ,␪2͔͒ P͑␪1,␪2͒ = Ќ Ќ Ќ Ќ ͓C͑␪1,␪2͒ + C͑␪1 ,␪2 ͒ + C͑␪1,␪2 ͒ + C͑␪1 ,␪2͔͒ and C͑␪1 ,␪2͒ is the coincidence rate measured at sites and . The measured value S=2.5564±0.0026 [11], obtainedA by Bintegrating the data over 180 s, corresponds to a violation as large as 213 standard deviations with respect to the limit value S=2 implied by local realistic theories [3,31,32]. As for the full set of measurements reported in the present paper, FIG. 3. Coincidence rate versus the position X of the beam the good performance of the apparatus was largely attribut- splitter in the Ou-Mandel experiment. In the inset the corresponding able to the high brightness of the source, because of which interferometric apparatus is shown. The selected state is ͉⌿͘ i␾ large sets of statistical data could be accumulated in short =͑1/ͱ2͉͒͑H1 ,V͘2 +e ͉V1 ,H2͒͘.

022321-4 Search for decoherence and CPT violation effects

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Decoherence and CPT violation Possible decoherence due quantum gravity effects (BH evaporation) (apparent loss of unitarity): Black hole information loss paradox => (“like candy rolling Possible decoherence near a black hole. S. Hawking (1975) on the tongue” by J. Wheeler )

Hawking [1] suggested that at a microscopic level, in a quantum gravity picture, non-trivial space-time fluctuations (generically space-time foam) could give rise to decoherence effects, which would necessarily entail a violation of CPT [2].

Modified Liouville – von Neumann equation for the density matrix of the kaon system with 3 new CPTV parameters α,β,γ [3]:

! t iH i H + L ; , , extra term inducing ρ ( ) = − ρ + ρ + (ρ α β γ) decoherence: !##"##$ QM pure state => mixed state [1] Hawking, Comm.Math.Phys.87 (1982) 395; [2] Wald, PR D21 (1980) 2742;[3] Ellis et. al, NP B241 (1984) 381; Ellis, Mavromatos et al. PRD53 (1996)3846; Handbook on kaon interferometry [hep-ph/0607322], M. Arzano PRD90 (2014) 024016 A. Di Domenico 14th Marcel Grossmann meeting July 12-18, 2015 Rome 29

Decoherence and CPT violation Possible decoherence due quantum gravity effects (BH evaporation) (apparent loss of unitarity): 10-35 m Black hole information loss paradox => (“like candy rolling Possible decoherence near a black hole. S. Hawking (1975) on the tongue” by J. Wheeler )

Modified Liouville – von Neumann equation for the density matrix of the kaon system with 3 new CPTV parameters α,β,γ [3]:

Modified Liouville – von Neumann equation for the density matrix of the kaon system with 3 new CPTV parameters α,β,γ [3]: at most one expects: ⎛ M 2 ⎞ ρ! t = −iHρ +iρH + + L ρ;α, β,γ α,β,γ = O⎜ K ⎟ ≈ 2×10−20 GeV ( ) !##"##$ ( ) ⎜ ⎟ ⎝ M PLANCK ⎠ QM [1] Hawking, Comm.Math.Phys.87 (1982) 395; [2] Wald, PR D21 (1980) 2742;[3] Ellis et. al, NP B241 (1984) 381; Ellis, Mavromatos et al. PRD53 (1996)3846; Handbook on kaon interferometry [hep-ph/0607322], M. Arzano PRD90 (2014) 024016 A. Di Domenico 14th Marcel Grossmann meeting July 12-18, 2015 Rome 31

+ - + - φ →KSKL→π π π π : decoherence and CPT violation Study of time evolution of single kaons decaying in π+π- and semileptonic final state CPLEAR PLB 364, 239 (1999) α = (−0.5 ± 2.8) ×10−17 GeV −19 β = (2.5 ± 2.3) ×10 GeV single γ = (1.1± 2.5) ×10−21 GeV kaons

In the complete positivity hypothesis α = γ , β = 0 € => only one independent parameter: γ The fit with I(π+π-,π+π-;Δt,γ) gives: KLOE result L=1.5 fb-1 γ = 0.7 ±1.2 ± 0.3 ×10−21 GeV entangled ( STAT SYST ) kaons PLB 642(2006) 315 Found. Phys. 40 (2010) 852

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€ + - + - φ →KSKL→π π π π : CPT violation in entangled K states In presence of decoherence and CPT violation induced by quantum gravity (CPT operator “ill-defined”) the definition of the particle-antiparticle states could be modified. This in turn could induce a breakdown of the correlations imposed by Bose statistics (EPR correlations) to the kaon state: [Bernabeu, et al. PRL 92 (2004) 131601, NPB744 (2006) 180]. I(π+π-, π+π-;Δt) (a.u.) i ∝ ( K 0 K 0 − K 0 K 0 ) +ω( K 0 K 0 + K 0 K 0 )

∝ ( KS KL − KL KS ) +ω( KS KS − KL KL ) ω = 3 ×10−3 2 at most one 2 ⎛ E M PLANCK ⎞ −5 −3 0 ω = O⎜ ⎟ ≈10 ⇒ ω ~ 10 φω = expects: ⎜ ⎟ ⎝ ΔΓ ⎠ € Δt/τS In some microscopic models of space-time foam arising from non-critical string theory: € [Bernabeu, Mavromatos, Sarkar PRD 74 (2006) 045014] ω ~10 −4 ÷10 −5

+ - The maximum sensitivity to ω is expected for f1=f2=π π All CPTV effects induced by QG (α,β,γ,ω) could be simultaneously disentangled.

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+ - + - φ →KSKL→π π π π : CPT violation in entangled K states

Im ω x10-2

Fit of I(π+π-,π+π-;Δt,ω): • Analysed data: 1.5 fb-1

KLOE result: PLB 642(2006) 315 Found. Phys. 40 (2010) 852 0

+3.0 −4 ℜω = (−1.6−2.1STAT ± 0.4 SYST ) ×10 +3.3 −4 ℑω = (−1.7−3.0STAT ±1.2SYST ) ×10 ω <1.0 ×10−3 at 95% C.L. 0 -2 In the B system [Alvarez, Bernabeu, Nebot JHEP 0611, 087]: Re ω x10 − 0.0084 ≤ ℜω ≤ 0.0100 at 95% C.L.

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CPT symmetry and Lorentz invariance test

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CPT and Lorentz invariance violation (SME)

" CPT theorem : Exact CPT invariance holds for any quantum field theory which assumes: (1) Lorentz invariance (2) Locality (3) Unitarity (i.e. conservation of probability). " “Anti-CPT theorem” (Greenberger 2002): Any unitary, local, point-particle quantum field theory that violates CPT invariance necessarily violates Lorentz invariance. " Kostelecky et al. developed a phenomenological effective model providing a framework for CPT and Lorentz violations, based on spontaneous breaking of CPT and Lorentz symmetry, which might happen in quantum gravity (e.g. in some models of string theory) Standard Model Extension (SME) [Kostelecky PRD61, 016002, PRD64, 076001] CPT violation in neutral kaons according to SME: • At first order CPTV appears only in mixing parameter δ (no direct CPTV in decay) • δ cannot be a constant (momentum dependence) ! ! ε = ε ±δ δ = isinφ eiφ SW γ Δa − β ⋅ Δa /Δm S ,L SW K ( 0 K ) q2 q1 µ where Δaµ = aµ - aµ are four parameters associated to SME lagrangian terms −aµqγ q for the valence quarks and related to CPT and Lorentz violation.

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The Earth as a moving laboratory

SUN

EARTH

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Search for CPTV and LV: results ! ! δ = isinφ eiφ SW γ Δa − β ⋅ Δa /Δm SW K ( 0 K ) Data divided in 4 sidereal time bins x 2 angular bins Simultaneous fit of the Δt distributions € to extract Δaµ parameters

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with L=1.7 fb-1 KLOE final result PLB 730 (2014) 89–94 −18 Δa0 = (−6.0 ± 7.7STAT ± 3.1SYST ) ×10 GeV −18 ΔaX = (0.9 ±1.5STAT ± 0.6SYST ) ×10 GeV a 2.0 1.5 0.5 10−18 GeV B meson system: Δ Y = (− ± STAT ± SYST ) × B B B -13 Δa x,y , (Δa 0 – 0.30 Δa Z ) ~O(10 GeV) a 3.1 1.7 0.6 10−18 GeV [Babar PRL 100 (2008) 131802] Δ Z = (− ± STAT ± SYST ) × D meson system: D D D -13 presently the most precise measurements Δa x,y , (Δa 0 – 0.6 Δa Z ) ~O(10 GeV) in the quark sector of the SME [Focus PLB 556 (2003) 7]

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€

Direct CPT symmetry test in neutral kaon transitions

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